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 $\delta c/c\sim (Dac\tau {)}^{-1/4}$ with ligand diffusivity $D$, receptor cross section $a$, and characteristic fluctuation timescale $\tau$, in stark contrast to the usual Berg and Purcell bound $\delta c/c\sim (DacT{)}^{-1/2}$ 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.

Figure 1

Numerical validations of analytical results. (a) The Gaussian ansatz (7)–(8) is validated by simulating the general equations for Bayesian filtering (5)–(6). The numerical solution approaches the Gaussian solution rapidly, as indicated by the decay of the Kullback-Leibler divergence $D_{\mathrm{KL}}\mathbf{(}P(\phi )\parallel P_{\text{Gaussian}}(\phi )\mathbf{)}=\int d\phi P(\phi )\ln [P(\phi )/P_{\text{Gaussian}}(\phi ))]$. We used $r\tau =Dac\tau =50$. (b) Concentration sensing error as a function of concentration. The error estimated from simulations follows closely the prediction from Eq. (13), which is expected to be valid for $4Dac\tau \gg 1$.

Figure 2

Performance of adaptive biochemical network in fluctuating ligand concentration. (a) Schematic of the biochemical network implementing optimal Bayesian filtering. The receptor-induced activation of the readout molecule $A^{*}$, as well as its deactivation are regulated by a second molecule $B^{*}$, which is made to scale like $\sqrt{A^{*}}$ using a mechanism of deactivation by dimerization (shaded box). (b) Simulation of the network readout $c_A(t)\propto A^{*}(t)$ in response to stochastic binding events in a fluctuating concentration field $c^{*}(t)$. The relative estimation error $⟨({\widehat{c}}_A-c^{*}{)}^2⟩/c^2$ behaves according to the theoretical bound $1/\sqrt{4Dac\tau }$ (inset).