GROMACS Reference Manual 2018.3

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GROMACS
Groningen Machine for Chemical Simulations

Reference Manual
Version 2018.3

GROMACS
Reference Manual
Version 2018.3
Contributions from
Emile Apol, Rossen Apostolov, Herman J.C. Berendsen,
Aldert van Buuren, Pär Bjelkmar, Rudi van Drunen,
Anton Feenstra, Sebastian Fritsch, Gerrit Groenhof,
Christoph Junghans, Jochen Hub, Peter Kasson,
Carsten Kutzner, Brad Lambeth, Per Larsson, Justin A. Lemkul,
Viveca Lindahl, Magnus Lundborg, Erik Marklund, Pieter Meulenhoff,
Teemu Murtola, Szilárd Páll, Sander Pronk,
Roland Schulz, Michael Shirts, Alfons Sijbers,
Peter Tieleman, Christian Wennberg and Maarten Wolf.

Mark Abraham, Berk Hess, David van der Spoel, and Erik
Lindahl.

c 1991–2000: Department of Biophysical Chemistry, University of Groningen.
Nijenborgh 4, 9747 AG Groningen, The Netherlands.

c 2001–2018: The GROMACS development teams at the Royal Institute of Technology and
Uppsala University, Sweden.
More information can be found on our website: www.gromacs.org.

Preface & Disclaimer
This manual is not complete and has no pretention to be so due to lack of time of the contributors
– our first priority is to improve the software. It is worked on continuously, which in some cases
might mean the information is not entirely correct.
Comments on form and content are welcome, please send them to one of the mailing lists (see
www.gromacs.org), or open an issue at redmine.gromacs.org. Corrections can also be made in the
GROMACS git source repository and uploaded to gerrit.gromacs.org.
We release an updated version of the manual whenever we release a new version of the software,
so in general it is a good idea to use a manual with the same major and minor release number as
your GROMACS installation.

On-line Resources
You can find more documentation and other material at our homepage www.gromacs.org. Among
other things there is an on-line reference, several GROMACS mailing lists with archives and
contributed topologies/force fields.

Citation information
When citing this document in any scientific publication please refer to it as:
M.J. Abraham, D. van der Spoel, E. Lindahl, B. Hess, and the GROMACS
development team, GROMACS User Manual version 2018.3, www.gromacs.org
(2018)
However, we prefer that you cite (some of) the GROMACS papers [1, 2, 3, 4, 5, 6, 7, 8] when
you publish your results. Any future development depends on academic research grants, since the
package is distributed as free software!

GROMACS is Free Software
The entire GROMACS package is available under the GNU Lesser General Public License (LGPL),
version 2.1. This means it’s free as in free speech, not just that you can use it without paying us money. You can redistribute GROMACS and/or modify it under the terms of the LGPL
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The GROMACS source code and and selected set of binary packages are available on our homepage, www.gromacs.org. Have fun.

Contents
1 Introduction

1.1

Computational Chemistry and Molecular Modeling . . . . . . . . . . . . . . . .

1.2

Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

Energy Minimization and Search Methods . . . . . . . . . . . . . . . . . . . . .

2 Definitions and Units

2.1

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

MD units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Reduced units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Mixed or Double precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Algorithms

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1

Some useful box types . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.2

Cut-off restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

The group concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.1

Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.2

Neighbor searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.3

Compute forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.4

The leap-frog integrator . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.5

The velocity Verlet integrator . . . . . . . . . . . . . . . . . . . . . . .

3.4.6

Understanding reversible integrators: The Trotter decomposition . . . . .

3.4.7

Multiple time stepping . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.8

Temperature coupling . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents
3.4.9

Pressure coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.10 The complete update algorithm . . . . . . . . . . . . . . . . . . . . . .

3.4.11 Output step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Shell molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5.1

Optimization of the shell positions . . . . . . . . . . . . . . . . . . . . .

Constraint algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6.1

SHAKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6.2

LINCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.8

Stochastic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.9

Brownian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.10 Energy Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.10.1 Steepest Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.10.2 Conjugate Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.10.3 L-BFGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.11 Normal-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.12 Free energy calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.12.1 Slow-growth methods . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.12.2 Thermodynamic integration . . . . . . . . . . . . . . . . . . . . . . . .

3.13 Replica exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.14 Essential Dynamics sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.15 Expanded Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.16 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.17 Domain decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.17.1 Coordinate and force communication . . . . . . . . . . . . . . . . . . .

3.17.2 Dynamic load balancing . . . . . . . . . . . . . . . . . . . . . . . . . .

3.17.3 Constraints in parallel . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.17.4 Interaction ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.17.5 Multiple-Program, Multiple-Data PME parallelization . . . . . . . . . .

3.17.6 Domain decomposition flow chart . . . . . . . . . . . . . . . . . . . . .

3.18 Implicit solvation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

3.6

4 Interaction function and force fields
4.1

Non-bonded interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67
67

Contents

4.2

4.3

4.4

4.5

vii
4.1.1

The Lennard-Jones interaction . . . . . . . . . . . . . . . . . . . . . . .

4.1.2

Buckingham potential . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.3

Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.4

Coulomb interaction with reaction field . . . . . . . . . . . . . . . . . .

4.1.5

Modified non-bonded interactions . . . . . . . . . . . . . . . . . . . . .

4.1.6

Modified short-range interactions with Ewald summation . . . . . . . . .

Bonded interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1

Bond stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.2

Morse potential bond stretching . . . . . . . . . . . . . . . . . . . . . .

4.2.3

Cubic bond stretching potential . . . . . . . . . . . . . . . . . . . . . .

4.2.4

FENE bond stretching potential . . . . . . . . . . . . . . . . . . . . . .

4.2.5

Harmonic angle potential . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.6

Cosine based angle potential . . . . . . . . . . . . . . . . . . . . . . . .

4.2.7

Restricted bending potential . . . . . . . . . . . . . . . . . . . . . . . .

4.2.8

Urey-Bradley potential . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.9

Bond-Bond cross term . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.10 Bond-Angle cross term . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.11 Quartic angle potential . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.12 Improper dihedrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.13 Proper dihedrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.14 Tabulated bonded interaction functions . . . . . . . . . . . . . . . . . .

Restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.1

Position restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.2

Flat-bottomed position restraints . . . . . . . . . . . . . . . . . . . . . .

4.3.3

Angle restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.4

Dihedral restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.5

Distance restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.6

Orientation restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.1

Simple polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2

Anharmonic polarization . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.3

Water polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.4

Thole polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Free energy interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

Contents
4.5.1
4.6

Soft-core interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.6.1

Exclusions and 1-4 Interactions. . . . . . . . . . . . . . . . . . . . . . . 102

4.6.2

Charge Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.6.3

Treatment of Cut-offs in the group scheme . . . . . . . . . . . . . . . . 103

4.7

Virtual interaction sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.8

Long Range Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.9

4.8.1

Ewald summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.8.2

PME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.8.3

P3M-AD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.8.4

Optimizing Fourier transforms and PME calculations . . . . . . . . . . . 110

Long Range Van der Waals interactions . . . . . . . . . . . . . . . . . . . . . . 110
4.9.1

Dispersion correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.9.2

Lennard-Jones PME . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.10 Force field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.10.1 GROMOS-96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.10.2 OPLS/AA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.10.3 AMBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.10.4 CHARMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.10.5 Coarse-grained force fields . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.10.6 MARTINI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.10.7 PLUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Topologies

119

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2

Particle type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3

5.4

5.2.1

Atom types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.2

Virtual sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Parameter files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3.1

Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.3.2

Non-bonded parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.3.3

Bonded parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Molecule definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.1

Moleculetype entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Contents

ix
5.4.2

Intermolecular interactions . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.4.3

Intramolecular pair interactions . . . . . . . . . . . . . . . . . . . . . . 125

5.4.4

Exclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.5

Implicit solvation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.6

Constraint algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.7

pdb2gmx input files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.8

5.9

5.7.1

Residue database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.7.2

Residue to building block database . . . . . . . . . . . . . . . . . . . . . 131

5.7.3

Atom renaming database . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.7.4

Hydrogen database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.7.5

Termini database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.7.6

Virtual site database . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.7.7

Special bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

File formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.8.1

Topology file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.8.2

Molecule.itp file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.8.3

Ifdef statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.8.4

Topologies for free energy calculations . . . . . . . . . . . . . . . . . . 148

5.8.5

Constraint forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.8.6

Coordinate file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Force field organization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.9.1

Force-field files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.9.2

Changing force-field parameters . . . . . . . . . . . . . . . . . . . . . . 153

5.9.3

Adding atom types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6 Special Topics

155

6.1

Free energy implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.2

Potential of mean force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.3

Non-equilibrium pulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.4

The pull code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.5

Adaptive biasing with AWH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.5.1

Basics of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.5.2

The initial stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.5.3

Choice of target distribution . . . . . . . . . . . . . . . . . . . . . . . . 166

Contents

6.6

6.5.4

Multiple independent or sharing biases . . . . . . . . . . . . . . . . . . 167

6.5.5

Reweighting and combining biased data . . . . . . . . . . . . . . . . . . 168

6.5.6

The friction metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.5.7

Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Enforced Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.6.1

Fixed Axis Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.6.2

Flexible Axis Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.6.3

Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.7

Electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.8

Computational Electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.8.1

6.9

Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Calculating a PMF using the free-energy code . . . . . . . . . . . . . . . . . . . 185

6.10 Removing fastest degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . 186
6.10.1 Hydrogen bond-angle vibrations . . . . . . . . . . . . . . . . . . . . . . 187
6.10.2 Out-of-plane vibrations in aromatic groups . . . . . . . . . . . . . . . . 189
6.11 Viscosity calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.12 Tabulated interaction functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.12.1 Cubic splines for potentials . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.12.2 User-specified potential functions . . . . . . . . . . . . . . . . . . . . . 192
6.13 Mixed Quantum-Classical simulation techniques . . . . . . . . . . . . . . . . . 193
6.13.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.13.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.13.3 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.13.4 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.14 Using VMD plug-ins for trajectory file I/O . . . . . . . . . . . . . . . . . . . . . 197
6.15 Interactive Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.15.1 Simulation input preparation . . . . . . . . . . . . . . . . . . . . . . . . 197
6.15.2 Starting the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.15.3 Connecting from VMD . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.16 Embedding proteins into the membranes . . . . . . . . . . . . . . . . . . . . . . 198
7 Run parameters and Programs

201

7.1

Online documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

7.2

File types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Contents
7.3

xi
Run Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8 Analysis
8.1

203

Using Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.1.1

Default Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

8.1.2

Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8.2

Looking at your trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8.3

General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8.4

Radial distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

8.5

Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

8.6

8.5.1

Theory of correlation functions . . . . . . . . . . . . . . . . . . . . . . 210

8.5.2

Using FFT for computation of the ACF . . . . . . . . . . . . . . . . . . 211

8.5.3

Special forms of the ACF . . . . . . . . . . . . . . . . . . . . . . . . . . 211

8.5.4

Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Curve fitting in GROMACS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.6.1

Sum of exponential functions . . . . . . . . . . . . . . . . . . . . . . . 212

8.6.2

Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

8.6.3

Interphase boundary demarcation . . . . . . . . . . . . . . . . . . . . . 213

8.6.4

Transverse current autocorrelation function . . . . . . . . . . . . . . . . 213

8.6.5

Viscosity estimation from pressure autocorrelation function . . . . . . . 213

8.7

Mean Square Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

8.8

Bonds/distances, angles and dihedrals . . . . . . . . . . . . . . . . . . . . . . . 214

8.9

Radius of gyration and distances . . . . . . . . . . . . . . . . . . . . . . . . . . 216

8.10 Root mean square deviations in structure . . . . . . . . . . . . . . . . . . . . . . 217
8.11 Covariance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.12 Dihedral principal component analysis . . . . . . . . . . . . . . . . . . . . . . . 220
8.13 Hydrogen bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.14 Protein-related items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.15 Interface-related items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
A Some implementation details

227

A.1 Single Sum Virial in GROMACS . . . . . . . . . . . . . . . . . . . . . . . . . . 227
A.1.1 Virial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
A.1.2 Virial from non-bonded forces . . . . . . . . . . . . . . . . . . . . . . . 228
A.1.3 The intra-molecular shift (mol-shift) . . . . . . . . . . . . . . . . . . . . 229

xii

Contents
A.1.4 Virial from Covalent Bonds . . . . . . . . . . . . . . . . . . . . . . . . 230
A.1.5 Virial from SHAKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
A.2 Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
A.2.1 Inner Loops for Water . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

B Averages and fluctuations

233

B.1 Formulae for averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
B.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
B.2.1

Part of a Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

B.2.2

Combining two simulations . . . . . . . . . . . . . . . . . . . . . . . . 235

B.2.3

Summing energy terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Bibliography

239

Index

253

Chapter 1
Introduction
1.1

Computational Chemistry and Molecular Modeling

GROMACS is an engine to perform molecular dynamics simulations and energy minimization.
These are two of the many techniques that belong to the realm of computational chemistry and
molecular modeling. Computational chemistry is just a name to indicate the use of computational
techniques in chemistry, ranging from quantum mechanics of molecules to dynamics of large
complex molecular aggregates. Molecular modeling indicates the general process of describing
complex chemical systems in terms of a realistic atomic model, with the goal being to understand and predict macroscopic properties based on detailed knowledge on an atomic scale. Often,
molecular modeling is used to design new materials, for which the accurate prediction of physical
properties of realistic systems is required.
Macroscopic physical properties can be distinguished by (a) static equilibrium properties, such
as the binding constant of an inhibitor to an enzyme, the average potential energy of a system, or
the radial distribution function of a liquid, and (b) dynamic or non-equilibrium properties, such
as the viscosity of a liquid, diffusion processes in membranes, the dynamics of phase changes,
reaction kinetics, or the dynamics of defects in crystals. The choice of technique depends on the
question asked and on the feasibility of the method to yield reliable results at the present state of
the art. Ideally, the (relativistic) time-dependent Schrödinger equation describes the properties of
molecular systems with high accuracy, but anything more complex than the equilibrium state of a
few atoms cannot be handled at this ab initio level. Thus, approximations are necessary; the higher
the complexity of a system and the longer the time span of the processes of interest is, the more
severe the required approximations are. At a certain point (reached very much earlier than one
would wish), the ab initio approach must be augmented or replaced by empirical parameterization
of the model used. Where simulations based on physical principles of atomic interactions still
fail due to the complexity of the system, molecular modeling is based entirely on a similarity
analysis of known structural and chemical data. The QSAR methods (Quantitative StructureActivity Relations) and many homology-based protein structure predictions belong to the latter
category.
Macroscopic properties are always ensemble averages over a representative statistical ensemble

Chapter 1. Introduction

(either equilibrium or non-equilibrium) of molecular systems. For molecular modeling, this has
two important consequences:
• The knowledge of a single structure, even if it is the structure of the global energy minimum, is not sufficient. It is necessary to generate a representative ensemble at a given
temperature, in order to compute macroscopic properties. But this is not enough to compute
thermodynamic equilibrium properties that are based on free energies, such as phase equilibria, binding constants, solubilities, relative stability of molecular conformations, etc. The
computation of free energies and thermodynamic potentials requires special extensions of
molecular simulation techniques.
• While molecular simulations, in principle, provide atomic details of the structures and motions, such details are often not relevant for the macroscopic properties of interest. This
opens the way to simplify the description of interactions and average over irrelevant details.
The science of statistical mechanics provides the theoretical framework for such simplifications. There is a hierarchy of methods ranging from considering groups of atoms as
one unit, describing motion in a reduced number of collective coordinates, averaging over
solvent molecules with potentials of mean force combined with stochastic dynamics [9],
to mesoscopic dynamics describing densities rather than atoms and fluxes as response to
thermodynamic gradients rather than velocities or accelerations as response to forces [10].
For the generation of a representative equilibrium ensemble two methods are available: (a) Monte
Carlo simulations and (b) Molecular Dynamics simulations. For the generation of non-equilibrium
ensembles and for the analysis of dynamic events, only the second method is appropriate. While
Monte Carlo simulations are more simple than MD (they do not require the computation of forces),
they do not yield significantly better statistics than MD in a given amount of computer time. Therefore, MD is the more universal technique. If a starting configuration is very far from equilibrium,
the forces may be excessively large and the MD simulation may fail. In those cases, a robust energy minimization is required. Another reason to perform an energy minimization is the removal
of all kinetic energy from the system: if several “snapshots” from dynamic simulations must be
compared, energy minimization reduces the thermal noise in the structures and potential energies
so that they can be compared better.

1.2

Molecular Dynamics Simulations

MD simulations solve Newton’s equations of motion for a system of N interacting atoms:
mi

∂ 2 ri
= F i , i = 1 . . . N.
∂t2

(1.1)

The forces are the negative derivatives of a potential function V (r 1 , r 2 , . . . , r N ):
Fi = −

∂V
∂r i

(1.2)

The equations are solved simultaneously in small time steps. The system is followed for some
time, taking care that the temperature and pressure remain at the required values, and the coordinates are written to an output file at regular intervals. The coordinates as a function of time

1.2. Molecular Dynamics Simulations

type of bond
C-H, O-H, N-H
C=C, C=O
HOH
C-C
H2 CX
CCC
O-H· · ·O
O-H· · ·O

3
type of
vibration
stretch
stretch
bending
stretch
sciss, rock
bending
libration
stretch

wavenumber
(cm−1 )
3000–3500
1700–2000
1600
1400–1600
1000–1500
800–1000
400– 700
50– 200

Table 1.1: Typical vibrational frequencies (wavenumbers) in molecules and hydrogen-bonded liquids. Compare kT /h = 200 cm−1 at 300 K.
represent a trajectory of the system. After initial changes, the system will usually reach an equilibrium state. By averaging over an equilibrium trajectory, many macroscopic properties can be
extracted from the output file.
It is useful at this point to consider the limitations of MD simulations. The user should be aware
of those limitations and always perform checks on known experimental properties to assess the
accuracy of the simulation. We list the approximations below.
The simulations are classical
Using Newton’s equation of motion automatically implies the use of classical mechanics to
describe the motion of atoms. This is all right for most atoms at normal temperatures, but
there are exceptions. Hydrogen atoms are quite light and the motion of protons is sometimes
of essential quantum mechanical character. For example, a proton may tunnel through a
potential barrier in the course of a transfer over a hydrogen bond. Such processes cannot be
properly treated by classical dynamics! Helium liquid at low temperature is another example
where classical mechanics breaks down. While helium may not deeply concern us, the high
frequency vibrations of covalent bonds should make us worry! The statistical mechanics of a
classical harmonic oscillator differs appreciably from that of a real quantum oscillator when
the resonance frequency ν approximates or exceeds kB T /h. Now at room temperature the
wavenumber σ = 1/λ = ν/c at which hν = kB T is approximately 200 cm−1 . Thus, all
frequencies higher than, say, 100 cm−1 may misbehave in classical simulations. This means
that practically all bond and bond-angle vibrations are suspect, and even hydrogen-bonded
motions as translational or librational H-bond vibrations are beyond the classical limit (see
Table 1.1). What can we do?
Well, apart from real quantum-dynamical simulations, we can do one of two things:
(a) If we perform MD simulations using harmonic oscillators for bonds, we should make
corrections to the total internal energy U = Ekin +Epot and specific heat CV (and to entropy
S and free energy A or G if those are calculated). The corrections to the energy and specific
heat of a one-dimensional oscillator with frequency ν are: [11]
U QM = U cl + kT

1
x
x−1+ x
2
e −1

(1.3)

Chapter 1. Introduction
!

CVQM

CVcl

x2 e x
−1 ,
(ex − 1)2

(1.4)

where x = hν/kT . The classical oscillator absorbs too much energy (kT ), while the highfrequency quantum oscillator is in its ground state at the zero-point energy level of 12 hν.
(b) We can treat the bonds (and bond angles) as constraints in the equations of motion. The
rationale behind this is that a quantum oscillator in its ground state resembles a constrained
bond more closely than a classical oscillator. A good practical reason for this choice is
that the algorithm can use larger time steps when the highest frequencies are removed. In
practice the time step can be made four times as large when bonds are constrained than
when they are oscillators [12]. GROMACS has this option for the bonds and bond angles.
The flexibility of the latter is rather essential to allow for the realistic motion and coverage
of configurational space [13].
Electrons are in the ground state
In MD we use a conservative force field that is a function of the positions of atoms only.
This means that the electronic motions are not considered: the electrons are supposed to
adjust their dynamics instantly when the atomic positions change (the Born-Oppenheimer
approximation), and remain in their ground state. This is really all right, almost always. But
of course, electron transfer processes and electronically excited states can not be treated.
Neither can chemical reactions be treated properly, but there are other reasons to shy away
from reactions for the time being.
Force fields are approximate
Force fields provide the forces. They are not really a part of the simulation method and
their parameters can be modified by the user as the need arises or knowledge improves.
But the form of the forces that can be used in a particular program is subject to limitations.
The force field that is incorporated in GROMACS is described in Chapter 4. In the present
version the force field is pair-additive (apart from long-range Coulomb forces), it cannot
incorporate polarizabilities, and it does not contain fine-tuning of bonded interactions. This
urges the inclusion of some limitations in this list below. For the rest it is quite useful and
fairly reliable for biologically-relevant macromolecules in aqueous solution!
The force field is pair-additive
This means that all non-bonded forces result from the sum of non-bonded pair interactions.
Non pair-additive interactions, the most important example of which is interaction through
atomic polarizability, are represented by effective pair potentials. Only average non pairadditive contributions are incorporated. This also means that the pair interactions are not
pure, i.e., they are not valid for isolated pairs or for situations that differ appreciably from the
test systems on which the models were parameterized. In fact, the effective pair potentials
are not that bad in practice. But the omission of polarizability also means that electrons in
atoms do not provide a dielectric constant as they should. For example, real liquid alkanes
have a dielectric constant of slightly more than 2, which reduce the long-range electrostatic
interaction between (partial) charges. Thus, the simulations will exaggerate the long-range
Coulomb terms. Luckily, the next item compensates this effect a bit.
Long-range interactions are cut off
In this version, GROMACS always uses a cut-off radius for the Lennard-Jones interactions

1.3. Energy Minimization and Search Methods

and sometimes for the Coulomb interactions as well. The “minimum-image convention”
used by GROMACS requires that only one image of each particle in the periodic boundary
conditions is considered for a pair interaction, so the cut-off radius cannot exceed half the
box size. That is still pretty big for large systems, and trouble is only expected for systems
containing charged particles. But then truly bad things can happen, like accumulation of
charges at the cut-off boundary or very wrong energies! For such systems, you should
consider using one of the implemented long-range electrostatic algorithms, such as particlemesh Ewald [14, 15].
Boundary conditions are unnatural
Since system size is small (even 10,000 particles is small), a cluster of particles will have a
lot of unwanted boundary with its environment (vacuum). We must avoid this condition if
we wish to simulate a bulk system. As such, we use periodic boundary conditions to avoid
real phase boundaries. Since liquids are not crystals, something unnatural remains. This
item is mentioned last because it is the least of the evils. For large systems, the errors are
small, but for small systems with a lot of internal spatial correlation, the periodic boundaries
may enhance internal correlation. In that case, beware of, and test, the influence of system
size. This is especially important when using lattice sums for long-range electrostatics, since
these are known to sometimes introduce extra ordering.

1.3

Energy Minimization and Search Methods

As mentioned in sec. 1.1, in many cases energy minimization is required. GROMACS provides a
number of methods for local energy minimization, as detailed in sec. 3.10.
The potential energy function of a (macro)molecular system is a very complex landscape (or hypersurface) in a large number of dimensions. It has one deepest point, the global minimum and
a very large number of local minima, where all derivatives of the potential energy function with
respect to the coordinates are zero and all second derivatives are non-negative. The matrix of
second derivatives, which is called the Hessian matrix, has non-negative eigenvalues; only the
collective coordinates that correspond to translation and rotation (for an isolated molecule) have
zero eigenvalues. In between the local minima there are saddle points, where the Hessian matrix
has only one negative eigenvalue. These points are the mountain passes through which the system
can migrate from one local minimum to another.
Knowledge of all local minima, including the global one, and of all saddle points would enable
us to describe the relevant structures and conformations and their free energies, as well as the
dynamics of structural transitions. Unfortunately, the dimensionality of the configurational space
and the number of local minima is so high that it is impossible to sample the space at a sufficient
number of points to obtain a complete survey. In particular, no minimization method exists that
guarantees the determination of the global minimum in any practical amount of time. Impractical
methods exist, some much faster than others [16]. However, given a starting configuration, it
is possible to find the nearest local minimum. “Nearest” in this context does not always imply
“nearest” in a geometrical sense (i.e., the least sum of square coordinate differences), but means the
minimum that can be reached by systematically moving down the steepest local gradient. Finding
this nearest local minimum is all that GROMACS can do for you, sorry! If you want to find other

Chapter 1. Introduction

minima and hope to discover the global minimum in the process, the best advice is to experiment
with temperature-coupled MD: run your system at a high temperature for a while and then quench
it slowly down to the required temperature; do this repeatedly! If something as a melting or glass
transition temperature exists, it is wise to stay for some time slightly below that temperature and
cool down slowly according to some clever scheme, a process called simulated annealing. Since
no physical truth is required, you can use your imagination to speed up this process. One trick
that often works is to make hydrogen atoms heavier (mass 10 or so): although that will slow
down the otherwise very rapid motions of hydrogen atoms, it will hardly influence the slower
motions in the system, while enabling you to increase the time step by a factor of 3 or 4. You can
also modify the potential energy function during the search procedure, e.g. by removing barriers
(remove dihedral angle functions or replace repulsive potentials by soft-core potentials [17]), but
always take care to restore the correct functions slowly. The best search method that allows rather
drastic structural changes is to allow excursions into four-dimensional space [18], but this requires
some extra programming beyond the standard capabilities of GROMACS.
Three possible energy minimization methods are:
• Those that require only function evaluations. Examples are the simplex method and its
variants. A step is made on the basis of the results of previous evaluations. If derivative
information is available, such methods are inferior to those that use this information.
• Those that use derivative information. Since the partial derivatives of the potential energy
with respect to all coordinates are known in MD programs (these are equal to minus the
forces) this class of methods is very suitable as modification of MD programs.
• Those that use second derivative information as well. These methods are superior in their
convergence properties near the minimum: a quadratic potential function is minimized in
one step! The problem is that for N particles a 3N × 3N matrix must be computed, stored,
and inverted. Apart from the extra programming to obtain second derivatives, for most
systems of interest this is beyond the available capacity. There are intermediate methods
that build up the Hessian matrix on the fly, but they also suffer from excessive storage
requirements. So GROMACS will shy away from this class of methods.
The steepest descent method, available in GROMACS, is of the second class. It simply takes a
step in the direction of the negative gradient (hence in the direction of the force), without any
consideration of the history built up in previous steps. The step size is adjusted such that the
search is fast, but the motion is always downhill. This is a simple and sturdy, but somewhat
stupid, method: its convergence can be quite slow, especially in the vicinity of the local minimum!
The faster-converging conjugate gradient method (see e.g. [19]) uses gradient information from
previous steps. In general, steepest descents will bring you close to the nearest local minimum
very quickly, while conjugate gradients brings you very close to the local minimum, but performs
worse far away from the minimum. GROMACS also supports the L-BFGS minimizer, which is
mostly comparable to conjugate gradient method, but in some cases converges faster.

Chapter 2
Definitions and Units
2.1

Notation

The following conventions for mathematical typesetting are used throughout this document:
Item
Notation
Example
Vector
Bold italic
ri
Vector Length Italic
ri
We define the lowercase subscripts i, j, k and l to denote particles: r i is the position vector of
particle i, and using this notation:
r ij = r j − r i

(2.1)

rij = |r ij |

(2.2)

The force on particle i is denoted by F i and
F ij = force on i exerted by j

(2.3)

Please note that we changed notation as of version 2.0 to r ij = r j − r i since this is the notation
commonly used. If you encounter an error, let us know.

2.2

MD units

GROMACS uses a consistent set of units that produce values in the vicinity of unity for most
relevant molecular quantities. Let us call them MD units. The basic units in this system are nm,
ps, K, electron charge (e) and atomic mass unit (u), see Table 2.1. The values used in GROMACS
are taken from the CODATA Internationally recommended 2010 values of fundamental physical
constants (see http://nist.gov).
Consistent with these units are a set of derived units, given in Table 2.2.
The electric conversion factor f =

1
4πεo

= 138.935 458 kJ mol−1 nm e−2 . It relates the mechan-

Chapter 2. Definitions and Units
Quantity
length
mass
time
charge
temperature

Symbol
r
m
t
q
T

Unit
nm = 10−9 m
u (unified atomic mass unit) = 1.660 538 921 × 10−27 kg
ps = 10−12 s
e = elementary charge = 1.602 176 565(×10−19 C
K

Table 2.1: Basic units used in GROMACS.
Quantity
energy
Force
pressure
velocity
dipole moment
electric potential
electric field

Symbol
E, V
F
p
v
µ
Φ
E

Unit
kJ mol−1
kJ mol−1 nm−1
bar
nm ps−1 = 1000 m s−1
e nm
kJ mol−1 e−1 = 0.010 364 269 19 Volt
kJ mol−1 nm−1 e−1 = 1.036 426 919 × 107 V m−1

Table 2.2: Derived units. Note that an additional conversion factor of 1028 a.m.u (≈16.6) is applied
to get bar instead of internal MD units in the energy and log files.
ical quantities to the electrical quantities as in
V =f

q2
q2
or F = f 2
r
r

(2.4)

Electric potentials Φ and electric fields E are intermediate quantities in the calculation of energies
and forces. They do not occur inside GROMACS. If they are used in evaluations, there is a choice
of equations and related units. We strongly recommend following the usual practice of including
the factor f in expressions that evaluate Φ and E:
Φ(r) = f

E(r) = f

X
j

qj
|r − r j |

(2.5)

(r − r j )
|r − r j |3

(2.6)

With these definitions, qΦ is an energy and qE is a force. The units are those given in Table 2.2:
about 10 mV for potential. Thus, the potential of an electronic charge at a distance of 1 nm equals
f ≈ 140 units ≈ 1.4 V. (exact value: 1.439 964 5 V)
Note that these units are mutually consistent; changing any of the units is likely to produce inconsistencies and is therefore strongly discouraged! In particular: if Å are used instead of nm, the unit
of time changes to 0.1 ps. If kcal mol−1 (= 4.184 kJ mol−1 ) is used instead of kJ mol−1 for energy,
the unit of time becomes 0.488882 ps and the unit of temperature changes to 4.184 K. But in both
cases all electrical energies go wrong, because they will still be computed in kJ mol−1 , expecting
nm as the unit of length. Although careful rescaling of charges may still yield consistency, it is
clear that such confusions must be rigidly avoided.

2.3. Reduced units
Symbol
NAV
R
kB
h
h̄
c

9
Name
Avogadro’s number
gas constant
Boltzmann’s constant
Planck’s constant
Dirac’s constant
velocity of light

Value
6.022 141 29 × 1023 mol−1
8.314 462 1 × 10−3 kJ mol−1 K−1
idem
0.399 031 271 kJ mol−1 ps
0.063 507 799 3 kJ mol−1 ps
299 792.458 nm ps−1

Table 2.3: Some Physical Constants
Quantity
Length
Mass
Time
Temperature
Energy
Force
Pressure
Velocity
Density

Symbol
r∗
m∗
t∗
T∗
E∗
F∗
P∗
v∗
ρ∗

Relation to SI
r σ −1
m M−1p
t σ −1 /M
kB T −1
E −1
F σ −1
Pp
σ 3 −1
v M/
N σ 3 V −1

Table 2.4: Reduced Lennard-Jones quantities

In terms of the MD units, the usual physical constants take on different values (see Table 2.3).
All quantities are per mol rather than per molecule. There is no distinction between Boltzmann’s
constant k and the gas constant R: their value is 0.008 314 462 1 kJ mol−1 K−1 .

2.3

Reduced units

When simulating Lennard-Jones (LJ) systems, it might be advantageous to use reduced units (i.e.,
setting ii = σii = mi = kB = 1 for one type of atoms). This is possible. When specifying
the input in reduced units, the output will also be in reduced units. The one exception is the
temperature, which is expressed in 0.008 314 462 1 reduced units. This is a consequence of using
Boltzmann’s constant in the evaluation of temperature in the code. Thus not T , but kB T , is the
reduced temperature. A GROMACS temperature T = 1 means a reduced temperature of 0.008 . . .
units; if a reduced temperature of 1 is required, the GROMACS temperature should be 120.272 36.
In Table 2.4 quantities are given for LJ potentials:
6 #

"
12

VLJ = 4

σ
r

−

σ
r

(2.7)

Chapter 2. Definitions and Units

2.4

Mixed or Double precision

GROMACS can be compiled in either mixed or double precision. Documentation of previous
GROMACS versions referred to “single precision”, but the implementation has made selective
use of double precision for many years. Using single precision for all variables would lead to a
significant reduction in accuracy. Although in “mixed precision” all state vectors, i.e. particle
coordinates, velocities and forces, are stored in single precision, critical variables are double precision. A typical example of the latter is the virial, which is a sum over all forces in the system,
which have varying signs. In addition, in many parts of the code we managed to avoid double precision for arithmetic, by paying attention to summation order or reorganization of mathematical
expressions. The default configuration uses mixed precision, but it is easy to turn on double precision by adding the option -DGMX_DOUBLE=on to cmake. Double precision will be 20 to 100%
slower than mixed precision depending on the architecture you are running on. Double precision
will use somewhat more memory and run input, energy and full-precision trajectory files will be
almost twice as large.
The energies in mixed precision are accurate up to the last decimal, the last one or two decimals
of the forces are non-significant. The virial is less accurate than the forces, since the virial is only
one order of magnitude larger than the size of each element in the sum over all atoms (sec. A.1).
In most cases this is not really a problem, since the fluctuations in the virial can be two orders
of magnitude larger than the average. Using cut-offs for the Coulomb interactions cause large
errors in the energies, forces, and virial. Even when using a reaction-field or lattice sum method,
the errors are larger than, or comparable to, the errors due to the partial use of single precision.
Since MD is chaotic, trajectories with very similar starting conditions will diverge rapidly, the
divergence is faster in mixed precision than in double precision.
For most simulations, mixed precision is accurate enough. In some cases double precision is
required to get reasonable results:
• normal mode analysis, for the conjugate gradient or l-bfgs minimization and the calculation
and diagonalization of the Hessian
• long-term energy conservation, especially for large systems

Chapter 3
Algorithms
3.1

Introduction

In this chapter we first give describe some general concepts used in GROMACS: periodic boundary conditions (sec. 3.2) and the group concept (sec. 3.3). The MD algorithm is described in
sec. 3.4: first a global form of the algorithm is given, which is refined in subsequent subsections.
The (simple) EM (Energy Minimization) algorithm is described in sec. 3.10. Some other algorithms for special purpose dynamics are described after this.
A few issues are of general interest. In all cases the system must be defined, consisting of
molecules. Molecules again consist of particles with defined interaction functions. The detailed
description of the topology of the molecules and of the force field and the calculation of forces is
given in chapter 4. In the present chapter we describe other aspects of the algorithm, such as pair
list generation, update of velocities and positions, coupling to external temperature and pressure,
conservation of constraints. The analysis of the data generated by an MD simulation is treated in
chapter 8.

3.2

Periodic boundary conditions

The classical way to minimize edge effects in a finite system is to apply periodic boundary conditions. The atoms of the system to be simulated are put into a space-filling box, which is surrounded
by translated copies of itself (Fig. 3.1). Thus there are no boundaries of the system; the artifact
caused by unwanted boundaries in an isolated cluster is now replaced by the artifact of periodic
conditions. If the system is crystalline, such boundary conditions are desired (although motions
are naturally restricted to periodic motions with wavelengths fitting into the box). If one wishes to
simulate non-periodic systems, such as liquids or solutions, the periodicity by itself causes errors.
The errors can be evaluated by comparing various system sizes; they are expected to be less severe
than the errors resulting from an unnatural boundary with vacuum.
There are several possible shapes for space-filling unit cells. Some, like the rhombic dodecahedron
and the truncated octahedron [20] are closer to being a sphere than a cube is, and are therefore

Chapter 3. Algorithms
y
j’

j’

i’

j’

i’
j’

i’

i
j’

i’
j’

i’

j’

i’

y
j’

j’

i’

i’
j’

i’
j

i’

j’

i
j’

i’

j’

i’
j’

i’

j’

i’

Figure 3.1: Periodic boundary conditions in two dimensions.
better suited to the study of an approximately spherical macromolecule in solution, since fewer
solvent molecules are required to fill the box given a minimum distance between macromolecular
images. At the same time, rhombic dodecahedra and truncated octahedra are special cases of
triclinic unit cells; the most general space-filling unit cells that comprise all possible space-filling
shapes [21]. For this reason, GROMACS is based on the triclinic unit cell.
GROMACS uses periodic boundary conditions, combined with the minimum image convention:
only one – the nearest – image of each particle is considered for short-range non-bonded interaction terms. For long-range electrostatic interactions this is not always accurate enough, and
GROMACS therefore also incorporates lattice sum methods such as Ewald Sum, PME and PPPM.
GROMACS supports triclinic boxes of any shape. The simulation box (unit cell) is defined by the
3 box vectors a,b and c. The box vectors must satisfy the following conditions:
ay = az = bz = 0
ax > 0,

by > 0,

cz > 0

(3.1)
(3.2)

1
1
1
ax , |cx | ≤ ax , |cy | ≤ by
(3.3)
2
2
2
Equations 3.1 can always be satisfied by rotating the box. Inequalities (3.2) and (3.3) can always
be satisfied by adding and subtracting box vectors.
|bx | ≤

Even when simulating using a triclinic box, GROMACS always keeps the particles in a brickshaped volume for efficiency, as illustrated in Fig. 3.1 for a 2-dimensional system. Therefore,
from the output trajectory it might seem that the simulation was done in a rectangular box. The
program trjconv can be used to convert the trajectory to a different unit-cell representation.

3.2. Periodic boundary conditions

Figure 3.2: A rhombic dodecahedron and truncated octahedron (arbitrary orientations).
box type

cubic
rhombic
dodecahedron
(xy-square)
rhombic
dodecahedron
(xy-hexagon)
truncated
octahedron

image
distance

box
volume

1
2

√

2 d3
0.707 d3
1
2

√

2 d3
0.707 d3
4
9

√

3 d3
0.770 d3

a
d
0
0
d
0
0
d
0
0
d
0
0

box vectors
b
c
0
0
d
0
0
d
1
0
2d
1
d
2d
√
1
0
2 2d
1
1
d
2
2d
√
√
1
1
2 3d
6 √3 d
1
0
3 6d
1
−√31 d
3d
√
2
1
3 2d
3 √2 d
1
0
3 6d

box vector angles
6 ac
6 ab
bc
90◦

90◦

60◦

90◦

60◦

71.53◦

109.47◦

71.53◦

Table 3.1: The cubic box, the rhombic dodecahedron and the truncated octahedron.
It is also possible to simulate without periodic boundary conditions, but it is usually more efficient
to simulate an isolated cluster of molecules in a large periodic box, since fast grid searching can
only be used in a periodic system.

3.2.1

Some useful box types

The three most useful box types for simulations of solvated systems are described in Table 3.1.
The rhombic dodecahedron (Fig. 3.2) is the smallest and most regular space-filling unit cell. Each
of the 12 image cells is at the same distance. The volume is 71% of the volume of a cube having
the same image distance. This saves about 29% of CPU-time when simulating a spherical or
flexible molecule in solvent. There are two different orientations of a rhombic dodecahedron that
satisfy equations 3.1, 3.2 and 3.3. The program editconf produces the orientation which has
a square intersection with the xy-plane. This orientation was chosen because the first two box
vectors coincide with the x and y-axis, which is easier to comprehend. The other orientation can

Chapter 3. Algorithms

be useful for simulations of membrane proteins. In this case the cross-section with the xy-plane is
a hexagon, which has an area which is 14% smaller than the area of a square with the same image
distance. The height of the box (cz ) should be changed to obtain an optimal spacing. This box
shape not only saves CPU time, it also results in a more uniform arrangement of the proteins.

3.2.2

Cut-off restrictions

The minimum image convention implies that the cut-off radius used to truncate non-bonded interactions may not exceed half the shortest box vector:
Rc <

1
min(kak, kbk, kck),
2

(3.4)

because otherwise more than one image would be within the cut-off distance of the force. When a
macromolecule, such as a protein, is studied in solution, this restriction alone is not sufficient: in
principle, a single solvent molecule should not be able to ‘see’ both sides of the macromolecule.
This means that the length of each box vector must exceed the length of the macromolecule in the
direction of that edge plus two times the cut-off radius Rc . It is, however, common to compromise
in this respect, and make the solvent layer somewhat smaller in order to reduce the computational
cost. For efficiency reasons the cut-off with triclinic boxes is more restricted. For grid search the
extra restriction is weak:
Rc < min(ax , by , cz )
(3.5)
For simple search the extra restriction is stronger:
Rc <

1
min(ax , by , cz )
2

(3.6)

Each unit cell (cubic, rectangular or triclinic) is surrounded by 26 translated images. A particular
image can therefore always be identified by an index pointing to one of 27 translation vectors and
constructed by applying a translation with the indexed vector (see 3.4.3). Restriction (3.5) ensures
that only 26 images need to be considered.

3.3

The group concept

The GROMACS MD and analysis programs use user-defined groups of atoms to perform certain
actions on. The maximum number of groups is 256, but each atom can only belong to six different
groups, one each of the following:
temperature-coupling group The temperature coupling parameters (reference temperature, time
constant, number of degrees of freedom, see 3.4.4) can be defined for each T-coupling group
separately. For example, in a solvated macromolecule the solvent (that tends to generate
more heating by force and integration errors) can be coupled with a shorter time constant to
a bath than is a macromolecule, or a surface can be kept cooler than an adsorbing molecule.
Many different T-coupling groups may be defined. See also center of mass groups below.

3.4. Molecular Dynamics

freeze group Atoms that belong to a freeze group are kept stationary in the dynamics. This is
useful during equilibration, e.g. to avoid badly placed solvent molecules giving unreasonable
kicks to protein atoms, although the same effect can also be obtained by putting a restraining
potential on the atoms that must be protected. The freeze option can be used, if desired, on
just one or two coordinates of an atom, thereby freezing the atoms in a plane or on a line.
When an atom is partially frozen, constraints will still be able to move it, even in a frozen
direction. A fully frozen atom can not be moved by constraints. Many freeze groups can
be defined. Frozen coordinates are unaffected by pressure scaling; in some cases this can
produce unwanted results, particularly when constraints are also used (in this case you will
get very large pressures). Accordingly, it is recommended to avoid combining freeze groups
with constraints and pressure coupling. For the sake of equilibration it could suffice to
start with freezing in a constant volume simulation, and afterward use position restraints in
conjunction with constant pressure.
accelerate group On each atom in an “accelerate group” an acceleration ag is imposed. This
is equivalent to an external force. This feature makes it possible to drive the system into
a non-equilibrium state and enables the performance of non-equilibrium MD and hence to
obtain transport properties.
energy-monitor group Mutual interactions between all energy-monitor groups are compiled during the simulation. This is done separately for Lennard-Jones and Coulomb terms. In principle up to 256 groups could be defined, but that would lead to 256×256 items! Better use
this concept sparingly.
All non-bonded interactions between pairs of energy-monitor groups can be excluded (see
details in the User Guide). Pairs of particles from excluded pairs of energy-monitor groups
are not put into the pair list. This can result in a significant speedup for simulations where
interactions within or between parts of the system are not required.
center of mass group In GROMACS the center of mass (COM) motion can be removed, for
either the complete system or for groups of atoms. The latter is useful, e.g. for systems
where there is limited friction (e.g. gas systems) to prevent center of mass motion to occur.
It makes sense to use the same groups for temperature coupling and center of mass motion
removal.
Compressed position output group In order to further reduce the size of the compressed trajectory file (.xtc or .tng), it is possible to store only a subset of all particles. All xcompression groups that are specified are saved, the rest are not. If no such groups are
specified, than all atoms are saved to the compressed trajectory file.
The use of groups in GROMACS tools is described in sec. 8.1.

