COLLECTIVE VARIABLES MODULE

Reference manual for GROMACS


Code version: 2023-12-05

Updated versions of this manual: [GROMACS] [LAMMPS] [NAMD] [Tinker-HP] [VMD]

Colvars logo
Alejandro Bernardin, Haochuan Chen, Jeffrey R. Comer, Giacomo Fiorin, Haohao Fu, Jérôme Hénin, Axel Kohlmeyer, Fabrizio Marinelli, Hubert Santuz, Joshua V. Vermaas, Andrew D. White

(PDF version)

Contents

 1 Overview
 2 Writing a Colvars configuration: a crash course
 3 Enabling and controlling the Colvars module in GROMACS
  3.1 Units in the Colvars module
  3.2 Running Colvars in GROMACS
  3.3 Configuration syntax used by the Colvars module
  3.4 Global keywords
  3.5 Input state file
   3.5.1 Restarting in GROMACS.
   3.5.2 Changing configuration upon restarting.
  3.6 Output files
  3.7 File formats
   3.7.1 Configuration and state files.
   3.7.2 Index (NDX) files
   3.7.3 XYZ coordinate files
   3.7.4 Grid files: multicolumn text format
   3.7.5 Output trajectory files
 4 Defining collective variables
  4.1 Choosing a function
  4.2 Distances
   4.2.1 distance: center-of-mass distance between two groups.
   4.2.2 distanceZ: projection of a distance vector on an axis.
   4.2.3 distanceXY: modulus of the projection of a distance vector on a plane.
   4.2.4 distanceVec: distance vector between two groups.
   4.2.5 distanceDir: distance unit vector between two groups.
   4.2.6 distanceInv: mean distance between two groups of atoms.
  4.3 Angles
   4.3.1 angle: angle between three groups.
   4.3.2 dipoleAngle: angle between two groups and dipole of a third group.
   4.3.3 dihedral: torsional angle between four groups.
   4.3.4 polarTheta: polar angle in spherical coordinates.
   4.3.5 polarPhi: azimuthal angle in spherical coordinates.
  4.4 Contacts
   4.4.1 coordNum: coordination number between two groups.
   4.4.2 selfCoordNum: coordination number between atoms within a group.
   4.4.3 hBond: hydrogen bond between two atoms.
  4.5 Collective metrics
   4.5.1 rmsd: root mean square displacement (RMSD) from reference positions.
   4.5.2 Advanced usage of the rmsd component.
   4.5.3 eigenvector: projection of the atomic coordinates on a vector.
   4.5.4 gyration: radius of gyration of a group of atoms.
   4.5.5 inertia: total moment of inertia of a group of atoms.
   4.5.6 dipoleMagnitude: dipole magnitude of a group of atoms.
   4.5.7 inertiaZ: total moment of inertia of a group of atoms around a chosen axis.
  4.6 Rotations
   4.6.1 orientation: orientation from reference coordinates.
   4.6.2 orientationAngle: angle of rotation from reference coordinates.
   4.6.3 orientationProj: cosine of the angle of rotation from reference coordinates.
   4.6.4 spinAngle: angle of rotation around a given axis.
   4.6.5 tilt: cosine of the rotation orthogonal to a given axis.
   4.6.6 eulerPhi: Roll angle from references coordinates.
   4.6.7 eulerTheta: Pitch angle from references coordinates.
   4.6.8 eulerPsi: Yaw angle from references coordinates.
  4.7 Raw data: building blocks for custom functions
   4.7.1 cartesian: vector of atomic Cartesian coordinates.
   4.7.2 distancePairs: set of pairwise distances between two groups.
  4.8 Geometric path collective variables
   4.8.1 gspath: progress along a path defined in atomic Cartesian coordinate space.
   4.8.2 gzpath: distance from a path defined in atomic Cartesian coordinate space.
   4.8.3 linearCombination: Helper CV to define a linear combination of other CVs
   4.8.4 customColvar: Helper CV to define a mathematical expression as CV from other CVs
   4.8.5 gspathCV: progress along a path defined in CV space.
   4.8.6 gzpathCV: distance from a path defined in CV space.
  4.9 Arithmetic path collective variables
   4.9.1 aspathCV: progress along a path defined in CV space.
   4.9.2 azpathCV: distance from a path defined in CV space.
   4.9.3 aspath: progress along a path defined in atomic Cartesian coordinate space.
   4.9.4 azpath: distance from a path defined in atomic Cartesian coordinate space.
  4.10 Dense neural network in CV space (MLCV)
  4.11 Shared keywords for all components
  4.12 Periodic components
  4.13 Non-scalar components
   4.13.1 Calculating total forces
  4.14 Linear and polynomial combinations of components
  4.15 Custom functions
  4.16 Defining grid parameters for a colvar
  4.17 Trajectory output
  4.18 Extended Lagrangian
  4.19 Multiple time-step variables
  4.20 Backward-compatibility
  4.21 Statistical analysis
 5 Selecting atoms
  5.1 Atom selection keywords
  5.2 Moving frame of reference.
  5.3 Treatment of periodic boundary conditions.
  5.4 Performance of a Colvars calculation based on group size.
 6 Biasing and analysis methods
  6.1 Thermodynamic integration
  6.2 Adaptive Biasing Force
   6.2.1 ABF requirements on collective variables
   6.2.2 Parameters for ABF
   6.2.3 Output files
   6.2.4 Multidimensional free energy surfaces
  6.3 Extended-system Adaptive Biasing Force (eABF)
   6.3.1 CZAR estimator of the free energy
  6.4 Metadynamics
   6.4.1 Treatment of the PMF boundaries
   6.4.2 Required metadynamics keywords
   6.4.3 Output files
   6.4.4 Performance optimization
   6.4.5 Ensemble-Biased Metadynamics
   6.4.6 Well-tempered metadynamics
   6.4.7 Multiple-walker metadynamics
  6.5 Harmonic restraints
   6.5.1 Moving restraints: steered molecular dynamics
   6.5.2 Moving restraints: umbrella sampling
   6.5.3 Changing force constant
  6.6 Computing the work of a changing restraint
  6.7 Harmonic wall restraints
  6.8 Linear restraints
  6.9 Adaptive Linear Bias/Experiment Directed Simulation
  6.10 Multidimensional histograms
   6.10.1 Defining grids for multidimensional histograms
   6.10.2 Output options for multi-dimensional histograms
   6.10.3 Histogramming vector variables
  6.11 Probability distribution-restraints
 7 Syntax changes from older versions
 8 Compilation notes

1 Overview

In molecular dynamics simulations, it is often useful to reduce the large number of degrees of freedom of a physical system into few parameters whose statistical distributions can be analyzed individually, or used to define biasing potentials to alter the dynamics of the system in a controlled manner. These have been called ‘order parameters', ‘collective variables', ‘(surrogate) reaction coordinates', and many other terms.

Here we use primarily the term ‘collective variable', often shortened to colvar, to indicate any differentiable function of atomic Cartesian coordinates, xi, with i between 1 and N, the total number of atoms:

ξ(t) = ξ(X(t)) = ξ (xi(t),xj(t),xk(t), ⁡),1 i,j,k ⁡ N (1)

This manual documents the collective variables module (Colvars), a software that provides an implementation for the functions ξ(X) with a focus on flexibility, robustness and high performance. The module is designed to perform multiple tasks concurrently during or after a simulation, the most common of which are:

Detailed explanations of the design of the Colvars module are provided in reference [1]. Please cite this reference whenever publishing work that makes use of this module, alongside any other publications for specific features being, according to the usage summary printed when running a Colvars-enabled MD simulation or analysis.

2 Writing a Colvars configuration: a crash course

The Colvars configuration is a plain text file or string that defines collective variables, biases, and general parameters of the Colvars module. It is passed to the module using back-end-specific commands documented in section 3.

Example: steering two groups of atoms away from each other. Now let us look at a complete, non-trivial configuration. Suppose that we want to run a steered MD experiment where a small molecule is pulled away from a protein binding site. In Colvars terms, this is done by applying a moving restraint to the distance between the two objects. The configuration will contain two blocks, one defining the distance variable (see section 4 and 4.2.1), and the other the moving harmonic restraint (6.5).

colvar {
  name dist
  distance {
    group1 { atomNumbersRange 42-55 }
    group2 { indexGroup C-alpha_15-30 }
  }
}

harmonic {
  colvars dist
  forceConstant 20.0
  centers 4.0         # initial distance
  targetCenters 15.0  # final distance
  targetNumSteps 500000
}

Reading this input in plain English: the variable here named dist consists in a distance function between the centers of two groups: the ligand (atoms 42 to 55) and the α-carbon atoms of residues 15 to 30 in the protein . To the “dist" variable, we apply a harmonic potential of force constant 20 kJ/mol/nm2, initially centered around a value of 4 nm, which will increase to 15 nm over 500,000 simulation steps.

The atom selection keywords are detailed in section 5.

Example: using multiple variables and multiple biasing/analysis methods together. A more complex example configuration is included below, showing how a variable may be constructed by combining multiple existing functions, and how multiple variables or multiple biases may be used concurrently. The colvar indicated below as “d" is defined as the difference between two distances (see 4.2): the first distance (d1) is taken between the center of mass of atoms 1 and 2 and that of atoms 3 to 5, the second (d2) between atom 7 and the center of mass of atoms 8 to 10 (see 5). The difference d = d1 d2 is obtained by multiplying the two by a coefficient C = +1 or C = 1, respectively (see 4.14). The colvar called “c" is the coordination number calculated between atoms 1 to 10 and atoms 11 to 20. A harmonic restraint (see 6.5) is applied to both d and c: to allow using the same force constant K, both d and c are scaled by their respective fluctuation widths wd and wc. The values of “c" are also recorded throughout the simulation as a joint 2-dimensional histogram (see 6.10).

colvar {
  # difference of two distances
  name d
  width 0.2 # estimated fluctuation width
  distance {
    componentCoeff 1.0
    group1 { atomNumbers 1 2 }
    group2 { atomNumbers 3 4 5 }
  }
  distance {
    componentCoeff -1.0
    group1 { atomNumbers 7 }
    group2 { atomNumbers 8 9 10 }
  }
}

colvar {
  name c
  coordNum {
    cutoff 6.0
    group1 { atomNumbersRange 1-10 }
    group2 { atomNumbersRange 11-20 }
    tolerance 1.0e-6
    pairListFrequency 1000
  }
}

harmonic {
  colvars d c
  centers 3.0 4.0
  forceConstant 5.0
}

histogram {
  colvars c
}

3 Enabling and controlling the Colvars module in GROMACS

Here, we document the syntax of the commands and parameters used to set up and use the Colvars module in GROMACS [2]. One of these parameters is the configuration file or the configuration text for the module itself, whose syntax is described in 3.3 and in the following sections.

3.1 Units in the Colvars module

The “internal units" of the Colvars module are the units in which values are expressed in the configuration file, and in which collective variable values, energies, etc. are expressed in the output and colvars trajectory files. Generally the Colvars module uses internally the same units as its back-end MD engine, with the exception of VMD, where different unit sets are supported to allow for easy setup, visualization and analysis of Colvars simulations performed with any simulation engine.

Note that angles are expressed in degrees, and derived quantities such as force constants are based on degrees as well. Some colvar components have default values, expressed in Ångström (Å) in this documentation. They are converted to the current length unit, if different from Å. Atomic coordinates read from XYZ files (and PDB files where applicable) are expected to be expressed in Ångström, no matter what unit system is in use by the back-end (GROMACS) or the Colvars Module. They are converted internally to the current length unit as needed. Note that force constants in harmonic and harmonicWalls biases (6.5) are rescaled according to the width parameter of colvars, so that they are formally in energy units, although if width is given its default value of 1.0, force constants are effectively expressed in kJ/mol/(colvar unit)2.

To avoid errors due to reading configuration files written in a different unit system, it can be specified within the input:

3.2 Running Colvars in GROMACS

Note: This section describes how to use Colvars in GROMACS versions until 2023, using patched GROMACS source code. This interface is now deprecated: whenever possible, please consider using GROMACS 2024 and later versions, which support and include Colvars natively.

To enable a Colvars-based calculation, just specify one or more Colvars configuration files with the -colvars command-line parameter to gmx mdrun:

gmx mdrun -s test.tpr -deffnm md -colvars variables.colvars.dat biases.colvars.dat 

Note that the standard extension of all files, including configuration files, have to be changed manually to end with “.dat" to satisfy GROMACS requirements on filenames.

Restarting from a previous simulation is done using the -colvars_restart parameter for the Colvars state file in conjunction with -cpi parameter for the checkpoint file (e.g. “state.cpt"). Altogether, the two files hold the required information to restart the simulation.

gmx mdrun -s test_restart.tpr -deffnm md -cpi state.cpt \
    -colvars config.colvars.dat -colvars_restart md.colvars.state.dat

Only checkpoint files generated by a Colvars-enabled GROMACS version are supported. It is also highly recommended to read checkpoint files that were written by the same major version of GROMACS (as is the case with unmodified GROMACS).

