扩张细菌菌落的动力学：连续介质建模与分析
Kinetics of an Expanding Bacterial Colony: Continuum Modeling and Analysis
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中文总结 AI 辅助
本研究针对硬质基质上扩张的细菌菌落，构建含反应-扩散方程、拉普拉斯方程的移动边界连续介质模型，分析其垂直与径向扩张特性，结果与实验及智能体模拟相符。
中文摘要 AI 辅助
我们研究细菌菌落于硬质基质上扩张的时空动力学。基质中的营养物质扩散进入菌落，被细菌细胞吸收以用于生长和分裂，使初始的单层菌落逐渐扩张为薄饼状菌落。营养物质的浓度决定了菌落内的局部细胞生长速率。质量守恒将该局部生长速率与速度关联，该速度通过达西定律近似为与压力梯度成正比。综上，生长的菌落被建模为移动边界问题，其中营养物质浓度和压力分别求解反应-扩散方程和拉普拉斯方程。我们针对不同几何设置分析该自洽移动边界模型：对于一般的三维圆柱对称模型，研究稳态营养物质浓度；构建并分析用于垂直扩张的一维模型，以及用于菌落径向扩张的二维圆盘模型。我们的分析发现，营养物质在菌落内耗尽，会减慢菌落的垂直扩张速度；营养物质浓度降至莫诺常数（单个细菌几乎无法生长的阈值）的垂直高度呈指数快速降低；我们还估算了渐近径向扩张速率；此外，我们确定营养物质浓度高于阈值（允许细菌生长及菌落径向扩张）的区域为固定厚度的环形外围区域。所有这些结果均与文献中报道的实验和基于智能体的模拟一致。
英文摘要
We study the spatiotemporal dynamics of the expansion of a bacterial colony on a hard substrate. Nutrient from the substrate diffuses into the colony and is taken up by the bacterial cells for them to grow and divide, expanding the initial monolayer and then pancake-shaped colony. The concentration of nutrient determines the local cell growth rate in the colony. Mass conservation relates such local growth rate with velocity which is approximated to be proportional to the pressure gradient by Darcy's law. Altogether, the growing colony is modeled as a moving-boundary problem with the nutrient concentration and pressure solving a reaction-diffusion equation and Laplace's equation, respectively. We analyze the self-consistent moving-boundary model with respect to different geometrical setting. For a general three-dimensional cylindrically symmetric model, we study the steady-state nutrient concentration. We construct and analyze a one-dimensional model for vertical expansion and a two-dimensional disk model for radial expansion of the colony. Our analysis finds that the nutrient depletes into the colony and slows down the vertical expansion of the colony. The vertical level where the nutrient concentration reaches the Monod constant, a threshold below which individual bacteria hardly grow, lowers down exponentially fast. We also estimate the asymptotic radial expansion rate. Moreover, we establish that the region where the nutrient concentration is above the threshold, allowing bacteria to grow and the colony to expand radially, is a ring-shaped peripheral region of fixed thickness. All these are consistent with experiment and agent-based simulations reported in literature.
