Estimation of the infinitesimal generator by square-root approximation

Abstract

For the analysis of molecular processes, the estimation of time-scales, i.e., tran- sition rates, is very important. Estimating the transition rates between molecular conformations is - from a mathematical point of view - an invariant subspace projec- tion problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is ap- proximated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the ge- ometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.