Asymmetric Random Walk Research Articles

Asymmetric Random Walk in a One-Dimensional Multizone Environment

We consider a random walk model in a one-dimensional environment, formed by several zones of finite width with the fixed transition probabilities. It is also assumed that the transitions to the left and right neighboring points have unequal probabilities. In continuous limit, we derive analytically the probability distribution function, which is mainly determined by a walker diffusion and drift and accounts perturbatively for interface effects between zones. It is used for computing the probability to find a walker in a given space-time point and the time dependence of the mean squared displacement of a walker, which reveals the transient anomalous diffusion. To justify our approach, the probability function is compared with the results of numerical simulations for a three-zone environment.

Critical time scale of coarse-graining entropy production.

We study coarse-grained entropy production in an asymmetric random walk system on a periodic one-dimensional lattice. In coarse-grained systems, the original dynamics are unavoidably destroyed, but the coarse-grained entropy production is not hidden below the critical time-scale separation. The hidden entropy production is rapidly increasing near the critical time-scale separation.

Asymmetric Lévy flights in the presence of absorbing boundaries

We consider a one-dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution ϕ(η) displaying asymmetric power-law tails (i.e. ϕ(η) ∼ c/ηα+1 for large positive jumps and ϕ(η) ∼ c/(γ|η|α+1) for large negative jumps, with 0 < α < 2). In the absence of boundaries and after a large number of steps n, the probability density function (PDF) of the walker position, xn, converges to an asymmetric Lévy stable law of stability index α and skewness parameter β = (γ − 1)/(γ + 1). In particular, the right tail of this PDF decays as . Much less is known when the walker is confined, or partially confined, in a region of the space. In this paper we first study the case of a walker constrained to move on the positive semi-axis and absorbed once it changes sign. In this case, the persistence exponent θ+, which characterizes the algebraic large time decay of the survival probability, can be computed exactly and we show that, if θ+ < 1, the tail of the PDF of the walker position decays as . This last result can be generalized in higher dimensions such as a two-dimensional random walker performing Lévy stable jumps and confined in a wedge with absorbing walls. Our results are corroborated by precise numerical simulations.

Modeling biological tissue growth: Discrete to continuum representations

There is much interest in building deterministic continuum models from discrete agent-based models governed by local stochastic rules where an agent represents a biological cell. In developmental biology, cells are able to move and undergo cell division on and within growing tissues. A growing tissue is itself made up of cells which undergo cell division, thereby providing a significant transport mechanism for other cells within it. We develop a discrete agent-based model where domain agents represent tissue cells. Each agent has the ability to undergo a proliferation event whereby an additional domain agent is incorporated into the lattice. If a probability distribution describes the waiting times between proliferation events for an individual agent, then the total length of the domain is a random variable. The average behavior of these stochastically proliferating agents defining the growing lattice is determined in terms of a Fokker-Planck equation, with an advection and diffusion term. The diffusion term differs from the one obtained Landman and Binder [J. Theor. Biol. 259, 541 (2009)] when the rate of growth of the domain is specified, but the choice of agents is random. This discrepancy is reconciled by determining a discrete-time master equation for this process and an associated asymmetric nonexclusion random walk, together with consideration of synchronous and asynchronous updating schemes. All theoretical results are confirmed with numerical simulations. This study furthers our understanding of the relationship between agent-based rules, their implementation, and their associated partial differential equations. Since tissue growth is a significant cellular transport mechanism during embryonic growth, it is important to use the correct partial differential equation description when combining with other cellular functions.

Asymmetric random walks and drift-diffusion

To obtain the drift-diffusion equation from an asymmetric random walk it has been required that the left and right jump probabilities are approximately equal. It is shown here that such a restriction is not needed. Using multiple scales the equation is derived, where the scales are based on the advection and diffusion scales.

Kinetic approaches to lactose operon induction and bimodality

The quasi-equilibrium approximation is acceptable when molecular interactions are fast enough compared to circuit dynamics, but is no longer allowed when cellular activities are governed by rare events. A typical example is the lactose operon (lac), one of the most famous paradigms of transcription regulation, for which several theories still coexist to describe its behaviors. The lac system is generally analyzed by using equilibrium constants, contradicting single-event hypotheses long suggested by Novick and Weiner (1957). Enzyme induction as an all-or-none phenomenon. Proc. Natl. Acad. Sci. USA 43, 553–566) and recently refined in the study of (Choi et al., 2008. A stochastic single-molecule event triggers phenotype switching of a bacterial cell. Science 322, 442–446). In the present report, a lac repressor (LacI)-mediated DNA immunoprecipitation experiment reveals that the natural LacI-lac DNA complex built in vivo is extremely tight and long-lived compared to the time scale of lac expression dynamics, which could functionally disconnect the abortive expression bursts and forbid using the standard modes of lac bistability. As alternatives, purely kinetic mechanisms are examined for their capacity to restrict induction through: (i) widely scattered derepression related to the arrival time variance of a predominantly backward asymmetric random walk and (ii) an induction threshold arising in a single window of derepression without recourse to nonlinear multimeric binding and Hill functions. Considering the complete disengagement of the lac repressor from the lac promoter as the probabilistic consequence of a transient stepwise mechanism, is sufficient to explain the sigmoidal lac responses as functions of time and of inducer concentration. This sigmoidal shape can be misleadingly interpreted as a phenomenon of equilibrium cooperativity classically used to explain bistability, but which has been reported to be weak in this system.

