Watkins J C, Woessner B
Department of Mathematics, Univesity of Southern California, Los Angeles 90089-1113.
Math Biosci. 1991 May;104(2):271-303. doi: 10.1016/0025-5564(91)90065-q.
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 physiology. 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.
开发了一个将随机微分方程应用于趋化性模型的程序。首先回顾了一些关于游动细菌和沿底物迁移细胞的趋化性实验模型和理论模型。在确定性系统的物理和生物模型中，有限差分方程常常被一个极限微分方程所取代，以便利用微积分使用上的简便性。这里为趋化性随机模型开发了一种类似但更复杂的方法。之所以能够进行这样的阐述，是因为概率论的最新研究成果使得随机微积分在扩散中的使用变得简便，并且随机差分方程收敛到随机微分方程具有广泛的适用性。随机微分方程为模型提供了有用的数据并提供了统计检验。在分析博亚尔斯基、诺布尔和彼得森在研究人类粒细胞时提出的一维模型时，我们从现象学考虑入手。在这种情况下，理论模型在于根据细胞生理学的理论考虑，确定哪种扩散最能近似细胞运动模型。给出了这样一个扩散近似定理，并讨论了自协方差与持续性之间的关系。随机微积分和扩散近似定理都在一维中进行了描述。最后，将这些工具扩展到多维模型，并应用于具有球对称性的三维实验装置。
