Persistent Random Walk Research Articles

Cattaneo-type subdiffusion-reaction equation.

Subdiffusion in a system in which mobile particles A can chemically react with static particles B according to the rule A+B→B is considered within a persistent random-walk model. This model, which assumes a correlation between successive steps of particles, provides hyperbolic Cattaneo normal diffusion or fractional subdiffusion equations. Starting with the difference equation, which describes a persistent random walk in a system with chemical reactions, using the generating function method and the continuous-time random-walk formalism, we will derive the Cattaneo-type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account. We will also find its solution over a long time limit. Based on the obtained results, we will find the Cattaneo-type subdiffusion-reaction equation in the case in which mobile particles of species A and B can chemically react according to a more complicated rule.

Protrusion Fluctuations Direct Cell Motion

Many physiological phenomena involve directional cell migration. It is usually attributed to chemical gradients in vivo. Recently, other cues have been shown to guide cells in vitro, including stiffness/adhesion gradients or micropatterned adhesive motifs. However, the cellular mechanism leading to these biased migrations remains unknown, and, often, even the direction of motion is unpredictable. In this study, we show the key role of fluctuating protrusions on ratchet-like structures in driving NIH3T3 cell migration. We identified the concept of efficient protrusion and an associated direction index. Our analysis of the protrusion statistics facilitated the quantitative prediction of cell trajectories in all investigated conditions. We varied the external cues by changing the adhesive patterns. We also modified the internal cues using drug treatments, which modified the protrusion activity. Stochasticity affects the short- and long-term steps. We developed a theoretical model showing that an asymmetry in the protrusion fluctuations is sufficient for predicting all measures associated with the long-term motion, which can be described as a biased persistent random walk.

Mean cover time of one-dimensional persistent random walks.

The cover time is defined as the time needed for a random walker to visit every site of a confined domain. Here, we focus on persistent random walks, which provide a minimal model of random walks with short-range memory. We derive the exact expression of the mean cover time of a one-dimensional lattice by such a persistent random walk, both for periodic and reflecting boundary conditions.

Emergence of collective modes and tri-dimensional structures from epithelial confinement.

Many in vivo processes, including morphogenesis or tumour maturation, involve small populations of cells within a spatially restricted region. However, the basic mechanisms underlying the dynamics of confined cell assemblies remain largely to be deciphered and would greatly benefit from well-controlled in vitro experiments. Here we show that confluent epithelial cells cultured on finite population-sized domains, exhibit collective low-frequency radial displacement modes as well as stochastic global rotation reversals. A simple mathematical model, in which cells are described as persistent random walkers that adapt their motion to that of their neighbours, captures the essential characteristics of these breathing oscillations. As these epithelia mature, a tri-dimensional peripheral cell cord develops at the domain edge by differential extrusion, as a result of the additional degrees of freedom of the border cells. These results demonstrate that epithelial confinement alone can induce morphogenesis-like processes including spontaneous collective pulsations and transition from 2D to 3D.

Swimming path statistics of an active Brownian particle with time-dependent self-propulsion

Typically, in the description of active Brownian particles, a constant effective propulsion force is assumed, which is then subjected to fluctuations in orientation and translation, leading to a persistent random walk with an enlarged long-time diffusion coefficient. Here, we generalize previous results for the swimming path statistics to a time-dependent, and thus in many situations more realistic, propulsion which is a prescribed input. We analytically calculate both the noise-free and the noise-averaged trajectories for time-periodic propulsion under the action of an additional torque. In the deterministic case, such an oscillatory microswimmer moves on closed paths that can be much more complicated than the commonly observed straight lines and circles. When exposed to random fluctuations, the mean trajectories turn out to be self-similar curves which bear the characteristics of their noise-free counterparts. Furthermore, we consider a propulsion force which scales in time t as ∝tα (with α = 0,1,2, …) and analyze the resulting superdiffusive behavior. Our predictions are verifiable for diffusiophoretic artificial microswimmers with prescribed propulsion protocols.

Renormalization group for quantum walks

We present a detailed introduction to the discrete-time quantum walk problem, in close analogy with the classical ordinary and persistent random walk. This approach facilitates a uniform application of the renormalization group that highlights similarities and differences between the classical and the quantum walk problem. Specifically, we discuss the renormalization group treatment for the mean-square displacement of a walker starting from a single site on the 1d-line for ordinary and persistent random walks and the quantum walk. We outline the significance of universality for quantum walks and the control this might provide for quantum algorithms. We use our RG method to verify that all 2-state quantum walks on the 1d-line are in the same universality class.

