Generalized Markov State Modeling Method for Nonequilibrium Biomolecular Dynamics: Exemplified on Amyloid β Conformational Dynamics Driven by an Oscillating Electric Field

Abstract

Markov state models (MSMs) have received an unabated increase in popularity in recent years, as they are very well suited for the identification and analysis of metastable states and related kinetics. However, the state-of-the-art Markov state modeling methods and tools enforce the fulfillment of a detailed balance condition, restricting their applicability to equilibrium MSMs. To date, they are unsuitable to deal with general dominant data structures including cyclic processes, which are essentially associated with nonequilibrium systems. To overcome this limitation, we developed a generalization of the common robust Perron Cluster Cluster Analysis (PCCA+) method, termed generalized PCCA (G-PCCA). This method handles equilibrium and nonequilibrium simulation data, utilizing Schur vectors instead of eigenvectors. G-PCCA is not limited to the detection of metastable states but enables the identification of dominant structures in a general sense, unraveling cyclic processes. This is exemplified by application of G-PCCA on nonequilibrium molecular dynamics data of the Amyloid β (1-40) peptide, periodically driven by an oscillating electric field.

CellRank for directed single-cell fate mapping

“…The aim of the GPCCA algorithm is to project the large transition matrix P onto a much smaller coarse-grained transition matrix P c , which describes transitions between macrostates of the biological system 22,23 . For the projected or embedded dynamics to be Markovian, we require the projection to be based on an invariant subspace of P (invariant subspace projection), i.e.…”

“…Since the GPCCA algorithm can not cope with complex vectors, we rely on real invariant subspaces of the matrix P for the projection. Such subspaces are provided by the real Schur decomposition of P 22,23 , where is orthogonal and is quasi-upper triangular 25 . R has 1-by-1 or 2-by-2 blocks on the diagonal, where the former are given by the real eigenvalues and the latter are associated with pairs of complex conjugate eigenvalues.…”

“…Depending on the constructed subspace, different dynamical properties of P will be projected onto P c . Choosing Schur vectors belonging to real eigenvalues close to 1, metastabilities are recovered, while for Schur vectors with complex eigenvalues close to the unit circle, cyclic dynamics are recovered 22,23 . Both options are available in CellRank, defaulting to the recovery of metastabilities.…”

“… for an invertible matrix . We obtain the projected transition matrix via a Galerkin projection 22,23 , …”

“…Following the original MATLAB implementation 22,23 , we wrote up GPCCA as a general algorithm in Python and included it in the 27 package, which is widely used for studying protein folding kinetics. From CellRank, we interface to for the GPCCA algorithm.…”