图灵斑图形成的离散-连续混合建模方法
A hybrid discrete-continuum approach to model Turing pattern formation
摘要：
自1952年引入以来，图灵的（前）模式理论（“形态发生的化学基础”）已被广泛应用于发育生物学的许多领域。相关的图案形成模型通常包括用于相互作用的化学物质（“形态发生剂”）的反应-扩散方程的系统，其在某些空间域中的异质分布充当细胞的模板，以通过例如由化学预图案诱导的分化或增殖来形成某种图案或结构。在这里，我们开发了一个混合离散连续建模框架，通过图灵机制的细胞模式的形成。在这个框架中，一个随机的基于个人的细胞运动和增殖模型相结合的反应扩散系统的浓度的一些形态。作为一个说明性的例子，我们专注于一个模型，其中的形态发生的动力学是由一个激活剂-抑制剂系统，产生图灵前模式。然后，细胞通过两种形式的化学依赖性细胞作用中的任一种与其局部区域的形态发生剂相互作用：趋化性和化学控制的增殖。我们首先考虑这样一个混合模型构成的静态空间域，然后转向增长域的情况下。在这两种情况下，我们正式推导出相应的确定性连续极限，并表明，有一个很好的定量匹配的空间模式所产生的随机个人为基础的模型和它的确定性连续对应，当足够大的细胞数量被认为是。本文的目的是提出一个概念的建模框架的基础上的想法的证明，目的是在未来的特定模式和形态发生过程的研究，然后应用相关的方法。
英文摘要：
Since its introduction in 1952, Turing's (pre-)pattern theory ("the chemical basis of morphogenesis") has been widely applied to a number of areas in developmental biology. The related pattern formation models normally comprise a system of reaction-diffusion equations for interacting chemical species ("morphogens"), whose heterogeneous distribution in some spatial domain acts as a template for cells to form some kind of pattern or structure through, for example, differentiation or proliferation induced by the chemical pre-pattern. Here we develop a hybrid discrete-continuum modelling framework for the formation of cellular patterns via the Turing mechanism. In this framework, a stochastic individual-based model of cell movement and proliferation is combined with a reaction-diffusion system for the concentrations of some morphogens. As an illustrative example, we focus on a model in which the dynamics of the morphogens are governed by an activator-inhibitor system that gives rise to Turing pre-patterns. The cells then interact with morphogens in their local area through either of two forms of chemically-dependent cell action: chemotaxis and chemically-controlled proliferation. We begin by considering such a hybrid model posed on static spatial domains, and then turn to the case of growing domains. In both cases, we formally derive the corresponding deterministic continuum limit and show that that there is an excellent quantitative match between the spatial patterns produced by the stochastic individual-based model and its deterministic continuum counterpart, when sufficiently large numbers of cells are considered. This paper is intended to present a proof of concept for the ideas underlying the modelling framework, with the aim to then apply the related methods to the study of specific patterning and morphogenetic processes in the future.
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