Properties of low-dimensional collective variables in the molecular dynamics of biopolymers

The description of the dynamics of a complex, high-dimensional system in terms of a low-dimensional set of collective variables Y can be fruitful if the low-dimensional representation satisfies a Langevin equation with drift and diffusion coefficients that depend only on Y. We present a computational scheme to evaluate whether a given collective variable provides a faithful low-dimensional representation of the dynamics of a high-dimensional system. The scheme is based on the framework of a finite-difference Langevin equation, similar to that used for molecular-dynamics simulations. This allows one to calculate the drift and diffusion coefficients in any point of the full-dimensional system. The width of the distribution of drift and diffusion coefficients in an ensemble of microscopic points at the same value of Y indicates to what extent the dynamics of Y is described by a simple Langevin equation. Using a simple protein model, we show that collective variables often used to describe biopolymers display a non-negligible width both in the drift and in the diffusion coefficients. We also show that the associated effective force is compatible with the equilibrium free energy calculated from a microscopic sampling, but it results in markedly different dynamical properties.