Physical Limit to Concentration Sensing in a Changing Environment

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 largely 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 nonlinear 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 similar to (Dac tau)(-1/4) with ligand diffusivity D, receptor cross section a, and characteristic fluctuation timescale tau, in stark contrast to the usual Berg and Purcell bound delta c/c similar to (DacT)(-1/2) for a perfect receptor sensing concentration during time T. We show how the bound can be achieved by a biochemical network downstream of the receptor that adapts the kinetics of signaling as a function of the square root of the sensed concentration.