TY - JOUR

T1 - Diffusion models for chemotaxis

T2 - a statistical analysis of noninteractive unicellular movement

AU - Watkins, Joseph C.

AU - Woessner, Birgit

N1 - Funding Information:
We both thank Moshe Pollak for seueral enthusiastic discussions and for always finding the appropriate statistical test. J.C. W. acknowledges support for research from the National Science Foundation in the form of grant DMS86 02029.

PY - 1991/5

Y1 - 1991/5

N2 - A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell phsyiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.

AB - A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell phsyiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.

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U2 - 10.1016/0025-5564(91)90065-Q

DO - 10.1016/0025-5564(91)90065-Q

M3 - Article

C2 - 1804464

AN - SCOPUS:0025805762

SN - 0025-5564

VL - 104

SP - 271

EP - 303

JO - Mathematical Biosciences

JF - Mathematical Biosciences

IS - 2

ER -