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Declaring and using collective variables

Each collective variable (colvar) is defined as a combination of one or more individual quantities, called components (see Figure 6). In most applications, only one is needed: in this case, the colvar and its component may be identified.

In the configuration file, each colvar is created by the keyword colvar, followed by its configuration options, usually between curly braces, colvar {...}. Each component is defined within the the colvar {...} block, with a specific keyword that identifies the functional form: for example, distance {...} defines a component of the type ``distance between two atom groups''.

To obtain the value of the colvar, $ \xi(\mathbf{r})$, its components $ q_i(\mathbf{r})$ are summed with the formula:

$\displaystyle \xi(\mathbf{r}) = \sum_i c_i [q_i(\mathbf{r})]^{n_i}$ (36)

where each component appears with a unique coefficient $ c_i$ (1.0 by default) the positive integer exponent $ n_i$ (1 by default). For information on setting these parameters, see 10.2.3.

General collective variable options

Colvar grid parameters

Boundary potentials (walls)

Trajectory output

Extended Lagrangian

Collective variable components

Each colvar is defined by one or more components (typically only one). Each component consists of a keyword identifying a functional form, and a definition block following that keyword, specifying the atoms involved and any additional parameters (cutoffs, ``reference'' values, ...).

The types of the components used in a colvar determine the properties of that colvar, and which biasing or analysis methods can be applied. In most cases, the colvar returns a real number, which is computed by one or more instances of the following components:

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.

The following keywords can be used within periodic components (and are illegal elsewhere):

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 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.

Non-scalar components.

When one of the following are used, the 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 $ \mathsf{q}$ is the same as the one represented by $ -\mathsf{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.

Calculating system forces.

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

Syntax of a component definition.

Most components make use of one or more atom groups, whose syntax of definition is by their name followed by a definition block like atoms {...}, or group1 {...} and group2 {...}. The contents of an atom group block are described in 10.2.4.

In the following, all the available component types are listed, along with their physical units and the limiting values, if any. Such limiting values can be used to define lowerBoundary and upperBoundary in the parent colvar.

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

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

The value returned is a positive number (in Å), 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).

Component 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. This component returns a number (in Å) 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 $ -b_{z}/2$ and $ b_{z}/2$, where $ b_{z}$ is the box length along $ z$ (this behavior is disabled if forceNoPBC is set).

Component 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 distanceZ, i.e. main, ref, either ref2 or axis, and oneSiteSystemForce. 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).

Component distanceVec: distance vector between two groups.

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

Component distanceDir: distance unit vector between two groups.

The distanceDir {...} block defines a distance unit vector component, which accepts the same keywords as distance: group1, group2, and forceNoPBC. It returns a 3-dimensional unit vector $ \mathbf{d} = (d_{x}, d_{y}, d_{z})$, with $ \vert\mathbf{d}\vert = 1$.

Component 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]$.

Component 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 colvar 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.

Component coordNum: coordination number between two groups.

The coordNum {...} block defines a coordination number (or number of contacts), which calculates the function $ (1-(d/d_0)^{n})/(1-(d/d_0)^{m})$, where $ d_0$ is the ``cutoff'' distance, and $ n$ and $ m$ are exponents that can control its long range behavior and stiffness [36]. This function is summed over all pairs of atoms in group1 and group2:

$\displaystyle C (\mathtt{group1}, \mathtt{group2}) \; = \; \sum_{i\in\mathtt{gr...
...}\vert/d_{0})^{n}}{ 1 - (\vert\mathbf{x}_{i}-\mathbf{x}_{j}\vert/d_{0})^{m} } }$ (37)

This colvar component accepts the same keywords as distance, group1 and group2. In addition to them, it recognizes the following keywords:

This component returns a dimensionless number, which ranges from approximately 0 (all interatomic distances much larger than the cutoff) to $ N_{\mathtt{group1}} * N_{\mathtt{group2}}$ (all distances within the cutoff), or $ N_{\mathtt{group1}}$ if group2CenterOnly is used. For performance reasons, at least one of group1 and group2 should be of limited size (unless group2CenterOnly is used), because the cost of the loop over all pairs grows as $ N_{\mathtt{group1}} * N_{\mathtt{group2}}$.

Component selfCoordNum: coordination number between atoms within a group.

The selfCoordNum {...} block defines a coordination number in much the same way as coordNum, but the function is summed over atom pairs within group1:

$\displaystyle C (\mathtt{group1}) \; = \; \sum_{i\in\mathtt{group1}}\sum_{j > i...
...}\vert/d_{0})^{n}}{ 1 - (\vert\mathbf{x}_{i}-\mathbf{x}_{j}\vert/d_{0})^{m} } }$ (38)

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.

