Limits to the precision of gradient sensing with spatial communication and temporal integration

Abstract

Gradient sensing requires at least two measurements at different points in space. These measurements must then be communicated to a common location to be compared, which is unavoidably noisy. Although much is known about the limits of measurement precision by cells, the limits placed by the communication are not understood. Motivated by recent experiments, we derive the fundamental limits to the precision of gradient sensing in a multicellular system, accounting for communication and temporal integration. The gradient is estimated by comparing a "local" and a "global" molecular reporter of the external concentration, where the global reporter is exchanged between neighboring cells. Using the fluctuation-dissipation framework, we find, in contrast to the case when communication is ignored, that precision saturates with the number of cells independently of the measurement time duration, because communication establishes a maximum length scale over which sensory information can be reliably conveyed. Surprisingly, we also find that precision is improved if the local reporter is exchanged between cells as well, albeit more slowly than the global reporter. The reason is that whereas exchange of the local reporter weakens the comparison, it decreases the measurement noise. We term such a model "regional excitation-global inhibition." Our results demonstrate that fundamental sensing limits are necessarily sharpened when the need to communicate information is taken into account.

Keywords: cell–cell communication; fluctuation–dissipation theorem; gradient sensing; linear response theory.

Fig. 1.

Spatially extended gradient sensing. (A) A chain of N compartments or cells is exposed to a linear profile of a diffusible chemical. (B) In an idealized detector, the two edge cells communicate their measurements perfectly and instantly. (C) In our model, bound receptors (R) activate local (X) and global reporter molecules (Y), and Y is exchanged between cells for the communication. Kinetic rates associated with various processes are indicated by Greek letters.

Fig. 2.

Precision of gradient sensing with temporal integration. Signal-to-noise ratio (SNR) vs. number of cells N is shown for the idealized detector (Eq. 2 with prefactor $1/\pi$) and for our model with communication (Eqs. 5, 6, 9, and 10). Whereas the SNR for the idealized detector increases indefinitely, the SNR for the model with communication saturates for $N\gg n_0$. The saturation level is bounded from above by the fundamental limit, Eq. 11. As shown, the bound is reached in the high-gain regime $\alpha /a^3\mu =\beta /\nu =100$, where intrinsic noise is negligible. Other parameters are $a=10$ μm, ${\overline{c}}_N=1$ nM, $g=1$ nM/mm, $D=50$ μm²/s, $\mu =\nu =1$ s⁻¹, and $n_0=\sqrt{\gamma /\nu }=10$, and the integration timescale is $T=10$ s.

Fig. 3.

Regional excitation–global inhibition (REGI). Signal-to-noise ratio (SNR) is enhanced by allowing both messengers to be exchanged between cells. The optimal enhancement over LEGI is substantial and occurs because exchange of the local species reduces measurement noise, but also reduces the signal. Parameters are as in Fig. 2, but with $N=100$, $\alpha /a^3\mu =\beta /\nu =5$, and several values of $n_y\equiv \sqrt{{\gamma }_y/\nu }$ as indicated.

Fig. S1.

Dependence of the optimal exchange rate ratio on system parameters in the regional excitation–global inhibition (REGI) strategy. The signal-to-noise ratio (SNR) has a maximum at a particular rate ratio ${\gamma }_x^{*}/{\gamma }_y$, which increases as a function of either gain factor, (A) $\alpha /a^3\mu$ or (B) $\beta /\nu$. Parameters are $a=10$ μm, ${\overline{c}}_N=1$ nM, $g=1$ nM/mm, $D=50$ μm²/s, $T=10$ s, $\mu =\nu =1$ s⁻¹, $N=100$, and $n_y=\sqrt{\gamma /\nu }=10$. In A, $\beta /\nu =5$, and $\alpha /a^3\mu$ is varied as indicated. In B, $\alpha /a^3\mu =5$, and $\beta /\nu$ is varied as indicated.