  Kinetic-transport equations that take into account the intra-cellular
pathways are now considered as the correct description of bacterial chemotaxis
by run and tumble. Recent mathematical studies have shown their interest and
their relations to more standard models. Macroscopic equations of Keller-Segel
type have been derived using parabolic scaling. Due to the randomness of
receptor methylation or intra-cellular chemical reactions, noise occurs in the
signaling pathways and affects the tumbling rate. Then, comes the question to
understand the role of an internal noise on the behavior of the full
population. In this paper we consider a kinetic model for chemotaxis which
includes biochemical pathway with noises. We show that under proper scaling and
conditions on the tumbling frequency as well as the form of noise, fractional
diffusion can arise in the macroscopic limits of the kinetic equation. This
gives a new mathematical theory about how long jumps can be due to the internal
noise of the bacteria.