This component returns a dimensionless number, which ranges from approximately 0 (all interatomic distances much larger than the cutoff) to $ N_{\mathtt{group1}} * (N_{\mathtt{group1}} - 1) / 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 $ N_{\mathtt{group1}}^2$.

Component hBond: hydrogen bond between two atoms.

The hBond {...} block defines a hydrogen bond, implemented as a coordination number (eq. 37) 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 an adimensional number, with values between 0 (acceptor and donor far outside the cutoff distance) and 1 (acceptor and donor much closer than the cutoff).

Component rmsd: root mean square displacement (RMSD) with respect to a reference structure.

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 $ \{ \mathbf{x}_1(t), \mathbf{x}_2(t), \ldots
\mathbf{x}_N(t) \}$, the colvar component rmsd calculates the optimal rotation $ U^{\{\mathbf{x}_{i}(t)\}\rightarrow\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}}$ that best superimposes the coordinates $ \{\mathbf{x}_{i}(t)\}$ onto a set of reference coordinates $ \{\mathbf{x}_{i}^{\mathrm{(ref)}}\}$. Both the current and the reference coordinates are centered on their centers of geometry, $ \mathbf{x}_{\mathrm{cog}}(t)$ and $ \mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}}$. The root mean square displacement is then defined as:

$\displaystyle { \mathrm{RMSD}(\{\mathbf{x}_{i}(t)\}, \{\mathbf{x}_{i}^{\mathrm{...
...{(ref)}} - \mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}} \right) \right\vert^{2} }$ (39)

The optimal rotation $ U^{\{\mathbf{x}_{i}(t)\}\rightarrow\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}}$ is calculated within the formalism developed in reference [18], which guarantees a continuous dependence of $ U^{\{\mathbf{x}_{i}(t)\}\rightarrow\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}}$ with respect to $ \{\mathbf{x}_{i}(t)\}$. The options for rmsd are: This component returns a positive real number (in Å).

Component 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 $ \mathbb{R}^{3n}$, where $ n$ is the number of atoms in the group. The computed quantity is the total projection:

$\displaystyle { p(\{\mathbf{x}_{i}(t)\}, \{\mathbf{x}_{i}^{\mathrm{(ref)}}\}) }...
...athrm{(ref)}} - \mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}}) \right)\mathrm{,} }$ (40)

where, as in the rmsd component, $ U$ is the optimal rotation matrix, $ \mathbf{x}_{\mathrm{cog}}(t)$ and $ \mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}}$ are the centers of geometry of the current and reference positions respectively, and $ \mathbf{v}_{i}$ are the components of the vector for each atom. Example choices for $ (\mathbf{v}_{i})$ are an eigenvector of the covariance matrix (essential mode), or a normal mode of the system. It is assumed that $ \sum_{i}\mathbf{v}_{i} = 0$: otherwise, the colvars module centers the $ \mathbf{v}_{i}$ automatically when reading them from the configuration.

As in the rmsd component, available options are atoms, refPositions or refPositionsFile, refPositionsCol and refPositionsColValue. In addition, the following are recognized:

This component returns a number (in Å), whose value ranges between the smallest and largest absolute positions in the unit cell during the simulations (see also distanceZ). Due to the normalization in eq. 40, this range does not depend on the number of atoms involved.

Component 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 $ \{ \mathbf{x}_1(t), \mathbf{x}_2(t), \ldots
\mathbf{x}_N(t) \}$ with respect to their center of geometry, $ \mathbf{x}_{\mathrm{cog}}(t)$:

$\displaystyle R_{\mathrm{gyr}} \; = \; \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left\vert\mathbf{x}_{i}(t) - \mathbf{x}_{\mathrm{cog}}(t)\right\vert^{2} }$ (41)

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

Component orientation: orientation from reference coordinates.

The block orientation {...} returns the same optimal rotation used in the rmsd component to superimpose the coordinates $ \{\mathbf{x}_{i}(t)\}$ onto a set of reference coordinates $ \{\mathbf{x}_{i}^{\mathrm{(ref)}}\}$. Such component returns a four dimensional vector $ \mathsf{q} = (q_0, q_1,
q_2, q_3)$, with $ \sum_{i} q_{i}^{2} = 1$; this quaternion expresses the optimal rotation $ \{\mathbf{x}_{i}(t)\} \rightarrow
\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}$ according to the formalism in reference [18]. The quaternion $ (q_0, q_1, q_2, q_3)$ can also be written as $ \left(\cos(\theta/2), \,
\sin(\theta/2)\mathbf{u}\right)$, where $ \theta$ is the angle and $ \mathbf{u}$ the normalized axis of rotation; for example, a rotation of 90$ ^{\circ}$ around the $ z$ axis should be expressed as ``(0.707, 0.0, 0.0, 0.707)''. The script quaternion2rmatrix.tcl provides Tcl functions for converting to and from a $ 4\times{}4$ rotation matrix in a format suitable for usage in VMD.

