School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK.
Department of Mathematical Sciences "G. L. Lagrange", Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, 10129 Torino, Italy.
Math Biosci Eng. 2020 Oct 29;17(6):7442-7479. doi: 10.3934/mbe.2020381.
Since its introduction in 1952, with a further refinement in 1972 by Gierer and Meinhardt, Turing's (pre-)pattern theory (the chemical basis of morphogenesis) has been widely applied to a number of areas in developmental biology, where evolving cell and tissue structures are naturally observed. 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 the 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.
自 1952 年引入以来，1972 年 Gierer 和 Meinhardt 进一步改进，图灵的（前）模式理论（形态发生的化学基础）已广泛应用于发育生物学的多个领域，在这些领域中，不断进化的细胞和组织结构是自然观察到的。相关的模式形成模型通常包括一个相互作用的化学物质（形态发生素）的反应扩散方程系统，其在某个空间域中的不均匀分布作为细胞通过例如由化学前模式诱导的分化或增殖来形成某种模式或结构的模板。在这里，我们通过图灵机制开发了一种用于通过细胞模式形成的混合离散连续建模框架。在该框架中，细胞运动和增殖的随机个体模型与一些形态发生素的浓度的反应扩散系统相结合。作为说明性示例，我们专注于其中形态发生素动力学由激活抑制剂系统控制的模型，该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与局部区域中的形态发生素相互作用：趋化性和化学控制增殖。我们首先考虑在静态空间域上的这种混合模型，然后转向生长域的情况。在这两种情况下，我们正式推导相应的确定性连续体极限，并表明，当考虑足够多的细胞时，由随机个体模型产生的空间模式与其确定性连续体对应物之间存在极好的定量匹配。本文旨在提出建模框架的基本思想的概念验证，目的是将来应用相关方法研究特定的模式形成和形态发生过程。
