scholarly journals A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations

1988 ◽  
Vol 106 (2) ◽  
pp. 303-309 ◽  
Author(s):  
RT Tranquillo ◽  
DA Lauffenburger ◽  
SH Zigmond
Keyword(s):  
Random Walk ◽  
Receptor Binding ◽  
Cell Movement ◽  
Movement Behavior ◽  
Random Motility ◽  
Concentration Gradients ◽  
Theoretical Understanding ◽  
Directional Bias ◽  
Persistence Time ◽  
Orientation Bias

Two central features of polymorphonuclear leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic "persistence time" between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an "orientation bias" characterizing the fraction of cells moving up the gradient. A coherent picture of cell movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper, we offer the possibility that "noise" in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we develop a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant/receptor binding. This model can simulate cell paths similar to those observed experimentally, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Furthermore, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. Thus, the concept of signal "noise" can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients.

scholarly journals Migration of individual microvessel endothelial cells: stochastic model and parameter measurement

1991 ◽  
Vol 99 (2) ◽  
pp. 419-430 ◽  
Author(s):  
C.L. Stokes ◽  
D.A. Lauffenburger ◽  
S.K. Williams
Keyword(s):  
Individual Cell ◽  
Mean Square Displacement ◽  
Cell Movement ◽  
Random Motility ◽  
Directional Bias ◽  
Decay Rate Constant ◽  
Persistence Time ◽  
Physiological Processes ◽  
The Mean ◽  
Alpha Beta

Analysis of cell motility effects in physiological processes can be facilitated by a mathematical model capable of simulating individual cell movement paths. A quantitative description of motility of individual cells would be useful, for example, in the study of the formation of new blood vessel networks in angiogenesis by microvessel endothelial cell (MEC) migration. In this paper we propose a stochastic mathematical model for the random motility and chemotaxis of single cells, and evaluate migration paths of MEC in terms of this model. In our model, cell velocity under random motility conditions is described as a persistent random walk using the Ornstein-Uhlenbeck (O-U) process. Two parameters quantify this process: the magnitude of random movement accelerations, alpha, and a decay rate constant for movement velocity, beta. Two other quantities often used in measurements of individual cell random motility properties—cell speed, S, and persistence time in velocity, Pv—can be defined in terms of the fundamental stochastic parameters alpha and beta by: S =square root (alpha/beta) and Pv = 1/beta. We account for chemotactic cell movement in chemoattractant gradients by adding a directional bias term to the O-U process. The magnitude of the directional bias is characterized by the chemotactic responsiveness, kappa. A critical advantage of the proposed model is that it can generate, using experimentally measured values of alpha, beta and kappa, computer simulations of theoretical individual cell paths for use in evaluating the role of cell migration in specific physiological processes. We have used the model to assess MEC migration in the presence of absence of the angiogenic stimulus acidic fibroblast growth factor (aFGF). Time-lapse video was used to observe and track the paths of cells moving in various media, and the mean square displacement was measured from these paths. To test the validity of the model, we compared the mean square displacement measurements of each cell with model predictions of that displacement. The comparison indicates that the O-U process provides a satisfactory description of the random migration at this level of comparison. Using nonlinear regression in these comparisons, we measured the magnitude of random accelerations, alpha, and the velocity decay rate constant, beta, for each cell path. We consequently obtained values for the derived quantities, speed and persistence time. In control medium, we find that alpha = 250 +/− 100 microns 2h-3 and beta = 0.22 +/− 0.03h-1, while in stimulus medium (control plus unpurified aFGF) alpha = 1900 +/− 720 microns 2h-3 and beta = 0.99 +/− 0.37h-1.(ABSTRACT TRUNCATED AT 400 WORDS)

scholarly journals Diamond aggregation

2010 ◽  
Vol 149 (2) ◽  
pp. 351-372
Author(s):  
WOUTER KAGER ◽  
LIONEL LEVINE
Keyword(s):  
Random Walk ◽  
Growth Model ◽  
Power Law ◽  
Simple Random Walk ◽  
Internal Diffusion ◽  
Diffusion Limited Aggregation ◽  
Directional Bias ◽  
Model Based ◽  
Diffusion Limited ◽  
Logarithmic Fluctuations

