Collective metrics

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 $\{ \mathbf{x}_1(t), \mathbf{x}_2(t), \ldots \mathbf{x}_N(t) \}$ , 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:

$${ \mathrm{RMSD}(\{\mathbf{x}_{i}(t)\}, \{\mathbf{x}_{i}^{\mathrm{... ...{(ref)}} - \mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}} \right) \right\vert^{2} }$$ (13.5)

The optimal rotation $U^{\{\mathbf{x}_{i}(t)\}\rightarrow\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}}$ is calculated within the formalism developed in reference [50], which guarantees a continuous dependence of $U^{\{\mathbf{x}_{i}(t)\}\rightarrow\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}}$ with respect to $\{\mathbf{x}_{i}(t)\}$ .

List of keywords (see also for additional options):

• atoms $\langle\,$ Atom group$\,\rangle$
  Context: rmsd
  Acceptable values: atoms {...} block
  Description: Defines the group of atoms of which the RMSD should be calculated. Optimal fit options (such as refPositions and rotateReference) should typically NOT be set within this block. Exceptions to this rule are the special cases discussed in the Advanced usage paragraph below.

• refPositions $\langle\,$ Reference coordinates$\,\rangle$
  Context: rmsd
  Acceptable values: space-separated list of (x, y, z) triplets
  Description: This option (mutually exclusive with refPositionsFile) sets the reference coordinates for RMSD calculation, and uses these to compute the roto-translational fit. It is functionally equivalent to the option refPositions in the atom group definition, which also supports more advanced fitting options.

• refPositionsFile $\langle\,$ Reference coordinates file$\,\rangle$
  Context: rmsd
  Acceptable values: UNIX filename
  Description: This option (mutually exclusive with refPositions) sets the reference coordinates for RMSD calculation, and uses these to compute the roto-translational fit. It is functionally equivalent to the option refPositionsFile in the atom group definition, which also supports more advanced fitting options.

• refPositionsCol $\langle\,$ PDB column containing atom flags$\,\rangle$
  Context: rmsd
  Acceptable values: O, B, X, Y, or Z
  Description: If refPositionsFile is a PDB file that contains all the atoms in the topology, this option may be provided to set which PDB field is used to flag the reference coordinates for atoms.

• refPositionsColValue $\langle\,$ Atom selection flag in the PDB column$\,\rangle$
  Context: rmsd
  Acceptable values: positive decimal
  Description: If defined, this value identifies in the PDB column refPositionsCol of the file refPositionsFile which atom positions are to be read. Otherwise, all positions with a non-zero value are read.

• atomPermutation $\langle\,$ Alternate ordering of atoms for RMSD computation$\,\rangle$
  Context: rmsd
  Acceptable values: List of atom numbers
  Description: If defined, this parameter defines a re-ordering (permutation) of the 1-based atom numbers that can be used to compute the RMSD, typically due to molecular symmetry. This parameter can be specified multiple times, each one defining a new permutation: the returned RMSD value is the minimum over the set of permutations. For example, if the atoms making up the group are 6, 7, 8, 9, and atoms 7, 8, and 9 are invariant by circular permutation (as the hydrogens in a CH3 group), a symmetry-adapted RMSD would be obtained by adding:
  atomPermutation 6 8 9 7
  atomPermutation 6 9 7 8
  Note that this does not affect the least-squares roto-translational fit, which is done using the topology ordering of atoms, and the reference positions in the order provided. Therefore, this feature is mostly useful when using custom fitting parameters within the atom group, such as fittingGroup, or when fitting is disabled altogether.

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

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 (rotateReference 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 (centerReference 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).

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:

$${ p(\{\mathbf{x}_{i}(t)\}, \{\mathbf{x}_{i}^{\mathrm{(ref)}}\}) }... ...athrm{(ref)}} - \mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}}) \right)\mathrm{,} }$$ (13.6)

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.

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)
• refPositions: see definition of refPositions (rmsd component)
• refPositionsFile: see definition of refPositionsFile (rmsd component)
• refPositionsCol: see definition of refPositionsCol (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue (rmsd component)

• vector $\langle\,$ Vector components$\,\rangle$
  Context: eigenvector
  Acceptable values: space-separated list of (x, y, z) triplets
  Description: This option (mutually exclusive with vectorFile) sets the values of the vector components.

• vectorFile $\langle\,$ file containing vector components$\,\rangle$
  Context: eigenvector
  Acceptable values: UNIX filename
  Description: This option (mutually exclusive with vector) sets the name of a coordinate file containing the vector components; the file is read according to the same format used for refPositionsFile. For a PDB file specifically, the components are read from the X, Y and Z fields. Note: The PDB file has limited precision and fixed-point numbers: in some cases, the vector components may not be accurately represented; a XYZ file should be used instead, containing floating-point numbers.

• vectorCol $\langle\,$ PDB column used to flag participating atoms$\,\rangle$
  Context: eigenvector
  Acceptable values: O or B
  Description: Analogous to atomsCol.

• vectorColValue $\langle\,$ Value used to flag participating atoms in the PDB file$\,\rangle$
  Context: eigenvector
  Acceptable values: positive decimal
  Description: Analogous to atomsColValue.

• differenceVector $\langle\,$ The $3n$ -dimensional vector is the difference between vector and refPositions$\,\rangle$
  Context: eigenvector
  Acceptable values: boolean
  Default value: off
  Description: If this option is on, the numbers provided by vector or vectorFile are interpreted as another set of positions, $\mathbf{x}_{i}'$ : the vector $\mathbf{v}_{i}$ is then defined as $\mathbf{v}_{i} = \left(\mathbf{x}_{i}' - \mathbf{x}_{i}^{\mathrm{(ref)}}\right)$ . This allows to conveniently define a colvar $\xi$ as a projection on the linear transformation between two sets of positions, ``A'' and ``B''. For convenience, the vector is also normalized so that $\xi = 0$ when the atoms are at the set of positions ``A'' and $\xi = 1$ at the set of positions ``B''.

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. 13.6, this range does not depend on the number of atoms involved.

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, $\mathbf{x}_{\mathrm{cog}}(t)$ :

$$R_{\mathrm{gyr}} \; = \; \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left\vert\mathbf{x}_{i}(t) - \mathbf{x}_{\mathrm{cog}}(t)\right\vert^{2} }$$ (13.7)

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

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)

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

$$I \; = \; \sum_{i=1}^{N} \left\vert\mathbf{x}_{i}(t) - \mathbf{x}_{\mathrm{cog}}(t)\right\vert^{2}$$ (13.8)

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 Å$^{2}$ .

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)

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 Å.

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)

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 $\mathbf{e}$ of the moment of inertia 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, $\mathbf{x}_{\mathrm{cog}}(t)$ :

$$I_{\mathbf{e}} \; = \; \sum_{i=1}^{N} \left(\left(\mathbf{x}_{i}(t) - \mathbf{x}_{\mathrm{cog}}(t)\right)\cdot\mathbf{e}\right)^{2}$$ (13.9)

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 Å$^{2}$ .

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)
• axis $\langle\,$ Projection axis (Å)$\,\rangle$
  Context: inertiaZ
  Acceptable values: (x, y, z) triplet
  Default value: (0.0, 0.0, 1.0)
  Description: The three components of this vector define (when normalized) the projection axis $\mathbf{e}$ .