The component accepts all the options of rmsd: atoms, refPositions, refPositionsFile and refPositionsCol, in addition to:

Hint: 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 throuh 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.

Component orientationAngle: angle of rotation from reference coordinates.

The block orientationAngle {...} accepts the same options as rmsd and orientation (atoms, refPositions, refPositionsFile and refPositionsCol), but it returns instead the angle of rotation $ \omega$ between the current and the reference positions. This angle is expressed in degrees within the range [0$ ^{\circ}$:180$ ^{\circ}$].

Component alpha: $ \alpha $-helix content of a protein segment.

The block alpha {...} defines the parameters to calculate the helical content of a segment of protein residues. The $ \alpha $-helical content across the $ N+1$ residues $ N_{0}$ to $ N_{0}+N$ is calculated by the formula:
$\displaystyle {
} \; = \; \; \; \;$     (42)
$\displaystyle \; \; \; \; {
\mathrm{N}^{(n+4)}\right) \mathrm{,}

where the score function for the $ \mathrm{C}_{\alpha} -
\mathrm{C}_{\alpha} - \mathrm{C}_{\alpha}$ angle is defined as:

$\displaystyle { \mathrm{angf}\left( \mathrm{C}_{\alpha}^{(n)}, \mathrm{C}_{\alp...
...eta_{0}\right)^{4} / \left(\Delta\theta_{\mathrm{tol}}\right)^{4}} \mathrm{,} }$ (43)

and the score function for the $ \mathrm{O}^{(n)} \leftrightarrow
\mathrm{N}^{(n+4)}$ hydrogen bond is defined through a hBond colvar component on the same atoms. The options recognized within the alpha {...} block are:

This component returns positive values, always comprised between 0 (lowest $ \alpha $-helical score) and 1 (highest $ \alpha $-helical score).

Component dihedralPC: protein dihedral pricipal component

The block dihedralPC {...} defines the parameters to calculate the projection of backbone dihedral angles within a protein segment onto a dihedral principal component, following the formalism of dihedral principal component analysis (dPCA) proposed by Mu et al.[52] and documented in detail by Altis et al.[2]. Given a peptide or protein segment of $ N$ residues, each with Ramachandran angles $ \phi_i$ and $ \psi_i$, dPCA rests on a variance/covariance analysis of the $ 4(N-1)$ variables $ \cos(\psi_1), \sin(\psi_1), \cos(\phi_2), \sin(\phi_2)
\cdots \cos(\phi_N), \sin(\phi_N)$. Note that angles $ \phi_1$ and $ \psi_N$ have little impact on chain conformation, and are therefore discarded, following the implementation of dPCA in the analysis software Carma.[26]

For a given principal component (eigenvector) of coefficients $ (k_i)_{1 \leq i \leq 4(N-1)}$, the projection of the current backbone conformation is:

$\displaystyle \xi = \sum_{n=1}^{N-1} k_{4n-3} \cos(\psi_n) + k_{4n-2} \sin (\psi_n) + k_{4n-1} \cos (\phi_{n+1}) + k_{4n} \sin(\phi_{n+1})$ (44)

dihedralPC expects the same parameters as the alpha component for defining the relevant residues (residueRange and psfSegID) in addition to the following:

Linear and polynomial combinations of components

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 (36)) 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 $ 1/N$.

Defining atom groups

Each component depends on one or more atom groups, which can be defined by different methods in the configuration file. Each atom group block is initiated by the name of the group itself within the component block, followed by the instructions to the colvar module on how to select the atoms involved. Here is an example configuration, for an atom group called myatoms, which makes use of the most common keywords:
# atom group definition
myatoms {
  # add atoms 1, 2 and 3 to this group (note: numbers start from 1)
  atomNumbers {
     1 2 3
  # add all the atoms with occupancy 2 in the file atoms.pdb
  atomsFile             atoms.pdb
  atomsCol              O
  atomsColValue         2.0
  # add all the C-alphas within residues 11 to 20 of segments "PR1" and "PR2"
  psfSegID              PR1 PR2
  atomNameResidueRange  CA 11-20
  atomNameResidueRange  CA 11-20

For any atom group, the available options are:

Note: to minimize the length of the NAMD standard output, messages in the atom group's configuration are not echoed by default. This can be overcome by the boolean keyword verboseOutput within the group.

Recommendations for using atom groups.

When defining the atom groups for a collective variable, these guidelines should be followed to avoid inconsistencies and performance losses:

Statistical analysis of individual collective variables

When the global keyword analysis is defined in the configuration file, calculations of statistical properties for individual colvars can be performed. At the moment, several types of time correlation functions, running averages and running standard deviations are available.

next up previous contents index
Next: Biasing and analysis methods Up: Collective Variable-based Calculations1 Previous: General parameters and input/output   Contents   Index