AbstractInternal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 for which the limiting shape is a diamond. Certain of these walks—those with a directional bias toward the origin—have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

scholarly journals Computational modelling of cell chain migration reveals mechanisms that sustain follow-the-leader behaviour

2012 ◽  
Vol 9 (72) ◽  
pp. 1576-1588 ◽  
Author(s):  
Michelle L. Wynn ◽  
Paul M. Kulesa ◽  
Santiago Schnell
Keyword(s):  
Cancer Metastasis ◽  
Computational Modelling ◽  
Cell Movement ◽  
Cell Contact ◽  
Model Parameters ◽  
Directional Bias ◽  
Agent Based ◽  
Chain Migration ◽  
A Chain ◽  
Cell Cell

Follow-the-leader chain migration is a striking cell migratory behaviour observed during vertebrate development, adult neurogenesis and cancer metastasis. Although cell–cell contact and extracellular matrix (ECM) cues have been proposed to promote this phenomenon, mechanisms that underlie chain migration persistence remain unclear. Here, we developed a quantitative agent-based modelling framework to test mechanistic hypotheses of chain migration persistence. We defined chain migration and its persistence based on evidence from the highly migratory neural crest model system, where cells within a chain extend and retract filopodia in short-lived cell contacts and move together as a collective. In our agent-based simulations, we began with a set of agents arranged as a chain and systematically probed the influence of model parameters to identify factors critical to the maintenance of the chain migration pattern. We discovered that chain migration persistence requires a high degree of directional bias in both lead and follower cells towards the target. Chain migration persistence was also promoted when lead cells maintained cell contact with followers, but not vice-versa. Finally, providing a path of least resistance in the ECM was not sufficient alone to drive chain persistence. Our results indicate that chain migration persistence depends on the interplay of directional cell movement and biased cell–cell contact.

scholarly journals Study of Pollutant Transport in Environmental Flows Using Depth-Averaged Random Walk Method

10.29007/gd96 ◽  
2018 ◽  
Author(s):  
Xuefei Wu ◽  
Fan Yang ◽  
Dongfang Liang
Keyword(s):  
Random Walk ◽  
Solute Transport ◽  
Analytical Solutions ◽  
Environmental Flows ◽  
Concentration Gradients ◽  
Bed Elevation ◽  
Numerical Predictions ◽  
Parametric Studies ◽  
Transport Problems ◽  
Sharp Concentration

A depth-averaged random walk scheme is applied to investigate the process of solute transport, including advection, diffusions and reaction. Firstly, the model is used to solve an instantaneous release problem in a uniform flow, for which analytical solutions exist. Its performance is examined by comparing numerical predictions with analytical solutions. The advantage of the random walk model includes high accuracy and small numerical diffusion. Extensive parametric studies are carried out to investigate the sensitivity of the predictions to the number of particles. The result reveals that the particle number influences the accuracy of the model significantly. Finally, the model is applied to track a pollutant cloud in the Thames Estuary, where the domain geometry and bed elevation are complex. The present model is free of fictitious oscillations close to sharp concentration gradients and displays encouraging efficiency and accuracy in solving the solute transport problems in a natural aquatic environment.

scholarly journals Three-dimensional random walk models of individual animal movement and their application to trap counts modelling

2020 ◽  
Author(s):  
Danish Ali Ahmed ◽  
Simon Benhamou ◽  
Michael Bonsall ◽  
Sergei Petrovskii
Keyword(s):  
Random Walk ◽  
Spatial Ecology ◽  
Animal Movement ◽  
Research Effort ◽  
Movement Pattern ◽  
Trapping Efficiency ◽  
Movement Direction ◽  
Short Term ◽  
Directional Bias ◽  
Cylindrical Trap

