Biasing molecular simulations with Gaussian processes using NAMD and Colvars

Juan M. Bello-Rivas

This is a TCL script for evaluating Gaussian processes in the context of molecular simulations using NAMD [1] and Colvars [2].

By the representer theorem [3], we know that a Gaussian process with radial basis function kernels is of the form:

f(x) = \sum_{i = 1}^N \alpha_i \, \exp\left\{ -\frac{d(x, x_i)^2}{2 \varepsilon^2} \right\},

for some \alpha_i \in \mathbb{R} and \varepsilon > 0. In the case of molecular simulations, it makes sense to take the function d(x, y) as the root mean square deviation (RMSD) between molecular conformations, which is obtained by finding the optimal rigid body motion that maps x to y.

The included file gaussian_process.tcl implements the evaluation of a Gaussian process and its Jacobian matrix to bias simulations. The user only has to provide the values of \varepsilon and \alpha_i for i = 1, ..., N. An example of its use can be found in example-namd.

[1] J. C. Phillips et al., “Scalable molecular dynamics on CPU and GPU architectures with NAMD,” J. Chem. Phys., vol. 153, no. 4, p. 044130, Jul. 2020, doi: 10.1063/5.0014475.

[2] G. Fiorin, M. L. Klein, and J. Hénin, “Using collective variables to drive molecular dynamics simulations,” Molecular Physics, vol. 111, no. 22–23, pp. 3345–3362, Dec. 2013, doi: 10.1080/00268976.2013.813594.

[3] T. Hofmann, B. Schölkopf, and A. J. Smola, “Kernel methods in machine learning,” Annals of Statistics, vol. 36, no. 3, pp. 1171–1220, 2008, doi: 10.1214/009053607000000677.