African Institute for Mathematical Sciences, 6 Melrose road, Muizenberg, 7945, South Africa.
Math Biosci Eng. 2013 Jun;10(3):821-42. doi: 10.3934/mbe.2013.10.821.
Biofilms are present in all natural, medical and industrial surroundings where bacteria live. Biofilm formation is a key factor in the growth and transport of both beneficial and harmful bacteria. While much is known about the later stages of biofilm formation, less is known about its initiation which is an important first step in the biofilm formation. In this paper, we develop a non-linear system of partial differential equations of Keller-Segel type model in one-dimensional space, which couples the dynamics of bacterial movement to that of the sensing molecules. In this case, bacteria perform a biased random walk towards the sensing molecules. We derive the boundary conditions of the adhesion of bacteria to a surface using zero-Dirichlet boundary conditions, while the equation describing sensing molecules at the interface needed particular conditions to be set. The numerical results show the profile of bacteria within the space and the time evolution of the density within the free-space and on the surface. Testing different parameter values indicate that significant amount of sensing molecules present on the surface leads to a faster bacterial movement toward the surface which is the first step of biofilm initiation. Our work gives rise to results that agree with the biological description of the early stages of biofilm formation.
生物膜存在于所有自然、医学和工业环境中，这些环境中都有细菌存在。生物膜的形成是有益和有害细菌生长和迁移的关键因素。虽然人们对生物膜形成的后期阶段了解较多，但对其初始阶段的了解较少，而初始阶段是生物膜形成的重要第一步。在本文中，我们在一维空间中开发了一个具有凯勒-塞格尔（Keller-Segel）类型模型的非线性偏微分方程组系统，该系统将细菌运动的动力学与感应分子的动力学结合在一起。在这种情况下，细菌会朝着感应分子进行偏向随机游走。我们使用零狄利克雷边界条件来推导细菌附着在表面的边界条件，而描述界面上感应分子的方程则需要设置特定的条件。数值结果显示了细菌在空间中的分布以及自由空间和表面上密度的时间演化。测试不同的参数值表明，表面上存在大量的感应分子会导致细菌更快地向表面移动，这是生物膜起始的第一步。我们的工作得出的结果与生物膜形成早期阶段的生物学描述相符。
