Online Resource 6 Rev.zip from From a discrete model of chemotaxis with volume-filling to a generalized Patlak–Keller–Segel model

We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak–Keller–Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this discrete model. The family of steady-state solutions of such a generalized PKS model are characterized and the conditions for the emergence of spatial patterns are studied via linear stability analysis. Moreover, we carry out a systematic quantitative comparison between numerical simulations of the discrete model and numerical solutions of the corresponding PKS model, both in one and in two spatial dimensions. The results obtained indicate that there is excellent quantitative agreement between the spatial patterns produced by the two models. Finally, we numerically show that the outcomes of the two models faithfully replicate those of the classical PKS model in a suitable asymptotic regime.

本文提出一种趋化性（chemotaxis）离散模型，将响应趋化因子（chemoattractant）的细胞建模为独立个体单元，其运动由一组可生成偏向性随机游走（biased random walk）的规则描述。为考量高细胞密度下（即体积填充效应，volume-filling）观测到的细胞运动能力变化，我们令细胞运动概率由细胞密度的衰减函数进行调制。本文严格证明，广为人知的Patlak-Keller-Segel（PKS）趋化模型的一般形式，可通过该离散模型的恰当连续极限（continuum limit）严格推导得到。我们对该广义PKS模型的稳态解（steady-state solution）族进行了特征刻画，并通过线性稳定性分析（linear stability analysis）研究了空间模式（spatial patterns）的形成条件。此外，我们分别在一维和二维空间维度下，对离散模型的数值模拟（numerical simulations）结果与对应PKS模型的数值解开展了系统性定量对比。所得结果表明，两类模型所生成的空间模式之间具备极佳的定量一致性。最后，我们通过数值模拟证明，在恰当的渐近区域中，两类模型的结果能够精准复现经典PKS模型的输出。