具有物种依赖运动的反应-亚扩散方程
Reaction-subdiffusion equations with species-dependent movement
摘要：
反应扩散方程是自然科学中最常见的数学模型之一，用于模拟反应与扩散运动相结合的系统。然而，而不是正常的扩散，异常亚扩散是在许多系统中观察到的，在细胞生物学中尤其普遍。描述一个包含一级反应和次扩散运动的系统的反应-次扩散方程是什么？在本文中，我们回答这个问题。我们推导出分数反应亚扩散方程描述了任意数量的分子物种，反应在一阶速率和移动subdiffusively与一般空间相关的扩散系数和漂移。重要的是，不同的物种可能有不同的扩散率和漂移，这与以前的方法，假设每个物种有相同的运动动力学这个问题。我们将含时分数阶Fokker-Planck方程的结果与分析随机开关演化方程的方法相结合，导出了该方程。此外，我们构建了随机描述的单个分子的确定性浓度遵循这些反应亚扩散方程。这种随机描述涉及从属的扩散过程，其动力学控制的从属马尔可夫跳过程。我们说明了我们的结果在几个例子中，并表明，反应子扩散方程的解决方案同意与随机模拟的单个分子。
英文摘要：
Reaction-diffusion equations are one of the most common mathematical models in the natural sciences and are used to model systems that combine reactions with diffusive motion. However, rather than normal diffusion, anomalous subdiffusion is observed in many systems and is especially prevalent in cell biology. What are the reaction-subdiffusion equations describing a system that involves first-order reactions and subdiffusive motion? In this paper, we answer this question. We derive fractional reaction-subdiffusion equations describing an arbitrary number of molecular species which react at first-order rates and move subdiffusively with general space-dependent diffusivities and drifts. Importantly, different species may have different diffusivities and drifts, which contrasts previous approaches to this question which assume that each species has the same movement dynamics. We derive the equations by combining results on time-dependent fractional Fokker-Planck equations with methods of analyzing stochastically switching evolution equations. Furthermore, we construct the stochastic description of individual molecules whose deterministic concentrations follow these reaction-subdiffusion equations. This stochastic description involves subordinating a diffusion process whose dynamics are controlled by a subordinated Markov jump process. We illustrate our results in several examples and show that solutions of the reaction-subdiffusion equations agree with stochastic simulations of individual molecules.
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