# Collective Variables¶

Collective variables (CVs) are arbitrary differentiable functions of the 3N Cartesian coordinates of the atoms in a simulation. They usually represent some chemically meaningful pathway along which advanced sampling can be performed. Listed below are the collective variables currently supported in SSAGES. In addition to specific properties for each CV, a name property can be defined for any CV which can be used to reference the CV from a method or other supported location.

"name" : "mycvname"


Specified names for CVs must be unique. It is possible to omit a name and reference a CV by its index as well.

## Angle¶

### Example¶

{
"type" : "Angle",
"atom_ids" : [0, 1, 2]
}


### Description¶

This CV calculates the bend angle, in radians, formed between three selected atoms $$i,j,k$$,

$\xi = \cos^{-1}\left(\frac{\mathbf{r}_{ij} \cdot \mathbf{r}_{kj}}{\Vert \mathbf{r}_{ij} \Vert \Vert \mathbf{r}_{kj} \Vert} \right).$

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "Angle".

"atom_ids"


Property atom_ids must contain three integers consisting of the atom ID forming the angle of interest.

## Box Volume¶

Warning

Non-orthorhombic boxes are currently not supported.

Note

Currently supported only in Gromacs and LAMMPS.

### Example¶

{
"type" : "BoxVolume"
}


### Description¶

This CV calculates the box volume as the determinant of the Parrinello-Rahman matrix $$\mathbf{H}$$,

$\xi = \det\left( H_{ij} \right)$

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "BoxVolume".

## Gyration Tensor¶

### Example¶

"type" : "GyrationTensor",
"atom_ids" : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
"component" : "shapeaniso"


### Description¶

This CV calculates components of the mass-weighted* gyration tensor defined as,

$S_{mn} = \frac{1}{N}\sum_{i=1}^{N}{r_m^i r_n^i}$

where $$r_m$$ is the coordinate of the $$m^{\mathrm{th}}$$ atom in the interial frame. With eigenvalues of $$\lambda_x^2, \lambda_y^2, \lambda_z^2$$, possible components to use as a CV include:

#### Radius of Gyration (Squared)¶

$R_g^2 = \lambda_x^2 + \lambda_y^2 + \lambda_z^2$

#### Principal Moment¶

$\lambda_i^2,\ i \in \{x,y,z\}$

#### Asphericity¶

$b = \lambda_z^2 - \frac{1}{2}\left(\lambda_x^2 + \lambda_y^2 \right)$

#### Acylindricity¶

$c = \lambda_y^2 - \lambda_x^2$

#### Shape Anisotropy¶

$\kappa^2 = \frac{3}{2}\frac{\lambda_x^4+\lambda_y^4+\lambda_z^4}{\left(\lambda_x^2+\lambda_y^2+\lambda_z^2\right)^2}-\frac{1}{2}$

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "GyrationTensor".

"atom_ids"


Property atom_ids must be an array of integers containing the atom IDs which will enter the calculation.

"component"


Property component must be a string defining the gyration tensor component of interest. Valid options are "Rg", "principal1", "principal2", "principal3", "asphericity", "acylindricity", or "shapeaniso".

## Particle Coordinate¶

### Example¶

{
"type" : "ParticleCoordinate",
"atom_ids" : [1, 5, 6, 10],
"dimension" : "x"
}


### Description¶

This CV calculates the $$x$$, $$y$$ or $$z$$ component center of mass of a group of atoms.

$\xi = \frac{1}{\sum_i{m^i}}\sum_{i=1}^{N}{r_\alpha^i}\ \ \ \alpha \in {x,y,z}$

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "ParticleCoordinate".

"atom_ids"


Property atom_ids must be an array of integers containing the atom IDs which will enter the calculation.

"dimension"


Property dimension must be a string defining the Cartesian component of interest "x", "y", or "z".

## Pairwise¶

### Example¶

{
"type" : "Pairwise",
"group1" : [1, 5],
"group2" : [2, 3, 4, 6, 7, 8],
"kernel" : {
"type" : "gaussian",
"mu" : 1.0,
"sigma" : 0.2
}
}


### Description¶

This CV calculates a variety of pairwise properties. The functions (kernels) used are continous analogs for otherwise discontinuous CVs. If parameters are chosen judiciously, these kernels can be used in place of some standard, discontinuous CVs. A Gaussian kernel can emulate a count of nearest neighbors; a switching function kernel can emulate a coordination number.