3.4

Molecular Dynamics

A global flow scheme for MD is given in Fig. 3.3. Each MD or EM run requires as input a set of
initial coordinates and – optionally – initial velocities of all particles involved. This chapter does
not describe how these are obtained; for the setup of an actual MD run check the online manual at
www.gromacs.org.

Chapter 3. Algorithms

THE GLOBAL MD ALGORITHM
1. Input initial conditions
Potential interaction V as a function of atom positions
Positions r of all atoms in the system
Velocities v of all atoms in the system
⇓
repeat 2,3,4 for the required number of steps:
2. Compute forces
The force on any atom
∂V
Fi = −
∂r i
is computed by calculating the force between non-bonded atom
pairs:
P
F i = j F ij
plus the forces due to bonded interactions (which may depend on 1,
2, 3, or 4 atoms), plus restraining and/or external forces.
The potential and kinetic energies and the pressure tensor may be
computed.
⇓
3. Update configuration
The movement of the atoms is simulated by numerically solving
Newton’s equations of motion
d2 r i
Fi
=
2
dt
mi
or
dr i
dv i
Fi
= vi;
=
dt
dt
mi
⇓
4. if required: Output step
write positions, velocities, energies, temperature, pressure, etc.

Figure 3.3: The global MD algorithm

3.4. Molecular Dynamics

Velocity

Figure 3.4: A Maxwell-Boltzmann velocity distribution, generated from random numbers.

3.4.1

Initial conditions

Topology and force field
The system topology, including a description of the force field, must be read in. Force fields and
topologies are described in chapter 4 and 5, respectively. All this information is static; it is never
modified during the run.

Coordinates and velocities
Then, before a run starts, the box size and the coordinates and velocities of all particles are required. The box size and shape is determined by three vectors (nine numbers) b1 , b2 , b3 , which
represent the three basis vectors of the periodic box.
If the run starts at t = t0 , the coordinates at t = t0 must be known. The leap-frog algorithm, the
default algorithm used to update the time step with ∆t (see 3.4.4), also requires that the velocities
at t = t0 − 21 ∆t are known. If velocities are not available, the program can generate initial atomic
velocities vi , i = 1 . . . 3N with a (Fig. 3.4) at a given absolute temperature T :
r

p(vi ) =

mi
mi vi2
exp −
2πkT
2kT

(3.7)

Chapter 3. Algorithms

where k is Boltzmann’s constant (see chapter 2). To accomplish this, normally distributed random
numbers are generated by adding twelve random numbers Rk in the range 0 ≤ Rk < 1 and
subtracting 6.0 fromptheir sum. The result is then multiplied by the standard deviation of the
velocity distribution kT /mi . Since the resulting total energy will not correspond exactly to the
required temperature T , a correction is made: first the center-of-mass motion is removed and then
all velocities are scaled so that the total energy corresponds exactly to T (see eqn. 3.18).
Center-of-mass motion
The center-of-mass velocity is normally set to zero at every step; there is (usually) no net external
force acting on the system and the center-of-mass velocity should remain constant. In practice,
however, the update algorithm introduces a very slow change in the center-of-mass velocity, and
therefore in the total kinetic energy of the system – especially when temperature coupling is used.
If such changes are not quenched, an appreciable center-of-mass motion can develop in long runs,
and the temperature will be significantly misinterpreted. Something similar may happen due to
overall rotational motion, but only when an isolated cluster is simulated. In periodic systems with
filled boxes, the overall rotational motion is coupled to other degrees of freedom and does not
cause such problems.

3.4.2

Neighbor searching

As mentioned in chapter 4, internal forces are either generated from fixed (static) lists, or from
dynamic lists. The latter consist of non-bonded interactions between any pair of particles. When
calculating the non-bonded forces, it is convenient to have all particles in a rectangular box. As
shown in Fig. 3.1, it is possible to transform a triclinic box into a rectangular box. The output
coordinates are always in a rectangular box, even when a dodecahedron or triclinic box was used
for the simulation. Equation 3.1 ensures that we can reset particles in a rectangular box by first
shifting them with box vector c, then with b and finally with a. Equations 3.3, 3.4 and 3.5 ensure
that we can find the 14 nearest triclinic images within a linear combination that does not involve
multiples of box vectors.
Pair lists generation
The non-bonded pair forces need to be calculated only for those pairs i, j for which the distance
rij between i and the nearest image of j is less than a given cut-off radius Rc . Some of the particle
pairs that fulfill this criterion are excluded, when their interaction is already fully accounted for by
bonded interactions. GROMACS employs a pair list that contains those particle pairs for which
non-bonded forces must be calculated. The pair list contains particles i, a displacement vector for
particle i, and all particles j that are within rlist of this particular image of particle i. The list
is updated every nstlist steps.
To make the neighbor list, all particles that are close (i.e. within the neighbor list cut-off) to a given
particle must be found. This searching, usually called neighbor search (NS) or pair search, involves
periodic boundary conditions and determining the image (see sec. 3.2). The search algorithm is
O(N ), although a simpler O(N 2 ) algorithm is still available under some conditions.

3.4. Molecular Dynamics

Cut-off schemes: group versus Verlet
From version 4.6, GROMACS supports two different cut-off scheme setups: the original one
based on particle groups and one using a Verlet buffer. There are some important differences
that affect results, performance and feature support. The group scheme can be made to work
(almost) like the Verlet scheme, but this will lead to a decrease in performance. The group scheme
is especially fast for water molecules, which are abundant in many simulations, but on the most
recent x86 processors, this advantage is negated by the better instruction-level parallelism available
in the Verlet-scheme implementation. The group scheme is deprecated in version 5.0, and will be
removed in a future version. For practical details of choosing and setting up cut-off schemes,
please see the User Guide.
In the group scheme, a neighbor list is generated consisting of pairs of groups of at least one
particle. These groups were originally charge groups (see sec. 3.4.2), but with a proper treatment
of long-range electrostatics, performance in unbuffered simulations is their only advantage. A
pair of groups is put into the neighbor list when their center of geometry is within the cut-off
distance. Interactions between all particle pairs (one from each charge group) are calculated for
a certain number of MD steps, until the neighbor list is updated. This setup is efficient, as the
neighbor search only checks distance between charge-group pair, not particle pairs (saves a factor
of 3×3 = 9 with a three-particle water model) and the non-bonded force kernels can be optimized
for, say, a water molecule “group”. Without explicit buffering, this setup leads to energy drift as
some particle pairs which are within the cut-off don’t interact and some outside the cut-off do
interact. This can be caused by
• particles moving across the cut-off between neighbor search steps, and/or
• for charge groups consisting of more than one particle, particle pairs moving in/out of the
cut-off when their charge group center of geometry distance is outside/inside of the cut-off.
Explicitly adding a buffer to the neighbor list will remove such artifacts, but this comes at a high
computational cost. How severe the artifacts are depends on the system, the properties in which
you are interested, and the cut-off setup.
The Verlet cut-off scheme uses a buffered pair list by default. It also uses clusters of particles, but
these are not static as in the group scheme. Rather, the clusters are defined spatially and consist
of 4 or 8 particles, which is convenient for stream computing, using e.g. SSE, AVX or CUDA on
GPUs. At neighbor search steps, a pair list is created with a Verlet buffer, ie. the pair-list cut-off
is larger than the interaction cut-off. In the non-bonded kernels, interactions are only computed
when a particle pair is within the cut-off distance at that particular time step. This ensures that
as particles move between pair search steps, forces between nearly all particles within the cut-off
distance are calculated. We say nearly all particles, because GROMACS uses a fixed pair list
update frequency for efficiency. A particle-pair, whose distance was outside the cut-off, could
possibly move enough during this fixed number of steps that its distance is now within the cutoff. This small chance results in a small energy drift, and the size of the chance depends on the
temperature. When temperature coupling is used, the buffer size can be determined automatically,
given a certain tolerance on the energy drift.
The Verlet cut-off scheme is implemented in a very efficient fashion based on clusters of particles.
The simplest example is a cluster size of 4 particles. The pair list is then constructed based on

Chapter 3. Algorithms

cluster pairs. The cluster-pair search is much faster searching based on particle pairs, because
4 × 4 = 16 particle pairs are put in the list at once. The non-bonded force calculation kernel can
then calculate many particle-pair interactions at once, which maps nicely to SIMD or SIMT units
on modern hardware, which can perform multiple floating operations at once. These non-bonded
kernels are much faster than the kernels used in the group scheme for most types of systems,
particularly on newer hardware.
Additionally, when the list buffer is determined automatically as described below, we also apply
dynamic pair list pruning. The pair list can be constructed infrequently, but that can lead to a lot
of pairs in the list that are outside the cut-off range for all or most of the life time of this pair
list. Such pairs can be pruned out by applying a cluster-pair kernel that only determines which
clusters are in range. Because of the way the non-bonded data is regularized in GROMACS, this
kernel is an order of magnitude faster than the search and the interaction kernel. On the GPU this
pruning is overlapped with the integration on the CPU, so it is free in most cases. Therefore we
can prune every 4-10 integration steps with little overhead and significantly reduce the number of
cluster pairs in the interaction kernel. This procedure is applied automatically, unless the user set
the pair-list buffer size manually.
Energy drift and pair-list buffering
For a canonical (NVT) ensemble, the average energy error caused by diffusion of j particles from
outside the pair-list cut-off r` to inside the interaction cut-off rc over the lifetime of the list can
be determined from the atomic displacements and the shape of the potential at the cut-off. The
displacement distribution along one dimension for a freely moving particle with mass m over time
t at temperature T is a Gaussian G(x) of zero mean and variance σ 2 = t2 kB T /m. For the distance
2 = t2 k T (1/m + 1/m ). Note that
between two particles, the variance changes to σ 2 = σ12
1
2
B
in practice particles usually interact with (bump into) other particles over time t and therefore the
real displacement distribution is much narrower. Given a non-bonded interaction cut-off distance
of rc and a pair-list cut-off r` = rc + rb for rb the Verlet buffer size, we can then write the average
energy error after time t for all missing pair interactions between a single i particle of type 1
surrounded by all j particles that are of type 2 with number density ρ2 , when the inter-particle
distance changes from r0 to rt , as:
h∆V i =

Z rc Z ∞
0

4πr02 ρ2 V

rt − r0
(rt )G
dr0 drt
σ

(3.8)

To evaluate this analytically, we need to make some approximations. First we replace V (rt ) by
a Taylor expansion around rc , then we can move the lower bound of the integral over r0 to −∞
which will simplify the result:
h∆V i ≈

Z rc Z ∞
−∞ r`

4πr02 ρ2 V 0 (rc )(rt − rc ) +
1
V 00 (rc ) (rt − rc )2 +
2
1
V 000 (rc ) (rt − rc )3 +
6

i r − r
t
0
O (rt − rc )4 G
dr0 drt
σ

(3.9)

3.4. Molecular Dynamics

Replacing the factor r02 by (r` + σ)2 , which results in a slight overestimate, allows us to calculate
the integrals analytically:
2

h∆V i ≈ 4π(r` + σ) ρ2

Z rc Z ∞ h
−∞ r`

V 0 (rc )(rt − rc ) +

1
V 00 (rc ) (rt − rc )2 +
2
i r − r
1
t
0
000
3
dr0 drt
(3.10)
V (rc ) (rt − rc ) G
6
σ

1
rb
rb
− (rb2 + σ 2 )E
+
= 4π(r` + σ)2 ρ2 V 0 (rc ) rb σG
2
σ
σ

1 00
rb
rb
− rb (rb2 + 3σ 2 )E
+
V (rc ) σ(rb2 + 2σ 2 )G
6
σ
σ

1 000
rb
V (rc ) rb σ(rb2 + 5σ 2 )G
24
σ

rb
(3.11)
− (rb4 + 6rb2 σ 2 + 3σ 4 )E
σ
√
where G(x) is a Gaussian distribution with 0 mean and unit variance and E(x) = 21 erfc(x/ 2).
We always want to achieve small energy error, so σ will be small compared to both rc and r` ,
thus the approximations in the equations above are good, since the Gaussian distribution decays
rapidly. The energy error needs to be averaged over all particle pair types and weighted with
the particle counts. In GROMACS we don’t allow cancellation of error between pair types, so
we average the absolute values. To obtain the average energy error per unit time, it needs to be
divided by the neighbor-list life time t = (nstlist − 1) × dt. The function can not be inverted
analytically, so we use bisection to obtain the buffer size rb for a target drift. Again we note that in
practice the error we usually be much smaller than this estimate, as in the condensed phase particle
displacements will be much smaller than for freely moving particles, which is the assumption used
here.
When (bond) constraints are present, some particles will have fewer degrees of freedom. This will
reduce the energy errors. For simplicity, we only consider one constraint per particle, the heaviest
particle in case a particle is involved in multiple constraints. This simplification overestimates the
displacement. The motion of a constrained particle is a superposition of the 3D motion of the
center of mass of both particles and a 2D rotation around the center of mass. The displacement in
an arbitrary direction of a particle with 2 degrees of freedom is not Gaussian, but rather follows
the complementary error function:
√

π
|r|
√ erfc √
(3.12)
2 2σ
2σ
where σ 2 is again t2 kB T /m. This distribution can no longer be integrated analytically to obtain
the energy error. But we can generate a tight upper bound using a scaled and shifted Gaussian
distribution (not shown). This Gaussian distribution can then be used to calculate the energy error
as described above. The rotation displacement around the center of mass can not be more than the
length of the arm. To take this into account, we scale σ in eqn. 3.12 (details not presented here) to
obtain an overestimate of the real displacement. This latter effect significantly reduces the buffer

Chapter 3. Algorithms
10

−2

drift per atom (kJ/mol/ps)

estimate 4x4
10

−3

−4

−5

−6

estimate 1x1

mixed precision
double precision
0

0.02

0.04
0.06
Verlet buffer (nm)

0.08

0.1

Figure 3.5: Energy drift per atom for an SPC/E water system at 300K with a time step of 2 fs and
a pair-list update period of 10 steps (pair-list life time: 18 fs). PME was used with ewald-rtol
set to 10−5 ; this parameter affects the shape of the potential at the cut-off. Error estimates due to
finite Verlet buffer size are shown for a 1 × 1 atom pair list and 4 × 4 atom pair list without and
with (dashed line) cancellation of positive and negative errors. Real energy drift is shown for simulations using double- and mixed-precision settings. Rounding errors in the SETTLE constraint
algorithm from the use of single precision causes the drift to become negative at large buffer size.
Note that at zero buffer size, the real drift is small because positive (H-H) and negative (O-H)
energy errors cancel.

size for longer neighborlist lifetimes in e.g. water, as constrained hydrogens are by far the fastest
particles, but they can not move further than 0.1 nm from the heavy atom they are connected to.
There is one important implementation detail that reduces the energy errors caused by the finite
Verlet buffer list size. The derivation above assumes a particle pair-list. However, the GROMACS
implementation uses a cluster pair-list for efficiency. The pair list consists of pairs of clusters of
4 particles in most cases, also called a 4 × 4 list, but the list can also be 4 × 8 (GPU CUDA
kernels and AVX 256-bit single precision kernels) or 4 × 2 (SSE double-precision kernels). This
means that the pair-list is effectively much larger than the corresponding 1 × 1 list. Thus slightly
beyond the pair-list cut-off there will still be a large fraction of particle pairs present in the list.
This fraction can be determined in a simulation and accurately estimated under some reasonable
assumptions. The fraction decreases with increasing pair-list range, meaning that a smaller buffer
can be used. For typical all-atom simulations with a cut-off of 0.9 nm this fraction is around 0.9,
which gives a reduction in the energy errors of a factor of 10. This reduction is taken into account
during the automatic Verlet buffer calculation and results in a smaller buffer size.
In Fig. 3.5 one can see that for small buffer sizes the drift of the total energy is much smaller
than the pair energy error tolerance, due to cancellation of errors. For larger buffer size, the error
estimate is a factor of 6 higher than drift of the total energy, or alternatively the buffer estimate is
0.024 nm too large. This is because the protons don’t move freely over 18 fs, but rather vibrate.

3.4. Molecular Dynamics

i’
111111111
000000000
000000000
111111111
000000000
111111111
j
000000
111111
000
111
0000
1111
000000
111111
0001111
111
0000
00000
11111
00000000000
11111111111
00000
11111
00000000000
11111111111
00000
11111
00000000000
11111111111
i
00000
11111
00000000000
11111111111
00000
11111
00000
11111
00000
11111
00000
11111
1111111111111111
0000000000000000

Figure 3.6: Grid search in two dimensions. The arrows are the box vectors.
Cut-off artifacts and switched interactions
With the Verlet scheme, the pair potentials are shifted to be zero at the cut-off, which makes the
potential the integral of the force. This is only possible in the group scheme if the shape of the
potential is such that its value is zero at the cut-off distance. However, there can still be energy
drift when the forces are non-zero at the cut-off. This effect is extremely small and often not
noticeable, as other integration errors (e.g. from constraints) may dominate. To completely avoid
cut-off artifacts, the non-bonded forces can be switched exactly to zero at some distance smaller
than the neighbor list cut-off (there are several ways to do this in GROMACS, see sec. 4.1.5). One
then has a buffer with the size equal to the neighbor list cut-off less the longest interaction cut-off.
Simple search
Due to eqns. 3.1 and 3.6, the vector r ij connecting images within the cut-off Rc can be found by
constructing:
r 000 = r j − r i
00

r ij

000

r − c ∗ round(rz000 /cz )
r 00 − b ∗ round(ry00 /by )
r 0 − a ∗ round(rx0 /ax )

(3.13)
(3.14)
(3.15)
(3.16)

When distances between two particles in a triclinic box are needed that do not obey eqn. 3.1, many
shifts of combinations of box vectors need to be considered to find the nearest image.
Grid search
The grid search is schematically depicted in Fig. 3.6. All particles are put on the NS grid, with the
smallest spacing ≥ Rc /2 in each of the directions. In the direction of each box vector, a particle

Chapter 3. Algorithms

i has three images. For each direction the image may be -1,0 or 1, corresponding to a translation
over -1, 0 or +1 box vector. We do not search the surrounding NS grid cells for neighbors of
i and then calculate the image, but rather construct the images first and then search neighbors
corresponding to that image of i. As Fig. 3.6 shows, some grid cells may be searched more than
once for different images of i. This is not a problem, since, due to the minimum image convention,
at most one image will “see” the j-particle. For every particle, fewer than 125 (53 ) neighboring
cells are searched. Therefore, the algorithm scales linearly with the number of particles. Although
the prefactor is large, the scaling behavior makes the algorithm far superior over the standard
O(N 2 ) algorithm when there are more than a few hundred particles. The grid search is equally
fast for rectangular and triclinic boxes. Thus for most protein and peptide simulations the rhombic
dodecahedron will be the preferred box shape.

Charge groups
Charge groups were originally introduced to reduce cut-off artifacts of Coulomb interactions.
When a plain cut-off is used, significant jumps in the potential and forces arise when atoms with
(partial) charges move in and out of the cut-off radius. When all chemical moieties have a net
charge of zero, these jumps can be reduced by moving groups of atoms with net charge zero,
called charge groups, in and out of the neighbor list. This reduces the cut-off effects from the
charge-charge level to the dipole-dipole level, which decay much faster. With the advent of full
range electrostatics methods, such as particle-mesh Ewald (sec. 4.8.2), the use of charge groups
is no longer required for accuracy. It might even have a slight negative effect on the accuracy or
efficiency, depending on how the neighbor list is made and the interactions are calculated.
But there is still an important reason for using “charge groups”: efficiency with the group cut-off
scheme. Where applicable, neighbor searching is carried out on the basis of charge groups which
are defined in the molecular topology. If the nearest image distance between the geometrical
centers of the atoms of two charge groups is less than the cut-off radius, all atom pairs between
the charge groups are included in the pair list. The neighbor searching for a water system, for
instance, is 32 = 9 times faster when each molecule is treated as a charge group. Also the highly
optimized water force loops (see sec. A.2.1) only work when all atoms in a water molecule form
a single charge group. Currently the name neighbor-search group would be more appropriate, but
the name charge group is retained for historical reasons. When developing a new force field, the
advice is to use charge groups of 3 to 4 atoms for optimal performance. For all-atom force fields
this is relatively easy, as one can simply put hydrogen atoms, and in some case oxygen atoms, in
the same charge group as the heavy atom they are connected to; for example: CH3 , CH2 , CH,
NH2 , NH, OH, CO2 , CO.
With the Verlet cut-off scheme, charge groups are ignored.

3.4.3

Compute forces

Potential energy
When forces are computed, the potential energy of each interaction term is computed as well. The
total potential energy is summed for various contributions, such as Lennard-Jones, Coulomb, and

3.4. Molecular Dynamics

bonded terms. It is also possible to compute these contributions for energy-monitor groups of
atoms that are separately defined (see sec. 3.3).
Kinetic energy and temperature
The temperature is given by the total kinetic energy of the N -particle system:
Ekin =

N
1X
mi vi2
2 i=1

(3.17)

From this the absolute temperature T can be computed using:
1
Ndf kT = Ekin
(3.18)
2
where k is Boltzmann’s constant and Ndf is the number of degrees of freedom which can be
computed from:
Ndf = 3N − Nc − Ncom
(3.19)
Here Nc is the number of constraints imposed on the system. When performing molecular dynamics Ncom = 3 additional degrees of freedom must be removed, because the three center-of-mass
velocities are constants of the motion, which are usually set to zero. When simulating in vacuo,
the rotation around the center of mass can also be removed, in this case Ncom = 6. When more
than one temperature-coupling group is used, the number of degrees of freedom for group i is:
i
Ndf
= (3N i − Nci )

3N − Nc − Ncom
3N − Nc

(3.20)

The kinetic energy can also be written as a tensor, which is necessary for pressure calculation in a
triclinic system, or systems where shear forces are imposed:
Ekin

N
1X
mi v i ⊗ v i
=
2 i

(3.21)

Pressure and virial
The pressure tensor P is calculated from the difference between kinetic energy Ekin and the virial
Ξ:
2
P = (Ekin − Ξ)
(3.22)
V
where V is the volume of the computational box. The scalar pressure P , which can be used for
pressure coupling in the case of isotropic systems, is computed as:
P = trace(P)/3

(3.23)

The virial Ξ tensor is defined as:
Ξ=−

1X
r ij ⊗ F ij
2 i rc , mdrun employs a smart algorithm to reduce the communication. Simply
communicating all charge groups within rmb would increase the amount of communication enormously. Therefore only charge-groups that are connected by bonded interactions to charge groups
which are not locally present are communicated. This leads to little extra communication, but also
to a slightly increased cost for the domain decomposition setup. In some cases, e.g. coarse-grained
simulations with a very short cut-off, one might want to set rmb by hand to reduce this cost.

3.17.5

Multiple-Program, Multiple-Data PME parallelization

Electrostatics interactions are long-range, therefore special algorithms are used to avoid summation over many atom pairs. In GROMACS this is usually PME (sec. 4.8.2). Since with PME all

Chapter 3. Algorithms
8 PP/PME ranks

6 PP ranks

2 PME ranks

Figure 3.14: Example of 8 ranks without (left) and with (right) MPMD. The PME communication
(red arrows) is much higher on the left than on the right. For MPMD additional PP - PME coordinate and force communication (blue arrows) is required, but the total communication complexity
is lower.

particles interact with each other, global communication is required. This will usually be the limiting factor for scaling with domain decomposition. To reduce the effect of this problem, we have
come up with a Multiple-Program, Multiple-Data approach [5]. Here, some ranks are selected to
do only the PME mesh calculation, while the other ranks, called particle-particle (PP) ranks, do
all the rest of the work. For rectangular boxes the optimal PP to PME rank ratio is usually 3:1,
for rhombic dodecahedra usually 2:1. When the number of PME ranks is reduced by a factor of
4, the number of communication calls is reduced by about a factor of 16. Or put differently, we
can now scale to 4 times more ranks. In addition, for modern 4 or 8 core machines in a network,
the effective network bandwidth for PME is quadrupled, since only a quarter of the cores will be
using the network connection on each machine during the PME calculations.
mdrun will by default interleave the PP and PME ranks. If the ranks are not number consecutively
inside the machines, one might want to use mdrun -ddorder pp_pme. For machines with a
real 3-D torus and proper communication software that assigns the ranks accordingly one should
use mdrun -ddorder cartesian.
To optimize the performance one should usually set up the cut-offs and the PME grid such that
the PME load is 25 to 33% of the total calculation load. grompp will print an estimate for this
load at the end and also mdrun calculates the same estimate to determine the optimal number
of PME ranks to use. For high parallelization it might be worthwhile to optimize the PME load
with the mdp settings and/or the number of PME ranks with the -npme option of mdrun. For
changing the electrostatics settings it is useful to know the accuracy of the electrostatics remains
nearly constant when the Coulomb cut-off and the PME grid spacing are scaled by the same factor.
Note that it is usually better to overestimate than to underestimate the number of PME ranks, since
the number of PME ranks is smaller than the number of PP ranks, which leads to less total waiting
time.
The PME domain decomposition can be 1-D or 2-D along the x and/or y axis. 2-D decomposition
is also known as pencil decomposition because of the shape of the domains at high parallelization.
1-D decomposition along the y axis can only be used when the PP decomposition has only 1 domain along x. 2-D PME decomposition has to have the number of domains along x equal to the
number of the PP decomposition. mdrun automatically chooses 1-D or 2-D PME decomposition
(when possible with the total given number of ranks), based on the minimum amount of commu-

3.18. Implicit solvation

nication for the coordinate redistribution in PME plus the communication for the grid overlap and
transposes. To avoid superfluous communication of coordinates and forces between the PP and
PME ranks, the number of DD cells in the x direction should ideally be the same or a multiple of
the number of PME ranks. By default, mdrun takes care of this issue.

3.17.6

Domain decomposition flow chart

In Fig. 3.15 a flow chart is shown for domain decomposition with all possible communication for
different algorithms. For simpler simulations, the same flow chart applies, without the algorithms
and communication for the algorithms that are not used.

3.18

Implicit solvation

Implicit solvent models provide an efficient way of representing the electrostatic effects of solvent
molecules, while saving a large piece of the computations involved in an accurate, aqueous description of the surrounding water in molecular dynamics simulations. Implicit solvation models
offer several advantages compared with explicit solvation, including eliminating the need for the
equilibration of water around the solute, and the absence of viscosity, which allows the protein to
more quickly explore conformational space.
Implicit solvent calculations in GROMACS can be done using the generalized Born-formalism,
and the Still [71], HCT [72], and OBC [73] models are available for calculating the Born radii.
Here, the free energy Gsolv of solvation is the sum of three terms, a solvent-solvent cavity term
(Gcav ), a solute-solvent van der Waals term (Gvdw ), and finally a solvent-solute electrostatics
polarization term (Gpol ).
The sum of Gcav and Gvdw corresponds to the (non-polar) free energy of solvation for a molecule
from which all charges have been removed, and is commonly called Gnp , calculated from the
total solvent accessible surface area multiplied with a surface tension. The total expression for the
solvation free energy then becomes:
Gsolv = Gnp + Gpol

(3.150)

Under the generalized Born model, Gpol is calculated from the generalized Born equation [71]:
n X
n
1 X
s
= 1−
i=1 j>i

qi qj

Gpol

2
rij

(3.151)

+ bi bj exp

2
−rij
4bi bj

In GROMACS, we have introduced the substitution [74]:
1
ci = √
bi

(3.152)

which makes it possible to introduce a cheap transformation to a new variable x when evaluating
each interaction, such that:

Chapter 3. Algorithms

Figure 3.15: Flow chart showing the algorithms and communication (arrows) for a standard MD
simulation with virtual sites, constraints and separate PME-mesh ranks.

3.18. Implicit solvation

rij
x= p
= rij ci cj
bi bj

(3.153)

In the end, the full re-formulation of 3.151 becomes:

Gpol = 1 −

n X
n

n
n
X
1 X
qi qj
1 X
p
ξ(x) = 1 −
qi ci
qj cj ξ(x)
i=1 j>i bi bj
i=1
j>i

(3.154)

The non-polar part (Gnp ) of Equation 3.150 is calculated directly from the Born radius of each
atom using a simple ACE type approximation by Schaefer et al. [75], including a simple loop
over all atoms. This requires only one extra solvation parameter, independent of atom type, but
differing slightly between the three Born radii models.

Chapter 3. Algorithms

Chapter 4
Interaction function and force
fields
To accommodate the potential functions used in some popular force fields (see 4.10), GROMACS
offers a choice of functions, both for non-bonded interaction and for dihedral interactions. They
are described in the appropriate subsections.
The potential functions can be subdivided into three parts
1. Non-bonded: Lennard-Jones or Buckingham, and Coulomb or modified Coulomb. The nonbonded interactions are computed on the basis of a neighbor list (a list of non-bonded atoms
within a certain radius), in which exclusions are already removed.
2. Bonded: covalent bond-stretching, angle-bending, improper dihedrals, and proper dihedrals.
These are computed on the basis of fixed lists.
3. Restraints: position restraints, angle restraints, distance restraints, orientation restraints and
dihedral restraints, all based on fixed lists.
4. Applied Forces: externally applied forces, see chapter 6.

4.1

Non-bonded interactions

Non-bonded interactions in GROMACS are pair-additive:
V (r 1 , . . . r N ) =

Vij (r ij );

(4.1)

i ∆φ


 0

for φ0 ≤ ∆φ

(4.84)

where ∆φ is a user defined angle and kdihr is the force constant. Note that in the input in topology
files, angles are given in degrees and force constants in kJ/mol/rad2 .

4.3. Restraints

89
15

−1

Vdisre (kJ mol )

0.1

0.2

0.3

0.4

0.5

r (nm)

Figure 4.15: Distance Restraint potential.

4.3.5

Distance restraints

Distance restraints add a penalty to the potential when the distance between specified pairs of
atoms exceeds a threshold value. They are normally used to impose experimental restraints from,
for instance, experiments in nuclear magnetic resonance (NMR), on the motion of the system.
Thus, MD can be used for structure refinement using NMR data. In GROMACS there are three
ways to impose restraints on pairs of atoms:
• Simple harmonic restraints: use [ bonds ] type 6 (see sec. 5.4.4).
• Piecewise linear/harmonic restraints: [ bonds ] type 10.
• Complex NMR distance restraints, optionally with pair, time and/or ensemble averaging.
The last two options will be detailed now.
The potential form for distance restraints is quadratic below a specified lower bound and between
two specified upper bounds, and linear beyond the largest bound (see Fig. 4.15).

Vdr (rij ) =

 1
2

2 kdr (rij − r0 )





 0

for

1

2


2 kdr (rij − r1 )



 1
2 kdr (r2

The forces are

Fi =

rij

< r0

for r0 ≤ rij

< r1

for r1 ≤ rij

< r2

(4.85)

− r1 )(2rij − r2 − r1 ) for r2 ≤ rij


r
−kdr (rij − r0 ) rijij







 0

r


−kdr (rij − r1 ) rijij





 −k (r − r ) r ij
1 rij
dr 2

rij

< r0

for r0 ≤ rij

< r1

for r1 ≤ rij

< r2

for

for r2 ≤ rij

(4.86)

Chapter 4. Interaction function and force fields

For restraints not derived from NMR data, this functionality will usually suffice and a section of
[ bonds ] type 10 can be used to apply individual restraints between pairs of atoms, see 5.8.1.
For applying restraints derived from NMR measurements, more complex functionality might be required, which is provided through the [ distance_restraints ] section and is described
below.
Time averaging
Distance restraints based on instantaneous distances can potentially reduce the fluctuations in a
molecule significantly. This problem can be overcome by restraining to a time averaged distance [96]. The forces with time averaging are:

r ij
a
r̄ij < r0

 −kdr (r̄ij − r0 ) rij for
Fi =






 0

a (r̄ − r ) r ij


−kdr
ij
1 rij





 −k a (r − r ) r ij
1 rij
dr 2

for r0 ≤ r̄ij

< r1

for r1 ≤ r̄ij

< r2

(4.87)

for r2 ≤ r̄ij

where r̄ij is given by an exponential running average with decay time τ :
−3
>−1/3
r̄ij = < rij

(4.88)

a is switched on slowly to compensate for the lack of history at the beginning
The force constant kdr
of the simulation:

t
a
kdr
= kdr 1 − exp −
(4.89)
τ
Because of the time averaging, we can no longer speak of a distance restraint potential.

This way an atom can satisfy two incompatible distance restraints on average by moving between
two positions. An example would be an amino acid side-chain that is rotating around its χ dihedral
angle, thereby coming close to various other groups. Such a mobile side chain can give rise to
multiple NOEs that can not be fulfilled by a single structure.
The computation of the time averaged distance in the mdrun program is done in the following
fashion:
r−3 ij (0) = rij (0)−3

i
−3 1 − exp − ∆t
r−3 ij (t) = r−3 ij (t − ∆t) exp − ∆t
+
r
(t)
ij
τ
τ

(4.90)

When a pair is within the bounds, it can still feel a force because the time averaged distance can
still be beyond a bound. To prevent the protons from being pulled too close together, a mixed
approach can be used. In this approach, the penalty is zero when the instantaneous distance is
within the bounds, otherwise the violation is the square root of the product of the instantaneous
violation and the time averaged violation:

q
r ij
a

for
rij < r0 and r̄ij < r0
k

 dr (rij − r0 )(r̄ij − r0 ) rij
Fi =




a min
−kdr







(rij − r1 )(r̄ij − r1 ), r2 − r1

r ij
rij

for

rij > r1 and r̄ij > r1

otherwise
(4.91)

4.3. Restraints

Averaging over multiple pairs
Sometimes it is unclear from experimental data which atom pair gives rise to a single NOE, in
other occasions it can be obvious that more than one pair contributes due to the symmetry of the
system, e.g. a methyl group with three protons. For such a group, it is not possible to distinguish
between the protons, therefore they should all be taken into account when calculating the distance
between this methyl group and another proton (or group of protons). Due to the physical nature of
magnetic resonance, the intensity of the NOE signal is inversely proportional to the sixth power
of the inter-atomic distance. Thus, when combining atom pairs, a fixed list of N restraints may be
taken together, where the apparent “distance” is given by:
rN (t) =

" N
X

#−1/6

r̄n (t)

−6

(4.92)

n=1

where we use rij or eqn. 4.88 for the r̄n . The rN of the instantaneous and time-averaged distances
can be combined to do a mixed restraining, as indicated above. As more pairs of protons contribute
to the same NOE signal, the intensity will increase, and the summed “distance” will be shorter than
any of its components due to the reciprocal summation.
There are two options for distributing the forces over the atom pairs. In the conservative option,
the force is defined as the derivative of the restraint potential with respect to the coordinates. This
results in a conservative potential when time averaging is not used. The force distribution over
the pairs is proportional to r−6 . This means that a close pair feels a much larger force than a
distant pair, which might lead to a molecule that is “too rigid.” The other option is an equal
force distribution. In this case each pair feels 1/N of the derivative of the restraint potential with
respect to rN . The advantage of this method is that more conformations might be sampled, but the
non-conservative nature of the forces can lead to local heating of the protons.
It is also possible to use ensemble averaging using multiple (protein) molecules. In this case the
bounds should be lowered as in:
r1
r2

= r1 ∗ M −1/6
= r2 ∗ M −1/6

(4.93)

where M is the number of molecules. The GROMACS preprocessor grompp can do this automatically when the appropriate option is given. The resulting “distance” is then used to calculate
the scalar force according to:

Fi =









0
r
kdr (rN − r1 ) rijij
r
kdr (r2 − r1 ) rijij

rN < r1
r1 ≤ rN < r2

(4.94)

rN ≥ r2

where i and j denote the atoms of all the pairs that contribute to the NOE signal.
Using distance restraints
A list of distance restrains based on NOE data can be added to a molecule definition in your
topology file, like in the following example:

Chapter 4. Interaction function and force fields

[ distance_restraints ]
; ai
aj
type
index
10
16
1
0
10
28
1
1
10
46
1
1
16
22
1
2
16
34
1
3

type’
1
1
1
1
1

low
0.0
0.0
0.0
0.0
0.0

up1
0.3
0.3
0.3
0.3
0.5

up2
0.4
0.4
0.4
0.4
0.6

fac
1.0
1.0
1.0
2.5
1.0

In this example a number of features can be found. In columns ai and aj you find the atom
numbers of the particles to be restrained. The type column should always be 1. As explained
in 4.3.5, multiple distances can contribute to a single NOE signal. In the topology this can be
set using the index column. In our example, the restraints 10-28 and 10-46 both have index 1,
therefore they are treated simultaneously. An extra requirement for treating restraints together is
that the restraints must be on successive lines, without any other intervening restraint. The type’
column will usually be 1, but can be set to 2 to obtain a distance restraint that will never be timeand ensemble-averaged; this can be useful for restraining hydrogen bonds. The columns low,
up1, and up2 hold the values of r0 , r1 , and r2 from eqn. 4.85. In some cases it can be useful
to have different force constants for some restraints; this is controlled by the column fac. The
force constant in the parameter file is multiplied by the value in the column fac for each restraint.
Information for each restraint is stored in the energy file and can be processed and plotted with
gmx nmr.

4.3.6

Orientation restraints

This section describes how orientations between vectors, as measured in certain NMR experiments, can be calculated and restrained in MD simulations. The presented refinement methodology and a comparison of results with and without time and ensemble averaging have been published [97].
Theory
In an NMR experiment, orientations of vectors can be measured when a molecule does not tumble completely isotropically in the solvent. Two examples of such orientation measurements are
residual dipolar couplings (between two nuclei) or chemical shift anisotropies. An observable for
a vector r i can be written as follows:
2
δi = tr(SDi )
3

(4.95)

where S is the dimensionless order tensor of the molecule. The tensor Di is given by:




3xx − 1 3xy
3xz
ci 

3yy − 1 3yz
Di =
 3xy

α
kr i k
3xz
3yz
3zz − 1
with: x =

ri,x
,
kr i k

ri,y
,
kr i k

ri,z
kr i k

(4.96)

(4.97)

4.3. Restraints

For a dipolar coupling r i is the vector connecting the two nuclei, α = 3 and the constant ci is
given by:
µ0 i i h̄
ci =
γ γ
(4.98)
4π 1 2 4π
where γ1i and γ2i are the gyromagnetic ratios of the two nuclei.
The order tensor is symmetric and has trace zero. Using a rotation matrix T it can be transformed
into the following form:




− 1 (1 − η)
0
0

 2
T
1
0
− 2 (1 + η) 0 
T ST = s 
0
0
1

(4.99)

where −1 ≤ s ≤ 1 and 0 ≤ η ≤ 1. s is called the order parameter and η the asymmetry of
the order tensor S. When the molecule tumbles isotropically in the solvent, s is zero, and no
orientational effects can be observed because all δi are zero.
Calculating orientations in a simulation
For reasons which are explained below, the D matrices are calculated which respect to a reference
orientation of the molecule. The orientation is defined by a rotation matrix R, which is needed to
least-squares fit the current coordinates of a selected set of atoms onto a reference conformation.
The reference conformation is the starting conformation of the simulation. In case of ensemble averaging, which will be treated later, the structure is taken from the first subsystem. The calculated
Dci matrix is given by:
Dci (t) = R(t)Di (t)RT (t)
(4.100)
The calculated orientation for vector i is given by:
2
δic (t) = tr(S(t)Dci (t))
3

(4.101)

The order tensor S(t) is usually unknown. A reasonable choice for the order tensor is the tensor
which minimizes the (weighted) mean square difference between the calculated and the observed
orientations:
!
M SD(t) =

N
X
i=1

−1 N
X

wi (δic (t) − δiexp )2

(4.102)

i=1

To properly combine different types of measurements, the unit of wi should be such that all terms
are dimensionless. This means the unit of wi is the unit of δi to the power −2. Note that scaling
all wi with a constant factor does not influence the order tensor.
Time averaging
Since the tensors Di fluctuate rapidly in time, much faster than can be observed in an experiment,
they should be averaged over time in the simulation. However, in a simulation the time and the
number of copies of a molecule are limited. Usually one can not obtain a converged average of the
Di tensors over all orientations of the molecule. If one assumes that the average orientations of

Chapter 4. Interaction function and force fields

the r i vectors within the molecule converge much faster than the tumbling time of the molecule,
the tensor can be averaged in an axis system that rotates with the molecule, as expressed by equation (4.100). The time-averaged tensors are calculated using an exponentially decaying memory
function:

Z t
t−u
c
Di (u) exp −
du
τ

Dai (t) = u=tZ0 t
(4.103)
t−u
exp −
du
τ
u=t0
Assuming that the order tensor S fluctuates slower than the Di , the time-averaged orientation can
be calculated as:
2
δia (t) = tr(S(t)Dai (t))
(4.104)
3
where the order tensor S(t) is calculated using expression (4.102) with δic (t) replaced by δia (t).
Restraining
The simulated structure can be restrained by applying a force proportional to the difference between the calculated and the experimental orientations. When no time averaging is applied, a
proper potential can be defined as:
N
1 X
wi (δic (t) − δiexp )2
V = k
2 i=1

(4.105)

where the unit of k is the unit of energy. Thus the effective force constant for restraint i is kwi .
The forces are given by minus the gradient of V . The force Fi working on vector r i is:
Fi (t) = −

dV
dr i

dδi (t)
dr i

2ci
2+α
T
T
T
= −kwi (δic (t) − δiexp )
2R
SRr
−
tr(R
SRr
r
)r
i
i i
i
krk2+α
krk2
= −kwi (δic (t) − δiexp )

Ensemble averaging
Ensemble averaging can be applied by simulating a system of M subsystems that each contain
an identical set of orientation restraints. The systems only interact via the orientation restraint
potential which is defined as:
N
1 X
V =M k
wi hδic (t) − δiexp i2
2 i=1

(4.106)

The force on vector r i,m in subsystem m is given by:
Fi,m (t) = −

c (t)
dδi,m
dV
= −kwi hδic (t) − δiexp i
dr i,m
dr i,m

(4.107)

4.3. Restraints

Time averaging
When using time averaging it is not possible to define a potential. We can still define a quantity
that gives a rough idea of the energy stored in the restraints:
N
1 X
V = M ka
wi hδia (t) − δiexp i2
2 i=1

(4.108)

The force constant ka is switched on slowly to compensate for the lack of history at times close to
t0 . It is exactly proportional to the amount of average that has been accumulated:
1
k =k
τ
a

Z t

t−u
exp −
du
τ
u=t0

(4.109)

What really matters is the definition of the force. It is chosen to be proportional to the square root
of the product of the time-averaged and the instantaneous deviation. Using only the time-averaged
deviation induces large oscillations. The force is given by:
Fi,m (t) =



 0

for a b ≤ 0

c (t)
a √ dδi,m
a

ab
 k wi
|a|
dr i,m

for a b > 0

(4.110)

a = hδia (t) − δiexp i
b = hδic (t) − δiexp i
Using orientation restraints
Orientation restraints can be added to a molecule definition in the topology file in the section
[ orientation_restraints ]. Here we give an example section containing five N-H
residual dipolar coupling restraints:
[ orientation_restraints ]
; ai
aj type exp. label
;
31
32
1
1
3
43
44
1
1
4
55
56
1
1
5
65
66
1
1
6
73
74
1
1
7

alpha
Hz
3
3
3
3
3

const.
nm^3
6.083
6.083
6.083
6.083
6.083

obs.
Hz
-6.73
-7.87
-7.13
-2.57
-2.10

weight
Hz^-2
1.0
1.0
1.0
1.0
1.0

The unit of the observable is Hz, but one can choose any other unit. In columns ai and aj you
find the atom numbers of the particles to be restrained. The type column should always be 1. The
exp. column denotes the experiment number, starting at 1. For each experiment a separate order
tensor S is optimized. The label should be a unique number larger than zero for each restraint. The
alpha column contains the power α that is used in equation (4.96) to calculate the orientation.
The const. column contains the constant ci used in the same equation. The constant should

Chapter 4. Interaction function and force fields

have the unit of the observable times nmα . The column obs. contains the observable, in any unit
you like. The last column contains the weights wi ; the unit should be the inverse of the square of
the unit of the observable.
Some parameters for orientation restraints can be specified in the grompp.mdp file, for a study
of the effect of different force constants and averaging times and ensemble averaging see [97].
Information for each restraint is stored in the energy file and can be processed and plotted with
gmx nmr.

