Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway
- 291 Downloads
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.
KeywordsKinetic-transport equations Chemotaxis Asymptotic analysis Run and tumble Biochemical pathway
Mathematics Subject Classification35B25 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.
- Endres RG (2013) Physical principles in sensing and signaling, with an introduction to modeling in biology. Oxford University Press, OxfordGoogle Scholar
- Hillen T, Painter K (2012) Transport and anisotropic diffusion models for movement in oriented habitats. In: Lewis MA, Maini P, Petrowskii S (eds) Dispersal, individual movement and spatial ecology: a mathematical perspective. Springer, Heidelberg, pp 177–222Google Scholar
- Saragosti J, Calvez V, Bournaveas N, Perthame B, Buguin A, Silberzan P (2011) Directional persistence of chemotactic bacteria in a traveling concentration wave. Proc Natl Acad Sci 108(39):16235-16240Google Scholar