Likelihood-Based Estimation of Multidimensional Langevin Models and Its Application to Biomolecular Dynamics

We crucially extend the novel method introduced in Horenko et al. [Multiscale Model. Simul., 5 (2006), pp. 802–827] for the identification of the most important metastable states of a system with complicated dynamical behavior from time series information. The approach represents the effective dynamics of the full system by a Markov jump process between metastable states and the dynamics within each of these metastable states by diffusions that include the classes called full and overdamped Langevin dynamics in biophysics (local SDEs). Its algorithmic realization combines hidden Markov models (for the unobserved jump process between the metastable states) with likelihood-based estimation of the parameters in the local linear SDEs based on discrete-time observations of the system. Despite the local linearity the approach is appropriate for handling nonlinear potentials with several dominant wells. Compared to Horenko et al. [Multiscale Model. Simul., 5 (2006), pp. 802–827], the algorithms proposed herein allow us to handle arbitrary dimensions instead of just one-dimensional local SDEs and to extend the class of diffusions considered. Moreover, the role of the fluctuation-dissipation relation in parameter estimation for Langevin models is discussed in detail. The performance of the algorithms is illustrated by numerical tests and by application to the molecular dynamics time series of a 12-Alanine molecule with implicit water.