# Collective Variables¶

Collective variables (CVs) are arbitrary differentiable functions of the $$3N$$ Cartesian coordinates of the atoms in a simulation. These usually represent some structurally, thermodynamically, or chemically meaningful quantity along which advanced sampling can be performed. Listed below are the collective variables currently supported in SSAGES, along with a brief description and information on how the syntax is implemented in SSAGES. In addition to specific properties for each CV, a name property may be defined for any CV which is used to reference the CV within the methods of SSAGES.

"name" : "mycvname"


The name specified for a CV must be unique. It is possible, however, to omit this and simply reference a CV by its numerical index.

Note

We tacitly assume we are working with spherical atoms that join together to form molecules.

## Angle¶

### Description¶

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

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

This can be helpful when probing the conformations of a molecule to understand its stable and metastable states. Angles do not have to be defined between bonded atoms.

### Example¶

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


Warning

The angle must be between three individual particles rather than the centers-of-mass of particle groups.

### 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.

## ANNCV¶

### Description¶

This CV takes scaled (specified by scaling_factor) Cartesian coordinates of a group of atoms (specified by atomids) as inputs to a neural network (its number of nodes, connection weights, and activation functions are specified by num_nodes, coeff_file, activations, respectively), computes one component (specified by out_index) of the final neural network outputs as the CV value. The coefficients of the neural network can be obtained from any feed-forward neural networks trained with Cartesian coordinates as inputs. Examples of the neural networks include the encoders of the autoencoders in the MESA framework, or State-free Reversible VAMPnets.

In the following example, we define an ANN CV which takes Cartesian coordinates of atoms [2, 5, 7, 9, 15, 17, 19], scaled by factor 0.5, as inputs to a neural network with node numbers [21, 40, 2] and activation functions ["Tanh", "Tanh"], and weights defined in file autoencoder_info_1.txt (which stores weights for the neural network), and outputs two components (marked as index 0 and 1) as CVs.

### Example¶

"CVs": [
{
"type": "ANNCV",
"atom_ids": [2, 5, 7, 9, 15, 17, 19],
"scaling_factor": 0.5,
"num_nodes": [21, 40, 2],
"activations": ["Tanh", "Tanh"],
"index": 0,
"coeff_file": "autoencoder_info_1.txt"
},
{
"type": "ANNCV",
"atom_ids": [2, 5, 7, 9, 15, 17, 19],
"scaling_factor": 0.5,
"num_nodes": [21, 40, 2],
"activations": ["Tanh", "Tanh"],
"index": 1,
"coeff_file": "autoencoder_info_1.txt"
}
]


### Options & Parameters¶

#### Required¶

"type": "ANNCV"


Selects this collective variable.

"atom_ids"


Property atom_ids must contain integers consisting of the atom ID for the inputs of ANN.

"scaling_factor"


Property scaling_factor is the scaling factor of the inputs.

"num_nodes"


Property num_nodes defines the number of nodes for each layer of the neural network.

"activations"


Property activations defines the activation functions for each layer of the neural network.

"coeff_file"


Property coeff_file defines the file which stores weights for the neural network.

"index"


Property index defines the output index we want to use for CV.

## Box Volume¶

### Description¶

The current volume of a simulation box is an important parameter determining the thermodynamic state. Constant-pressure simulations where volume information is recorded may be reweighted according to standard methods [5]. This CV calculates the box volume as the determinant of the Parrinello-Rahman matrix $$\mathbf{H}$$,

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

### Example¶

{
"type" : "BoxVolume"
}


Warning

Non-orthorhombic boxes are currently not supported. Only Gromacs and LAMMPS are currently supported

### Options & Parameters¶

#### Required¶

"type"


Property type must be set to string "BoxVolume".