4.4

Polarization

Polarization can be treated by GROMACS by attaching shell (Drude) particles to atoms and/or
virtual sites. The energy of the shell particle is then minimized at each time step in order to remain
on the Born-Oppenheimer surface.

4.4.1

Simple polarization

This is implemented as a harmonic potential with equilibrium distance 0. The input given in the
topology file is the polarizability α (in GROMACS units) as follows:
[ polarization ]
; Atom i j type
1
2 1

alpha
0.001

in this case the polarizability volume is 0.001 nm3 (or 1 Å3 ). In order to compute the harmonic
force constant kcs (where cs stands for core-shell), the following is used [45]:
kcs =

qs2
α

(4.111)

where qs is the charge on the shell particle.

4.4.2

Anharmonic polarization

For the development of the Drude force field by Roux and McKerell [98] it was found that some
particles can overpolarize and this was fixed by introducing a higher order term in the polarization
energy:
kcs 2
2 rcs

Vpol =
=

kcs 2
2 rcs

+ khyp (rcs −

δ)4

rcs ≤ δ

(4.112)

rcs > δ

(4.113)

where δ is a user-defined constant that is set to 0.02 nm for anions in the Drude force field [99].
Since this original introduction it has also been used in other atom types [98].
[ polarization ]
;Atom i j
type
1
2
2

alpha (nm^3)
0.001786

delta
0.02

khyp
16.736e8

The above force constant khyp corresponds to 4·108 kcal/mol/nm4 , hence the strange number.

4.5. Free energy interactions

4.4.3

Water polarization

A special potential for water that allows anisotropic polarization of a single shell particle [45].

4.4.4

Thole polarization

Based on early work by Thole [100], Roux and coworkers have implemented potentials for molecules
like ethanol [101, 102, 103]. Within such molecules, there are intra-molecular interactions between shell particles, however these must be screened because full Coulomb would be too strong.
The potential between two shell particles i and j is:
qi qj
r̄ij
1− 1+
rij
2

Vthole =

exp−r̄ij

(4.114)

Note that there is a sign error in Equation 1 of Noskov et al. [103]:
r̄ij = a

rij
(αi αj )1/6

(4.115)

where a is a magic (dimensionless) constant, usually chosen to be 2.6 [103]; αi and αj are the
polarizabilities of the respective shell particles.

4.5

Free energy interactions

This section describes the λ-dependence of the potentials used for free energy calculations (see
sec. 3.12). All common types of potentials and constraints can be interpolated smoothly from state
A (λ = 0) to state B (λ = 1) and vice versa. All bonded interactions are interpolated by linear
interpolation of the interaction parameters. Non-bonded interactions can be interpolated linearly
or via soft-core interactions.
Starting in GROMACS 4.6, λ is a vector, allowing different components of the free energy transformation to be carried out at different rates. Coulomb, Lennard-Jones, bonded, and restraint terms
can all be controlled independently, as described in the .mdp options.

Harmonic potentials
The example given here is for the bond potential, which is harmonic in GROMACS. However,
these equations apply to the angle potential and the improper dihedral potential as well.
Vb =
∂Vb
∂λ

ih
i2
1h
B
(1 − λ)kbA + λkbB b − (1 − λ)bA
0 − λb0
2
h
i2
1 B
B
(kb − kbA ) b − (1 − λ)bA
+
0 + λb0
2
h

B
A
B
(bA
0 − b0 ) b − (1 − λ)b0 − λb0

(1 − λ)kbA + λkbB

(4.116)

(4.117)

Chapter 4. Interaction function and force fields

GROMOS-96 bonds and angles
Fourth-power bond stretching and cosine-based angle potentials are interpolated by linear interpolation of the force constant and the equilibrium position. Formulas are not given here.
Proper dihedrals
For the proper dihedrals, the equations are somewhat more complicated:
Vd =
∂Vd
∂λ

(1 − λ)kdA + λkdB

B
1 + cos nφ φ − (1 − λ)φA
s − λφs

B
= (kdB − kdA ) 1 + cos nφ φ − (1 − λ)φA
s − λφs

(4.118)

A
A
B
A
B
(φB
s − φs ) (1 − λ)kd − λkd sin nφ φ − (1 − λ)φs − λφs

(4.119)

Note: that the multiplicity nφ can not be parameterized because the function should remain periodic on the interval [0, 2π].
Tabulated bonded interactions
For tabulated bonded interactions only the force constant can interpolated:
V
∂V
∂λ

= ((1 − λ)k A + λk B ) f

(4.120)

= (k B − k A ) f

(4.121)

Coulomb interaction
The Coulomb interaction between two particles of which the charge varies with λ is:
Vc =
∂Vc
∂λ
where f =

1
4πε0

f
εrf rij
f
εrf rij

(1 − λ)qiA qjA + λ qiB qjB

−qiA qjA + qiB qjB

(4.122)
(4.123)

= 138.935 458 (see chapter 2).

Coulomb interaction with reaction field
The Coulomb interaction including a reaction field, between two particles of which the charge
varies with λ is:
"

Vc
∂Vc
∂λ

h
i
1
2
= f
+ krf rij
− crf (1 − λ)qiA qjA + λ qiB qjB
rij
"

(4.124)

h
i
1
2
= f
+ krf rij
− crf −qiA qjA + qiB qjB
rij

(4.125)

Note that the constants krf and crf are defined using the dielectric constant εrf of the medium
(see sec. 4.1.4).

4.5. Free energy interactions

Lennard-Jones interaction
For the Lennard-Jones interaction between two particles of which the atom type varies with λ we
can write:
VLJ

A + λ CB
(1 − λ)C6A + λ C6B
(1 − λ)C12
12
−
12
6
rij
rij

(4.126)

∂VLJ
∂λ

B − CA
C6B − C6A
C12
12
−
12
6
rij
rij

(4.127)

It should be noted that it is also possible to express a pathway from state A to state B using σ and
(see eqn. 4.5). It may seem to make sense physically to vary the force field parameters σ and
rather than the derived parameters C12 and C6 . However, the difference between the pathways in
parameter space is not large, and the free energy itself does not depend on the pathway, so we use
the simple formulation presented above.
Kinetic Energy
When the mass of a particle changes, there is also a contribution of the kinetic energy to the free
energy (note that we can not write the momentum p as mv, since that would result in the sign of
∂Ek
∂λ being incorrect [104]):
1
p2
2 (1 − λ)mA + λmB
1
p2 (mB − mA )
= −
2 ((1 − λ)mA + λmB )2

Ek =
∂Ek
∂λ

(4.128)
(4.129)

after taking the derivative, we can insert p = mv, such that:
∂Ek
1
= − v 2 (mB − mA )
∂λ
2

(4.130)

Constraints
The constraints are formally part of the Hamiltonian, and therefore they give a contribution to the
free energy. In GROMACS this can be calculated using the LINCS or the SHAKE algorithm. If
we have k = 1 . . . K constraint equations gk for LINCS, then
gk = |r k | − dk

(4.131)

where r k is the displacement vector between two particles and dk is the constraint distance between the two particles. We can express the fact that the constraint distance has a λ dependency
by
B
dk = (1 − λ)dA
(4.132)
k + λdk
Thus the λ-dependent constraint equation is

B
gk = |r k | − (1 − λ)dA
k + λdk .

(4.133)

100

Chapter 4. Interaction function and force fields
5
LJ, α=0
LJ, α=1.5
LJ, α=2
3/r, α=0
3/r, α=1.5
3/r, α=2

Vsc

3
2
1
0
−1

0.5

1.5
r

2.5

A = C B = C B = 1.
Figure 4.16: Soft-core interactions at λ = 0.5, with p = 2 and C6A = C12
6
12

The (zero) contribution G to the Hamiltonian from the constraints (using Lagrange multipliers λk ,
which are logically distinct from the free-energy λ) is
G =

K
X

λk gk

(4.134)

∂G ∂dk
∂dk ∂λ

(4.135)

∂G
∂λ

= −

K
X

A
λk dB
k − dk

(4.136)

For SHAKE, the constraint equations are
gk = r 2k − d2k

(4.137)

with dk as before, so
∂G
∂λ

4.5.1

= −2

K
X

A
λk dB
k − dk

(4.138)

Soft-core interactions

In a free-energy calculation where particles grow out of nothing, or particles disappear, using the
the simple linear interpolation of the Lennard-Jones and Coulomb potentials as described in Equations 4.127 and 4.125 may lead to poor convergence. When the particles have nearly disappeared,
or are close to appearing (at λ close to 0 or 1), the interaction energy will be weak enough for
particles to get very close to each other, leading to large fluctuations in the measured values of
∂V /∂λ (which, because of the simple linear interpolation, depends on the potentials at both the
endpoints of λ).

4.5. Free energy interactions

101

To circumvent these problems, the singularities in the potentials need to be removed. This can be
done by modifying the regular Lennard-Jones and Coulomb potentials with “soft-core” potentials
that limit the energies and forces involved at λ values between 0 and 1, but not at λ = 0 or 1.
In GROMACS the soft-core potentials Vsc are shifted versions of the regular potentials, so that the
singularity in the potential and its derivatives at r = 0 is never reached:
Vsc (r) = (1 − λ)V A (rA ) + λV B (rB )

(4.139)

rA =

6 p
ασA
λ + r6

rB =

6
ασB
(1 − λ)p + r6

(4.140)
1
6

(4.141)

where V A and V B are the normal “hard core” Van der Waals or electrostatic potentials in state A
(λ = 0) and state B (λ = 1) respectively, α is the soft-core parameter (set with sc_alpha in
the .mdp file), p is the soft-core λ power (set with sc_power), σ is the radius of the interaction,
which is (C12 /C6 )1/6 or an input parameter (sc_sigma) when C6 or C12 is zero.
For intermediate λ, rA and rB alter the interactions very little for r > α1/6 σ and quickly switch
the soft-core interaction to an almost constant value for smaller r (Fig. 4.16). The force is:
∂Vsc (r)
r
Fsc (r) = −
= (1 − λ)F A (rA )
∂r
rA

r
+ λF (rB )
rB
B

(4.142)

where F A and F B are the “hard core” forces. The contribution to the derivative of the free energy
is:
∂Vsc (r)
∂λ

= V B (rB ) − V A (rA ) + (1 − λ)

∂V A (rA ) ∂rA
∂V B (rB ) ∂rB
+λ
∂rA
∂λ
∂rB
∂λ

= V B (rB ) − V A (rA ) +
i
pα h B
−5 6 p−1
−5 6
σA λ
(4.143)
σB (1 − λ)p−1 − (1 − λ)F A (rA )rA
λF (rB )rB
6
The original GROMOS Lennard-Jones soft-core function [105] uses p = 2, but p = 1 gives a
smoother ∂H/∂λ curve.
Another issue that should be considered is the soft-core effect of hydrogens without Lennard-Jones
interaction. Their soft-core σ is set with sc-sigma in the .mdp file. These hydrogens produce
peaks in ∂H/∂λ at λ is 0 and/or 1 for p = 1 and close to 0 and/or 1 with p = 2. Lowering
sc-sigma will decrease this effect, but it will also increase the interactions with hydrogens
relative to the other interactions in the soft-core state.
When soft-core potentials are selected (by setting sc-alpha >0), and the Coulomb and LennardJones potentials are turned on or off sequentially, then the Coulombic interaction is turned off
linearly, rather than using soft-core interactions, which should be less statistically noisy in most
cases. This behavior can be overwritten by using the mdp option sc-coul to yes. Note that the
sc-coul is only taken into account when lambda states are used, not with couple-lambda0 /
couple-lambda1, and you can still turn off soft-core interactions by setting sc-alpha=0.
Additionally, the soft-core interaction potential is only applied when either the A or B state has
zero interaction potential. If both A and B states have nonzero interaction potential, default linear
scaling described above is used. When both Coulombic and Lennard-Jones interactions are turned

102

Chapter 4. Interaction function and force fields

i+1

i+3

i+2

i+4

Figure 4.17: Atoms along an alkane chain.
off simultaneously, a soft-core potential is used, and a hydrogen is being introduced or deleted,
the sigma is set to sc-sigma-min, which itself defaults to sc-sigma-default.
Recently, a new formulation of the soft-core approach has been derived that in most cases gives
lower and more even statistical variance than the standard soft-core path described above. [106,
107] Specifically, we have:
Vsc (r) = (1 − λ)V A (rA ) + λV B (rB )

1
48

rA =

48 p
ασA
λ + r48

rB =

48
ασB
(1 − λ)p + r48

(4.144)
(4.145)

1
48

(4.146)

This “1-1-48” path is also implemented in GROMACS. Note that for this path the soft core α
should satisfy 0.001 < α < 0.003, rather than α ≈ 0.5.

4.6
4.6.1

Methods
Exclusions and 1-4 Interactions.

Atoms within a molecule that are close by in the chain, i.e. atoms that are covalently bonded,
or linked by one or two atoms are called first neighbors, second neighbors and third neighbors,
respectively (see Fig. 4.17). Since the interactions of atom i with atoms i+1 and i+2 are mainly
quantum mechanical, they can not be modeled by a Lennard-Jones potential. Instead it is assumed
that these interactions are adequately modeled by a harmonic bond term or constraint (i, i+1) and
a harmonic angle term (i, i+2). The first and second neighbors (atoms i+1 and i+2) are therefore
excluded from the Lennard-Jones interaction list of atom i; atoms i+1 and i+2 are called exclusions
of atom i.
For third neighbors, the normal Lennard-Jones repulsion is sometimes still too strong, which
means that when applied to a molecule, the molecule would deform or break due to the internal
strain. This is especially the case for carbon-carbon interactions in a cis-conformation (e.g. cisbutane). Therefore, for some of these interactions, the Lennard-Jones repulsion has been reduced
in the GROMOS force field, which is implemented by keeping a separate list of 1-4 and normal Lennard-Jones parameters. In other force fields, such as OPLS [108], the standard LennardJones parameters are reduced by a factor of two, but in that case also the dispersion (r−6 ) and the
Coulomb interaction are scaled. GROMACS can use either of these methods.

4.6. Methods

4.6.2

103

Charge Groups

In principle, the force calculation in MD is an O(N 2 ) problem. Therefore, we apply a cut-off for
non-bonded force (NBF) calculations; only the particles within a certain distance of each other
are interacting. This reduces the cost to O(N ) (typically 100N to 200N ) of the NBF. It also
introduces an error, which is, in most cases, acceptable, except when applying the cut-off implies
the creation of charges, in which case you should consider using the lattice sum methods provided
by GROMACS.
Consider a water molecule interacting with another atom. If we would apply a plain cut-off on an
atom-atom basis we might include the atom-oxygen interaction (with a charge of −0.82) without
the compensating charge of the protons, and as a result, induce a large dipole moment over the
system. Therefore, we have to keep groups of atoms with total charge 0 together. These groups are
called charge groups. Note that with a proper treatment of long-range electrostatics (e.g. particlemesh Ewald (sec. 4.8.2), keeping charge groups together is not required.

4.6.3

Treatment of Cut-offs in the group scheme

GROMACS is quite flexible in treating cut-offs, which implies there can be quite a number of
parameters to set. These parameters are set in the input file for grompp. There are two sort of
parameters that affect the cut-off interactions; you can select which type of interaction to use in
each case, and which cut-offs should be used in the neighbor searching.
For both Coulomb and van der Waals interactions there are interaction type selectors (termed
vdwtype and coulombtype) and two parameters, for a total of six non-bonded interaction
parameters. See the User Guide for a complete description of these parameters.
In the group cut-off scheme, all of the interaction functions in Table 4.2 require that neighbor
searching be done with a radius at least as large as the rc specified for the functional form, because
of the use of charge groups. The extra radius is typically of the order of 0.25 nm (roughly the
largest distance between two atoms in a charge group plus the distance a charge group can diffuse
within neighbor list updates).

Coulomb

VdW

Type
Plain cut-off
Reaction field
Shift function
Switch function
Plain cut-off
Shift function
Switch function

Parameters
rc , εr
rc , εrf
r1 , rc , εr
r1 , rc , εr
rc
r1 , rc
r1 , rc

Table 4.2: Parameters for the different functional forms of the non-bonded interactions.

104

Chapter 4. Interaction function and force fields

θ
a

1-a

|d |

|b |

3fd

|c |

3out

3fad

4fdn

Figure 4.18: The six different types of virtual site construction in GROMACS. The constructing
atoms are shown as black circles, the virtual sites in gray.

4.7

Virtual interaction sites

Virtual interaction sites (called dummy atoms in GROMACS versions before 3.3) can be used in
GROMACS in a number of ways. We write the position of the virtual site r s as a function of the
positions of other particles r i : r s = f (r 1 ..r n ). The virtual site, which may carry charge or be
involved in other interactions, can now be used in the force calculation. The force acting on the
virtual site must be redistributed over the particles with mass in a consistent way. A good way to
do this can be found in ref. [109]. We can write the potential energy as:
V = V (r s , r 1 , . . . , r n ) = V ∗ (r 1 , . . . , r n )

(4.147)

The force on the particle i is then:
Fi = −

∂V ∗
∂V
∂V ∂r s
=−
−
= F direct
+ F 0i
i
∂r i
∂r i ∂r s ∂r i

(4.148)

The first term is the normal force. The second term is the force on particle i due to the virtual site,
which can be written in tensor notation:




0
Fi = 




∂xs
∂xi
∂xs
∂yi
∂xs
∂zi

∂ys
∂xi
∂ys
∂yi
∂ys
∂zi

∂zs
∂xi
∂zs
∂yi
∂zs
∂zi





Fs




(4.149)

where F s is the force on the virtual site and xs , ys and zs are the coordinates of the virtual site. In
this way, the total force and the total torque are conserved [109].
The computation of the virial (eqn. 3.24) is non-trivial when virtual sites are used. Since the virial
involves a summation over all the atoms (rather than virtual sites), the forces must be redistributed
from the virtual sites to the atoms (using eqn. 4.149) before computation of the virial. In some
special cases where the forces on the atoms can be written as a linear combination of the forces on
the virtual sites (types 2 and 3 below) there is no difference between computing the virial before
and after the redistribution of forces. However, in the general case redistribution should be done
first.
There are six ways to construct virtual sites from surrounding atoms in GROMACS, which we
classify by the number of constructing atoms. Note that all site types mentioned can be constructed

4.7. Virtual interaction sites

105

from types 3fd (normalized, in-plane) and 3out (non-normalized, out of plane). However, the
amount of computation involved increases sharply along this list, so we strongly recommended
using the first adequate virtual site type that will be sufficient for a certain purpose. Fig. 4.18
depicts 6 of the available virtual site constructions. The conceptually simplest construction types
are linear combinations:
rs =

N
X

wi r i

(4.150)

i=1

The force is then redistributed using the same weights:
F 0i = wi F s

(4.151)

The types of virtual sites supported in GROMACS are given in the list below. Constructing atoms
in virtual sites can be virtual sites themselves, but only if they are higher in the list, i.e. virtual
sites can be constructed from “particles” that are simpler virtual sites.
2. As a linear combination of two atoms (Fig. 4.18 2):
wi = 1 − a , wj = a

(4.152)

In this case the virtual site is on the line through atoms i and j.
3. As a linear combination of three atoms (Fig. 4.18 3):
wi = 1 − a − b , wj = a , wk = b

(4.153)

In this case the virtual site is in the plane of the other three particles.
3fd. In the plane of three atoms, with a fixed distance (Fig. 4.18 3fd):
rs = ri + b

r ij + ar jk
|r ij + ar jk |

(4.154)

In this case the virtual site is in the plane of the other three particles at a distance of |b| from
i. The force on particles i, j and k due to the force on the virtual site can be computed as:
F 0i

F 0j

= (1 − a)γ(F s − p)

F 0k

F s − γ(F s − p)
aγ(F s − p)

b
|r ij + ar jk |
r is · F s
p=
r is
r is · r is

γ=
where

(4.155)

3fad. In the plane of three atoms, with a fixed angle and distance (Fig. 4.18 3fad):
r s = r i + d cos θ

r ij · r jk
r ij
r⊥
+ d sin θ
where r ⊥ = r jk −
r ij
|r ij |
|r ⊥ |
r ij · r ij

(4.156)

In this case the virtual site is in the plane of the other three particles at a distance of |d| from
i at an angle of α with r ij . Atom k defines the plane and the direction of the angle. Note
that in this case b and α must be specified, instead of a and b (see also sec. 5.2.2). The force

106

Chapter 4. Interaction function and force fields
on particles i, j and k due to the force on the virtual site can be computed as (with r ⊥ as
defined in eqn. 4.156):
F 0i
F 0j

= Fs −

d cos θ
F1 +
|r ij |

d sin θ
|r ⊥ |

r ij · r jk
F2 + F3
r ij · r ij

d cos θ
F1 −
|r ij |

d sin θ
|r ⊥ |

r ij · r jk
F2 + F3
F2 +
r ij · r ij

d sin θ
F2
|r ⊥ |

F 0k =
where F 1 = F s −

r ij · F s
r⊥ · F s
r ij · F s
r ij , F 2 = F 1 −
r⊥
r ⊥ and F 3 =
r ij · r ij
r⊥ · r⊥
r ij · r ij
(4.157)

3out. As a non-linear combination of three atoms, out of plane (Fig. 4.18 3out):
r s = r i + ar ij + br ik + c(r ij × r ik )

(4.158)

This enables the construction of virtual sites out of the plane of the other atoms. The force
on particles i, j and k due to the force on the virtual site can be computed as:
F 0j

a
−c zik c yik


a
−c xik  F s
=  c zik
−c yik c xik
a

F 0k

b

=  −c zij
c yij

F 0i

= F s − F 0j − F 0k





c zij
b
−c xij

−c yij

c xij  F s
b


(4.159)

4fdn. From four atoms, with a fixed distance, see separate Fig. 4.19. This construction is a bit
complex, in particular since the previous type (4fd) could be unstable which forced us to
introduce a more elaborate construction:

rja = a rik − rij = a (xk − xi ) − (xj − xi )
rjb = b ril − rij = b (xl − xi ) − (xj − xi )
rm = rja × rjb
rm
xs = xi + c
|rm |

(4.160)

In this case the virtual site is at a distance of |c| from i, while a and b are parameters. Note
that the vectors rik and rij are not normalized to save floating-point operations. The force
on particles i, j, k and l due to the force on the virtual site are computed through chain rule
derivatives of the construction expression. This is exact and conserves energy, but it does

4.8. Long Range Electrostatics

107

xi
xk
xl

rjb

rja
xj

Figure 4.19: The new 4fdn virtual site construction, which is stable even when all constructing
atoms are in the same plane.
lead to relatively lengthy expressions that we do not include here (over 200 floating-point
operations). The interested reader can look at the source code in vsite.c. Fortunately,
this vsite type is normally only used for chiral centers such as Cα atoms in proteins.
The new 4fdn construct is identified with a ‘type’ value of 2 in the topology. The earlier
4fd type is still supported internally (‘type’ value 1), but it should not be used for new
simulations. All current GROMACS tools will automatically generate type 4fdn instead.
N. A linear combination of N atoms with relative weights ai . The weight for atom i is:


wi = a i 

N
X

−1

aj 

(4.161)

j=1

There are three options for setting the weights:
COG center of geometry: equal weights
COM center of mass: ai is the mass of atom i; when in free-energy simulations the mass of
the atom is changed, only the mass of the A-state is used for the weight
COW center of weights: ai is defined by the user

4.8
4.8.1

Long Range Electrostatics
Ewald summation

The total electrostatic energy of N particles and their periodic images is given by
V =

N X
N
f XXXX
qi qj
.
2 nx ny nz ∗ i j rij,n

(4.162)

108

Chapter 4. Interaction function and force fields

(nx , ny , nz ) = n is the box index vector, and the star indicates that terms with i = j should be
omitted when (nx , ny , nz ) = (0, 0, 0). The distance rij,n is the real distance between the charges
and not the minimum-image. This sum is conditionally convergent, but very slow.
Ewald summation was first introduced as a method to calculate long-range interactions of the periodic images in crystals [110]. The idea is to convert the single slowly-converging sum eqn. 4.162
into two quickly-converging terms and a constant term:
V

= Vdir + Vrec + V0

(4.163)

Vdir =

N XXX
fX
erfc(βrij,n )
qi qj
2 i,j nx ny nz ∗
rij,n

Vrec =

N
X X X exp −(πm/β)2 + 2πim · (ri − rj )
f X
qi qj
2πV i,j
m2
mx my mz ∗

(4.164)

N
fβ X
V0 = − √
q2,
π i i

(4.165)

(4.166)

where β is a parameter that determines the relative weight of the direct and reciprocal sums and
m = (mx , my , mz ). In this way we can use a short cut-off (of the order of 1 nm) in the direct
space sum and a short cut-off in the reciprocal space sum (e.g. 10 wave vectors in each direction).
Unfortunately, the computational cost of the reciprocal part of the sum increases as N 2 (or N 3/2
with a slightly better algorithm) and it is therefore not realistic for use in large systems.
Using Ewald
Don’t use Ewald unless you are absolutely sure this is what you want - for almost all cases the PME
method below will perform much better. If you still want to employ classical Ewald summation
enter this in your .mdp file, if the side of your box is about 3 nm:
coulombtype
rvdw
rlist
rcoulomb
fourierspacing
ewald-rtol

=
=
=
=
=
=

Ewald
0.9
0.9
0.9
0.6
1e-5

The ratio of the box dimensions and the fourierspacing parameter determines the highest
magnitude of wave vectors mx , my , mz to use in each direction. With a 3-nm cubic box this example would use 11 wave vectors (from −5 to 5) in each direction. The ewald-rtol parameter
is the relative strength of the electrostatic interaction at the cut-off. Decreasing this gives you a
more accurate direct sum, but a less accurate reciprocal sum.

4.8.2

PME

Particle-mesh Ewald is a method proposed by Tom Darden [14] to improve the performance of
the reciprocal sum. Instead of directly summing wave vectors, the charges are assigned to a grid

4.8. Long Range Electrostatics

109

using interpolation. The implementation in GROMACS uses cardinal B-spline interpolation [15],
which is referred to as smooth PME (SPME). The grid is then Fourier transformed with a 3D FFT
algorithm and the reciprocal energy term obtained by a single sum over the grid in k-space.
The potential at the grid points is calculated by inverse transformation, and by using the interpolation factors we get the forces on each atom.
The PME algorithm scales as N log(N ), and is substantially faster than ordinary Ewald summation on medium to large systems. On very small systems it might still be better to use Ewald
to avoid the overhead in setting up grids and transforms. For the parallelization of PME see the
section on MPMD PME (3.17.5).
With the Verlet cut-off scheme, the PME direct space potential is shifted by a constant such that
the potential is zero at the cut-off. This shift is small and since the net system charge is close to
zero, the total shift is very small, unlike in the case of the Lennard-Jones potential where all shifts
add up. We apply the shift anyhow, such that the potential is the exact integral of the force.
Using PME
As an example for using Particle-mesh Ewald summation in GROMACS, specify the following
lines in your .mdp file:
coulombtype
rvdw
rlist
rcoulomb
fourierspacing
pme-order
ewald-rtol

=
=
=
=
=
=
=

PME
0.9
0.9
0.9
0.12
4
1e-5

In this case the fourierspacing parameter determines the maximum spacing for the FFT grid
(i.e. minimum number of grid points), and pme-order controls the interpolation order. Using
fourth-order (cubic) interpolation and this spacing should give electrostatic energies accurate to
about 5 · 10−3 . Since the Lennard-Jones energies are not this accurate it might even be possible to
increase this spacing slightly.
Pressure scaling works with PME, but be aware of the fact that anisotropic scaling can introduce
artificial ordering in some systems.

4.8.3

P3M-AD

The Particle-Particle Particle-Mesh methods of Hockney & Eastwood can also be applied in GROMACS for the treatment of long range electrostatic interactions [111]. Although the P3M method
was the first efficient long-range electrostatics method for molecular simulation, the smooth PME
(SPME) method has largely replaced P3M as the method of choice in atomistic simulations. One
performance disadvantage of the original P3M method was that it required 3 3D-FFT back transforms to obtain the forces on the particles. But this is not required for P3M and the forces can be

110

Chapter 4. Interaction function and force fields

derived through analytical differentiation of the potential, as done in PME. The resulting method is
termed P3M-AD. The only remaining difference between P3M-AD and PME is the optimization
of the lattice Green influence function for error minimization that P3M uses. However, in 2012
it has been shown that the SPME influence function can be modified to obtain P3M [112]. This
means that the advantage of error minimization in P3M-AD can be used at the same computational
cost and with the same code as PME, just by adding a few lines to modify the influence function.
However, at optimal parameter setting the effect of error minimization in P3M-AD is less than
10%. P3M-AD does show large accuracy gains with interlaced (also known as staggered) grids,
but that is not supported in GROMACS (yet).
P3M is used in GROMACS with exactly the same options as used with PME by selecting the
electrostatics type:
coulombtype

4.8.4

= P3M-AD

Optimizing Fourier transforms and PME calculations

It is recommended to optimize the parameters for calculation of electrostatic interaction such as
PME grid dimensions and cut-off radii. This is particularly relevant to do before launching long
production runs.
gmx mdrun will automatically do a lot of PME optimization, and GROMACS also includes a
special tool, gmx tune_pme, which automates the process of selecting the optimal number of
PME-only ranks.

4.9

Long Range Van der Waals interactions

4.9.1

Dispersion correction

In this section, we derive long-range corrections due to the use of a cut-off for Lennard-Jones
or Buckingham interactions. We assume that the cut-off is so long that the repulsion term can
safely be neglected, and therefore only the dispersion term is taken into account. Due to the
nature of the dispersion interaction (we are truncating a potential proportional to −r−6 ), energy
and pressure corrections are both negative. While the energy correction is usually small, it may be
important for free energy calculations where differences between two different Hamiltonians are
considered. In contrast, the pressure correction is very large and can not be neglected under any
circumstances where a correct pressure is required, especially for any NPT simulations. Although
it is, in principle, possible to parameterize a force field such that the pressure is close to the desired
experimental value without correction, such a method makes the parameterization dependent on
the cut-off and is therefore undesirable.
Energy
The long-range contribution of the dispersion interaction to the virial can be derived analytically, if
we assume a homogeneous system beyond the cut-off distance rc . The dispersion energy between

4.9. Long Range Van der Waals interactions

111

two particles is written as:
−6
V (rij ) = −C6 rij

(4.167)

−8
F ij = −6 C6 rij
r ij

(4.168)

and the corresponding force is:
In a periodic system it is not easy to calculate the full potentials, so usually a cut-off is applied,
which can be abrupt or smooth. We will call the potential and force with cut-off Vc and F c . The
long-range contribution to the dispersion energy in a system with N particles and particle density
ρ = N/V is:
Z ∞
1
Vlr = N ρ
4πr2 g(r) (V (r) − Vc (r)) dr
(4.169)
2
0
We will integrate this for the shift function, which is the most general form of van der Waals
interaction available in GROMACS. The shift function has a constant difference S from 0 to r1
and is 0 beyond the cut-off distance rc . We can integrate eqn. 4.169, assuming that the density
in the sphere within r1 is equal to the global density and the radial distribution function g(r) is 1
beyond r1 :
Vlr =
=

rc
∞
r1
1
4πr2 V (r) dr
4πr2 g(r) C6 S dr + ρ
4πr2 (V (r) − Vc (r)) dr + ρ
N ρ
2
r
rc
0

Z rc 1
1
4
4
3
2
−3
N
πρr1 − 1 C6 S + ρ
4πr (V (r) − Vc (r)) dr − πN ρ C6 rc
(4.170)
2
3
3
r1

where the term −1 corrects for the self-interaction. For a plain cut-off we only need to assume
that g(r) is 1 beyond rc and the correction reduces to [113]:
2
Vlr = − πN ρ C6 rc−3
3

(4.171)

If we consider, for example, a box of pure water, simulated with a cut-off of 0.9 nm and a density
of 1 g cm−3 this correction is −0.75 kJ mol−1 per molecule.
For a homogeneous mixture we need to define an average dispersion constant:
hC6 i =

N
N X
X
2
C6 (i, j)
N (N − 1) i j>i

(4.172)

In GROMACS, excluded pairs of atoms do not contribute to the average.
In the case of inhomogeneous simulation systems, e.g. a system with a lipid interface, the energy
correction can be applied if hC6 i for both components is comparable.
Virial and pressure
The scalar virial of the system due to the dispersion interaction between two particles i and j is
given by:
1
−6
Ξ = − r ij · F ij = 3 C6 rij
(4.173)
2
The pressure is given by:
2
P =
(Ekin − Ξ)
(4.174)
3V

112

Chapter 4. Interaction function and force fields

The long-range correction to the virial is given by:
Ξlr

1
= Nρ
2

Z ∞

4πr2 g(r)(Ξ − Ξc ) dr

(4.175)

We can again integrate the long-range contribution to the virial assuming g(r) is 1 beyond r1 :
∞
rc
1
−6
4πr2 3 C6 rij
dr
4πr2 (Ξ − Ξc ) dr +
Nρ
2
rc
r1

Z rc
1
2
−3
4πr (Ξ − Ξc ) dr + 4πC6 rc
Nρ
2
r1

Ξlr =
=

(4.176)

For a plain cut-off the correction to the pressure is [113]:
4
Plr = − πC6 ρ2 rc−3
3

(4.177)

Using the same example of a water box, the correction to the virial is 0.75 kJ mol−1 per molecule,
the corresponding correction to the pressure for SPC water is approximately −280 bar.
For homogeneous mixtures, we can again use the average dispersion constant hC6 i (eqn. 4.172):
4
Plr = − π hC6 i ρ2 rc−3
3

(4.178)

For inhomogeneous systems, eqn. 4.178 can be applied under the same restriction as holds for the
energy (see sec. 4.9.1).

4.9.2

Lennard-Jones PME

In order to treat systems, using Lennard-Jones potentials, that are non-homogeneous outside of the
cut-off distance, we can instead use the Particle-mesh Ewald method as discussed for electrostatics
above. In this case the modified Ewald equations become
V

= Vdir + Vrec + V0

Vdir = −

Vrec =

N X X X ij
1X
C6 g(βrij,n )
2 i,j nx ny nz ∗
rij,n 6

3
N
X
π 2 β3 X X X
f (π|m|/β) ×
C6ij exp [−2πim · (ri − rj )]
2V mx my mz ∗
i,j

V0 = −

N
β6 X
C ii
12 i 6

(4.179)
(4.180)

(4.181)

(4.182)

where m = (mx , my , mz ), β is the parameter determining the weight between direct and reciprocal space, and C6ij is the combined dispersion parameter for particle i and j. The star indicates that
terms with i = j should be omitted when ((nx , ny , nz ) = (0, 0, 0)), and rij,n is the real distance

4.9. Long Range Van der Waals interactions

113

between the particles. Following the derivation by Essmann [15], the functions f and g introduced
above are defined as
h
i
√
f (x) = 1/3 (1 − 2x2 )exp(−x2 ) + 2x3 π erfc(x)
(4.183)
g(x) = exp(−x2 )(1 + x2 +

x4
).
2

(4.184)

The above methodology works fine as long as the dispersion parameters can be combined geometrically (eqn. 4.6) in the same way as the charges for electrostatics

ij
C6,geom
= C6ii C6jj

1/2

(4.185)

For Lorentz-Berthelot combination rules (eqn. 4.7), the reciprocal part of this sum has to be calculated seven times due to the splitting of the dispersion parameter according to
ij
C6,L−B
= (σi + σj )6 =

6
X

(6−n)

Pn σin σj

(4.186)

n=0

for Pn the Pascal triangle coefficients. This introduces a non-negligible cost to the reciprocal part,
requiring seven separate FFTs, and therefore this has been the limiting factor in previous attempts
to implement LJ-PME. A solution to this problem is to use geometrical combination rules in order
to calculate an approximate interaction parameter for the reciprocal part of the potential, yielding
a total interaction of
recip
V (r < rc ) = C6dir g(βr)r−6 + C6,geom
[1 − g(βr)]r−6

Direct space

}

Reciprocal space

}

recip
recip
r−6 + C6dir − C6,geom
g(βr)r−6
= C6,geom

(4.187)

recip
V (r > rc ) = C6,geom
[1 − g(βr)]r−6 .

Reciprocal space

(4.188)

}

This will preserve a well-defined Hamiltonian and significantly increase the performance of the
simulations. The approximation does introduce some errors, but since the difference is located in
the interactions calculated in reciprocal space, the effect will be very small compared to the total
interaction energy. In a simulation of a lipid bilayer, using a cut-off of 1.0 nm, the relative error
in total dispersion energy was below 0.5%. A more thorough discussion of this can be found in
[114].
In GROMACS we now perform the proper calculation of this interaction by subtracting, from the
direct-space interactions, the contribution made by the approximate potential that is used in the
reciprocal part
Vdir = C6dir r−6 − C6recip [1 − g(βr)]r−6 .
(4.189)
This potential will reduce to the expression in eqn. 4.180 when C6dir = C6recip , and the total
interaction is given by
V (r < rc ) = C6dir r−6 − C6recip [1 − g(βr)]r−6 + C6recip [1 − g(βr)]r−6
|

Direct space

= C6dir r−6
V (r > rc ) =

C6recip [1

}

Reciprocal space

}

(4.190)
− g(βr)]r

−6

(4.191)

114

Chapter 4. Interaction function and force fields

For the case when C6dir 6= C6recip this will retain an unmodified LJ force up to the cut-off, and
the error is an order of magnitude smaller than in simulations where the direct-space interactions
do not account for the approximation used in reciprocal space. When using a VdW interaction
modifier of potential-shift, the constant

−C6dir + C6recip [1 − g(βrc )] rc−6

(4.192)

is added to eqn. 4.190 in order to ensure that the potential is continuous at the cutoff. Note
that, in the same way as eqn. 4.189, this degenerates into the expected −C6 g(βrc )rc−6 when
C6dir = C6recip . In addition to this, a long-range dispersion correction can be applied to correct for
the approximation using a combination rule in reciprocal space. This correction assumes, as for
the cut-off LJ potential, a uniform particle distribution. But since the error of the combination rule
approximation is very small this long-range correction is not necessary in most cases. Also note
that this homogenous correction does not correct the surface tension, which is an inhomogeneous
property.
Using LJ-PME
As an example for using Particle-mesh Ewald summation for Lennard-Jones interactions in GROMACS, specify the following lines in your .mdp file:
vdwtype
rvdw
vdw-modifier
rlist
rcoulomb
fourierspacing
pme-order
ewald-rtol-lj
lj-pme-comb-rule

=
=
=
=
=
=
=
=
=

PME
0.9
Potential-Shift
0.9
0.9
0.12
4
0.001
geometric

The same Fourier grid and interpolation order are used if both LJ-PME and electrostatic PME are
active, so the settings for fourierspacing and pme-order are common to both. ewald-rtol-lj
controls the splitting between direct and reciprocal space in the same way as ewald-rtol. In addition to this, the combination rule to be used in reciprocal space is determined by lj-pme-comb-rule.
If the current force field uses Lorentz-Berthelot combination rules, it is possible to set lj-pme-comb-rule
= geometric in order to gain a significant increase in performance for a small loss in accuracy.
The details of this approximation can be found in the section above.
Note that the use of a complete long-range dispersion correction means that as with Coulomb
PME, rvdw is now a free parameter in the method, rather than being necessarily restricted by the
force-field parameterization scheme. Thus it is now possible to optimize the cutoff, spacing, order
and tolerance terms for accuracy and best performance.
Naturally, the use of LJ-PME rather than LJ cut-off adds computation and communication done
for the reciprocal-space part, so for best performance in balancing the load of parallel simulations
using PME-only ranks, more such ranks should be used. It may be possible to improve upon the
automatic load-balancing used by mdrun.

4.10. Force field

4.10

115

Force field

A force field is built up from two distinct components:
• The set of equations (called the potential functions) used to generate the potential energies
and their derivatives, the forces. These are described in detail in the previous chapter.
• The parameters used in this set of equations. These are not given in this manual, but in the
data files corresponding to your GROMACS distribution.
Within one set of equations various sets of parameters can be used. Care must be taken that the
combination of equations and parameters form a consistent set. It is in general dangerous to make
ad hoc changes in a subset of parameters, because the various contributions to the total force are
usually interdependent. This means in principle that every change should be documented, verified
by comparison to experimental data and published in a peer-reviewed journal before it can be used.
GROMACS 2018.3 includes several force fields, and additional ones are available on the website.
If you do not know which one to select we recommend GROMOS-96 for united-atom setups and
OPLS-AA/L for all-atom parameters. That said, we describe the available options in some detail.
All-hydrogen force field
The GROMOS-87-based all-hydrogen force field is almost identical to the normal GROMOS-87
force field, since the extra hydrogens have no Lennard-Jones interaction and zero charge. The
only differences are in the bond angle and improper dihedral angle terms. This force field is only
useful when you need the exact hydrogen positions, for instance for distance restraints derived
from NMR measurements. When citing this force field please read the previous paragraph.

