一类混合元胞自动机细胞集落生长模型的稳定性分析
Stability Analysis of a Hybrid Cellular Automaton Model of Cell Colony Growth
摘要：
细菌、肿瘤细胞和真菌的细胞集落在营养有限的生长条件下，表现出复杂的分支生长模式。为了研究这一现象，我们提出了一个简单的混合元胞自动机模型的细胞集落生长。在该模型中，集落的生长受到细胞消耗的营养物的限制，如果它低于一定的阈值，则会抑制细胞分裂。使用这个模型，我们已经研究了细胞的营养消耗率如何影响菌落的生长动力学。我们发现，对于低消费率的殖民地采取伊甸园般的形态，而较高的消费率的殖民地的形态是分支的分形几何。这些发现与以前的结果是一致的，但这里提出的模型的简单性允许系统的线性稳定性分析。通过观察到的殖民地的局部增长是成比例的营养通量，我们推导出一个近似的色散关系的殖民地接口的增长。这种分散关系表明，生长的稳定性取决于营养物质渗透到菌落中的程度。对于低营养消耗率的渗透距离是大的，这稳定的增长，而对于高消耗率的渗透距离是小的，这导致不稳定的分支生长。当穿透距离消失的色散关系减少到一个描述没有紫外正则化的拉普拉斯增长。通过测量平均分支宽度如何依赖于细胞的消耗速率来验证色散关系，并且理论与模拟之间显示出良好的一致性。
英文摘要：
Cell colonies of bacteria, tumour cells and fungi, under nutrient limited growth conditions, exhibit complex branched growth patterns. In order to investigate this phenomenon we present a simple hybrid cellular automaton model of cell colony growth. In the model the growth of the colony is limited by a nutrient that is consumed by the cells and which inhibits cell division if it falls below a certain threshold. Using this model we have investigated how the nutrient consumption rate of the cells affects the growth dynamics of the colony. We found that for low consumption rates the colony takes on a Eden-like morphology, while for higher consumption rates the morphology of the colony is branched with a fractal geometry. These findings are in agreement with previous results, but the simplicity of the model presented here allows for a linear stability analysis of the system. By observing that the local growth of the colony is proportional to the flux of the nutrient we derive an approximate dispersion relation for the growth of the colony interface. This dispersion relation shows that the stability of the growth depends on how far the nutrient penetrates into the colony. For low nutrient consumption rates the penetration distance is large, which stabilises the growth, while for high consumption rates the penetration distance is small, which leads to unstable branched growth. When the penetration distance vanishes the dispersion relation is reduced to the one describing Laplacian growth without ultra-violet regularisation. The dispersion relation was verified by measuring how the average branch width depends on the consumption rate of the cells and shows good agreement between theory and simulations.
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