Biased Random Walk Research Articles

Regulation of Pom cluster dynamics in Myxococcus xanthus.

Precise positioning of the cell division site is essential for the correct segregation of the genetic material into the two daughter cells. In the bacterium Myxococcus xanthus, the proteins PomX and PomY form a cluster on the chromosome that performs a biased random walk to midcell and positively regulates cell division there. PomZ, an ATPase, is necessary for tethering of the cluster to the nucleoid and regulates its movement towards midcell. It has remained unclear how the cluster dynamics change when the biochemical parameters, such as the attachment rates of PomZ dimers to the nucleoid and the cluster, the ATP hydrolysis rate of PomZ or the mobility of PomZ interacting with the nucleoid and cluster, are varied. To answer these questions, we investigate a one-dimensional model that includes the nucleoid, the Pom cluster and PomZ proteins. We find that a mechanism based on the diffusive PomZ fluxes on the nucleoid into the cluster can explain the latter’s midnucleoid localization for a broad parameter range. Furthermore, there is an ATP hydrolysis rate that minimizes the time the cluster needs to reach midnucleoid. If the dynamics of PomZ on the nucleoid is slow relative to the cluster’s velocity, we observe oscillatory cluster movements around midnucleoid. To understand midnucleoid localization, we developed a semi-analytical approach that dissects the net movement of the cluster into its components: the difference in PomZ fluxes into the cluster from either side, the force exerted by a single PomZ dimer on the cluster and the effective friction coefficient of the cluster. Importantly, we predict that the Pom cluster oscillates around midnucleoid if the diffusivity of PomZ on the nucleoid is reduced. A similar approach to that applied here may also prove useful for cargo localization in ParABS systems.

Ergodicity, rejuvenation, enhancement, and slow relaxation of diffusion in biased continuous-time random walks.

Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we report on an interesting phenomenon for the enhancement of diffusion by the start of the measurement in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case, the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. Moreover, the diffusivity can be increased by the aging time when the initial ensemble is not in equilibrium. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance; i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay unexpectedly slowly. Our findings provide a rejuvenation phenomenon in the superdiffusive regime, that is, the diffusivity for a nonequilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed.

Biased random walks on the interlacement set

We study a biased random walk on the interlacement set of $\mathbb{Z}^d$ for $d\geq 3$. Although the walk is always transient, we can show, in the case $d=3$, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.

The origins of Zipf's meaning‐frequency law

In his pioneering research, G.K. Zipf observed that more frequent words tend to have more meanings, and showed that the number of meanings of a word grows as the square root of its frequency. He derived this relationship from two assumptions: that words follow Zipf's law for word frequencies (a power law dependency between frequency and rank) and Zipf's law of meaning distribution (a power law dependency between number of meanings and rank). Here we show that a single assumption on the joint probability of a word and a meaning suffices to infer Zipf's meaning‐frequency law or relaxed versions. Interestingly, this assumption can be justified as the outcome of a biased random walk in the process of mental exploration.

Concerted pulsatile and graded neural dynamics enables efficient chemotaxis in C. elegans

The ability of animals to effectively locate and navigate toward food sources is central for survival. Here, using C. elegans nematodes, we reveal the neural mechanism underlying efficient navigation in chemical gradients. This mechanism relies on the activity of two types of chemosensory neurons: one (AWA) coding gradients via stochastic pulsatile dynamics, and the second (AWCON) coding the gradients deterministically in a graded manner. The pulsatile dynamics of the AWA neuron adapts to the magnitude of the gradient derivative, allowing animals to take trajectories better oriented toward the target. The robust response of AWCON to negative derivatives promotes immediate turns, thus alleviating the costs incurred by erroneous turns dictated by the AWA neuron. This mechanism empowers an efficient navigation strategy that outperforms the classical biased-random walk strategy. This general mechanism thus may be applicable to other sensory modalities for efficient gradient-based navigation.