Abstract Background: Random walks (RWs) have proved to be a powerful modelling tool in ecology, particularly in the study of animal movement. An application of RW concerns trapping which is the predominant sampling method to date in insect ecology, invasive species, and agricultural pest management. A lot of research effort has been directed towards modelling ground-dwelling insects by simulating their movement in 2D, and computing pitfall trap counts, but comparatively very little for flying insects with 3D elevated traps. Methods: We introduce the mathematics behind 3D RWs and present key metrics such as the mean squared displacement (MSD) and path sinuosity, which are already well known in 2D. We develop the mathematical theory behind the 3D correlated random walk (CRW) which involves short-term directional persistence and the 3D Biased random walk (BRW) which introduces a long-term directional bias in the movement so that there is an overall preferred movement direction. In this study, we consider three types of shape of 3D traps, which are commonly used in ecological field studies; a spheroidal trap, a cylindrical trap and a rectangular cuboidal trap. By simulating movement in 3D space, we investigated the effect of 3D trap shapes and sizes and of movement diffusion on trapping efficiency. Results: We found that there is a non-linear dependence of trap counts on the trap surface area or volume, but the effect of volume appeared to be a simple consequence of changes in area. Nevertheless, there is a slight but clear hierarchy of trap shapes in terms of capture efficiency, with the spheroidal trap retaining more counts than a cylinder, followed by the cuboidal type for a given area. We also showed that there is no effect of short-term persistence when diffusion is kept constant, but trap counts significantly decrease with increasing diffusion. Conclusion: Our results provide a better understanding of the interplay between the movement pattern, trap geometry and impacts on trapping efficiency, which leads to improved trap count interpretations, and more broadly, has implications for spatial ecology and population dynamics.

Analysis of the Movements of Myogenic Cells during Muscle Organogenesis in Vitro

1974 ◽  
Vol 52 (4) ◽  
pp. 901-905
Author(s):  
P. B. Noble ◽  
S. C. Peterson
Keyword(s):  
Random Walk ◽  
Cell Movement ◽  
Two Dimensional ◽  
Myogenic Cells ◽  
Organogenesis In Vitro ◽  
Random Walk Theory ◽  
Two Phases

The paths taken by myoblasts during the proliferation and fusion phases have been analyzed. It was found that, although the criteria for the two-dimensional random-walk theory were not satisfied, analyses of related parameters showed important differences between cell movement in these two phases. It appears that cell movement just prior to fusion is characterized by longer steps and a smaller angle of departure from the previous direction of motion when compared with cells in the proliferative cells.

scholarly journals Nonmotile Single-Cell Migration as a Random Walk in Nonuniformity: The “Extreme Dumping Limit” for Cell-to-Cell Communications

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Grigorios P. Panotopoulos ◽  
Sebastian Aguayo ◽  
Ziyad S. Haidar
Keyword(s):  
Stochastic Process ◽  
Cell Migration ◽  
Random Walk ◽  
Single Cell ◽  
Brownian Particle ◽  
Cell Movement ◽  
Planck Equation ◽  
Equation Of Motion ◽  
Higher Moments ◽  
Cell Responses

In the present work, we model single-cell movement as a random walk in an external potential observed within the extreme dumping limit, which we define herein as the extreme nonuniform behavior observed for cell responses and cell-to-cell communications. Starting from the Newton–Langevin equation of motion, we solve the corresponding Fokker–Planck equation to compute higher moments of the displacement of the cell, and then we build certain quantities that can be measurable experimentally. We show that, each time, the dynamics depend on the external force applied, leading to predictions distinct from the standard results of a free Brownian particle. Our findings demonstrate that cell migration viewed as a stochastic process is still compatible with biological and experimental observations without the need to rely on more complicated or sophisticated models proposed previously in the literature.