For the output files, the prefix of Colvars output files will use the -defnnm option of mdrun, or be the same as the GROMACS log file if the option is not set. Otherwise, the prefix will be output.

3.3 Configuration syntax used by the Colvars module

Configuration for the Colvars module is passed using an external file. Configuration lines follow the format “keyword value" or “keyword { ... }", where the keyword and its value must be separated by one or more space characters. The following formatting rules apply:

3.4 Global keywords

The following keywords are available in the global context of the Colvars configuration, i.e. they are not nested inside other keywords:

3.5 Input state file

Several of the sampling methods implemented in Colvars are time- or history-dependent, i.e. they work by accumulating data as a simulation progresses, and use these data to determine their biasing forces. If the simulation engine uses a checkpoint or restart file (as GROMACS and LAMMPS do), any data needed by Colvars are embedded into that file. Otherwise, a dedicated state file can be loaded into Colvars directly.

When a dedicated Colvars state file is used, it may be in either one of two formats:

In either format, the state file contains accumulated data as well as the step number at the end of the run. The step number read from a state file overrides any value that GROMACS provides, and will be incremented if the simulation proceeds. This means that the step number used internally by Colvars may not always match the step number reported by GROMACS.

3.5.1 Restarting in GROMACS.

For the GROMACS patched versions until 2023, Colvars can be restarted by providing the -colvars_restart option to gmx mdrun:

gmx mdrun -s test_restart.tpr -deffnm md -cpi state.cpt \
    -colvars config.colvars.dat -colvars_restart md.colvars.state.dat

3.5.2 Changing configuration upon restarting.

In some cases, it is useful to modify the configuration of variables or biases between consecutive runs, for example by adding or removing a restraint. Some special provisions will happen in that case. When a state file is loaded, no information is available about any newly added variable or bias, which will thus remain uninitialized until the first compute step. Conversely, any information that the state file may contain about variables or biases that are no longer defined will be silently ignored. Please note that these checks are performed based only on the names of variables and biases: it is your responsibility to ensure that these names have consistent definitions between runs.

The flexibility just described carries some limitations: namely, it is only supported when reading text-format Colvars state files. Instead, restarting from binary files after a configuration change will trigger an error. It is also important to remind that when switching to a different build of GROMACS, the binary format may change slightly, even if the release version is the same.

To work around the potential issues just described, a text-format Colvars state file should be loaded.

3.6 Output files

When the output prefix output is defined (in GROMACS, this is automatically set to the value of the -gflag of mdrun), the following output files are written during a simulation run:

3.7 File formats

This section summarizes the file formats of various files that Colvars may be reading or writing.

3.7.1 Configuration and state files.

Configuration files are text files that are generally read as input by GROMACS. Starting from version 2017-02-01, changes in newline encodings are handled transparently, i.e. it is possible to typeset a configuration file in Windows (CR-LF newlines) and then use it with Linux or macOS (LF-only newlines).

Formatted state files, although not written manually, follow otherwise the same text format as configuration files. Binary state files can only be read by the Colvars code itself.

3.7.2 Index (NDX) files

For atom selections that cannot be specified only by using internal Colvars keywords, external index files may also be used following the NDX format used in GROMACS:

[ group_1_name ]
  i1  i2  i3  i4  ...
  ...             ...  iN
[ group_2_name ]
  ...

where i1 through iN are 1-based indices. Each group name may not contain spaces or tabs: otherwise, a parsing error will be raised.

Multiple index files may be provided to Colvars, each using the keyword indexFile. Two index files may contain groups with the same names, however these must also represent identical atom selections, i.e. the same sequence of indices including order.

Note that although GROMACS reads index files natively, indexFile uses a Colvars internal function. Thus, index files loaded into GROMACS and Colvars do not need to coincide, but it is recommended that they do for simplicity.

Other than with GROMACS, an index group may also be generated from the VMD command-line interface, using the helper function write_index_group provided in the colvartools folder:

source colvartools/write_index_group.tcl
set sel [atomselect top "resname XXX and not hydrogen"]
write_index_group indexfile.ndx $sel "Ligand"

3.7.3 XYZ coordinate files

XYZ coordinate files are text files with the extension “.xyz". They are read by the Colvars module using an internal reader, and expect the following format:

N
Comment line
E 1 x1 y1 z1
E2 x2 y2 z2
EN xN yN zN

where N is the number of atomic coordinates in the file and Ei is the chemical element of the i-th atom. Because Ei is not used in Colvars, any string that does not contain tabs or spaces is acceptable.

Note: all XYZ coordinates are assumed to be expressed in Å units; it is advisable to obtain them from a PDB file, or using VMD and/or the Colvars Dashboard to avoid inconstencies with GROMACSńm units.

An XYZ file may contain either one of the following scenarios:

  1. The file contains as many coordinates as the atoms that they are being read for: all coordinates will be read from the file following the same order as the atoms appear in the selection generated using the keywords listed in section 5. (Note that the order is guaranteed only if a single type of selection keyword is used one or more times, and not guaranteed when different types of selection keywords are used.)
  2. The file contains more coordinates than needed, and it is assumed to contain coordinates for the entire system: only coordinates that match the numeric indices of the selected atoms are read, in order of increasing number.

XYZ-file coordinates are read directly by Colvars and stored internally as double-precision floating point numbers.

3.7.4 Grid files: multicolumn text format

Many simulation methods and analysis tools write files that contain functions of the collective variables tabulated on a grid (e.g. potentials of mean force or multidimensional histograms) for the purpose of analyzing results. Such files are produced by ABF (6.2), metadynamics (6.4), multidimensional histograms (6.10), as well as any restraint with optional thermodynamic integration support (6.1).

In some cases, these files may also be read as input of a new simulation. Suitable input files for this purpose are typically generated as output files of previous simulations, or directly by the user in the specific case of ensemble-biased metadynamics (6.4.5). This section explains the “multicolumn" format used by these files. For a multidimensional function f(ξ1, ξ2, …) the multicolumn grid format is defined as follows:

# Ncv
# min(ξ1) width(ξ1) npoints(ξ1) periodic(ξ1)
# min(ξ2) width(ξ2) npoints(ξ2) periodic(ξ2)
#
# min(ξNcv) width(ξNcv) npoints(ξNcv) periodic(ξNcv)
ξ11 ξ21 ξNcv1 f(ξ11, ξ21, …, ξNcv1)
ξ11 ξ21 ξNcv2 f(ξ11, ξ21, …, ξNcv2)

Lines beginning with the character “#" are the header of the file. Ncv is the number of collective variables sampled by the grid. For each variable ξi, min(ξi) is the lowest value sampled by the grid (i.e. the left-most boundary of the grid along ξi), width(ξi) is the width of each grid step along ξi, npoints(ξi) is the number of points and periodic(ξi) is a flag whose value is 1 or 0 depending on whether the grid is periodic along ξi. In most situations:

How the grid's boundaries affect the sequence of points depends on how the contents of the file were computed. In many cases, such as histograms and PMFs computed by metadynamics (6.4.5), the values of ξi in the first few columns correspond to the midpoints of the corresponding bins, i.e. ξ11 = min(ξi)+width(ξi)2. However, there is a slightly different format in PMF files computed by ABF (6.2) or other biases that use thermodynamic integration (6.1). In these cases, it is free-energy gradients that are accumulated on an (npoints)-long grid along each variable ξ: after these gradients are integrated, the resulting PMF is discretized on a slightly larger grid with (npoints+1) points along ξ (unless the interval is periodic). Therefore, the grid's outer edges extend by width(ξi)2 above and below the specified boundaries, so that for instance min(ξi) in the header appears to be shifted back by width(ξi)2 compared to what would be expected. Please keep this difference in mind when comparing PMFs computed by different methods.

After the header, the rest of the file contains values of the tabulated function f(ξ1, ξ2, …ξNcv), one for each line. The first Ncv columns contain values of ξ1, ξ2, …ξNcv and the last column contains the value of the function f. Points are sorted in ascending order with the fastest-changing values at the right (“C-style" order). Each sweep of the right-most variable ξNcv is terminated by an empty line. For two dimensional grid files, this allows quick visualization by programs such as GNUplot.

Example 1: multicolumn text file for a one-dimensional histogram with lowerBoundary = 15, upperBoundary = 48 and width = 0.1.

# 1
# 15 0.1 330 0
15.05 6.14012e-07
15.15 7.47644e-07
47.85 1.65944e-06
47.95 1.46712e-06

Example 2: multicolumn text file for a two-dimensional histogram of two dihedral angles (periodic interval with 6 bins):

# 2
# -180.0 6.0 30 1
# -180.0 6.0 30 1
-177.0 -177.0 8.97117e-06
-177.0 -171.0 1.53525e-06
-177.0 177.0 2.442956-06
-171.0 -177.0 2.04702e-05

3.7.5 Output trajectory files

The Colvars trajectory file (with a suffix .colvars.traj) is a plain text file (scientific notation with 14-digit precision) whose columns represent quantities such as colvar values, applied forces, or individual restraints' energies. Under most scenarios, plotting or analyzing this file is straightforward: for example, the following contains a variable “A" and the energy of a restraint “rA":

#       step   A                     E_rA
           0    1.42467449615693e+01  6.30982865292123e+02
         100    1.42282559728026e+01  6.20640585041317e+02

Occasionally, if the Colvars configuration is changed mid-run certain quantities may be added or removed, changing the column layout. Labels in comment lines can assist in such cases: for example, consider the trajectory above with the addition of a second variable, “B", after 10,000 steps:

#       step   A                     E_rA
           0    1.42467449615693e+01  6.30982865292123e+02
         100    1.42282559728026e+01  6.20640585041317e+02

#       step   A                     B                     E_rA              
       10000    1.38136915830229e+01  9.99574098859265e-01  4.11184644791030e+02
       10100    1.36437184346326e+01  9.99574091957314e-01  3.37726286543895e+02

Analyzing the above file with standard tools is possible, but laborious: a convenience script is provided for this and related purposes. It may be used either as a command-line tool or as a Python module:

>>> from plot_colvars_traj import Colvars_traj
>>> traj = Colvars_traj('test.colvars.traj')
>>> print(traj['A'].steps, traj['A'].values)
[    0   100  ...  10000 10100] [14.246745 14.228256 ... 13.813692 13.643718]
>>> print(traj['B'].steps, traj['B'].values)
[10000 10100] [0.999574  0.9995741]

4 Defining collective variables

A collective variable is defined by the keyword colvar followed by its configuration options contained within curly braces:

colvar {
  name xi
   <other options >
  function_name {
     <parameters >
     <atom selection >
  }
}

There are multiple ways of defining a variable:

Choosing a component (function) is the only parameter strictly required to define a collective variable. It is also highly recommended to specify a name for the variable:

4.1 Choosing a function

In this context, the function that computes a colvar is called a component. A component's choice and definition consists of including in the variable's configuration a keyword indicating the type of function (e.g. rmsd), followed by a definition block specifying the atoms involved (see 5) and any additional parameters (cutoffs, “reference" values, …). At least one component must be chosen to define a variable: if none of the keywords listed below is found, an error is raised.

The following components implement functions with a scalar value (i.e. a real number):

Some components do not return scalar, but vector values:

The types of components used in a colvar (scalar or not) determine the properties of that colvar, and particularly which biasing or analysis methods can be applied.

What if “X" is not listed? If a function type is not available on this list, it may be possible to define it as a polynomial superposition of existing ones (see 4.14), a custom function (see 4.15).

In the rest of this section, all available component types are listed, along with their physical units and their ranges of values, if limited. Such ranges are often used to define automatically default sampling intervals, for example by setting the parameters lowerBoundary and upperBoundary in the parent colvar.

For each type of component, the available configurations keywords are listed: when two components share certain keywords, the second component references to the documentation of the first one that uses that keyword. The very few keywords that are available for all types of components are listed in a separate section 4.11.

4.2 Distances

4.2.1 distance: center-of-mass distance between two groups.

The distance {...} block defines a distance component between the two atom groups, group1 and group2.

List of keywords (see also 4.14 for additional options):

The value returned is a positive number (in nm), ranging from 0 to the largest possible interatomic distance within the chosen boundary conditions (with PBCs, the minimum image convention is used unless the forceNoPBC option is set).

4.2.2 distanceZ: projection of a distance vector on an axis.

The distanceZ {...} block defines a distance projection component, which can be seen as measuring the distance between two groups projected onto an axis, or the position of a group along such an axis. The axis can be defined using either one reference group and a constant vector, or dynamically based on two reference groups. One of the groups can be set to a dummy atom to allow the use of an absolute Cartesian coordinate.

List of keywords (see also 4.14 for additional options):

This component returns a number (in nm) whose range is determined by the chosen boundary conditions. For instance, if the z axis is used in a simulation with periodic boundaries, the returned value ranges between bz2 and bz2, where bz is the box length along z (this behavior is disabled if forceNoPBC is set).

4.2.3 distanceXY: modulus of the projection of a distance vector on a plane.