4.10.1

GROMOS-96

GROMACS supports the GROMOS-96 force fields [82]. All parameters for the 43A1, 43A2
(development, improved alkane dihedrals), 45A3, 53A5, and 53A6 parameter sets are included.
All standard building blocks are included and topologies can be built automatically by pdb2gmx.
The GROMOS-96 force field is a further development of the GROMOS-87 force field. It has
improvements over the GROMOS-87 force field for proteins and small molecules. Note that
the sugar parameters present in 53A6 do correspond to those published in 2004[115], which are
different from those present in 45A4, which is not included in GROMACS at this time. The 45A4
parameter set corresponds to a later revision of these parameters. The GROMOS-96 force field is
not, however, recommended for use with long alkanes and lipids. The GROMOS-96 force field
differs from the GROMOS-87 force field in a few respects:
• the force field parameters
• the parameters for the bonded interactions are not linked to atom types
• a fourth power bond stretching potential (4.2.1)

116

Chapter 4. Interaction function and force fields
• an angle potential based on the cosine of the angle (4.2.6)

There are two differences in implementation between GROMACS and GROMOS-96 which can
lead to slightly different results when simulating the same system with both packages:
• in GROMOS-96 neighbor searching for solvents is performed on the first atom of the solvent
molecule. This is not implemented in GROMACS, but the difference with searching by
centers of charge groups is very small
• the virial in GROMOS-96 is molecule-based. This is not implemented in GROMACS,
which uses atomic virials
The GROMOS-96 force field was parameterized with a Lennard-Jones cut-off of 1.4 nm, so be
sure to use a Lennard-Jones cut-off (rvdw) of at least 1.4. A larger cut-off is possible because the
Lennard-Jones potential and forces are almost zero beyond 1.4 nm.
GROMOS-96 files
GROMACS can read and write GROMOS-96 coordinate and trajectory files. These files should
have the extension .g96. Such a file can be a GROMOS-96 initial/final configuration file, a
coordinate trajectory file, or a combination of both. The file is fixed format; all floats are written
as 15.9, and as such, files can get huge. GROMACS supports the following data blocks in the
given order:
• Header block:
TITLE (mandatory)
• Frame blocks:
TIMESTEP (optional)
POSITION/POSITIONRED (mandatory)
VELOCITY/VELOCITYRED (optional)
BOX (optional)
See the GROMOS-96 manual [82] for a complete description of the blocks. Note that all GROMACS programs can read compressed (.Z) or gzipped (.gz) files.

4.10.2

OPLS/AA

4.10.3

AMBER

GROMACS provides native support for the following AMBER force fields:
• AMBER94 [116]

4.10. Force field

117

• AMBER96 [117]
• AMBER99 [118]
• AMBER99SB [119]
• AMBER99SB-ILDN [120]
• AMBER03 [121]
• AMBERGS [122]

4.10.4

CHARMM

GROMACS supports the CHARMM force field for proteins [123, 124], lipids [125] and nucleic
acids [126, 127]. The protein parameters (and to some extent the lipid and nucleic acid parameters)
were thoroughly tested – both by comparing potential energies between the port and the standard
parameter set in the CHARMM molecular simulation package, as well by how the protein force
field behaves together with GROMACS-specific techniques such as virtual sites (enabling long
time steps) and a fast implicit solvent recently implemented [74] – and the details and results are
presented in the paper by Bjelkmar et al. [128]. The nucleic acid parameters, as well as the ones
for HEME, were converted and tested by Michel Cuendet.
When selecting the CHARMM force field in pdb2gmx the default option is to use CMAP (for
torsional correction map). To exclude CMAP, use -nocmap. The basic form of the CMAP term
implemented in GROMACS is a function of the φ and ψ backbone torsion angles. This term is
defined in the .rtp file by a [ cmap ] statement at the end of each residue supporting CMAP.
The following five atom names define the two torsional angles. Atoms 1-4 define φ, and atoms
2-5 define ψ. The corresponding atom types are then matched to the correct CMAP type in the
cmap.itp file that contains the correction maps.
A port of the CHARMM36 force field for use with GROMACS is also available at http://
mackerell.umaryland.edu/charmm_ff.shtml#gromacs.
For branched polymers or other topologies not supported by pdb2gmx, it is possible to use TopoTools [129] to generate a GROMACS top file.

4.10.5

Coarse-grained force fields

Coarse-graining is a systematic way of reducing the number of degrees of freedom representing
a system of interest. To achieve this, typically whole groups of atoms are represented by single
beads and the coarse-grained force fields describes their effective interactions. Depending on the
choice of parameterization, the functional form of such an interaction can be complicated and
often tabulated potentials are used.
Coarse-grained models are designed to reproduce certain properties of a reference system. This
can be either a full atomistic model or even experimental data. Depending on the properties to
reproduce there are different methods to derive such force fields. An incomplete list of methods is
given below:

118

Chapter 4. Interaction function and force fields
• Conserving free energies
– Simplex method
– MARTINI force field (see next section)
• Conserving distributions (like the radial distribution function), so-called structure-based
coarse-graining
– (iterative) Boltzmann inversion
– Inverse Monte Carlo
• Conversing forces
– Force matching

Note that coarse-grained potentials are state dependent (e.g. temperature, density,...) and should
be re-parametrized depending on the system of interest and the simulation conditions. This can
for example be done using the Versatile Object-oriented Toolkit for Coarse-Graining Applications
(VOTCA) [130]. The package was designed to assists in systematic coarse-graining, provides
implementations for most of the algorithms mentioned above and has a well tested interface to
GROMACS. It is available as open source and further information can be found at www.votca.org.

4.10.6

MARTINI

The MARTINI force field is a coarse-grain parameter set that allows for the construction of many
systems, including proteins and membranes.

4.10.7

PLUM

The PLUM force field [131] is an example of a solvent-free protein-membrane model for which
the membrane was derived from structure-based coarse-graining [132]. A GROMACS implementation can be found at code.google.com/p/plumx.

Chapter 5
Topologies
5.1

Introduction

GROMACS must know on which atoms and combinations of atoms the various contributions to
the potential functions (see chapter 4) must act. It must also know what parameters must be
applied to the various functions. All this is described in the topology file *.top, which lists the
constant attributes of each atom. There are many more atom types than elements, but only atom
types present in biological systems are parameterized in the force field, plus some metals, ions and
silicon. The bonded and special interactions are determined by fixed lists that are included in the
topology file. Certain non-bonded interactions must be excluded (first and second neighbors), as
these are already treated in bonded interactions. In addition, there are dynamic attributes of atoms
- their positions, velocities and forces. These do not strictly belong to the molecular topology,
and are stored in the coordinate file *.gro (positions and velocities), or trajectory file *.trr
(positions, velocities, forces).
This chapter describes the setup of the topology file, the *.top file and the database files: what
the parameters stand for and how/where to change them if needed. First, all file formats are
explained. Section 5.9.1 describes the organization of the files in each force field.
Note: if you construct your own topologies, we encourage you to upload them to our topology
archive at www.gromacs.org! Just imagine how thankful you’d have been if your topology had
been available there before you started. The same goes for new force fields or modified versions
of the standard force fields - contribute them to the force field archive!

5.2

Particle type

In GROMACS, there are three types of particles, see Table 5.1. Only regular atoms and virtual
interaction sites are used in GROMACS; shells are necessary for polarizable models like the ShellWater models [45].

120

Chapter 5. Topologies
Particle
atoms
shells
virtual interaction sites

Symbol
A
S
V (or D)

Table 5.1: Particle types in GROMACS

5.2.1

Atom types

Each force field defines a set of atom types, which have a characteristic name or number, and mass
(in a.m.u.). These listings are found in the atomtypes.atp file (.atp = atom type parameter
file). Therefore, it is in this file that you can begin to change and/or add an atom type. A sample
from the gromos43a1.ff force field is listed below.
O
OM
OA
OW
N
NT
NL
NR
NZ
NE
C
CH1
CH2
CH3

15.99940
15.99940
15.99940
15.99940
14.00670
14.00670
14.00670
14.00670
14.00670
14.00670
12.01100
13.01900
14.02700
15.03500

;
;
;
;
;
;
;
;
;
;
;
;
;
;

carbonyl oxygen (C=O)
carboxyl oxygen (CO-)
hydroxyl, sugar or ester oxygen
water oxygen
peptide nitrogen (N or NH)
terminal nitrogen (NH2)
terminal nitrogen (NH3)
aromatic nitrogen
Arg NH (NH2)
Arg NE (NH)
bare carbon
aliphatic or sugar CH-group
aliphatic or sugar CH2-group
aliphatic CH3-group

Note: GROMACS makes use of the atom types as a name, not as a number (as e.g. in GROMOS).

5.2.2

Virtual sites

Some force fields use virtual interaction sites (interaction sites that are constructed from other
particle positions) on which certain interactions are located (e.g. on benzene rings, to reproduce
the correct quadrupole). This is described in sec. 4.7.
To make virtual sites in your system, you should include a section [ virtual_sites? ] (for
backward compatibility the old name [ dummies? ] can also be used) in your topology file,
where the ‘?’ stands for the number constructing particles for the virtual site. This will be ‘2’ for
type 2, ‘3’ for types 3, 3fd, 3fad and 3out and ‘4’ for type 4fdn. The last of these replace an older
4fd type (with the ‘type’ value 1) that could occasionally be unstable; while it is still supported
internally in the code, the old 4fd type should not be used in new input files. The different types
are explained in sec. 4.7.
Parameters for type 2 should look like this:
[ virtual_sites2 ]

5.2. Particle type
; Site
5

from
1

121

funct
1

a
0.7439756

funct
1

a
0.7439756

b
0.128012

funct
2

a
0.5

d
-0.105

funct
3

theta
120

d
0.5

funct
4

a
-0.4

b
-0.4

for type 3 like this:
[ virtual_sites3 ]
; Site from
5
1
2

for type 3fd like this:
[ virtual_sites3 ]
; Site from
5
1
2

for type 3fad like this:
[ virtual_sites3 ]
; Site from
5
1
2

for type 3out like this:
[ virtual_sites3 ]
; Site from
5
1
2

c
6.9281

for type 4fdn like this:
[ virtual_sites4 ]
; Site from
5
1
2

funct
2

a
1.0

b
0.9

c
0.105

This will result in the construction of a virtual site, number 5 (first column ‘Site’), based on the
positions of the atoms whose indices are 1 and 2 or 1, 2 and 3 or 1, 2, 3 and 4 (next two, three
or four columns ‘from’) following the rules determined by the function number (next column
‘funct’) with the parameters specified (last one, two or three columns ‘a b . .’). Obviously,
the atom numbers (including virtual site number) depend on the molecule. It may be instructive
to study the topologies for TIP4P or TIP5P water models that are included with the GROMACS
distribution.
Note that if any constant bonded interactions are defined between virtual sites and/or normal
atoms, they will be removed by grompp (unless the option tt -normvsbds is used). This removal of bonded interactions is done after generating exclusions, as the generation of exclusions
is based on “chemically” bonded interactions.
Virtual sites can be constructed in a more generic way using basic geometric parameters. The directive that can be used is [ virtual_sitesn ]. Required parameters are listed in Table 5.5.
An example entry for defining a virtual site at the center of geometry of a given set of atoms might
be:
[ virtual_sitesn ]
; Site
funct
from
5
1
1

122

Chapter 5. Topologies
Property
Type
Mass
Charge
epsilon
sigma

Symbol
m
q

Unit
a.m.u.
electron
kJ/mol
nm

Table 5.2: Static atom type properties in GROMACS

5.3
5.3.1

Parameter files
Atoms

The static properties (see Table 5.2 assigned to the atom types are assigned based on data in several
places. The mass is listed in atomtypes.atp (see 5.2.1), whereas the charge is listed in *.rtp
(.rtp = residue topology parameter file, see 5.7.1). This implies that the charges are only defined
in the building blocks of amino acids, nucleic acids or otherwise, as defined by the user. When
generating a topology (*.top) using the pdb2gmx program, the information from these files is
combined.

5.3.2

Non-bonded parameters

The non-bonded parameters consist of the van der Waals parameters V (c6 or σ, depending on the
combination rule) and W (c12 or ), as listed in the file ffnonbonded.itp, where ptype is
the particle type (see Table 5.1). As with the bonded parameters, entries in [ *type ] directives
are applied to their counterparts in the topology file. Missing parameters generate warnings, except
as noted below in section 5.4.3.
[ atomtypes ]
;name
at.num
O
8
OM
8
.....

mass
15.99940
15.99940

[ nonbond_params ]
; i
j func
V(c6)
O
O
1 0.22617E-02
O
OA
1 0.22617E-02
.....

charge
0.000
0.000

ptype
A
A

V(c6)
0.22617E-02
0.22617E-02

W(c12)
0.74158E-06
0.74158E-06

W(c12)
0.74158E-06
0.13807E-05

Note that most of the included force fields also include the at.num. column, but this same information is implied in the OPLS-AA bond_type column. The interpretation of the parameters
V and W depends on the combination rule that was chosen in the [ defaults ] section of the
topology file (see 5.8.1):
(6)

for combination rule 1 :

Vii = Ci
= 4 i σi6 [ kJ mol−1 nm6 ]
(5.1)
(12)
Wii = Ci
= 4 i σi12 [ kJ mol−1 nm12 ]

5.3. Parameter files

123
Vii = σi [ nm ]
Wii = i [ kJ mol−1 ]

for combination rules 2 and 3 :

(5.2)

Some or all combinations for different atom types can be given in the [ nonbond_params ]
section, again with parameters V and W as defined above. Any combination that is not given will
be computed from the parameters for the corresponding atom types, according to the combination
rule:
(6)

Cij

for combination rules 1 and 3 :

(12)
Cij

σij
ij

for combination rule 2 :

(12) (12)
Ci Cj

(6)

= 21 (σi + σj )
√
=
i j

(5.3)

(5.4)

When σ and need to be supplied (rules 2 and 3), it would seem it is impossible to have a non-zero
C 12 combined with a zero C 6 parameter. However, providing a negative σ will do exactly that,
such that C 6 is set to zero and C 12 is calculated normally. This situation represents a special case
in reading the value of σ, and nothing more.
There is only one set of combination rules for Buckingham potentials:
1/2
= (Aii
Ajj )

= 2/ B1ii + B1jj

Aij
Bij
Cij

5.3.3

= (Cii Cjj )

(5.5)

1/2

Bonded parameters

The bonded parameters (i.e. bonds, bond angles, improper and proper dihedrals) are listed in
ffbonded.itp. The entries in this database describe, respectively, the atom types in the interactions, the type of the interaction, and the parameters associated with that interaction. These
parameters are then read by grompp when processing a topology and applied to the relevant
bonded parameters, i.e. bondtypes are applied to entries in the [ bonds ] directive, etc.
Any bonded parameter that is missing from the relevant [ *type ] directive generates a fatal
error. The types of interactions are listed in Table 5.5. Example excerpts from such files follow:
[ bondtypes ]
; i
j func
C
O
1
C
OM
1
......

b0
0.12300
0.12500

[ angletypes ]
; i
j
k func
HO
OA
C
1
HO
OA CH1
1
......
[ dihedraltypes ]

kb
502080.
418400.

th0
109.500
109.500

cth
397.480
397.480

124

Chapter 5. Topologies

; i
l func
NR5* NR5
2
NR5* NR5*
2
......

q0
0.000
0.000

cq
167.360
167.360

[ dihedraltypes ]
; j
k func
phi0
C
OA
1
180.000
C
N
1
180.000
......

cp
16.736
33.472

[ dihedraltypes ]
;
; Ryckaert-Bellemans Dihedrals
;
; aj
ak
funct
CP2
CP2
3
9.2789

12.156

mult
2
2

-13.120 -3.0597 26.240

-31.495

In the ffbonded.itp file, you can add bonded parameters. If you want to include parameters
for new atom types, make sure you define them in atomtypes.atp as well.
For most interaction types, bonded parameters are searched and assigned using an exact match for
all type names and allowing only a single set of parameters. The exception to this rule are dihedral
parameters. For [ dihedraltypes ] wildcard atom type names can be specified with the
letter X in one or more of the four positions. Thus one can for example assign proper dihedral
parameters based on the types of the middle two atoms. The parameters for the entry with the
most exact matches, i.e. the least wildcard matches, will be used. Note that GROMACS versions
older than 5.1.3 used the first match, which means that a full match would be ignored if it is
preceded by an entry that matches on wildcards. Thus it is suggested to put wildcard entries at the
end, in case someone might use a forcefield with older versions of GROMACS. In addition there
is a dihedral type 9 which adds the possibility of assigning multiple dihedral potentials, useful
for combining terms with different multiplicities. The different dihedral potential parameter sets
should be on directly adjacent lines in the [ dihedraltypes ] section.

5.4

Molecule definition

5.4.1

Moleculetype entries

An organizational structure that usually corresponds to molecules is the [ moleculetype ]
entry. This entry serves two main purposes. One is to give structure to the topology file(s), usually corresponding to real molecules. This makes the topology easier to read and writing it less
labor intensive. A second purpose is computational efficiency. The system definition that is kept
in memory is proportional in size of the moleculetype definitions. If a molecule is present
in 100000 copies, this saves a factor of 100000 in memory, which means the system usually fits
in cache, which can improve performance tremendously. Interactions that correspond to chemical
bonds, that generate exclusions, can only be defined between atoms within a moleculetype. It
is allowed to have multiple molecules which are not covalently bonded in one moleculetype

5.4. Molecule definition

125

definition. Molecules can be made infinitely long by connecting to themselves over periodic
boundaries. When such periodic molecules are present, an option in the mdp file needs to be
set to tell GROMACS not to attempt to make molecules that are broken over periodic boundaries
whole again.

5.4.2

Intermolecular interactions

In some cases, one would like atoms in different molecules to also interact with other interactions than the usual non-bonded interactions. This is often the case in binding studies. When
the molecules are covalently bound, e.g. a ligand binding covalently to a protein, they are effectively one molecule and they should be defined in one [ moleculetype ] entry. Note that
pdb2gmx has an option to put two or more molecules in one [ moleculetype ] entry. When
molecules are not covalently bound, it is much more convenient to use separate moleculetype
definitions and specify the intermolecular interactions in the [ intermolecular_interactions]
section. In this section, which is placed at the end of the topology (see Table 5.4), normal bonded
interactions can be specified using global atom indices. The only restrictions are that no interactions can be used that generates exclusions and no constraints can be used.

5.4.3

Intramolecular pair interactions

Extra Lennard-Jones and electrostatic interactions between pairs of atoms in a molecule can be
added in the [ pairs ] section of a molecule definition. The parameters for these interactions
can be set independently from the non-bonded interaction parameters. In the GROMOS force
fields, pairs are only used to modify the 1-4 interactions (interactions of atoms separated by three
bonds). In these force fields the 1-4 interactions are excluded from the non-bonded interactions
(see sec. 5.4.4).

[ pairtypes ]
; i
j func
cs6
O
O
1 0.22617E-02
O
OM
1 0.22617E-02
.....

cs12 ; THESE ARE 1-4 INTERACTIONS
0.74158E-06
0.74158E-06

The pair interaction parameters for the atom types in ffnonbonded.itp are listed in the
[ pairtypes ] section. The GROMOS force fields list all these interaction parameters explicitly, but this section might be empty for force fields like OPLS that calculate the 1-4 interactions by
uniformly scaling the parameters. Pair parameters that are not present in the [ pairtypes ]
section are only generated when gen-pairs is set to “yes” in the [ defaults ] directive of
forcefield.itp (see 5.8.1). When gen-pairs is set to “no,” grompp will give a warning
for each pair type for which no parameters are given.
The normal pair interactions, intended for 1-4 interactions, have function type 1. Function type 2
and the [ pairs_nb ] are intended for free-energy simulations. When determining hydration
free energies, the solute needs to be decoupled from the solvent. This can be done by adding a
B-state topology (see sec. 3.12) that uses zero for all solute non-bonded parameters, i.e. charges

126

Chapter 5. Topologies

and LJ parameters. However, the free energy difference between the A and B states is not the total
hydration free energy. One has to add the free energy for reintroducing the internal Coulomb and
LJ interactions in the solute when in vacuum. This second step can be combined with the first step
when the Coulomb and LJ interactions within the solute are not modified. For this purpose, there is
a pairs function type 2, which is identical to function type 1, except that the B-state parameters are
always identical to the A-state parameters. For searching the parameters in the [ pairtypes ]
section, no distinction is made between function type 1 and 2. The pairs section [ pairs_nb ]
is intended to replace the non-bonded interaction. It uses the unscaled charges and the non-bonded
LJ parameters; it also only uses the A-state parameters. Note that one should add exclusions for
all atom pairs listed in [ pairs_nb ], otherwise such pairs will also end up in the normal
neighbor lists.
Alternatively, this same behavior can be achieved without ever touching the topology, by using
the couple-moltype, couple-lambda0, couple-lambda1, and couple-intramol
keywords. See sections sec. 3.12 and sec. 6.1 for more information.
All three pair types always use plain Coulomb interactions, even when Reaction-field, PME, Ewald
or shifted Coulomb interactions are selected for the non-bonded interactions. Energies for types
1 and 2 are written to the energy and log file in separate “LJ-14” and “Coulomb-14” entries per
energy group pair. Energies for [ pairs_nb ] are added to the “LJ-(SR)” and “Coulomb(SR)” terms.

5.4.4

Exclusions

The exclusions for non-bonded interactions are generated by grompp for neighboring atoms up
to a certain number of bonds away, as defined in the [ moleculetype ] section in the topology file (see 5.8.1). Particles are considered bonded when they are connected by “chemical”
bonds ([ bonds ] types 1 to 5, 7 or 8) or constraints ([ constraints ] type 1). Type 5
[ bonds ] can be used to create a connection between two atoms without creating an interaction. There is a harmonic interaction ([ bonds ] type 6) that does not connect the atoms by a
chemical bond. There is also a second constraint type ([ constraints ] type 2) that fixes
the distance, but does not connect the atoms by a chemical bond. For a complete list of all these
interactions, see Table 5.5.
Extra exclusions within a molecule can be added manually in a [ exclusions ] section. Each
line should start with one atom index, followed by one or more atom indices. All non-bonded
interactions between the first atom and the other atoms will be excluded.
When all non-bonded interactions within or between groups of atoms need to be excluded, is it
more convenient and much more efficient to use energy monitor group exclusions (see sec. 3.3).

5.5

Implicit solvation parameters

Starting with GROMACS 4.5, implicit solvent is supported. A section in the topology has been
introduced to list those parameters:
[ implicit_genborn_params ]

5.6. Constraint algorithms
; Atomtype
NH1
N
H
CT1

sar
0.155
0.155
0.1
0.180

127
st
1
1
1
1

pi
1.028
1
1
1.276

gbr
0.17063
0.155
0.115
0.190

hct
0.79
0.79
0.85
0.72

;
;
;
;

N
Proline backbone N
H
C

In this example the atom type is listed first, followed by five numbers, and a comment (following
a semicolon).
Values in columns 1-3 are not currently used. They pertain to more elaborate surface area algorithms, the one from Qiu et al. [71] in particular. Column 4 contains the atomic van der Waals
radii, which are used in computing the Born radii. The dielectric offset is specified in the *.mdp
file, and gets subtracted from the input van der Waals radii for the different Born radii methods, as
described by Onufriev et al. [73]. Column 5 is the scale factor for the HCT and OBC models. The
values are taken from the Tinker implementation of the HCT pairwise scaling method [72]. This
method has been modified such that the scaling factors have been adjusted to minimize differences
between analytical surface areas and GB using the HCT algorithm. The scaling is further modified
in that it is not applied pairwise as proposed by Hawkins et al. [72], but on a per-atom (rather than
a per-pair) basis.

5.6

Constraint algorithms

Constraints are defined in the [ constraints ] section. The format is two atom numbers
followed by the function type, which can be 1 or 2, and the constraint distance. The only difference between the two types is that type 1 is used for generating exclusions and type 2 is not (see
sec. 5.4.4). The distances are constrained using the LINCS or the SHAKE algorithm, which can
be selected in the *.mdp file. Both types of constraints can be perturbed in free-energy calculations by adding a second constraint distance (see 5.8.5). Several types of bonds and angles (see
Table 5.5) can be converted automatically to constraints by grompp. There are several options
for this in the *.mdp file.
We have also implemented the SETTLE algorithm [47], which is an analytical solution of SHAKE,
specifically for water. SETTLE can be selected in the topology file. See, for instance, the SPC
molecule definition:
[ moleculetype ]
; molname
nrexcl
SOL
1
[ atoms ]
; nr
at type res nr
1
OW
1
2
HW
1
3
HW
1

ren nm
SOL
SOL
SOL

[ settles ]
; OW
funct
1
1

dhh
0.16333

doh
0.1

at nm
OW1
HW2
HW3

cg nr
1
1
1

charge
-0.82
0.41
0.41

128

Chapter 5. Topologies

[ exclusions ]
1
2
2
1
3
1

3
3
2

The [ settles ] directive defines the first atom of the water molecule. The settle funct is
always 1, and the distance between O-H and H-H distances must be given. Note that the algorithm
can also be used for TIP3P and TIP4P [133]. TIP3P just has another geometry. TIP4P has a virtual
site, but since that is generated it does not need to be shaken (nor stirred).

5.7

pdb2gmx input files

The GROMACS program pdb2gmx generates a topology for the input coordinate file. Several
formats are supported for that coordinate file, but *.pdb is the most commonly-used format
(hence the name pdb2gmx). pdb2gmx searches for force fields in sub-directories of the GROMACS share/top directory and your working directory. Force fields are recognized from the
file forcefield.itp in a directory with the extension .ff. The file forcefield.doc
may be present, and if so, its first line will be used by pdb2gmx to present a short description to
the user to help in choosing a force field. Otherwise, the user can choose a force field with the
-ff xxx command-line argument to pdb2gmx, which indicates that a force field in a xxx.ff
directory is desired. pdb2gmx will search first in the working directory, then in the GROMACS
share/top directory, and use the first matching xxx.ff directory found.
Two general files are read by pdb2gmx: an atom type file (extension .atp, see 5.2.1) from the
force-field directory, and a file called residuetypes.dat from either the working directory, or
the GROMACS share/top directory. residuetypes.dat determines which residue names
are considered protein, DNA, RNA, water, and ions.
pdb2gmx can read one or multiple databases with topological information for different types of
molecules. A set of files belonging to one database should have the same basename, preferably
telling something about the type of molecules (e.g. aminoacids, rna, dna). The possible files are:
• .rtp
• .r2b (optional)
• .arn (optional)
• .hdb (optional)
• .n.tdb (optional)
• .c.tdb (optional)
Only the .rtp file, which contains the topologies of the building blocks, is mandatory. Information from other files will only be used for building blocks that come from an .rtp file with
the same base name. The user can add building blocks to a force field by having additional files

5.7. pdb2gmx input files

129

with the same base name in their working directory. By default, only extra building blocks can be
defined, but calling pdb2gmx with the -rtpo option will allow building blocks in a local file to
replace the default ones in the force field.

5.7.1

Residue database

The files holding the residue databases have the extension .rtp. Originally this file contained
building blocks (amino acids) for proteins, and is the GROMACS interpretation of the rt37c4.dat
file of GROMOS. So the residue database file contains information (bonds, charges, charge groups,
and improper dihedrals) for a frequently-used building block. It is better not to change this file
because it is standard input for pdb2gmx, but if changes are needed make them in the *.top file
(see 5.8.1), or in a .rtp file in the working directory as explained in sec. 5.7. Defining topologies
of new small molecules is probably easier by writing an include topology file *.itp directly.
This will be discussed in section 5.8.2. When adding a new protein residue to the database, don’t
forget to add the residue name to the residuetypes.dat file, so that grompp, make_ndx
and analysis tools can recognize the residue as a protein residue (see 8.1.1).
The .rtp files are only used by pdb2gmx. As mentioned before, the only extra information this
program needs from the .rtp database is bonds, charges of atoms, charge groups, and improper
dihedrals, because the rest is read from the coordinate input file. Some proteins contain residues
that are not standard, but are listed in the coordinate file. You have to construct a building block for
this “strange” residue, otherwise you will not obtain a *.top file. This also holds for molecules
in the coordinate file such as ligands, polyatomic ions, crystallization co-solvents, etc. The residue
database is constructed in the following way:
[ bondedtypes ]
; bonds angles
1
1
[ GLY ]

; mandatory
dihedrals impropers
1
2 ; mandatory

; mandatory

[ atoms ] ;
; name type
N
N
H
H
CA
CH2
C
C
O
O

mandatory
charge chargegroup
-0.280
0
0.280
0
0.000
1
0.380
2
-0.380
2

[ bonds ] ; optional
;atom1 atom2
b0
N
H
N
CA
CA
C
C
O
-C
N
[ exclusions ]
;atom1 atom2

; optional

130

Chapter 5. Topologies

[ angles ] ; optional
;atom1 atom2 atom3
th0

cth

[ dihedrals ] ; optional
;atom1 atom2 atom3 atom4
phi0
[ impropers ] ; optional
;atom1 atom2 atom3 atom4
N
-C
CA
H
-C
-CA
N
-O

mult

[ ZN ]
[ atoms ]
ZN
ZN

2.000

The file is free format; the only restriction is that there can be at most one entry on a line. The first
field in the file is the [ bondedtypes ] field, which is followed by four numbers, indicating
the interaction type for bonds, angles, dihedrals, and improper dihedrals. The file contains residue
entries, which consist of atoms and (optionally) bonds, angles, dihedrals, and impropers. The
charge group codes denote the charge group numbers. Atoms in the same charge group should
always be ordered consecutively. When using the hydrogen database with pdb2gmx for adding
missing hydrogens (see 5.7.4), the atom names defined in the .rtp entry should correspond exactly to the naming convention used in the hydrogen database. The atom names in the bonded
interaction can be preceded by a minus or a plus, indicating that the atom is in the preceding or
following residue respectively. Explicit parameters added to bonds, angles, dihedrals, and impropers override the standard parameters in the .itp files. This should only be used in special
cases. Instead of parameters, a string can be added for each bonded interaction. This is used in
GROMOS-96 .rtp files. These strings are copied to the topology file and can be replaced by
force-field parameters by the C-preprocessor in grompp using #define statements.
pdb2gmx automatically generates all angles. This means that for most force fields the [ angles ]
field is only useful for overriding .itp parameters. For the GROMOS-96 force field the interaction number of all angles needs to be specified.
pdb2gmx automatically generates one proper dihedral for every rotatable bond, preferably on
heavy atoms. When the [ dihedrals ] field is used, no other dihedrals will be generated for
the bonds corresponding to the specified dihedrals. It is possible to put more than one dihedral
function on a rotatable bond. In the case of CHARMM27 FF pdb2gmx can add correction maps
to the dihedrals using the default -cmap option. Please refer to 4.10.4 for more information.
pdb2gmx sets the number of exclusions to 3, which means that interactions between atoms connected by at most 3 bonds are excluded. Pair interactions are generated for all pairs of atoms
that are separated by 3 bonds (except pairs of hydrogens). When more interactions need to be
excluded, or some pair interactions should not be generated, an [ exclusions ] field can be
added, followed by pairs of atom names on separate lines. All non-bonded and pair interactions
between these atoms will be excluded.

5.7. pdb2gmx input files
ARG
ARGN
ASP
ASPH
CYS
CYS2
GLU
GLUH
HISD
HISE
HISH
HIS1
LYSN
LYS
HEME

131
protonated arginine
neutral arginine
negatively charged aspartic acid
neutral aspartic acid
neutral cysteine
cysteine with sulfur bound to another cysteine or a heme
negatively charged glutamic acid
neutral glutamic acid
neutral histidine with Nδ protonated
neutral histidine with N protonated
positive histidine with both Nδ and N protonated
histidine bound to a heme
neutral lysine
protonated lysine
heme

Table 5.3: Internal GROMACS residue naming convention.

5.7.2

Residue to building block database

Each force field has its own naming convention for residues. Most residues have consistent naming, but some, especially those with different protonation states, can have many different names.
The .r2b files are used to convert standard residue names to the force-field build block names. If
no .r2b is present in the force-field directory or a residue is not listed, the building block name is
assumed to be identical to the residue name. The .r2b can contain 2 or 5 columns. The 2-column
format has the residue name in the first column and the building block name in the second. The
5-column format has 3 additional columns with the building block for the residue occurring in
the N-terminus, C-terminus and both termini at the same time (single residue molecule). This is
useful for, for instance, the AMBER force fields. If one or more of the terminal versions are not
present, a dash should be entered in the corresponding column.
There is a GROMACS naming convention for residues which is only apparent (except for the
pdb2gmx code) through the .r2b file and specbond.dat files. This convention is only of
importance when you are adding residue types to an .rtp file. The convention is listed in Table 5.3. For special bonds with, for instance, a heme group, the GROMACS naming convention
is introduced through specbond.dat (see 5.7.7), which can subsequently be translated by the
.r2b file, if required.

5.7.3

Atom renaming database

Force fields often use atom names that do not follow IUPAC or PDB convention. The .arn
database is used to translate the atom names in the coordinate file to the force-field names. Atoms
that are not listed keep their names. The file has three columns: the building block name, the
old atom name, and the new atom name, respectively. The residue name supports question-mark
wildcards that match a single character.

132

Chapter 5. Topologies

An additional general atom renaming file called xlateat.dat is present in the share/top
directory, which translates common non-standard atom names in the coordinate file to IUPAC/PDB
convention. Thus, when writing force-field files, you can assume standard atom names and no
further atom name translation is required, except for translating from standard atom names to the
force-field ones.

5.7.4

Hydrogen database

The hydrogen database is stored in .hdb files. It contains information for the pdb2gmx program
on how to connect hydrogen atoms to existing atoms. In versions of the database before GROMACS 3.3, hydrogen atoms were named after the atom they are connected to: the first letter of
the atom name was replaced by an ‘H.’ In the versions from 3.3 onwards, the H atom has to be
listed explicitly, because the old behavior was protein-specific and hence could not be generalized
to other molecules. If more than one hydrogen atom is connected to the same atom, a number will
be added to the end of the hydrogen atom name. For example, adding two hydrogen atoms to ND2
(in asparagine), the hydrogen atoms will be named HD21 and HD22. This is important since atom
naming in the .rtp file (see 5.7.1) must be the same. The format of the hydrogen database is as
follows:
; res
ALA
ARG

# additions
# H add type
1
1
1
4
1
2
1
1
2
3
2
3

-C

H
HE
HH1
HH2

N
NE
NH1
NH2

CA
CD
CZ
CZ

C
CZ
NE
NE

On the first line we see the residue name (ALA or ARG) and the number of kinds of hydrogen
atoms that may be added to this residue by the hydrogen database. After that follows one line for
each addition, on which we see:
• The number of H atoms added
• The method for adding H atoms, which can be any of:
1 one planar hydrogen, e.g. rings or peptide bond
One hydrogen atom (n) is generated, lying in the plane of atoms (i,j,k) on the plane
bisecting angle (j-i-k) at a distance of 0.1 nm from atom i, such that the angles (n-i-j)
and (n-i-k) are > 90o .
2 one single hydrogen, e.g. hydroxyl
One hydrogen atom (n) is generated at a distance of 0.1 nm from atom i, such that
angle (n-i-j)=109.5 degrees and dihedral (n-i-j-k)=trans.
3 two planar hydrogens, e.g. ethylene -C=CH2 , or amide -C(=O)NH2
Two hydrogens (n1,n2) are generated at a distance of 0.1 nm from atom i, such that

5.7. pdb2gmx input files

133

angle (n1-i-j)=(n2-i-j)=120 degrees and dihedral (n1-i-j-k)=cis and (n2-i-j-k)=trans,
such that names are according to IUPAC standards [134].
4 two or three tetrahedral hydrogens, e.g. -CH3
Three (n1,n2,n3) or two (n1,n2) hydrogens are generated at a distance of 0.1 nm from
atom i, such that angle (n1-i-j)=(n2-i-j)=(n3-i-j)=109.47o , dihedral (n1-i-j-k)=trans,
(n2-i-j-k)=trans+120 and (n3-i-j-k)=trans+240o .
5 one tetrahedral hydrogen, e.g. C3 CH
One hydrogen atom (n0 ) is generated at a distance of 0.1 nm from atom i in tetrahedral
conformation such that angle (n0 -i-j)=(n0 -i-k)=(n0 -i-l)=109.47o .
6 two tetrahedral hydrogens, e.g. C-CH2 -C
Two hydrogen atoms (n1,n2) are generated at a distance of 0.1 nm from atom i in
tetrahedral conformation on the plane bisecting angle j-i-k with angle (n1-i-n2)=(n1-ij)=(n1-i-k)=109.47o .
7 two water hydrogens
Two hydrogens are generated around atom i according to SPC [85] water geometry.
The symmetry axis will alternate between three coordinate axes in both directions.
10 three water “hydrogens”
Two hydrogens are generated around atom i according to SPC [85] water geometry.
The symmetry axis will alternate between three coordinate axes in both directions.
In addition, an extra particle is generated on the position of the oxygen with the first
letter of the name replaced by ‘M’. This is for use with four-atom water models such
as TIP4P [133].
11 four water “hydrogens”
Same as above, except that two additional particles are generated on the position of
the oxygen, with names ‘LP1’ and ‘LP2.’ This is for use with five-atom water models
such as TIP5P [135].
• The name of the new H atom (or its prefix, e.g. HD2 for the asparagine example given
earlier).
• Three or four control atoms (i,j,k,l), where the first always is the atom to which the H atoms
are connected. The other two or three depend on the code selected. For water, there is only
one control atom.
Some more exotic cases can be approximately constructed from the above tools, and with suitable use of energy minimization are good enough for beginning MD simulations. For example
secondary amine hydrogen, nitrenyl hydrogen (C=NH) and even ethynyl hydrogen could be approximately constructed using method 2 above for hydroxyl hydrogen.

5.7.5

Termini database

The termini databases are stored in aminoacids.n.tdb and aminoacids.c.tdb for the Nand C-termini respectively. They contain information for the pdb2gmx program on how to connect new atoms to existing ones, which atoms should be removed or changed, and which bonded
interactions should be added. Their format is as follows (from gromos43a1.ff/aminoacids.c.tdb):

134

Chapter 5. Topologies

[ None ]
[ COO- ]
[ replace ]
C C C 12.011 0.27
O O1 OM 15.9994 -0.635
OXT O2 OM 15.9994 -0.635
[ add ]
2 8 O C CA N
OM 15.9994 -0.635
[ bonds ]
C O1 gb_5
C O2 gb_5
[ angles ]
O1 C O2 ga_37
CA C O1 ga_21
CA C O2 ga_21
[ dihedrals ]
N CA C O2 gd_20
[ impropers ]
C CA O2 O1 gi_1

The file is organized in blocks, each with a header specifying the name of the block. These
blocks correspond to different types of termini that can be added to a molecule. In this example
[ COO- ] is the first block, corresponding to changing the terminal carbon atom into a deprotonated carboxyl group. [ None ] is the second terminus type, corresponding to a terminus
that leaves the molecule as it is. Block names cannot be any of the following: replace, add,
delete, bonds, angles, dihedrals, impropers. Doing so would interfere with the parameters of the block, and would probably also be very confusing to human readers.
For each block the following options are present:
• [ replace ]
Replace an existing atom by one with a different atom type, atom name, charge, and/or
mass. This entry can be used to replace an atom that is present both in the input coordinates
and in the .rtp database, but also to only rename an atom in the input coordinates such
that it matches the name in the force field. In the latter case, there should also be a corresponding [ add ] section present that gives instructions to add the same atom, such that
the position in the sequence and the bonding is known. Such an atom can be present in the
input coordinates and kept, or not present and constructed by pdb2gmx. For each atom to
be replaced on line should be entered with the following fields:
– name of the atom to be replaced
– new atom name (optional)
– new atom type
– new mass
– new charge

5.7. pdb2gmx input files

135

• [ add ]
Add new atoms. For each (group of) added atom(s), a two-line entry is necessary. The
first line contains the same fields as an entry in the hydrogen database (name of the new
atom, number of atoms, type of addition, control atoms, see 5.7.4), but the possible types of
addition are extended by two more, specifically for C-terminal additions:
8 two carboxyl oxygens, -COO−
Two oxygens (n1,n2) are generated according to rule 3, at a distance of 0.136 nm from
atom i and an angle (n1-i-j)=(n2-i-j)=117 degrees
9 carboxyl oxygens and hydrogen, -COOH
Two oxygens (n1,n2) are generated according to rule 3, at distances of 0.123 nm and
0.125 nm from atom i for n1 and n2, respectively, and angles (n1-i-j)=121 and (n2-ij)=115 degrees. One hydrogen (n0 ) is generated around n2 according to rule 2, where
n-i-j and n-i-j-k should be read as n0 -n2-i and n0 -n2-i-j, respectively.
After this line, another line follows that specifies the details of the added atom(s), in the
same way as for replacing atoms, i.e.:
– atom type
– mass
– charge
– charge group (optional)
Like in the hydrogen database (see 5.7.1), when more than one atom is connected to an
existing one, a number will be appended to the end of the atom name. Note that, like in the
hydrogen database, the atom name is now on the same line as the control atoms, whereas it
was at the beginning of the second line prior to GROMACS version 3.3. When the charge
group field is left out, the added atom will have the same charge group number as the atom
that it is bonded to.
• [ delete ]
Delete existing atoms. One atom name per line.
• [ bonds ], [ angles ], [ dihedrals ] and [ impropers ]
Add additional bonded parameters. The format is identical to that used in the *.rtp file,
see 5.7.1.

5.7.6

Virtual site database

Since we cannot rely on the positions of hydrogens in input files, we need a special input file
to decide the geometries and parameters with which to add virtual site hydrogens. For more
complex virtual site constructs (e.g. when entire aromatic side chains are made rigid) we also need
information about the equilibrium bond lengths and angles for all atoms in the side chain. This
information is specified in the .vsd file for each force field. Just as for the termini, there is one
such file for each class of residues in the .rtp file.
The virtual site database is not really a very simple list of information. The first couple of sections
specify which mass centers (typically called MCH3 /MNH3 ) to use for CH3 , NH3 , and NH2 groups.

136

Chapter 5. Topologies

Depending on the equilibrium bond lengths and angles between the hydrogens and heavy atoms
we need to apply slightly different constraint distances between these mass centers. Note that we
do not have to specify the actual parameters (that is automatic), just the type of mass center to use.
To accomplish this, there are three sections names [ CH3 ], [ NH3 ], and [ NH2 ]. For each
of these we expect three columns. The first column is the atom type bound to the 2/3 hydrogens,
the second column is the next heavy atom type which this is bound, and the third column the type
of mass center to use. As a special case, in the [ NH2 ] section it is also possible to specify
planar in the second column, which will use a different construction without mass center. There
are currently different opinions in some force fields whether an NH2 group should be planar or
not, but we try hard to stick to the default equilibrium parameters of the force field.
The second part of the virtual site database contains explicit equilibrium bond lengths and angles
for pairs/triplets of atoms in aromatic side chains. These entries are currently read by specific
routines in the virtual site generation code, so if you would like to extend it e.g. to nucleic acids
you would also need to write new code there. These sections are named after the short amino
acid names ([ PHE ], [ TYR ], [ TRP ], [ HID ], [ HIE ], [ HIP ]), and simply
contain 2 or 3 columns with atom names, followed by a number specifying the bond length (in
nm) or angle (in degrees). Note that these are approximations of the equilibrated geometry for the
entire molecule, which might not be identical to the equilibrium value for a single bond/angle if
the molecule is strained.

5.7.7

Special bonds

The primary mechanism used by pdb2gmx to generate inter-residue bonds relies on head-totail linking of backbone atoms in different residues to build a macromolecule. In some cases
(e.g. disulfide bonds, a heme group, branched polymers), it is necessary to create inter-residue
bonds that do not lie on the backbone. The file specbond.dat takes care of this function.
It is necessary that the residues belong to the same [ moleculetype ]. The -merge and
-chainsep functions of pdb2gmx can be useful when managing special inter-residue bonds
between different chains.
The first line of specbond.dat indicates the number of entries that are in the file. If you
add a new entry, be sure to increment this number. The remaining lines in the file provide the
specifications for creating bonds. The format of the lines is as follows:
resA atomA nbondsA resB atomB nbondsB length newresA newresB
The columns indicate:
1. resA The name of residue A that participates in the bond.
2. atomA The name of the atom in residue A that forms the bond.
3. nbondsA The total number of bonds atomA can form.
4. resB The name of residue B that participates in the bond.
5. atomB The name of the atom in residue B that forms the bond.
6. nbondsB The total number of bonds atomB can form.

