Collective Variables
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Distance
Angle
Coordination
Shape
Special / Path
Energy
Algebra
Introduction
In the framework of potential of mean force (PMF) calculations, collective variables (CVs) define the reduced space in which the free energy is calculated. The PMF is expressed as a function of a selected set of collective variables , while the remaining degrees of freedom are not treated explicitly in this representation but are averaged over statistically by a computer simulation. Thus, represents the free energy associated with a given value of the collective variables and includes contributions from all microscopic configurations (ideal case) compatible with that value.
The vector is the set of collective variables used in the calculation. Collective variables (CVs) are user-defined functions of the molecular geometry that describe selected, usually low-dimensional, features of a molecular system. They are used to monitor or bias processes that are difficult to sample directly in ordinary molecular dynamics simulations, such as conformational changes, ligand binding, ion transfer, or chemical transformations.
A collective variable maps the full set of atomic coordinates, x, onto one scalar quantity:
where x represents the Cartesian coordinates of the system, and is the value of the collective variable. Common examples include distances, angles, torsions, coordination numbers, RMSD-like measures, and more complex path-based descriptors.
The choice and number of collective variables are critical for the efficiency and reliability of a PMF calculation. In principle, adding more collective variables can provide a more complete description of the process under study. In practice, however, the computational cost grows rapidly, often exponentially, with the dimensionality of the collective-variable space. Each additional variable increases the amount of sampling required to obtain a converged free energy surface. For this reason, PMF calculations are usually performed in a low-dimensional space, most commonly using one or two collective variables, while three or more variables are used only in special cases.
A well-chosen collective variable should capture the slow or physically relevant degrees of freedom of the process under study, distinguish important metastable states along the reaction pathway, and avoid unnecessary complexity. Poorly chosen collective variables may lead to slow convergence, hidden barriers, hysteresis, or misleading free energy profiles.
Selecting appropriate collective variables is usually a very difficult problem.
Supported CVs
PMFLib supports the following collective variables. The majority of collective variables are supported by all free-energy methods. The exception is POT because of its gradient limitation. The number of collective variables is not technically limited, but biasing methods should employ as few as possible.
| Distance | |||||
| Type | Description | Type | Description | ||
| DD | distance difference | DIS | distance between two points | ||
| MDIS | minimum atom-pair distance | MDISG | minimum distance between two atom groups | ||
| ODIS | distance between a plane and a point | ||||
| OPOS | absolute position in an oriented coordinate system | OPOS2 | absolute position in a superimposed coordinate system | ||
| ORAD | radial distance in an oriented coordinate system | ORAD2 | radial distance in a superimposed coordinate system | ||
| POS | absolute position | RAD | radial distance | ||
| Angle | |||||
| Type | Description | Type | Description | ||
| ANG | angle defined by three points | ANG2 | angle between two vectors defined by four points | ||
| AXANG | angle between two principal axes | AXANG2 | angle between two reference axes | ||
| DIH | dihedral angle defined by four points | DIH2 | dihedral angle defined by three vectors and six points | ||
| PANG | angle between two planes | PVANG | angle between a plane and a vector | ||
| Coordination Numbers | |||||
| Type | Description | Type | Description | ||
| CNFF | atom-atom coordination number calculated using a Fermi-like function | CNGFF | point-atom coordination number calculated using a Fermi-like function | ||
| CNSW | atom-atom coordination number calculated using a switching function | CNGSW | point-atom coordination number calculated using a switching function | ||
| CNRF | atom-atom coordination number calculated using a rational function | CNGRF | point-atom coordination number calculated using a rational function | ||
| Shape | |||||
| Type | Description | Type | Description | ||
| ACYL | acylindricity | PMGTD | difference between two principal moments of the gyration tensor | ||
| ASPH | asphericity | PMOGT | principal moment of the gyration tensor | ||
| SANIS | relative shape anisotropy | RMSDT | root-mean-square deviation from a target structure | ||
| RGYR | radius of gyration | PLANE | root-mean-square deviation from a plane | ||
| EVEC | projection onto an essential vector | ||||
| Nucleic Acids | |||||
| Type | Description | Type | Description | ||
| NALBPP | local base-pair parameters | NALSTP | local base-pair step parameters | ||
| NASBPP | simple base-pair parameters | NASSTP | simple base-pair step parameters | ||
| NASBPPOLD | legacy simple base-pair parameters | NASSTPOLD | legacy simple base-pair step parameters | ||
| NASBO | simple base-pair opening | NAPBO | primitive base-pair opening | ||
| NABEND | nucleic acid bending | ||||
| Ring Puckering | |||||
| Type | Description | Type | Description | ||
| PUCK6Q | total puckering amplitude of a six-membered ring | PUCK5Q | total puckering amplitude of a five-membered ring | ||
| PUCK6P | puckering azimuthal angle of a six-membered ring | PUCK5P | puckering azimuthal angle of a five-membered ring | ||
| PUCK6T | puckering polar angle of a six-membered ring | ||||
| Special / Path | |||||
| Type | Description | Type | Description | ||
| PTPATHS | path-progress coordinate for point-based paths | PTPATHS2P | path-progress coordinate along a line generated from two points | ||
| PTPATHS2PN | path-progress coordinate along bidirectional line from origin + direction | PTPATHZ | path-distance coordinate for point-based paths | ||
| PATHS | path-progress coordinate | PATHZ | path-distance coordinate | ||
| WORMPOS | parametric position of an axle inside a macrocycle | WORMANG | orientation of an axle inside a macrocycle | ||
| Energy | |||||
| Type | Description | Type | Description | ||
| POT | potential energy | ||||
| Algebra | |||||
| Type | Description | Type | Description | ||
| ADD | sum of two collective variables | SUB | difference between two collective variables | ||
| MULT | product of two collective variables | DIV | ratio of two collective variables | ||
| FSWITCH | Fermi-like switching function | RSWITCH | rational switching function | ||
Specification
The CV definition can be provided either directly in the input file or in an external file. The selected mode depends on the configuration in the [files] section.
If the value of the fcvsdef key specifies the name of an input-file group, for example a string beginning with the "{" character, the CV definition is read from this group in the input file. Otherwise, the value of fcvsdef is interpreted as the name of an external file containing the CV definition. In this case, the CV definition must be placed in the {MAIN} group, which is implicit and may be omitted from the file.
A single group or file may define multiple collective variables. Each collective variable is specified in a separate section. The section name determines the CV type. Each section must contain the name key, which defines a unique name of the CV, together with additional keys specific to the selected CV type.