Graph transformation method for calculating waiting times in Markov chains

Trygubenko, Semen A.; Wales, David J.

doi:10.1063/1.2198806

Cited by 52 publications

(87 citation statements)

References 59 publications

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“…The relative free energy of the basin , taking basin as reference, is . Besides, the expected waiting time to escape from to any adjacent basin is [34]. Other magnitudes, such as first-passage time for inter-basins transitions and other rate constants relaxing the local equilibrium condition [19],[34] can also be computed from the original CMN.…”

Section: Methodsmentioning

confidence: 99%

Exploring the Free Energy Landscape: From Dynamics to Networks and Back

et al. 2009

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Knowledge of the Free Energy Landscape topology is the essential key to understanding many biochemical processes. The determination of the conformers of a protein and their basins of attraction takes a central role for studying molecular isomerization reactions. In this work, we present a novel framework to unveil the features of a Free Energy Landscape answering questions such as how many meta-stable conformers there are, what the hierarchical relationship among them is, or what the structure and kinetics of the transition paths are. Exploring the landscape by molecular dynamics simulations, the microscopic data of the trajectory are encoded into a Conformational Markov Network. The structure of this graph reveals the regions of the conformational space corresponding to the basins of attraction. In addition, handling the Conformational Markov Network, relevant kinetic magnitudes as dwell times and rate constants, or hierarchical relationships among basins, completes the global picture of the landscape. We show the power of the analysis studying a toy model of a funnel-like potential and computing efficiently the conformers of a short peptide, dialanine, paving the way to a systematic study of the Free Energy Landscape in large peptides.

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Section: Methodsmentioning

confidence: 99%

Exploring the Free Energy Landscape: From Dynamics to Networks and Back

et al. 2009

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“…To the best of our knowledge, this is the most successful approach as far as computing reaction rates in LJ 38 is concerned. 38 However, the numerical methods involved are quite elaborate, require considerable expertise, and have a number of drawbacks, all deriving from the fact that it is based on the harmonic superposition approximation and the theory of thermally activated processes. It thus requires any intermediate minima between the two basins to be equilibrated and this is only possible for small enough systems at low temperatures.…”

Section: A Lj 38 Clustermentioning

confidence: 99%

Simulating structural transitions by direct transition current sampling: The example of LJ38

Picciani

Athènes

Kurchan

et al. 2011

The Journal of Chemical Physics

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Reaction paths and probabilities are inferred, in a usual Monte Carlo or molecular dynamic simulation, directly from the evolution of the positions of the particles. The process becomes time-consuming in many interesting cases in which the transition probabilities are small. A radically different approach consists of setting up a computation scheme where the object whose time evolution is simulated is the transition current itself. The relevant timescale for such a computation is the one needed for the transition probability rate to reach a stationary level, and this is usually substantially shorter than the passage time of an individual system. As an example, we show, in the context of the "benchmark" case of 38 particles interacting via the Lennard-Jones potential ("LJ(38)" cluster), how this method may be used to explore the reactions that take place between different phases, recovering efficiently known results, and uncovering new ones with small computational effort.

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“…In other applications, methods for path analysis and path sampling have been developed, for example discrete path sampling databases for discrete time Markov chains [32], or where the probability of paths, rather than that of trajectories of discrete Markov processes can be used to analyse behaviour [30]. In [12], a method for transition path sampling is presented for protein folding, where the Markov chain has absorbing states.…”

Section: Introductionmentioning

confidence: 99%

Constrained approximation of effective generators for multiscale stochastic reaction networks and application to conditioned path sampling

Cotter

2016

Journal of Computational Physics

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Efficient analysis and simulation of multiscale stochastic systems of chemical kinetics is an ongoing area for research, and is the source of many theoretical and computational challenges. In this paper, we present a significant improvement to the constrained approach, which is a method for computing effective dynamics of slowly changing quantities in these systems, but which does not rely on the quasi-steady-state assumption (QSSA). The QSSA can cause errors in the estimation of effective dynamics for systems where the difference in timescales between the "fast" and "slow" variables is not so pronounced.This new application of the constrained approach allows us to compute the effective generator of the slow variables, without the need for expensive stochastic simulations. This is achieved by finding the null space of the generator of the constrained system. For complex systems where this is not possible, or where the constrained subsystem is itself multiscale, the constrained approach can then be applied iteratively. This results in breaking the problem down into finding the solutions to many small eigenvalue problems, which can be efficiently solved using standard methods.Since this methodology does not rely on the quasi steady-state assumption, the effective dynamics that are approximated are highly accurate, and in the case of systems with only monomolecular reactions, are exact. We will demonstrate this with some numerics, and also use the effective generators to sample paths of the slow variables which are conditioned on their endpoints, a task which would be computationally intractable for the generator of the full system.

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Graph transformation method for calculating waiting times in Markov chains

Cited by 52 publications

References 59 publications

Exploring the Free Energy Landscape: From Dynamics to Networks and Back

Exploring the Free Energy Landscape: From Dynamics to Networks and Back

Simulating structural transitions by direct transition current sampling: The example of LJ38

Constrained approximation of effective generators for multiscale stochastic reaction networks and application to conditioned path sampling

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