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Figure 1

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The cell signaling network.

(a) The time-varying ligand concentration is modeled as a memoryless (Markovian) signal with mean $\bar{L}$ , variance $σ_{L}^{2}$ , and correlation time $τ_{L} = λ^{- 1}$ . A free ligand molecule L (light blue circle) can bind at rate k₁ to a free receptor R (magenta protein) on the cell membrane (black line), forming the complex RL, and unbind at rate k₂ from RL. The correlation time of the receptor state is $τ_{c}$ . The complex RL catalyzes the phosphorylation reaction, driven by adenosine triphosphate (ATP) conversion, of a downstream readout from the unphosphorylated (inactive) state x to the phosphorylated (active) state $x^{*}$ , with rate $k_{f}$ . The phosphorylated readout then spontaneously decays to the x state with rate $k_{r}$ . Microscopic reverse reactions of each signaling pathway are represented by dashed arrows. The relaxation time of the push–pull network is $τ_{r}$ . (b) Free-energy landscape of a readout molecule across the activation/deactivation reactions. Fuel turnover, provided by ATP conversion, drives the activation (phosphorylation) reaction characterized by the forward rate $k_{f}$ and its microscopic reverse rate $k_{- f}$ (green arrows). Associated with this activation reaction is a free-energy drop $Δ μ_{1} = \log \frac{k_{f} \bar{x}}{k_{- f} {\bar{x}}^{*}}$ . The deactivation pathway corresponds to the spontaneous release of the inorganic phosphate; it is characterized by the rate $k_{r}$ and its microscopic reverse $k_{- r}$ (blue arrows) and corresponds to a free-energy drop $Δ μ_{2} = \log \frac{k_{r} {\bar{x}}^{*}}{k_{- r} \bar{x}}$ .

Figure 2

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The precision of estimating a time-varying ligand concentration L.

(a) The cell estimates the current ligand concentration $L = L (t)$ by estimating the average receptor occupancy $p_{τ_{r}}$ over the past integration time $τ_{r}$ and by inverting the dynamic input–output relation $p_{τ_{r}} (L)$ (black solid line). The latter describes the mapping between the current concentration $L (t)$ of the time-varying signal and the average receptor occupancy $p_{τ_{r}}$ over the past $τ_{r}$ , see also (b); it depends on the timescale $τ_{L}$ of the input signal and hence differs from the conventional static input–output relation $p (L_{s})$ , which describes the mapping between the average receptor occupancy and a static ligand concentration $L_{s}$ that does not vary in time (gray solid line). The squared error in the estimate of the concentration ${(δ \hat{L})}^{2} = σ_{{\hat{p}}_{τ_{r}}}^{2} / {\tilde{g}}_{L \to p_{τ_{r}}}^{2}$ depends on the variance $σ_{{\hat{p}}_{τ_{r}}}^{2}$ in the estimate of the average receptor occupancy ${\hat{p}}_{τ_{r}}$ and the dynamic gain ${\tilde{g}}_{L \to p_{τ_{r}}}$ , which is the slope of $p_{τ_{r}} (L)$ . Only in the limit $τ_{c}, τ_{r} ≪ τ_{L}$ , does $p_{τ_{r}} (L)$ reduce to (the linearized form of) $p (L_{s})$ and does the dynamic gain ${\tilde{g}}_{L \to p_{τ_{r}}}$ become the static gain $g_{L \to p}$ , which is the slope of $p (L_{s})$ at the average ligand concentration $\bar{L}$ . The input distribution, shown in blue, has width $σ_{L}$ . (b) The average receptor occupancy $p_{τ_{r}}$ over the past integration time $τ_{r}$ is estimated via the downstream network, which is modeled as a device that discretely samples the ligand-binding state of the receptor via its readout molecules x (Govern and Ten Wolde, 2014a); the fraction of modified readout molecules provides an estimate of $p_{τ_{r}}$ . The sensing error has two contributions (Equation 6): sampling error and dynamical error. The sampling error arises from the error in the estimate of $p_{τ_{r}}$ that is due to the stochasticity of the sampling process; it depends on the number of samples, their independence, and their accuracy. (c) The dynamical error arises because the current ligand concentration $L (t)$ is estimated via the average receptor occupancy $p_{τ_{r}}$ over the past integration time $τ_{r}$ : the latter depends on the ligand concentration in the past $τ_{r}$ , which will, in general, deviate from the current concentration. Two different input trajectories (L₁ in blue, L₂ in green) ending at time t at the same value $L (t)$ (red dot) lead to different estimates of $L (t)$ due to their different average receptor occupancy ( $p_{τ_{r}, 1} > p_{τ_{r}, 2}$ ) in the past $τ_{r}$ .