5.8. File formats

137

7. length The reference length for the bond. If atomA and atomB are not within length
± 10% in the coordinate file supplied to pdb2gmx, no bond will be formed.
8. newresA The new name of residue A, if necessary. Some force fields use e.g. CYS2 for a
cysteine in a disulfide or heme linkage.
9. newresB The new name of residue B, likewise.

5.8

File formats

5.8.1

Topology file

The topology file is built following the GROMACS specification for a molecular topology. A
*.top file can be generated by pdb2gmx. All possible entries in the topology file are listed
in Tables 5.4 and 5.5. Also tabulated are: all the units of the parameters, which interactions
can be perturbed for free energy calculations, which bonded interactions are used by grompp for
generating exclusions, and which bonded interactions can be converted to constraints by grompp.

138

Chapter 5. Topologies

Parameters
interaction
type
mandatory

directive

#
at.

f.
tp

parameters

F. E.

defaults

mandatory

atomtypes

LJ
Buckingham

bondtypes
pairtypes
angletypes
dihedraltypes(∗)
constrainttypes
nonbond_params
nonbond_params

non-bonded function type;
combination rule(cr) ;
generate pairs (no/yes);
fudge LJ (); fudge QQ ()
atom type; m (u); q (e); particle type;
V(cr) ; W(cr)
(see Table 5.5, directive bonds)
(see Table 5.5, directive pairs)
(see Table 5.5, directive angles)
(see Table 5.5, directive dihedrals)
(see Table 5.5, directive constraints)
2
1 V (cr) ; W (cr)
2
2 a (kJ mol−1 ); b (nm−1 );
c6 (kJ mol−1 nm6 )

Molecule definition(s)
mandatory
mandatory

moleculetype
atoms

(nrexcl)

molecule name; nex
atom type; residue number;
residue name; atom name;
charge group number; q (e); m (u)

type
q, m

intra-molecular interaction and geometry definitions as described in Table 5.5

System
mandatory
mandatory

system
molecules

system name
molecule name; number of molecules

Inter-molecular interactions
optional
intermolecular_interactions
one or more bonded interactions as described in Table 5.5, with two or more atoms,
no interactions that generate exclusions, no constraints, use global atom numbers
‘# at’ is the required number of atom type indices for this directive
‘f. tp’ is the value used to select this function type
‘F. E.’ indicates which of the parameters can be interpolated in free energy calculations
(cr) the combination rule determines the type of LJ parameters, see 5.3.2
(∗) for dihedraltypes one can specify 4 atoms or the inner (outer for improper) 2 atoms
(nrexcl) exclude neighbors n bonds away for non-bonded interactions
ex
For free energy calculations, type, q and m or no parameters should be added
for topology ‘B’ (λ = 1) on the same line, after the normal parameters.
Table 5.4: The topology (*.top) file.

Name of interaction

Topology file directive

bond
G96 bond
Morse
cubic bond
connection
harmonic potential
FENE bond
tabulated bond
tabulated bondk
restraint potential
extra LJ or Coulomb
extra LJ or Coulomb
extra LJ or Coulomb
angle
G96 angle
cross bond-bond
cross bond-angle
Urey-Bradley

bonds§ ,¶
bonds§ ,¶
bonds§ ,¶
bonds§ ,¶
bonds§
bonds
bonds§
bonds§
bonds
bonds
pairs
pairs
pairs_nb
angles¶
angles¶
angles
angles
angles¶

quartic angle

angles¶

num.
atoms∗
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3

func.
type†
1
2
3
4
5
6
7
8
9
10
1
2
1
1
2
3
4
5

Order of parameters and their units
b0 (nm); kb (kJ mol−1 nm−2 )
b0 (nm); kb (kJ mol−1 nm−4 )
b0 (nm); D (kJ mol−1 ); β (nm−1 )
b0 (nm); Ci=2,3 (kJ mol−1 nm−i )
b0 (nm); kb (kJ mol−1 nm−2 )
bm (nm); kb (kJ mol−1 nm−2 )
table number (≥ 0); k (kJ mol−1 )
table number (≥ 0); k (kJ mol−1 )
low, up1 , up2 (nm); kdr (kJ mol−1 nm−2 )
V ∗∗ ; W ∗∗
fudge QQ (); qi , qj (e), V ∗∗ ; W ∗∗
qi , qj (e); V ∗∗ ; W ∗∗
θ0 (deg); kθ (kJ mol−1 rad−2 )
θ0 (deg); kθ (kJ mol−1 )
r1e , r2e (nm); krr0 (kJ mol−1 nm−2 )
r1e , r2e r3e (nm); krθ (kJ mol−1 nm−2 )
θ0 (deg); kθ (kJ mol−1 rad−2 ); r13 (nm);
kU B (kJ mol−1 nm−2 )
θ0 (deg); Ci=0,1,2,3,4 (kJ mol−1 rad−i )

use in
F.E.?‡
all
all
all

all
k
k
all
all

all
all

all

Crossreferences
4.2.1
4.2.1
4.2.2
4.2.3
5.4.4
4.2.1,5.4.4
4.2.4
4.2.14
4.2.14,5.4.4
4.3.5
5.4.3
5.4.3
5.4.3
4.2.5
4.2.6
4.2.9
4.2.10
4.2.8

5.8. File formats

Table 5.5: Details of [ moleculetype ] directives

4.2.11

∗

The required number of atom indices for this directive
The index to use to select this function type
‡
Indicates which of the parameters can be interpolated in free energy calculations
§
This interaction type will be used by grompp for generating exclusions
¶
This interaction type can be converted to constraints by grompp
∗∗
The combination rule determines the type of LJ parameters, see 5.3.2
k
No connection, and so no exclusions, are generated for this interaction
†

139

140

Table 5.5: Details of [ moleculetype ] directives
Topology file directive

num.
atoms∗
3
3
4
4
4
4
4
4
4
4
4

func.
type†
8
10
1
2
3
4
5
8
9
10
11

tabulated angle
restricted bending potential
proper dihedral
improper dihedral
Ryckaert-Bellemans dihedral
periodic improper dihedral
Fourier dihedral
tabulated dihedral
proper dihedral (multiple)
restricted dihedral
combined
bending-torsion
potential
exclusions
constraint
constraintk
SETTLE
2-body virtual site
3-body virtual site
3-body virtual site (fd)
3-body virtual site (fad)
3-body virtual site (out)
4-body virtual site (fdn)
N-body virtual site (COG)
N-body virtual site (COM)
N-body virtual site (COW)

angles
angles
dihedrals
dihedrals
dihedrals
dihedrals
dihedrals
dihedrals
dihedrals
dihedrals
dihedrals
exclusions
constraints§
constraints
settles
virtual_sites2
virtual_sites3
virtual_sites3
virtual_sites3
virtual_sites3
virtual_sites4
virtual_sitesn
virtual_sitesn
virtual_sitesn

1
2
2
1
3
4
4
4
4
5
1
1
1

1
2
1
1
1
2
3
4
2
1
2
3

position restraint

position_restraints

Order of parameters and their units
table number (≥ 0); k (kJ mol−1 )
θ0 (deg); kθ (kJ mol−1 )
φs (deg); kφ (kJ mol−1 ); multiplicity
ξ0 (deg); kξ (kJ mol−1 rad−2 )
C0 , C1 , C2 , C3 , C4 , C5 (kJ mol−1 )
φs (deg); kφ (kJ mol−1 ); multiplicity
C1 , C2 , C3 , C4 (kJ mol−1 )
table number (≥ 0); k (kJ mol−1 )
φs (deg); kφ (kJ mol−1 ); multiplicity
φ0 (deg); kφ (kJ mol−1 )
a0 , a1 , a2 , a3 , a4 (kJ mol−1 )
one or more atom indices
b0 (nm)
b0 (nm)
dOH , dHH (nm)
a ()
a, b ()
a (); d (nm)
θ (deg); d (nm)
a, b (); c (nm−1 )
a, b (); c (nm)
one or more constructing atom indices
one or more constructing atom indices
one or more pairs consisting of
constructing atom index and weight
kx , ky , kz (kJ mol−1 nm−2 )

use in
F.E.?‡
k
φ, k
all
all
φ, k
all
k
φ, k

all
all

all

Crossreferences
4.2.14
4.2.7
4.2.13
4.2.12
4.2.13
4.2.12
4.2.13
4.2.14
4.2.13
4.2.13
4.2.13
5.4.4
4.5,5.6
4.5,5.6,5.4.4
3.6.1,5.6
4.7
4.7
4.7
4.7
4.7
4.7
4.7
4.7
4.7
4.3.1

Chapter 5. Topologies

Name of interaction

num.
atoms∗
1

func.
type†
2

dihedral_restraints

2
4

1
1

orientation restraint

orientation_restraints

angle restraint
angle restraint (z)

angle_restraints

4
2

1
1

flat-bottomed position
straint
distance restraint
dihedral restraint

Topology file directive
re-

position_restraints

distance_restraints

angle_restraints_z

Order of parameters and their units

use in
F.E.?‡

g, r (nm), k (kJ mol−1 nm−2 )
type; label; low, up1 , up2 (nm); weight ()
φ0 (deg); ∆φ (deg); kdihr
(kJ mol−1 rad−2 )
exp.; label; α; c (U nmα ); obs. (U);
weight (U−1 )
θ0 (deg); kc (kJ mol−1 ); multiplicity
θ0 (deg); kc (kJ mol−1 ); multiplicity

all

Crossreferences
4.3.2

5.8. File formats

Table 5.5: Details of [ moleculetype ] directives

4.3.5
4.3.4
4.3.6

θ, k
θ, k

4.3.3
4.3.3

141

142

Chapter 5. Topologies

Description of the file layout:
• Semicolon (;) and newline characters surround comments
• On a line ending with \ the newline character is ignored.
• Directives are surrounded by [ and ]
• The topology hierarchy (which must be followed) consists of three levels:
– the parameter level, which defines certain force-field specifications (see Table 5.4)
– the molecule level, which should contain one or more molecule definitions (see Table 5.5)
– the system level, containing only system-specific information ([ system ] and
[ molecules ])
• Items should be separated by spaces or tabs, not commas
• Atoms in molecules should be numbered consecutively starting at 1
• Atoms in the same charge group must be listed consecutively
• The file is parsed only once, which implies that no forward references can be treated: items
must be defined before they can be used
• Exclusions can be generated from the bonds or overridden manually
• The bonded force types can be generated from the atom types or overridden per bond
• It is possible to apply multiple bonded interactions of the same type on the same atoms
• Descriptive comment lines and empty lines are highly recommended
• Starting with GROMACS version 3.1.3, all directives at the parameter level can be used
multiple times and there are no restrictions on the order, except that an atom type needs to
be defined before it can be used in other parameter definitions
• If parameters for a certain interaction are defined multiple times for the same combination
of atom types the last definition is used; starting with GROMACS version 3.1.3 grompp
generates a warning for parameter redefinitions with different values
• Using one of the [ atoms ], [ bonds ], [ pairs ], [ angles ], etc. without
having used [ moleculetype ] before is meaningless and generates a warning
• Using [ molecules ] without having used [ system ] before is meaningless and
generates a warning.
• After [ system ] the only allowed directive is [ molecules ]
• Using an unknown string in [ ] causes all the data until the next directive to be ignored
and generates a warning

5.8. File formats

143

Here is an example of a topology file, urea.top:
;
;
Example topology file
;
; The force-field files to be included
#include "amber99.ff/forcefield.itp"
[ moleculetype ]
; name nrexcl
Urea
3
[ atoms ]
1 C 1
2 O 1
3 N 1
4 H 1
5 H 1
6 N 1
7 H 1
8 H 1

URE
URE
URE
URE
URE
URE
URE
URE

C
O
N1
H11
H12
N2
H21
H22

1
2
3
4
5
6
7
8

0.880229
-0.613359
-0.923545
0.395055
0.395055
-0.923545
0.395055
0.395055

12.01000
16.00000
14.01000
1.00800
1.00800
14.01000
1.00800
1.00800

;
;
;
;
;
;
;
;

amber
amber
amber
amber
amber
amber
amber
amber

[ bonds ]
1 2
1 3
1
6
3 4
3 5
6 7
6 8
[ dihedrals ]
;
ai
aj
2
1
2
1
2
1
2
1
3
1
3
1
6
1
6
1

ak
3
3
6
6
6
6
3
3

[ dihedrals ]
3
6
1
4
1
7

1
3
6

[
;
;
;

al funct
4
9
5
9
7
9
8
9
7
9
8
9
4
9
5
9

2
5
8

definition

4
4
4

position_restraints ]
you wouldn’t normally use this for a molecule like Urea,
but we include it here for didactic purposes
ai
funct
fc
1
1
1000
1000
1000 ; Restrain to a point

C
O
N
H
H
N
H
H

type
type
type
type
type
type
type
type

144

Chapter 5. Topologies
2
3

1
1

1000
1000

0
0

[ dihedral_restraints ]
; ai
aj
ak
al type
3
6
1
2
1
1
4
3
5
1

1000 ; Restrain to a line (Y-axis)
0 ; Restrain to a plane (Y-Z-plane)

phi
180
180

dphi
0
0

fc
10
10

; Include TIP3P water topology
#include "amber99/tip3p.itp"
[ system ]
Urea in Water
[ molecules ]
;molecule name
Urea
SOL

nr.
1
1000

Here follows the explanatory text.
#include "amber99.ff/forcefield.itp" : this includes the information for the force
field you are using, including bonded and non-bonded parameters. This example uses the AMBER99 force field, but your simulation may use a different force field. grompp will automatically
go and find this file and copy-and-paste its content.
That content can be seen in
share/top/amber99.ff/forcefield.itp, and it is
#define _FF_AMBER
#define _FF_AMBER99
[ defaults ]
; nbfunc
1

comb-rule
2

gen-pairs
yes

fudgeLJ fudgeQQ
0.5
0.8333

#include "ffnonbonded.itp"
#include "ffbonded.itp"
#include "gbsa.itp"

The two #define statements set up the conditions so that future parts of the topology can know
that the AMBER 99 force field is in use.
[ defaults ] :
• nbfunc is the non-bonded function type. Use 1 (Lennard-Jones) or 2 (Buckingham)
• comb-rule is the number of the combination rule (see 5.3.2).
• gen-pairs is for pair generation. The default is ‘no’, i.e. get 1-4 parameters from the
pairtypes list. When parameters are not present in the list, stop with a fatal error. Setting
‘yes’ generates 1-4 parameters that are not present in the pair list from normal LennardJones parameters using fudgeLJ

5.8. File formats

145

• fudgeLJ is the factor by which to multiply Lennard-Jones 1-4 interactions, default 1
• fudgeQQ is the factor by which to multiply electrostatic 1-4 interactions, default 1
• N is the power for the repulsion term in a 6-N potential (with nonbonded-type LennardJones only), starting with GROMACS version 4.5, mdrun also reads and applies N , for
values not equal to 12 tabulated interaction functions are used (in older version you would
have to use user tabulated interactions).
Note that gen-pairs, fudgeLJ, fudgeQQ, and N are optional. fudgeLJ is only used when
generate pairs is set to ‘yes’, and fudgeQQ is always used. However, if you want to specify N
you need to give a value for the other parameters as well.
Then some other #include statements add in the large amount of data needed to describe the
rest of the force field. We will skip these and return to urea.top. There we will see
[ moleculetype ] : defines the name of your molecule in this *.top and nrexcl = 3 stands
for excluding non-bonded interactions between atoms that are no further than 3 bonds away.
[ atoms ] : defines the molecule, where nr and type are fixed, the rest is user defined. So
atom can be named as you like, cgnr made larger or smaller (if possible, the total charge of a
charge group should be zero), and charges can be changed here too.
[ bonds ] : no comment.
[ pairs ] : LJ and Coulomb 1-4 interactions
[ angles ] : no comment
[ dihedrals ] : in this case there are 9 proper dihedrals (funct = 1), 3 improper (funct =
4) and no Ryckaert-Bellemans type dihedrals. If you want to include Ryckaert-Bellemans type
dihedrals in a topology, do the following (in case of e.g. decane):
[ dihedrals ]
; ai
aj
1
2
2
3

ak
3
4

al funct
4
3
5
3

In the original implementation of the potential for alkanes [136] no 1-4 interactions were used,
which means that in order to implement that particular force field you need to remove the 1-4
interactions from the [ pairs ] section of your topology. In most modern force fields, like
OPLS/AA or Amber the rules are different, and the Ryckaert-Bellemans potential is used as a
cosine series in combination with 1-4 interactions.
[ position_restraints ] : harmonically restrain the selected particles to reference positions (4.3.1). The reference positions are read from a separate coordinate file by grompp.
[ dihedral_restraints ] : restrain selected dihedrals to a reference value. The implementation of dihedral restraints is described in section 4.3.4 of the manual. The parameters specified in the [dihedral_restraints] directive are as follows:
• type has only one possible value which is 1
• phi is the value of φ0 in eqn. 4.83 and eqn. 4.84 of the manual.

146

Chapter 5. Topologies
• dphi is the value of ∆φ in eqn. 4.84 of the manual.
• fc is the force constant kdihr in eqn. 4.84 of the manual.

#include "tip3p.itp" : includes a topology file that was already constructed (see section 5.8.2).
[ system ] : title of your system, user-defined
[ molecules ] : this defines the total number of (sub)molecules in your system that are defined in this *.top. In this example file, it stands for 1 urea molecule dissolved in 1000 water
molecules. The molecule type SOL is defined in the tip3p.itp file. Each name here must
correspond to a name given with [ moleculetype ] earlier in the topology. The order of the
blocks of molecule types and the numbers of such molecules must match the coordinate file that
accompanies the topology when supplied to grompp. The blocks of molecules do not need to
be contiguous, but some tools (e.g. genion) may act only on the first or last such block of a
particular molecule type. Also, these blocks have nothing to do with the definition of groups (see
sec. 3.3 and sec. 8.1).

5.8.2

Molecule.itp file

If you construct a topology file you will use frequently (like the water molecule, tip3p.itp,
which is already constructed for you) it is good to make a molecule.itp file. This only lists
the information of one particular molecule and allows you to re-use the [ moleculetype ]
in multiple systems without re-invoking pdb2gmx or manually copying and pasting. An example
urea.itp follows:
[ moleculetype ]
; molname nrexcl
URE 3
[ atoms ]
1 C 1
...
8 H 1

URE

0.880229

12.01000

; amber C

type

URE

H22

0.395055

1.00800

; amber H

type

[ bonds ]
1 2
...
6 8
[ dihedrals ]
;
ai
aj
2
1
...
6
1
[ dihedrals ]
3
6
1
4
1
7

ak
3

al funct
4
9

1
3
6

2
5
8

4
4
4

definition

5.8. File formats

147

Using *.itp files results in a very short *.top file:
;
;
Example topology file
;
; The force field files to be included
#include "amber99.ff/forcefield.itp"
#include "urea.itp"
; Include TIP3P water topology
#include "amber99/tip3p.itp"
[ system ]
Urea in Water
[ molecules ]
;molecule name
Urea
SOL

5.8.3

nr.
1
1000

Ifdef statements

A very powerful feature in GROMACS is the use of #ifdef statements in your *.top file. By
making use of this statement, and associated #define statements like were seen in
amber99.ff/forcefield.itp earlier, different parameters for one molecule can be used
in the same *.top file. An example is given for TFE, where there is an option to use different
charges on the atoms: charges derived by De Loof et al. [137] or by Van Buuren and Berendsen [138]. In fact, you can use much of the functionality of the C preprocessor, cpp, because
grompp contains similar pre-processing functions to scan the file. The way to make use of the
#ifdef option is as follows:
• either use the option define = -DDeLoof in the *.mdp file (containing grompp input
parameters), or use the line #define DeLoof early in your *.top or *.itp file; and
• put the #ifdef statements in your *.top, as shown below:
...

[ atoms ]
; nr
type
resnr
#ifdef DeLoof
; Use Charges from DeLoof
1
C
1
2
F
1
3
F
1
4
F
1

residu

atom

cgnr

charge

TFE
TFE
TFE
TFE

C
F
F
F

1
1
1
1

0.74
-0.25
-0.25
-0.25

mass

148

Chapter 5. Topologies

5
CH2
1
6
OA
1
7
HO
1
#else
; Use Charges from VanBuuren
1
C
1
2
F
1
3
F
1
4
F
1
5
CH2
1
6
OA
1
7
HO
1
#endif

TFE
TFE
TFE

CH2
OA
HO

1
1
1

0.25
-0.65
0.41

TFE
TFE
TFE
TFE
TFE
TFE
TFE

C
F
F
F
CH2
OA
HO

1
1
1
1
1
1
1

0.59
-0.2
-0.2
-0.2
0.26
-0.55
0.3

[ bonds ]
; ai
aj funct
c0
c1
6
7
1 1.000000e-01 3.138000e+05
1
2
1 1.360000e-01 4.184000e+05
1
3
1 1.360000e-01 4.184000e+05
1
4
1 1.360000e-01 4.184000e+05
1
5
1 1.530000e-01 3.347000e+05
5
6
1 1.430000e-01 3.347000e+05
...

This mechanism is used by pdb2gmx to implement optional position restraints (4.3.1) by #includeing an .itp file whose contents will be meaningful only if a particular #define is set (and
spelled correctly!)

5.8.4

Topologies for free energy calculations

Free energy differences between two systems, A and B, can be calculated as described in sec. 3.12.
Systems A and B are described by topologies consisting of the same number of molecules with
the same number of atoms. Masses and non-bonded interactions can be perturbed by adding B
parameters under the [ atoms ] directive. Bonded interactions can be perturbed by adding B
parameters to the bonded types or the bonded interactions. The parameters that can be perturbed
are listed in Tables 5.4 and 5.5. The λ-dependence of the interactions is described in section
sec. 4.5. The bonded parameters that are used (on the line of the bonded interaction definition, or
the ones looked up on atom types in the bonded type lists) is explained in Table 5.6. In most cases,
things should work intuitively. When the A and B atom types in a bonded interaction are not all
identical and parameters are not present for the B-state, either on the line or in the bonded types,
grompp uses the A-state parameters and issues a warning. For free energy calculations, all or no
parameters for topology B (λ = 1) should be added on the same line, after the normal parameters,
in the same order as the normal parameters. From GROMACS 4.6 onward, if λ is treated as a
vector, then the bonded-lambdas component controls all bonded terms that are not explicitly
labeled as restraints. Restrain terms are controlled by the restraint-lambdas component.
Below is an example of a topology which changes from 200 propanols to 200 pentanes using the
GROMOS-96 force field.

5.8. File formats

149

B-state atom types
all identical to
A-state atom types

yes

parameters
on line
A
B
+AB −
+A +B
−
−
−
−
−
−
+AB −
+A +B
−
−
−
−
−
−
−
−
−
−

parameters in bonded types
A atom types B atom types
A
B
A
B
x
x
x
x
−
−
+AB
−
+A
+B
x
x
x
x
x
x
x
x
−
−
x
x
+AB
−
−
−
+A
+B
−
−
+A
x
+B
−
+A
x
+
+B

message

error

warning
error
warning
warning

Table 5.6: The bonded parameters that are used for free energy topologies, on the line of the
bonded interaction definition or looked up in the bond types section based on atom types. A and
B indicate the parameters used for state A and B respectively, + and − indicate the (non-)presence
of parameters in the topology, x indicates that the presence has no influence.

; Include force field parameters
#include "gromos43a1.ff/forcefield.itp"
[ moleculetype ]
; Name
PropPent

nrexcl
3

[ atoms ]
; nr type resnr residue atom cgnr
1
H
1
PROP
PH
1
2
OA
1
PROP
PO
1
3 CH2
1
PROP
PC1
1
4 CH2
1
PROP
PC2
2
5 CH3
1
PROP
PC3
2
[ bonds ]
; ai
aj funct
1
2
2
2
3
2
3
4
2
4
5
2
[ pairs ]
; ai
aj funct
1
4
1
2
5
1

par_A
gb_1
gb_17
gb_26
gb_26

par_B
gb_26
gb_26
gb_26

charge
0.398
-0.548
0.150
0.000
0.000

mass typeB chargeB massB
1.008 CH3
0.0 15.035
15.9994 CH2
0.0 14.027
14.027 CH2
0.0 14.027
14.027
15.035

150

Chapter 5. Topologies

[ angles ]
; ai
aj
1
2
2
3
3
4

ak funct
3
2
4
2
5
2

[ dihedrals ]
; ai
aj
1
2
2
3

ak
3
4

par_A
ga_11
ga_14
ga_14

al funct
4
1
5
1

par_B
ga_14
ga_14
ga_14

par_A
gd_12
gd_17

par_B
gd_17
gd_17

[ system ]
; Name
Propanol to Pentane
[ molecules ]
; Compound
PropPent

#mols
200

Atoms that are not perturbed, PC2 and PC3, do not need B-state parameter specifications, since the
B parameters will be copied from the A parameters. Bonded interactions between atoms that are
not perturbed do not need B parameter specifications, as is the case for the last bond in the example
topology. Topologies using the OPLS/AA force field need no bonded parameters at all, since both
the A and B parameters are determined by the atom types. Non-bonded interactions involving one
or two perturbed atoms use the free-energy perturbation functional forms. Non-bonded interactions between two non-perturbed atoms use the normal functional forms. This means that when,
for instance, only the charge of a particle is perturbed, its Lennard-Jones interactions will also be
affected when lambda is not equal to zero or one.
Note that this topology uses the GROMOS-96 force field, in which the bonded interactions are not
determined by the atom types. The bonded interaction strings are converted by the C-preprocessor.
The force-field parameter files contain lines like:
#define gb_26
#define gd_17

5.8.5

0.1530
0.000

7.1500e+06
5.86

Constraint forces

The constraint force between two atoms in one molecule can be calculated with the free energy
perturbation code by adding a constraint between the two atoms, with a different length in the A
and B topology. When the B length is 1 nm longer than the A length and lambda is kept constant at zero, the derivative of the Hamiltonian with respect to lambda is the constraint force. For
constraints between molecules, the pull code can be used, see sec. 6.4. Below is an example for
calculating the constraint force at 0.7 nm between two methanes in water, by combining the two
methanes into one “molecule.” Note that the definition of a “molecule” in GROMACS does not
necessarily correspond to the chemical definition of a molecule. In GROMACS, a “molecule” can

5.8. File formats

151

be defined as any group of atoms that one wishes to consider simultaneously. The added constraint
is of function type 2, which means that it is not used for generating exclusions (see sec. 5.4.4). Note
that the constraint free energy term is included in the derivative term, and is specifically included
in the bonded-lambdas component. However, the free energy for changing constraints is not
included in the potential energy differences used for BAR and MBAR, as this requires reevaluating
the energy at each of the constraint components. This functionality is planned for later versions.

; Include force-field parameters
#include "gromos43a1.ff/forcefield.itp"
[ moleculetype ]
; Name
Methanes

nrexcl
1

[ atoms ]
; nr
type
resnr residu
1
CH4
1
CH4
2
CH4
1
CH4
[ constraints ]
; ai
aj funct
length_A
1
2
2
0.7

atom
C1
C2

cgnr
1
2

charge
0
0

mass
16.043
16.043

length_B
1.7

#include "gromos43a1.ff/spc.itp"
[ system ]
; Name
Methanes in Water
[ molecules ]
; Compound
Methanes
SOL

5.8.6

#mols
1
2002

Coordinate file

Files with the .gro file extension contain a molecular structure in GROMOS-87 format. A sample
piece is included below:
MD of 2 waters, reformat step,
6
1WATER OW1
1
0.126
1WATER HW2
2
0.190
1WATER HW3
3
0.177
2WATER OW1
4
1.275
2WATER HW2
5
1.337
2WATER HW3
6
1.326
1.82060
1.82060
1.82060

PA aug-91
1.624
1.661
1.568
0.053
0.002
0.120

1.679 0.1227 -0.0580 0.0434
1.747 0.8085 0.3191 -0.7791
1.613 -0.9045 -2.6469 1.3180
0.622 0.2519 0.3140 -0.1734
0.680 -1.0641 -1.1349 0.0257
0.568 1.9427 -0.8216 -0.0244

152

Chapter 5. Topologies

This format is fixed, i.e. all columns are in a fixed position. If you want to read such a file in your
own program without using the GROMACS libraries you can use the following formats:
C-format: "%5i%5s%5s%5i%8.3f%8.3f%8.3f%8.4f%8.4f%8.4f"
Or to be more precise, with title etc. it looks like this:
"%s\n", Title
"%5d\n", natoms
for (i=0; (i.ff
directories in the $GMXLIB/share/gromacs/top sub-directory and/or the working directory. The information regarding the location of the force field files is printed by pdb2gmx so
you can easily keep track of which version of a force field is being called, in case you have made
modifications in one location or another. The force fields included with GROMACS are:
• AMBER03 protein, nucleic AMBER94 (Duan et al., J. Comp. Chem. 24, 1999-2012, 2003)
• AMBER94 force field (Cornell et al., JACS 117, 5179-5197, 1995)
• AMBER96 protein, nucleic AMBER94 (Kollman et al., Acc. Chem. Res. 29, 461-469, 1996)
• AMBER99 protein, nucleic AMBER94 (Wang et al., J. Comp. Chem. 21, 1049-1074, 2000)
• AMBER99SB protein, nucleic AMBER94 (Hornak et al., Proteins 65, 712-725, 2006)
• AMBER99SB-ILDN protein, nucleic AMBER94 (Lindorff-Larsen et al., Proteins 78, 1950-58, 2010)
• AMBERGS force field (Garcia & Sanbonmatsu, PNAS 99, 2782-2787, 2002)
• CHARMM27 all-atom force field (CHARM22 plus CMAP for proteins)
• GROMOS96 43a1 force field
• GROMOS96 43a2 force field (improved alkane dihedrals)
• GROMOS96 45a3 force field (Schuler JCC 2001 22 1205)

5.9. Force field organization

153

• GROMOS96 53a5 force field (JCC 2004 vol 25 pag 1656)
• GROMOS96 53a6 force field (JCC 2004 vol 25 pag 1656)
• GROMOS96 54a7 force field (Eur. Biophys. J. (2011), 40„ 843-856, DOI: 10.1007/s00249-0110700-9)
• OPLS-AA/L all-atom force field (2001 aminoacid dihedrals)

A force field is included at the beginning of a topology file with an #include statement followed
by .ff/forcefield.itp. This statement includes the force-field file, which, in turn,
may include other force-field files. All the force fields are organized in the same way. An example
of the amber99.ff/forcefield.itp was shown in 5.8.1.
For each force field, there several files which are only used by pdb2gmx. These are: residue
databases (.rtp, see 5.7.1) the hydrogen database (.hdb, see 5.7.4), two termini databases
(.n.tdb and .c.tdb, see 5.7.5) and the atom type database (.atp, see 5.2.1), which contains
only the masses. Other optional files are described in sec. 5.7.

5.9.2

Changing force-field parameters

If one wants to change the parameters of few bonded interactions in a molecule, this is most easily
accomplished by typing the parameters behind the definition of the bonded interaction directly in
the *.top file under the [ moleculetype ] section (see 5.8.1 for the format and units). If
one wants to change the parameters for all instances of a certain interaction one can change them
in the force-field file or add a new [ ???types ] section after including the force field. When
parameters for a certain interaction are defined multiple times, the last definition is used. As of
GROMACS version 3.1.3, a warning is generated when parameters are redefined with a different
value. Changing the Lennard-Jones parameters of an atom type is not recommended, because in
the GROMOS force fields the Lennard-Jones parameters for several combinations of atom types
are not generated according to the standard combination rules. Such combinations (and possibly
others that do follow the combination rules) are defined in the [ nonbond_params ] section,
and changing the Lennard-Jones parameters of an atom type has no effect on these combinations.

5.9.3

Adding atom types

As of GROMACS version 3.1.3, atom types can be added in an extra [ atomtypes ] section
after the the inclusion of the normal force field. After the definition of the new atom type(s), additional non-bonded and pair parameters can be defined. In pre-3.1.3 versions of GROMACS, the
new atom types needed to be added in the [ atomtypes ] section of the force-field files, because all non-bonded parameters above the last [ atomtypes ] section would be overwritten
using the standard combination rules.

154

Chapter 5. Topologies

Chapter 6
Special Topics
6.1

Free energy implementation

For free energy calculations, there are two things that must be specified; the end states, and the
pathway connecting the end states. The end states can be specified in two ways. The most straightforward is through the specification of end states in the topology file. Most potential forms support
both an A state and a B state. Whenever both states are specified, then the A state corresponds to
the initial free energy state, and the B state corresponds to the final state.
In some cases, the end state can also be defined in some cases without altering the topology,
solely through the .mdp file, through the use of the couple-moltype,couple-lambda0,
couple-lambda1, and couple-intramol mdp keywords. Any molecule type selected in
couple-moltype will automatically have a B state implicitly constructed (and the A state redefined) according to the couple-lambda keywords. couple-lambda0 and couple-lambda1
define the non-bonded parameters that are present in the A state (couple-lambda0) and the B
state (couple-lambda1). The choices are ’q’,’vdw’, and ’vdw-q’; these indicate the Coulombic, van der Waals, or both parameters that are turned on in the respective state.
Once the end states are defined, then the path between the end states has to be defined. This path
is defined solely in the .mdp file. Starting in 4.6, λ is a vector of components, with Coulombic,
van der Waals, bonded, restraint, and mass components all able to be adjusted independently. This
makes it possible to turn off the Coulombic term linearly, and then the van der Waals using soft
core, all in the same simulation. This is especially useful for replica exchange or expanded ensemble simulations, where it is important to sample all the way from interacting to non-interacting
states in the same simulation to improve sampling.
fep-lambdas is the default array of λ values ranging from 0 to 1. All of the other lambda arrays
use the values in this array if they are not specified. The previous behavior, where the pathway
is controlled by a single λ variable, can be preserved by using only fep-lambdas to define the
pathway.
For example, if you wanted to first to change the Coulombic terms, then the van der Waals terms,
changing bonded at the same time rate as the van der Waals, but changing the restraints throughout

156

Chapter 6. Special Topics

the first two-thirds of the simulation, then you could use this λ vector:
coul-lambdas
vdw-lambdas
bonded-lambdas
restraint-lambdas

=
=
=
=

0.0
0.0
0.0
0.0

0.2
0.0
0.0
0.0

0.5
0.0
0.0
0.1

1.0
0.0
0.0
0.2

1.0
0.4
0.4
0.3

1.0
0.5
0.5
0.5

1.0
0.6
0.6
0.7

1.0
0.7
0.7
1.0

1.0
0.8
0.8
1.0

1.0
1.0
1.0
1.0

This is also equivalent to:
fep-lambdas
coul-lambdas
restraint-lambdas

= 0.0 0.0 0.0 0.0 0.4 0.5 0.6 0.7 0.8 1.0
= 0.0 0.2 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0
= 0.0 0.0 0.1 0.2 0.3 0.5 0.7 1.0 1.0 1.0

The fep-lambda array, in this case, is being used as the default to fill in the bonded and van
der Waals λ arrays. Usually, it’s best to fill in all arrays explicitly, just to make sure things are
properly assigned.
If you want to turn on only restraints going from A to B, then it would be:
restraint-lambdas

= 0.0 0.1 0.2 0.4 0.6 1.0

and all of the other components of the λ vector would be left in the A state.
To compute free energies with a vector λ using thermodynamic integration, then the TI equation
becomes vector equation:
Z
∆F = h∇Hi · d~λ
(6.1)
or for finite differences:
∆F ≈

Z X

h∇Hi · ∆λ

(6.2)

The external pymbar script downloaded from https://SimTK.org/home/pymbar can compute this
integral automatically from the GROMACS dhdl.xvg output.

6.2

Potential of mean force

A potential of mean force (PMF) is a potential that is obtained by integrating the mean force from
an ensemble of configurations. In GROMACS, there are several different methods to calculate the
mean force. Each method has its limitations, which are listed below.
• pull code: between the centers of mass of molecules or groups of molecules.
• AWH code: currently acts on coordinates provided by the pull code.
• free-energy code with harmonic bonds or constraints: between single atoms.
• free-energy code with position restraints: changing the conformation of a relatively immobile group of atoms.

6.3. Non-equilibrium pulling

157

• pull code in limited cases: between groups of atoms that are part of a larger molecule for
which the bonds are constrained with SHAKE or LINCS. If the pull group if relatively large,
the pull code can be used.
The pull and free-energy code a described in more detail in the following two sections.
Entropic effects
When a distance between two atoms or the centers of mass of two groups is constrained or restrained, there will be a purely entropic contribution to the PMF due to the rotation of the two
groups [139]. For a system of two non-interacting masses the potential of mean force is:
Vpmf (r) = −(nc − 1)kB T log(r)

(6.3)

where nc is the number of dimensions in which the constraint works (i.e. nc = 3 for a normal constraint and nc = 1 when only the z-direction is constrained). Whether one needs to correct for this
contribution depends on what the PMF should represent. When one wants to pull a substrate into a
protein, this entropic term indeed contributes to the work to get the substrate into the protein. But
when calculating a PMF between two solutes in a solvent, for the purpose of simulating without
solvent, the entropic contribution should be removed. Note that this term can be significant; when
at 300K the distance is halved, the contribution is 3.5 kJ mol−1 .

6.3

Non-equilibrium pulling

When the distance between two groups is changed continuously, work is applied to the system,
which means that the system is no longer in equilibrium. Although in the limit of very slow pulling
the system is again in equilibrium, for many systems this limit is not reachable within reasonable
computational time. However, one can use the Jarzynski relation [140] to obtain the equilibrium
free-energy difference ∆G between two distances from many non-equilibrium simulations:
D

∆GAB = −kB T log e−βWAB

E
A

(6.4)

where WAB is the work performed to force the system along one path from state A to B, the
angular bracket denotes averaging over a canonical ensemble of the initial state A and β = 1/kB T .

6.4

The pull code

The pull code applies forces or constraints between the centers of mass of one or more pairs of
groups of atoms. Each pull reaction coordinate is called a “coordinate” and it operates on usually
two, but sometimes more, pull groups. A pull group can be part of one or more pull coordinates.
Furthermore, a coordinate can also operate on a single group and an absolute reference position in
space. The distance between a pair of groups can be determined in 1, 2 or 3 dimensions, or can
be along a user-defined vector. The reference distance can be constant or can change linearly with
time. Normally all atoms are weighted by their mass, but an additional weighting factor can also
be used.

158

Chapter 6. Special Topics

z link
Vrup

z spring

Figure 6.1: Schematic picture of pulling a lipid out of a lipid bilayer with umbrella pulling. Vrup
is the velocity at which the spring is retracted, Zlink is the atom to which the spring is attached
and Zspring is the location of the spring.
Several different pull types, i.e. ways to apply the pull force, are supported, and in all cases the
reference distance can be constant or linearly changing with time.
1. Umbrella pulling A harmonic potential is applied between the centers of mass of two
groups. Thus, the force is proportional to the displacement.
2. Constraint pulling The distance between the centers of mass of two groups is constrained.
The constraint force can be written to a file. This method uses the SHAKE algorithm but
only needs 1 iteration to be exact if only two groups are constrained.
3. Constant force pulling A constant force is applied between the centers of mass of two
groups. Thus, the potential is linear. In this case there is no reference distance of pull rate.
4. Flat bottom pulling Like umbrella pulling, but the potential and force are zero for coordinate values below (pull-coord?-type = flat-bottom) or above (pull-coord?-type
= flat-bottom-high) a reference value. This is useful for restraining e.g. the distance
between two molecules to a certain region.
In addition, there are different types of reaction coordinates, so-called pull geometries. These are
set with the .mdp option pull-coord?-geometry.
Definition of the center of mass
In GROMACS, there are three ways to define the center of mass of a group. The standard way
is a “plain” center of mass, possibly with additional weighting factors. With periodic boundary
conditions it is no longer possible to uniquely define the center of mass of a group of atoms.
Therefore, a reference atom is used. For determining the center of mass, for all other atoms in the
group, the closest periodic image to the reference atom is used. This uniquely defines the center of
mass. By default, the middle (determined by the order in the topology) atom is used as a reference
atom, but the user can also select any other atom if it would be closer to center of the group.