The large deviation function for entropy production: the optimal trajectory and the role of fluctuations

We study the large deviation function for the entropy production rate in two driven one-dimensional systems: the asymmetric random walk on a discrete lattice and Brownian motion in a continuous periodic potential. We compare two approaches: using the Donsker–Varadhan theory and using the Freidlin–Wentzell theory. We show that the wings of the large deviation function are dominated by a single optimal trajectory: either in the forward direction (positive rate) or in the backward direction (negative rate). The joining of the two branches at zero entropy production implies a non-differentiability and thus the appearance of a ‘kink’. However, around zero entropy production, many trajectories contribute and thus the ‘kink’ is smeared out.

Mean-field theory of a plastic network of integrate-and-fire neurons

We consider a noise-driven network of integrate-and-fire neurons. The network evolves as result of the activities of the neurons following spike-timing-dependent plasticity rules. We apply a self-consistent mean-field theory to the system to obtain the mean activity level for the system as a function of the mean synaptic weight, which predicts a first-order transition and hysteresis between a noise-dominated regime and a regime of persistent neural activity. Assuming Poisson firing statistics for the neurons, the plasticity dynamics of a synapse under the influence of the mean-field environment can be mapped to the dynamics of an asymmetric random walk in synaptic-weight space. Using a master equation for small steps, we predict a narrow distribution of synaptic weights that scales with the square root of the plasticity rate for the stationary state of the system given plausible physiological parameter values describing neural transmission and plasticity. The dependence of the distribution on the synaptic weight of the mean-field environment allows us to determine the mean synaptic weight self-consistently. The effect of fluctuations in the total synaptic conductance and plasticity step sizes are also considered. Such fluctuations result in a smoothing of the first-order transition for low number of afferent synapses per neuron and a broadening of the synaptic-weight distribution, respectively.

Asymmetric one-dimensional random walks

For one-dimensional surface diffusion in the presence of fields, movement of particles has to be considered as a random walk in which jumps to the right occur at a rate different from jumps to the left. Moments of the displacement distribution are worked out for such a one-dimensional walk to nearest-neighbor sites as well as by longer jumps to second nearest neighbors. The actual distribution of displacements, and how it changes as the asymmetry of the jump rates changes, is also examined, as this provides important information about the participating jump processes. We show that deriving the third moment gives a clear indication of the asymmetry in the random walk.

Large deviation function for entropy production in driven one-dimensional systems

The large deviation function for entropy production is calculated by solving a time-independent eigenvalue problem for a particle driven along a periodic potential. In an intermediate force regime, the large deviation function shows pronounced deviations from a Gaussian behavior with a characteristic "kink" at zero entropy production. Such a feature can also be extracted from the analytical solution of the asymmetric random walk to which the driven particle can be mapped in a certain parameter range.

Propagators and related descriptors for non-Markovian asymmetric random walks with and without boundaries

There are many current applications of the continuous-time random walk (CTRW), particularly in describing kinetic and transport processes in different chemical and biophysical phenomena. We derive exact solutions for the Laplace transforms of the propagators for non-Markovian asymmetric one-dimensional CTRW's in an infinite space and in the presence of an absorbing boundary. The former is used to produce exact results for the Laplace transforms of the first two moments of the displacement of the random walker, the asymptotic behavior of the moments as t-->infinity, and the effective diffusion constant. We show that in the infinite space, the propagator satisfies a relation that can be interpreted as a generalized fluctuation theorem since it reduces to the conventional fluctuation theorem at large times. Based on the Laplace transform of the propagator in the presence of an absorbing boundary, we derive the Laplace transform of the survival probability of the random walker, which is then used to find the mean lifetime for terminated trajectories of the random walk.

On the Local Times of Transient Random Walks

We study the properties of the local and occupation times of certain transient random walks. First, our recent results concerning simple symmetric random walk in higher dimension are surveyed, then we start to establish similar results for simple asymmetric random walk on the line.

