Ward John P, King John R, Koerber Adrian J, Croft Julie M, Sockett R Elizabeth, Williams Paul
Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK.
J Math Biol. 2003 Jul;47(1):23-55. doi: 10.1007/s00285-002-0190-6.
We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions.
我们开发数学模型来研究细菌生物膜的形成、生长和群体感应活性。该模型的生长方面基于这样一种假设：在发育中的生物膜内，细菌细胞连续体的生长会产生运动，这种运动由一个速度场来描述。沃德等人（2001年）提出的一个描述群体感应的模型——细菌通过使用群体感应分子（QSMs）来监测自身种群密度的过程——与生长模型相结合。由此产生的非线性偏微分方程组通过数值方法求解，得到的结果在定性上与实验结果一致。通过假设均匀初始条件得出的解析解表明，在很长时间内，生物膜随时间呈代数增长；建立了与实验数据一致的生物膜生物量线性增长标准。分析表明，在生物学上现实的极限情况下，生物膜中无活性和活性群体感应之间存在分岔。该模型还预测，群体感应行为的行波可能在特定时间范围内出现；而行波分析揭示了一系列可能的行波速度，数值解表明，通过线性化确定的最小波速在广泛的初始条件下都能实现。
