Identification of slow molecular order parameters for Markov model construction

Abstract

A goal in the kinetic characterization of a macromolecular system is the description of its slow relaxation processes via (i) identification of the structural changes involved in these processes and (ii) estimation of the rates or timescales at which these slow processes occur. Most of the approaches to this task, including Markov models, master-equation models, and kinetic network models, start by discretizing the high-dimensional state space and then characterize relaxation processes in terms of the eigenvectors and eigenvalues of a discrete transition matrix. The practical success of such an approach depends very much on the ability to finely discretize the slow order parameters. How can this task be achieved in a high-dimensional configuration space without relying on subjective guesses of the slow order parameters? In this paper, we use the variational principle of conformation dynamics to derive an optimal way of identifying the "slow subspace" of a large set of prior order parameters - either generic internal coordinates or a user-defined set of parameters. Using a variational formulation of conformational dynamics, it is shown that an existing method-the time-lagged independent component analysis-provides the optional solution to this problem. In addition, optimal indicators-order parameters indicating the progress of the slow transitions and thus may serve as reaction coordinates-are readily identified. We demonstrate that the slow subspace is well suited to construct accurate kinetic models of two sets of molecular dynamics simulations, the 6-residue fluorescent peptide MR121-GSGSW and the 30-residue intrinsically disordered peptide kinase inducible domain (KID). The identified optimal indicators reveal the structural changes associated with the slow processes of the molecular system under analysis.

We are grateful to Thomas Weikl (MPI Potsdam) for advice and support. G.P.H. acknowledges support from German Science Foundation DFG fund NO 825-3. F.P. acknowledges funding from the Max Planck Society. T.G. gratefully acknowledges former support from the “Beatriu de Pinós” scheme of the Agència de Gestió d’Ajuts Universitaris i de Recerca (Generalitat de Catalunya). G.D.F. acknowledges support from the Ramón y Cajal scheme and support by the Spanish Ministry of Science and Innovation (Ref. BIO2011-27450). F.N. acknowledges funding from DFG center Matheon and ERC starting grant 307494 pcCell.