6.4. The pull code

159

dc
dc
c

Figure 6.2: Comparison of a plain center of mass reference group versus a cylinder reference
group applied to interface systems. C is the reference group. The circles represent the center of
mass of two groups plus the reference group, dc is the reference distance.
For a layered system, for instance a lipid bilayer, it may be of interest to calculate the PMF of a
lipid as function of its distance from the whole bilayer. The whole bilayer can be taken as reference
group in that case, but it might also be of interest to define the reaction coordinate for the PMF
more locally. The .mdp option pull-coord?-geometry = cylinder does not use all
the atoms of the reference group, but instead dynamically only those within a cylinder with radius
pull-cylinder-r around the pull vector going through the pull group. This only works for
distances defined in one dimension, and the cylinder is oriented with its long axis along this one
dimension. To avoid jumps in the pull force, contributions of atoms are weighted as a function of
distance (in addition to the mass weighting):
w(r < rcyl ) = 1 − 2
w(r ≥ rcyl ) = 0

r
rcyl

(6.5)
(6.6)

Note that the radial dependence on the weight causes a radial force on both cylinder group and
the other pull group. This is an undesirable, but unavoidable effect. To minimize this effect, the
cylinder radius should be chosen sufficiently large. The effective mass is 0.47 times that of a
cylinder with uniform weights and equal to the mass of uniform cylinder of 0.79 times the radius.
For a group of molecules in a periodic system, a plain reference group might not be well-defined.
An example is a water slab that is connected periodically in x and y, but has two liquid-vapor
interfaces along z. In such a setup, water molecules can evaporate from the liquid and they will
move through the vapor, through the periodic boundary, to the other interface. Such a system is
inherently periodic and there is no proper way of defining a “plain” center of mass along z. A
proper solution is to using a cosine shaped weighting profile for all atoms in the reference group.
The profile is a cosine with a single period in the unit cell. Its phase is optimized to give the
maximum sum of weights, including mass weighting. This provides a unique and continuous
reference position that is nearly identical to the plain center of mass position in case all atoms are

160

Chapter 6. Special Topics

all within a half of the unit-cell length. See ref [141] for details.
When relative weights wi are used during the calculations, either by supplying weights in the input
or due to cylinder geometry or due to cosine weighting, the weights need to be scaled to conserve
momentum:
,
wi0

= wi

N
X

wj mj

j=1

N
X

wj2 mj

(6.7)

j=1

where mj is the mass of atom j of the group. The mass of the group, required for calculating the
constraint force, is:
M=

N
X

wi0 mi

(6.8)

i=1

The definition of the weighted center of mass is:
r com =

N
X

wi0 mi r i

(6.9)

i=1

From the centers of mass the AFM, constraint, or umbrella force F com on each group can be
calculated. The force on the center of mass of a group is redistributed to the atoms as follows:
Fi =

wi0 mi
F com
M

(6.10)

Definition of the pull direction
The most common setup is to pull along the direction of the vector containing the two pull groups,
this is selected with pull-coord?-geometry = distance. You might want to pull along
a certain vector instead, which is selected with pull-coord?-geometry = direction.
But this can cause unwanted torque forces in the system, unless you pull against a reference group
with (nearly) fixed orientation, e.g. a membrane protein embedded in a membrane along x/y while
pulling along z. If your reference group does not have a fixed orientation, you should probably
use pull-coord?-geometry = direction-relative, see Fig. 6.3. Since the potential
now depends on the coordinates of two additional groups defining the orientation, the torque forces
will work on these two groups.
Definition of the angle and dihedral pull geometries
Four pull groups are required for pull-coord?-geometry = angle. In the same way
as for geometries with two groups, each consecutive pair of groups i and i + 1 define a vector
connecting the COMs of groups i and i + 1. The angle is defined as the angle between the two
resulting vectors. E.g., the .mdp option pull-coord?-groups = 1 2 2 4 defines the
angle between the vector from the COM of group 1 to the COM of group 2 and the vector from
the COM of group 2 to the COM of group 4. The angle takes values in the closed interval [0,
180] deg. For pull-coord?-geometry = angle-axis the angle is defined with respect
to a reference axis given by pull-coord?-vec and only two groups need to be given. The
dihedral geometry requires six pull groups. These pair up in the same way as described above and

6.5. Adaptive biasing with AWH

161

2
dp
1

Figure 6.3: The pull setup for geometry direction-relative. The “normal” pull groups
are 1 and 2. Groups 3 and 4 define the pull direction and thus the direction of the normal pull
forces (red). This leads to reaction forces (blue) on groups 3 and 4, which are perpendicular to the
pull direction. Their magnitude is given by the “normal” pull force times the ratio of dp and the
distance between groups 3 and 4.
so define three vectors. The dihedral angle is defined as the angle between the two planes spanned
by the two first and the two last vectors. Equivalently, the dihedral angle can be seen as the angle
between the first and the third vector when these vectors are projected onto a plane normal to the
second vector (the axis vector). As an example, consider a dihedral angle involving four groups:
1, 5, 8 and 9. Here, the .mdp option pull-coord?-groups = 8 1 1 5 5 9 specifies the
three vectors that define the dihedral angle: the first vector is the COM distance vector from group
8 to 1, the second vector is the COM distance vector from group 1 to 5, and the third vector is
the COM distance vector from group 5 to 9. The dihedral angle takes values in the interval (-180,
180] deg and has periodic boundaries.
Limitations
There is one theoretical limitation: strictly speaking, constraint forces can only be calculated
between groups that are not connected by constraints to the rest of the system. If a group contains
part of a molecule of which the bond lengths are constrained, the pull constraint and LINCS
or SHAKE bond constraint algorithms should be iterated simultaneously. This is not done in
GROMACS. This means that for simulations with constraints = all-bonds in the .mdp
file pulling is, strictly speaking, limited to whole molecules or groups of molecules. In some
cases this limitation can be avoided by using the free energy code, see sec. 6.9. In practice, the
errors caused by not iterating the two constraint algorithms can be negligible when the pull group
consists of a large amount of atoms and/or the pull force is small. In such cases, the constraint
correction displacement of the pull group is small compared to the bond lengths.

6.5

Adaptive biasing with AWH

The accelerated weight histogram method (AWH) [142] calculates the PMF along a reaction coordinate by adding an adaptively determined biasing potential. AWH flattens free energy barriers
along the reaction coordinate by applying a history-dependent potential to the system that “fills
up” free energy minima. This is similar in spirit to other adaptive biasing potential methods, e.g.
the Wang-Landau [143], local elevation [144] and metadynamics [145] methods. The initial sam-

162

Chapter 6. Special Topics

pling stage of AWH makes the method robust against the choice of input parameters. Furthermore,
the target distribution along the reaction coordinate may be chosen freely.

6.5.1

Basics of the method

Rather than biasing the reaction coordinate ξ(x) directly, AWH acts on a reference coordinate λ.
The reaction coordinate ξ(x) is coupled to λ with a harmonic potential
1
Q(ξ, λ) = βk(ξ − λ)2 ,
2

(6.11)

so that for large force constants k, ξ ≈ λ. Note the use of dimensionless energies for compatibility
with previously published work. Units of energy are obtained by multiplication with kB T = 1/β.
In the simulation, λ samples the user-defined sampling interval I. For a multidimensional reaction
coordinate ξ, the sampling interval is the Cartesian product I = Πµ Iµ (a rectangular domain). The
connection between atom coordinates and λ is established through the extended ensemble [68],
P (x, λ) =

1 g(λ)−Q(ξ(x),λ)−V (x)
e
,
Z

(6.12)

where g(λ) is a bias function (a free variable) and V (x) is the unbiased potential energy of the
system. The distribution along λ can be tuned to be any predefined target distribution ρ(λ) (often
chosen to be flat) by choosing g(λ) wisely. This is evident from
P (λ) =

P (x, λ)dx =

1 g(λ)
e
Z

e−Q(ξ(x),λ)−V (x) dx ≡

1 g(λ)−F (λ)
e
,
Z

(6.13)

where F (λ) is the free energy
F (λ) = − ln

e−Q(ξ(x),λ)−V (x) dx.

(6.14)

Being the convolution of the PMF with the Gaussian defined by the harmonic potential, F (λ) is
a smoothened version of the PMF. Eq. 6.13 shows that in order to obtain P (λ) = ρ(λ), F (λ)
needs to be determined accurately. Thus, AWH adaptively calculates F (λ) and simultaneously
converges P (λ) toward ρ(λ).
The free energy update
AWH is initialized with an estimate of the free energy F0 (λ). At regular time intervals this estimate
is updated using data collected in between the updates. At update n, the applied bias gn (λ) is a
function of the current free energy estimate Fn (λ) and target distribution ρn (λ),
gn (λ) = ln ρn (λ) + Fn (λ),

(6.15)

which is consistent with Eq. 6.13. Note that also the target distribution may be updated during the
simulation (see examples in section 6.5.3). Substituting this choice of g = gn back into Eq. 6.13
yields the simple free energy update
∆Fn (λ) = F (λ) − Fn (λ) = − ln

Pn (λ)
,
ρn (λ)

(6.16)

6.5. Adaptive biasing with AWH

163

which would yield a better estimate Fn+1 = Fn + ∆Fn , assuming Pn (λ) can be measured accurately. AWH estimates Pn (λ) by regularly calculating the conditional distribution
egn (λ)−Q(ξ(x),λ)
.
gn (λ0 )−Q(ξ(x),λ0 )
λ0 e

ωn (λ|x) ≡ Pn (λ|x) = P

(6.17)
R

Accumulating these probability weights yields t ω(λ|x(t)) ∼ Pn (λ), where Pn (λ|x)Pn (x)dx =
Pn (λ) has been used. The ωn (λ|x) weights are thus the samples of the AWH method. With the
limited amount of sampling one has in practice, update scheme 6.16 yields very noisy results.
AWH instead applies a free energy update that has the same form but which can be applied repeatedly with limited and localized sampling,
P

Wn (λ) + t ωn (λ|x(t))
P
∆Fn = − ln
.
Wn (λ) + t ρn (λ))
P

(6.18)

Here Wn (λ) is the reference weight histogram representing prior sampling. The update for W (λ),
disregarding the initial stage (see section 6.5.2), is
Wn+1 (λ) = Wn (λ) +

ρn (λ).

(6.19)

Thus, the weight histogram equals the targeted, “ideal” history of samples. There are two important things to note about the free energy update. First, sampling is driven away from oversampled,
currently local regions. For such λ values, ωn (λ) > ρn (λ) and ∆Fn (λ) < 0, which by Eq. 6.15
implies ∆gn (λ) < 0 (assuming ∆ρn ≡ 0). Thus, the probability to sample λ decreases after the
P
update (see Eq. 6.13). Secondly, the normalization of the histogram Nn = λ Wn (λ), determines
the update size |∆F (λ)|. For instance, for a single√sample ω(λ|x), the shape of the update is
approximately a Gaussian function of width σ = 1/ βk and height ∝ 1/Nn [142],
1 − 1 βk(ξ(x)−λ)2
|∆Fn (λ)| ∝
e 2
.
(6.20)
Nn
Therefore, as samples accumulate in W (λ) and Nn grows, the updates get smaller, allowing for
the free energy to converge.
Note that quantity of interest to the user is not F (λ) but the PMF Φ(ξ). Φ(ξ) is extracted by
reweighting samples ξ(t) on the fly [142] (see also section 6.5.5) and will converge at the same
rate as F (λ), see Fig. 6.4. The PMF will be written to output (see section 6.5.7).
Applying the bias to the system
The bias potential can be applied to the system in two ways. Either by applying a harmonic
potential centered at λ(t), which is sampled using (rejection-free) Monte-Carlo sampling from
the conditional distribution ωn (λ|x(t)) = Pn (λ|x(t)), see Eq. 6.17. This is also called Gibbs
sampling or independence sampling. Alternatively, and by default in the code, the following
convolved bias potential can be applied,
Un (ξ) = − ln

egn (λ)−Q(ξ,λ) dλ.

(6.21)

These two approaches are equivalent in the sense that they give rise to the same biased probabilities
Pn (x) (cf. 6.12) while the dynamics are clearly different in the two cases. This choice does not
affect the internals of the AWH algorithm, only what force and potential AWH returns to the MD
engine.

164

Chapter 6. Special Topics

Sampling interval

Final stage
Time

∼ 1/t
0

ln(1/γ)

PMF Φ(ξ)

Log of sample weight, lns
0

Time

Time
Exact PMF
1st covering
2nd
3rd

lns(t)

slope ∝ ln[(N + ∆N)/N]

1/(N0 γm )

10 kB T

Initial stage
0

1/N(t)

Update size 1/N

Reaction coordinate ξ

ξ(t)

Reaction coordinate ξ

Figure 6.4: AWH evolution in time for a Brownian particle in a double-well potential. The reaction
coordinate ξ(t) traverses the sampling interval multiple times in the initial stage before exiting
and entering the final stage (top left). In the final stage, the dynamics of ξ becomes increasingly
diffusive. The times of covering are shown as ×-markers of different colors. At these times
the free energy update size ∼ 1/N , where N is the size of the weight histogram, is decreased by
scaling N by a factor of γ = 3 (top right). In the final stage, N grows at the sampling rate and thus
1/N ∼ 1/t. The exit from the final stage is determined on the fly by ensuring that the effective
sample weight s of data collected in the final stage exceeds that of initial stage data (bottom left;
note that ln s(t) is plotted). An estimate of the PMF is also extracted from the simulation (bottom
right), which after exiting the initial stage should estimate global free energy differences fairly
accurately.

6.5. Adaptive biasing with AWH

6.5.2

165

The initial stage

Initially, when the bias potential is far from optimal, samples will be highly correlated. In such
cases, letting W (λ) accumulate samples as prescribed by Eq. 6.19, entails a too rapid decay of
the free energy update size. This motivates splitting the simulation into an initial stage where
the weight histogram grows according to a more restrictive and robust protocol, and a final stage
where the the weight histogram grows linearly at the sampling rate (Eq. 6.19). The AWH initial
stage takes inspiration from the well-known Wang-Landau algorithm [143], although there are
differences in the details.
In the initial stage the update size is kept constant (by keeping Nn constant) until a transition
across the sampling interval has been detected, a “covering”. For the definition of a covering, see
Eq. 6.22 below. After a covering has occurred, Nn is scaled up by a constant “growth factor” γ,
chosen heuristically as γ = 3. Thus, in the initial stage Nn is set dynamically as Nn = γ m N0 ,
where m is the number of coverings. Since the update size scales as 1/N ( Eq. 6.20) this leads to
a close to exponential decay of the update size in the initial stage, see Fig. 6.4.
The update size directly determines the rate of change of Fn (λ) and hence, from Eq. 6.15, also the
rate of change of the bias funcion gn (λ) Thus initially, when Nn is kept small and updates large,
the system will be driven along the reaction coordinate by the constantly fluctuating bias. If N0
is set small enough, the first transition will typically be fast because of the large update size and
will quickly give a first rough estimate of the free energy. The second transition, using N1 = γN0
refines this estimate further. Thus, rather than very carefully filling free energy minima using a
small initial update size, the sampling interval is sweeped back-and-forth multiple times, using a
wide range of update sizes, see Fig. 6.4. This way, the initial stage also makes AWH robust against
the choice of N0 .
The covering criterion
In the general case of a multidimensional reaction coordinate λ = (λµ ), the sampling interval I is
considered covered when all dimensions have been covered. A dimension d is covered if all points
λµ in the one-dimensional sampling interval Iµ have been “visited”. Finally, a point λµ ∈ Iµ
has been visited if there is at least one point λ∗ ∈ I with λ∗µ = λµ that since the last covering
has accumulated probability weight corresponding to the peak of a multidimensional Gaussian
distribution
Y ∆λµ
√
∆W (λ∗ ) ≥ wpeak ≡
.
(6.22)
2πσk
µ
p

Here, ∆λµ is the point spacing of the discretized Iµ and σk = 1/ βkµ (where kµ is the force
constant) is the Gaussian width.
Exit from the initial stage
For longer times, when major free energy barriers have largely been flattened by the converging
bias potential, the histogram W (λ) should grow at the actual sampling rate and the initial stage
needs to be exited [146]. There are multiple reasonable (heuristic) ways of determining when
this transition should take place. One option is to postulate that the number of samples in the

166

Chapter 6. Special Topics

weight histogram Nn should never exceed the actual number of collected samples, and exit the
initial stage when this condition breaks [142]. In the initial stage, N grows close to exponentially
while the collected number of samples grows linearly, so an exit will surely occur eventually. Here
we instead apply an exit criterion based on the observation that “artifically” keeping N constant
while continuing to collect samples corresponds to scaling down the relative weight of old samples
relative to new ones. Similarly, the subsequent scaling up of N by a factor γ corresponds to scaling
up the weight of old data. Briefly, the exit criterion is devised such that the weight of a sample
collected after the initial stage is always larger or equal to the weight of a sample collected during
the initial stage, see Fig. 6.4. This is consistent with scaling down early, noisy data.
The initial stage exit criterion will now be described in detail. We start out at the beginning of a
covering stage, so that N has just been scaled by γ and is now kept constant. Thus, the first sample
of this stage has the weight s = 1/γ relative to the last sample of the previous covering stage. We
assume that ∆N samples are collected and added to W for each update . To keep N constant, W
needs to be scaled down by a factor N/(N + ∆N ) after every update. Equivalently, this means
that new data is scaled up relative to old data by the inverse factor. Thus, after ∆n updates a new
sample has the relative weight s = (1/γ)[(Nn + ∆N )/Nn ]∆n . Now assume covering occurs at
this time. To continue to the next covering stage, N should be scaled by γ, which corresponds
to again multiplying s by 1/γ. If at this point s ≥ γ, then after rescaling s ≥ 1; i.e. overall the
relative weight of a new sample relative to an old sample is still growing fast. If on the contrary
s < γ, and this defines the exit from the initial stage, then the initial stage is over and from now N
simply grows at the sampling rate (see Eq. 6.19). To really ensure that s ≥ 1 holds before exiting,
so that samples after the exit have at least the sample weight of older samples, the last covering
stage is extended by a sufficient number of updates.

6.5.3

Choice of target distribution

The target distribution ρ(λ) is traditionally chosen to be uniform
ρconst (λ) = const.

(6.23)

This choice exactly flattens F (λ) in user-defined sampling interval I. Generally, ρ(λ) = 0, λ ∈
/ I.
In certain cases other choices may be preferable. For instance, in the multidimensional case the
rectangular sampling interval is likely to contain regions of very high free energy, e.g. where
atoms are clashing. To exclude such regions, ρ(λ) can specified by the following function of the
free energy
1
ρcut (λ) ∝
,
(6.24)
F
1 + e (λ)−Fcut
where Fcut is a free energy cutoff (relative to minλ F (λ)). Thus, regions of the sampling interval
where F (λ) > Fcut will be exponentially suppressed (in a smooth fashion). Alternatively, very
high free energy regions could be avoided while still flattening more moderate free energy barriers
by targeting a Boltzmann distribution corresponding to scaling β = 1/kB T by a factor 0 < sβ <
1,
ρBoltz (λ) ∝ e−sβ F (λ) ,
(6.25)
The parameter sβ determines to what degree the free energy landscape is flattened; the lower sβ ,
the flatter. Note that both ρcut (λ) and ρBoltz (λ) depend on F (λ), which needs to be substituted

6.5. Adaptive biasing with AWH

167

by the current best estimate Fn (λ). Thus, the target distribution is also updated (consistently with
Eq. 6.15).
There is in fact an alternative approach to obtaining ρBoltz (λ) as the limiting target distribution
in AWH, which is particular in the way the weight histogram W (λ) and the target distribution
ρ are updated and coupled to each other. This yields an evolution of the bias potential which is
very similar to that of well-tempered metadynamics [147], see [142] for details. Because of the
popularity and success of well-tempered metadynamics, this is a special case worth considering.
In this case ρ is a function of the reference weight histogram
ρBoltz,loc (λ) ∝ W (λ),

(6.26)

and the update of the weight histogram is modified (cf. Eq. 6.19)
Wn+1 (λ) = Wn (λ) + sβ

ω(λ|x(t)).

(6.27)

Thus, here the weight histogram equals the real history of samples, but scaled by sβ . This target
distribution is called local Boltzmann since W is only modified locally, where sampling has taken
place. We see that when sβ ≈ 0 the histogram essentially does not grow and the size of the
free energy update will stay at a constant value (as in the original formulation of metadynamics).
Thus, the free energy estimate will not converge, but continue to fluctuate around the correct value.
This illustrates the inherent coupling between the convergence and choice of target distribution for
this special choice of target. Furthermore note that when using ρ = ρBoltz,loc there is no initial
stage (section 6.5.2). The rescaling of the weight histogram applied in the initial stage is a global
operation, which is incompatible ρBoltz,loc only depending locally on the sampling history.
Lastly, the target distribution can be modulated by arbitrary probability weights
ρ(λ) = ρ0 (λ)wuser (λ).

(6.28)

where wuser (λ) is provided by user data and in principle ρ0 (λ) can be any of the target distributions
mentioned above.

6.5.4

Multiple independent or sharing biases

Multiple independent bias potentials may be applied within one simulation. This only makes
sense if the biased coordinates ξ (1) , ξ (2) , . . . evolve essentially independently from one another.
A typical example of this would be when applying an independent bias to each monomer of a
protein. Furthermore, multiple AWH simulations can be launched in parallel, each with a (set of)
indepedendent biases.
If the defined sampling interval is large relative to the diffusion time of the reaction coordinate,
traversing the sampling interval multiple times as is required by the initial stage (section 6.5.2) may
take an infeasible mount of simulation time. In these cases it could be advantageous to parallelize
the work and have a group of multiple “walkers” ξ (i) (t) share a single bias potential. This can be
achieved by collecting samples from all ξ (i) of the same sharing group into a single histogram and
update a common free energy estimate. Samples can be shared between walkers within the simulation and/or between multiple simulations. However, currently only sharing between simulations
is supported in the code while all biases within a simulation are independent.

168

Chapter 6. Special Topics

Note that when attempting to shorten the simulation time by using bias-sharing walkers, care
must be taken to ensure the simulations are still long enough to properly explore and equilibrate
all regions of the sampling interval. To begin, the walkers in a group should be decorrelated
and distributed approximately according to the target distribution before starting to refine the free
energy. This can be achieved e.g. by “equilibrating” the shared weight histogram before letting it
grow; for instance, W (λ)/N ≈ ρ(λ) with some tolerance.
Furthermore, the “covering” or transition criterion of the initial stage should to be generalized to
detect when the sampling interval has been collectively traversed. One alternative is to just use
the same criterion as for a single walker (but now with more samples), see Eq. 6.22. However, in
contrast to the single walker case this does not ensure that any real transitions across the sampling
interval has taken place; in principle all walkers could be sampling only very locally and still
cover the whole interval. Just as with a standard umbrella sampling procedure, the free energy
may appear to be converged while in reality simulations sampling closeby λ values are sampling
disconnected regions of phase space. A stricter criterion, which helps avoid such issues, is to
require that before a simulation marks a point λµ along dimension µ as visited, and shares this
with the other walkers, also all points within a certain diameter Dcover should have been visited
(i.e.fulfill Eq. 6.22). Increasing Dcover increases robustness, but may slow down convergence. For
the maximum value of Dcover , equal to the length of the sampling interval, the sampling interval
is considered covered when at least one walker has independently traversed the sampling interval.

6.5.5

Reweighting and combining biased data

Often one may want to, post-simulation, calculate the unbiased PMF Φ(u) of another variable
u(x). Φ(u) can be estimated using ξ-biased data by reweighting (“unbiasing”) the trajectory using
the bias potential Un(t) , see Eq. 6.21. Essentially, one bins the biased data along u and removes
the effect of Un(t) by dividing the weight of samples u(t) by e−Un(t) (ξ(t)) ,
Φ̂(u) = − ln

1u (u(t))eUn(t) (ξ(t) Zn(t) .

(6.29)

Here the indicator function 1u denotes the binning procedure: 1Ru (u0 ) = 1 if u0 falls into the bin
labeled by u and 0 otherwise. The normalization factor Zn = e−Φ(ξ)−Un (ξ) dξ is the partition
function of the extended ensemble. As can be seen Zn depends on Φ(ξ), the PMF of the (biased)
reaction coordinate ξ (which is calculated and written to file by the AWH simulation). It is advisable to use only final stage data in the reweighting procedure due to the rapid change of the bias
potential during the initial stage. If one would include initial stage data, one should use the sample
weights that are inferred by the repeated rescaling of the histogram in the initial stage, for the sake
of consistency. Initial stage samples would then in any case be heavily scaled down relative to final
stage samples. Note that Eq. 6.29 can also be used to combine data from multiple simulations (by
adding another sum also over the trajectory set). Furthermore, when multiple independent AWH
biases have generated a set of PMF estimates {Φ̂(i) (ξ)}, a combined best estimate Φ̂(ξ) can be
obtained by applying self-consistent exponential averaging. More details on this procedure and a
derivation of Eq. 6.29 (using slightly different notation) can be found in [148].

6.5. Adaptive biasing with AWH

6.5.6

169

The friction metric

During the AWH simulation, the following time-integrated force correlation function is calculated,
Z ∞
hδFµ (x(t), λ)δFν (x(0), λ)ω(λ|x(t))ω(λ|x(0))i
ηµν (λ) = β
dt.
(6.30)
hω 2 (λ|x)i
0
Here Fµ (x, λ) = kµ (ξµ (x) − λµ ) is the force along dimension µ from an harmonic potential
centered at λ and δFµ (x, λ) = Fµ (x, λ) − hFµ (x, λ)i is the deviation of the force. The factors ω(λ|x(t)), see Eq. 6.17, reweight the samples. ηµν (λ) is a friction tensor [149]. Its matrix elements are inversely proportional to local diffusion coefficients. A measure of sampling
(in)efficiency at each λ is given by
1

η 2 (λ) =

det ηµν (λ).

(6.31)

A large value of η 2 (λ) indicates slow dynamics and long correlation times, which may require
more sampling.

6.5.7

Usage

AWH stores data in the energy file (.edr) with a frequency set by the user. The data – the PMF,
the convolved bias, distributions of the λ and ξ coordinates, etc. – can be extracted after the
simulation using the gmx awh tool. Furthermore, the trajectory of the reaction coordinate ξ(t) is
printed to the pull output file pullx.xvg. The log file (.log) also contains information; check
for messages starting with “awh”, they will tell you about covering and potential sampling issues.
Setting the initial update size
The initial value of the weight histogram size N sets the initial update size (and the rate of change
of the bias). When N is kept constant, like in the initial stage, the average variance of the free
energy scales as ε2 ∼ 1/(N D) [142], for a simple model system with constant diffusion D along
the reaction coordinate. This provides a ballpark estimate used by AWH to initialize N in terms
of more meaningful quantities
1
1
=
∼ Dε20 .
(6.32)
N0
N0 (ε0 , D)
Essentially, this tells us that a slower system (small D) requires more samples (larger N 0 ) to
attain the same level of accuracy (ε0 ) at a given sampling rate. Conversely, for a system of given
diffusion, how to choose the initial biasing rate depends on how good the initial accuracy is. Both
the initial error ε0 and the diffusion D only need to be roughly estimated or guessed. In the typical
case, one would only tweak the D parameter, and use a default value for ε0 . For good convergence,
D should be chosen as large as possible (while maintaining a stable system) giving large initial
bias updates and fast initial transitions. Choosing D too small can lead to slow initial convergence.
It may be a good idea to run a short trial simulation and after the first covering check the maximum
free energy difference of the PMF estimate. If this is much larger than the expected magnitude of
the free energy barriers that should be crossed, then the system is probably being pulled too hard
and D should be decreased. ε0 on the other hand, would only be tweaked when starting an AWH
simulation using a fairly accurate guess of the PMF as input.

170

Chapter 6. Special Topics

Tips for efficient sampling
The force constant k should be larger than the curvature of the PMF landscape. If this is not
the case, the distributions of the reaction coordinate ξ and the reference coordinate λ, will differ
significantly and warnings will be printed in the log file. One can choose k as large as the time step
supports. This will neccessarily increase the number of points of the discretized sampling interval
I. In general however, it should not affect the performance of the simulation noticeably because
the AWH update is implemented such that only sampled points are accessed at free energy update
time.
As with any method, the choice of reaction coordinate(s) is critical. If a single reaction coordinate
does not suffice, identifying a second reaction coordinate and sampling the two-dimensional landscape may help. In this case, using a target distribution with a free energy cutoff (see Eq. 6.24)
might be required to avoid sampling uninteresting regions of very high free energy. Obtaining
accurate free energies for reaction coordinates of much higher dimensionality than 3 or possibly 4
is generally not feasible.
Monitoring the transition rate of ξ(t), across the sampling interval is also advisable. For reliable
statistics (e.g. when reweighting the trajectory as described in section 6.5.5), one would generally
want to observe at least a few transitions after having exited the initial stage. Furthermore, if
the dynamics of the reaction coordinate suddenly changes, this may be a sign of e.g. a reaction
coordinate problem.
Difficult regions of sampling may also be detected by calculating the friction tensor ηµν (λ) in
1
the sampling interval, see section 6.5.6. ηµν (λ) as well as the sampling efficiency measure η 2 (λ)
1
(Eq. 6.31) are written to the energy file and can be extracted with gmx awh. A high peak in η 2 (λ)
indicates that this region requires longer time to sample properly.

6.6

Enforced Rotation

This module can be used to enforce the rotation of a group of atoms, as e.g. a protein subunit.
There are a variety of rotation potentials, among them complex ones that allow flexible adaptations
of both the rotated subunit as well as the local rotation axis during the simulation. An example
application can be found in ref. [150].

6.6.1

Fixed Axis Rotation

Stationary Axis with an Isotropic Potential
In the fixed axis approach (see Fig. 6.5B), torque on a group of N atoms with positions xi (denoted
“rotation group”) is applied by rotating a reference set of atomic positions – usually their initial
positions y 0i – at a constant angular velocity ω around an axis defined by a direction vector v̂ and
a pivot point u. To that aim, each atom with position xi is attracted by a “virtual spring” potential
to its moving reference position y i = Ω(t)(y 0i − u), where Ω(t) is a matrix that describes the

6.6. Enforced Rotation

171

Figure 6.5: Comparison of fixed and flexible axis rotation. A: Rotating the sketched shape inside
the white tubular cavity can create artifacts when a fixed rotation axis (dashed) is used. More
realistically, the shape would revolve like a flexible pipe-cleaner (dotted) inside the bearing (gray).
B: Fixed rotation around an axis v with a pivot point specified by the vector u. C: Subdividing
the rotating fragment into slabs with separate rotation axes (↑) and pivot points (•) for each slab
allows for flexibility. The distance between two slabs with indices n and n + 1 is ∆x.
rotation around the axis. In the simplest case, the “springs” are described by a harmonic potential,
V iso =

N
h
i2
kX
wi Ω(t)(y 0i − u) − (xi − u) ,
2 i=1

(6.33)

with optional mass-weighted prefactors wi = N mi /M with total mass M =
rotation matrix Ω(t) is


i=1 mi .

The



cos ωt + vx2 ξ
vx vy ξ − vz sin ωt vx vz ξ + vy sin ωt


cos ωt + vy2 ξ
vy vz ξ − vx sin ωt  ,
Ω(t) =  vx vy ξ + vz sin ωt
2
vx vz ξ − vy sin ωt vy vz ξ + vx sin ωt
cos ωt + vz ξ
where vx , vy , and vz are the components of the normalized rotation vector v̂, and ξ := 1−cos(ωt).
As illustrated in Fig. 6.6A for a single atom j, the rotation matrix Ω(t) operates on the initial
reference positions y 0j = xj (t0 ) of atom j at t = t0 . At a later time t, the reference position has
rotated away from its initial place (along the blue dashed line), resulting in the force
h

F jiso = −∇j V iso = k wj Ω(t)(y 0j − u) − (xj − u) ,

(6.34)

which is directed towards the reference position.
Pivot-Free Isotropic Potential
Instead of a fixed pivot vector u this potential uses the center of mass xc of the rotation group as
pivot for the rotation axis,
xc =

N
1 X
mi xi
M i=1

and

y 0c =

N
1 X
mi y 0i ,
M i=1

(6.35)

172

Chapter 6. Special Topics

V iso

V rm , V flex

V rm2 , V flex2 ( = 0 nm2 )

V rm2 , V flex2 ( = 0.01 nm2 )

Figure 6.6: Selection of different rotation potentials and definition of notation. All four potentials
V (color coded) are shown for a single atom at position xj (t). A: Isotropic potential V iso , B:
radial motion potential V rm and flexible potential V flex , C–D: radial motion 2 potential V rm2
and flexible 2 potential V flex2 for 0 = 0 nm2 (C) and 0 = 0.01 nm2 (D). The rotation axis is
perpendicular to the plane and marked by ⊗. The light gray contours indicate Boltzmann factors
e−V /(kB T ) in the xj -plane for T = 300 K and k = 200 kJ/(mol·nm2 ). The green arrow shows
the direction of the force F j acting on atom j; the blue dashed line indicates the motion of the
reference position.

6.6. Enforced Rotation

173

which yields the “pivot-free” isotropic potential
V iso-pf =

N
h
i2
kX
wi Ω(t)(y 0i − y 0c ) − (xi − xc ) ,
2 i=1

with forces

(6.36)

Fjiso-pf = k wj Ω(t)(y 0j − y 0c ) − (xj − xc ) .

(6.37)

Without mass-weighting, the pivot xc is the geometrical center of the group.
Parallel Motion Potential Variant
The forces generated by the isotropic potentials (eqns. 6.33 and 6.36) also contain components
parallel to the rotation axis and thereby restrain motions along the axis of either the whole rotation group (in case of V iso ) or within the rotation group (in case of V iso-pf ). For cases where
unrestrained motion along the axis is preferred, we have implemented a “parallel motion” variant
by eliminating all components parallel to the rotation axis for the potential. This is achieved by
projecting the distance vectors between reference and actual positions
r i = Ω(t)(y 0i − u) − (xi − u)

(6.38)

onto the plane perpendicular to the rotation vector,
r⊥
i := r i − (r i · v̂)v̂ ,

(6.39)

yielding
V pm =
=

N
kX
2
wi (r ⊥
i )
2 i=1
N
n
kX
wi Ω(t)(y 0i − u) − (xi − u)
2 i=1

−

o o2

Ω(t)(y 0i − u) − (xi − u) · v̂ v̂

(6.40)

and similarly
F jpm = k wj r ⊥
j .

(6.41)

Pivot-Free Parallel Motion Potential
Replacing in eqn. 6.40 the fixed pivot u by the center of mass xc yields the pivot-free variant of
the parallel motion potential. With
si = Ω(t)(y 0i − y 0c ) − (xi − xc )

(6.42)

the respective potential and forces are
N
kX
2
wi (s⊥
i ) ,
2 i=1

V pm-pf

F jpm-pf

= k w j s⊥
j .

(6.43)
(6.44)

174

Chapter 6. Special Topics

Radial Motion Potential
In the above variants, the minimum of the rotation potential is either a single point at the reference
position y i (for the isotropic potentials) or a single line through y i parallel to the rotation axis (for
the parallel motion potentials). As a result, radial forces restrict radial motions of the atoms. The
two subsequent types of rotation potentials, V rm and V rm2 , drastically reduce or even eliminate
this effect. The first variant, V rm (Fig. 6.6B), eliminates all force components parallel to the vector
connecting the reference atom and the rotation axis,
N
kX
wi [pi · (xi − u)]2 ,
2 i=1

V rm =
with
pi :=

(6.45)

v̂ × Ω(t)(y 0i − u)
.
kv̂ × Ω(t)(y 0i − u)k

(6.46)

This variant depends only on the distance pi · (xi − u) of atom i from the plane spanned by v̂ and
Ω(t)(y 0i − u). The resulting force is
h

Fjrm = −k wj pj · (xj − u) pj .

(6.47)

Pivot-Free Radial Motion Potential
Proceeding similar to the pivot-free isotropic potential yields a pivot-free version of the above
potential. With
v̂ × Ω(t)(y 0i − y 0c )
q i :=
,
(6.48)
kv̂ × Ω(t)(y 0i − y 0c )k
the potential and force for the pivot-free variant of the radial motion potential read
V

rm-pf

Fjrm-pf

N
kX
wi [q i · (xi − xc )]2 ,
2 i=1

(6.49)

N
mj X
= −k wj q j · (xj − xc ) q j + k
wi [q i · (xi − xc )] q i .
M i=1

(6.50)

Radial Motion 2 Alternative Potential
As seen in Fig. 6.6B, the force resulting from V rm still contains a small, second-order radial
component. In most cases, this perturbation is tolerable; if not, the following alternative, V rm2 ,
fully eliminates the radial contribution to the force, as depicted in Fig. 6.6C,
V

rm2

N
kX
(v̂ × (xi − u)) · Ω(t)(y 0i − u)
=
wi
2 i=1
kv̂ × (xi − u)k2 + 0

(6.51)

where a small parameter 0 has been introduced to avoid singularities. For 0 = 0 nm2 , the equipotential planes are spanned by xi − u and v̂, yielding a force perpendicular to xi − u, thus not
contracting or expanding structural parts that moved away from or toward the rotation axis.

6.6. Enforced Rotation

175

Choosing a small positive 0 (e.g., 0 = 0.01 nm2 , Fig. 6.6D) in the denominator of eqn. 6.51 yields
a well-defined potential and continuous forces also close to the rotation axis, which is not the case
for 0 = 0 nm2 (Fig. 6.6C). With
r i := Ω(t)(y 0i − u)
v̂ × (xi − u)
si :=
≡ Ψi v̂ × (xi − u)
kv̂ × (xi − u)k
1
Ψ∗i :=
kv̂ × (xi − u)k2 + 0

(6.52)
(6.53)
(6.54)

the force on atom j reads
(

F jrm2

= −k

wj (sj · rj )

" ∗
Ψ

Ψj∗2
rj − 3 (sj · rj )sj
Ψj
Ψj

× v̂.

(6.55)

Pivot-Free Radial Motion 2 Potential
The pivot-free variant of the above potential is
V rm2-pf =

N
kX
(v̂ × (xi − xc )) · Ω(t)(y 0i − y c )
wi
2 i=1
kv̂ × (xi − xc )k2 + 0

(6.56)

With
r i := Ω(t)(y 0i − y c )
v̂ × (xi − xc )
si :=
≡ Ψi v̂ × (xi − xc )
kv̂ × (xi − xc )k
1
Ψ∗i :=
kv̂ × (xi − xc )k2 + 0

(6.57)
(6.58)
(6.59)

the force on atom j reads
(

F jrm2-pf

= −k

wj (sj · rj )

mj
+k
M

6.6.2

(N
X
i=1

" ∗
Ψ

Ψj∗2
rj − 3 (sj · rj )sj
Ψj
Ψj
j

wi (si · r i )

× v̂

Ψ∗i
Ψ∗2
r i − i3 (si · r i ) si
Ψi
Ψi

× v̂ .

(6.60)

Flexible Axis Rotation

As sketched in Fig. 6.5A–B, the rigid body behavior of the fixed axis rotation scheme is a drawback
for many applications. In particular, deformations of the rotation group are suppressed when the
equilibrium atom positions directly depend on the reference positions. To avoid this limitation,
eqns. 6.50 and 6.56 will now be generalized towards a “flexible axis” as sketched in Fig. 6.5C. This
will be achieved by subdividing the rotation group into a set of equidistant slabs perpendicular to
the rotation vector, and by applying a separate rotation potential to each of these slabs. Fig. 6.5C
shows the midplanes of the slabs as dotted straight lines and the centers as thick black dots.

176

Chapter 6. Special Topics

Figure 6.7: Gaussian functions gn centered at n ∆x for a slab distance ∆x = 1.5 nm and n ≥ −2.
Gaussian function g0 is highlighted in bold; the dashed line depicts the sum of the shown Gaussian
functions.
To avoid discontinuities in the potential and in the forces, we define “soft slabs” by weighing the
contributions of each slab n to the total potential function V flex by a Gaussian function
!

β 2 (xi )
gn (xi ) = Γ exp − n 2
,
2σ

(6.61)

centered at the midplane of the nth slab. Here σ is the width of the Gaussian function, ∆x the
distance between adjacent slabs, and
βn (xi ) := xi · v̂ − n ∆x .

(6.62)

A most convenient choice is σ = 0.7∆x and
(n − 14 )2
exp −
1/Γ =
2 · 0.72
n∈Z
X

≈ 1.75464 ,

which yields a nearly constant sum, essentially independent of xi (dashed line in Fig. 6.7), i.e.,
X

gn (xi ) = 1 + (xi ) ,

(6.63)

n∈Z

with |(xi )| < 1.3 · 10−4 . This choice also implies that the individual contributions to the force
from the slabs add up to unity such that no further normalization is required.
To each slab center xnc , all atoms contribute by their Gaussian-weighted (optionally also massweighted) position vectors gn (xi ) xi . The instantaneous slab centers xnc are calculated from the
current positions xi ,
PN
i=1 gn (xi ) mi xi
xnc = P
,
(6.64)
N
i=1 gn (xi ) mi
while the reference centers y nc are calculated from the reference positions y 0i ,
y nc

0
0
i=1 gn (y i ) mi y i
0
i=1 gn (y i ) mi

(6.65)

Due to the rapid decay of gn , each slab will essentially involve contributions from atoms located
within ≈ 3∆x from the slab center only.

6.6. Enforced Rotation

177

Flexible Axis Potential
We consider two flexible axis variants. For the first variant, the slab segmentation procedure with
Gaussian weighting is applied to the radial motion potential (eqn. 6.50 / Fig. 6.6B), yielding as the
contribution of slab n
V

N
kX
=
wi gn (xi ) [q ni · (xi − xnc )]2 ,
2 i=1

and a total potential function
V flex =

Vn.

(6.66)

Note that the global center of mass xc used in eqn. 6.50 is now replaced by xnc , the center of mass
of the slab. With
v̂ × Ω(t)(y 0i − y nc )
kv̂ × Ω(t)(y 0i − y nc )k
:= q ni · (xi − xnc ) ,

q ni :=

(6.67)

bni

(6.68)

the resulting force on atom j reads
F jflex = − k wj

gn (xj ) bnj q nj − bnj

+ k mj

X
n

βn (xj )
v̂
2σ 2

N
gn (xj ) X
βn (xj ) n
wi gn (xi ) bni q ni −
[q i · (xj − xnc )] v̂ . (6.69)
2
σ
g
(x
)
h n h i=1

Note that for V flex , as defined, the slabs are fixed in space and so are the reference centers y nc . If
during the simulation the rotation group moves too far in v direction, it may enter a region where
– due to the lack of nearby reference positions – no reference slab centers are defined, rendering
the potential evaluation impossible. We therefore have included a slightly modified version of
this potential that avoids this problem by attaching the midplane of slab n = 0 to the center of
mass of the rotation group, yielding slabs that move with the rotation group. This is achieved by
subtracting the center of mass xc of the group from the positions,
x̃i = xi − xc , and ỹ 0i = y 0i − y 0c ,

(6.70)

such that
V

N
k XX
v̂ × Ω(t)(ỹ 0i − ỹ nc )
wi gn (x̃i )
· (x̃i − x̃nc )
2 n i=1
kv̂ × Ω(t)(ỹ 0i − ỹ nc )k

flex-t

(6.71)

To simplify the force derivation, and for efficiency reasons, we here assume xc to be constant, and
thus ∂xc /∂x = ∂xc /∂y = ∂xc /∂z = 0. The resulting force error is small (of order O(1/N ) or
O(mj /M ) if mass-weighting is applied) and can therefore be tolerated. With this assumption, the
forces F flex-t have the same form as eqn. 6.69.

178

Chapter 6. Special Topics

Flexible Axis 2 Alternative Potential
In this second variant, slab segmentation is applied to V rm2 (eqn. 6.56), resulting in a flexible axis
potential without radial force contributions (Fig. 6.6C),
V

flex2

N X
kX
(v̂ × (xi − xnc )) · Ω(t)(y 0i − y nc )
=
wi gn (xi )
2 i=1 n
kv̂ × (xi − xnc )k2 + 0

(6.72)

With
r ni := Ω(t)(y 0i − y nc )
v̂ × (xi − xnc )
sni :=
≡ ψi v̂ × (xi − xnc )
kv̂ × (xi − xnc )k
1
ψi∗ :=
kv̂ × (xi − xnc )k2 + 0
gn (xj ) mj
Wjn := P
h gn (xh ) mh
S

N
X

wi gn (xi ) (sni

r ni )

i=1

ψi∗ n ψi∗2 n n n
r − 3 (si · r i ) si
ψi i
ψi

(6.73)
(6.74)
(6.75)
(6.76)
#

(6.77)

the force on atom j reads
(

F jflex2

= −k

wj gn (xj )

(snj

rjn )

" ∗
ψ
j

ψj

)

(

Wjn S n

k
+
2

(

× v̂ − k

(
X
n

rjn

ψj∗2
− 3 (snj · rjn ) sjn
ψj

× v̂

)

Wjn

βn (xj ) 1 n
s · S n v̂
σ 2 ψj j

βn (xj ) ψj∗ n n 2
wj gn (xj )
(s · r ) v̂.
σ 2 ψj2 j j
)

X
n

(6.78)

Applying transformation (6.70) yields a “translation-tolerant” version of the flexible 2 potential,
V flex2-t . Again, assuming that ∂xc /∂x, ∂xc /∂y, ∂xc /∂z are small, the resulting equations for
V flex2-t and F flex2-t are similar to those of V flex2 and F flex2 .

6.6.3

Usage

To apply enforced rotation, the particles i that are to be subjected to one of the rotation potentials
are defined via index groups rot-group0, rot-group1, etc., in the .mdp input file. The
reference positions y 0i are read from a special .trr file provided to grompp. If no such file is
found, xi (t = 0) are used as reference positions and written to .trr such that they can be used for
subsequent setups. All parameters of the potentials such as k, 0 , etc. (Table 6.1) are provided as
.mdp parameters; rot-type selects the type of the potential. The option rot-massw allows
to choose whether or not to use mass-weighted averaging. For the flexible potentials, a cutoff
value gnmin (typically gnmin = 0.001) makes shure that only significant contributions to V and F
are evaluated, i.e. terms with gn (x) < gnmin are omitted. Table 6.2 summarizes observables that
are written to additional output files and which are described below.

