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The Diffusion Limit of Transport Equations II: Chemotaxis Equations
"... this paper we use the diusionlimit expansion of transport equations developed earlier [23] to study the limiting equation under avariety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modication of the turning rate, the movement speed or t ..."
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Cited by 28 (4 self)
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this paper we use the diusionlimit expansion of transport equations developed earlier [23] to study the limiting equation under avariety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modication of the turning rate, the movement speed or the preferred direction of movement. Depending on the strength of the bias, it leads to anisotropic diusion, to a drift term in the ux or to both, in the parabolic limit. We show that the classical chemotaxis equation  whichwe call the PatlakKellerSegelAlt (PKSA) equation  only arises when the bias is suciently small. Using this general framework, we derive phenomenological models for chemotaxis of agellated bacteria, of slime molds and of myxobacteria. We also show that certain results derived earlier for onedimensional motion can easily be generalized to two or threedimensional motion as well. ## ############# The linear transport equation @ @t p(x; v; t)+v ##p(x; v; t)=#p(x; v; t)+ # # T(v;v # )p(x; v # ;t)dv # ; (1.1) in which p(x; v; t) represents the density of particles at spatial position x # IR # moving with velocity v # V # IR # at time t # 0, arises when the movement of biological organisms is modeled byavelocityjump process [38]. Here the turning rate may be space or velocitydependent, but in other contexts it may also depend on internal variables that evolve in space and time, in which case (1.1) must be generalized. The turning kernel or turn angle distribution T (v; v # ) gives the probabilityofavelocity jump from v # to v if a jump occurs: in general it may also be spacedependent or depend on internal variables. In the present formulation we assume that the `decision' to turn as reected in is not coupled to th...
From individual to collective behavior in bacterial chemotaxis
 SIAM J Appl Math
, 2004
"... Abstract Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individ ..."
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Cited by 25 (5 self)
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Abstract Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time and/or spacedependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time we obtain a single secondorder hyperbolic equation for the populations density, and using a parabolic scaling we obtain the classical chemotaxis equation, wherein the chemotactic sensitivity is now a known function of parameters of the internal dynamics. Numerical simulations show that the solutions of the macroscopic equations agree very well with the results of Monte Carlo simulations of individual movement. 1. Introduction. The
From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, Multiscale Model
 Simul
, 2005
"... Abstract. The collective behavior of bacterial populations provides an example of how celllevel decision making translates into populationlevel behavior and illustrates clearly the difficult multiscale mathematical problem of incorporating individuallevel behavior into populationlevel models. He ..."
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Cited by 18 (10 self)
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Abstract. The collective behavior of bacterial populations provides an example of how celllevel decision making translates into populationlevel behavior and illustrates clearly the difficult multiscale mathematical problem of incorporating individuallevel behavior into populationlevel models. Here we focus on the flagellated bacterium E. coli, for which a great deal is known about signal detection, transduction, and celllevel swimming behavior. We review the biological background on individual and populationlevel processes and discuss the velocityjump approach used for describing populationlevel behavior based on individuallevel intracellular processes. In particular, we generalize the momentbased approach to macroscopic equations used earlier [R. Erban and H. G. Othmer, SIAM J. Appl. Math., 65 (2004), pp. 361–391] to higher dimensions and show how aspects of the signal transduction and response enter into the macroscopic equations. We also discuss computational issues surrounding the bacterial pattern formation problem and technical issues involved in the derivation of macroscopic equations.
MULTISCALE MODELS OF TAXISDRIVEN PATTERNING IN BACTERIAL POPULATIONS
"... Abstract. Spatiallydistributed populations of various types of bacteria often display intricate spatial patterns that are thought to result from the cellular response to gradients of nutrients or other attractants. In the past decade a great deal has been learned about signal transduction, metaboli ..."
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Cited by 10 (5 self)
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Abstract. Spatiallydistributed populations of various types of bacteria often display intricate spatial patterns that are thought to result from the cellular response to gradients of nutrients or other attractants. In the past decade a great deal has been learned about signal transduction, metabolism and movement in E. coli and other bacteria, but translating the individuallevel behavior into populationlevel dynamics is still a challenging problem. However it is computationally impractical to use a strictly cellbased model to understand patterning in growing populations, since the total number of cells may reach 10 12 − 10 15 in some experiments. In the past phenomenological equations such as the PatlakKellerSegel equations have been used in modeling the cell movement that is involved in the formation of such patterns, but the question remains as to how the microscopic behavior can be correctly described by a macroscopic equation. Significant progress has been made for bacterial species that employ a ‘runandtumble ’ strategy of movement, in that macroscopic equations based on simplified schemes for signal transduction and turning behavior have been derived [14, 13]. Here we extend previous work in a number of directions: (i) we relax a number of the assumptions on the attractant gradient made in previous derivations, (ii) we use a more general turning rate function that better describes the biological behavior, and (iii) we incorporate the effect of hydrodynamic forces that arise when cells swim in close proximity to a surface. We also develop a new approach to solving the moment equations derived from the transport equation to obtain macroscopic equations to any desired order. Numerical examples show that the solution of the lowestorder macroscopic equation agrees well with the solution obtained from a Monte Carlo simulation of cell movement under a variety of temporal protocols for the signal. We also apply the method to derive equations of chemotactic movement that are governed by multiple chemotactic signals.
