Reaction paths based on mean first-passage times

Park, Sang‐Hyun; Sener, Melih; Lu, Deyu; Schulten, Klaus

doi:10.1063/1.1570396

Cited by 77 publications

(89 citation statements)

References 30 publications

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“…Therefore, there are two possible paths to move from x L to x R : A direct path, and one that passes through the central well x C . Properties of (2.1) with this potential were studied by various authors (see [41,17] and references therein).…”

Section: Examplesmentioning

confidence: 99%

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

Coifman¹,

Kevrekidis²,

Lafon³

et al. 2008

Multiscale Model. Simul.

262

336

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Abstract.The concise representation of complex high dimensional stochastic systems via a few reduced coordinates is an important problem in computational physics, chemistry and biology. In this paper we use the first few eigenfunctions of the backward Fokker-Planck diffusion operator as a coarse grained low dimensional representation for the long term evolution of a stochastic system, and show that they are optimal under a certain mean squared error criterion. We denote the mapping from physical space to these eigenfunctions as the diffusion map. While in high dimensional systems these eigenfunctions are difficult to compute numerically by conventional methods such as finite differences or finite elements, we describe a simple computational data-driven method to approximate them from a large set of simulated data. Our method is based on defining an appropriately weighted graph on the set of simulated data, and computing the first few eigenvectors and eigenvalues of the corresponding random walk matrix on this graph. Thus, our algorithm incorporates the local geometry and density at each point into a global picture that merges in a natural way data from different simulation runs. Furthermore, we describe lifting and restriction operators between the diffusion map space and the original space. These operators facilitate the description of the coarse-grained dynamics, possibly in the form of a low-dimensional effective free energy surface parameterized by the diffusion map reduction coordinates. They also enable a systematic exploration of such effective free energy surfaces through the design of additional "intelligently biased" computational experiments. We conclude by demonstrating our method on a few examples.Key words. Diffusion maps, dimensional reduction, stochastic dynamical systems, Fokker Planck operator, metastable states, normalized graph Laplacian. AMS subject classifications. 60H10, 60J60, 62M051. Introduction. Systems of stochastic differential equations (SDE's) are commonly used as models for the time evolution of many chemical, physical and biological systems of interacting particles [22,45,52]. There are two main approaches to the study of such systems. The first is by detailed Brownian Dynamics (BD) or other stochastic simulations, which follow the motion of each particle (or more generally variable) in the system and generate one or more long trajectories. The second is via analysis of the time evolution of the probability densities of these trajectories using the numerical solution of the corresponding time dependent Fokker-Planck (FP) partial differential equation.For typical high dimensional systems, both approaches suffer from severe limitations, when applied directly. The main limitation of standard BD simulations is the scale gap between the atomistic time scale of single particle motions, at which the SDE's are formulated, and the macroscopic time scales that characterize the long term evolution and equilibration of these systems. This scale gap puts severe constraints on detailed simulat...

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Section: Examplesmentioning

confidence: 99%

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

Coifman¹,

Kevrekidis²,

Lafon³

et al. 2008

Multiscale Model. Simul.

262

336

View full text Add to dashboard Cite

show abstract

“…The backward equation method has been widely used to calculate mean first-passage time and other average values (16)(17)(18)(19). However, this method has limited value for obtaining distribution functions such as dwell-time distributions.…”

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confidence: 99%

Extending the absorbing boundary method to fit dwell-time distributions of molecular motors with complex kinetic pathways

Liao¹,

Spudich

Parker

et al. 2007

Proc. Natl. Acad. Sci. U.S.A.

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Dwell-time distributions, waiting-time distributions, and distributions of pause durations are widely reported for molecular motors based on single-molecule biophysical experiments. These distributions provide important information concerning the functional mechanisms of enzymes and their underlying kinetic and mechanical processes. We have extended the absorbing boundary method to simulate dwell-time distributions of complex kinetic schemes, which include cyclic, branching, and reverse transitions typically observed in molecular motors. This extended absorbing boundary method allows global fitting of dwell-time distributions for enzymes subject to different experimental conditions. We applied the extended absorbing boundary method to experimental dwell-time distributions of single-headed myosin V, and were able to use a single kinetic scheme to fit dwell-time distributions observed under different ligand concentrations and different directions of optical trap forces. The ability to use a single kinetic scheme to fit dwell-time distributions arising from a variety of experimental conditions is important for identifying a mechanochemical model of a molecular motor. This efficient method can be used to study dwell-time distributions for a broad class of molecular motors, including kinesin, RNA polymerase, helicase, F1 ATPase, and to examine conformational dynamics of other enzymes such as ion channels.myosin ͉ single molecule ͉ waiting-time distributions ͉ pause durations T he movements of molecular motors are frequently recorded in biophysical experiments using tools such as optical trap assays and single molecule fluorescence assays. Due to transitions of conformational and chemical states, the motions of molecular motors usually show alternations between an enzyme's moving and pausing behaviors. For example, doubleheaded myosin VI moving on an actin filament demonstrates a mechanical stepping behavior ( Fig. 1a) (1), whereas the events of actin binding and unbinding of a single-headed myosin V construct show alternate phases of oscillating and pausing (Fig. 1b) (2). Collecting a large number of these events and performing statistical analysis provides insight into the kinetics and mechanochemical characteristics of an individual molecule.Dwell events arise from pauses in mechanical stepping (Fig. 1a) or stochastic delays before a binding-unbinding transition (Fig. 1b). Each dwell represents an experimentally observable event, which can be a single conformational or chemical state, or a collective number of states. The dwell time, or the duration of a dwell, is determined by the probability of exiting a dwell after the time entering it. Histograms of dwell times, also called dwell-time distributions, waiting-time distributions, or distributions of pause durations, are widely used to decipher single molecule kinetics.The dynamics of an enzyme's activity can be modeled as a number of kinetic states using kinetic equations (3) or mechanical coordinates using diffusion-based (Fokker-Planck) equations (4). For quest...

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“…that is a well-known test example for studying rare events (see, e.g., [21,35,31]). As the left panel of Figure 3.5 shows, the potential (3.8) has two deep minima approximately at (±1,0), a shallow minimum approximately at (0,1.5), three saddle points approximately at (±0.6,1.1),(−1.4,0) and a maximum at (0,0.5).…”

Section: Numerical Resultsmentioning

confidence: 99%

A structure-preserving numerical discretization of reversible diffusions

Latorre

Metzner

Hartmann

et al. 2011

Communications in Mathematical Sciences

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Abstract. We propose a numerical discretization scheme for the infinitesimal generator of a diffusion process based on a finite volume approximation. The resulting discrete-space operator can be interpreted as a jump process on the mesh whose invariant distribution is precisely the cell approximation of the Boltzmann invariant measure and preserves the detailed balance property of the original stochastic process. Moreover this approximation is robust in the sense that these properties remain valid independently of the grid size.

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Reaction paths based on mean first-passage times

Cited by 77 publications

References 30 publications

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

Extending the absorbing boundary method to fit dwell-time distributions of molecular motors with complex kinetic pathways

A structure-preserving numerical discretization of reversible diffusions

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