  Run-and-tumble chemotaxis is one of the representative search strategies of
an odor source via sensing its spatial gradient. The optimal ways of sensing
and control in the run-and-tumble chemotaxis have been analyzed theoretically
to elucidate the efficiency of strategies implemented in organisms. However,
because of theoretical difficulties, most of attempts have been limited only to
either linear or deterministic analysis even though real biological chemotactic
systems involve considerable stochasticity and nonlinearity in their sensory
processes and controlled responses. In this paper, by combining the theories of
optimal filtering and Kullback-Leibler control of partially observed Markov
decision process (POMDP), we derive the optimal and fully nonlinear strategy
for controlling run-and-tumble motion depending on noisy sensing of ligand
gradient. The derived optimal strategy consists of the optimal filtering
dynamics to estimate the run-direction from noisy sensory input and the control
function to regulate the motor output. We further show that this optimal
strategy can be associated naturally with a standard biochemical model and
experimental data of the Escherichia coli's chemotaxis. These results
demonstrate that our theoretical framework can work as a basis for analyzing
the efficiency and optimality of run-and-tumble chemotaxis.
