Extracting Markov Models of Peptide Conformational Dynamics from Simulation Data

Abstract

A high-dimensional time series obtained by simulating a complex and stochastic dynamical system (like a peptide in solution) may code an underlying multiple-state Markov process. We present a computational approach to most plausibly identify and reconstruct this process from the simulated trajectory. Using a mixture of normal distributions we first construct a maximum likelihood estimate of the point density associated with this time series and thus obtain a density-oriented partition of the data space. This discretization allows us to estimate the transfer operator as a matrix of moderate dimension at sufficient statistics. A nonlinear dynamics involving that matrix and, alternatively, a deterministic coarse-graining procedure are employed to construct respective hierarchies of Markov models, from which the model most plausibly mapping the generating stochastic process is selected by consideration of certain observables. Within both procedures the data are classified in terms of prototypical points, the conformations, marking the various Markov states. As a typical example, the approach is applied to analyze the conformational dynamics of a tripeptide in solution. The corresponding high-dimensional time series has been obtained from an extended molecular dynamics simulation.

Estimation and uncertainty of reversible Markov models

“…Moreover, (4) implies that π is the equilibrium probability vector of P(τ). By defining the diagonal matrix Π = diag(π 1 , .…”

“…[1][2][3][4][5][6][7] These models allow us to analyze the longest-living (metastable) sets of structures, 8 the effective transition rates between them, 9,10 the kinetic relaxation processes and their relationship to equilibrium kinetics experiments, 7,[11][12][13][14] and the thermodynamics and kinetics over multiple thermodynamic states. [15][16][17][18] A key advantage of MSMs is that they are estimated from conditional transition statistics between states, and they thus do not require the data to be in global equilibrium across all states.…”

A comparative analysis of clustering algorithms: O2 migration in truncated hemoglobin I from transition networks

“…[5][6][7][8][9] KMC is particularly suitable for situations in which the time scale separation between different motions of interest is large, such as in protein folding. TNs have found several applications in protein folding, [10][11][12][13][14][15][16][17][18] enzyme catalysis, 19,20 ligand migration, 21 and studies of electron spin resonance. 22 For a transition network analysis, the original data set needs to be clustered.…”

Long‐Time Protein Folding Dynamics from Short‐Time Molecular Dynamics Simulations

“…Unfortunately, the number of minima grows rapidly with increasing system size, mak-ing the procedure prohibitively expensive for larger proteins or systems containing explicit solvent molecules. Other work [11,50,46,1,47,40] has focused on the construction of discrete-or continuous-time Markovian models to describe dynamics between a small number of states. These models, however, have yet to demonstrate that they can adequately describe the dynamics on timescales much longer than the trajectories from which the models were constructed; no attempt is made to compare the dynamics predicted by the model with long trajectories of explicitly solvated systems.…”