scholarly journals Three-dimensional random walk models of individual animal movement and their application to trap counts modelling

2020 ◽  
Author(s):  
Danish Ali Ahmed ◽  
Simon Benhamou ◽  
Michael Bonsall ◽  
Sergei Petrovskii
Keyword(s):  
Random Walk ◽  
Spatial Ecology ◽  
Animal Movement ◽  
Research Effort ◽  
Movement Pattern ◽  
Trapping Efficiency ◽  
Movement Direction ◽  
Short Term ◽  
Directional Bias ◽  
Cylindrical Trap

Abstract Background: Random walks (RWs) have proved to be a powerful modelling tool in ecology, particularly in the study of animal movement. An application of RW concerns trapping which is the predominant sampling method to date in insect ecology, invasive species, and agricultural pest management. A lot of research effort has been directed towards modelling ground-dwelling insects by simulating their movement in 2D, and computing pitfall trap counts, but comparatively very little for flying insects with 3D elevated traps. Methods: We introduce the mathematics behind 3D RWs and present key metrics such as the mean squared displacement (MSD) and path sinuosity, which are already well known in 2D. We develop the mathematical theory behind the 3D correlated random walk (CRW) which involves short-term directional persistence and the 3D Biased random walk (BRW) which introduces a long-term directional bias in the movement so that there is an overall preferred movement direction. In this study, we consider three types of shape of 3D traps, which are commonly used in ecological field studies; a spheroidal trap, a cylindrical trap and a rectangular cuboidal trap. By simulating movement in 3D space, we investigated the effect of 3D trap shapes and sizes and of movement diffusion on trapping efficiency. Results: We found that there is a non-linear dependence of trap counts on the trap surface area or volume, but the effect of volume appeared to be a simple consequence of changes in area. Nevertheless, there is a slight but clear hierarchy of trap shapes in terms of capture efficiency, with the spheroidal trap retaining more counts than a cylinder, followed by the cuboidal type for a given area. We also showed that there is no effect of short-term persistence when diffusion is kept constant, but trap counts significantly decrease with increasing diffusion . Conclusion: Our results provide a better understanding of the interplay between the movement pattern, trap geometry and impacts on trapping efficiency, which leads to improved trap count interpretations, and more broadly, has implications for spatial ecology and population dynamics.

scholarly journals Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias

2015 ◽  
Vol 12 (106) ◽  
pp. 20150228 ◽  
Author(s):  
Rachelle N. Binny ◽  
Michael J. Plank ◽  
Alex James
Keyword(s):  
Spatial Structure ◽  
Mean Field ◽  
Cell Movement ◽  
Population Level ◽  
Average Density ◽  
Small Scale ◽  
Directional Bias ◽  
Field Approach ◽  
Spatial Moment ◽  
Migrating Cells

The ability of cells to undergo collective movement plays a fundamental role in tissue repair, development and cancer. Interactions occurring at the level of individual cells may lead to the development of spatial structure which will affect the dynamics of migrating cells at a population level. Models that try to predict population-level behaviour often take a mean-field approach, which assumes that individuals interact with one another in proportion to their average density and ignores the presence of any small-scale spatial structure. In this work, we develop a lattice-free individual-based model (IBM) that uses random walk theory to model the stochastic interactions occurring at the scale of individual migrating cells. We incorporate a mechanism for local directional bias such that an individual's direction of movement is dependent on the degree of cell crowding in its neighbourhood. As an alternative to the mean-field approach, we also employ spatial moment theory to develop a population-level model which accounts for spatial structure and predicts how these individual-level interactions propagate to the scale of the whole population. The IBM is used to derive an equation for dynamics of the second spatial moment (the average density of pairs of cells) which incorporates the neighbour-dependent directional bias, and we solve this numerically for a spatially homogeneous case.

Sign in / Sign up

Export Citation Format

Share Document