$\xi = \sum_{i \in A}\sum_{i \in B}{f_{ij}}$

where $$f_{ij}$$ is a pairwise function for atoms $$i$$ and $$j$$. are at a distance of the center of the Gaussian, $$r_{ij}=\mu$$, and decreases to zero as the distance deviates away from $$\mu$$.

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "Pairwise".

"group1"


Property group1 must be an array of integers containing the atom IDs in the first set.

"group2"


Property group2 must be an array of integers containing the atom IDs in the second set.

Note

Atoms can exist in both group1 and group2 simultaneously. Contacts are automatically skipped if $$i = j$$.

"kernel"


Property kernel must be an object defining the properties of the pairwise kernel function and its associated properties.

#### Pairwise Kernels¶

##### Gaussian Function¶

The Gaussian function is defined as:

$g_{ij} = e^{-\frac{\left(r_{ij} - \mu\right)^2}{2 sigma^2}}.$
###### Properties¶
"mu"


Property mu is required and must be numeric.

"sigma"


Property sigma is required and must be numeric.

##### Rational Switching Function¶

The rational switching function is defined as:

$s_{ij} = \frac{1-\left(\frac{r_{ij} - d_0}{r_0}\right)^n}{1-\left(\frac{r_{ij} - d_0}{r_0}\right)^m}.$
###### Properties¶
"type"


Property type must be set to string "rationalswitch".

"d0"


Property d0 is required and must be numeric.

"r0"


Property r0 is required and must be numeric.

"n"


Property n is required and must be an integer.

"m"


Property m is required and must be an integer.

## Particle Position¶

### Example¶

{
"type" : "ParticlePosition",
"atom_ids" : [1, 5, 6, 10],
"fix" : [true, false, true],
"position" : [3.51, 6.66, 2.14]
}


### Description¶

This CV calculates the distance of the center of mass of a group of atoms from a particular point in Cartesian space.

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "ParticlePosition".

"atom_ids"


Property atom_ids must be an array of integers containing the atom IDs which will enter the calculation.

"position"


Property position must be a 3-dimensional array of numbers defining the reference point in the simulation box.

"fix"


Property fix must be a 3-dimensional array of booleans specifying the components of the distance vector to include in the calculation.

## Particle Separation¶

### Example¶

{
"type" : "ParticleSeparation",
"group1" : [1],
"group2" : [5, 6, 10]
}


### Description¶

This CV calculates the distance between the centers of mass of two groups of atoms.

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "ParticleSeparation".

"group1"


Property group1 must be an array of integers containing the atom ID(s) which make up the first group of atoms. The CV will calculate the distance between the center of mass of this group and the group defined by property group2.

"group2"


Property group2 must be an array of integers containing the atom ID(s) which make up the second group of atoms. The CV will calculate the distance between the center of mass of this group and the group defined by property group1.

#### Optional¶

"dimension"


Property dimension is a 3-dimensional array of booleans specifying which Cartesian components to include in the calculation. If left unspecified, all three xyz components will be used.

## Polymer Rouse Modes¶

### Example¶

{
"type": "RouseMode",
"mode": 1,
"groups":  [
[ 1, 2, 3, 4, 5],
[ 6, 7, 8, 9,10],
[11,12,13,14,15],
[16,17,18,19,20],
[21,22,23,24,25],
[26,27,28,29,30],
[31,32,33,34,35],
[36,37,38,39,40],
[41,42,43,44,45],
[46,47,48,49,50]
]
}


### Description¶

This CV calculates the magnitude of a given Rouse mode for a set of atoms as

$X_p = \sqrt{\mathbf{X}_p\cdot\mathbf{X}_p},$

with the :math: p th Rouse mode defined as

$\mathbf{X}_p = \sqrt{\frac{c_p}{N}}\sum_{i=1}^N \mathbf{R}_i \cos \Bigl[\frac{p\pi}{N}\bigl(i-\frac{1}{2}\bigr) \Bigr],$

where :math: N is the number of groups or beads comprising the polymer, :math: mathbf{R}_i is the center-of-mass of the :math: i th bead, and :math: c_p is a constant equal to 1 for :math: p=0 and equal to 2 for :math: p=1,cdots,N-1.

