Simpson Matthew J, Baker Ruth E
Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 May;83(5 Pt 1):051922. doi: 10.1103/PhysRevE.83.051922. Epub 2011 May 26.
In the exclusion-process literature, mean-field models are often derived by assuming that the occupancy status of lattice sites is independent. Although this assumption is questionable, it is the foundation of many mean-field models. In this work we develop methods to relax the independence assumption for a range of discrete exclusion-process-based mechanisms motivated by applications from cell biology. Previous investigations that focused on relaxing the independence assumption have been limited to studying initially uniform populations and ignored any spatial variations. By ignoring spatial variations these previous studies were greatly simplified due to translational invariance of the lattice. These previous corrected mean-field models could not be applied to many important problems in cell biology such as invasion waves of cells that are characterized by moving fronts. Here we propose generalized methods that relax the independence assumption for spatially inhomogeneous problems, leading to corrected mean-field descriptions of a range of exclusion-process-based models that incorporate (i) unbiased motility, (ii) biased motility, and (iii) unbiased motility with agent birth and death processes. The corrected mean-field models derived here are applicable to spatially variable processes including invasion wave-type problems. We show that there can be large deviations between simulation data and traditional mean-field models based on invoking the independence assumption. Furthermore, we show that the corrected mean-field models give an improved match to the simulation data in all cases considered.
在排除过程文献中，平均场模型通常是通过假设晶格位点的占据状态是独立的而推导出来的。尽管这个假设存在问题，但它是许多平均场模型的基础。在这项工作中，我们开发了一些方法，以放宽对一系列基于离散排除过程的机制的独立性假设，这些机制是由细胞生物学中的应用所激发的。以前专注于放宽独立性假设的研究仅限于研究初始均匀的群体，并且忽略了任何空间变化。由于晶格的平移不变性，通过忽略空间变化，这些先前的研究被大大简化了。这些先前的修正平均场模型无法应用于细胞生物学中的许多重要问题，例如以移动前沿为特征的细胞入侵波。在这里，我们提出了广义方法，放宽了对空间非均匀问题的独立性假设，从而得到了一系列基于排除过程的模型的修正平均场描述，这些模型包括：（i）无偏运动性，（ii）有偏运动性，以及（iii）带有个体出生和死亡过程的无偏运动性。这里推导的修正平均场模型适用于包括入侵波类型问题在内的空间可变过程。我们表明，基于独立性假设的模拟数据与传统平均场模型之间可能存在很大偏差。此外，我们表明，在所有考虑的情况下，修正平均场模型与模拟数据的匹配度都有所提高。