The distanceXY {...} block defines a distance projected on a plane, and accepts the same keywords as the component distanceZ, i.e. main, ref, either ref2 or axis, and oneSiteTotalForce. It returns the norm of the projection of the distance vector between main and ref onto the plane orthogonal to the axis. The axis is defined using the axis parameter or as the vector joining ref and ref2 (see distanceZ above).

List of keywords (see also 4.14 for additional options):

4.2.4 distanceVec: distance vector between two groups.

The distanceVec {...} block defines a distance vector component, which accepts the same keywords as the component distance: group1, group2, and forceNoPBC. Its value is the 3-vector joining the centers of mass of group1 and group2.

List of keywords (see also 4.14 for additional options):

4.2.5 distanceDir: distance unit vector between two groups.

The distanceDir {...} block defines a distance unit vector component, which accepts the same keywords as the component distance: group1, group2, and forceNoPBC. It returns a 3-dimensional unit vector d = (dx,dy,dz), with |d| = 1.

List of keywords (see also 4.14 for additional options):

4.2.6 distanceInv: mean distance between two groups of atoms.

The distanceInv {...} block defines a generalized mean distance between two groups of atoms 1 and 2, where each distance is taken to the power n:

d1,2[n] = ( 1 N1N2i,jdijn)1n (2)

where dij is the distance between atoms i and j in groups 1 and 2 respectively, and n is an even integer.

List of keywords (see also 4.14 for additional options):

This component returns a number ranging from 0 to the largest possible distance within the chosen boundary conditions.

4.3 Angles

4.3.1 angle: angle between three groups.

The angle {...} block defines an angle, and contains the three blocks group1, group2 and group3, defining the three groups. It returns an angle (in degrees) within the interval [0 : 180].

List of keywords (see also 4.14 for additional options):

4.3.2 dipoleAngle: angle between two groups and dipole of a third group.

The dipoleAngle {...} block defines an angle, and contains the three blocks group1, group2 and group3, defining the three groups, being group1 the group where dipole is calculated. It returns an angle (in degrees) within the interval [0 : 180].

List of keywords (see also 4.14 for additional options):

4.3.3 dihedral: torsional angle between four groups.

The dihedral {...} block defines a torsional angle, and contains the blocks group1, group2, group3 and group4, defining the four groups. It returns an angle (in degrees) within the interval [180 : 180]. The Colvars module calculates all the distances between two angles taking into account periodicity. For instance, reference values for restraints or range boundaries can be defined by using any real number of choice.

List of keywords (see also 4.14 for additional options):

4.3.4 polarTheta: polar angle in spherical coordinates.

The polarTheta {...} block defines the polar angle in spherical coordinates, for the center of mass of a group of atoms described by the block atoms. It returns an angle (in degrees) within the interval [0 : 180]. To obtain spherical coordinates in a frame of reference tied to another group of atoms, use the fittingGroup (5.2) option within the atoms block. An example is provided in file examples/11_polar_angles.in of the Colvars public repository.

List of keywords (see also 4.14 for additional options):

4.3.5 polarPhi: azimuthal angle in spherical coordinates.

The polarPhi {...} block defines the azimuthal angle in spherical coordinates, for the center of mass of a group of atoms described by the block atoms. It returns an angle (in degrees) within the interval [180 : 180]. The Colvars module calculates all the distances between two angles taking into account periodicity. For instance, reference values for restraints or range boundaries can be defined by using any real number of choice. To obtain spherical coordinates in a frame of reference tied to another group of atoms, use the fittingGroup (5.2) option within the atoms block. An example is provided in file examples/11_polar_angles.in of the Colvars public repository.

List of keywords (see also 4.14 for additional options):

4.4 Contacts

4.4.1 coordNum: coordination number between two groups.

The coordNum {...} block defines a coordination number (or number of contacts), which calculates the function (1(dd0)n)(1(dd0)m), where d0 is the “cutoff" distance, and n and m are exponents that can control its long range behavior and stiffness [3]. This function is summed over all pairs of atoms in group1 and group2:

C(group1,group2) = igroup1jgroup21(|xixj|d0)n 1(|xixj|d0)m (3)

List of keywords (see also 4.14 for additional options):

This component returns a dimensionless number, which ranges from approximately 0 (all interatomic distances are much larger than the cutoff) to Ngroup1×Ngroup2 (all distances are less than the cutoff), or Ngroup1 if group2CenterOnly is used. For performance reasons, at least one of group1 and group2 should be of limited size or group2CenterOnly should be used: the cost of the loop over all pairs grows as Ngroup1×Ngroup2. Setting tolerance > 0 ameliorates this to some degree, although every pair is still checked to regenerate the pair list.

4.4.2 selfCoordNum: coordination number between atoms within a group.

The selfCoordNum {...} block defines a coordination number similarly to the component coordNum, but the function is summed over atom pairs within group1:

C(group1) = igroup1j>i 1(|xixj|d0)n 1(|xixj|d0)m (4)

The keywords accepted by selfCoordNum are a subset of those accepted by coordNum, namely group1 (here defining all of the atoms to be considered), cutoff, expNumer, and expDenom.

List of keywords (see also 4.14 for additional options):

This component returns a dimensionless number, which ranges from approximately 0 (all interatomic distances much larger than the cutoff) to Ngroup1×(Ngroup11)2 (all distances within the cutoff). For performance reasons, group1 should be of limited size, because the cost of the loop over all pairs grows as Ngroup12.

4.4.3 hBond: hydrogen bond between two atoms.

The hBond {...} block defines a hydrogen bond, implemented as a coordination number (eq. 3) between the donor and the acceptor atoms. Therefore, it accepts the same options cutoff (with a different default value of 3.3 Å), expNumer (with a default value of 6) and expDenom (with a default value of 8). Unlike coordNum, it requires two atom numbers, acceptor and donor, to be defined. It returns a dimensionless number, with values between 0 (acceptor and donor far outside the cutoff distance) and 1 (acceptor and donor much closer than the cutoff).

List of keywords (see also 4.14 for additional options):

4.5 Collective metrics

4.5.1 rmsd: root mean square displacement (RMSD) from reference positions.

The block rmsd {...} defines the root mean square replacement (RMSD) of a group of atoms with respect to a reference structure. For each set of coordinates {x1(t),x2(t), &ApplyFunction;xN(t)}, the colvar component rmsd calculates the optimal rotation U{xi(t)}{xi(ref)} that best superimposes the coordinates {xi(t)} onto a set of reference coordinates {xi(ref)}. Both the current and the reference coordinates are centered on their centers of geometry, xcog(t) and xcog(ref). The root mean square displacement is then defined as:

RMSD ( {xi (t)}, {xi(ref)}) = 1 Ni=1N |U (xi(t)xcog(t)) (xi(ref) xcog(ref))|2 (5)

The optimal rotation U{xi(t)}{xi(ref)} is calculated within the formalism developed in reference [4], which guarantees a continuous dependence of U{xi(t)}{xi(ref)} with respect to {xi(t)}.

List of keywords (see also 4.14 for additional options):

This component returns a positive real number (in nm).

4.5.2 Advanced usage of the rmsd component.

In the standard usage as described above, the rmsd component calculates a minimum RMSD, that is, current coordinates are optimally fitted onto the same reference coordinates that are used to compute the RMSD value. The fit itself is handled by the atom group object, whose parameters are automatically set by the rmsd component. For very specific applications, however, it may be useful to control the fitting process separately from the definition of the reference coordinates, to evaluate various types of non-minimal RMSD values. This can be achieved by setting the related options (refPositions, etc.) explicitly in the atom group block. This allows for the following non-standard cases:

  1. applying the optimal translation, but no rotation (rotateToReference off), to bias or restrain the shape and orientation, but not the position of the atom group;
  2. applying the optimal rotation, but no translation (centerToReference off), to bias or restrain the shape and position, but not the orientation of the atom group;
  3. disabling the application of optimal roto-translations, which lets the RMSD component describe the deviation of atoms from fixed positions in the laboratory frame: this allows for custom positional restraints within the Colvars module;
  4. fitting the atomic positions to different reference coordinates than those used in the RMSD calculation itself (by specifying refPositions or refPositionsFile within the atom group as well as within the rmsd block);
  5. applying the optimal rotation and/or translation from a separate atom group, defined through fittingGroup: the RMSD then reflects the deviation from reference coordinates in a separate, moving reference frame (see example in the section on fittingGroup).

4.5.3 eigenvector: projection of the atomic coordinates on a vector.

The block eigenvector {...} defines the projection of the coordinates of a group of atoms (or more precisely, their deviations from the reference coordinates) onto a vector in 3n, where n is the number of atoms in the group. The computed quantity is the total projection:

p ( {xi (t)}, {xi(ref)}) = i=1nv i (U(xi(t)xcog(t))(xi(ref) x cog(ref))), (6)

where, as in the rmsd component, U is the optimal rotation matrix, xcog(t) and xcog(ref) are the centers of geometry of the current and reference positions respectively, and vi are the components of the vector for each atom. Example choices for (vi) are an eigenvector of the covariance matrix (essential mode), or a normal mode of the system. It is assumed that &ApplyFunction;ivi = 0: otherwise, the Colvars module centers the vi automatically when reading them from the configuration.

List of keywords (see also 4.14 for additional options):

4.5.4 gyration: radius of gyration of a group of atoms.

The block gyration {...} defines the parameters for calculating the radius of gyration of a group of atomic positions {x1(t),x2(t), &ApplyFunction;xN(t)} with respect to their center of geometry, xcog(t):

Rgyr = 1 Ni=1N |xi(t)xcog(t)|2 (7)

This component must contain one atoms {...} block to define the atom group, and returns a positive number, expressed in nm.

List of keywords (see also 4.14 for additional options):

4.5.5 inertia: total moment of inertia of a group of atoms.

The block inertia {...} defines the parameters for calculating the total moment of inertia of a group of atomic positions {x1(t),x2(t), &ApplyFunction;xN(t)} with respect to their center of geometry, xcog(t):

I = i=1N |x i(t)xcog(t)|2 (8)

Note that all atomic masses are set to 1 for simplicity. This component must contain one atoms {...} block to define the atom group, and returns a positive number, expressed in nm2.

List of keywords (see also 4.14 for additional options):

4.5.6 dipoleMagnitude: dipole magnitude of a group of atoms.

The dipoleMagnitude {...} block defines the dipole magnitude of a group of atoms (norm of the dipole moment's vector), being atoms the group where dipole magnitude is calculated. It returns the magnitude in elementary charge e times nm.

List of keywords (see also 4.14 for additional options):

4.5.7 inertiaZ: total moment of inertia of a group of atoms around a chosen axis.

The block inertiaZ {...} defines the parameters for calculating the component along the axis e of the moment of inertia of a group of atomic positions {x1(t),x2(t), &ApplyFunction;xN(t)} with respect to their center of geometry, xcog(t):

Ie = i=1N ( (x i(t)xcog(t))e)2 (9)

Note that all atomic masses are set to 1 for simplicity. This component must contain one atoms {...} block to define the atom group, and returns a positive number, expressed in nm2.

List of keywords (see also 4.14 for additional options):

4.6 Rotations

The variables discussed in this section quantify the rotations of macromolecules (or other quasi-rigid objects) from a given set of reference coordinates to the current coordinates. Such rotations are computed following the same method used for best-fit RMSDs (see rmsd and fittingGroup). The underlying mathematical formalism is described in reference [4], and the implementation in reference [1].

The first of the functions described is the orientation, which describes the full rotation as a unit quaternion q = (q0,q1,q2,q3), i.e. 4 numbers with one constraint (3 degrees of freedom). The quaternion q is one of only two representations that are both complete and accurate, the other being a 3×3 unit matrix with 3 independent parameters. Although q is used internally in the Colvars module for features such as the rmsd function and the fittingGroup option, its direct use as a collective variable is more difficult, and mostly limited to fixed or moving restraints.

The two functions orientationAngle and orientationProj, with the latter being the cosine of the former, represent the amplitude of the full rotation described by q, regardless of the direction of its axis. As one-dimensional scalar variables, both orientationAngle and orientationProj are a much reduced simplification of the full rotation. However, they can be used in a variety of methods including both restraints and PMF computations.

A slightly more complete parametrization is achieved by decomposing the full rotation into the two parameters, tilt and spinAngle. These quantify the amplitudes of two independent sub-rotations away from a certain axis e, and around the same axis e, respectively. The axis e is chosen by the user, and is by default the Z axis: under that choice, tilt is equivalent to the sine of the Euler “pitch" angle 𝜃, and spinAngle to the sum of the other two angles, (ϕ +ψ). This parameterization is mathematically well defined for almost all full rotations, including small ones when the current coordinates are almost completely aligned with the reference ones. However, a mathematical singularity prevents using the spinAngle function near configurations where the value of tilt tilt is -1 (i.e. a -180 rotation around an axis orthogonal to e). For these reasons, tilt and spinAngle are useful when the allowed rotations are known to have approximately the same axis, and differ only in the magnitude of the corresponding angle. In this use case, spinAngle measures the angle of the sub-rotation around the chosen axis e, whereas tilt measures the dot product between e and the actual axis of the full rotation.

