List of available colvar components

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 13.4.4 for additional options):

• group1 $\langle\,$ First group of atoms$\,\rangle$
  Context: distance
  Acceptable values: Block group1 {...}
  Description: First group of atoms.

• group2: analogous to group1

• forceNoPBC $\langle\,$ Calculate absolute rather than minimum-image distance?$\,\rangle$
  Context: distance
  Acceptable values: boolean
  Default value: no
  Description: By default, in calculations with periodic boundary conditions, the distance component returns the distance according to the minimum-image convention. If this parameter is set to yes, PBC will be ignored and the distance between the coordinates as maintained internally will be used. This is only useful in a limited number of special cases, e.g. to describe the distance between remote points of a single macromolecule, which cannot be split across periodic cell boundaries, and for which the minimum-image distance might give the wrong result because of a relatively small periodic cell.

• oneSiteTotalForce $\langle\,$ Measure total force on group 1 only?$\,\rangle$
  Context: angle, dipoleAngle, dihedral
  Acceptable values: boolean
  Default value: no
  Description: If this is set to yes, the total force is measured along a vector field (see equation (13.19) in section 13.5.1) that only involves atoms of group1. This option is only useful for ABF, or custom biases that compute total forces. See section 13.5.1 for details.

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).

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.

List of keywords (see also 13.4.4 for additional options):

• main $\langle\,$ Main group of atoms$\,\rangle$
  Context: distanceZ
  Acceptable values: Block main {...}
  Description: Group of atoms whose position ${\mbox{\boldmath {$r$}}}$ is measured.

• ref $\langle\,$ Reference group of atoms$\,\rangle$
  Context: distanceZ
  Acceptable values: Block ref {...}
  Description: Reference group of atoms. The position of its center of mass is noted ${\mbox{\boldmath {$r$}}}_1$ below.

• ref2 $\langle\,$ Secondary reference group$\,\rangle$
  Context: distanceZ
  Acceptable values: Block ref2 {...}
  Default value: none
  Description: Optional group of reference atoms, whose position ${\mbox{\boldmath {$r$}}}_2$ can be used to define a dynamic projection axis: ${\mbox{\boldmath {$e$}}}=(\Vert {\mbox{\boldmath {$r$}}}_2 - {\mbox{\boldmath ... ...}_1\Vert)^{-1} \times ({\mbox{\boldmath {$r$}}}_2 - {\mbox{\boldmath {$r$}}}_1)$ . In this case, the origin is ${\mbox{\boldmath {$r$}}}_m = 1/2 ({\mbox{\boldmath {$r$}}}_1+{\mbox{\boldmath {$r$}}}_2)$ , and the value of the component is ${\mbox{\boldmath {$e$}}} \cdot ({\mbox{\boldmath {$r$}}}-{\mbox{\boldmath {$r$}}}_m)$ .

• axis $\langle\,$ Projection axis (Å)$\,\rangle$
  Context: distanceZ
  Acceptable values: (x, y, z) triplet
  Default value: (0.0, 0.0, 1.0)
  Description: The three components of this vector define a projection axis ${\mbox{\boldmath {$e$}}}$ for the distance vector ${\mbox{\boldmath {$r$}}} - {\mbox{\boldmath {$r$}}}_1$ joining the centers of groups ref and main. The value of the component is then ${\mbox{\boldmath {$e$}}} \cdot ({\mbox{\boldmath {$r$}}}-{\mbox{\boldmath {$r$}}}_1)$ . The vector should be written as three components separated by commas and enclosed in parentheses.

• forceNoPBC: see definition of forceNoPBC (distance component)

• oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)

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).

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 13.4.4 for additional options):

• main: see definition of main (distanceZ component)
• ref: see definition of ref (distanceZ component)
• ref2: see definition of ref2 (distanceZ component)
• axis: see definition of axis (distanceZ component)
• forceNoPBC: see definition of forceNoPBC (distance component)
• oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)

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 13.4.4 for additional options):

• group1: see definition of group1 (distance component)
• group2: analogous to group1
• oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)

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 $\mathbf{d} = (d_{x}, d_{y}, d_{z})$ , with $\vert\mathbf{d}\vert = 1$ .

List of keywords (see also 13.4.4 for additional options):

• group1: see definition of group1 (distance component)
• group2: analogous to group1
• oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)

distanceInv: mean distance between two groups of atoms.

