Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway

TL;DR

This paper derives a bacterial run-and-tumble kinetic equation from a biochemical pathway model, connecting intracellular molecular content to chemotactic behavior and explaining how gradient-based responses emerge from molecular-level dynamics.

Contribution

It provides a rigorous derivation of the classical run-and-tumble equation from a biochemical pathway model, linking intracellular processes to population-level chemotactic models.

Findings

Derived the run-and-tumble kinetic equation from biochemical pathways.

Connected molecular content to tumbling frequency and chemotactic response.

Included effects of receptor methylation noise in the model.

Abstract

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.