Lastly, the traditional Euler angles are also available as the functions eulerPhi, eulerTheta and eulerPsi. Altogether, these are sufficient to represent all three degrees of freedom of a full rotation. However, they also suffer from the potential “gimbal lock" problem, which emerges whenever 𝜃 ±90, which includes also the case where the full rotation is small. Under such conditions, the angles ϕ and ψ are both ill-defined and cannot be used as collective variables. For these reasons, it is highly recommended that Euler angles are used only in simulations where their range of applicability is known ahead of time, and excludes configurations where 𝜃 ±90 altogether.

4.6.1 orientation: orientation from reference coordinates.

The block orientation {...} returns the same optimal rotation used in the rmsd component to superimpose the coordinates {xi(t)} onto a set of reference coordinates {xi(ref)}. Such component returns a four dimensional vector q = (q0,q1,q2,q3), with &ApplyFunction;iqi2 = 1; this quaternion expresses the optimal rotation {xi(t)}{xi(ref)} according to the formalism in reference [4]. The quaternion (q0,q1,q2,q3) can also be written as (cos &ApplyFunction; (𝜃2),sin &ApplyFunction; (𝜃2)u), where 𝜃 is the angle and u the normalized axis of rotation; for example, a rotation of 90 around the z axis is expressed as “(0.707, 0.0, 0.0, 0.707)". The script quaternion2rmatrix.tcl provides Tcl functions for converting to and from a 4×4 rotation matrix in a format suitable for usage in VMD.

As for the component rmsd, the available options are atoms, refPositionsFile and refPositions.

Note: refPositions and refPositionsFile define the set of positions from which the optimal rotation is calculated, but this rotation is not applied to the coordinates of the atoms involved: it is used instead to define the variable itself.

List of keywords (see also 4.14 for additional options):

Tip: stopping the rotation of a protein. To stop the rotation of an elongated macromolecule in solution (and use an anisotropic box to save water molecules), it is possible to define a colvar with an orientation component, and restrain it through the harmonic bias around the identity rotation, (1.0, 0.0, 0.0, 0.0). Only the overall orientation of the macromolecule is affected, and not its internal degrees of freedom. The user should also take care that the macromolecule is composed by a single chain, or disable wrapAll otherwise.

4.6.2 orientationAngle: angle of rotation from reference coordinates.

The block orientationAngle {...} accepts the same base options as the component orientation: atoms, refPositions, refPositionsFile. The returned value is the angle of rotation 𝜃 between the current and the reference positions. This angle is expressed in degrees within the range [0:180].

List of keywords (see also 4.14 for additional options):

4.6.3 orientationProj: cosine of the angle of rotation from reference coordinates.

The block orientationProj {...} accepts the same base options as the component orientation: atoms, refPositions, refPositionsFile. The returned value is the cosine of the angle of rotation 𝜃 between the current and the reference positions. The range of values is [-1:1].

List of keywords (see also 4.14 for additional options):

4.6.4 spinAngle: angle of rotation around a given axis.

The complete rotation described by orientation can optionally be decomposed into two sub-rotations: one is a “spin" rotation around e, and the other a “tilt" rotation around an axis orthogonal to e. The component spinAngle measures the angle of the “spin" sub-rotation around e.

List of keywords (see also 4.14 for additional options):

The component spinAngle returns an angle (in degrees) within the periodic interval [180 : 180].

Note: the value of spinAngle is a continuous function almost everywhere, with the exception of configurations with the corresponding “tilt" angle equal to 180 (i.e. the tilt component is equal to 1): in those cases, spinAngle is undefined. If such configurations are expected, consider defining a tilt colvar using the same axis e, and restraining it with a lower wall away from 1.

4.6.5 tilt: cosine of the rotation orthogonal to a given axis.

The component tilt measures the cosine of the angle of the “tilt" sub-rotation, which combined with the “spin" sub-rotation provides the complete rotation of a group of atoms. The cosine of the tilt angle rather than the tilt angle itself is implemented, because the latter is unevenly distributed even for an isotropic system: consider as an analogy the angle 𝜃 in the spherical coordinate system. The component tilt relies on the same options as spinAngle, including the definition of the axis e. The values of tilt are real numbers in the interval [1 : 1]: the value 1 represents an orientation fully parallel to e (tilt angle = 0), and the value 1 represents an anti-parallel orientation.

List of keywords (see also 4.14 for additional options):

4.6.6 eulerPhi: Roll angle from references coordinates.

Assuming the axes of the original frame are denoted as x, y, z and the axes of the rotated frame as X, Y, Z, the line of nodes, N, can be defined as the intersection of the plane xy and XY. The axis perpendicular to N and z is defined as P. In this case, as illustrated in the figure below, the complete rotation described by orientation can optionally be decomposed into three Euler angles:

PIC

Although Euler angles are more straightforward to use than quaternions, they are also potentially subject to the “gimbal lock" problem:
https://en.wikipedia.org/wiki/Gimbal_lock
which arises whenever 𝜃 ±90, including the common case when the simulated coordinates are near the reference coordinates. Therefore, a safe use of Euler angles as collective variables requires the use of restraints to avoid such singularities, such as done in reference [6] and in the protein-ligand binding NAMD tutorial.

The eulerPhi component accepts exactly the same options as orientation, and measures the rotation angle from the x axis to the N axis. This angle is expressed in degrees within the periodic range [180 : 180].

List of keywords (see also 4.14 for additional options):

4.6.7 eulerTheta: Pitch angle from references coordinates.

This component accepts exactly the same options as orientation, and measures the rotation angle from the P axis to the Z axis. This angle is expressed in degrees within the range [90 : 90].

Warning: When this angle reaches 90 or 90, the definition of orientation by euler angles suffers from the gimbal lock issue. You may need to apply a restraint to keep eulerTheta away from the singularities.

List of keywords (see also 4.14 for additional options):

4.6.8 eulerPsi: Yaw angle from references coordinates.

This component accepts exactly the same options as orientation, and measures the rotation angle from the N axis to the X axis. This angle is expressed in degrees within the periodic range [180 : 180].

List of keywords (see also 4.14 for additional options):

4.7 Raw data: building blocks for custom functions

4.7.1 cartesian: vector of atomic Cartesian coordinates.

The cartesian {...} block defines a component returning a flat vector containing the Cartesian coordinates of all participating atoms, in the order (x1,y1,z1,,xn,yn,zn).

List of keywords (see also 4.14 for additional options):

4.7.2 distancePairs: set of pairwise distances between two groups.

The distancePairs {...} block defines a N1 ×N2-dimensional variable that includes all mutual distances between the atoms of two groups.

List of keywords (see also 4.14 for additional options):

This component returns a N1 ×N2-dimensional vector of numbers, each ranging from 0 to the largest possible distance within the chosen boundary conditions.

4.8 Geometric path collective variables

The geometric path collective variables define the progress along a path, s, and the distance from the path, z. These CVs are proposed by Leines and Ensing[7] , which differ from that[8] proposed by Branduardi et al., and utilize a set of geometric algorithms. The path is defined as a series of frames in the atomic Cartesian coordinate space or the CV space. s and z are computed as

s = m M ± 1 2M ((v 1 v 3 )2 |v 3 |2 (|v 1 |2 |v 2 |2 )(v1 v3) |v3|2 1) (10)
z = (v 1 + 1 2 ((v 1 v 3 )2 |v 3 |2 (|v 1 |2 |v 2 |2 )(v1 v3) |v3|2 1)v4)2 (11)

where v1 = smz is the vector connecting the current position to the closest frame, v2 = z sm1 is the vector connecting the second closest frame to the current position, v3 = sm+1 sm is the vector connecting the closest frame to the third closest frame, and v4 = smsm1 is the vector connecting the second closest frame to the closest frame. m and M are the current index of the closest frame and the total number of frames, respectively. If the current position is on the left of the closest reference frame, the ± in s turns to the positive sign. Otherwise it turns to the negative sign.

The equations above assume: (i) the frames are equidistant and (ii) the second and the third closest frames are neighbouring to the closest frame. When these assumptions are not satisfied, this set of path CV should be used carefully.

4.8.1 gspath: progress along a path defined in atomic Cartesian coordinate space.

In the gspath {...} and the gzpath {...} block all vectors, namely z and sk are defined in atomic Cartesian coordinate space. More specifically, z = [r1,,rn], where ri is the i-th atom specified in the atoms block. sk = [rk,1,,rk,n], where rk,i means the i-th atom of the k-th reference frame.

List of keywords (see also 4.14 for additional options):

4.8.2 gzpath: distance from a path defined in atomic Cartesian coordinate space.

List of keywords (see also 4.14 for additional options):

The usage of gzpath and gspath is illustrated below:

colvar {
  # Progress along the path
  name gs
  # The path is defined by 5 reference frames (from string-00.pdb to string-04.pdb)
  # Use atomic coordinate from atoms 1, 2 and 3 to compute the path
  gspath {
    atoms {atomnumbers { 1 2 3 }}
    refPositionsFile1 string-00.pdb
    refPositionsFile2 string-01.pdb
    refPositionsFile3 string-02.pdb
    refPositionsFile4 string-03.pdb
    refPositionsFile5 string-04.pdb
  }
}
colvar {
  # Distance from the path
  name gz
  # The path is defined by 5 reference frames (from string-00.pdb to string-04.pdb)
  # Use atomic coordinate from atoms 1, 2 and 3 to compute the path
  gzpath {
    atoms {atomnumbers { 1 2 3 }}
    refPositionsFile1 string-00.pdb
    refPositionsFile2 string-01.pdb
    refPositionsFile3 string-02.pdb
    refPositionsFile4 string-03.pdb
    refPositionsFile5 string-04.pdb
  }
}

4.8.3 linearCombination: Helper CV to define a linear combination of other CVs

This is a helper CV which can be defined as a linear combination of other CVs. It maybe useful when you want to define the gspathCV {...} and the gzpathCV {...} as combinations of other CVs. Total forces (required by ABF) of this CV are not available.

4.8.4 customColvar: Helper CV to define a mathematical expression as CV from other CVs

This is a helper CV which can be defined as a mathematical expression (see 4.15) of other CVs by using customFunction. Currently only the scalar type of customFunction is supported. If customFunction is not provided, this component falls back to linearCombination. It maybe useful when you want to define the gspathCV {...}, the gzpathCV {...} and NeuralNetwork {...} as combinations of other CVs. Total forces (required by ABF) of this CV are not available.

4.8.5 gspathCV: progress along a path defined in CV space.

In the gspathCV {...} and the gzpathCV {...} block all vectors, namely z and sk are defined in CV space. More specifically, z = [ξ1,,ξn], where ξi is the i-th CV. sk = [ξk,1,,ξk,n], where ξk,i means the i-th CV of the k-th reference frame. It should be note that these two CVs requires the pathFile option, which specifies a path file. Each line in the path file contains a set of space-seperated CV value of the reference frame. The sequence of reference frames matches the sequence of the lines.

List of keywords (see also 4.14 for additional options):

4.8.6 gzpathCV: distance from a path defined in CV space.

List of keywords (see also 4.14 for additional options):

The usage of gzpathCV and gspathCV is illustrated below:

colvar {
  # Progress along the path
  name gs
  # Path defined by the CV space of two dihedral angles
  gspathCV {
    pathFile ./path.txt
    dihedral {
      name 001
      group1 {atomNumbers {5}}
      group2 {atomNumbers {7}}
      group3 {atomNumbers {9}}
      group4 {atomNumbers {15}}
    }
    dihedral {
      name 002
      group1 {atomNumbers {7}}
      group2 {atomNumbers {9}}
      group3 {atomNumbers {15}}
      group4 {atomNumbers {17}}
    }
  }
}

colvar {
  # Distance from the path
  name gz
  gzpathCV {
    pathFile ./path.txt
    dihedral {
      name 001
      group1 {atomNumbers {5}}
      group2 {atomNumbers {7}}
      group3 {atomNumbers {9}}
      group4 {atomNumbers {15}}
    }
    dihedral {
      name 002
      group1 {atomNumbers {7}}
      group2 {atomNumbers {9}}
      group3 {atomNumbers {15}}
      group4 {atomNumbers {17}}
    }
  }
}

4.9 Arithmetic path collective variables

The arithmetic path collective variable in CV space uses a similar formula as the one proposed by Branduardi[8] et al., except that it computes s and z in CV space instead of RMSDs in Cartesian space. Moreover, this implementation allows different coefficients for each CV components as described in [9]. Assuming a path is composed of N reference frames and defined in an M-dimensional CV space, then the equations of s and z of the path are

s = 1 N 1 i=0N1iexp &ApplyFunction; (λj=1Mcj2 (xjxi,j)2) i=0N1exp &ApplyFunction; (λj=1Mcj2 (xjxi,j)2) (12)
z = 1 λln &ApplyFunction; (i=0N1exp &ApplyFunction; (λ j=1Mc j2 (x jxi,j)2)) (13)

where cj is the coefficient(weight) of the j-th CV, xi,j is the value of j-th CV of i-th reference frame and xj is the value of j-th CV of current frame. λ is a parameter to smooth the variation of s and z. It should be noted that the index i ranges from 0 to N 1, and the definition of s is normalized by 1(N 1). Consequently, the scope of s is [0 : 1].