## Gyration Tensor¶

### Description¶

This CV calculates quantities derived from the symmetric mass-weighted gyration tensor of a group of $$N$$ atoms defined as,

$\mathbf{S} = \frac{1}{\sum_{i=1}^{N}{m_i}}\sum_{i=1}^{N}{m_i \left( \mathbf{r}_i - \mathbf{r}_\mathrm{COM}\right) \otimes \left( \mathbf{r}_i - \mathbf{r}_\mathrm{COM}\right)}$

where $$m_i$$ is the mass and $$\mathbf{r}_i$$ is the vector of coordinates of the $$i^{\mathrm{th}}$$ atom, $$\mathbf{r}_\mathrm{COM}$$ is the vector of the center of mass of all $$N$$ atoms in the group, and $$\otimes$$ is the outer, or tensor, product.

The eigenvalues of the radius of gyration tensor are particularly useful as collective variables which quantify the conformation of a molecule (such as a long polymer) or the shape of a given assembly of molecules. With eigenvalues of $$\lambda_x^2,~\lambda_y^2,~\lambda_z^2$$ (in increasing order) defined in the frame of the principal axes of inertia, the following quantities may be computed:

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

#### 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¶

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

### Example¶

This example computes the shape anisotropy of a ten-atom group.

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


### 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".

#### Optional¶

"dimension"


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

## Particle Coordinate¶

### Description¶

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

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

### Example¶

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


### 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¶

### 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 $$\mu$$.

### Example¶

This example uses a Gaussian pairwise kernel to compute contributions from contact-type interactions between two atoms of size 1.0.

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


### 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:

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

This type of kernel is useful to select between conformations which have a different position of (e.g.) neighbors and next nearest neighbors in a particle cluster. Selection of particle separations approximates a math:delta distribution.

###### 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:

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

This quantity is useful for measuring how many atoms in group 2 occupy a spherical shell around atoms in group 1. The form is chosen so that the variable is continuous and differentiable. Through tuning $$n$$ and $$m$$ this can be made arbitrarily close to a Heaviside switching function.

###### 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],
"dimension" : [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-element array of numbers defining the reference point in the simulation box.

#### Optional¶

"dimension"


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

## Particle Separation¶

### Description¶

This CV calculates the distance between the centers of mass of two groups of atoms. The variable is unsigned, as the distance is a magnitude.

### Example¶

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


### 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-element 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¶

### 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. This CV can be helpful to bias the conformations of both moderate-size and long-chain proteins and polymers.

### 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]
]
}


### 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¶

### Description¶

This CV calculates the dihedral angle, in radians, formed by four atoms $$i,j,k,l$$. It is computed as in [3],

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

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

Warning

The torsional angle can only be defined between four atoms rather than four groups of atoms.

### Example¶

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


### 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¶

### 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 switching functions applied to the RMSD rather than to coordinates; in future versions, the user will be able to choose custom parameters for the switching function.

Note

Unlike the simpler CVs discussed above, this one takes atomic labels in the form of a reference PDB structure. This is true of all protein-like CVs below which compare to a reference structure.

Warning

Since the definition of this CV uses nanometers as a unit length, you must specify the unitconv parameter, as outlined below, in order to apply this CV when that is not the base unit of length.

### Example¶

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


### 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¶

### 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 switching functions applied to the RMSD rather than to coordinates; in future versions, the user will be able to choose custom parameters for the switching function.

Note

Unlike the simpler CVs discussed above, this one takes atomic labels in the form of a reference PDB structure. This is true of all protein-like CVs below which compare to a reference structure.

Warning

Since the definition of this CV uses nanometers as a unit length, you must specify the unitconv parameter, as outlined below, in order to apply this CV when that is not the base unit of length.

### Example¶

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


### 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¶

### 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 switching functions applied to the RMSD rather than to coordinates; in future versions, the user will be able to choose custom parameters for the switching function.

Note

Unlike the simpler CVs discussed above, this one takes atomic labels in the form of a reference PDB structure. This is true of all protein-like CVs below which compare to a reference structure.

Warning

Since the definition of this CV uses nanometers as a unit length, you must specify the unitconv parameter, as outlined below, in order to apply this CV when that is not the base unit of length.

### Example¶

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


### 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.