Metropolis\u2013Hastings Monte Carlo Method for Neutron Emissivity Tomographic Inversion in Tokamak Plasma

The article presents a new approach to plasma neutron emissivity reconstruction based on the Metropolis–Hastings Monte Carlo algorithm. The algorithm is based on a biased random walk. A dedicated computer code generates pseudo-random samples within the domain that contains the solution. The properly chosen objective function ensures convergence to the desired solution. The method allows to overcome the problem of trapping in local minima by accepting, with some finite probability, changes in the direction opposite to that indicated by the objective function. The developed method was tested using a synthetic data set. The results show that the method provides reliable reconstructions and thus can be used as an alternative or complementary method to the commonly applied ones. The method capabilities were also tested against noise present in the line-integrated projections.

A convection–diffusion model for gang territoriality

We present an agent-based model to simulate gang territorial development motivated by graffiti marking on a two-dimensional discrete lattice. For simplicity, we assume that there are two rival gangs present, and they compete for territory. In this model, agents represent gang members and move according to a biased random walk, adding graffiti with some probability as they move and preferentially avoiding the other gang’s graffiti. All agent interactions are indirect, with the interactions occurring through the graffiti field. We show numerically that as parameters vary, a phase transition occurs between a well-mixed state and a well-segregated state. The numerical results show that system mass, decay rate and graffiti rate influence the critical parameter. From the discrete model, we derive a continuum system of convection–diffusion equations for territorial development. Using the continuum equations, we perform a linear stability analysis to determine the stability of the equilibrium solutions and we find that we can determine the precise location of the phase transition in parameter space as a function of the system mass and the graffiti creation and decay rates.

A quenched central limit theorem for biased random walks on supercritical Galton–Watson trees

AbstractIn this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

Impact of diffusive motion on anomalous dispersion in structured disordered media: From correlated Lévy flights to continuous time random walks.

We elucidate the impact of diffusive motion on the nature of anomalous dispersion in layered and fibrous disordered media. We consider two types of disorder characterized by quenched random velocities and quenched random retardation properties. Purely advective particle motion is ballistic in both disorder models. This changes dramatically in the presence of transverse diffusion, which leads to dimension-dependent disorder sampling. For d≤3 dimensions, heavy-tailed velocity distributions render large-scale particle motion a correlated Lévy flight, while transport in the quenched random retardation model behaves as a biased continuous time random walk with correlated time increments.

Using strain dynamics for fracture warning of shaft lining

Understanding the dynamic of shaft strain is of great importance for fracture warning of shaft lining. This study detects the statistical pattern of strain series during the fracture process of shaft lining by using the Rescaled Range Analysis (R/S) method, and two types of algorithms are employed for the comparison purpose to estimate the Hurst exponent. It is shown that the dynamical behavior of shaft lining system is a biased random walk in the long-term scale with Hurst exponent close to 0.6. This paper initiates the direction for reconsidering the fracture warnings of shaft lining based on the behavior of relative strain series. In addition, the rolling time window approach for the short scale analysis is adopted in this work to understand the time-varying correlation of cumulative strains and relative strains, which gives some deep insights into the fracture predictions.

Current fluctuations in periodically driven systems

Small nonequelibrium systems driven by an external periodic protocol can be described by Markov processes with time-periodic transition rates. In general, current fluctuations in such small systems are large and may play a crucial role. We develop a theoretical formalism to evaluate the rate of such large deviations in periodically driven systems. We show that the scaled cumulant generating function that characterizes current fluctuations is given by a maximal Floquet exponent. Comparing deterministic protocols with stochastic protocols, we show that, with respect to large deviations, systems driven by a stochastic protocol with an infinitely large number of jumps are equivalent to systems driven by deterministic protocols. Our results are illustrated with three case studies: a two-state model for a heat engine, a three-state model for a molecular pump, and a biased random walk with a time-periodic affinity.

Decision making improves sperm chemotaxis in the presence of noise.