4.9.1 aspathCV: progress along a path defined in CV space.

This colvar component computes the s variable.

List of keywords (see also 4.14 for additional options):

4.9.2 azpathCV: distance from a path defined in CV space.

This colvar component computes the z variable. Options are the same as in 4.9.1.

The usage of azpathCV and aspathCV is illustrated below:

colvar {
  # Progress along the path
  name as
  # Path defined by the CV space of two dihedral angles
  aspathCV {
    pathFile ./path.txt
    weights {1.0 1.0}
    lambda 0.005
    dihedral {
      name 001
      group1 {atomNumbers {5}}
      group2 {atomNumbers {7}}
      group3 {atomNumbers {9}}
      group4 {atomNumbers {15}}
    }
    dihedral {
      name 002
      group1 {atomNumbers {7}}
      group2 {atomNumbers {9}}
      group3 {atomNumbers {15}}
      group4 {atomNumbers {17}}
    }
  }
}

colvar {
  # Distance from the path
  name az
  azpathCV {
    pathFile ./path.txt
    weights {1.0 1.0}
    lambda 0.005
    dihedral {
      name 001
      group1 {atomNumbers {5}}
      group2 {atomNumbers {7}}
      group3 {atomNumbers {9}}
      group4 {atomNumbers {15}}
    }
    dihedral {
      name 002
      group1 {atomNumbers {7}}
      group2 {atomNumbers {9}}
      group3 {atomNumbers {15}}
      group4 {atomNumbers {17}}
    }
  }
}

4.9.3 aspath: progress along a path defined in atomic Cartesian coordinate space.

This CV computes a special case of Eq. 12, where xj is the j-th atomic position, xi,j is the j-th atomic position of the i-th reference frame. The subtraction xjxi,j is actually calculated as xjRixi,j, where Ri is a 3x3 rotation matrix that minimizes the RMSD between the current atomic positions of simulation and the i-th reference frame. Bold xj is used since an atomic position is a vector.

List of keywords (see also 4.14 for additional options):

4.9.4 azpath: distance from a path defined in atomic Cartesian coordinate space.

Similar to aspath, this CV computes a special case of Eq. 13, and shares the same options as aspath.

The usage of azpath and aspath is illustrated below:

colvar {
  # Progress along the path
  name as
  # The path is defined by 5 reference frames (from string-00.pdb to string-04.pdb)
  # Use atomic coordinate from atoms 1, 2 and 3 to compute the path
  aspath {
    atoms {atomnumbers { 1 2 3 }}
    refPositionsFile1 string-00.pdb
    refPositionsFile2 string-01.pdb
    refPositionsFile3 string-02.pdb
    refPositionsFile4 string-03.pdb
    refPositionsFile5 string-04.pdb
  }
}
colvar {
  # Distance from the path
  name az
  # The path is defined by 5 reference frames (from string-00.pdb to string-04.pdb)
  # Use atomic coordinate from atoms 1, 2 and 3 to compute the path
  azpath {
    atoms {atomnumbers { 1 2 3 }}
    refPositionsFile1 string-00.pdb
    refPositionsFile2 string-01.pdb
    refPositionsFile3 string-02.pdb
    refPositionsFile4 string-03.pdb
    refPositionsFile5 string-04.pdb
  }
}

4.10 Dense neural network in CV space (MLCV)

This colvar component computes a non-linear combination of other scalar colvar components, where the transformation is defined by a dense neural network.[10] The network can be optimized using any framework, and its parameters are provided to Colvars in plain text files, as detailed below. An example Python script to export the parameters of a TensorFlow model is provided in colvartools/extract_weights_biases.py in the Colvars source tree.


Dense neural network

Figure 1: Graphical representation of an example dense neural network with two hidden layers y1 and y2. Both the input layer x and the output layer z have two nodes. The input nodes can be any existing scalar CVs.

The output of the j-th node of a k-th layer that has Nk nodes is computed by

yk,j = fk (i=1Nk1w (k,j),(k1,i)yk1,i+bk,j), (14)

where fk is the activation function of the k-th layer, w(k,j),(k1,i) is the weight of j-th node with respect to the i-th output of previous layer, and bk,j is the bias of j-th node of k-th layer.

List of keywords (see also 4.14 for additional options):

An example of configuration using NeuralNetwork is shown below:

colvar {
  # Define a neural network with 2 layers
  # The inputs are two torsion angles
  # and the first node at the output layer is used as the final CV
  name nn_output_1
  NeuralNetwork {
    output_component 0
    layer1_WeightsFile dense_1_weights.txt
    layer1_BiasesFile dense_1_biases.txt
    layer1_activation tanh
    layer2_WeightsFile dense_2_weights.txt
    layer2_BiasesFile dense_2_biases.txt
    layer2_activation tanh
    # The component coefficient is used for normalization
    componentCoeff 180.0
    dihedral {
      name 001
      # normalization factor 1.0/180.0
      componentCoeff 0.00555555555555555556
      group1 {atomNumbers {5}}
      group2 {atomNumbers {7}}
      group3 {atomNumbers {9}}
      group4 {atomNumbers {15}}
    }
    dihedral {
      name 002
      # normalization factor 1.0/180.0
      componentCoeff 0.00555555555555555556
      group1 {atomNumbers {7}}
      group2 {atomNumbers {9}}
      group3 {atomNumbers {15}}
      group4 {atomNumbers {17}}
    }
  }
}

4.11 Shared keywords for all components

The following options can be used for any of the above colvar components in order to obtain a polynomial combination.

4.12 Periodic components

The following components returns real numbers that lie in a periodic interval:

In certain conditions, distanceZ can also be periodic, namely when periodic boundary conditions (PBCs) are defined in the simulation and distanceZ's axis is parallel to a unit cell vector.

In addition, a custom scalar colvar may be periodic depending on its user-defined expression. It will only be treated as such by the Colvars module if the period is specified using the period keyword, while wrapAround is optional.

The following keywords can be used within periodic components, or within custom variables (4.15)).

Internally, all differences between two values of a periodic colvar follow the minimum image convention: they are calculated based on the two periodic images that are closest to each other.

Note: linear or polynomial combinations of periodic components (see 4.14) may become meaningless when components cross the periodic boundary. Use such combinations carefully: estimate the range of possible values of each component in a given simulation, and make use of wrapAround to limit this problem whenever possible.

4.13 Non-scalar components

When one of the following components are used, the defined colvar returns a value that is not a scalar number:

The distance between two 3-dimensional unit vectors is computed as the angle between them. The distance between two quaternions is computed as the angle between the two 4-dimensional unit vectors: because the orientation represented by q is the same as the one represented by q, distances between two quaternions are computed considering the closest of the two symmetric images.

Non-scalar components carry the following restrictions:

Note: while these restrictions apply to individual colvars based on non-scalar components, no limit is set to the number of scalar colvars. To compute multi-dimensional histograms and PMFs, use sets of scalar colvars of arbitrary size.

4.13.1 Calculating total forces

In addition to the restrictions due to the type of value computed (scalar or non-scalar), a final restriction can arise when calculating total force (outputTotalForce option or application of a abf bias). total forces are available currently only for the following components: distance, distanceZ, distanceXY, angle, dihedral, rmsd, eigenvector and gyration.

4.14 Linear and polynomial combinations of components

To extend the set of possible definitions of colvars ξ(r), multiple components qi(r) can be summed with the formula:

ξ (r) =ici[qi(r)]ni (15)

where each component appears with a unique coefficient ci (1.0 by default) the positive integer exponent ni (1 by default).

Any set of components can be combined within a colvar, provided that they return the same type of values (scalar, unit vector, vector, or quaternion). By default, the colvar is the sum of its components. Linear or polynomial combinations (following equation (15)) can be obtained by setting the following parameters, which are common to all components:

Example: To define the average of a colvar across different parts of the system, simply define within the same colvar block a series of components of the same type (applied to different atom groups), and assign to each component a componentCoeff of 1N.

4.15 Custom functions

Collective variables may be defined by specifying a custom function of multiple components, i.e. an analytical expression that is more general than the linear combinations described in 4.14. Such expression is parsed and calculated by Lepton, the lightweight expression parser written by Peter Eastman (https://simtk.org/projects/lepton) that produces efficient evaluation routines for both the expression and its derivatives. Although Lepton is generally available in most applications and builds where Colvars is included, it is best to check section 8 to confirm.

The expression may use the collective variable components as variables, referred to by their user-defined name. Scalar elements of vector components may be accessed by appending a 1-based index to their name, as in the example below. When implementing generic functions of Cartesian coordinates rather than functions of existing components, the cartesian component may be particularly useful. A scalar-valued custom variable may be manually defined as periodic by providing the keyword period, and the optional keyword wrapAround, with the same meaning as in periodic components (see 4.12 for details). A vector variable may be defined by specifying the customFunction parameter several times: each expression defines one scalar element of the vector colvar, as in this example:

colvar {
  name custom

  # A 2-dimensional vector function of a scalar x and a 3-vector r
  customFunction cos(x) * (r1 + r2 + r3)
  customFunction sqrt(r1 * r2)

  distance {
    name x
    group1 { atomNumbers 1 }
    group2 { atomNumbers 50 }
  }
  distanceVec {
    name r
    group1 { atomNumbers 10 11 12 }
    group2 { atomNumbers 20 21 22 }
  }
}

Numeric constants may be given in either decimal or exponential form (e.g. 3.12e-2). An expression may be followed by definitions for intermediate values that appear in the expression, separated by semicolons. For example, the expression:
a^2 + a*b + b^2; a = a1 + a2; b = b1 + b2
is exactly equivalent to:
(a1 + a2)^2 + (a1 + a2) * (b1 + b2) + (b1 + b2)^2.
The definition of an intermediate value may itself involve other intermediate values. All uses of a value must appear before that value's definition.

Lepton supports the usual arithmetic operators +, -, *, /, and ^ (power), as well as the following functions:



sqrt Square root
exp Exponential
log Natural logarithm
erf Error function
erfc Complementary error function


sin Sine (angle in radians)
cos Cosine (angle in radians)
sec Secant (angle in radians)
csc Cosecant (angle in radians)
tan Tangent (angle in radians)
cot Cotangent (angle in radians)
asin Inverse sine (in radians)
acos Inverse cosine (in radians)
atan Inverse tangent (in radians)
atan2 Two-argument inverse tangent (in radians)


sinh Hyperbolic sine
cosh Hyperbolic cosine
tanh Hyperbolic tangent


abs Absolute value
floor Floor
ceil Ceiling
min Minimum of two values
max Maximum of two values
delta delta(x) = 1 if x = 0, 0 otherwise
step step(x) = 0 if x < 0, 1 if x >= 0
select select(x,y,z) = z if x = 0, y otherwise


4.16 Defining grid parameters for a colvar

Many algorithms require the definition of two boundaries and a bin width for each colvar, which are necessary to compute discrete “states" for a collective variable's otherwise continuous values. The following keywords define these parameters for a specific variable, and will be used by all bias that refer to that variable unless otherwise specified.

4.17 Trajectory output

4.18 Extended Lagrangian

The following options enable extended-system dynamics, where a colvar is coupled to an additional degree of freedom (fictitious particle) by a harmonic spring. This extended coordinate masks the colvar and replaces it transparently from the perspective of biasing and analysis methods. Biasing forces are then applied to the extended degree of freedom, and the actual geometric colvar (function of Cartesian coordinates) only feels the force from the harmonic spring. This is particularly useful when combined with an abf bias to perform eABF simulations (6.3).

Note that for some biases (harmonicWalls, histogram), this masking behavior is controlled by the keyword bypassExtendedLagrangian. Specifically for harmonicWalls, the default behavior is to bypass extended Lagrangian coordinates and act directly on the actual colvars.

4.19 Multiple time-step variables

4.20 Backward-compatibility

4.21 Statistical analysis

Run-time calculations of statistical properties that depend explicitly on time can be performed for individual collective variables. Currently, several types of time correlation functions, running averages and running standard deviations are implemented. For run-time computation of histograms, please see the histogram bias (6.10).

5 Selecting atoms

To define collective variables, atoms are usually selected as groups. Each group is defined using an identifying keyword that is unique in the context of the specific colvar component (e.g. for a distance component, the two groups are identified by the group1 and group2 keywords).

The group's identifying keyword is followed by a brace-delimited block containing selection keywords and other parameters, one of which is name:

Other keywords are documented in the following sections.

