Multicellular sensing at a feedback-induced critical point

Feedback in sensory biochemical networks can give rise to bifurcations in cells' behavioral response. These bifurcations share many properties with thermodynamic critical points. Evidence suggests that biological systems may operate near these critical points, but the functional benefit of doing so remains poorly understood. Here we investigate a simple biochemical model with nonlinear feedback and multicellular communication to determine if criticality provides a functional benefit in terms of the ability to gain information about a stochastic chemical signal. We find that when signal fluctuations are slow, the mutual information between the signal and the intracellular readout is maximized at criticality, because the benefit of high signal susceptibility outweighs the detriment of high readout noise. When cells communicate, criticality gives rise to long-range correlations in readout molecule number among cells. Consequently, we find that communication increases the mutual information between a given cell's readout and the spatial average of the signal across the population. Finally, we find that both with and without communication, the sensory benefits of criticality compete with critical slowing down, such that the information rate, as opposed to the information itself, is minimized at the critical point. Our results reveal the costs and benefits of feedback-induced criticality for multicellular sensing.

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Figure 1

Illustration of the model. Each site on a 1D lattice is a well-mixed cell and its immediate environment. A ligand binds to a receptor and produces a readout molecule that is subject to nonlinear feedback and may be exchanged between cells. The ligand is spatially uniform and has its own fluctuating dynamics.

Figure 2

Sensory information for a single cell. (a) Signal-to-noise ratio for the readout in the slow ligand regime with $k_{\ell }^{-}/k_1^{-}=5\times {10}^{-3}$. (b) Mutual information between ligand and readout. Blue (yellow): Linear noise approximation Eq. (8) [Eq. (B10)] with $k_{\ell }^{-}/k_1^{-}=0$ (0.1). Red (purple): Gillespie simulations with $k_{\ell }^{-}/k_1^{-}=5\times {10}^{-3}$ (0.1). (c) Information rate between ligand and readout using linear noise approximation with $k_{\ell }^{-}/k_1^{-}=5\times {10}^{-3}$. All curves have $h=0,x_c={10}^3$, and $\overline{\ell }=500$.

Figure 3

Long-range correlations in the multicellular system. (a) Correlation length $\xi$, numerically integrated from simulations using the trapezoidal rule, scales with reduced temperature $\theta$ with mean-field exponent $\nu =1/2$. Rolloff at $\theta =0$ is due to finite-size effects. Here $k_{\ell }^{-}/k_1^{-}={\gamma }^{\prime }/k_1^{-}=5\times {10}^{-3},\gamma /k_1^{-}=1,h_i=0,\overline{\ell }=150,x_c=300$, and $N=20$ cells. (b) Critical correlation length increases with communication strength $\gamma$, approaching system size $N/2$. Parameters as in (a) with $\theta =0$.

Figure 4

Mutual information between a single cell's readout and the spatial average of the ligand fluctuations. This was obtained from Gillespie simulations with $N=10,h_i=0,x_c=300,\overline{\ell }=150$, and $k_{\ell }^{-}/k_1^{-}={\gamma }^{\prime }/k_{\ell }^{-}=5\times {10}^{-3}$. This is maximized near criticality as the communication strength increases.

Figure 5

Information rate between a single cell's readout and the spatial average of the ligand fluctuations. This was obtained from the cross-spectrum under the linear noise approximation and numerically integrating. This plot has $N=10,h_i=0,x_c=300,\overline{\ell }=150$, and $k_{\ell }^{-}/k_1^{-}={\gamma }^{\prime }/k_1^{-}=1$. The rate is maximized if we increase the effective temperature and communication strength.