Treloar Katrina K, Simpson Matthew J, McCue Scott W
School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland 4001, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Dec;84(6 Pt 1):061920. doi: 10.1103/PhysRevE.84.061920. Epub 2011 Dec 28.
Velocity-jump processes are discrete random-walk models that have many applications including the study of biological and ecological collective motion. In particular, velocity-jump models are often used to represent a type of persistent motion, known as a run and tumble, that is exhibited by some isolated bacteria cells. All previous velocity-jump processes are noninteracting, which means that crowding effects and agent-to-agent interactions are neglected. By neglecting these agent-to-agent interactions, traditional velocity-jump models are only applicable to very dilute systems. Our work is motivated by the fact that many applications in cell biology, such as wound healing, cancer invasion, and development, often involve tissues that are densely packed with cells where cell-to-cell contact and crowding effects can be important. To describe these kinds of high-cell-density problems using a velocity-jump process we introduce three different classes of crowding interactions into a one-dimensional model. Simulation data and averaging arguments lead to a suite of continuum descriptions of the interacting velocity-jump processes. We show that the resulting systems of hyperbolic partial differential equations predict the mean behavior of the stochastic simulations very well.
速度跳跃过程是离散随机游走模型，有许多应用，包括对生物和生态集体运动的研究。特别地，速度跳跃模型常被用于表示一种持续运动，即“跑与翻”，一些孤立的细菌细胞会表现出这种运动。之前所有的速度跳跃过程都是非相互作用的，这意味着拥挤效应和个体间相互作用被忽略了。由于忽略了这些个体间相互作用，传统的速度跳跃模型仅适用于非常稀疏的系统。我们开展这项工作的动机是，细胞生物学中的许多应用，如伤口愈合、癌症侵袭和发育，通常涉及细胞密集堆积的组织，其中细胞间接触和拥挤效应可能很重要。为了用速度跳跃过程描述这类高细胞密度问题，我们在一维模型中引入了三类不同的拥挤相互作用。模拟数据和平均论证得出了相互作用速度跳跃过程的一系列连续统描述。我们表明，由此产生的双曲型偏微分方程组能很好地预测随机模拟的平均行为。
