  We study a simple run-and-tumble random walk whose switching frequency from
run mode to tumble mode and the reverse depend on a stochastic signal. We
consider a particularly sharp, step-like dependence, where the run to tumble
switching probability jumps from zero to one as the signal crosses a particular
value (say y_1 ) from below. Similarly, tumble to run switching probability
also shows a jump like this as the signal crosses another value (y_2 < y_1 )
from above. We are interested in characterizing the effect of signaling noise
on the long time behavior of the random walker. We consider two different
time-evolutions of the stochastic signal. In one case, the signal dynamics is
an independent stochastic process and does not depend on the run-and-tumble
motion. In this case we can analytically calculate the mean value and the
complete distribution function of the run duration and tumble duration. In the
second case, we assume that the signal dynamics is influenced by the spatial
location of the random walker. For this system, we numerically measure the
steady state position distribution of the random walker. We discuss some
similarities and differences between our system and E.coli chemotaxis, which is
another well-known run-and-tumble motion encountered in nature.
