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 which 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 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 which 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 results in markedly different dynamical properties.