To navigate their surroundings, cells rely on sensory input that is corrupted by noise. In cells performing chemotaxis, such noise arises from the stochastic binding of signalling molecules at low chemoattractant concentrations. We reveal a fundamental relationship between the speed of chemotactic steering and the strength of directional fluctuations that result from the amplification of noise in a chemical input signal. This relation implies a trade-off between steering that is slow and reliable, and steering that is fast but less reliable. We show that dynamic switching between these two modes of steering can substantially increase the probability to find a target, such as an egg to be found by sperm cells. This decision making confers no advantage in the absence of noise, but is beneficial when chemical signals are detectable, yet characterized by low signal-to-noise ratios. The latter applies at intermediate distances from a target, where signalling molecules are diluted, thus defining a ‘noise zone’ that cells have to cross. Our results explain decision making observed in recent experiments on sea urchin sperm chemotaxis. More generally, our theory demonstrates how decision making enables chemotactic agents to cope with high levels of noise in gradient sensing by dynamically adjusting the persistence length of a biased random walk.

Invariance principle for biased bootstrap random walks

Our main goal is to study a class of processes whose increments are generated via a cellular automata rule. Given the increments of a simple biased random walk, a new sequence of (dependent) Bernoulli random variables is produced. It is built, from the original sequence, according to a cellular automata rule. Equipped with these two sequences, we construct two more according to the same cellular automata rule. The construction is repeated a fixed number of times yielding an infinite array ({−K,…,K}×N) of (dependent) Bernoulli random variables. Taking partial sums of these sequences, we obtain a (2K+1)-dimensional process whose increments belong to the state space {−1,1}2K+1.The aim of the paper is to study the long term behaviour of this process. In particular, we establish transience/recurrence properties and prove an invariance principle. The limiting behaviour of these processes depends strongly on the direction of the iteration, and exhibits few surprising features. This work is motivated by an earlier investigation (see Collevecchio et al. (2015)), in which the starting sequence is symmetric, and by the related work Ferrari et al. (2000).

Biased Random Walk Using Stochastic Switching of Nanomagnets: Application to SAT Solver

Random walk (RW)-based local search algorithms are highly popular for solving combinatorial optimization problems such as the satisfiability (SAT) problem. The RW algorithm tries to solve the SAT problem by flipping a randomly chosen variable to minimize the number of unsatisfied clauses for a given problem. In this paper, we propose a biased RW (BRW) based on stochastic magnetization switching dynamics of nanomagnets in the presence of thermal noise. The controllable stochastic switching behavior of nanomagnets is used to flip the current state of the variable of interest based on the assigned probability. Here, the flipping probabilities are assigned to the variables responsible for unsatisfied clauses (instead of deterministic flipping in traditional algorithms). The stochasticity of individual units of the proposed hardware SAT solver based on a magnetic tunnel junction lying on top of a heavy metal layer enables parallel search of the solution space, which results in rapid convergence in comparison to the baseline RW algorithm. A device-circuit-algorithm cosimulation framework (benchmarked to experimental measurements of a magnetic stack) is used to assess the efficiency of the proposal. In comparison to the baseline RW algorithm, stochastic magnetization switching-driven BRW achieves ~94% reduction in search time while consuming ~30 pJ energy per iteration.

Critical properties of unlimited gliding: Unexpected flocking behavior driven by the exchange of information.

Inspired by albatrosses that use thermal lifts to fly across oceans we develop a simple model of gliders that serves us to study theoretical limitations of unlimited exploration of the Earth. Our studies, grounded in physical theory of continuous percolation and biased random walks, allow us to identify a variety of percolation transitions, which are understood as providing potentially unlimited movement through a space in a specified direction. We discover an unexpected phenomenon of self-organization of gliders in clusters, which resembles the flock organization of birds. This self-organization is intriguing, as it occurs thanks to exchange of information only and without any particular rules that could favor the clustering of the gliders (in contrast to the causes well known in literature, like, for example, attractive forces used in the Vicsek-type models or fitness functions used in evolutionary computation).

