How input noise limits biochemical sensing in ultrasensitive systems

Many biological processes are regulated by molecular devices that respond in an ultrasensitive fashion to upstream signals. An important question is whether such ultrasensitivity improves or limits its ability to read out the (noisy) input stimuli. Here, we develop a simple model to study the statistical properties of ultrasensitive signaling systems. We demonstrate that the output sensory noise is always bounded, in contrast to earlier theories using the small noise approximation, which tends to overestimate the impact of noise in ultrasensitive pathways. Our analysis also shows that the apparent sensitivity of the system is ultimately constrained by the input signal-to-noise ratio. Thus, ultrasensitivity can improve the precision of biochemical sensing only to a finite extent. This corresponds to a new limit for ultrasensitive signaling systems, which is strictly tighter than the Berg-Purcell limit.

Figure 1

(a) The input-output relationship $\langle Y_i\rangle$ versus $\langle X\rangle =c$. The output $\langle Y_i\rangle$ obtained from simulations (symbols) can be approximated by the Hill function $\tilde{f}\left(c\right)$ with the apparent Hill coefficient $\tilde{h}$ depending on the Fano factor ${\sigma }_x^2/c$. (b) The distribution of $Z_T$ in a simulation with ${\tau }_y=1$, ${\tau }_x=10$, and $T=100$. The solid red line is the fitting of a Gaussian distribution (with standard deviation 0.11).

Figure 2

(a) The correlation functions ${\mathcal{A}}_y$, ${\mathcal{C}}_y$, and ${\mathcal{A}}_x$ versus the time lag $\Delta t$. Parameters used in this simulation: $h=5$, $c=K_d$, ${\sigma }_x/c=0.2$, ${\tau }_y=1$, and ${\tau }_x=10$. (b) The correlation coefficient $\rho$ versus the Hill coefficient $h$ for different levels of input noise ${\sigma }_x/c$. (c) ${\mathcal{A}}_y<{\mathcal{A}}_x$, regardless of $h$. (d) ${\mathcal{A}}_y<{\mathcal{A}}_x$, regardless of ${\sigma }_x/c$.

Figure 3

(a) ${\sigma }_{z,T}^2$ versus ${\sigma }_x/c$ for a single sensor with $h=5$ and $h=10$. In these simulations, we used ${\tau }_y=1$, ${\tau }_x=10$, and $T=100$. By Eq. (12), the upper bound of ${\sigma }_{z,T}^2$ for $N=1$ is ${\sigma }_y^2{\xi }_x^T\approx 0.05$ at $c=K_d$. (b) ${\sigma }_{z,T}^2$ versus ${\sigma }_x/c$ for two sensors ($N=2$) with $h=10$ at $c=K_d$. Here, the upper and lower bounds of ${\sigma }_{z,T}^2$ are ${\sigma }_y^2{\xi }_x^T(1+\rho )/2$ and ${\sigma }_y^2{\xi }_x^T\rho$, respectively. Both bounds (dotted lines) depend on the correlation coefficient $\rho$ which is shown in Fig. 2.

Figure 4

(a) The input-output relationship, $\langle Y_i\rangle =\Gamma (\beta c,\beta K_d)$, can be approximated by the Hill equation, $c^{{\tilde{h}}_{\infty }}/(c^{{\tilde{h}}_{\infty }}+K_d^{{\tilde{h}}_{\infty }})$, with ${\tilde{h}}_{\infty }$ given by Eq. (19). (b) The limit Hill coefficient, ${\tilde{h}}_{\infty }$, as a function of the signal-to-noise ratio ${\alpha }^{*}=\beta c^{*}={(c^{*}/{\sigma }_x)}^2$ at $c^{*}=K_d+1/\beta$ for fixed $\beta =1$.

Figure 5

(a) $\tilde{h}$ versus $h$ for two different values of $\beta$: $\beta =1$ (solid red line) and $\beta =1/3$ (dashed blue line). (b) $\delta c/c$ versus $h$ at $c=K_d$ for $\beta =1$ (solid red line) and $\beta =1/3$ (dashed blue line). The dotted lines correspond to the limit values of $\tilde{h}$ in (a) and $\delta c/c$ in (b) under the limit $h\to \infty$. We have used $K_d=100$, ${\tau }_y=1$, ${\tau }_x=10$, and $T=100$ in numerical simulations.

Figure 6

A numerical comparison between the lower bound (solid blue line) in Eq. (22) and the lower bound (dashed red line) in Eq. (26). The dotted green line corresponds to the upper bound of $\delta c/c$ in Eq. (21) for a single sensor ($N=1$) with $\tilde{h}=10$. The black symbols represent the Monte Carlo simulation results of $\delta c/c$ by a sensor with the apparent sensitivity $\tilde{h}\approx 10$ under the Poisson noise (i.e., $\beta =1$) at different mean concentration levels. The sensing error $\delta c/c$ was calculated using the error propagation formula in Eq. (21). These black symbols fall between the bounds as indicated by Eq. (27). Other parameters used here include $K_d=100$, ${\tau }_y=1$, ${\tau }_x=10$, and $T=100$.