The distanceInv {...} block defines a generalized mean distance between two groups of atoms 1 and 2, weighted with exponent $1/n$ :

$$d_{\mathrm{1,2}}^{[n]} \; = \; \left(\frac{1}{N_{\mathrm{1}}N_{\m... ...}\sum_{i,j} \left(\frac{1}{\Vert\mathbf{d}^{ij}\Vert}\right)^{n} \right)^{-1/n}$$ (13.2)

where $\Vert\mathbf{d}^{ij}\Vert$ 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 13.4.4 for additional options):

• group1: see definition of group1 (distance component)
• group2: analogous to group1
• oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)
• exponent $\langle\,$ Exponent $n$ in equation 13.2$\,\rangle$
  Context: distanceInv
  Acceptable values: positive even integer
  Default value: 6
  Description: Defines the exponent to which the individual distances are elevated before averaging. The default value of 6 is useful for example to applying restraints based on NOE-measured distances.

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

distancePairs: set of pairwise distances between two groups.

The distancePairs {...} block defines a $N_{\mathrm{1}}\times{}N_{\mathrm{2}}$ -dimensional variable that includes all mutual distances between the atoms of two groups. This can be useful, for example, to develop a new variable defined over two groups, by using the scriptedFunction feature.

List of keywords (see also 13.4.4 for additional options):

• group1: see definition of group1 (distance component)
• group2: analogous to group1
• forceNoPBC: see definition of forceNoPBC (distance component)

This component returns a $N_{\mathrm{1}}\times{}N_{\mathrm{2}}$ -dimensional vector of numbers, each ranging from 0 to the largest possible distance within the chosen boundary conditions.

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 $(x_1, y_1, z_1, \cdots, x_n, y_n, z_n)$ .

List of keywords (see also 13.4.4 for additional options):

• atoms $\langle\,$ Group of atoms$\,\rangle$
  Context: cartesian
  Acceptable values: Block atoms {...}
  Description: Defines the atoms whose coordinates make up the value of the component. If rotateReference or centerReference are defined, coordinates are evaluated within the moving frame of reference.

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 13.4.4 for additional options):

• group1: see definition of group1 (distance component)
• group2: analogous to group1
• group3: analogous to group1
• oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)

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 13.4.4 for additional options):

• group1: see definition of group1 (distance component)
• group2: analogous to group1
• group3: analogous to group1
• oneSiteTotalForce: see definition of oneSiteTotalForce (distance 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 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 13.4.4 for additional options):

• group1: see definition of group1 (distance component)
• group2: analogous to group1
• group3: analogous to group1
• group4: analogous to group1
• oneSiteTotalForce: see definition of oneSiteTotalForce (distance 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 [45]. This function is summed over all pairs of atoms in group1 and group2:

$$C (\mathtt{group1}, \mathtt{group2}) \; = \; \sum_{i\in\mathtt{gr... ...}\vert/d_{0})^{n}}{ 1 - (\vert\mathbf{x}_{i}-\mathbf{x}_{j}\vert/d_{0})^{m} } }$$ (13.3)

List of keywords (see also 13.4.4 for additional options):

• group1: see definition of group1 (distance component)

• group2: analogous to group1

• cutoff $\langle\,$ ``Interaction'' distance (Å)$\,\rangle$
  Context: coordNum
  Acceptable values: positive decimal
  Default value: 4.0
  Description: This number defines the switching distance to define an interatomic contact: for $d \ll d_0$ , the switching function $(1-(d/d_0)^{n})/(1-(d/d_0)^{m})$ is close to 1, at $d = d_0$ it has a value of $n/m$ ($1/2$ with the default $n$ and $m$ ), and at $d \gg d_0$ it goes to zero approximately like $d^{m-n}$ . Hence, for a proper behavior, $m$ must be larger than $n$ .

• cutoff3 $\langle\,$ Reference distance vector (Å)$\,\rangle$
  Context: coordNum
  Acceptable values: ``(x, y, z)'' triplet of positive decimals
  Default value: (4.0, 4.0, 4.0)
  Description: The three components of this vector define three different cutoffs $d_{0}$ for each direction. This option is mutually exclusive with cutoff.

• expNumer $\langle\,$ Numerator exponent$\,\rangle$
  Context: coordNum
  Acceptable values: positive even integer
  Default value: 6
  Description: This number defines the $n$ exponent for the switching function.

• expDenom $\langle\,$ Denominator exponent$\,\rangle$
  Context: coordNum
  Acceptable values: positive even integer
  Default value: 12
  Description: This number defines the $m$ exponent for the switching function.

• group2CenterOnly $\langle\,$ Use only group2's center of mass$\,\rangle$
  Context: coordNum
  Acceptable values: boolean
  Default value: off
  Description: If this option is on, only contacts between each atoms in group1 and the center of mass of group2 are calculated (by default, the sum extends over all pairs of atoms in group1 and group2). If group2 is a dummyAtom, this option is set to yes by default.

