  Cells adapt to changing environments by sensing ligand concentrations using
specific receptors. The accuracy of sensing is ultimately limited by the finite
number of ligand molecules bound by receptors. Previously derived physical
limits to sensing accuracy have assumed that the concentration was constant and
ignored its temporal fluctuations. We formulate the problem of concentration
sensing in a strongly fluctuating environment as a non-linear field-theoretic
problem, for which we find an excellent approximate Gaussian solution. We
derive a new physical bound on the relative error in concentration $c$ which
scales as $\delta c/c \sim (Dac\tau)^{-1/4}$ with ligand diffusivity $D$,
receptor cross-section $a$, and characteristic fluctuation time scale $\tau$,
in stark contrast with the usual Berg and Purcell bound $\delta c/c \sim
(DacT)^{-1/2}$ for a perfect receptor sensing concentration during time $T$. We
show how the bound can be achieved by a simple biochemical network downstream
the receptor that adapts the kinetics of signaling as a function of the square
root of the sensed concentration.
