  The ability to monitor nutrient and other environmental conditions with high
sensitivity is crucial for cell growth and survival. Sensory adaptation allows
a cell to recover its sensitivity after a transient response to a shift in the
strength of extracellular stimulus. The working principles of adaptation have
been established previously based on rate equations which do not consider
fluctuations in a thermal environment. Recently, G. Lan et al. (Nature Phys.,
8:422-8, 2012) performed a detailed analysis of a stochastic model for the E.
coli sensory network. They showed that accurate adaptation is possible only
when the system operates in a nonequilibrium steady-state (NESS). They further
proposed an energy-speed-accuracy (ESA) trade-off relation. We present here
analytic results on the NESS of the model through a mapping to a
one-dimensional birth-death process. An exact expression for the entropy
production rate is also derived. Based on these results, we are able to discuss
the ESA relation in a more general setting. Our study suggests that the
adaptation error can be reduced exponentially as the methylation range
increases. Finally, we show that a nonequilibrium phase transition exists in
the infinite methylation range limit, despite the fact that the model contains
only two discrete variables.
