Mayo Manuel, Soto Rodrigo
Universidad de Sevilla, Física Teórica, Apartado de Correos 1065, E-41080 Sevilla, Spain.
Universidad de Chile, Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Avenida Blanco Encalada 2008, Santiago, Chile.
Phys Rev E. 2025 May;111(5-1):054409. doi: 10.1103/PhysRevE.111.054409.
Bacterial chemotaxis for the case of Escherichia coli is controlled by methylation of chemoreceptors, which in a biochemical pathway regulates the concentration of the CheY-P protein that finally controls the tumbling rate. As a consequence, the tumbling rate adjusts to changes in the concentration of relevant chemicals, such as to produce a biased random walk toward chemoattractants or against the repellers. Methylation is a slow process, implying that the internal concentration of CheY-P is not instantaneously adapted to the environment, and therefore the tumbling rate presents a memory. This implies that the Keller-Segel equations used to describe chemotaxis at the macroscopic scale, which assume a local relation between the bacterial flux and the chemical gradient, cannot be fully valid as memory and the associated nonlocal response are not taken into account. To derive the new equations that replace the Keller-Segel ones, we use a kinetic approach, in which a kinetic equation for the bacterial transport is written considering the dynamics of the protein concentration. When memory is large, the protein concentration field must be considered a relevant variable on equal foot as the bacterial density. Working out in detail the Chapman-Enskog method, the dynamical equations for these fields are obtained, which have the form of reaction-diffusion equations with flux and source terms depending on the gradients on the chemical signal. Also, the transport coefficients are obtained entirely in terms o the microscopic dynamics, showing important symmetry properties and giving their values of the case of E. coli. Solving the equations for an inhomogeneous signal it is shown that the response is nonlocal, with a smoothing length as large as 170µm for E. coli. The homogeneous response and the relaxational dynamics are also studied in detail. For completeness, the case of small memory is also studied, in which case the Chapman-Enskog method reproduces the Keller-Segel equations, with explicit expressions for the transport coefficients.
大肠杆菌的细菌趋化作用由化学感受器的甲基化控制，在生化途径中，这调节了最终控制翻滚速率的CheY-P蛋白的浓度。因此，翻滚速率会根据相关化学物质浓度的变化进行调整，例如朝着化学引诱剂或远离排斥剂产生有偏向性的随机游动。甲基化是一个缓慢的过程，这意味着CheY-P的内部浓度不会立即适应环境，因此翻滚速率具有记忆性。这意味着用于在宏观尺度描述趋化作用的Keller-Segel方程，由于其假设细菌通量与化学梯度之间存在局部关系，在未考虑记忆和相关非局部响应的情况下，并不完全有效。为了推导取代Keller-Segel方程的新方程，我们采用了一种动力学方法，其中在考虑蛋白质浓度动态的情况下写出细菌运输的动力学方程。当记忆效应较大时，蛋白质浓度场必须被视为与细菌密度同等重要的相关变量。通过详细运用Chapman-Enskog方法，得到了这些场的动力学方程，其形式为反应扩散方程，通量和源项取决于化学信号的梯度。此外，传输系数完全由微观动力学得出，显示出重要的对称性质，并给出了大肠杆菌情况下的取值。求解非均匀信号的方程表明，响应是非局部的，对于大肠杆菌，平滑长度高达170µm。还详细研究了均匀响应和弛豫动力学。为了完整起见，也研究了记忆效应较小的情况，在这种情况下，Chapman-Enskog方法重现了Keller-Segel方程，并给出了传输系数的明确表达式。