Figure 3

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Receptors $R_{T}$ , readout molecules $X_{T}$ , and $\dot{w}$ fundamentally limit sensing, and there exists an optimal integration time $τ_{r}$ that depends on which of the resources is limiting.

(a, b) $R_{T}$ , $X_{T}$ , and $\dot{w}$ are fundamental resources, with no trade-offs between them. Plotted is the maximum mutual information $I_{\max} = 1 / 2 \ln (1 + {SNR}_{\max})$ , obtained by minimizing Equation 6 over p and $τ_{r}$ , for different combinations of (a) $X_{T}$ and $R_{T}$ in the irreversible limit $q \to 1$ and (b) $\dot{w}$ and $R_{T}$ for two different values of $Δ μ$ . The sensing precision is bounded by the limiting resource, $R_{T}$ (solid gray lines, Equation 8 with $h = R_{T} / τ_{r} / τ_{c}$ ), $X_{T}$ (dashed gray line, Equation 8 with $h = X_{T}$ , panel a), or $\dot{w}$ (dashed gray lines, Equation 8 with $h = β \dot{w} τ_{r}$ or $h = \dot{w} τ_{r} / (Δ μ / 4)$ , panel b). (c) $I_{\max}$ as a function of $τ_{r}$ for different values of $R_{T}$ in the Berg–Purcell limit ( $q \to 1$ and $X_{T} \to \infty$ ). There exists an optimal integration time $τ_{r}^{opt}$ that maximizes the sensing precision; $τ_{r}^{opt}$ decreases with $R_{T}$ . (d) In this limit, $τ_{r}^{opt}$ depends non-monotonically on the receptor–ligand correlation time $τ_{c}$ : it first increases with $τ_{c}$ to sustain time-averaging, but then drops when $τ_{r}^{opt} / τ_{c}$ becomes of order unity and time-averaging is no longer effective (see inset). (e) $τ_{r}^{opt}$ as a function of $X_{T}$ for different values of $R_{T}$ . When $X_{T} < R_{T}$ , time averaging is not possible and the optimal system is an instantaneous responder, $τ_{r}^{opt} \to 0$ ; when $X_{T} ≫ R_{T}$ , the system reaches the Berg–Purcell regime in which $I_{\max}$ is limited by $R_{T}$ rather than $X_{T}$ (see panel a). (f) $τ_{r}^{opt}$ and $X_{T}$ as a function of $\dot{w}$ . When the power $\dot{w} \sim X_{T} / τ_{r}$ is limiting, the sampling error dominates and $τ_{r}^{opt}$ equals $τ_{L}$ to maximize $X_{T}$ , minimizing the sampling error; $τ_{r}^{opt}$ then decreases to trade part of the decrease in the sampling error for a reduction in the dynamical error such that both decrease; when the sampling interval $Δ \sim τ_{r} R_{T} / X_{T}$ becomes comparable to $τ_{c}$ , in the region marked by the yellow bar, the sampling error is no longer limited by $X_{T}$ , such that $τ_{r}$ now limits both sources of error; the two sources can therefore no longer be decreased simultaneously by increasing $\dot{w} \sim X_{T} / τ_{r}$ ; the system has entered the Berg–Purcell regime, where $τ_{r}^{opt}$ is determined by $R_{T}$ rather than $\dot{w}$ (see panel b). Parameter values unless specified: $τ_{c} / τ_{L} = 10^{- 2}; σ_{L} / {\bar{L}}_{T} = 10^{- 2}$ .