6.6. Enforced Rotation

179

Table 6.1: Parameters used by the various rotation potentials. x’s indicate which parameter is
actually used for a given potential.
k

parameter
.mdp input variable name
unit
fixed axis potentials:
isotropic
V iso
— pivot-free
V iso-pf
parallel motion V pm
— pivot-free
V pm-pf
radial motion
V rm
— pivot-free
V rm-pf
radial motion 2 V rm2
— pivot-free
V rm2-pf
flexible axis potentials:
flexible
V flex
— transl. tol.
V flex-t
flexible 2
V flex2
— transl. tol.
V flex2-t

eqn.
(6.33)
(6.36)
(6.40)
(6.44)
(6.45)
(6.50)
(6.51)
(6.56)
eqn.
(6.66)
(6.71)
(6.72)
-

v̂

vec pivot

∆x

gnmin

rate

eps

◦ /ps

nm2

x
x
x
x
x
x
x
x

x
x
x
x
-

x
x
x
x
x
x
x
x

x
x

x
x
x
x

x
x

x
x
x
x

kJ
mol·nm2

x
x
x
x
x
x
x
x
x
x
x
x

slab-dist min-gauss

Table 6.2: Quantities recorded in output files during enforced rotation simulations. All slab-wise
data is written every nstsout steps, other rotation data every nstrout steps.
quantity
V (t)
θref (t)
θav (t)
θfit (t), θfit (t, n)
y 0 (n), x0 (t, n)
τ (t)
τ (t, n)

unit
kJ/mol
degrees
degrees
degrees
nm
kJ/mol
kJ/mol

equation
see 6.1
θref (t) = ωt
(6.79)
(6.81)
(6.64, 6.65)
(6.82)
(6.82)

output file
rotation
rotation
rotation
rotangles
rotslabs
rotation
rottorque

fixed
x
x
x
x
-

flexible
x
x
x
x
x

180

Chapter 6. Special Topics

Angle of Rotation Groups: Fixed Axis
For fixed axis rotation, the average angle θav (t) of the group relative to the reference group is
determined via the distance-weighted angular deviation of all rotation group atoms from their
reference positions,
,
θav =

N
X

ri θi

i=1

N
X

ri .

(6.79)

i=1

Here, ri is the distance of the reference position to the rotation axis, and the difference angles θi
are determined from the atomic positions, projected onto a plane perpendicular to the rotation axis
through pivot point u (see eqn. 6.39 for the definition of ⊥),
cos θi =

(y i − u)⊥ · (xi − u)⊥
.
k(y i − u)⊥ · (xi − u)⊥ k

(6.80)

The sign of θav is chosen such that θav > 0 if the actual structure rotates ahead of the reference.
Angle of Rotation Groups: Flexible Axis
For flexible axis rotation, two outputs are provided, the angle of the entire rotation group, and
separate angles for the segments in the slabs. The angle of the entire rotation group is determined
by an RMSD fit of xi to the reference positions y 0i at t = 0, yielding θfit as the angle by which
the reference has to be rotated around v̂ for the optimal fit,
!

RMSD(xi , Ω(θfit )y 0i ) = min .

(6.81)

To determine the local angle for each slab n, both reference and actual positions are weighted with
the Gaussian function of slab n, and θfit (t, n) is calculated as in eqn. 6.81) from the Gaussianweighted positions.
For all angles, the .mdp input option rot-fit-method controls whether a normal RMSD fit
is performed or whether for the fit each position xi is put at the same distance to the rotation axis
as its reference counterpart y 0i . In the latter case, the RMSD measures only angular differences,
not radial ones.
Angle Determination by Searching the Energy Minimum
Alternatively, for rot-fit-method = potential, the angle of the rotation group is determined as the angle for which the rotation potential energy is minimal. Therefore, the used rotation
potential is additionally evaluated for a set of angles around the current reference angle. In this
case, the rotangles.log output file contains the values of the rotation potential at the chosen
set of angles, while rotation.xvg lists the angle with minimal potential energy.
Torque
The torque τ (t) exerted by the rotation potential is calculated for fixed axis rotation via
τ (t) =

N
X
i=1

r i (t) × fi⊥ (t),

(6.82)

6.7. Electric fields

181

where r i (t) is the distance vector from the rotation axis to xi (t) and fi⊥ (t) is the force component
perpendicular to r i (t) and v̂. For flexible axis rotation, torques τn are calculated for each slab
using the local rotation axis of the slab and the Gaussian-weighted positions.

6.7

Electric fields

A pulsed and oscillating electric field can be applied according to:
#

(t − t0 )2
cos [ω(t − t0 )]
E(t) = E0 exp −
2σ 2

(6.83)

where E0 is the field strength, the angular frequency ω = 2πc/λ, t0 is the time at of the peak in
the field strength and σ is the with of the pulse. Special cases occur when σ = 0 (non-pulsed field)
and for ω is 0 (static field).
This simulated laser-pulse was applied to simulations of melting ice [151]. A pulsed electric field
may look ike Fig. 6.8. In the supporting information of that paper the impact of an applied electric
field on a system under periodic boundary conditions is analyzed. It is described that the effective
electric field under PBC is larger than the applied field, by a factor depending on the size of the box
and the dielectric properties of molecules in the box. For a system with static dielectric properties
this factor can be corrected for. But for a system where the dielectric varies over time, for example
a membrane protein with a pore that opens and closes during the simulatippn, this way of applying
an electric field is not useful. In such cases one can use the computational electrophysiology
protocol described in the next section (sec. 6.8).
2

Electric field (V/nm)

-1

-2
0

0.5

1.5

Time (ps)

Figure 6.8: A simulated laser pulse in GROMACS.
Electric fields are applied when the following options are specified in the grompp.mdp file. You
specify, in order, E0 , ω, t0 and σ:
electric-field-x = 0.04 0

yields a static field with E0 = 0.04 V/nm in the X-direction. In contrast,

182

Chapter 6. Special Topics

electric-field-x = 2.0

150

yields an oscillating electric field with E0 = 2 V/nm, ω = 150/ps and t0 = 5 ps. Finally
electric-field-x = 2.0

150

yields an pulsed-oscillating electric field with E0 = 2 V/nm, ω = 150/ps and t0 = 5 ps and σ = 1 ps.
Read more in ref. [151]. Note that the input file format is changed from the undocumented older
version. A figure like Fig. 6.8 may be produced by passing the -field option to gmx mdrun.

6.8

Computational Electrophysiology

The Computational Electrophysiology (CompEL) protocol [152] allows the simulation of ion flux
through membrane channels, driven by transmembrane potentials or ion concentration gradients.
Just as in real cells, CompEL establishes transmembrane potentials by sustaining a small imbalance of charges ∆q across the membrane, which gives rise to a potential difference ∆U according
to the membrane capacitance:
∆U = ∆q/Cmembrane
(6.84)
The transmembrane electric field and concentration gradients are controlled by .mdp options,
which allow the user to set reference counts for the ions on either side of the membrane. If a difference between the actual and the reference numbers persists over a certain time span, specified
by the user, a number of ion/water pairs are exchanged between the compartments until the reference numbers are restored. Alongside the calculation of channel conductance and ion selectivity,
CompEL simulations also enable determination of the channel reversal potential, an important
characteristic obtained in electrophysiology experiments.
In a CompEL setup, the simulation system is divided into two compartments A and B with independent ion concentrations. This is best achieved by using double bilayer systems with a copy (or
copies) of the channel/pore of interest in each bilayer (Fig. 6.9 A, B). If the channel axes point in
the same direction, channel flux is observed simultaneously at positive and negative potentials in
this way, which is for instance important for studying channel rectification.
The potential difference ∆U across the membrane is easily calculated with the gmx potential
utility. By this, the potential drop along z or the pore axis is exactly known in each time interval
of the simulation (Fig. 6.9 C). Type and number of ions ni of charge qi , traversing the channel in
the simulation, are written to the swapions.xvg output file, from which the average channel
conductance G in each interval ∆t is determined by:
i n i qi

∆t ∆U

(6.85)

The ion selectivity is calculated as the number flux ratio of different species. Best results are
obtained by averaging these values over several overlapping time intervals.
The calculation of reversal potentials is best achieved using a small set of simulations in which
a given transmembrane concentration gradient is complemented with small ion imbalances of
varying magnitude. For example, if one compartment contains 1 M salt and the other 0.1 M, and

6.8. Computational Electrophysiology

183

B
channel 0

0.4 0.8

U [V]

+1.0

offset A

U

2 nm

A
channel 1

-1.0

q ref
0e
4e
8e
12 e

Figure 6.9: Typical double-membrane setup for CompEL simulations (A, B). Ion / water molecule
exchanges will be performed as needed between the two light blue volumes around the dotted
black lines (A). Plot (C) shows the potential difference ∆U resulting from the selected charge
imbalance ∆qref between the compartments.
given charge neutrality otherwise, a set of simulations with ∆q = 0 e, ∆q = 2 e, ∆q = 4 e could
be used. Fitting a straight line through the current-voltage relationship of all obtained I-U pairs
near zero current will then yield Urev .

6.8.1

Usage

The following .mdp options control the CompEL protocol:
swapcoords
= Z
swap-frequency = 100

; Swap positions: no, X, Y, Z
; Swap attempt frequency

Choose Z if your membrane is in the xy-plane (Fig. 6.9). Ions will be exchanged between compartments depending on their z-positions alone. swap-frequency determines how often a
swap attempt will be made. This step requires that the positions of the split groups, the ions, and
possibly the solvent molecules are communicated between the parallel processes, so if chosen too
small it can decrease the simulation performance. The Position swapping entry in the cycle
and time accounting table at the end of the md.log file summarizes the amount of runtime spent
in the swap module.
split-group0
split-group1
massw-split0
massw-split1

=
=
=
=

channel0 ; Defines compartment boundary
channel1 ; Defines other compartment boundary
no
; use mass-weighted center?
no

184

Chapter 6. Special Topics

split-group0 and split-group1 are two index groups that define the boundaries between
the two compartments, which are usually the centers of the channels. If massw-split0 or
massw-split1 are set to yes, the center of mass of each index group is used as boundary, here
in z-direction. Otherwise, the geometrical centers will be used (× in Fig. 6.9 A). If, such as here,
a membrane channel is selected as split group, the center of the channel will define the dividing
plane between the compartments (dashed horizontal lines). All index groups must be defined in
the index file.
If, to restore the requested ion counts, an ion from one compartment has to be exchanged with a
water molecule from the other compartment, then those molecules are swapped which have the
largest distance to the compartment-defining boundaries (dashed horizontal lines). Depending on
the ion concentration, this effectively results in exchanges of molecules between the light blue
volumes. If a channel is very asymmetric in z-direction and would extend into one of the swap
volumes, one can offset the swap exchange plane with the bulk-offset parameter. A value
of 0.0 means no offset b, values −1.0 < b < 0 move the swap exchange plane closer to the
lower, values 0 < b < 1.0 closer to the upper membrane. Fig. 6.9 A (left) depicts that for the A
compartment.
solvent-group
iontypes
iontype0-name
iontype0-in-A
iontype0-in-B
iontype1-name
iontype1-in-A
iontype1-in-B
iontype2-name
iontype2-in-A
iontype2-in-B

=
=
=
=
=
=
=
=
=
=
=

SOL
3
NA
51
35
K
10
38
CL
-1
-1

;
;
;
;
;

Group containing the solvent molecules
Number of different ion types to control
Group name of the ion type
Reference count of ions of type 0 in A
Reference count of ions of type 0 in B

The group name of solvent molecules acting as exchange partners for the ions has to be set with
solvent-group. The number of different ionic species under control of the CompEL protocol
is given by the iontypes parameter, while iontype0-name gives the name of the index
group containing the atoms of this ionic species. The reference number of ions of this type can
be set with the iontype0-in-A and iontype0-in-B options for compartments A and B,
respectively. Obviously, the sum of iontype0-in-A and iontype0-in-B needs to equal
the number of ions in the group defined by iontype0-name. A reference number of -1 means:
use the number of ions as found at the beginning of the simulation as the reference value.
coupl-steps
threshold

= 10
= 1

; Average over these many swap steps
; Do not swap if < threshold

If coupl-steps is set to 1, then the momentary ion distribution determines whether ions are
exchanged. coupl-steps > 1 will use the time-average of ion distributions over the selected
number of attempt steps instead. This can be useful, for example, when ions diffuse near compartment boundaries, which would lead to numerous unproductive ion exchanges. A threshold of
1 means that a swap is performed if the average ion count in a compartment differs by at least 1

6.9. Calculating a PMF using the free-energy code

185

from the requested values. Higher thresholds will lead to toleration of larger differences. Ions are
exchanged until the requested number ± the threshold is reached.
cyl0-r
cyl0-up
cyl0-down
cyl1-r
cyl1-up
cyl1-down

=
=
=
=
=
=

5.0
0.75
0.75
5.0
0.75
0.75

;
;
;
;

Split cylinder
Split cylinder
Split cylinder
same for other

0 radius (nm)
0 upper extension (nm)
0 lower extension (nm)
channel

The cylinder options are used to define virtual geometric cylinders around the channel’s pore to
track how many ions of which type have passed each channel. Ions will be counted as having
traveled through a channel according to the definition of the channel’s cylinder radius, upper and
lower extension, relative to the location of the respective split group. This will not affect the
actual flux or exchange, but will provide you with the ion permeation numbers across each of the
channels. Note that an ion can only be counted as passing through a particular channel if it is
detected within the defined split cylinder in a swap step. If swap-frequency is chosen too
high, a particular ion may be detected in compartment A in one swap step, and in compartment B
in the following swap step, so it will be unclear through which of the channels it has passed.
A double-layered system for CompEL simulations can be easily prepared by duplicating an existing membrane/channel MD system in the direction of the membrane normal (typically z) with gmx
editconf -translate 0 0 , where l_z is the box length in that direction. If you
have already defined index groups for the channel for the single-layered system, gmx make_ndx
-n index.ndx -twin will provide you with the groups for the double-layered system.
To suppress large fluctuations of the membranes along the swap direction, it may be useful to
apply a harmonic potential (acting only in the swap dimension) between each of the two channel
and/or bilayer centers using umbrella pulling (see section 6.4).

Multimeric channels
If a split group consists of more than one molecule, the correct PBC image of all molecules with
respect to each other has to be chosen such that the channel center can be correctly determined.
GROMACS assumes that the starting structure in the .tpr file has the correct PBC representation.
Set the following environment variable to check whether that is the case:
• GMX_COMPELDUMP: output the starting structure after it has been made whole to .pdb
file.

6.9

Calculating a PMF using the free-energy code

The free-energy coupling-parameter approach (see sec. 3.12) provides several ways to calculate
potentials of mean force. A potential of mean force between two atoms can be calculated by
connecting them with a harmonic potential or a constraint. For this purpose there are special
potentials that avoid the generation of extra exclusions, see sec. 5.4.4. When the position of the

186

Chapter 6. Special Topics

minimum or the constraint length is 1 nm more in state B than in state A, the restraint or constraint
force is given by ∂H/∂λ. The distance between the atoms can be changed as a function of λ and
time by setting delta-lambda in the .mdp file. The results should be identical (although not
numerically due to the different implementations) to the results of the pull code with umbrella
sampling and constraint pulling. Unlike the pull code, the free energy code can also handle atoms
that are connected by constraints.
Potentials of mean force can also be calculated using position restraints. With position restraints,
atoms can be linked to a position in space with a harmonic potential (see 4.3.1). These positions
can be made a function of the coupling parameter λ. The positions for the A and the B states are
supplied to grompp with the -r and -rb options, respectively. One could use this approach to
do targeted MD; note that we do not encourage the use of targeted MD for proteins. A protein
can be forced from one conformation to another by using these conformations as position restraint
coordinates for state A and B. One can then slowly change λ from 0 to 1. The main drawback of
this approach is that the conformational freedom of the protein is severely limited by the position
restraints, independent of the change from state A to B. Also, the protein is forced from state A to
B in an almost straight line, whereas the real pathway might be very different. An example of a
more fruitful application is a solid system or a liquid confined between walls where one wants to
measure the force required to change the separation between the boundaries or walls. Because the
boundaries (or walls) already need to be fixed, the position restraints do not limit the system in its
sampling.

6.10

Removing fastest degrees of freedom

The maximum time step in MD simulations is limited by the smallest oscillation period that can
be found in the simulated system. Bond-stretching vibrations are in their quantum-mechanical
ground state and are therefore better represented by a constraint instead of a harmonic potential.
For the remaining degrees of freedom, the shortest oscillation period (as measured from a simulation) is 13 fs for bond-angle vibrations involving hydrogen atoms. Taking as a guideline that
with a Verlet (leap-frog) integration scheme a minimum of 5 numerical integration steps should be
performed per period of a harmonic oscillation in order to integrate it with reasonable accuracy,
the maximum time step will be about 3 fs. Disregarding these very fast oscillations of period 13 fs,
the next shortest periods are around 20 fs, which will allow a maximum time step of about 4 fs.
Removing the bond-angle degrees of freedom from hydrogen atoms can best be done by defining
them as virtual interaction sites instead of normal atoms. Whereas a normal atom is connected
to the molecule with bonds, angles and dihedrals, a virtual site’s position is calculated from the
position of three nearby heavy atoms in a predefined manner (see also sec. 4.7). For the hydrogens
in water and in hydroxyl, sulfhydryl, or amine groups, no degrees of freedom can be removed,
because rotational freedom should be preserved. The only other option available to slow down
these motions is to increase the mass of the hydrogen atoms at the expense of the mass of the
connected heavy atom. This will increase the moment of inertia of the water molecules and the
hydroxyl, sulfhydryl, or amine groups, without affecting the equilibrium properties of the system
and without affecting the dynamical properties too much. These constructions will shortly be
described in sec. 6.10.1 and have previously been described in full detail [153].

6.10. Removing fastest degrees of freedom

187

111
000
000
111
000
111

1111
0000
0000
1111
A

111
000
000
111

111
000
1111
0000
0000
1111

1111
0000
0000
1111

111
000
000
111
000
111

1111
0000
0000
1111
0000
1111

1111
0000
0000
1111
C

Figure 6.10: The different types of virtual site constructions used for hydrogen atoms. The atoms
used in the construction of the virtual site(s) are depicted as black circles, virtual sites as gray
ones. Hydrogens are smaller than heavy atoms. A: fixed bond angle, note that here the hydrogen
is not a virtual site; B: in the plane of three atoms, with fixed distance; C: in the plane of three
atoms, with fixed angle and distance; D: construction for amine groups (-NH2 or -NH+
3 ), see text
for details.
Using both virtual sites and modified masses, the next bottleneck is likely to be formed by the
improper dihedrals (which are used to preserve planarity or chirality of molecular groups) and the
peptide dihedrals. The peptide dihedral cannot be changed without affecting the physical behavior
of the protein. The improper dihedrals that preserve planarity mostly deal with aromatic residues.
Bonds, angles, and dihedrals in these residues can also be replaced with somewhat elaborate virtual
site constructions.
All modifications described in this section can be performed using the GROMACS topology building tool pdb2gmx. Separate options exist to increase hydrogen masses, virtualize all hydrogen
atoms, or also virtualize all aromatic residues. Note that when all hydrogen atoms are virtualized, those inside the aromatic residues will be virtualized as well, i.e. hydrogens in the aromatic
residues are treated differently depending on the treatment of the aromatic residues.
Parameters for the virtual site constructions for the hydrogen atoms are inferred from the forcefield parameters (vis. bond lengths and angles) directly by grompp while processing the topology
file. The constructions for the aromatic residues are based on the bond lengths and angles for the
geometry as described in the force fields, but these parameters are hard-coded into pdb2gmx due
to the complex nature of the construction needed for a whole aromatic group.

6.10.1

Hydrogen bond-angle vibrations

Construction of virtual sites
The goal of defining hydrogen atoms as virtual sites is to remove all high-frequency degrees of
freedom from them. In some cases, not all degrees of freedom of a hydrogen atom should be
removed, e.g. in the case of hydroxyl or amine groups the rotational freedom of the hydrogen
atom(s) should be preserved. Care should be taken that no unwanted correlations are introduced
by the construction of virtual sites, e.g. bond-angle vibration between the constructing atoms could
translate into hydrogen bond-length vibration. Additionally, since virtual sites are by definition
massless, in order to preserve total system mass, the mass of each hydrogen atom that is treated as
virtual site should be added to the bonded heavy atom.

188

Chapter 6. Special Topics

ε
δ

111
000
000
111

1111
0000
0000
1111
δ

Phe

δ
γ

1111
0000
0000
1111

111
000
000
111

000
ε111

Tyr

1111
0000
0000
1111

1111
0000
0000
1111
0000
1111
ε

111
000
000
111

11
00
00
11

1111
0000
0000
1111
0000
1111

Trp

111
000
000
111

ζ
δ

111
000
000
111
000
111

δ
ε

ε111
000

111
000
000
111

111
000
000ε
111

000
γ111
111
000
000
111

His

Figure 6.11: The different types of virtual site constructions used for aromatic residues. The atoms
used in the construction of the virtual site(s) are depicted as black circles, virtual sites as gray ones.
Hydrogens are smaller than heavy atoms. A: phenylalanine; B: tyrosine (note that the hydroxyl
hydrogen is not a virtual site); C: tryptophan; D: histidine.
Taking into account these considerations, the hydrogen atoms in a protein naturally fall into several
categories, each requiring a different approach (see also Fig. 6.10).
• hydroxyl (-OH) or sulfhydryl (-SH) hydrogen: The only internal degree of freedom in a
hydroxyl group that can be constrained is the bending of the C-O-H angle. This angle is
fixed by defining an additional bond of appropriate length, see Fig. 6.10A. Doing so removes
the high-frequency angle bending, but leaves the dihedral rotational freedom. The same goes
for a sulfhydryl group. Note that in these cases the hydrogen is not treated as a virtual site.
• single amine or amide (-NH-) and aromatic hydrogens (-CH-): The position of these hydrogens cannot be constructed from a linear combination of bond vectors, because of the
flexibility of the angle between the heavy atoms. Instead, the hydrogen atom is positioned
at a fixed distance from the bonded heavy atom on a line going through the bonded heavy
atom and a point on the line through both second bonded atoms, see Fig. 6.10B.
• planar amine (-NH2 ) hydrogens: The method used for the single amide hydrogen is not well
suited for planar amine groups, because no suitable two heavy atoms can be found to define
the direction of the hydrogen atoms. Instead, the hydrogen is constructed at a fixed distance
from the nitrogen atom, with a fixed angle to the carbon atom, in the plane defined by one
of the other heavy atoms, see Fig. 6.10C.
• amine group (umbrella -NH2 or -NH+
3 ) hydrogens: Amine hydrogens with rotational freedom cannot be constructed as virtual sites from the heavy atoms they are connected to, since
this would result in loss of the rotational freedom of the amine group. To preserve the rotational freedom while removing the hydrogen bond-angle degrees of freedom, two “dummy
masses” are constructed with the same total mass, moment of inertia (for rotation around
the C-N bond) and center of mass as the amine group. These dummy masses have no interaction with any other atom, except for the fact that they are connected to the carbon and
to each other, resulting in a rigid triangle. From these three particles, the positions of the
nitrogen and hydrogen atoms are constructed as linear combinations of the two carbon-mass
vectors and their outer product, resulting in an amine group with rotational freedom intact,
but without other internal degrees of freedom. See Fig. 6.10D.

6.11. Viscosity calculation

6.10.2

189

Out-of-plane vibrations in aromatic groups

The planar arrangements in the side chains of the aromatic residues lends itself perfectly to a
virtual-site construction, giving a perfectly planar group without the inherently unstable constraints that are necessary to keep normal atoms in a plane. The basic approach is to define three
atoms or dummy masses with constraints between them to fix the geometry and create the rest of
the atoms as simple virtual sites type (see sec. 4.7) from these three. Each of the aromatic residues
require a different approach:
• Phenylalanine: Cγ , C1 , and C2 are kept as normal atoms, but with each a mass of one
third the total mass of the phenyl group. See Fig. 6.10A.
• Tyrosine: The ring is treated identically to the phenylalanine ring. Additionally, constraints
are defined between C1 , C2 , and Oη . The original improper dihedral angles will keep both
triangles (one for the ring and one with Oη ) in a plane, but due to the larger moments of
inertia this construction will be much more stable. The bond-angle in the hydroxyl group
will be constrained by a constraint between Cγ and Hη . Note that the hydrogen is not treated
as a virtual site. See Fig. 6.10B.
• Tryptophan: Cβ is kept as a normal atom and two dummy masses are created at the center
of mass of each of the rings, each with a mass equal to the total mass of the respective ring
(Cδ2 and C2 are each counted half for each ring). This keeps the overall center of mass and
the moment of inertia almost (but not quite) equal to what it was. See Fig. 6.10C.
• Histidine: Cγ , C1 and N2 are kept as normal atoms, but with masses redistributed such
that the center of mass of the ring is preserved. See Fig. 6.10D.

6.11

Viscosity calculation

The shear viscosity is a property of liquids that can be determined easily by experiment. It is useful
for parameterizing a force field because it is a kinetic property, while most other properties which
are used for parameterization are thermodynamic. The viscosity is also an important property,
since it influences the rates of conformational changes of molecules solvated in the liquid.
The viscosity can be calculated from an equilibrium simulation using an Einstein relation:
1 V
d
η=
lim
2 kB T t→∞ dt

t0 +t

Pxz (t )dt
t0

2 +

(6.86)
t0

This can be done with gmx energy. This method converges very slowly [154], and as such a
nanosecond simulation might not be long enough for an accurate determination of the viscosity.
The result is very dependent on the treatment of the electrostatics. Using a (short) cut-off results
in large noise on the off-diagonal pressure elements, which can increase the calculated viscosity
by an order of magnitude.
GROMACS also has a non-equilibrium method for determining the viscosity [154]. This makes
use of the fact that energy, which is fed into system by external forces, is dissipated through viscous

190

Chapter 6. Special Topics

friction. The generated heat is removed by coupling to a heat bath. For a Newtonian liquid adding
a small force will result in a velocity gradient according to the following equation:
ax (z) +

η ∂ 2 vx (z)
=0
ρ ∂z 2

(6.87)

Here we have applied an acceleration ax (z) in the x-direction, which is a function of the zcoordinate. In GROMACS the acceleration profile is:

ax (z) = A cos

2πz
lz

(6.88)

where lz is the height of the box. The generated velocity profile is:
2πz
vx (z) = V cos
lz

lz
2π

A
lz
η= ρ
V
2π

ρ
V =A
η

(6.89)

(6.90)

The viscosity can be calculated from A and V :

(6.91)

In the simulation V is defined as:
N
X

V =

2πz
mi vi,x 2 cos
lz
i=1

N
X

(6.92)

i=1

The generated velocity profile is not coupled to the heat bath. Moreover, the velocity profile is
excluded from the kinetic energy. One would like V to be as large as possible to get good statistics.
However, the shear rate should not be so high that the system gets too far from equilibrium. The
maximum shear rate occurs where the cosine is zero, the rate being:
shmax = max
z

∂vx (z)
ρ lz
=A
∂z
η 2π

(6.93)

For a simulation with: η = 10−3 [kg m−1 s−1 ], ρ = 103 [kg m−3 ] and lz = 2π [nm], shmax =
1 [ps nm−1 ] A. This shear rate should be smaller than one over the longest correlation time in
the system. For most liquids, this will be the rotation correlation time, which is around 10 ps. In
this case, A should be smaller than 0.1 [nm ps−2 ]. When the shear rate is too high, the observed
viscosity will be too low. Because V is proportional to the square of the box height, the optimal
box is elongated in the z-direction. In general, a simulation length of 100 ps is enough to obtain
an accurate value for the viscosity.
The heat generated by the viscous friction is removed by coupling to a heat bath. Because this
coupling is not instantaneous the real temperature of the liquid will be slightly lower than the

6.12. Tabulated interaction functions

191

observed temperature. Berendsen derived this temperature shift [31], which can be written in
terms of the shear rate as:
ητ
sh2
(6.94)
Ts =
2ρ Cv max
where τ is the coupling time for the Berendsen thermostat and Cv is the heat capacity. Using
the values of the example above, τ = 10−13 [s] and Cv = 2 · 103 [J kg−1 K−1 ], we get: Ts =
25 [K ps−2 ] sh2max . When we want the shear rate to be smaller than 1/10 [ps−1 ], Ts is smaller than
0.25 [K], which is negligible.
Note that the system has to build up the velocity profile when starting from an equilibrium state.
This build-up time is of the order of the correlation time of the liquid.
Two quantities are written to the energy file, along with their averages and fluctuations: V and
1/η, as obtained from (6.91).

6.12

Tabulated interaction functions

6.12.1

Cubic splines for potentials

In some of the inner loops of GROMACS, look-up tables are used for computation of potential
and forces. The tables are interpolated using a cubic spline algorithm. There are separate tables
for electrostatic, dispersion, and repulsion interactions, but for the sake of caching performance
these have been combined into a single array. The cubic spline interpolation for xi ≤ x < xi+1
looks like this:
Vs (x) = A0 + A1 + A2 2 + A3 3
(6.95)
where the table spacing h and fraction are given by:
h = xi+1 − xi

(6.96)

= (x − xi )/h

(6.97)

so that 0 ≤ < 1. From this, we can calculate the derivative in order to determine the forces:
dVs (x) d
= −(A1 + 2A2 + 3A3 2 )/h
(6.98)
d dx
The four coefficients are determined from the four conditions that Vs and −Vs0 at both ends of each
interval should match the exact potential V and force −V 0 . This results in the following errors for
each interval:
−Vs0 (x) = −

h4
+ O(h5 )
(6.99)
384
h3
|Vs0 − V 0 |max = V 0000 √ + O(h4 )
(6.100)
72 3
h2
|Vs00 − V 00 |max = V 0000 + O(h3 )
(6.101)
12
V and V’ are continuous, while V” is the first discontinuous derivative. The number of points
per nanometer is 500 and 2000 for mixed- and double-precision versions of GROMACS, respectively. This means that the errors in the potential and force will usually be smaller than the mixed
precision accuracy.
|Vs − V |max = V 0000

192

Chapter 6. Special Topics

GROMACS stores A0 , A1 , A2 and A3 . The force routines get a table with these four parameters
and a scaling factor s that is equal to the number of points per nm. (Note that h is s−1 ). The
algorithm goes a little something like this:
1. Calculate distance vector (r ij ) and distance rij
2. Multiply rij by s and truncate to an integer value n0 to get a table index
3. Calculate fractional component ( = srij − n0 ) and 2
4. Do the interpolation to calculate the potential V and the scalar force f
5. Calculate the vector force F by multiplying f with r ij
Note that table look-up is significantly slower than computation of the most simple Lennard-Jones
and Coulomb interaction. However, it is much faster than the shifted Coulomb function used in
conjunction with the PPPM method. Finally, it is much easier to modify a table for the potential
(and get a graphical representation of it) than to modify the inner loops of the MD program.

6.12.2

User-specified potential functions

You can also use your own potential functions without editing the GROMACS code. The potential
function should be according to the following equation
V (rij ) =

qi qj
f (rij ) + C6 g(rij ) + C12 h(rij )
4π0

(6.102)

where f , g, and h are user defined functions. Note that if g(r) represents a normal dispersion
interaction, g(r) should be < 0. C6 , C12 and the charges are read from the topology. Also note
that combination rules are only supported for Lennard-Jones and Buckingham, and that your tables
should match the parameters in the binary topology.
When you add the following lines in your .mdp file:
rlist
coulombtype
rcoulomb
vdwtype
rvdw

=
=
=
=
=

1.0
User
1.0
User
1.0

mdrun will read a single non-bonded table file, or multiple when energygrp-table is set (see
below). The name of the file(s) can be set with the mdrun option -table. The table file should
contain seven columns of table look-up data in the order: x, f (x), −f 0 (x), g(x), −g 0 (x), h(x),
−h0 (x). The x should run from 0 to rc + 1 (the value of table_extension can be changed
in the .mdp file). You can choose the spacing you like; for the standard tables GROMACS uses a
spacing of 0.002 and 0.0005 nm when you run in mixed and double precision, respectively. In this
context, rc denotes the maximum of the two cut-offs rvdw and rcoulomb (see above). These
variables need not be the same (and need not be 1.0 either). Some functions used for potentials
contain a singularity at x = 0, but since atoms are normally not closer to each other than 0.1 nm,

6.13. Mixed Quantum-Classical simulation techniques

193

the function value at x = 0 is not important. Finally, it is also possible to combine a standard
Coulomb with a modified LJ potential (or vice versa). One then specifies e.g. coulombtype =
Cut-off or coulombtype = PME, combined with vdwtype = User. The table file must
always contain the 7 columns however, and meaningful data (i.e. not zeroes) must be entered in
all columns. A number of pre-built table files can be found in the GMXLIB directory for 6-8, 6-9,
6-10, 6-11, and 6-12 Lennard-Jones potentials combined with a normal Coulomb.
If you want to have different functional forms between different groups of atoms, this can be
set through energy groups. Different tables can be used for non-bonded interactions between
different energy groups pairs through the .mdp option energygrp-table (see details in the
User Guide). Atoms that should interact with a different potential should be put into different
energy groups. Between group pairs which are not listed in energygrp-table, the normal
user tables will be used. This makes it easy to use a different functional form between a few types
of atoms.

6.13

Mixed Quantum-Classical simulation techniques

In a molecular mechanics (MM) force field, the influence of electrons is expressed by empirical
parameters that are assigned on the basis of experimental data, or on the basis of results from
high-level quantum chemistry calculations. These are valid for the ground state of a given covalent
structure, and the MM approximation is usually sufficiently accurate for ground-state processes
in which the overall connectivity between the atoms in the system remains unchanged. However,
for processes in which the connectivity does change, such as chemical reactions, or processes that
involve multiple electronic states, such as photochemical conversions, electrons can no longer be
ignored, and a quantum mechanical description is required for at least those parts of the system in
which the reaction takes place.
One approach to the simulation of chemical reactions in solution, or in enzymes, is to use a combination of quantum mechanics (QM) and molecular mechanics (MM). The reacting parts of the
system are treated quantum mechanically, with the remainder being modeled using the force field.
The current version of GROMACS provides interfaces to several popular Quantum Chemistry
packages (MOPAC [155], GAMESS-UK [156], Gaussian [157] and CPMD [158]).
GROMACS interactions between the two subsystems are either handled as described by Field et
al. [159] or within the ONIOM approach by Morokuma and coworkers [160, 161].

6.13.1

Overview

Two approaches for describing the interactions between the QM and MM subsystems are supported in this version:
1. Electronic Embedding The electrostatic interactions between the electrons of the QM region and the MM atoms and between the QM nuclei and the MM atoms are included in the
Hamiltonian for the QM subsystem:
H QM/M M = HeQM −

n X
M
X
e2 QJ
i

4π0 riJ

N X
M
X
e2 ZA QJ
A

eπ0 RAJ

(6.103)

194

Chapter 6. Special Topics
where n and N are the number of electrons and nuclei in the QM region, respectively,
and M is the number of charged MM atoms. The first term on the right hand side is the
original electronic Hamiltonian of an isolated QM system. The first of the double sums is
the total electrostatic interaction between the QM electrons and the MM atoms. The total
electrostatic interaction of the QM nuclei with the MM atoms is given by the second double
sum. Bonded interactions between QM and MM atoms are described at the MM level by the
appropriate force-field terms. Chemical bonds that connect the two subsystems are capped
by a hydrogen atom to complete the valence of the QM region. The force on this atom,
which is present in the QM region only, is distributed over the two atoms of the bond. The
cap atom is usually referred to as a link atom.

2. ONIOM In the ONIOM approach, the energy and gradients are first evaluated for the isolated QM subsystem at the desired level of ab initio theory. Subsequently, the energy and
gradients of the total system, including the QM region, are computed using the molecular
mechanics force field and added to the energy and gradients calculated for the isolated QM
subsystem. Finally, in order to correct for counting the interactions inside the QM region
twice, a molecular mechanics calculation is performed on the isolated QM subsystem and
the energy and gradients are subtracted. This leads to the following expression for the total
QM/MM energy (and gradients likewise):
MM
− EIM M ,
Etot = EIQM + EI+II

(6.104)

where the subscripts I and II refer to the QM and MM subsystems, respectively. The superscripts indicate at what level of theory the energies are computed. The ONIOM scheme
has the advantage that it is not restricted to a two-layer QM/MM description, but can easily
handle more than two layers, with each layer described at a different level of theory.

6.13.2

Usage

To make use of the QM/MM functionality in GROMACS, one needs to:
1. introduce link atoms at the QM/MM boundary, if needed;
2. specify which atoms are to be treated at a QM level;
3. specify the QM level, basis set, type of QM/MM interface and so on.
Adding link atoms
At the bond that connects the QM and MM subsystems, a link atoms is introduced. In GROMACS
the link atom has special atomtype, called LA. This atomtype is treated as a hydrogen atom in the
QM calculation, and as a virtual site in the force-field calculation. The link atoms, if any, are part
of the system, but have no interaction with any other atom, except that the QM force working on
it is distributed over the two atoms of the bond. In the topology, the link atom (LA), therefore, is
defined as a virtual site atom:

6.13. Mixed Quantum-Classical simulation techniques

195

[ virtual_sites2 ]
LA QMatom MMatom 1 0.65

See sec. 5.2.2 for more details on how virtual sites are treated. The link atom is replaced at every
step of the simulation.
In addition, the bond itself is replaced by a constraint:
[ constraints ]
QMatom MMatom 2 0.153

Note that, because in our system the QM/MM bond is a carbon-carbon bond (0.153 nm), we use
a constraint length of 0.153 nm, and dummy position of 0.65. The latter is the ratio between the
ideal C-H bond length and the ideal C-C bond length. With this ratio, the link atom is always
0.1 nm away from the QMatom, consistent with the carbon-hydrogen bond length. If the QM and
MM subsystems are connected by a different kind of bond, a different constraint and a different
dummy position, appropriate for that bond type, are required.
Specifying the QM atoms
Atoms that should be treated at a QM level of theory, including the link atoms, are added to the
index file. In addition, the chemical bonds between the atoms in the QM region are to be defined
as connect bonds (bond type 5) in the topology file:
[ bonds ]
QMatom1 QMatom2 5
QMatom2 QMatom3 5

Specifying the QM/MM simulation parameters
In the .mdp file, the following parameters control a QM/MM simulation.
QMMM = no
If this is set to yes, a QM/MM simulation is requested. Several groups of atoms can be
described at different QM levels separately. These are specified in the QMMM-grps field
separated by spaces. The level of ab initio theory at which the groups are described is
specified by QMmethod and QMbasis Fields. Describing the groups at different levels of
theory is only possible with the ONIOM QM/MM scheme, specified by QMMMscheme.
QMMM-grps =
groups to be described at the QM level
QMMMscheme = normal
Options are normal and ONIOM. This selects the QM/MM interface. normal implies
that the QM subsystem is electronically embedded in the MM subsystem. There can only
be one QMMM-grps that is modeled at the QMmethod and QMbasis level of ab initio

196

Chapter 6. Special Topics
theory. The rest of the system is described at the MM level. The QM and MM subsystems
interact as follows: MM point charges are included in the QM one-electron Hamiltonian
and all Lennard-Jones interactions are described at the MM level. If ONIOM is selected, the
interaction between the subsystem is described using the ONIOM method by Morokuma
and co-workers. There can be more than one QMMM-grps each modeled at a different level
of QM theory (QMmethod and QMbasis).

QMmethod =
Method used to compute the energy and gradients on the QM atoms. Available methods are AM1, PM3, RHF, UHF, DFT, B3LYP, MP2, CASSCF, MMVB and CPMD. For
CASSCF, the number of electrons and orbitals included in the active space is specified by
CASelectrons and CASorbitals. For CPMD, the plane-wave cut-off is specified by
the planewavecutoff keyword.
QMbasis =
Gaussian basis set used to expand the electronic wave-function. Only Gaussian basis sets
are currently available, i.e. STO-3G, 3-21G, 3-21G*, 3-21+G*, 6-21G, 6-31G, 6-31G*,
6-31+G*, and 6-311G. For CPMD, which uses plane wave expansion rather than atomcentered basis functions, the planewavecutoff keyword controls the plane wave expansion.
QMcharge =
The total charge in e of the QMMM-grps. In case there are more than one QMMM-grps, the
total charge of each ONIOM layer needs to be specified separately.
QMmult =
The multiplicity of the QMMM-grps. In case there are more than one QMMM-grps, the
multiplicity of each ONIOM layer needs to be specified separately.
CASorbitals =
The number of orbitals to be included in the active space when doing a CASSCF computation.
CASelectrons =
The number of electrons to be included in the active space when doing a CASSCF computation.
SH = no
If this is set to yes, a QM/MM MD simulation on the excited state-potential energy surface
and enforce a diabatic hop to the ground-state when the system hits the conical intersection
hyperline in the course the simulation. This option only works in combination with the
CASSCF method.

6.13.3

Output

The energies and gradients computed in the QM calculation are added to those computed by GROMACS. In the .edr file there is a section for the total QM energy.

6.14. Using VMD plug-ins for trajectory file I/O

6.13.4

197

Future developments

Several features are currently under development to increase the accuracy of the QM/MM interface. One useful feature is the use of delocalized MM charges in the QM computations. The most
important benefit of using such smeared-out charges is that the Coulombic potential has a finite
value at interatomic distances. In the point charge representation, the partially-charged MM atoms
close to the QM region tend to “over-polarize” the QM system, which leads to artifacts in the
calculation.
What is needed as well is a transition state optimizer.

6.14

Using VMD plug-ins for trajectory file I/O

GROMACS tools are able to use the plug-ins found in an existing installation of VMD in order
to read and write trajectory files in formats that are not native to GROMACS. You will be able to
supply an AMBER DCD-format trajectory filename directly to GROMACS tools, for example.
This requires a VMD installation not older than version 1.8, that your system provides the dlopen
function so that programs can determine at run time what plug-ins exist, and that you build shared
libraries when building GROMACS. CMake will find the vmd executable in your path, and from
it, or the environment variable VMDDIR at configuration or run time, locate the plug-ins. Alternatively, the VMD_PLUGIN_PATH can be used at run time to specify a path where these plug-ins
can be found. Note that these plug-ins are in a binary format, and that format must match the
architecture of the machine attempting to use them.

6.15

Interactive Molecular Dynamics

GROMACS supports the interactive molecular dynamics (IMD) protocol as implemented by VMD
to control a running simulation in NAMD. IMD allows to monitor a running GROMACS simulation from a VMD client. In addition, the user can interact with the simulation by pulling on
atoms, residues or fragments with a mouse or a force-feedback device. Additional information
about the GROMACS implementation and an exemplary GROMACS IMD system can be found
on this homepage.

6.15.1

Simulation input preparation

The GROMACS implementation allows transmission and interaction with a part of the running
simulation only, e.g. in cases where no water molecules should be transmitted or pulled. The
group is specified via the .mdp option IMD-group. When IMD-group is empty, the IMD
protocol is disabled and cannot be enabled via the switches in mdrun. To interact with the entire
system, IMD-group can be set to System. When using grompp, a .gro file to be used as
VMD input is written out (-imd switch of grompp).