Persistent random walk with exclusion

Modelling the propagation of a pulse in a dense milieu poses fundamental challenges at the theoretical and applied levels. To this aim, in this paper we generalize the telegraph equation to non-ideal conditions by extending the concept of persistent random walk to account for spatial exclusion effects. This is achieved by introducing an explicit constraint in the hopping rates, that weights the occupancy of the target sites. We derive the mean-field equations, which display nonlinear terms that are important at high density. We compute the evolution of the mean square displacement (MSD) for pulses belonging to a specific class of spatially symmetric initial conditions. The MSD still displays a transition from ballistic to diffusive behaviour. We derive an analytical formula for the effective velocity of the ballistic stage, which is shown to depend in a nontrivial fashion upon both the density (area) and the shape of the initial pulse. After a density-dependent crossover time, nonlinear terms become negligible and normal diffusive behaviour is recovered at long times.

A visualization tool of 3-D time-varying data for the simulation of tissue growth

Data Visualization affords us the ability to explore the spatial and temporal domains of many time-varying phenomena. In this article, we describe our application of visualization to a three-dimensional simulation model for tissue growth. We review the different components of the model where cellular automata is used to model populations of cells that execute persistent random walks, collide, and proliferate until they reach confluence. We then describe the system architecture of the developed visualization tool, the employed rendering techniques, and the related prototyping interfaces. We also discuss some of the visualization results obtained thus far that are pertinent to enhancing the validity of the computational model. This visualization tool could be useful in facilitating the research of scientists by providing them with meaningful means to interpret and analyze simulation data and to compare them to experimental results. Our objective in this work is to develop computer-aided design solutions that support the simulation of tissue growth and its design exploration.

Reconstruction of a persistent random walk from exit time distributions

In this paper, we study the inverse problem of reconstructing the spatially dependent transition rate F (x) of a one-dimensional Broadwell process from exit time distributions. In such a process, an advecting particle is assumed to undergo transitions between states with constant positive (+v) and negative (−v) velocities. The goal is to reconstruct the transition rate function F (x) from the exit time distributions out of a finite interval. Using the associated backward equation, we compute the distribution of exit times and its Laplace transform, given a fixed starting position and velocity. We propose two methods (called ‘t’ and ‘s’) for finding F (x). In both methods, we represent F (x) as a linear combination of polynomials and repeatedly solve the backward equation to minimize the difference between its solution and given first exit time data. In the t-method we work in the time domain, using exit times directly and leveraging a novel series solution for the exit time distribution. In the s-method, we work with the Laplace-transformed equation and Laplace-transformed exit times. Noisy data is generated using a custom-designed algorithm to simulate the trajectories of a Broadwell process. In most cases we can find 4 coefficients to within O(10−1) accuracy from O(104) exit times, with the t-method method slightly out-performing the s-method. We also explore the effectiveness of our algorithms for a fixed number of exit times under different advection speeds and find that optimal reconstruction occurs when v = O(1).

Three-dimensional hMSC motility within peptide-functionalized PEG-based hydrogels of varying adhesivity and crosslinking density

Human mesenchymal stem cell (hMSC) migration and recruitment play a critical role during bone fracture healing. Within the complex three-dimensional (3-D) in vivo microenvironment, hMSC migration is regulated through a myriad of extracellular cues. Here, we use a thiol–ene photopolymerized hydrogel to recapitulate structural and bioactive inputs in a tunable manner to understand their role in regulating 3-D hMSC migration. Specifically, peptide-functionalized poly(ethylene glycol) hydrogels were used to encapsulate hMSC while varying the crosslinking density, from 0.18±0.02 to 1.60±0.04mM, and the adhesive ligand density, from 0.001 to 1.0mM. Using live-cell videomicroscopy, migratory cell paths were tracked and fitted to a Persistent Random Walk model. It was shown that hMSC migrating through the lowest crosslinking density and highest adhesivity had more sustained polarization, higher migrating speeds (17.6±0.9μmh−1) and higher cell spreading (elliptical form factor=3.9±0.2). However, manipulation of these material properties did not significantly affect migration persistence. Further, there was a monotonic increase in cell speed and spreading with increasing adhesivity that showed a lack of the biphasic trend seen in 2-D cell migration. Immunohistochemistry showed well-formed actin fibers and β1 integrin staining at the ends of stress fibers. This thiol–ene platform provides a highly tunable substrate to characterize 3-D hMSC migration that can be applied as an implantable cell carrier platform or for the recruitment of endogenous hMSC in vivo.

