Nonlinearity, Fluctuations, and Response in Sensory Systems

The statistics of fluctuations in biological sensing pathways and its relation to the response to environmental stimuli is investigated. We focus on bacterial chemotaxis, where detailed experiments and reliable models are available. We consider allosteric models of receptors’ activity and derive analytically their steady-state probability distribution and correlation times. By using fluctuation relations, we then relate appropriate steady-state correlations to the response of the system to step and ramp stimuli of arbitrary amplitudes. We show that the combined effect of nonlinearity and fluctuations generically yields a complex nonlinear response at the single sensing unit and at the whole-cell level. Such responses display a nonexponential decay with a broad range of time scales. Slow, ineffective responses are associated to signaling units locked into poorly performing states. However, the nonlinear response reduces to a nearly exponential one for an appropriate range of the kinetic parameters. This provides a systematic explanation for the relation between fluctuation and response observed in recent experiments.

Figure 1

Equilibrium statistics of a receptor cluster. (a) Average activity as a function of the ramp rate $\lambda$ (FRET data from 14). The solid line is Eq. (3) with $k_r=0.06{\mathrm{s}}^{-1}$, $k_b=0.12{\mathrm{s}}^{-1}$. (b) Standard deviation $⟨⟨a^2⟩{⟩}_{\lambda }^{1/2}$, where we used the equality $I(2,0)/I(1,0)=[(q+1)/(q+s+1)](k_b/k_r{)}^{q+1}{}_{2}F_1(q+2,q+s+1;q+s+2;1-k_b/k_r)$ and Eq. (3). (c) Integral correlation time ${\tau }_c^{(\lambda )}$ from Eq. (4) vs the ramp rate $\lambda$. (d) Normalized correlation functions computed numerically for $\lambda =0$ and $\lambda =\alpha k_r/(2N)$ (lower and upper curve) compared with the exponential approximations $\exp (-t/{\tau }_c^{(\lambda )})$. In all panels and subsequent figures, the number of receptor dimers in the allosteric pool $N=6$ and the free-energy parameter $\alpha =1$.

Figure 2

Response to a ramp. (a) Integral response time ${\tau }_r^{(\lambda )}$ (red curve) and differential response time ${\tilde{\tau }}_r^{(\lambda )}$ (green horizontal line) as a function of the ramp rate $\lambda$. The mean-field approximations ${\tau }_{r,\mathrm{MF}}^{(\lambda )}$ (purple curve) and ${\tilde{\tau }}_{r,\mathrm{MF}}^{(\lambda )}$ (blue horizontal line) lie well below the exact values. (b) Response times multiplied by the ramp rate as functions of the mean-field free energy change $\Delta f_{\mathrm{MF}}=\log (s/q)$. Colors are as in panel (a). The experimental points shown as circles are FRET data with the response time estimated by an exponential fit [14].

Figure 3

Average activity vs time for a ramp protocol. (a) $\lambda =-0.005{\mathrm{s}}^{-1}$: deviations from an exponential behavior are confined to large delays. This value of $\lambda$ corresponds to the minimum difference between integral and differential times [see Fig. 2a]. (b) $\lambda =0.005{\mathrm{s}}^{-1}$: deviations from an exponential behavior (fits with the differential and the integral time are shown in green and red as well as the least-square fit in thin black line) are substantial and the wide spectrum of time scales generated by nonlinearities is manifest.

Figure 4

Response vs correlation times. The filled circle and square mark the average activity $⟨a{⟩}_0=1/3$ and $⟨a{⟩}_0=1/2$. Solid line is the analytical result from Eqs. (4, 9). Dashed line is the curve ${\tau }_r={\tau }_c$ that would hold for a linear, Gaussian process (Ornstein-Uhlenbeck). Circles are obtained from data on single-motor responses to a small step of attractant (10 nM of aspartate) [9]. Inset: An enlarged version of the main figure that shows the departure from near linearity for smaller values of the average activity (larger values of $k_b$).