  Signal transduction in biological cells is effected by signaling pathways
that typically include multiple feedback loops. Here we analyze information
transfer through a prototypical signaling module with biochemical feedback. The
module switches stochastically between an inactive and active state; the input
to the module governs the activation rate while the output (i.e., the product
concentration) perturbs the inactivation rate. Using a novel perturbative
approach, we compute the rate with which information about the input is gained
from observation of the output. We obtain an explicit analytical result valid
to first order in feedback strength and to second order in the strength of
input. The total information gained during an extended time interval is found
to depend on the feedback strength only through the total number of
activation/inactivation events.