Non-Backtracking Centrality Based Random Walk on Networks

Random walks are a fundamental tool for analyzing realistic complex networked systems and implementing randomized algorithms to solve diverse problems such as searching and sampling. For many real applications, their actual effect and convenience depend on the properties (e.g. stationary distribution and hitting time) of random walks, with biased random walks often outperforming traditional unbiased random walks (TURW). In this paper, we present a new class of biased random walks, non-backtracking centrality based random walks (NBCRW) on a network, where the walker prefers to jump to neighbors with high non-backtracking centrality that has some advantages over eigenvector centrality. We study some properties of the non-backtracking matrix of a network, on the basis of which we propose a theoretical framework for fast computation of the transition probabilities, stationary distribution, and hitting times for NBCRW on the network. Within the paradigm, we study NBCRW on some model and real networks and compare the results with those corresponding to TURW and maximal entropy random walks (MERW), with the latter being biased random walks based on eigenvector centrality. We show that the behaviors of stationary distribution and hitting times for NBCRW widely differ from those associated with TURW and MERW, especially for heterogeneous networks.

Stability and Change in Kelp Forest Habitats at San Nicolas Island

Kelp forest communities are highly variable over space and time. Despite this complexity it has been suggested that kelp forest communities can be classified into one of 2 states: kelp dominated or sea urchin dominated. It has been further hypothesized that these represent “alternate stable states” because a site can remain in either of these states for decades before some perturbation causes a rapid shift to the other state. Our research group has maintained a subtidal community monitoring program for 38 years at San Nicolas Island consisting of twice-annual scuba-based surveys at 6 sites distributed within 4 regions around the island. Three types of perturbations are thought to be relevant to subtidal community dynamics at San Nicolas: (1) physical disturbances in the form of major storm and El Nino/ Southern Oscillation (ENSO) events; (2) invertebrate diseases, which periodically decimate urchin populations; and (3) the reintroduction and subsequent increase of sea otters (Enhydra lutris nereis). These 3 perturbations differ in spatial and temporal specificity; physical disturbances and disease outbreaks occur periodically and could affect all 4 regions, while sea otter predation has been concentrated primarily at the West End sites over the last 15 years. The different types of perturbations and the duration of the time series at the kelp forests at San Nicolas make the data set ideal for testing the “alternate stable state” hypothesis. We use nonmetric multidimensional scaling (NMDS) to examine spatial and temporal patterns of community similarity at the 4 regions. In particular, we evaluate support for the existence of stable states, which are represented on NMDS plots as distinct spatial clusters. Community dynamics at each site approximated a biased random walk in NMDS space, with one or more basins of attraction and occasional jumps between basins. We found evidence for alternative stable states at some sites, and we show that transitions from one stable state to another may be influenced by interactions between multiple perturbations.

Effective dynamics of a random walker on a heterogeneous ring: Exact results

In this paper, by considering a biased random walker hopping on a one-dimensional lattice with a ring geometry, we investigate the fluctuations of the speed of the random walker. We assume that the lattice is heterogeneous i.e. the hopping rate of the random walker between the first and the last lattice sites is different from the hopping rate of the random walker between the other links of the lattice. Assuming that the average speed of the random walker in the steady-state is v∗, we have been able to find the unconditional effective dynamics of the random walker where the absolute value of the average speed of the random walker is −v∗. Using a perturbative method in the large system-size limit, we have also been able to show that the effective hopping rates of the random walker near the defective link are highly site-dependent.

Scaling limits for sub-ballistic biased random walks in random conductances

We consider biased random walks in positive random conductances on the $d$-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional law of large numbers for the position of the walker, properly rescaled. Moreover, we state a functional central limit theorem where an atypical process, related to the fractional kinetics, appears in the limit.

Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model

We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed overline{hbox {v}} of the walk. Namely, there exists some critical value lambda _{hbox {c}}>0 such that overline{hbox {v}}>0 if lambda in (0,lambda _{hbox {c}}) and overline{hbox {v}}=0 if lambda ge lambda _{hbox {c}}. We show that the speed overline{hbox {v}} is continuous in lambda on (0,infty ) and differentiable on (0,lambda _{hbox {c}}/2). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of overline{hbox {v}} on (0,lambda _{hbox {c}}/2), we require and prove a central limit theorem for the biased random walk. Additionally, we prove that the central limit theorem fails to hold for lambda ge lambda _{hbox {c}}/2.