In the example below, the gyration component uses the identifying keyword atoms to define its associated group, which is defined based on the index group named “Protein-H". Optionally, the group is also given the unique name “my_protein", so that atom groups defined later in the Colvars configuration may refer to it.

colvar {
  name rgyr
  gyration {
    atoms {
      name my_protein
      indexGroup Protein-H
    }
  }
}

5.1 Atom selection keywords

Selection keywords may be used individually or in combination with each other, and each can be repeated any number of times. Selection is incremental: each keyword adds the corresponding atoms to the selection, so that different sets of atoms can be combined. However, atoms included by multiple keywords are only counted once. Below is an example configuration for an atom group called “atoms". Note: this is an unusually varied combination of selection keywords, demonstrating how they can be combined together: most simulations only use one of them.

atoms {

  # add atoms 1 and 3 to this group (note: first atom in the system is 1)
  atomNumbers {
    1 3
  }

  # add atoms starting from 20 up to and including 50
  atomNumbersRange 20-50
  # add index group (requires a .ndx file to be provided globally)
  indexGroup Water
}

The resulting selection includes atoms 1 and 3, those between 20 and 50, and those in the index group called “Water". The indices of this group are read from the file provided by the global keyword indexFile.

The complete list of selection keywords available in GROMACS is:

5.2 Moving frame of reference.

The following options define an automatic calculation of an optimal translation (centerToReference) or optimal rotation (rotateToReference), that superimposes the positions of this group to a provided set of reference coordinates. Alternately, centerToOrigin applies a translation to place the geometric center of the group at (0, 0, 0). This can allow, for example, to effectively remove from certain colvars the effects of molecular tumbling and of diffusion. Given the set of atomic positions xi, the colvar ξ can be defined on a set of roto-translated positions xi = R(xixC)+xref. xC is the geometric center of the xi, R is the optimal rotation matrix to the reference positions and xref is the geometric center of the reference positions.

Components that are defined based on pairwise distances are naturally invariant under global roto-translations. Other components are instead affected by global rotations or translations: however, they can be made invariant if they are expressed in the frame of reference of a chosen group of atoms, using the centerToReference and rotateToReference options. Finally, a few components are defined by convention using a roto-translated frame (e.g. the minimal RMSD): for these components, centerToReference and rotateToReference are enabled by default. In typical applications, the default settings result in the expected behavior.

Warning on rotating frames of reference and periodic boundary conditions. rotateToReference affects coordinates that depend on minimum-image distances in periodic boundary conditions (PBC). After rotation of the coordinates, the periodic cell vectors become irrelevant: the rotated system is effectively non-periodic. A safe way to handle this is to ensure that the relevant inter-group distance vectors remain smaller than the half-size of the periodic cell. If this is not desirable, one should avoid the rotating frame of reference, and apply orientational restraints to the reference group instead, in order to keep the orientation of the reference group consistent with the orientation of the periodic cell.

Warning on rotating frames of reference and ABF. Note that centerToReference and rotateToReference may affect the Jacobian derivative of colvar components in a way that is not taken into account by default. Be careful when using these options in ABF simulations or when using total force values.

The following options have default values appropriate for the vast majority of applications, and are only provided to support rare, special cases.

5.3 Treatment of periodic boundary conditions.

In simulations with periodic boundary conditions (PBCs), Colvars computes all distances between two points following the nearest-image convention, using PBC parameters provided by GROMACS. However, many common variables rely on a consistent definition of the center of mass or geometry of a group of atoms. This requires the use of unwrapped coordinates, which are not subject to “jumps" when they diffuse across periodic boundaries.

Internally, GROMACS wraps individual atom coordinates into a single periodic cell, which may break the calculation of some variables if their atom groups become split across PBCs. To prevent this, Colvars unwraps coordinates throughout the simulation, by assuming that each atom group is intact in the initial coordinates, and canceling any later jumps across the periodic box. This information is propagated across restarts using a checkpoint (cpt) file.

Whenever preparing a new simulation input with gmx grompp, users should provide input coordinates such that the atoms involved in collective variables will not be artificially moved across the boundary conditions, but occupy their relevant positions relative to each other -- usually the nearest ones. Unwrapped coordinates are communicated between replicas when GROMACS is used for replica-exchange simulations. Thus, Colvars is compatible with native replica-exchange in GROMACS.

In general, internal coordinate wrapping by GROMACS does not affect the calculation of colvars if each atom group satisfies one or more of the following:

  1. it is composed by only one atom;
  2. it is used by a colvar component which does not make use of its center of geometry, but only of pairwise distances (distanceInv, coordNum, hBond, alpha, dihedralPC);
  3. it is used by a colvar component that ignores the ill-defined Cartesian components of its center of mass (such as the x and y components of a membrane's center of mass modeled with distanceZ).

5.4 Performance of a Colvars calculation based on group size.

In simulations performed with message-passing programs (such as NAMD, LAMMPS or GROMACS), the calculation of energy and forces is distributed (i.e., parallelized) across multiple nodes, as well as over the processor cores of each node. When Colvars is enabled, certain atomic coordinates are collected on a single node, where the calculation of collective variables and of their biases is executed. This means that for simulations over large numbers of nodes, a Colvars calculation may produce a significant overhead, coming from the costs of transmitting atomic coordinates to one node and of processing them.

Performance can be improved in multiple ways:

6 Biasing and analysis methods

A biasing or analysis method can be applied to existing collective variables by using the following configuration:

<biastype > {
  name  <name >
  colvars  <xi1 >  <xi2 > ...
   <parameters >
}

The keyword <biastype > indicates the method of choice. There can be multiple instances of the same method, e.g. using multiple harmonic blocks allows defining multiple restraints.

All biasing and analysis methods implemented recognize the following options:

6.1 Thermodynamic integration

The methods implemented here provide a variety of estimators of conformational free-energies. These are carried out at run-time, or with the use of post-processing tools over the generated output files. The specifics of each estimator are discussed in the documentation of each biasing or analysis method.

A special case is the traditional thermodynamic integration (TI) method, used for example to compute potentials of mean force (PMFs). Most types of restraints (6.5, 6.7, 6.8, ...) as well as metadynamics (6.4) can optionally use TI alongside their own estimator, based on the keywords documented below.

In adaptive biasing force (ABF) (6.2) the above keywords are not recognized, because their functionality is either included already (conventional ABF) or not available (extended-system ABF).

6.2 Adaptive Biasing Force

For a full description of the Adaptive Biasing Force method, see reference [12]. For details about this implementation, see references [13] and [14]. When publishing research that makes use of this functionality, please cite references [12] and [14].

An alternate usage of this feature is the application of custom tabulated biasing potentials to one or more colvars. See inputPrefix and updateBias below.

Combining ABF with the extended Lagrangian feature (4.18) of the variables produces the extended-system ABF variant of the method (6.3).

ABF is based on the thermodynamic integration (TI) scheme for computing free energy profiles. The free energy as a function of a set of collective variables ξ = (ξi)i[1,n] is defined from the canonical distribution of ξ, 𝒫(ξ):

A (ξ) = 1 βln &ApplyFunction; 𝒫(ξ)+A0 (16)

In the TI formalism, the free energy is obtained from its gradient, which is generally calculated in the form of the average of a force Fξ exerted on ξ, taken over an iso-ξ surface:

&ApplyFunction;ξA (ξ) = Fξξ (17)

Several formulae that take the form of (17) have been proposed. This implementation relies partly on the classic formulation [15], and partly on a more versatile scheme originating in a work by Ruiz-Montero et al. [16], generalized by den Otter [17] and extended to multiple variables by Ciccotti et al. [18]. Consider a system subject to constraints of the form σk(x) = 0. Let (vi)i[1,n] be arbitrarily chosen vector fields (3N 3N) verifying, for all i, j, and k:

vi &ApplyFunction;xξj = δij (18) vi &ApplyFunction;xσk = 0 (19)

then the following holds [18]:

A ξi = vi &ApplyFunction;xV kBT &ApplyFunction;xviξ (20)

where V is the potential energy function. vi can be interpreted as the direction along which the force acting on variable ξi is measured, whereas the second term in the average corresponds to the geometric entropy contribution that appears as a Jacobian correction in the classic formalism [15]. Condition (18) states that the direction along which the total force on ξi is measured is orthogonal to the gradient of ξj, which means that the force measured on ξi does not act on ξj.

Equation (19) implies that constraint forces are orthogonal to the directions along which the free energy gradient is measured, so that the measurement is effectively performed on unconstrained degrees of freedom.

In the framework of ABF, Fξ is accumulated in bins of finite size δξ, thereby providing an estimate of the free energy gradient according to equation (17). The biasing force applied along the collective variables to overcome free energy barriers is calculated as:

FABF = α(N ξ)× &ApplyFunction;xA~(ξ) (21)

where &ApplyFunction;xA~ denotes the current estimate of the free energy gradient at the current point ξ in the collective variable subspace, and α(Nξ) is a scaling factor that is ramped from 0 to 1 as the local number of samples Nξ increases to prevent non-equilibrium effects in the early phase of the simulation, when the gradient estimate has a large variance. See the fullSamples parameter below for details.

As sampling of the phase space proceeds, the estimate &ApplyFunction;xA~ is progressively refined. The biasing force introduced in the equations of motion guarantees that in the bin centered around ξ, the forces acting along the selected collective variables average to zero over time. Eventually, as the underlying free energy surface is canceled by the adaptive bias, evolution of the system along ξ is governed mainly by diffusion. Although this implementation of ABF can in principle be used in arbitrary dimension, a higher-dimension collective variable space is likely to be difficult to sample and visualize. Most commonly, the number of variables is one or two, sometimes three.

6.2.1 ABF requirements on collective variables

The following conditions must be met for an ABF simulation to be possible and to produce an accurate estimate of the free energy profile. Note that these requirements do not apply when using the extended-system ABF method (6.3).

  1. Only linear combinations of colvar components can be used in ABF calculations.
  2. Availability of total forces is necessary. The following colvar components can be used in ABF calculations: distance, distance_xy, distance_z, angle, dihedral, gyration, rmsd and eigenvector. Atom groups may not be replaced by dummy atoms, unless they are excluded from the force measurement by specifying oneSiteTotalForce, if available.
  3. Mutual orthogonality of colvars. In a multidimensional ABF calculation, equation (18) must be satisfied for any two colvars ξi and ξj. Various cases fulfill this orthogonality condition:
    • ξi and ξj are based on non-overlapping sets of atoms.
    • atoms involved in the force measurement on ξi do not participate in the definition of ξj. This can be obtained using the option oneSiteTotalForce of the distance, angle, and dihedral components (example: Ramachandran angles ϕ, ψ).
    • ξi and ξj are orthogonal by construction. Useful cases are the sum and difference of two components, or distance_z and distance_xy using the same axis.
  4. Mutual orthogonality of components: when several components are combined into a colvar, it is assumed that their vectors vi (equation (20)) are mutually orthogonal. The cases described for colvars in the previous paragraph apply.
  5. Orthogonality of colvars and constraints: equation 19 can be satisfied in two simple ways, if either no constrained atoms are involved in the force measurement (see point 3 above) or pairs of atoms joined by a constrained bond are part of an atom group which only intervenes through its center (center of mass or geometric center) in the force measurement. In the latter case, the contributions of the two atoms to the left-hand side of equation 19 cancel out. For example, all atoms of a rigid TIP3P water molecule can safely be included in an atom group used in a distance component.

6.2.2 Parameters for ABF

ABF depends on parameters from each collective variable to define the grid on which free energy gradients are computed: see 4.16 for detauls. Other parameters to control the ABF runtime can be set in the ABF configuration block:

6.2.3 Output files

The ABF bias produces the following files, all in multicolumn text format (3.7.4):

Also in the case of one-dimensional calculations, the ABF bias can report its current energy via outputEnergy; in higher dimensions, such computation is not implemented and the energy reported is zero.

If several ABF biases are defined concurrently, their name is inserted to produce unique filenames for output, as in output.abf1.grad. This should not be done routinely and could lead to meaningless results: only do it if you know what you are doing!

If the colvar space has been partitioned into sections (windows) in which independent ABF simulations have been run, the resulting data can be merged using the inputPrefix option described above (a run of 0 steps is enough).

6.2.4 Multidimensional free energy surfaces

The ABF method only produces an estimate of the free energy gradient. The free energy surface itself can be computed depending on the value of integrate and related options.

In dimension 4 or greater, integrating the discretized gradient becomes non-trivial. The standalone utility abf_integrate is provided to perform that task. Because 4D ABF calculations are uncommon, this tool is practically deprecated by the Poisson integration described above.

abf_integrate reads the gradient data and uses it to perform a Monte-Carlo (M-C) simulation in discretized collective variable space (specifically, on the same grid used by ABF to discretize the free energy gradient). By default, a history-dependent bias (similar in spirit to metadynamics) is used: at each M-C step, the bias at the current position is incremented by a preset amount (the hill height). Upon convergence, this bias counteracts optimally the underlying gradient; it is negated to obtain the estimate of the free energy surface.

abf_integrate is invoked using the command-line:
abf_integrate <gradient_file> [-n <nsteps>] [-t <temp>] [-m (0|1)] [-h <hill_height>] [-f <factor>]

The gradient file name is provided first, followed by other parameters in any order. They are described below, with their default value in square brackets:

Using the default values of all parameters should give reasonable results in most cases.

abf_integrate produces the following output files:

Note: Typically, the “deviation" vector field does not vanish as the integration converges. This happens because the numerical estimate of the gradient does not exactly derive from a potential, due to numerical approximations used to obtain it (finite sampling and discretization on a grid). See Ref.[19] for details.

