A square root approximation of transition rates for a Markov State Model

Please always quote using this URN: urn:nbn:de:0297-zib-42195
  • Trajectory- or mesh-based methods for analyzing the dynamical behavior of large molecules tend to be impractical due to the curse of dimensionality - their computational cost increases exponentially with the size of the molecule. We propose a method to break the curse by a novel square root approximation of transition rates, Monte Carlo quadrature and a discretization approach based on solving linear programs. With randomly sampled points on the molecular energy landscape and randomly generated discretizations of the molecular configuration space as our initial data, we construct a matrix describing the transition rates between adjacent discretization regions. This transition rate matrix yields a Markov State Model of the molecular dynamics. We use Perron cluster analysis and coarse-graining techniques in order to identify metastable sets in configuration space and approximate the transition rates between the metastable sets. Application of our method to a simple energy landscape on a two-dimensional configuration space provides proof of concept and an example for which we compare the performance of different discretizations. We show that the computational cost of our method grows only polynomially with the size of the molecule. However, finding discretizations of higher-dimensional configuration spaces in which metastable sets can be identified remains a challenge.

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Author:Han Cheng Lie, Konstantin Fackeldey, Marcus Weber
Document Type:ZIB-Report
Tag:Markov State Models; Markov chains; Voronoi; linear programming; meshfree methods; metastability
MSC-Classification:60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Jxx Markov processes / 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Jxx Markov processes / 60J22 Computational methods in Markov chains [See also 65C40]
82-XX STATISTICAL MECHANICS, STRUCTURE OF MATTER / 82Bxx Equilibrium statistical mechanics / 82B80 Numerical methods (Monte Carlo, series resummation, etc.) [See also 65-XX, 81T80]
CCS-Classification:G. Mathematics of Computing
PACS-Classification:30.00.00 ATOMIC AND MOLECULAR PHYSICS
Date of first Publication:2013/08/28
Series (Serial Number):ZIB-Report (13-43)
ISSN:1438-0064
Published in:Appeared in: SIAM J. Matrix Anal. Appl. 34 (2013) pp. 738 - 756
DOI:https://doi.org/10.1137/120899959