The intersection of theory and application in elucidating pattern formation in developmental biology
, 2009
"... Abstract. We discuss theoretical and experimental approaches to three distinct developmental systems that illustrate how theory can influence experimental work and viceversa. The chosen systems – Drosophila melanogaster, bacterial pattern formation, and pigmentation patterns – illustrate the fundam ..."
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Cited by 4 (1 self)
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Abstract. We discuss theoretical and experimental approaches to three distinct developmental systems that illustrate how theory can influence experimental work and viceversa. The chosen systems – Drosophila melanogaster, bacterial pattern formation, and pigmentation patterns – illustrate the fundamental physical processes of signaling, growth and cell division, and cell movement involved in pattern formation and development. These systems exemplify the current state of theoretical and experimental understanding of how these processes produce the observed patterns, and illustrate how theoretical and experimental approaches can interact to lead to a better understanding of development. As John Bonner said long ago ‘We have arrived at the stage where models are useful to suggest experiments, and the facts of the experiments in turn lead to new and improved models that suggest new experiments. By this rocking back and forth between the reality of experimental facts and the dream world of hypotheses, we can move slowly toward a satisfactory solution of the major problems of developmental biology.’
doi:10.1016/S00928240(03)000302 Chemotactic Signaling, Microglia, and Alzheimer’s Disease Senile Plaques: Is There a Connection?
"... Chemotactic cells known as microglia are involved in the inflammation associated with pathology in Alzheimer’s disease (AD). We investigate conditions that lead to aggregation of microglia and formation of local accumulations of chemicals observed in AD senile plaques. We develop a model for chemota ..."
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Cited by 3 (0 self)
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Chemotactic cells known as microglia are involved in the inflammation associated with pathology in Alzheimer’s disease (AD). We investigate conditions that lead to aggregation of microglia and formation of local accumulations of chemicals observed in AD senile plaques. We develop a model for chemotaxis in response to a combination of chemoattractant and chemorepellent signaling chemicals. Linear stability analysis and numerical simulations of the model predict that periodic patterns in cell and chemical distributions can evolve under local attraction, longranged repulsion, and other constraints on concentrations and diffusion coefficients of the chemotactic signals. Using biological parameters from the literature, we compare and discuss the applicability of this model to actual processes in AD. c ○ 2003 Society for Mathematical Biology. Published by Elsevier Science Ltd. All rights reserved. ∗Author to whom correspondence should be addressed. † Reprint address. ‡ Maternity leave.
Understanding the immune response in tuberculosis using different mathematical models and biological scales
 SIAM Journal of Multiscale Modeling & Simulation
, 2005
"... Abstract. The use of different mathematical tools to study biological processes is necessary to capture effects occurring at different scales. Here we study as an example the immune response to infection with the bacteria Mycobacterium tuberculosis, the causative agent of tuberculosis (TB). Immune r ..."
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Cited by 2 (0 self)
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Abstract. The use of different mathematical tools to study biological processes is necessary to capture effects occurring at different scales. Here we study as an example the immune response to infection with the bacteria Mycobacterium tuberculosis, the causative agent of tuberculosis (TB). Immune responses are both global (lymph nodes, blood, and spleen) as well as local (site of infection) in nature. Interestingly, the immune response in TB at the site of infection results in the formation of spherical structures comprised of cells, bacteria, and effector molecules known as granulomas. In this work, we use four different mathematical tools to explore both the global immune response as well as the more local one (granuloma formation) and compare and contrast results obtained using these methods. Applying a range of approaches from continuous deterministic models to discrete stochastic ones allows us to make predictions and suggest hypotheses about the underlying biology that might otherwise go unnoticed. The tools developed and applied here are also applicable in other settings such as tumor modeling.
The Mathematical Analysis of Biological Aggregation and Dispersal: Progress, Problems and Perspectives
"... Abstract Motile organisms alter their movement in response to signals in their environment for a variety of reasons, such as to find food or mates or to escape danger. In populations of individuals this often this leads to largescale pattern formation in the form of coherent movement or localized a ..."
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Cited by 1 (1 self)
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Abstract Motile organisms alter their movement in response to signals in their environment for a variety of reasons, such as to find food or mates or to escape danger. In populations of individuals this often this leads to largescale pattern formation in the form of coherent movement or localized aggregates of individuals, and an important question is how the individuallevel decisions are translated into populationlevel behavior. Mathematical models are frequently developed for a populationlevel description, and while these are often phenomenological, it is important to understand how individuallevel properties can be correctly embedded in the populationlevel models. We discuss several classes of models that are used to describe individual movement and indicate how they can be translated into populationlevel models. 1
DOI 10.1007/s0028500802013 Mathematical Biology A user’s guide to PDE models for chemotaxis
"... Abstract Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the unde ..."
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Abstract Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The KellerSegel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225– 234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “autoaggregation”, has led to its prominence as a mechanism for selforganisation of biological systems. This phenomenon has been shown to lead to finitetime blowup under certain formulations of the model, and a large body of work has been devoted to determining when blowup occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.