Calculation of the distribution of eigenvalues and eigenvectors in Markovian state models for molecular dynamics.

Markovian state models (MSMs) are a convenient and efficient means to compactly describe the kinetics of a molecular system as well as a formalism for using many short simulations to predict long time scale behavior. Building a MSM consists of grouping the conformations into states and estimating the transition probabilities between these states. In a previous paper, we described an efficient method for calculating the uncertainty due to finite sampling in the mean first passage time between two states. In this paper, we extend the uncertainty analysis to derive similar closed-form solutions for the distributions of the eigenvalues and eigenvectors of the transition matrix, quantities that have numerous applications when using the model. We demonstrate the accuracy of the distributions on a six-state model of the terminally blocked alanine peptide. We also show how to significantly reduce the total number of simulations necessary to build a model with a given precision using these uncertainty estimates for the blocked alanine system and for a 2454-state MSM for the dynamics of the villin headpiece.

[1]  F. Young.  Biochemistry , 1955, The Indian Medical Gazette.

[2]  Calyampudi R. Rao,et al.  Linear statistical inference and its applications , 1965 .

[3]  C. R. Rao,et al.  Linear Statistical Inference and its Applications , 1968 .

[4]  D. Rose,et al.  Partitioning, tearing and modification of sparse linear systems , 1974 .

[5]  Calyampudi R. Rao,et al.  Linear Statistical Inference and Its Applications. , 1975 .

[6]  David Chandler,et al.  Statistical mechanics of isomerization dynamics in liquids and the transition state approximation , 1978 .

[7]  Jimmie D. Doll,et al.  Dynamical aspects of precursor state kinetics , 1981 .

[8]  Kishore Singhal,et al.  Computer Methods for Circuit Analysis and Design , 1983 .

[9]  Arthur F. Voter,et al.  Dynamical corrections to transition state theory for multistate systems: Surface self-diffusion in the rare-event regime , 1985 .

[10]  C. Brooks,et al.  Statistical clustering techniques for the analysis of long molecular dynamics trajectories: analysis of 2.2-ns trajectories of YPGDV. , 1993, Biochemistry.

[11]  P. Tavan,et al.  Molecular dynamics of conformational substates for a simplified protein model , 1994 .

[12]  N. L. Johnson,et al.  Discrete Multivariate Distributions , 1998 .

[13]  Axel Ruhe,et al.  Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils , 1998, SIAM J. Sci. Comput..

[14]  Chao Yang,et al.  ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , 1998, Software, environments, tools.

[15]  P. Deuflhard,et al.  A Direct Approach to Conformational Dynamics Based on Hybrid Monte Carlo , 1999 .

[16]  Vijay S. Pande,et al.  Screen Savers of the World Unite! , 2000, Science.

[17]  Vijay S Pande,et al.  Foldamer simulations: novel computational methods and applications to poly-phenylacetylene oligomers. , 2004, The Journal of chemical physics.

[18]  Vijay S Pande,et al.  Using path sampling to build better Markovian state models: predicting the folding rate and mechanism of a tryptophan zipper beta hairpin. , 2004, The Journal of chemical physics.

[19]  William Swope,et al.  Describing Protein Folding Kinetics by Molecular Dynamics Simulations. 2. Example Applications to Alanine Dipeptide and a β-Hairpin Peptide† , 2004 .

[20]  William Swope,et al.  Describing Protein Folding Kinetics by Molecular Dynamics Simulations. 1. Theory , 2004 .

[21]  P. Deuflhard,et al.  Robust Perron cluster analysis in conformation dynamics , 2005 .

[22]  I. Kevrekidis,et al.  Coarse master equation from Bayesian analysis of replica molecular dynamics simulations. , 2005, The journal of physical chemistry. B.

[23]  N. L. Johnson,et al.  Continuous Multivariate Distributions: Models and Applications , 2005 .

[24]  Vijay S Pande,et al.  Foldamer dynamics expressed via Markov state models. I. Explicit solvent molecular-dynamics simulations in acetonitrile, chloroform, methanol, and water. , 2005, The Journal of chemical physics.

[25]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[26]  Paul Tavan,et al.  Extracting Markov Models of Peptide Conformational Dynamics from Simulation Data. , 2005, Journal of chemical theory and computation.

[27]  V. Pande,et al.  Error analysis and efficient sampling in Markovian state models for molecular dynamics. , 2005, The Journal of chemical physics.

[28]  R. Levy,et al.  Protein folding pathways from replica exchange simulations and a kinetic network model. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[29]  Peter M. Kasson,et al.  Ensemble molecular dynamics yields submillisecond kinetics and intermediates of membrane fusion , 2006, Proceedings of the National Academy of Sciences.

[30]  John D. Chodera,et al.  Long-Time Protein Folding Dynamics from Short-Time Molecular Dynamics Simulations , 2006, Multiscale Model. Simul..

[31]  Wei Yang,et al.  Conformational sampling via a self-regulating effective energy surface. , 2006, The Journal of chemical physics.

[32]  K. Dill,et al.  Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics. , 2007, The Journal of chemical physics.

[33]  Jeremy C. Smith,et al.  Hierarchical analysis of conformational dynamics in biomolecules: transition networks of metastable states. , 2007, The Journal of chemical physics.

[34]  N. L. Johnson,et al.  Continuous Multivariate Distributions, Volume 1: Models and Applications , 2019 .