McKane Alan J, Biancalani Tommaso, Rogers Tim
Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK,
Bull Math Biol. 2014 Apr;76(4):895-921. doi: 10.1007/s11538-013-9827-4. Epub 2013 Mar 8.
We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.
我们回顾了生物系统中随机性建模背后的数学形式。从根据系统的基本组成部分对其进行描述开始，我们推导出了控制动力学的介观方程，这些方程推广了更为人熟知的宏观方程。我们将这种形式应用于分析在受生物启发的模型中观察到的两种特定的噪声诱导现象。在第一个例子中，我们展示了图灵不稳定性的随机放大如何产生空间和时间模式，这些模式可以在线性噪声近似下得到理解。第二个例子涉及细胞极性的自发出现，我们通过利用时间尺度的分离取得了分析进展。
