The Berg-Purcell limit revisited

Abstract

Biological systems often have to measure extremely low concentrations of chemicals with high precision. When dealing with such small numbers of molecules, the inevitable randomness of physical transport processes and binding reactions will limit the precision with which measurements can be made. An important question is what the lower bound on the noise would be in such measurements. Using the theory of diffusion-influenced reactions, we derive an analytical expression for the precision of concentration estimates that are obtained by monitoring the state of a receptor to which a diffusing ligand can bind. The variance in the estimate consists of two terms, one resulting from the intrinsic binding kinetics and the other from the diffusive arrival of ligand at the receptor. The latter term is identical to the fundamental limit derived by Berg and Purcell (Biophys. J., 1977), but disagrees with a more recent expression by Bialek and Setayeshgar. Comparing the theoretical predictions against results from particle-based simulations confirms the accuracy of the resulting expression and reaffirms the fundamental limit established by Berg and Purcell.

Figure 1

The power spectrum of the receptor state P_n(ω) for c = 0.4 μM. The simulation results (black circles) agree well with the theoretical prediction of Eq. 13 (solid red line). At high frequencies ω > 1/τ_m = D/σ², the effect of diffusion is negligible and the receptor dynamics is that of a Markovian switching process with intrinsic rates k_ac and k_d (dashed red line), while at low frequencies it is that of a Markovian switching process with effective rates k_onc and k_off, respectively (solid gray line). The zero-frequency limit determines the precision of the concentration estimate. Parameters: $\overline{n}=0.5$, D = 1 μm² s⁻¹, σ = 10 nm, L = 1 μm, and k_a = 552 μM⁻¹ s⁻¹. To see this figure in color, go online.

Figure 2

The zero-frequency limit of the power spectrum as a function of the average receptor occupancy $\overline{n}$ for c = 0.4 μM; $\overline{n}$ is varied by changing k_d. It is seen that the agreement between the theoretical prediction of Eq. 13 and the simulation results is very good (red line). In contrast, the prediction of Bialek and Setayeshgar (9) (black line) differs markedly from our results. Parameters: see Fig. 1. To see this figure in color, go online.

Figure 3

Cartoon of the coarse-grained model. (a) A typical time trace of the receptor state n(t) of the original system. (b) Time trace of the coarse-grained model. (Top-left cartoon; red) a successful and an unsuccessful binding trajectory; (blue) a trajectory in which a ligand molecule undergoes a number of rounds of receptor dissociation and rebinding before it escapes into the bulk. The key observation is that the time a molecule spends near the receptor is very short on the timescale at which molecules arrive from the bulk. This makes it possible to integrate out the receptor rebindings and the unsuccessful arrivals of molecules from the bulk, giving the two-state model of Eq. 19. Fig. S1 in the Supporting Material quantifies the timescale separation. To see this figure in color, go online.