Collective Variable-based Calculations¹

In today's molecular dynamics simulations, it is often useful to reduce the large number of degrees of freedom of a physical system into few parameters whose statistical distributions can be analyzed individually, or used to define biasing potentials to alter the dynamics of the system in a controlled manner. These have been called `order parameters', `collective variables', `(surrogate) reaction coordinates', and many other terms. Here we use primarily the term `collective variable' (shortened to colvar), which indicates any differentiable function of atomic Cartesian coordinates, ${\mbox{\boldmath {$x$}}}_{i}$ , with $i$ between $1$ and $N$ , the total number of atoms:

$$\xi(t) \; = \xi\left({\mbox{\boldmath {$x$}}}_{i}(t), {\mbox{\bol... ...mbox{\boldmath {$x$}}}_{k}(t), \ldots \right)\;, \;\; 1 \leq i,j,k\ldots \leq N$$ (35)

The colvars module in NAMD may be used in both MD simulations and energy minimization runs. The module is designed to perform multiple tasks concurrently, the most common of which are:

• apply restraints or biasing potentials to multiple colvars, tailored on the system by choosing from a wide set of basis functions, without limitations on their number or on the number of atoms involved; while this can in principle be done through a TclForces script, using the colvars is both easier and computationally more efficient;

• calculate potentials of mean force (PMFs) along any set of colvars, using different enhanced sampling methods, such as Adaptive Biasing Force (ABF), metadynamics, steered MD and umbrella sampling; variants of these methods that make use of an ensemble of replicas are supported as well;

• calculate statistical properties of the colvars, such as running averages and standard deviations, correlation functions of pairs of colvars, and multidimensional histograms, without the need to save very large trajectory files ;
• analyze an existing MD trajectory in terms of colvars: use NAMD's coorfile read command, and perform a 0-timestep run for each set of coordinates, as illustrated in 16.

To briefly illustrate the flexibility of the colvars module, Figure 6 shows an example of a non-trivial configuration (the corresponding input can be found in 10.1.2).

Figure 6: Graphical representation of a collective variables configuration The colvar called ``$d$ '' is defined as the difference between two distances: the first distance ($d_{1}$ ) is taken between the center of mass of atoms 1 and 2 and that of atoms 3 to 5, the second ($d_{2}$ ) between atom 7 and the center of mass of atoms 8 to 10. The difference $d = d_{1} - d_{2}$ is obtained by multiplying the two by a coefficient $C = +1$ or $C = -1$ , respectively. The colvar called ``$c$ '' is the coordination number calculated between atoms 1 to 10 and atoms 11 to 20. A harmonic restraint is applied to both $d$ and $c$ : to allow using the same force constant $K$ , both $d$ and $c$ are scaled by their respective fluctuation widths $w_d$ and $w_c$ . A third colvar ``alpha'' is defined as the $\alpha$ -helical content of residues 1 to 10. The values of ``$c$ '' and ``alpha'' are also recorded throughout the simulation as a joint 2-dimensional histogram.

Detailed explanations of the design of the colvars module are provided in reference [22]. Please cite this reference whenever publishing work that makes use of this module.