Chen K C, Ford R M, Cummings P T
Department of Chemical Engineering, University of Virginia, Charlottesville, VA 22903-2442, USA.
J Theor Biol. 1998 Dec 21;195(4):481-504. doi: 10.1006/jtbi.1998.0808.
Mathematical models considering motile bacterial transport within a geometrically restrictive cylindrical tube were developed. Two macroscopic transport parameters, the random motility coefficient as a self-diffusion coefficient of the cell population and the chemotactic velocity as a chemical-induced velocity, were derived. The three-dimensional cell balance equation was reduced to forms similar to Segel's one-dimensional phenomenological cell balance equations with additional modifications for bacteria-wall interactions. Two conceptually different approaches accounting for such interactions were presented. The first approach parallels treatments in the gas kinetic theory by viewing bacterial interactions with walls as collisions and subsequent diffusive/specular reflections, which led to the Bosanquet formula for the bacterial diffusion coefficient. Based on the experimental observation that bacterial swimming motion is guided by a straight tube, the second approach considered modifications in the bacterial swimming orientation as a consequence of various long-range interactions with the tube surface. A phenomenological turning model capable of aligning bacterial motion along a tube axis was proposed. The model predicts that under the geometrical restriction of a small cylindrical tube, the macroscopic bacterial transport resulting from the proposed turning model can exhibit behavior that ranges from dimensionally reduced diffusion to pure wave propagation, depending on the influence of the tube diameter on the reversal probability in the bacterial swimming motion. Our theoretical model provides explicit equations that explain how such a transition can occur. The predicted results were then qualitatively compared with experimental data from the literature. As a preliminary comparison, we concluded that bacterial transport in cylindrical tubes of diameter 10 micrometers remains in the mode of a dimensionally reduced diffusion, and shifts to a wave motion when the tube diameter decreases to 6 micrometers.
建立了考虑细菌在几何形状受限的圆柱形管内运动性传输的数学模型。推导了两个宏观传输参数，即作为细胞群体自扩散系数的随机运动系数和作为化学诱导速度的趋化速度。三维细胞平衡方程被简化为与塞格尔一维唯象细胞平衡方程相似的形式，并对细菌与管壁的相互作用进行了额外修正。提出了两种在概念上不同的处理这种相互作用的方法。第一种方法类似于气体动力学理论中的处理方式，将细菌与壁的相互作用视为碰撞以及随后的扩散/镜面反射，这导致了细菌扩散系数的博赞奎特公式。基于细菌游动受直管引导的实验观察，第二种方法考虑了由于与管表面的各种长程相互作用导致的细菌游动方向的改变。提出了一种能够使细菌运动沿管轴排列的唯象转向模型。该模型预测，在小直径圆柱形管的几何限制下，根据管直径对细菌游动中反转概率的影响，由所提出的转向模型产生的宏观细菌传输行为可以从维度降低的扩散到纯波传播。我们的理论模型提供了明确的方程来解释这种转变是如何发生的。然后将预测结果与文献中的实验数据进行了定性比较。作为初步比较，我们得出结论，直径为10微米的圆柱形管中的细菌传输仍处于维度降低的扩散模式，而当管直径减小到6微米时，会转变为波动运动。