This component returns a dimensionless number, which ranges from approximately 0 (all interatomic distances are much larger than the cutoff) to $N_{\mathtt{group1}} \times N_{\mathtt{group2}}$ (all distances are less than 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 or group2CenterOnly should be used: the cost of the loop over all pairs grows as $N_{\mathtt{group1}} \times N_{\mathtt{group2}}$ .

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:

$$C (\mathtt{group1}) \; = \; \sum_{i\in\mathtt{group1}}\sum_{j > i... ...}\vert/d_{0})^{n}}{ 1 - (\vert\mathbf{x}_{i}-\mathbf{x}_{j}\vert/d_{0})^{m} } }$$ (13.4)

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 13.4.4 for additional options):

• group1: see definition of group1 (coordNum component)
• group2: analogous to group1
• cutoff: see definition of cutoff (coordNum component)
• cutoff3: see definition of cutoff3 (coordNum component)
• expNumer: see definition of expNumer (coordNum component)
• expDenom: see definition of expDenom (coordNum component)

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

hBond: hydrogen bond between two atoms.

The hBond {...} block defines a hydrogen bond, implemented as a coordination number (eq. 13.3) 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 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).

List of keywords (see also 13.4.4 for additional options):

• acceptor $\langle\,$ Number of the acceptor atom$\,\rangle$
  Context: hBond
  Acceptable values: positive integer
  Description: Number that uses the same convention as atomNumbers.
• donor: analogous to acceptor
• cutoff: see definition of cutoff (coordNum component)
  Note: default value is 3.3 Å.
• expNumer: see definition of expNumer (coordNum component)
  Note: default value is 6.
• expDenom: see definition of expDenom (coordNum component)
  Note: default value is 8.

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 [46], 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 13.4.4 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. If only centerReference is on, the list can be a single (x, y, z) triplet; if also rotateReference is on, the list should be as long as the atom group. This option is independent from that with the same keyword within the atoms {...} block (see 13.3.2). The latter (and related fitting options for the atom group) are normally not needed, and should be omitted altogether except for advanced usage cases.

• refPositionsFile $\langle\,$ Reference coordinates file$\,\rangle$
  Context: rmsd
  Acceptable values: UNIX filename
  Description: This option (mutually exclusive with refPositions) sets the file name for the reference coordinates to be compared with. The format is the same as that provided by refPositionsFile within an atom group's definition (see 13.3.2).

• 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.

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 (translateReference 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 decribe 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;
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.

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 13.4.4 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 $\langle\,$ PDB file containing vector components$\,\rangle$
  Context: eigenvector
  Acceptable values: UNIX filename
  Description: This option (mutually exclusive with vector) sets the name of a PDB file where the vector components will be read from the X, Y, and Z fields. Note: The PDB file has limited precision and fixed point numbers: in some cases, the vector may not be accurately represented, and vector should be used instead.

• 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 13.4.4 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 13.4.4 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 13.4.4 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 $\mathbf{e}$ .

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 [46]. 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 is expressed as ``(0.707, 0.0, 0.0, 0.707)''. The script quaternion2rmatrix.tcl provides Tcl functions for converting to and from a $4\times{}4$ rotation matrix in a format suitable for usage in VMD.

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

Note: refPositionsand 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 13.4.4 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)

• closestToQuaternion $\langle\,$ Reference rotation$\,\rangle$
  Context: orientation
  Acceptable values: ``(q0, q1, q2, q3)'' quadruplet
  Default value: (1.0, 0.0, 0.0, 0.0) (``null'' rotation)
  Description: Between the two equivalent quaternions $(q_0, q_1, q_2, q_3)$ and $(-q_0, -q_1, -q_2, -q_3)$ , the closer to (1.0, 0.0, 0.0, 0.0) is chosen. This simplifies the visualization of the colvar trajectory when samples values are a smaller subset of all possible rotations. Note: this only affects the output, never the dynamics.

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 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.

orientationAngle: angle of rotation from reference coordinates.

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

List of keywords (see also 13.4.4 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)

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, refPositionsCol and refPositionsColValue. The returned value is the cosine of the angle of rotation $\theta$ between the current and the reference positions. The range of values is [-1:1].

List of keywords (see also 13.4.4 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)

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 13.4.4 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)
• axis $\langle\,$ Special rotation axis (Å)$\,\rangle$
  Context: tilt
  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 special rotation axis used to calculate the tilt and spinAngle components.

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$^\circ$ (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 $-1$ .

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 $\theta$ 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$^\circ$ ), and the value $-1$ represents an anti-parallel orientation.