### Required¶

"type"


Property mode must be set to string "RouseMode".

"groups"


Property groups is an array of arrays containing the atom IDs (as integers) that comprise the discretized polymer beads. The number of groups provided implicitly defines :math: N, the number of polymer beads.

"mode"


Property mode is an integer indicating the index of the desired Rouse mode. Valid values range from 0 up to one less than the number of groups, or 0,cdots, N-1.

## Torsional Angle¶

### Example¶

{
"type" : "Torsional",
"atom_ids" : [1, 5, 6, 10]
}


### Description¶

This CV calculates the dihedral angle, in radians, formed by four atoms $$i,j,k,l$$. It is computed as,

$\xi = \tan^{-1}\left( \frac{\left[(r_{lk} \times r_{jk}) \times (r_{ij} \times r_{jk}) \right] \cdot \frac{r_{jk}}{\Vert r_{jk}\Vert}}{(r_{lk} \times r_{jk}) \cdot (r_{ij} \times r_{jk}) } \right).$

Specifically, the function atan2 is used for the inverse tangent calculation to yield a four-quadrant angle.

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "Torsional".

"atom_ids"


Property atom_ids must be an array of 4 integers containing the atom IDs which form the dihedral.

## Alpha Helix RMSD¶

### Example¶

{
"type" : "AlphaRMSD",
"residue_ids" : [3, 21],
"reference" : "reference_structure.pdb",
"unitconv" : 10
}


### Description¶

This CV calculates alpha helix character by comparision to an “ideal” alpha helix structure composed of 6 amino acids. This is computed by performing a summation over all possible sequences of 6 consecutive amino acids in the segment of interest:

$\xi = \sum_i \frac{1 - \left(\frac{r_i}{0.1\text{ nm}}\right)^8}{1 - (\frac{r_i}{0.1\text{ nm}})^{12}}$

where $$r_i$$ is the pairwise RMSD calculated between the backbone atoms in the 6 amino acid sequence and the ideal reference structure. 5 backbone atoms are used for each amino acid, so each pairwise RMSD is calculated between two sets of 30 atoms. In the case of glycine, the HA1 atom is used in place of CB backbone atom.

Note

Note that this CV is basically a summation of a switching function; in the future the user will be able to choose custom parameters for the switching function.

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "AlphaRMSD".

"residue_ids"


Property residue_ids must be an array of two integers designating the range of amino acids for which to calculate the CV. The indices of the amino acids must match those from the reference structure provided in the property reference. The smaller index must be listed first, and the range must span at least 6 amino acids.

"reference"


Property reference must be a string containing the name of a reference pdb structure. This reference pdb structure is used along with the residue range defined in residue_ids to check for alpha helix character. For now, all residues in the system must be numbered in increasing order, even if they belong to separate chains. For example, if your system has two chains of 20 amino acids each, the first amino acid in the second chain should be numbered 21.

#### Optional¶

"unitconv"


Property unitconv must be numeric. This factor is used to reconcile the internal MD units for your engine and the units used in the ideal alpha helix reference structure. If your engine uses units of nanometers, this can be ignored. Otherwise, unitconv must be set to the equivalent number of length units in your MD engine equal to 1 nm. For example, if your default unit length is in angstroms, unitconv will be set to 10.

## Anti Beta RMSD¶

### Example¶

{
"type" : "AntiBetaRMSD",
"residue_ids" : [3, 21],
"reference" : "reference_structure.pdb",
"unitconv" : 10,
"mode" : 0
}


### Description¶

This CV calculates anti beta-sheet character by comparision to an “ideal” anti beta-sheet structure composed of 6 amino acids. This is computed by performing a summation over all possible sequences of 6 amino acids, consisting of two segments of 3 consecutive amino acids each, in the region of interest.