6.3 Extended-system Adaptive Biasing Force (eABF)

Extended-system ABF (eABF) is a variant of ABF (6.2) where the bias is not applied directly to the collective variable, but to an extended coordinate (“fictitious variable") λ that evolves dynamically according to Newtonian or Langevin dynamics. Such an extended coordinate is enabled for a given colvar using the extendedLagrangian and associated keywords (4.18). The theory of eABF and the present implementation are documented in detail in reference [20].

Defining an ABF bias on a colvar wherein the extendedLagrangian option is active will perform eABF automatically; there is no dedicated option.

The extended variable λ is coupled to the colvar z = ξ(q) by the harmonic potential (k2)(zλ)2. Under eABF dynamics, the adaptive bias on λ is the running estimate of the average spring force:

Fbias(λ) = k(λ z) λ (23)

where the angle brackets indicate a canonical average conditioned by λ = λ. At long simulation times, eABF produces a flat histogram of the extended variable λ, and a flattened histogram of ξ, whose exact shape depends on the strength of the coupling as defined by extendedFluctuation in the colvar. Coupling should be somewhat loose for faster exploration and convergence, but strong enough that the bias does help overcome barriers along the colvar ξ.[20] Distribution of the colvar may be assessed by plotting its histogram, which is written to the output.zcount file in every eABF simulation. Note that a histogram bias (6.10) applied to an extended-Lagrangian colvar will access the extended degree of freedom λ, not the original colvar ξ; however, the joint histogram may be explicitly requested by listing the name of the colvar twice in a row within the colvars parameter of the histogram block.

The eABF PMF is that of the coordinate λ, it is not exactly the free energy profile of ξ. That quantity can be calculated based on the CZAR estimator.

6.3.1 CZAR estimator of the free energy

The corrected z-averaged restraint (CZAR) estimator is described in detail in reference [20]. It is computed automatically in eABF simulations, regardless of the number of colvars involved. Note that ABF may also be applied on a combination of extended and non-extended colvars; in that case, CZAR still provides an unbiased estimate of the free energy gradient.

CZAR estimates the free energy gradient as:

A(z) = 1 β dln &ApplyFunction; ρ~(z) dz +k(λzz). (24)

where z = ξ(q) is the colvar, λ is the extended variable harmonically coupled to z with a force constant k, and ρ~(z) is the observed distribution (histogram) of z, affected by the eABF bias.

Parameters for the CZAR estimator are:

Similar to ABF, the CZAR estimator produces two output files in multicolumn text format (3.7.4):

The sampling histogram associated with the CZAR estimator is the z-histogram, which is written in the file output.zcount.

6.4 Metadynamics

The metadynamics method uses a history-dependent potential [21] that generalizes to any type of colvars the conformational flooding [22] and local elevation [23] methods, originally formulated to use as colvars the principal components of a covariance matrix or a set of dihedral angles, respectively. The metadynamics potential on the colvars ξ = (ξ1,ξ2, &ApplyFunction;,ξNcv) is defined as:

Vmeta (ξ (t)) = t=δt,2δt,t<tW i=1Ncv exp &ApplyFunction; ((ξi(t)ξi(t))2 2σξi2 ), (25)

where Vmeta is the history-dependent potential acting on the current values of the colvars ξ, and depends only parametrically on the previous values of the colvars. Vmeta is constructed as a sum of Ncv-dimensional repulsive Gaussian “hills", whose height is a chosen energy constant W, and whose centers are the previously explored configurations (ξ(δt),ξ(2δt), &ApplyFunction;).

During the simulation, the system evolves towards the nearest minimum of the “effective" potential of mean force Ã(ξ), which is the sum of the “real" underlying potential of mean force A(ξ) and the the metadynamics potential, Vmeta(ξ). Therefore, at any given time the probability of observing the configuration ξ is proportional to exp &ApplyFunction; (Ã(ξ)κBT ): this is also the probability that a new Gaussian “hill" is added at that configuration. If the simulation is run for a sufficiently long time, each local minimum is canceled out by the sum of the Gaussian “hills". At that stage the “effective" potential of mean force Ã(ξ) is constant, and Vmeta(ξ) is an estimator of the “real" potential of mean force A(ξ), save for an additive constant:

A(ξ) Vmeta(ξ)+K (26)

Such estimate of the free energy can be provided by enabling writeFreeEnergyFile. Assuming that the set of collective variables includes all relevant degrees of freedom, the predicted error of the estimate is a simple function of the correlation times of the colvars τξi, and of the user-defined parameters W, σξi and δt [24]. In typical applications, a good rule of thumb can be to choose the ratio Wδt much smaller than κBT τξ, where τξ is the longest among ξ's correlation times: σξi then dictates the resolution of the calculated PMF.

If the metadynamics parameters are chosen correctly, after an equilibration time, te, the estimator provided by eq. 26 oscillates on time around the “real" free energy, thereby a better estimate of the latter can be obtained as the time average of the bias potential after te [2526]:

A (ξ) = 1 ttottetettot Vmeta(ξ,t)dt (27)

where te is the time after which the bias potential grows (approximately) evenly during the simulation and ttot is the total simulation time. The free energy calculated according to eq. 27 can thus be obtained averaging on time multiple time-dependent free energy estimates, that can be printed out through the keyword keepFreeEnergyFiles. An alternative is to obtain the free energy profiles by summing the hills added during the simulation; the hills trajectory can be printed out by enabling the option writeHillsTrajectory.

6.4.1 Treatment of the PMF boundaries

In typical scenarios the Gaussian hills of a metadynamics potential are interpolated and summed together onto a grid, which is much more efficient than computing each hill independently at every step (the keyword useGrids is on by default). This numerical approximation typically yields negligible errors in the resulting PMF [1]. However, due to the finite thickness of the Gaussian function, the metadynamics potential would suddenly vanish each time a variable exceeds its grid boundaries.

To avoid such discontinuity the Colvars metadynamics code will keep an explicit copy of each hill that straddles a grid's boundary, and will use it to compute metadynamics forces outside the grid. This measure is taken to protect the accuracy and stability of a metadynamics simulation, except in cases of “natural" boundaries (for example, the [0 : 180] interval of an angle colvar) or when the flags hardLowerBoundary and hardUpperBoundary are explicitly set by the user. Unfortunately, processing explicit hills alongside the potential and force grids could easily become inefficient, slowing down the simulation and increasing the state file's size.

In general, it is a good idea to define a repulsive potential to avoid hills from coming too close to the grid's boundaries, for example as a harmonicWalls restraint (see 6.7).

Example: Using harmonic walls to protect the grid's boundaries.

colvar {
  name r
  distance { ... }
  upperBoundary 15.0
  width 0.2
}

metadynamics {
  name meta_r
  colvars r
  hillWeight 0.001
  hillWidth 2.0
}

harmonicWalls {
  name wall_r
  colvars r
  upperWalls 13.0
  upperWallConstant 2.0
}

In the colvar r, the distance function used has a lowerBoundary automatically set to 0 by default, thus the keyword lowerBoundary itself is not mandatory and hardLowerBoundary is set to yes internally. However, upperBoundary does not have such a “natural" choice of value. The metadynamics potential meta_r will individually process any hill whose center is too close to the upperBoundary, more precisely within fewer grid points than 6 times the Gaussian σ parameter plus one. It goes without saying that if the colvar r represents a distance between two freely-moving molecules, it will cross this “threshold" rather frequently.

In this example, where the value of hillWidth (2σ) amounts to 2 grid points, the threshold is 6+1 = 7 grid points away from upperBoundary. In explicit units, the width of r is wr = 0.2 Å, and the threshold is 15.0 - 7×0.2 = 13.6 Å.

The wall_r restraint included in the example prevents this: the position of its upperWall is 13 Å, i.e. 3 grid points below the buffer's threshold (13.6 Å). For the chosen value of upperWallConstant, the energy of the wall_r bias at r = rupper = 13.6 Å is:

E = 1 2k (rrupper wr )2 = 1 22.0 (3)2 = 9kcalmol

which results in a relative probability exp &ApplyFunction; (EκBT ) 3×107 that r crosses the threshold. The probability that r exceeds upperBoundary, which is further away, has also become vanishingly small. At that point, you may want to set hardUpperBoundary to yes for r, and let meta_r know that no special treatment near the grid's boundaries will be needed.

What is the impact of the wall restraint onto the PMF? Not a very complicated one: the PMF reconstructed by metadynamics will simply show a sharp increase in free-energy where the wall potential kicks in (r  > 13 Å). You may then choose between using the PMF only up until that point and discard the rest, or subtracting the energy of the harmonicWalls restraint from the PMF itself. Keep in mind, however, that the statistical convergence of metadynamics may be less accurate where the wall potential is strong.

In summary, although it would be simpler to set the wall's position upperWall and the grid's boundary upperBoundary to the same number, the finite width of the Gaussian hills calls for setting the former strictly within the latter.

6.4.2 Required metadynamics keywords

To enable a metadynamics-based calculation, a metadynamics {...} block must be included in the Colvars configuration file.

By default, metadynamics bias energy and forces will be recorded onto a grid, the parameters of which can be defined within the definition of each colvar, as described in 4.16.

Other required keywords will be specified within the metadynamics block: these are colvars (the names of the variables involved), hillWeight (the weight parameter W), and the widths 2σ of the Gaussian hills in each dimension that can be given either as the single dimensionless parameter hillWidth, or explicitly for each colvar with gaussianSigmas.

6.4.3 Output files

When interpolating grids are enabled (default behavior), the PMF is written by default every colvarsRestartFrequency steps to the file output.pmf in multicolumn text format (3.7.4). The following two options allow to disable or control this behavior and to track statistical convergence:

6.4.4 Performance optimization

The following options control the computational cost of metadynamics calculations, but do not affect results. Default values are chosen to minimize such cost with no loss of accuracy.

6.4.5 Ensemble-Biased Metadynamics

The ensemble-biased metadynamics (EBMetaD) approach [27] is designed to reproduce a target probability distribution along selected collective variables. Standard metadynamics can be seen as a special case of EBMetaD with a flat distribution as target. This is achieved by weighing the Gaussian functions used in the metadynamics approach by the inverse of the target probability distribution:

VEBmetaD (ξ (t)) = t=δt,2δt,t<t W exp &ApplyFunction; (Sρ)ρexp(ξ(t))i=1Ncv exp &ApplyFunction; ((ξi(t)ξi(t))2 2σξi2 ), (28)

where ρexp(ξ) is the target probability distribution and Sρ = &ApplyFunction;ρexp(ξ)log &ApplyFunction; ρexp(ξ)dξ its corresponding differential entropy. The method is designed so that during the simulation the resulting distribution of the collective variable ξ converges to ρexp(ξ). A practical application of EBMetaD is to reproduce an “experimental" probability distribution, for example the distance distribution between spectroscopic labels inferred from Förster resonance energy transfer (FRET) or double electron-electron resonance (DEER) experiments [27].

The PMF along ξ can be estimated from the bias potential and the target ditribution [27]:

A(ξ) VEBmetaD(ξ)κBT log &ApplyFunction; ρexp(ξ) (29)

and obtained by enabling writeFreeEnergyFile. Similarly to eq. 27, a more accurate estimate of the free energy can be obtained by averaging (after an equilibration time) multiple time-dependent free energy estimates (see keepFreeEnergyFiles).

The following additional options define the configuration for the ensemble-biased metadynamics approach:

As with standard metadynamics, multidimensional probability distributions can be targeted using a single metadynamics block using multiple colvars and a multidimensional target distribution file (see 3.7.4). Instead, multiple probability distributions on different variables can be targeted separately in the same simulation by introducing multiple metadynamics blocks with the ebMeta option.

Example: EBmetaD configuration for a single variable.

colvar {
  name r
  distance {
    group1 { atomNumbers 991 992 }
    group2 { atomNumbers 1762 1763 }
  }
  upperBoundary 100.0
  width 0.1
}

metadynamics {
  name ebmeta
  colvars r
  hillWeight 0.01
  hillWidth 3.0
  ebMeta on
  targetDistFile targetdist1.dat
  ebMetaEquilSteps 500000
}

where targetdist1.dat is a text file in “multicolumn" format (3.7.4) with the same width as the variable r (0.1 in this case):

# 1
# 0.0 0.1 1000 0
0.05 0.0012
0.15 0.0014
99.95 0.0010

Tip: Besides setting a meaningful value for targetDistMinVal, the exploration of unphysically low values of the target distribution (which would lead to very large hills and possibly numerical instabilities) can be also prevented by restricting sampling to a given interval, using e.g. harmonicWalls restraint (6.7).

