Fundamental Limits on Sensing Chemical Concentrations with Linear Biochemical Networks

Living cells often need to extract information from biochemical signals that are noisy. We study how accurately cells can measure chemical concentrations with signaling networks that are linear. For stationary signals of long duration, they can reach, but not beat, the Berg-Purcell limit, which relies on uniformly averaging in time the fluctuations in the input signal. For short times or nonstationary signals, however, they can beat the Berg-Purcell limit, by nonuniformly time averaging the input. We derive the optimal weighting function for time averaging and use it to provide the fundamental limit of measuring chemical concentrations with linear signaling networks.

Figure 1

Responding to noisy environments. (a) The environment changes instantaneously at time $t=0$, and the number of bound receptors, $S(t)$, adjusts instantaneously. $S(t)$ is stationary between time 0 and the time $T_o$, when the cell responds. The signaling network is either in a steady state by time $T_o$, independent of the initial condition (b), or in a basal state at time $t=0$ (c). $X(t)$ denotes the number of $X$ molecules at time $t$. The solid and dashed lines in panel (b) represent different initial conditions.

Figure 2

Extracting information from noisy input signals with linear signaling networks. (a),(c),(e),(g) The weighting functions corresponding to different signaling networks are not uniform. (b),(d),(f),(h) The ability of a signaling network to measure ligand concentration depends on its weighting function. The typical error (variance) in the estimate of ligand concentration is plotted as a percentage increase over the error of an estimate based on uniform weighting, assumed in the Berg-Purcell limit [Eq. (1) with $T=T_o$]. (a) Reversible, one-level cascades selectively amplify late ($t=T_o$) values of the signal, (b) leading to worse performance than the uniform average. (c) Irreversible, $N$-level cascades amplify early ($t=0$) values of the signal, (d) leading to worse performance than the uniform average. (e) The optimal weighting function, given by Eq. (4), averages the signal, selectively amplifying less correlated values. The delta functions are truncated for illustration. (f) The optimal weighting function outperforms the uniform average. (g) A signaling network consisting of two branches, which selectively amplify late ($t=T_o$) (left branch) and early ($t=0$) (right branch) values of the signal, approximates the optimal weighting function ($k_1=4.4k_3{k}_4{T}_o$; $k_2=20/T_o$; $k_4=0.35k_3$; $\overline{f}$ independent of $k_3$,$k_5$; $k_6\gg T_o^{-1}$). (h) The network in (f) can outperform the uniform average.