Closed-Form Solutions for Continuous Time Random Walks on Finite Chains

Flomenbom, Ophir; Klafter, J.

doi:10.1103/physrevlett.95.098105

Cited by 27 publications

(49 citation statements)

References 43 publications

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“…Recent studies in single-molecule spectroscopy and enzymology have invigorated a classic subject, the continuous time random walks (CTRW), which was developed and extensively studied many years ago by Montroll and coworkers [3,4]. See [5][6][7] for the recent work that motivated CTRW in connection to single motor proteins and single-enzyme kinetics.Applying stochastic models to molecular procsses requires serious considerations of the microscopic reversibility. When a closed molecular system with fluctuations reaches its stationary state, it is necessarily a chemical equilibrium with zero flux in any part of the system.…”

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confidence: 99%

“…An alternative starting point is a generalized master equation (GME) which replaces the rate constants in the standard master equation with time-dependent memory kernels K ij (t) [6,18]. K ij (t) will be defined and discussed later.…”

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confidence: 99%

“…Equation (5) (5) is the starting point of several recent studies on single-molecule kinetics [5,6], replacing the standard master equation for Markov processes. Detailed balance.…”

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Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

Qian

Wang

2006

Europhys. Lett.

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Abstract. -Continuous time random walk (CTRW) is studied with a new dynamic equation based on the age-structure of states. For a CTRW in a closed molecular system, two necessary conditions for microscopic reversibility are introduced: 1) independence of transition direction and waiting time for every state and 2) detailed balance among the transition probabilities. Together they are also sufficient condition. For a CTRW in an open system with explicit chemical energy input 1) still holds while 2) breaks down. Hence, CTRW models not satisfying 1) are either inconsistent with thermodynamics or cannot attain equilibrium due to hidden dissipation in non-Markovian states. Each CTRW defines a unique corresponding Markov process (cMP). The steady-state distribution of a CTRW equals that of the corresponding Markov process, and the two systems have the same steady-state flux, the same exit probabilities and the same mean trapping times. Mechanicity is discussed; a paradox observed by Kolomeisky and Fisher (J. Chem. Phys., 113 (2000) 10867) is resolved.Stochastic models are the theoretical basis for describing dynamics of biological, chemical and physical processes at the single-molecule level [1,2]. Recent studies in single-molecule spectroscopy and enzymology have invigorated a classic subject, the continuous time random walks (CTRW), which was developed and extensively studied many years ago by Montroll and coworkers [3,4]. See [5][6][7] for the recent work that motivated CTRW in connection to single motor proteins and single-enzyme kinetics.Applying stochastic models to molecular procsses requires serious considerations of the microscopic reversibility. When a closed molecular system with fluctuations reaches its stationary state, it is necessarily a chemical equilibrium with zero flux in any part of the system. This realization led to the introduction of detailed balance (DB) as a necessary condition for Markov kinetic models of closed system [8]. The celebrated fluctuation-dissipation relation [9] is in fact an explicit expression of detailed balance in the Langevin dynamics [10]. When a molecular system is in an open environment with sustained chemical input and dissipation, detailed balance breaks down [2, 11], which gives rise to free energy transduction, such as in c EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx

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Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

Qian

Wang

2006

Europhys. Lett.

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“…Unfortunately even for an NCPP the n-fold convolution cannot be computed as easily as for p X (x, t) in Eq. (18). However, the characteristic function of the quadratic variation can be written as…”

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confidence: 99%

“…ψ(τ ) = exp(−τ /γ t )/γ t [16,17]. An uncoupled CTRW belongs to the class of semi-Markov processes [17,18,19,20], i.e. for any A ⊂ R d and t > 0 we have P (S n ∈ A, τ n ≤ t | S 0 , .…”

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Stochastic calculus for uncoupled continuous-time random walks

et al. 2009

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The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, that includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy α-stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional FokkerPlanck equation, that generalize the standard diffusion equation solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE, and check it by Monte Carlo.

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Making it Possible: Constructing a Reliable Mechanism from a Finite Trajectory

Flomenbom¹

2011

Single‐Molecule Biophysics

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Deducing an underlying multi-substate on-off kinetic scheme (KS) from the statistical properties of a two-state trajectory is the aim of many experiments in biophysics and chemistry, such as, ion-channel recordings, enzymatic activity and structural dynamics of bio-molecules. Doing so is almost always impossible, as the mapping of a KS into a two-state trajectory leads to the loss of information about the KS (almost always). Here, we present the optimal way to solve this problem. It is based on unique forms of reduced dimensions (RD). RD forms are on-off networks with connections only between substates of different states, where the connections can have multiexponential waiting time probability density functions (WT-PDFs). A RD form has the simplest topology that can reproduce a given data. In theory, only a single RD form can be constructed from the full data (hence its uniqueness), still this task is not easy when dealing with finite data. For doing so, a toolbox made of known statistical methods in data analysis and new statistical methods and numerical algorithms developed for this problem is presented. Our toolbox is self-contained: it builds a mechanism based only on the information it extracts from the data. The implementation of the toolbox on the data is fast. The toolbox is automated and is available for academic research upon electronic request. KEYWORDS Two-state trajectories: analysis of-, mechanisms from-, canonical forms of-Single molecules: trajectories of-, time traces of-, analysis of-

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Closed-Form Solutions for Continuous Time Random Walks on Finite Chains

Cited by 27 publications

References 43 publications

Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

Stochastic calculus for uncoupled continuous-time random walks

Making it Possible: Constructing a Reliable Mechanism from a Finite Trajectory

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