List of keywords (see also 13.4.4 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)
• axis: see definition of axis (spinAngle 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:

$${ \alpha\left( \mathrm{C}_{\alpha}^{(N_{0})}, \mathrm{O}^{(N_{0})... ...thrm{N}^{(N_{0}+N)}, \mathrm{C}_{\alpha}^{(N_{0}+N)} \right) } \; = \; \; \; \;$$ (13.10)

$$\; \; \; \; { \frac{1}{2(N-2)} \sum_{n=N_{0}}^{N_{0}+N-2} \mathrm... ...-4} \mathrm{hbf}\left( \mathrm{O}^{(n)}, \mathrm{N}^{(n+4)}\right) \mathrm{,} }$$

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

$${ \mathrm{angf}\left( \mathrm{C}_{\alpha}^{(n)}, \mathrm{C}_{\alp... ...eta_{0}\right)^{4} / \left(\Delta\theta_{\mathrm{tol}}\right)^{4}} \mathrm{,} }$$ (13.11)

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.

List of keywords (see also 13.4.4 for additional options):

• residueRange $\langle\,$ Potential $\alpha$ -helical residues$\,\rangle$
  Context: alpha
  Acceptable values: ``$<$ Initial residue number$>$ -$<$ Final residue number$>$ ''
  Description: This option specifies the range of residues on which this component should be defined. The Colvars module looks for the atoms within these residues named ``CA'', ``N'' and ``O'', and raises an error if any of those atoms is not found.

• psfSegID $\langle\,$ PSF segment identifier$\,\rangle$
  Context: alpha
  Acceptable values: string (max 4 characters)
  Description: This option sets the PSF segment identifier for the residues specified in residueRange. This option is only required when PSF topologies are used.

• hBondCoeff $\langle\,$ Coefficient for the hydrogen bond term$\,\rangle$
  Context: alpha
  Acceptable values: positive between 0 and 1
  Default value: 0.5
  Description: This number specifies the contribution to the total value from the hydrogen bond terms. 0 disables the hydrogen bond terms, 1 disables the angle terms.

• angleRef $\langle\,$ Reference $\mathrm{C}_{\alpha} - \mathrm{C}_{\alpha} - \mathrm{C}_{\alpha}$ angle$\,\rangle$
  Context: alpha
  Acceptable values: positive decimal
  Default value: 88$^{\circ}$
  Description: This option sets the reference angle used in the score function (13.12).

• angleTol $\langle\,$ Tolerance in the $\mathrm{C}_{\alpha} - \mathrm{C}_{\alpha} - \mathrm{C}_{\alpha}$ angle$\,\rangle$
  Context: alpha
  Acceptable values: positive decimal
  Default value: 15$^{\circ}$
  Description: This option sets the angle tolerance used in the score function (13.12).

• hBondCutoff $\langle\,$ Hydrogen bond cutoff$\,\rangle$
  Context: alpha
  Acceptable values: positive decimal
  Default value: 3.3 Å
  Description: Equivalent to the cutoff option in the hBond component.

• hBondExpNumer $\langle\,$ Hydrogen bond numerator exponent$\,\rangle$
  Context: alpha
  Acceptable values: positive integer
  Default value: 6
  Description: Equivalent to the expNumer option in the hBond component.

• hBondExpDenom $\langle\,$ Hydrogen bond denominator exponent$\,\rangle$
  Context: alpha
  Acceptable values: positive integer
  Default value: 8
  Description: Equivalent to the expDenom option in the hBond component.

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

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.[47] and documented in detail by Altis et al.[48]. 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.[49]

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:

$$\xi = \sum_{n=1}^{N-1} k_{4n-3} \cos(\psi_n) + k_{4n-2} \sin (\psi_n) + k_{4n-1} \cos (\phi_{n+1}) + k_{4n} \sin(\phi_{n+1})$$ (13.12)

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

List of keywords (see also 13.4.4 for additional options):

• residueRange: see definition of residueRange (alpha component)

• psfSegID: see definition of psfSegID (alpha component)

• hBondCoeff: see definition of hBondCoeff (alpha component)

• angleRef: see definition of angleRef (alpha component)

• angleTol: see definition of angleTol (alpha component)

• hBondCutoff: see definition of hBondCutoff (alpha component)

• hBondExpNumer: see definition of hBondExpNumer (alpha component)

• hBondExpDenom: see definition of hBondExpDenom (alpha component)

• vectorFile $\langle\,$ File containing dihedral PCA eigenvector(s)$\,\rangle$
  Context: dihedralPC
  Acceptable values: file name
  Description: A text file containing the coefficients of dihedral PCA eigenvectors on the cosine and sine coordinates. The vectors should be arranged in columns, as in the files output by Carma.[49]

• vectorNumber $\langle\,$ File containing dihedralPCA eigenvector(s)$\,\rangle$
  Context: dihedralPC
  Acceptable values: positive integer
  Description: Number of the eigenvector to be used for this component.