Woolley Thomas E, Baker Ruth E, Gaffney Eamonn A, Maini Philip K
Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, OX1 3LB, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Aug;84(2 Pt 1):021915. doi: 10.1103/PhysRevE.84.021915. Epub 2011 Aug 10.
A central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity. Generally, the mathematical treatments of biological systems have used continuum, mean-field hypotheses for their constituent parts, which ignores any sources of intrinsic stochastic effects. In this paper we consider a stochastic space-jump process as a description of diffusion, i.e., particles are able to undergo a random walk on a discretized domain. By developing analytical Fourier methods we are able to probe this probabilistic framework, which gives us insight into the patterning potential of diffusive systems. Further, an alternative description of domain growth is introduced, with which we are able to rigorously link the mean-field and stochastic descriptions. Finally, through combining these ideas, it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean-field steady state.
发育生物学中的一个核心挑战是理解稳健的时空异质性是如何产生的。一般来说，对生物系统的数学处理方法对其组成部分采用连续统、平均场假设，这忽略了任何内在随机效应的来源。在本文中，我们考虑一种随机空间跳跃过程作为扩散的一种描述，即粒子能够在离散域上进行随机游走。通过开发解析傅里叶方法，我们能够探究这个概率框架，这使我们深入了解扩散系统的模式形成潜力。此外，还引入了一种对域增长的替代描述，通过它我们能够严格地将平均场描述和随机描述联系起来。最后，通过结合这些想法，结果表明，在确定性增长域上的这种扩散随机描述能够支持远离确定性平均场稳态的状态的成核。
