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In this paper we use an individual-based model and its associated kineticequation to study the generation of long jumps in the motion of E. coli. Thesemodels relate the run-and-tumble process to the intracellular reaction wherethe intrinsic noise plays a central role. Compared with the previous work in[13] in which the parametric assumptions are mainly for mathematicalconvenience and not well-suited for either numerical simulation or comparisonwith experimental results, our current paper make use of biologicallymeaningful pathways and tumbling kernels. Moreover, using the individual-basedmodel we can now perform numerical simulations. Power-law decay of the runlength, which corresponds to Levy-type motions, are observed in our numericalresults. The particular decay rate agrees quantitatively with the analyticalresult. We also rigorously recover the fractional diffusion equation as thelimit of the kinetic model.

Mathematical Modelling and Analysis of Fractional Diffusion Induced by Intracellular Noise