6.4.6 Well-tempered metadynamics

The following options define the configuration for the “well-tempered" metadynamics approach [28]:

6.4.7 Multiple-walker metadynamics

Metadynamics calculations can be performed concurrently by multiple replicas that share a common history. This variant of the method is called multiple-walker metadynamics [29]: the Gaussian hills of all replicas are periodically combined into a single biasing potential, intended to converge to a single PMF.

In the implementation here described [1], replicas communicate through files. This arrangement allows launching the replicas either (1) as a bundle (i.e. a single job in a cluster's queueing system) or (2) as fully independent runs (i.e. as separate jobs for the queueing system). One advantage of the use case (1) is that an identical Colvars configuration can be used for all replicas (otherwise, replicaID needs to be manually set to a different string for each replica). However, the use case (2) is less demanding in terms of high-performance computing resources: a typical scenario would be a computer cluster (including virtual servers from a cloud provider) where not all nodes are connected to each other at high speed, and thus each replica runs on a small group of nodes or a single node.

Whichever way the replicas are started (coupled or not), a shared filesystem is needed so that each replica can read the files created by the others: paths to these files are stored in the shared file replicasRegistry. This file, and those listed in it, are read every replicaUpdateFrequency steps. Each time the Colvars state file is written (for example, colvarsRestartFrequency steps), the file named:
output.colvars.name.replicaID.state
is written as well; this file contains only the state of the metadynamics bias, which the other replicas will read in turn. In between the times when this file is modified/replaced, new hills are also temporarily written to the file named:
output.colvars.name.replicaID.hills
Both files are only used for communication, and may be deleted after the replica begins writing files with a new output.

Example: Multiple-walker metadynamics with file-based communication.

metadynamics {
  name mymtd
  colvars x
  hillWeight 0.001
  newHillFrequency 1000
  hillWidth 3.0
  
  multipleReplicas on
  replicasRegistry /shared-folder/mymtd-replicas.txt
  replicaUpdateFrequency 50000 # Best if larger than newHillFrequency
}

The following are the multiple-walkers related options:

6.5 Harmonic restraints

The harmonic biasing method may be used to enforce fixed or moving restraints, including variants of Steered and Targeted MD. Within energy minimization runs, it allows for restrained minimization, e.g. to calculate relaxed potential energy surfaces. In the context of the Colvars module, harmonic potentials are meant according to their textbook definition:

V (ξ ) = 1 2k (ξ ξ0 wξ )2 (30)

There are two noteworthy aspects of this expression:

  1. Because the standard coefficient of 12 of the harmonic potential is included, this expression differs from harmonic bond and angle potentials historically used in common force fields, where the factor was typically omitted resulting in a non-standard definition of the force constant.
  2. The variable ξ is not only centered at ξ0, but is also scaled by its characteristic length scale wξ (keyword width). The resulting dimensionless variable z = (ξ ξ0)wξ is typically easier to treat numerically: for example, when the forces typically experienced by ξ are much smaller than kwξ and k is chosen equal to κBT (thermal energy), the resulting probability distribution of z is approximately a Gaussian with mean equal to 0 and standard deviation equal to 1.

    This property can be used for setting the force constant in umbrella-sampling ensemble runs: if the restraint centers are chosen in increments of wξ, the resulting distributions of ξ are most often optimally overlapped. In regions where the underlying free-energy landscape induces highly skewed distributions of ξ, additional windows may be added as needed, with spacings finer than wξ.

Beyond one dimension, the use of a scaled harmonic potential also allows a standard definition of a multi-dimensional restraint with a unified force constant:

V (ξ1, &ApplyFunction;,ξM) = 1 2ki=1M (ξiξ0 wξ )2 (31)

If one-dimensional or homogeneous multi-dimensional restraints are defined, and there are no other uses for the parameter wξ, width can be left at its default value of 1.

A harmonic restraint is defined by a harmonic {...} block, which may contain the following keywords:

Tip: A complex set of restraints can be applied to a system, by defining several colvars, and applying one or more harmonic restraints to different groups of colvars. In some cases, dozens of colvars can be defined, but their value may not be relevant: to limit the size of the colvars trajectory file, it may be wise to disable outputValue for such “ancillary" variables, and leave it enabled only for “relevant" ones.

6.5.1 Moving restraints: steered molecular dynamics

The following options allow to change gradually the centers of the harmonic restraints during a simulations. When the centers are changed continuously, a steered MD in a collective variable space is carried out.

Note on restarting moving restraint simulations: Information about the current step and stage of a simulation with moving restraints is stored in the restart file (state file). Thus, such simulations can be run in several chunks, and restarted directly using the same colvars configuration file. In case of a restart, the values of parameters such as targetCenters, targetNumSteps, etc. should not be changed manually.

6.5.2 Moving restraints: umbrella sampling

The centers of the harmonic restraints can also be changed in discrete stages: in this cases a one-dimensional umbrella sampling simulation is performed. The sampling windows in simulation are calculated in sequence. The colvars trajectory file may then be used both to evaluate the correlation times between consecutive windows, and to calculate the frequency distribution of the colvar of interest in each window. Furthermore, frequency distributions on a predefined grid can be automatically obtained by using the histogram bias (see 6.10).

To activate an umbrella sampling simulation, the same keywords as in the previous section can be used, with the addition of the following:

6.5.3 Changing force constant

The force constant of the harmonic restraint may also be changed to equilibrate [30].

6.6 Computing the work of a changing restraint

If the restraint centers or force constant are changed continuosly (targetNumStages undefined) it is possible to record the net work performed by the changing restraint:

6.7 Harmonic wall restraints

The harmonicWalls {...} bias is closely related to the harmonic bias (see 6.5), with the following two differences: (i) instead of a center a lower wall and/or an upper wall are defined, outside of which the bias implements a half-harmonic potential;

V (ξ ) = { 1 2k (ξξupper wξ ) 2 ifξ > ξupper 0 ifξlower ξ ξupper 1 2k (ξξlower wξ ) 2 ifξ < ξlower (32)

where ξlower and ξupper are the lower and upper wall thresholds, respectively; (ii) because an interval between two walls is defined, only scalar variables can be used (but any number of variables can be defined, and the wall bias is intrinsically multi-dimensional).

Note: this bias replaces the keywords lowerWall, lowerWallConstant, upperWall and upperWallConstant defined in the colvar context. Those keywords are deprecated.

The harmonicWalls bias implements the following options:

Example 1: harmonic walls for one variable with two different force constants.

harmonicWalls {
  name mywalls
  colvars dist
  lowerWalls 22.0
  upperWalls 38.0
  lowerWallConstant 2.0
  upperWallConstant 10.0
}

Example 2: harmonic walls for two variables with a single force constant.

harmonicWalls {
  name mywalls
  colvars phi psi
  lowerWalls -180.0 0.0
  upperWalls 0.0 180.0
  forceConstant 5.0
}

6.8 Linear restraints

The linear keyword defines a linear potential:

V (ξ ) = k (ξ ξ0 wξ ) (33)

whose force is simply given by the constant kwξ itself:

f(ξ) = kwξ (34)

This type of bias is therefore most useful in situations where a constant force is desired. As all other restraints, it can be defined on one or more CVs, with each contribution added to the total potential and the parameters wξ determining the relative magnitude for each.

Example: A possible use case of the linear bias is mimicking a constant electric field acting on a specific particle, or the center of mass of many particles. In the following example, a linear restraint is applied on a distanceZ variable (4.2.2), generating a constant force parallel to the Z axis of magnitude 5 kJ/mol/nm:

colvar {
  name z
  distanceZ {
    ...
  }
}

linear {
  colvars z
  forceConstant 5.0
  centers 0.0
}

Another useful application of a linear restraint is to enforce experimental constraints in a simulation, with a lower non-equilibrium work than e.g. harmonic restraints [31]. There is generally a unique strength of bias for each CV center, which means you must know the bias force constant specifically for the center of the CV. This force constant may be found by using experiment directed simulation described in section 6.9.

6.9 Adaptive Linear Bias/Experiment Directed Simulation

Experiment directed simulation applies a linear bias with a changing force constant. Please cite White and Voth [32] when using this feature. As opposed to that reference, the force constant here is scaled by the width corresponding to the biased colvar. In White and Voth, each force constant is scaled by the colvars set center. The bias converges to a linear bias, after which it will be the minimal possible bias. You may also stop the simulation, take the median of the force constants (ForceConst) found in the colvars trajectory file, and then apply a linear bias with that constant. All the notes about units described in sections 6.8 and 6.5 apply here as well. This is not a valid simulation of any particular statistical ensemble and is only an optimization algorithm until the bias has converged.

6.10 Multidimensional histograms

The histogram feature is used to record the distribution of a set of collective variables in the form of a N-dimensional histogram. Defining such a histogram is generally useful for analysis purposes, but it has no effect on the simulation.

Example 1: the two-dimensional histogram of a distance and an angle can be generated using the configuration below. The histogram code requires that each variable is a scalar number that is confined within a pre-defined interval. The interval's boundaries may be specified by hand (e.g. through lowerBoundary and upperBoundary in the variable definition), or auto-detected based on the type of function. In this example, the lower boundary for the distance variable “r" is automatically set to zero, and interval for the three-body angle “theta" is 0 and 180: however, that an upper boundary for the distance “r" still needs to be specified manually. The grid spacings for the two variables are 0.2 nmand 3.0, respectively.

colvar {
  name r
  width 0.2
  upperBoundary 20.0
  distance { ... }
}

colvar {
  name theta
  width 3.0
  dihedral { ... }
}

histogram {
  name hist2d
  colvars r theta
}

Example 2: This example is similar to the previous one, but with the important difference that the parameters for the histogram's grid are defined explicitly for this histogram instance. Therefore, this histogram's grid may differ from the one defined from parameters embedded in the colvar { ... } block (for example, narrower intervals and finer grid spacings may be selected).

colvar {
  name r
  upperBoundary 20.0
  distance { ... }
}

colvar {
  name theta
  dihedral { ... }
}

histogram {
  name hist2d
  colvars r theta
  histogramGrid {
    widths  0.1 1.0
    lowerBoundaries   2.0 30.0
    upperBoundaries  10.0 90.0
  }
}

The standard keywords below are used to control the histogram's computation and to select the variables that are sampled. See also 6.10.1 for keywords used to define the grid, 6.10.2 for output parameters and 6.10.3 for more advanced keywords.

6.10.1 Defining grids for multidimensional histograms

Grid parameters for the histogram may be provided at the level of the individual variables, or via a dedicated configuration block histogramGrid { …} inside the configuration of this histogram. The options supported inside this block are:

6.10.2 Output options for multi-dimensional histograms

The accumulated histogram is written in the Colvars state file, allowing for its accumulation across continued runs. Additionally, the following files are written depending on the histogram's dimensionality:

As with any other biasing and analysis method, when a histogram is applied to an extended-system colvar (4.18), it accesses the value of the extended coordinate rather than that of the actual colvar. This can be overridden by enabling the bypassExtendedLagrangian option. A joint histogram of the actual colvar and the extended coordinate may be collected by specifying the colvar name twice in a row in the colvars parameter (e.g. colvars myColvar myColvar): the first instance will be understood as the actual colvar, and the second, as the extended coordinate.

6.11 Probability distribution-restraints

The histogramRestraint bias implements a continuous potential of many variables (or of a single high-dimensional variable) aimed at reproducing a one-dimensional statistical distribution that is provided by the user. The M variables (ξ1, &ApplyFunction;,ξM) are interpreted as multiple observations of a random variable ξ with unknown probability distribution. The potential is minimized when the histogram h(ξ), estimated as a sum of Gaussian functions centered at (ξ1, &ApplyFunction;,ξM), is equal to the reference histogram h0(ξ):

V (ξ1, &ApplyFunction;,ξM) = 1 2k (h(ξ)h0(ξ))2dξ (35)
h (ξ ) = 1 M2πσ2i=1Mexp &ApplyFunction; ((ξ ξi)2 2σ2 ) (36)

When used in combination with a distancePairs multi-dimensional variable, this bias implements the refinement algorithm against ESR/DEER experiments published by Shen et al [33].

This bias behaves similarly to the histogram bias with the gatherVectorColvars option, with the important difference that all variables are gathered, resulting in a one-dimensional histogram. Future versions will include support for multi-dimensional histograms.

The list of options is as follows:

7 Syntax changes from older versions

The following is a list of syntax changes in Colvars since its first release. Many of the older keywords are still recognized by the current code, thanks to specific compatibility code. This is not a list of new features: its primary purpose is to make you aware of those improvements that affect the use of old configuration files with new versions of the code.

Note: if you are using any of the NAMD and VMD tutorials:
https://www.ks.uiuc.edu/Training/Tutorials/
please be aware that several of these tutorials are not actively maintained: for those cases, this list will help you reconcile any inconsistencies.

8 Compilation notes

The Colvars module is typically built using the recipes of each supported software package: for this reason, no installation instructions are needed, and the vast majority of the features described in this manual are supported in the most common builds of each package.

This section lists the few cases where the choice of compilation settings affects the availability features in the Colvars module.

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