In contrast to normal diffusion, there is no canonical model for reactions between chemical species which move by anomalous subdiffusion. Indeed, the type of mesoscopic equation describing reaction-subdiffusion systems depends on subtle assumptions about the microscopic behavior of individual molecules. Furthermore, the correspondence between mesoscopic and microscopic models is not well understood. In this paper, we study the subdiffusion-limited model, which is defined by mesoscopic equations with fractional derivatives applied to both the movement and the reaction terms. Assuming that the reaction terms are affine functions, we show that the solution to the fractional system is the expectation of a random time change of the solution to the corresponding integer order system. This result yields a simple and explicit algebraic relationship between the fractional and integer order solutions in Laplace space. We then find the microscopic Langevin description of individual molecules that corresponds to such mesoscopic equations and give a computer simulation method to generate their stochastic trajectories. This analysis identifies some precise microscopic conditions that dictate when this type of mesoscopic model is or is not appropriate. We apply our results to several scenarios in cell biology which, despite the ubiquity of subdiffusion in cellular environments, have been modeled almost exclusively by normal diffusion. Specifically, we consider subdiffusive models of morphogen gradient formation, fluctuating mobility, and fluorescence recovery after photobleaching (FRAP) experiments. We also apply our results to fractional ordinary differential equations.

中文翻译：

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仿射反应的受限子扩散的子扩散方程：解，随机路径和应用

与正常扩散相反，由于异常亚扩散而移动的化学物质之间没有标准的反应模型。实际上，描述反应-扩散系统的介观方程的类型取决于有关单个分子微观行为的微妙假设。此外，对介观模型和微观模型之间的对应关系还不太了解。在本文中，我们研究了子扩散受限模型，该模型由介观方程定义，并且分数阶导数同时应用于运动和反应项。假设反应项是仿射函数，我们表明分数系统的解是对相应整数阶系统解的随机时间变化的期望。该结果在拉普拉斯空间中的分数阶和整数阶解之间产生了简单且明确的代数关系。然后，我们找到了与此类介观方程相对应的单个分子的微观Langevin描述，并给出了一种计算机模拟方法来生成其随机轨迹。该分析确定了一些精确的微观条件，这些条件决定了这种介观模型何时合适。我们将结果应用于细胞生物学中的几种情况，尽管在细胞环境中普遍存在亚扩散，但几乎只能通过正常扩散对其进行建模。具体来说，我们考虑了光致漂白（FRAP）实验后，形态发生子梯度形成，波动迁移率和荧光恢复的亚扩散模型。