Hitting times for independent random walks on ℤd

We consider a system of asymmetric independent random walks on Z d , denoted by {η t . t e R}, stationary under the product Poisson measure V p of marginal density p > 0. We fix a pattern A, an increasing local event, and denote by τ the hitting time of A. By using a loss network representation of our system, at small density, we obtain a coupling between the laws of η t conditioned on (r > t} for all times t. When d ≥ 3, this provides bounds on the rate of convergence of the law of η t conditioned on (r > t} toward its limiting probability measure as t tends to infinity. We also treat the case where the initial measure is close to V ρ without being product.

Rate Constant in Far-from-Equilibrium States of a Replicating System with Mutually Catalyzing Chemicals

As a first step to study reaction dynamics in far-from-equilibrium open systems, we propose a stochastic protocell model in which two mutually catalyzing chemicals are replicating depending on the external flow of energy resources J. This model exhibits an Arrhenius type reaction; furthermore, it produces a non-Arrhenius reaction that exhibits a power-law reaction rate with regard to the activation energy. These dependences are explained using the dynamics of J; the asymmetric random walk of J results in the Arrhenius equation and conservation of J results in a power-law dependence. Further, we find that the discreteness of molecules results in the power change. Effects of cell divisions are also discussed in our model.

On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients

In this paper, we provide a scheme for simulating one-dimensional processes generated by divergence or non-divergence form operators with discontinuous coefficients. We use a space bijection to transform such a process in another one that behaves locally like a Skew Brownian motion. Indeed the behavior of the Skew Brownian motion can easily be approached by an asymmetric random walk.

Mean first-passage and residence times of random walks on asymmetric disordered chains

An algebraic derivation is presented which yields the exact solution of the mean first-passage and mean residence times of a one-dimensional asymmetric random walk for quenched disorder. Two models of disorder are analytically treated. Both, absorbing-absorbing and reflecting-absorbing boundaries are considered. Particularly, the interplay between asymmetry and disorder is studied.

Asymmetric Random Walk in a Reaction Intermediate of Homologous Recombination

At an intermediate step of the homologous recombination between two double-stranded DNA molecules, a point (often called Holliday structure) connecting two strands coming from two recombining partners migrates along the homologous region. Assuming random walk of a connecting point, we previously explained the dependence of recombination frequency on the homology length observed in vivo. In this model, the random walk was assumed to be symmetric in that the forward transition rate equals the backward one. According to observations in vitro, however, catalysed migration appears unidirectional. Taking into account possible asymmetry, we thus reformulate our random walk model to reexamine the observations in vivo . We also derive some theoretical results to analyse dynamic processes observed in vitro.

Number of times a site is visited in two-dimensional random walks.

In this paper, formulas are derived to compute the mean number of times a site has been visited in a random walk on a two-dimensional lattice. Asymmetric random walks are considered, with or without drift, for different boundary conditions. It is shown that in case of absorbing boundaries the mean number of visits reaches stationary values over the lattice; comparisons with a Monte Carlo simulation are also presented.

Simple stochastic model of spontaneous imbibition in Hele‐Shaw cells

AbstractThe imbibition of water in oil in Hele‐Shaw cells is described as an asymmetric random walk of water in a globally disordered medium. The disorder is modeled by an external, nonwhite, dichotomic noise, and the resulting random walk is described in terms of an effective Markovian master equation. The solution to this equation is given, and the mean‐squared‐displacement (MSD) of fluid particles in an effective ordered medium is calculated. This same quantity was also measured in squared cells with a fixed separation, where Soltrol 170 was displaced by twice‐distilled water. The flow generated by spontaneous displacement of oil presented three sequential stages, each with a characteristic speed and advancing front structure. In the first and third stages viscous fingering was not detected, but in the intermediate development stage it was observed. From the recorded time evolution of the front, its average displacement speed and the MSD of water in oil as a function of time were determined. Imbibition of water shows an enhanced diffusion regime in the first stage where MSD varies with t2, without any external noise effect. This is followed by a simple diffusion behavior in an effective medium in the second stage, where the external noise effect is large. In the third stage Gaussian diffusion prevails again. Numerical regression techniques were used to determine the best values of model parameters that fit experimental values for different stages, so that the sum of squared errors between theoretical predictions and experimental measurements is minimized. Good agreement was found with average errors of 14.3%, 5.9% and 0.6%, respectively.

Clark–Ocone formulas and Poincaré inequalities on the discrete cube

We establish Poincaré inequalities for the continuous time random walk on the cube {−1,+1} d . A first method is based on the study of cylindrical functionals. A Poincaré inequality is proved for these functionals and extended to arbitrary functionals. A second method is based on martingale representation formulas. A whole family of Clark–Ocone formulas is then available, which leads to the corresponding family of Poincaré inequalities. These various inequalities are compared through examples. We also show that the cylindrical method extends to some asymmetric continous time random walks on {−1,+1} d .