Modeling Killer Cell Migration and Search Strategies

During the immune response, cytotoxic T lymphocytes (CTL) and Natural Killer cells (NK cells) patrol the human body. Upon recognition of virus-infected or cancerous target cells, the killer cells release lytic granules at the immunological synapse and kill the targets. For optimal immune function, the migration machinery has to be adjusted in order to maximize the search efficiency.We used in vitro time lapse microscopy on primary human CTL and NK cells to study their migration in a two-dimensional geometry. In the absence of target cells, CTL alternate between a mobile and a stationary state. The preference for each state depends on the external calcium concentration. On a short time scale (seconds/minutes), killer cell migration is characterized by directional persistence, while migration on a longer time scale (minutes/hours) is random. A mathematical model of a persistent random walker can accurately describe killer cell migration and reveals an optimum for target search efficiency. This optimum depends on the length of persistent motion in one direction and the size of the search space. We believe that this model is very useful to understand how regulation of killer cell migration can serve to optimize the immune response.

Parallelization of a Bio-inspired Computational Model for the Simulation of 3-D Multicellular Tissue Growth

The use of parallelism may overcome some of the constraints imposed by single processor computing systems. Besides offering faster solutions, applications that are parallelized can solve bigger or more complex problems. For instance, simulations can be run at finer resolutions while physical phenomena can be potentially modeled more realistically. We describe in this paper the development of a bio-inspired parallel algorithm used in the three-dimensional simulation of multicellular tissue growth. We report on the different components of the model where cellular automata is used to model different types of cell populations that execute persistent random walks on the computational grid, collide, and proliferate until they reach confluence. We also discuss the main issues encountered in the parallelization of the model and its implementation on a parallel machine.

Diffusion coefficients for periodically induced multi-step persistent walks on regular lattices

We present a generalization of our formalism for the computation of diffusion coefficients of multi-step persistent random walks on regular lattices to walks which include zero-displacement states. This situation is especially relevant to systems where tracer particles move across potential barriers as a result of the action of a periodic forcing whose period sets the timescale between transitions.

Optimizing Persistent Random Searches

We consider a minimal model of persistent random searcher with a short range memory. We calculate exactly for such a searcher the mean first-passage time to a target in a bounded domain and find that it admits a nontrivial minimum as function of the persistence length. This reveals an optimal search strategy which differs markedly from the simple ballistic motion obtained in the case of Poisson distributed targets. Our results show that the distribution of targets plays a crucial role in the random search problem. In particular, in the biologically relevant cases of either a single target or regular patterns of targets, we find that, in strong contrast to repeated statements in the literature, persistent random walks with exponential distribution of excursion lengths can minimize the search time, and in that sense perform better than any Levy walk.

Vibrated polar disks: spontaneous motion, binary collisions, and collective dynamics

We study the spontaneous motion, binary collisions, and collective dynamics of “polar disks”, i.e. specially designed particles which, when vibrated between two horizontal plates, move coherently along a direction strongly correlated to their intrinsic polarity. The motion of our particles, although nominally three-dimensional and complicated, is well accounted for by a two-dimensional persistent random walk. Their binary collisions are spatiotemporally extended events during which multiple actual collisions happen, yielding a weak average effective alignment. We show that this well-controlled, “dry active matter” system can display collective motion with orientationally-ordered regions of the order of the system size. We provide evidence of strong number density fluctuations in the most ordered regimes observed. These results are discussed in the light of the limitations of our system, notably those due to the inevitable presence of walls.

Passive control of cell locomotion using micropatterns: the effect of micropattern geometry on the migratory behavior of adherent cells