Figure 4

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The optimal integration time for the chemotaxis system of E. coli.

(a) The optimal integration time $τ_{r}^{opt}$ , obtained by numerically optimizing Equation 8 with $h = R_{T} τ_{r} / τ_{c}$ , as a function of the relative strength of the input noise, $σ_{L} / \bar{L}$ , for two different copy numbers $R_{T}$ of the receptor–CheA complexes; for an exponential gradient with length scale x₀, the relative noise strength $σ_{L} / \bar{L} ≃ l / x_{0}$ , where $l \approx 50 μ m$ is the run length of E. coli. It is seen that $τ_{r}^{opt}$ increases as $σ_{L} / \bar{L}$ decreases because the relative importance of the sampling error compared to the dynamical error increases. The figure also shows that $τ_{r}^{opt}$ decreases as $R_{T}$ is increased because that allows for more instantaneous measurements (see also Figure 3). The red bar indicates the range of the estimated integration time of E. coli, $50 m s < τ_{r} < 500 m s$ , based on its attractant and repellent response, respectively (Sourjik and Berg, 2002), divided by the input timescale $τ_{L} \approx 1 s$ based on its typical run time of about a second (Berg and Brown, 1972; Taute et al., 2015). The panel indicates that E. coli has been optimized to detect shallow concentration gradients. (b) The signal-to-noise ratio ${SNR}_{τ_{L}} = {(σ_{L} / δ \hat{L})}^{2} τ_{L} / τ_{r}$ , with ${(σ_{L} / δ \hat{L})}^{2} = SNR$ given by Equation 6, as a function of $σ_{L} / \bar{L} ≃ l / x_{0}$ . To be able to detect the gradient, the ${SNR}_{τ_{L}}$ must exceed unity. The panel shows that the shallowest gradient that E. coli can detect (marked with dashed red line) has, for $R_{T} = 10^{4}$ , a length scale of $x_{0} \approx 25000 μ m$ (corresponding to $σ_{L} / \bar{L} \approx 2 \times 10^{- 3}$ ), which is consistent with experiments based on ramp responses (Shimizu et al., 2010). Other parameter: receptor–ligand-binding correlation time $τ_{c} = 10 m s$ (Vaknin and Berg, 2007; Danielson et al., 1994).

Figure 5

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The benefit of a sensing system depends on the sensing precision it can achieve and the cost of making it.

The Pareto front characterizes the trade-off between the maximal sensing precision, quantified by the maximal mutual information $I_{\max} (x^{*}; L)$ , and the cost of making the sensing system, $C = R_{T} + c_{X} X_{T}$ , where $c_{X}$ is the relative cost of making a readout versus a receptor protein, here taken to be $c_{X} = 1$ . System designs below the Pareto front are suboptimal and can be improved by reducing the cost, that is, the number of proteins, and / or increasing the sensing precision. The optimal systems at the Pareto front obey, to a good approximation, the allocation principle Equation 12. The Pareto front, formed by the maximal value $I_{\max} (x^{*}; L)$ of $I (x^{*}; L) = 1 / 2 \ln (1 + SNR)$ as a function of C, is obtained by minimizing Equation 6 over $p, τ_{r}, R_{T}, X_{T}$ subject to the constraint $C = R_{T} + X_{T}$ ; the quality parameter is $q^{opt} \approx 0.76$ corresponding to $Δ μ^{o p t} \approx 4 k_{B} T$ ; $τ_{c} / τ_{L} = 10^{- 2}$ ; $σ_{L} / {\bar{L}}_{T} = 10^{- 2}$ .

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Giulia Malaguti
Pieter Rein ten Wolde

(2021)

Theory for the optimal detection of time-varying signals in cellular sensing systems

eLife 10:e62574.

https://doi.org/10.7554/eLife.62574

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