$\xi = \sum_i \frac{1 - \left(\frac{r_i}{0.1\text{ nm}}\right)^8}{1 - (\frac{r_i}{0.1\text{ nm}})^{12}}$

where $$r_i$$ is the pairwise RMSD calculated between the backbone atoms in the 6 amino acid sequence and the ideal reference structure. 5 backbone atoms are used for each amino acid, so each pairwise RMSD is calculated between two sets of 30 atoms. In the case of glycine, the HA1 atom is used in place of CB backbone atom.

Note

Note that this CV is basically a summation of a switching function; in the future the user will be able to choose custom parameters for the switching function.

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "AntiBetaRMSD".

"residue_ids"


Property residue_ids must be an array of two integers designating the range of amino acids for which to calculate the CV. The indices of the amino acids must match those from the reference structure provided in the property reference. The smaller index must be listed first, and the range must span at least 6 amino acids.

"reference"


Property reference must be a string containing the name of a reference pdb structure. This reference pdb structure is used along with the residue range defined in residue_ids to check for anti beta-sheet character. For now, all residues in the system must be numbered in increasing order, even if they belong to separate chains. For example, if your system has two chains of 20 amino acids each, the first amino acid in the second chain should be numbered 21.

#### Optional¶

"unitconv"


Property unitconv must be numeric. This factor is used to reconcile the internal MD units for your engine and the units used in the ideal anti beta-sheet reference structure. If your engine uses units of nanometers, this can be ignored. Otherwise, unitconv must be set to the equivalent number of length units in your MD engine equal to 1 nm. For example, if your default unit length is in angstroms, unitconv will be set to 10.

"mode"


Property mode is an integer specifying whether to calculate beta-sheets formed only between residues on the same chain (intra) or only between residues on separate chains (inter). If mode is set to 0, both modes will be used. A value of 1 selects for the intra mode; a value of 2 selects for inter mode.

## Parallel Beta RMSD¶

### Example¶

{
"type" : "ParallelBetaRMSD",
"residue_ids" : [3, 21],
"reference" : "reference_structure.pdb",
"unitconv" : 10,
"mode" : 0
}


### Description¶

This CV calculates anti beta-sheet character by comparision to an “ideal” parallel beta-sheet structure composed of 6 amino acids. This is computed by performing a summation over all possible sequences of 6 amino acids, consisting of two segments of 3 consecutive amino acids each, in the region of interest.

$\xi = \sum_i \frac{1 - \left(\frac{r_i}{0.1\text{ nm}}\right)^8}{1 - (\frac{r_i}{0.1\text{ nm}})^{12}}$

where $$r_i$$ is the pairwise RMSD calculated between the backbone atoms in the 6 amino acid sequence and the ideal reference structure. 5 backbone atoms are used for each amino acid, so each pairwise RMSD is calculated between two sets of 30 atoms. In the case of glycine, the HA1 atom is used in place of CB backbone atom.

Note

Note that this CV is basically a summation of a switching function; in the future the user will be able to choose custom parameters for the switching function.

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "ParallelBetaRMSD".

"residue_ids"


Property residue_ids must be an array of two integers designating the range of amino acids for which to calculate the CV. The indices of the amino acids must match those from the reference structure provided in the property reference. The smaller index must be listed first, and the range must span at least 6 amino acids.

"reference"


Property reference must be a string containing the name of a reference pdb structure. This reference pdb structure is used along with the residue range defined in residue_ids to check for parallel beta-sheet character. For now, all residues in the system must be numbered in increasing order, even if they belong to separate chains. For example, if your system has two chains of 20 amino acids each, the first amino acid in the second chain should be numbered 21.

#### Optional¶

"unitconv"


Property unitconv must be numeric. This factor is used to reconcile the internal MD units for your engine and the units used in the ideal parallel beta-sheet reference structure. If your engine uses units of nanometers, this can be ignored. Otherwise, unitconv must be set to the equivalent number of length units in your MD engine equal to 1 nm. For example, if your default unit length is in angstroms, unitconv will be set to 10.

"mode"


Property mode is an integer specifying whether to calculate beta-sheets formed only between residues on the same chain (intra) or only between residues on separate chains (inter). If mode is set to 0, both modes will be used. A value of 1 selects for the intra mode; a value of 2 selects for inter mode.