随机图灵模式：基于隔室的方法分析
Stochastic Turing patterns: analysis of compartment-based approaches
摘要：
在反应扩散系统中可以观察到图灵斑图，其中化学物种具有不同的扩散常数。近年来，一些研究调查了噪声对图灵模式的影响，并表明观察到随机图灵模式的参数区域可以大于确定性模型预测的参数区域，确定性模型是用物种浓度的偏微分方程来描述的。一个常见的随机反应扩散方法是写在基于隔室（基于格子）的模型，其中感兴趣的域被划分为人工隔室和模拟每个隔室中的分子数量。本文研究了随机图灵斑图对区间大小的依赖性。以前已经表明（对于相对简单的系统），建模者不应该选择太小或太大的隔室尺寸，并且最佳隔室尺寸取决于扩散常数。考虑到这些结果，我们提出并研究了一个基于隔室的图灵模式模型，其中每个化学物种使用不同的一组隔室来描述。它示出的参数区域的空间模式的形式是不同的经典的确定性偏微分方程为基础的模型所获得的区域，但它们也是不同的随机反应扩散模型，使用一套单一的隔间所有的化学物种所获得的结果。特别是，有人认为，一些以前报道的结果的影响，噪声在生物系统中的图灵模式需要重新解释。
英文摘要：
Turing patterns can be observed in reaction-diffusion systems where chemical species have different diffusion constants. In recent years, several studies investigated the effects of noise on Turing patterns and showed that the parameter regimes, for which stochastic Turing patterns are observed, can be larger than the parameter regimes predicted by deterministic models, which are written in terms of partial differential equations for species concentrations. A common stochastic reaction-diffusion approach is written in terms of compartment-based (lattice-based) models, where the domain of interest is divided into artificial compartments and the number of molecules in each compartment is simulated. In this paper, the dependence of stochastic Turing patterns on the compartment size is investigated. It has previously been shown (for relatively simpler systems) that a modeller should not choose compartment sizes which are too small or too large, and that the optimal compartment size depends on the diffusion constant. Taking these results into account, we propose and study a compartment-based model of Turing patterns where each chemical species is described using a different set of compartments. It is shown that the parameter regions where spatial patterns form are different from the regions obtained by classical deterministic PDE-based models, but they are also different from the results obtained for the stochastic reaction-diffusion models which use a single set of compartments for all chemical species. In particular, it is argued that some previously reported results on the effect of noise on Turing patterns in biological systems need to be reinterpreted.
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