Chemical Sensing by Nonequilibrium Cooperative Receptors

Cooperativity arising from local interactions in equilibrium receptor systems provides gain, but does not increase sensory performance, as measured by the signal-to-noise ratio (SNR) due to a fundamental tradeoff between gain and intrinsic noise. Here we allow sensing to be a nonequilibrium process and show that energy dissipation cannot circumvent the fundamental tradeoff, so that the SNR is still optimal for independent receptors. For systems requiring high gain, nonequilibrium 2D-coupled receptors maximize the SNR, revealing a new design principle for biological sensors.

Figure 1

Single receptor. (a) Schematic diagram of interconversion among the four states of a single receptor: inactive and unbound, inactive and bound, active and bound, and active and unbound. (b) In the limit of fast ligand binding and unbinding, the receptor dynamics reduces to switching between two states, active and inactive.

Figure 2

1D- and 2D-coupled receptors. (a) Schematic diagram of 4-state cycle for a chain of 1D-coupled receptors. White-up and black-down arrows denote active and inactive receptors, respectively. Optimal $\mathrm{SNR}/[{\tau }_{\mathrm{avg}}(\Delta f{)}^2]$ as a function of gain $R/N$ for (b) 1D- and (c) 2D-coupled receptors for equilibrium (black) and nonequilibrium (red $N=9$ and blue $N=25$). (d) $\mathrm{SNR}/[{\tau }_{\mathrm{avg}}(\Delta f{)}^2]$ as a function of linear system size $N^{1/2}$ for gain $R/N=5$ for equilibrium (black) and nonequilibrium with ${\gamma }_J=-0.5$ (red). Error bars show standard errors of the mean and curves are exponential fits. The inset shows the residuals from the fits.

Figure 3

SNR, MFPT, and effective potentials at fixed gain. (a) $\mathrm{SNR}/[{\tau }_{\mathrm{avg}}(\Delta f{)}^2]$ versus $1/\mathrm{MFPT}$ for 2D-coupled receptors with $N=25$ for gain $R/N=0.7N$ as ${\gamma }_J$ is increased from zero (starting at left). (b) $\mathrm{SNR}/[{\tau }_{\mathrm{avg}}(\Delta f{)}^2]$ versus normalized gain $R/N^2$ for 1D- and 2D-coupled receptors and their mappings to one-step processes (dashed lines). Inset shows schematic of one-step process with transition probabilities [21]. (c) “Potentials,” $-\log p(a)$, as functions of normalized and symmetrized activity $a=2A/N-1$ for Ising lattices with $N=25$ for gain $R/N=0.7N$ and the optimal one-step process (dashed curve). (d) MFPT as a function of $R/N^2(=\mathrm{gain}/N)$ for Ising lattices with $N=25$. Inset shows the diffusion coefficients $D(a)$ for gain $R/N=0.7N$.