Journal of Mathematical Biology

, Volume 73, Issue 5, pp 1161–1178 | Cite as

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



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.


Kinetic-transport equations Chemotaxis Asymptotic analysis Run and tumble Biochemical pathway 

Mathematics Subject Classification

35B25 82C40 92C17 



Benoît Perthame and Nicolas Vauchelet are partially supported by ANR Kibord-ANR-13-BS01-0004 funded by the French Ministry of Research. Min Tang is partially supported by NSF of Shanghai under Grant 12ZR1445400, NSFC 11301336 and 91330203, and Shanghai Pujiang Program 13PJ140700.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis Lions UMR CNRS 7598 and INRIA ParisSorbonne Université, UPMC Univ Paris 06ParisFrance
  2. 2.Department of Mathematics, Institute of Natural SciencesShanghai Jiao Tong UniversityShanghaiChina

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