198

6.15.2

Chapter 6. Special Topics

Starting the simulation

Communication between VMD and GROMACS is achieved via TCP sockets and thus enables
controlling an mdrun running locally or on a remote cluster. The port for the connection can be
specified with the -imdport switch of mdrun, 8888 is the default. If a port number of 0 or
smaller is provided, GROMACS automatically assigns a free port to use with IMD.
Every N steps, the mdrun client receives the applied forces from VMD and sends the new positions to the client. VMD permits increasing or decreasing the communication frequency interactively. By default, the simulation starts and runs even if no IMD client is connected. This behavior
is changed by the -imdwait switch of mdrun. After startup and whenever the client has disconnected, the integration stops until reconnection of the client. When the -imdterm switch is
used, the simulation can be terminated by pressing the stop button in VMD. This is disabled by
default. Finally, to allow interacting with the simulation (i.e. pulling from VMD) the -imdpull
switch has to be used. Therefore, a simulation can only be monitored but not influenced from the
VMD client when none of -imdwait, -imdterm or -imdpull are set. However, since the
IMD protocol requires no authentication, it is not advisable to run simulations on a host directly
reachable from an insecure environment. Secure shell forwarding of TCP can be used to connect
to running simulations not directly reachable from the interacting host. Note that the IMD command line switches of mdrun are hidden by default and show up in the help text only with gmx
mdrun -h -hidden.

6.15.3

Connecting from VMD

In VMD, first the structure corresponding to the IMD group has to be loaded (File → New
Molecule). Then the IMD connection window has to be used (Extensions → Simulation → IMD
Connect (NAMD)). In the IMD connection window, hostname and port have to be specified and
followed by pressing Connect. Detach Sim allows disconnecting without terminating the simulation, while Stop Sim ends the simulation on the next neighbor searching step (if allowed by
-imdterm).
The timestep transfer rate allows adjusting the communication frequency between simulation and
IMD client. Setting the keep rate loads every N th frame into VMD instead of discarding them
when a new one is received. The displayed energies are in SI units in contrast to energies displayed
from NAMD simulations.

6.16

Embedding proteins into the membranes

GROMACS is capable of inserting the protein into pre-equilibrated lipid bilayers with minimal perturbation of the lipids using the method, which was initially described as a ProtSqueeze
technique,[162] and later implemented as g_membed tool.[163] Currently the functionality of g_membed is available in mdrun as described in the user guide.
This method works by first artificially shrinking the protein in the xy-plane, then it removes lipids
that overlap with that much smaller core. Then the protein atoms are gradually resized back to
their initial configuration, using normal dynamics for the rest of the system, so the lipids adapt to

6.16. Embedding proteins into the membranes
the protein. Further lipids are removed as required.

199

200

Chapter 6. Special Topics

Chapter 7
Run parameters and
Programs
7.1

Online documentation

More documentation is available online from the GROMACS web site, http://manual.
gromacs.org/documentation.
In addition, we install standard UNIX man pages for all the programs. If you have sourced the
GMXRC script in the GROMACS binary directory for your host they should already be present in
your MANPATH environment variable, and you should be able to type e.g. man gmx-grompp.
You can also use the -h flag on the command line (e.g. gmx grompp -h) to see the same
information, as well as gmx help grompp. The list of all programs are available from gmx
help.

7.2

File types

Table 7.1 lists the file types used by GROMACS along with a short description, and you can find
a more detail description for each file in your HTML reference, or in our online version.
GROMACS files written in XDR format can be read on any architecture with GROMACS version
1.6 or later if the configuration script found the XDR libraries on your system. They should always
be present on UNIX since they are necessary for NFS support.

7.3

Run Parameters

The descriptions of .mdp parameters can be found at http://manual.gromacs.org/current/
mdp-options.html or in your installation at share/gromacs/html/mdp-options.html

202

Default
Name
Ext.
atomtp.atp
eiwit.brk
state.cpt
nnnice.dat
user.dlg
sam.edi
sam.edo
ener.edr
ener.edr
ener.ene
eiwit.ent
plot.eps
conf.esp
conf.g96
conf.gro
conf.gro
out.gro
polar.hdb
topinc.itp
run.log
ps.m2p
ss.map
ss.mat
grompp.mdp
hessian.mtx
index.ndx
hello.out
eiwit.pdb
residue.rtp
doc.tex
topol.top
topol.tpr
topol.tpr
topol.tpr
traj.trr
traj.trr
root.xpm
traj.xtc
traj.xtc
traj.xtc
graph.xvg

Chapter 7. Run parameters and Programs

Type
Asc
Asc
xdr
Asc
Asc
Asc
Asc
xdr
Bin
Asc
Asc
Asc
Asc
Asc

Asc
Asc
Asc
Asc
Asc
Asc
Asc
Bin
Asc
Asc
Asc
Asc
Asc
Asc

xdr

Default
Option
-f

-f
-c
-c
-c
-c
-o

-l

-f
-m
-n
-o
-f
-o
-p
-s
-s
-s

xdr
Asc
-f
-f
xdr
Asc

-o

Description
Atomtype file used by pdb2gmx
Brookhaven data bank file
Checkpoint file
Generic data file
Dialog Box data for ngmx
ED sampling input
ED sampling output
Generic energy: edr ene
Energy file in portable xdr format
Energy file
Entry in the protein date bank
Encapsulated PostScript (tm) file
Coordinate file in ESPResSo format
Coordinate file in Gromos-96 format
Coordinate file in Gromos-87 format
Structure: gro g96 pdb esp tpr
Structure: gro g96 pdb esp
Hydrogen data base
Include file for topology
Log file
Input file for mat2ps
File that maps matrix data to colors
Matrix Data file
grompp input file with MD parameters
Hessian matrix
Index file
Generic output file
Protein data bank file
Residue Type file used by pdb2gmx
LaTeX file
Topology file
Generic run input: tpr
Structure+mass(db): tpr gro g96 pdb
Portable xdr run input file
Full precision trajectory: trr cpt
Trajectory in portable xdr format
X PixMap compatible matrix file
Trajec., input: xtc trr tng cpt gro g96 pdb
Trajectory, output: xtc trr tng gro g96 pdb
Compressed trajectory (portable xdr format)
xvgr/xmgr file

Table 7.1: The GROMACS file types.

Chapter 8
Analysis
In this chapter different ways of analyzing your trajectory are described. The names of the corresponding analysis programs are given. Specific information on the in- and output of these programs can be found in the online manual at www.gromacs.org. The output files are often produced
as finished Grace/Xmgr graphs.
First, in sec. 8.1, the group concept in analysis is explained. 8.1.2 explains a newer concept of
dynamic selections, which is currently supported by a few tools. Then, the different analysis tools
are presented.

8.1

Using Groups

gmx make_ndx, gmx mk_angndx, gmx select
In chapter 3, it was explained how groups of atoms can be used in mdrun (see sec. 3.3). In most
analysis programs, groups of atoms must also be chosen. Most programs can generate several
default index groups, but groups can always be read from an index file. Let’s consider the example
of a simulation of a binary mixture of components A and B. When we want to calculate the radial
distribution function (RDF) gAB (r) of A with respect to B, we have to calculate:
4πr2 gAB (r) = V

NA X
NB
X

P (r)

(8.1)

i∈A j∈B

where V is the volume and P (r) is the probability of finding a B atom at distance r from an A
atom.
By having the user define the atom numbers for groups A and B in a simple file, we can calculate
this gAB in the most general way, without having to make any assumptions in the RDF program
about the type of particles.
Groups can therefore consist of a series of atom numbers, but in some cases also of molecule
numbers. It is also possible to specify a series of angles by triples of atom numbers, dihedrals
by quadruples of atom numbers and bonds or vectors (in a molecule) by pairs of atom numbers.

204

Chapter 8. Analysis

When appropriate the type of index file will be specified for the following analysis programs. To
help creating such index files (index.ndx), there are a couple of programs to generate them,
using either your input configuration or the topology. To generate an index file consisting of a
series of atom numbers (as in the example of gAB ), use gmx make_ndx or gmx select. To
generate an index file with angles or dihedrals, use gmx mk_angndx. Of course you can also
make them by hand. The general format is presented here:
[ Oxygen ]
1
4
[ Hydrogen ]
2
3
8
9

First, the group name is written between square brackets. The following atom numbers may be
spread out over as many lines as you like. The atom numbering starts at 1.
Each tool that can use groups will offer the available alternatives for the user to choose. That
choice can be made with the number of the group, or its name. In fact, the first few letters of the
group name will suffice if that will distinguish the group from all others. There are ways to use
Unix shell features to choose group names on the command line, rather than interactively. Consult
www.gromacs.org for suggestions.

8.1.1

Default Groups

When no index file is supplied to analysis tools or grompp, a number of default groups are
generated to choose from:
System
all atoms in the system
Protein
all protein atoms
Protein-H
protein atoms excluding hydrogens
C-alpha
Cα atoms
Backbone
protein backbone atoms; N, Cα and C
MainChain
protein main chain atoms: N, Cα , C and O, including oxygens in C-terminus
MainChain+Cb
protein main chain atoms including Cβ

8.1. Using Groups

205

MainChain+H
protein main chain atoms including backbone amide hydrogens and hydrogens on the Nterminus
SideChain
protein side chain atoms; that is all atoms except N, Cα , C, O, backbone amide hydrogens,
oxygens in C-terminus and hydrogens on the N-terminus
SideChain-H
protein side chain atoms excluding all hydrogens
Prot-Masses
protein atoms excluding dummy masses (as used in virtual site constructions of NH3 groups
and tryptophan side-chains), see also sec. 5.2.2; this group is only included when it differs
from the “Protein” group
Non-Protein
all non-protein atoms
DNA
all DNA atoms
RNA
all RNA atoms
Water
water molecules (names like SOL, WAT, HOH, etc.) See residuetypes.dat for a full
listing
non-Water
anything not covered by the Water group
Ion
any name matching an Ion entry in residuetypes.dat
Water_and_Ions
combination of the Water and Ions groups
molecule_name
for all residues/molecules which are not recognized as protein, DNA, or RNA; one group
per residue/molecule name is generated
Other
all atoms which are neither protein, DNA, nor RNA.
Empty groups will not be generated. Most of the groups only contain protein atoms. An atom is
considered a protein atom if its residue name is listed in the residuetypes.dat file and is
listed as a “Protein” entry. The process for determinding DNA, RNA, etc. is analogous. If you
need to modify these classifications, then you can copy the file from the library directory into your
working directory and edit the local copy.

206

Chapter 8. Analysis

8.1.2

Selections

gmx select
Currently, a few analysis tools support an extended concept of (dynamic) selections. There are
three main differences to traditional index groups:
• The selections are specified as text instead of reading fixed atom indices from a file, using
a syntax similar to VMD. The text can be entered interactively, provided on the command
line, or from a file.
• The selections are not restricted to atoms, but can also specify that the analysis is to be
performed on, e.g., center-of-mass positions of a group of atoms. Some tools may not
support selections that do not evaluate to single atoms, e.g., if they require information that
is available only for single atoms, like atom names or types.
• The selections can be dynamic, i.e., evaluate to different atoms for different trajectory
frames. This allows analyzing only a subset of the system that satisfies some geometric
criteria.
As an example of a simple selection, resname ABC and within 2 of resname DEF
selects all atoms in residues named ABC that are within 2 nm of any atom in a residue named
DEF.
Tools that accept selections can also use traditional index files similarly to older tools: it is possible
to give an .ndx file to the tool, and directly select a group from the index file as a selection, either
by group number or by group name. The index groups can also be used as a part of a more
complicated selection.
To get started, you can run gmx select with a single structure, and use the interactive prompt
to try out different selections. The tool provides, among others, output options -on and -ofpdb
to write out the selected atoms to an index file and to a .pdb file, respectively. This does not allow
testing selections that evaluate to center-of-mass positions, but other selections can be tested and
the result examined.
The detailed syntax and the individual keywords that can be used in selections can be accessed by
typing help in the interactive prompt of any selection-enabled tool, as well as with gmx help
selections. The help is divided into subtopics that can be accessed with, e.g., help syntax
/ gmx help selections syntax. Some individual selection keywords have extended help
as well, which can be accessed with, e.g., help keywords within.
The interactive prompt does not currently provide much editing capabilities. If you need them,
you can run the program under rlwrap.
For tools that do not yet support the selection syntax, you can use gmx select -on to generate
static index groups to pass to the tool. However, this only allows for a small subset (only the first
bullet from the above list) of the flexibility that fully selection-aware tools offer.
It is also possible to write your own analysis tools to take advantage of the flexibility of these
selections: see the template.cpp file in the share/gromacs/template directory of your
installation for an example.

8.2. Looking at your trajectory

207

Figure 8.1: The window of gmx view showing a box of water.

8.2

Looking at your trajectory

gmx view
Before analyzing your trajectory it is often informative to look at your trajectory first. GROMACS
comes with a simple trajectory viewer gmx view; the advantage with this one is that it does not
require OpenGL, which usually isn’t present on e.g. supercomputers. It is also possible to generate
a hard-copy in Encapsulated Postscript format (see Fig. 8.1). If you want a faster and more fancy
viewer there are several programs that can read the GROMACS trajectory formats – have a look
at our homepage (www.gromacs.org) for updated links.

8.3

General properties

gmx energy, gmx traj
To analyze some or all energies and other properties, such as total pressure, pressure tensor,
density, box-volume and box-sizes, use the program gmx energy. A choice can be made from
a list a set of energies, like potential, kinetic or total energy, or individual contributions, like
Lennard-Jones or dihedral energies.
The center-of-mass velocity, defined as
vcom =

N
1 X
m i vi
M i=1

(8.2)

with M = N
i=1 mi the total mass of the system, can be monitored in time by the program gmx
traj -com -ov. It is however recommended to remove the center-of-mass velocity every step
(see chapter 3)!
P

208

Chapter 8. Analysis
e
r+dr

θ+dθ

r+dr
r

Figure 8.2: Definition of slices in gmx rdf: A. gAB (r). B. gAB (r, θ). The slices are colored gray.
C. Normalization hρB ilocal . D. Normalization hρB ilocal, θ . Normalization volumes are colored
gray.

8.4

Radial distribution functions

gmx rdf
The radial distribution function (RDF) or pair correlation function gAB (r) between particles of
type A and B is defined in the following way:
gAB (r) =
=

hρB (r)i
hρB ilocal
NA X
NB
1
1 X
δ(rij − r)
hρB ilocal NA i∈A j∈B 4πr2

(8.3)

with hρB (r)i the particle density of type B at a distance r around particles A, and hρB ilocal the
particle density of type B averaged over all spheres around particles A with radius rmax (see
Fig. 8.2C).
Usually the value of rmax is half of the box length. The averaging is also performed in time. In
practice the analysis program gmx rdf divides the system into spherical slices (from r to r + dr,
see Fig. 8.2A) and makes a histogram in stead of the δ-function. An example of the RDF of
oxygen-oxygen in SPC water [85] is given in Fig. 8.3.
With gmx rdf it is also possible to calculate an angle dependent rdf gAB (r, θ), where the angle
θ is defined with respect to a certain laboratory axis e, see Fig. 8.2B.
gAB (r, θ) =
cos(θij ) =

NA X
NB
1
1 X
δ(rij − r)δ(θij − θ)
hρB ilocal, θ NA i∈A j∈B
2πr2 sin(θ)
rij · e
krij k kek

(8.4)
(8.5)

8.4. Radial distribution functions

209

2.5

g(r)

1.5

0.5

0
0

0.2

0.4

0.6

0.8

r (nm)

Figure 8.3: gOO (r) for Oxygen-Oxygen of SPC-water.

210

Chapter 8. Analysis

This gAB (r, θ) is useful for analyzing anisotropic systems. Note that in this case the normalization
hρB ilocal, θ is the average density in all angle slices from θ to θ+dθ up to rmax , so angle dependent,
see Fig. 8.2D.

8.5
8.5.1

Correlation functions
Theory of correlation functions

The theory of correlation functions is well established [113]. We describe here the implementation of the various correlation function flavors in the GROMACS code. The definition of the
autocorrelation function (ACF) Cf (t) for a property f (t) is:
Cf (t) = hf (ξ)f (ξ + t)iξ

(8.6)

where the notation on the right hand side indicates averaging over ξ, i.e. over time origins. It is
also possible to compute cross-correlation function from two properties f (t) and g(t):
Cf g (t) = hf (ξ)g(ξ + t)iξ

(8.7)

however, in GROMACS there is no standard mechanism to do this (note: you can use the xmgr
program to compute cross correlations). The integral of the correlation function over time is the
correlation time τf :
Z
∞

τf =

Cf (t)dt

(8.8)

In practice, correlation functions are calculated based on data points with discrete time intervals
∆t, so that the ACF from an MD simulation is:
Cf (j∆t) =

NX
−1−j
1
f (i∆t)f ((i + j)∆t)
N − j i=0

(8.9)

where N is the number of available time frames for the calculation. The resulting ACF is obviously
only available at time points with the same interval ∆t. Since, for many applications, it is necessary
to know the short time behavior of the ACF (e.g. the first 10 ps) this often means that we have to
save the data with intervals much shorter than the time scale of interest. Another implication of
eqn. 8.9 is that in principle we can not compute all points of the ACF with the same accuracy, since
we have N − 1 data points for Cf (∆t) but only 1 for Cf ((N − 1)∆t). However, if we decide to
compute only an ACF of length M ∆t, where M ≤ N/2 we can compute all points with the same
statistical accuracy:
X
1 N −1−M
Cf (j∆t) =
f (i∆t)f ((i + j)∆t)
(8.10)
M i=0
Here of course j < M . M is sometimes referred to as the time lag of the correlation function.
When we decide to do this, we intentionally do not use all the available points for very short time
intervals (j << M ), but it makes it easier to interpret the results. Another aspect that may not be
neglected when computing ACFs from simulation is that usually the time origins ξ (eqn. 8.6) are
not statistically independent, which may introduce a bias in the results. This can be tested using a

8.5. Correlation functions

211

block-averaging procedure, where only time origins with a spacing at least the length of the time
lag are included, e.g. using k time origins with spacing of M ∆t (where kM ≤ N ):
X
1 k−1
f (iM ∆t)f ((iM + j)∆t)
k i=0

Cf (j∆t) =

(8.11)

However, one needs very long simulations to get good accuracy this way, because there are many
fewer points that contribute to the ACF.

8.5.2

Using FFT for computation of the ACF

The computational cost for calculating an ACF according to eqn. 8.9 is proportional to N 2 , which
is considerable. However, this can be improved by using fast Fourier transforms to do the convolution [113].

8.5.3

Special forms of the ACF

There are some important varieties on the ACF, e.g. the ACF of a vector p:
Cp (t) =

Z ∞

Pn (cos 6 (p(ξ), p(ξ + t)) dξ

(8.12)

where Pn (x) is the nth order Legendre polynomial 1 . Such correlation times can actually be obtained experimentally using e.g. NMR or other relaxation experiments. GROMACS can compute
correlations using the 1st and 2nd order Legendre polynomial (eqn. 8.12). This can also be used
for rotational autocorrelation (gmx rotacf) and dipole autocorrelation (gmx dipoles).
In order to study torsion angle dynamics, we define a dihedral autocorrelation function as [164]:
C(t) = hcos(θ(τ ) − θ(τ + t))iτ

(8.13)

Note that this is not a product of two functions as is generally used for correlation functions, but
it may be rewritten as the sum of two products:
C(t) = hcos(θ(τ )) cos(θ(τ + t)) + sin(θ(τ )) sin(θ(τ + t))iτ

8.5.4

(8.14)

Some Applications

The program gmx velacc calculates the velocity autocorrelation function.
Cv (τ ) = hv i (τ ) · v i (0)ii∈A

(8.15)

The self diffusion coefficient can be calculated using the Green-Kubo relation [113]:
DA
1

1
=
3

Z ∞

P0 (x) = 1, P1 (x) = x, P2 (x) = (3x2 − 1)/2

hvi (t) · vi (0)ii∈A dt

(8.16)

212

Chapter 8. Analysis

which is just the integral of the velocity autocorrelation function. There is a widely-held belief
that the velocity ACF converges faster than the mean square displacement (sec. 8.7), which can
also be used for the computation of diffusion constants. However, Allen & Tildesley [113] warn
us that the long-time contribution to the velocity ACF can not be ignored, so care must be taken.
Another important quantity is the dipole correlation time. The dipole correlation function for
particles of type A is calculated as follows by gmx dipoles:
Cµ (τ ) = hµi (τ ) · µi (0)ii∈A

(8.17)

with µi = j∈i rj qj . The dipole correlation time can be computed using eqn. 8.8. For some
applications see [165].
P

The viscosity of a liquid can be related to the correlation time of the Pressure tensor P [166, 167].
gmx energy can compute the viscosity, but this is not very accurate [154], and actually the
values do not converge.

8.6

Curve fitting in GROMACS

8.6.1

Sum of exponential functions

Sometimes it is useful to fit a curve to an analytical function, for example in the case of autocorrelation functions with noisy tails. GROMACS is not a general purpose curve-fitting tool however
and therefore GROMACS only supports a limited number of functions. Table 8.1 lists the available options with the corresponding command-line options. The underlying routines for fitting use
the Levenberg-Marquardt algorithm as implemented in the lmfit package [168] (a bare-bones
version of which is included in GROMACS in which an option for error-weighted fitting was
implemented).
Table 8.1: Overview of fitting functions supported in (most) analysis tools that compute autocorrelation functions. The “Note” column describes properties of the output parameters.
Command
Functional form f (t)
Note
line option
exp
e−t/a0
aexp
a1 e−t/a0
exp_exp
a1 e−t/a0 + (1 − a1 )e−t/a2
a2 ≥ a0 ≥ 0
exp5
a1 e−t/a0 + a3 e−t/a2 + a4
a2 ≥ a0 ≥ 0
exp7
a1 e−t/a0 + a3 e−t/a2 + a5 e−t/a4 + a6
a4 ≥ a2 ≥ a0 ≥ 0
−t/a
0 + a e−t/a2 + a e−t/a4 + a e−t/a6 + a
exp9
a1 e
a
8
6 ≥ a4 ≥ a2 ≥ a0 ≥ 0
3
5
7

8.6.2

Error estimation

Under the hood GROMACS implements some more fitting functions, namely a function to estimate the error in time-correlated data due to Hess [154]:

τ2 −t/τ2
τ1 −t/τ1
ε2 (t) = ατ1 1 +
e
− 1 + (1 − α)τ2 1 +
e
−1
(8.18)
t
t

8.6. Curve fitting in GROMACS

213

where τ1 and τ2 are time constants (with τ2 ≥ τ1 ) and α usually is close to 1 (in the fitting
procedure it is enforced that 0 ≤ α ≤ 1). This is used in gmx analyze for error estimation
using
s
2(ατ1 + (1 − α)τ2 )
lim ε(t) = σ
(8.19)
t→∞
T
where σ is the standard deviation of the data set and T is the total simulation time [154].

8.6.3

Interphase boundary demarcation

In order to determine the position and width of an interface, Steen-Sæthre et al. fitted a density
profile to the following function
f (x) =

a0 + a1 a0 − a1
−
erf
2
2

x − a2
a23

(8.20)

where a0 and a1 are densities of different phases, x is the coordinate normal to the interface, a2 is
the position of the interface and a3 is the width of the interface [169]. This is implemented in gmx
densorder.

8.6.4

Transverse current autocorrelation function

In order to establish the transverse current autocorrelation function (useful for computing viscosity [170]) the following function is fitted:

f (x) = e−ν cosh(ων) +
with ν = x/(2a0 ) and ω =

8.6.5

sinh(ων)
ω

(8.21)

√
1 − a1 . This is implemented in gmx tcaf.

Viscosity estimation from pressure autocorrelation function

The viscosity is a notoriously difficult property to extract from simulations [154, 171]. It is in
principle possible to determine it by integrating the pressure autocorrelation function [166], however this is often hampered by the noisy tail of the ACF. A workaround to this is fitting the ACF
to the following function [172]:
βf

f (t)/f (0) = (1 − C)cos(ωt)e−(t/τf )

βs

+ Ce−(t/τs )

(8.22)

where ω is the frequency of rapid pressure oscillations (mainly due to bonded forces in molecular simulations), τf and βf are the time constant and exponent of fast relaxation in a stretchedexponential approximation, τs and βs are constants for slow relaxation and C is the pre-factor that
determines the weight between fast and slow relaxation. After a fit, the integral of the function
f (t) is used to compute the viscosity:
η=

Z ∞

kB T

f (t)dt

(8.23)

This equation has been applied to computing the bulk and shear viscosity using different elements
from the pressure tensor [173]. This is implemented in gmx viscosity.

214

Chapter 8. Analysis

Mean Square Displacement
-5

-1

D = 3.5027 (10 cm s )
4000.0

-5

-1

MSD (10 cm s )

3000.0

2000.0

1000.0

0.0
0.0

50.0

100.0

150.0

Time (ps)

Figure 8.4: Mean Square Displacement of SPC-water.

8.7

Mean Square Displacement

gmx msd
To determine the self diffusion coefficient DA of particles of type A, one can use the Einstein
relation [113]:
lim hkri (t) − ri (0)k2 ii∈A = 6DA t
(8.24)
t→∞

This mean square displacement and DA are calculated by the program gmx msd. Normally
an index file containing atom numbers is used and the MSD is averaged over these atoms. For
molecules consisting of more than one atom, ri can be taken as the center of mass positions of
the molecules. In that case, you should use an index file with molecule numbers. The results will
be nearly identical to averaging over atoms, however. The gmx msd program can also be used
for calculating diffusion in one or two dimensions. This is useful for studying lateral diffusion on
interfaces.
An example of the mean square displacement of SPC water is given in Fig. 8.4.

8.8

Bonds/distances, angles and dihedrals

gmx distance, gmx angle, gmx gangle
To monitor specific bonds in your modules, or more generally distances between points, the program gmx distance can calculate distances as a function of time, as well as the distribution of
the distance. With a traditional index file, the groups should consist of pairs of atom numbers, for
example:
[ bonds_1 ]
1
2
3
4

8.8. Bonds/distances, angles and dihedrals

215

A
φ=0

B
φ=0

Figure 8.5: Dihedral conventions: A. “Biochemical convention”. B. “Polymer convention”.
9

[ bonds_2 ]
12
13
Selections are also supported, with first two positions defining the first distance, second pair of
positions defining the second distance and so on. You can calculate the distances between CA and
CB atoms in all your residues (assuming that every residue either has both atoms, or neither) using
a selection such as:
name CA CB
The selections also allow more generic distances to be computed. For example, to compute the
distances between centers of mass of two residues, you can use:
com of resname AAA plus com of resname BBB
The program gmx angle calculates the distribution of angles and dihedrals in time. It also gives
the average angle or dihedral. The index file consists of triplets or quadruples of atom numbers:
[ angles ]
1
2
2
3
3
4

3
4
5

[ dihedrals ]
1
2
3
2
3
5

4
5

For the dihedral angles you can use either the “biochemical convention” (φ = 0 ≡ cis) or “polymer convention” (φ = 0 ≡ trans), see Fig. 8.5.
The program gmx gangle provides a selection-enabled version to compute angles. This tool can
also compute angles and dihedrals, but does not support all the options of gmx angle, such as
autocorrelation or other time series analyses. In addition, it supports angles between two vectors,
a vector and a plane, two planes (defined by 2 or 3 points, respectively), a vector/plane and the

216

Chapter 8. Analysis

Figure 8.6: Angle options of gmx gangle: A. Angle between two vectors. B. Angle between
two planes. C. Angle between a vector and the z axis. D. Angle between a vector and the normal of
a sphere. Also other combinations are supported: planes and vectors can be used interchangeably.
z axis, or a vector/plane and the normal of a sphere (determined by a single position). Also the
angle between a vector/plane compared to its position in the first frame is supported. For planes,
gmx gangle uses the normal vector perpendicular to the plane. See Fig. 8.6A, B, C) for the
definitions.

8.9

Radius of gyration and distances

gmx gyrate, gmx distance, gmx mindist, gmx mdmat, gmx pairdist, gmx
xpm2ps
To have a rough measure for the compactness of a structure, you can calculate the radius of gyration with the program gmx gyrate as follows:
2
i kri k mi

Rg =

!1
2

(8.25)

i mi

where mi is the mass of atom i and ri the position of atom i with respect to the center of mass of
the molecule. It is especially useful to characterize polymer solutions and proteins. The program
will also provide the radius of gyration around the coordinate axis (or, optionally, principal axes)
by only summing the radii components orthogonal to each axis, for instance
P

Rg,x = 

2 + r2
ri,y
i,z mi
i mi

 12


(8.26)

Sometimes it is interesting to plot the distance between two atoms, or the minimum distance between two groups of atoms (e.g.: protein side-chains in a salt bridge). To calculate these distances
between certain groups there are several possibilities:
• The distance between the geometrical centers of two groups can be calculated with the program
gmx distance, as explained in sec. 8.8.
• The minimum distance between two groups of atoms during time can be calculated with the
program gmx mindist. It also calculates the number of contacts between these groups
within a certain radius rmax .

8.10. Root mean square deviations in structure

217

90
80

t=0 ps

70
60
50
40
30
21
21

Residue Number
0

Distance (nm)

1.2

Figure 8.7: A minimum distance matrix for a peptide [174].
• gmx pairdist is a selection-enabled version of gmx mindist.
• To monitor the minimum distances between amino acid residues within a (protein) molecule,
you can use the program gmx mdmat. This minimum distance between two residues Ai
and Aj is defined as the smallest distance between any pair of atoms (i ∈ Ai , j ∈ Aj ). The
output is a symmetrical matrix of smallest distances between all residues. To visualize this
matrix, you can use a program such as xv. If you want to view the axes and legend or if
you want to print the matrix, you can convert it with xpm2ps into a Postscript picture, see
Fig. 8.7.
Plotting these matrices for different time-frames, one can analyze changes in the structure,
and e.g. forming of salt bridges.

8.10

Root mean square deviations in structure

gmx rms, gmx rmsdist
The root mean square deviation (RM SD) of certain atoms in a molecule with respect to a reference structure can be calculated with the program gmx rms by least-square fitting the structure
to the reference structure (t2 = 0) and subsequently calculating the RM SD (eqn. 8.27).
"

RM SD(t1 , t2 ) =

N
1 X
mi kri (t1 ) − ri (t2 )k2
M i=1

# 12

(8.27)

where M = N
i=1 mi and ri (t) is the position of atom i at time t. Note that fitting does not
have to use the same atoms as the calculation of the RM SD; e.g. a protein is usually fitted on
P

218

Chapter 8. Analysis

the backbone atoms (N,Cα ,C), but the RM SD can be computed of the backbone or of the whole
protein.
Instead of comparing the structures to the initial structure at time t = 0 (so for example a crystal
structure), one can also calculate eqn. 8.27 with a structure at time t2 = t1 − τ . This gives some
insight in the mobility as a function of τ . A matrix can also be made with the RM SD as a function
of t1 and t2 , which gives a nice graphical interpretation of a trajectory. If there are transitions in a
trajectory, they will clearly show up in such a matrix.
Alternatively the RM SD can be computed using a fit-free method with the program gmx rmsdist:
1



2
N X
N
1 X
2

RM SD(t) =
krij (t) − rij (0)k
N 2 i=1 j=1

(8.28)

where the distance rij between atoms at time t is compared with the distance between the same
atoms at time 0.

8.11

Covariance analysis

Covariance analysis, also called principal component analysis or essential dynamics [175], can
find correlated motions. It uses the covariance matrix C of the atomic coordinates:

1
2

Cij = Mii (xi − hxi i)Mjj (xj − hxj i)

(8.29)

where M is a diagonal matrix containing the masses of the atoms (mass-weighted analysis) or
the unit matrix (non-mass weighted analysis). C is a symmetric 3N × 3N matrix, which can be
diagonalized with an orthonormal transformation matrix R:
RT CR = diag(λ1 , λ2 , . . . , λ3N ) where λ1 ≥ λ2 ≥ . . . ≥ λ3N

(8.30)

The columns of R are the eigenvectors, also called principal or essential modes. R defines a
transformation to a new coordinate system. The trajectory can be projected on the principal modes
to give the principal components pi (t):
1

p(t) = RT M 2 (x(t) − hxi)

(8.31)

The eigenvalue λi is the mean square fluctuation of principal component i. The first few principal
modes often describe collective, global motions in the system. The trajectory can be filtered along
one (or more) principal modes. For one principal mode i this goes as follows:
1

xf (t) = hxi + M − 2 R∗i pi (t)

(8.32)

When the analysis is performed on a macromolecule, one often wants to remove the overall rotation and translation to look at the internal motion only. This can be achieved by least square fitting
to a reference structure. Care has to be taken that the reference structure is representative for the
ensemble, since the choice of reference structure influences the covariance matrix.

8.11. Covariance analysis

219

One should always check if the principal modes are well defined. If the first principal component
resembles a half cosine and the second resembles a full cosine, you might be filtering noise (see
below). A good way to check the relevance of the first few principal modes is to calculate the
overlap of the sampling between the first and second half of the simulation. Note that this can
only be done when the same reference structure is used for the two halves.
A good measure for the overlap has been defined in [176]. The elements of the covariance matrix
are proportional to the square of the displacement, so we need to take the square root of the matrix
to examine the extent of sampling. The square root can be calculated from the eigenvalues λi and
the eigenvectors, which are the columns of the rotation matrix R. For a symmetric and diagonallydominant matrix A of size 3N × 3N the square root can be calculated as:
1

2
A 2 = R diag(λ12 , λ22 , . . . , λ3N
) RT

(8.33)

It can be verified easily that the product of this matrix with itself gives A. Now we can define a
difference d between covariance matrices A and B as follows:
d(A, B) =
=
=

A2 − B 2

(8.34)
1

tr A + B − 2A 2 B 2

(8.35)


1
N q
N
N X

2 2
X
X

λA + λB − 2
λA λB RA · RB 
i

(8.36)

i=1 j=1

i=1

where tr is the trace of a matrix. We can now define the overlap s as:
d(A, B)
s(A, B) = 1 − √
trA + trB

(8.37)

The overlap is 1 if and only if matrices A and B are identical. It is 0 when the sampled subspaces
are completely orthogonal.
A commonly-used measure is the subspace overlap of the first few eigenvectors of covariance
matrices. The overlap of the subspace spanned by m orthonormal vectors w1 , . . . , wm with a
reference subspace spanned by n orthonormal vectors v1 , . . . , vn can be quantified as follows:
overlap(v, w) =

n X
m
1X
(vi · wj )2
n i=1 j=1

(8.38)

The overlap will increase with increasing m and will be 1 when set v is a subspace of set w.
The disadvantage of this method is that it does not take the eigenvalues into account. All eigenvectors are weighted equally, and when degenerate subspaces are present (equal eigenvalues), the
calculated overlap will be too low.
Another useful check is the cosine content. It has been proven that the the principal components
of random diffusion are cosines with the number of periods equal to half the principal component
index [177, 176]. The eigenvalues are proportional to the index to the power −2. The cosine
content is defined as:
2
T

Z T
0

iπt
cos
T

pi (t)dt

Z T
0

!−1

p2i (t)dt

(8.39)

220

Chapter 8. Analysis

D
α

Figure 8.8: Geometrical Hydrogen bond criterion.
When the cosine content of the first few principal components is close to 1, the largest fluctuations
are not connected with the potential, but with random diffusion.
The covariance matrix is built and diagonalized by gmx covar. The principal components and
overlap (and many more things) can be plotted and analyzed with gmx anaeig. The cosine
content can be calculated with gmx analyze.

8.12

Dihedral principal component analysis

gmx angle, gmx covar, gmx anaeig
Principal component analysis can be performed in dihedral space [178] using GROMACS. You
start by defining the dihedral angles of interest in an index file, either using gmx mk_angndx
or otherwise. Then you use the gmx angle program with the -or flag to produce a new .trr
file containing the cosine and sine of each dihedral angle in two coordinates, respectively. That
is, in the .trr file you will have a series of numbers corresponding to: cos(φ1 ), sin(φ1 ), cos(φ2 ),
sin(φ2 ), ..., cos(φn ), sin(φn ), and the array is padded with zeros, if necessary. Then you can use
this .trr file as input for the gmx covar program and perform principal component analysis
as usual. For this to work you will need to generate a reference file (.tpr, .gro, .pdb etc.)
containing the same number of “atoms” as the new .trr file, that is for n dihedrals you need
2n/3 atoms (rounded up if not an integer number). You should use the -nofit option for gmx
covar since the coordinates in the dummy reference file do not correspond in any way to the
information in the .trr file. Analysis of the results is done using gmx anaeig.

8.13

Hydrogen bonds

gmx hbond
The program gmx hbond analyzes
D and acceptors A. To determine if
Fig. 8.8:
r
α

the hydrogen bonds (H-bonds) between all possible donors
an H-bond exists, a geometrical criterion is used, see also
≤ rHB = 0.35 nm
≤ αHB = 30o

(8.40)

The value of rHB = 0.35 nm corresponds to the first minimum of the RDF of SPC water (see also
Fig. 8.3).
The program gmx hbond analyzes all hydrogen bonds existing between two groups of atoms
(which must be either identical or non-overlapping) or in specified donor-hydrogen-acceptor triplets,

8.13. Hydrogen bonds

221

D
H
(2)
(1)
O
H

A
H

(2)

Figure 8.9: Insertion of water into an H-bond. (1) Normal H-bond between two residues. (2)
H-bonding bridge via a water molecule.
in the following ways:
• Donor-Acceptor distance (r) distribution of all H-bonds
• Hydrogen-Donor-Acceptor angle (α) distribution of all H-bonds
• The total number of H-bonds in each time frame
• The number of H-bonds in time between residues, divided into groups n-n+i where n and
n+i stand for residue numbers and i goes from 0 to 6. The group for i = 6 also includes
all H-bonds for i > 6. These groups include the n-n+3, n-n+4 and n-n+5 H-bonds, which
provide a measure for the formation of α-helices or β-turns or strands.
• The lifetime of the H-bonds is calculated from the average over all autocorrelation functions
of the existence functions (either 0 or 1) of all H-bonds:
C(τ ) = hsi (t) si (t + τ )i

(8.41)

with si (t) = {0, 1} for H-bond i at time t. The integral of C(τ ) gives a rough estimate of
the average H-bond lifetime τHB :
τHB =

Z ∞

C(τ )dτ

(8.42)

Both the integral and the complete autocorrelation function C(τ ) will be output, so that more
sophisticated analysis (e.g. using multi-exponential fits) can be used to get better estimates
for τHB . A more complete analysis is given in ref. [179]; one of the more fancy option is
the Luzar and Chandler analysis of hydrogen bond kinetics [180, 181].
• An H-bond existence map can be generated of dimensions # H-bonds×# frames. The ordering is identical to the index file (see below), but reversed, meaning that the last triplet in the
index file corresponds to the first row of the existence map.
• Index groups are output containing the analyzed groups, all donor-hydrogen atom pairs
and acceptor atoms in these groups, donor-hydrogen-acceptor triplets involved in hydrogen
bonds between the analyzed groups and all solvent atoms involved in insertion.

222

8.14

Chapter 8. Analysis

Protein-related items

gmx do_dssp, gmx rama, gmx wheel
To analyze structural changes of a protein, you can calculate the radius of gyration or the minimum
residue distances over time (see sec. 8.9), or calculate the RMSD (sec. 8.10).
You can also look at the changing of secondary structure elements during your run. For this,
you can use the program gmx do_dssp, which is an interface for the commercial program
DSSP [182]. For further information, see the DSSP manual. A typical output plot of gmx do_dssp is given in Fig. 8.10.
One other important analysis of proteins is the so-called Ramachandran plot. This is the projection
of the structure on the two dihedral angles φ and ψ of the protein backbone, see Fig. 8.11.
To evaluate this Ramachandran plot you can use the program gmx rama. A typical output is
given in Fig. 8.12.
When studying α-helices it is useful to have a helical wheel projection of your peptide, to see
whether a peptide is amphipathic. This can be done using the gmx wheel program. Two examples are plotted in Fig. 8.13.

8.15

Interface-related items

gmx order, gmx density, gmx potential, gmx traj
When simulating molecules with long carbon tails, it can be interesting to calculate their average
orientation. There are several flavors of order parameters, most of which are related. The program
gmx order can calculate order parameters using the equation:
3
1
Sz = hcos2 θz i −
(8.43)
2
2
where θz is the angle between the z-axis of the simulation box and the molecular axis under
consideration. The latter is defined as the vector from Cn−1 to Cn+1 . The parameters Sx and Sy are
defined in the same way. The brackets imply averaging over time and molecules. Order parameters
can vary between 1 (full order along the interface normal) and −1/2 (full order perpendicular to
the normal), with a value of zero in the case of isotropic orientation.
The program can do two things for you. It can calculate the order parameter for each CH2 segment
separately, for any of three axes, or it can divide the box in slices and calculate the average value
of the order parameter per segment in one slice. The first method gives an idea of the ordering of
a molecule from head to tail, the second method gives an idea of the ordering as function of the
box length.
The electrostatic potential (ψ) across the interface can be computed from a trajectory by evaluating
the double integral of the charge density (ρ(z)):
ψ(z) − ψ(−∞) = −

Z z
−∞

Z z0
−∞

ρ(z 00 )dz 00 /0

(8.44)

where the position z = −∞ is far enough in the bulk phase such that the field is zero. With this
method, it is possible to “split” the total potential into separate contributions from lipid and water

Residue

8.15. Interface-related items

223

15
10
5
1
0

100

200

300

400

500

600

700

800

900

1000

Time (ps)
Coil

Bend

Turn

A-Helix

B-Bridge

Figure 8.10: Analysis of the secondary structure elements of a peptide in time.
N

H
R

C
O

Cα
φ

O
N

Figure 8.11: Definition of the dihedral angles φ and ψ of the protein backbone.

224

Chapter 8. Analysis

Ramachandran Plot
180.0

120.0

Psi

60.0

0.0

–60.0

–120.0

–180.0
–180.0 –120.0 –60.0

0.0
Phi

60.0

120.0

180.0

18
OPR

N-

GLU-2
5-

Figure 8.12: Ramachandran plot of a small protein.

LY
-2

-17+

ARG

PHE

LYS-24+

HPr-A

-22

HIS-15+

ALA

A-

VAL-23

27
+

THR-1

S-

AL
A

-26

-1

Figure 8.13: Helical wheel projection of the N-terminal helix of HPr.

8.15. Interface-related items

225

molecules. The program gmx potential divides the box in slices and sums all charges of the
atoms in each slice. It then integrates this charge density to give the electric field, which is in turn
integrated to give the potential. Charge density, electric field, and potential are written to xvgr
input files.
The program gmx traj is a very simple analysis program. All it does is print the coordinates,
velocities, or forces of selected atoms. It can also calculate the center of mass of one or more
molecules and print the coordinates of the center of mass to three files. By itself, this is probably
not a very useful analysis, but having the coordinates of selected molecules or atoms can be very
handy for further analysis, not only in interfacial systems.
The program gmx density calculates the mass density of groups and gives a plot of the density
against a box axis. This is useful for looking at the distribution of groups or atoms across the
interface.

226

Chapter 8. Analysis

Appendix A
Some implementation details
In this chapter we will present some implementation details. This is far from complete, but we
deemed it necessary to clarify some things that would otherwise be hard to understand.

A.1

Single Sum Virial in GROMACS

The virial Ξ can be written in full tensor form as:
Ξ = −

N
1 X
r ij ⊗ F ij
2 i
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