Perthame Benoît, Tang Min, Vauchelet Nicolas
Laboratoire Jacques-Louis Lions UMR CNRS 7598 and INRIA Paris, Sorbonne Université, UPMC Univ Paris 06, Inria, 75005, Paris, France.
Department of Mathematics, Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China.
J Math Biol. 2016 Nov;73(5):1161-1178. doi: 10.1007/s00285-016-0985-5. Epub 2016 Mar 18.
Kinetic-transport equations are, by now, standard models to describe the dynamics of populations of bacteria moving by run-and-tumble. Experimental observations show that bacteria increase their run duration when encountering an increasing gradient of chemotactic molecules. This led to a first class of models which heuristically include tumbling frequencies depending on the path-wise gradient of chemotactic signal. More recently, the biochemical pathways regulating the flagellar motors were uncovered. This knowledge gave rise to a second class of kinetic-transport equations, that takes into account an intra-cellular molecular content and which relates the tumbling frequency to this information. It turns out that the tumbling frequency depends on the chemotactic signal, and not on its gradient. For these two classes of models, macroscopic equations of Keller-Segel type, have been derived using diffusion or hyperbolic rescaling. We complete this program by showing how the first class of equations can be derived from the second class with molecular content after appropriate rescaling. The main difficulty is to explain why the path-wise gradient of chemotactic signal can arise in this asymptotic process. Randomness of receptor methylation events can be included, and our approach can be used to compute the tumbling frequency in presence of such a noise.
目前，动力学输运方程是描述通过“游动-翻滚”方式运动的细菌群体动力学的标准模型。实验观察表明，细菌在遇到趋化分子浓度不断增加的梯度时，会延长其游动持续时间。这导致了第一类模型的出现，这类模型启发式地将翻滚频率与趋化信号的路径梯度联系起来。最近，调节鞭毛马达的生化途径被揭示。这一知识催生了第二类动力学输运方程，该方程考虑了细胞内分子含量，并将翻滚频率与该信息联系起来。结果表明，翻滚频率取决于趋化信号，而不是其梯度。对于这两类模型，已经使用扩散或双曲重标度推导出了凯勒-西格尔型的宏观方程。我们通过展示在适当重标度后，如何从具有分子含量的第二类方程推导出第一类方程来完成这个程序。主要困难在于解释在这个渐近过程中，趋化信号的路径梯度是如何出现的。可以纳入受体甲基化事件的随机性，并且我们的方法可用于计算存在这种噪声时的翻滚频率。