Directed cell migration is critical to a variety of biological and physiological processes. Although simple topographical patterns such as parallel grooves and three-dimensional post arrays have been studied to guide cell migration, the effect of the dimensions and shape of micropatterns, which respectively represent the amount and gradient of physical spatial cues, on cell migration has not yet been fully explored. This motivates a quantitative characterization of cell migration in response to micropatterns having different widths and divergence angles. The changes in the migratory (and even locational) behavior of adherent cells, when the cells are exposed to physical spatial cues imposed by the micropatterns, are explored here using a microfabricated biological platform, nicknamed the "Rome platform". The Rome platform, made of a biocompatible, ultraviolet (UV) curable polymer (ORMOCOMP), consists of 3 μm thick micropatterns with different widths of 3 to 75 μm and different divergence angles of 0.5 to 5.0°. The migration paths through which NIH 3T3 fibroblasts move on the micropatterns are analyzed with a persistent random walk model, thus quantifying the effect of the divergence angle of micropatterns on cell migratory characteristics such as cell migration speed, directional persistence time, and random motility coefficient. The effect of the width of micropatterns on cell migratory characteristics is also extensively investigated. Cell migration direction is manipulated by creating the gradient of physical spatial cues (that is, divergence angle of micropatterns), while cell migration speed is controlled by modulating the amount of them (namely, width of micropatterns). In short, the amount and gradient of physical spatial cues imposed by changing the width and divergence angle of micropatterns make it possible to control the rate and direction of cell migration in a passive way. These results offer a potential for reducing the healing time of open wounds with a smart wound dressing engraved with micropatterns (or microscaffolds).

Chemotaxis in Densely Populated Tissue Determines Germinal Center Anatomy and Cell Motility: A New Paradigm for the Development of Complex Tissues

Germinal centers (GCs) are complex dynamic structures that form within lymph nodes as an essential process in the humoral immune response. They represent a paradigm for studying the regulation of cell movement in the development of complex anatomical structures. We have developed a simulation of a modified cyclic re-entry model of GC dynamics which successfully employs chemotaxis to recapitulate the anatomy of the primary follicle and the development of a mature GC, including correctly structured mantle, dark and light zones. We then show that correct single cell movement dynamics (including persistent random walk and inter-zonal crossing) arise from this simulation as purely emergent properties. The major insight of our study is that chemotaxis can only achieve this when constrained by the known biological properties that cells are incompressible, exist in a densely packed environment, and must therefore compete for space. It is this interplay of chemotaxis and competition for limited space that generates all the complex and biologically accurate behaviors described here. Thus, from a single simple mechanism that is well documented in the biological literature, we can explain both higher level structure and single cell movement behaviors. To our knowledge this is the first GC model that is able to recapitulate both correctly detailed anatomy and single cell movement. This mechanism may have wide application for modeling other biological systems where cells undergo complex patterns of movement to produce defined anatomical structures with sharp tissue boundaries.

Correlation of cell membrane dynamics and cell motility

BackgroundEssential events of cell development and homeostasis are revealed by the associated changes of cell morphology and therefore have been widely used as a key indicator of physiological states and molecular pathways affecting various cellular functions via cytoskeleton. Cell motility is a complex phenomenon primarily driven by the actin network, which plays an important role in shaping the morphology of the cells. Most of the morphology based features are approximated from cell periphery but its dynamics have received none to scant attention. We aim to bridge the gap between membrane dynamics and cell states from the perspective of whole cell movement by identifying cell edge patterns and its correlation with cell dynamics.ResultsWe present a systematic study to extract, classify, and compare cell dynamics in terms of cell motility and edge activity. Cell motility features extracted by fitting a persistent random walk were used to identify the initial set of cell subpopulations. We propose algorithms to extract edge features along the entire cell periphery such as protrusion and retraction velocity. These constitute a unique set of multivariate time-lapse edge features that are then used to profile subclasses of cell dynamics by unsupervised clustering.ConclusionsBy comparing membrane dynamic patterns exhibited by each subclass of cells, correlated trends of edge and cell movements were identified. Our findings are consistent with published literature and we also identified that motility patterns are influenced by edge features from initial time points compared to later sampling intervals.

Diffusive transport of light in two-dimensional granular materials

We study photon diffusion in a two-dimensional random packing of monodisperse disks as a simple model of granular material. We apply ray optics approximation to set up a persistent random walk for the photons. We employ Fresnel's intensity reflectance with its rich dependence on the incidence angle and polarization state of the light. We present an analytic expression for the transport-mean-free path l* in terms of the refractive indices of grains and host medium, grain radius, and packing fraction. We perform numerical simulations to examine our analytical result.

Scaling laws in the diffusion limited aggregation of persistent random walkers

We investigate the diffusion limited aggregation of particles executing persistent random walks. The scaling properties of both random walks and large aggregates are presented. The aggregates exhibit a crossover between ballistic and diffusion limited aggregation models. A non-trivial scaling relation ξ∼ℓ1.25 between the characteristic size ξ, in which the cluster undergoes a morphological transition, and the persistence length ℓ, between ballistic and diffusive regimes of the random walk, is observed.