Receptor time integration via discrete sampling

Living cells can measure chemical concentrations with remarkable accuracy, even though these measurements are inherently noisy due to the stochastic binding of the ligand to the receptor. A widely used mechanism for reducing the sensing error is to increase the effective number of measurements via receptor time integration. This mechanism is implemented via the signaling network downstream of the receptor, yet how it is implemented optimally given constraints on cellular resources such as protein copies and time remains unknown. To address this question, we employ our sampling framework [Govern and ten Wolde, Proc. Natl. Acad. Sci. USA 111, 17486 (2014)] and extend it here to time-varying ligand concentrations. This framework starts from the observation that the signaling network implements the mechanism of time integration by discretely sampling the ligand-binding state of the receptor and storing these states into chemical modification states of the readout molecules downstream. It reveals that the sensing error has two distinct contributions: a sampling error, which is determined by the number of samples, their independence, and their accuracy, and a dynamical error, which depends on the timescale that these samples are generated. We test our previously identified design principle, which states that in an optimally designed system the number of receptors and their integration time, which determine the number of independent concentration measurements at the receptor level, equals the number of readout proteins, which store these measurements. We show that this principle is robust to the dynamics of the input and the relative costs of the receptor and readout proteins: these resources are fundamental and cannot compensate each other.

Figure 1

The precision of estimating a time-varying ligand concentration $L\left(t\right)$. (a) The cell signaling network measures the ligand concentration via receptors on its surface. (b) The dynamic input-output relation $p_{{\tau }_{\mathrm{r}}}\left(L\right)$. The cell employs the push-pull network to estimate the average receptor occupancy $p_{{\tau }_{\mathrm{r}}}$ over the past integration time ${\tau }_{\mathrm{r}}$ [see (b)] and then infers the current ligand concentration $L\left(t\right)$ by inverting $p_{{\tau }_{\mathrm{r}}}\left(L\right)$. The error in the estimate of the concentration ${\left(\delta \widehat{L}\right)}^2={\sigma }_{{\widehat{p}}_{{\tau }_r}}^2/{\tilde{g}}_{L\to p_{{\tau }_{\mathrm{r}}}}^2$ depends on the variance ${\sigma }_{{\widehat{p}}_{{\tau }_r}}^2$ in the estimate of the average receptor occupancy ${\widehat{p}}_{{\tau }_r}$ and the dynamic gain ${\tilde{g}}_{L\to p_{{\tau }_{\mathrm{r}}}}$, the slope of $p_{{\tau }_{\mathrm{r}}}\left(L\right)$, which determines how the error in ${\widehat{p}}_{{\tau }_r}$ propagates to that in $\delta \widehat{L}$. The input distribution has width ${\sigma }_L^2$. (c) The push-pull network as a sampling device. The network discretely samples the ligand-binding state of the receptor via the readout molecules $x$ [15] and the fraction of modified readout molecules gives the estimate of $p_{{\tau }_{\mathrm{r}}}$; see Eq. (5). The sensing error has two contributions [Eq. (26)]: the sampling error and the dynamical error. The sampling error arises from the noise in the sampling of the receptor and depends on the number of samples, their independence, and their accuracy. (d) Origin of the dynamical error. The current ligand concentration $L\left(t\right)$ is estimated via the average receptor occupancy $p_{{\tau }_{\mathrm{r}}}$ in the past ${\tau }_{\mathrm{r}}$ [see (a)], but the latter depends on the ligand concentration in the past ${\tau }_{\mathrm{r}}$, which deviates from the current concentration that the cell aims to estimate. Two different input trajectories ($L_1$ in blue, $L_2$ in green) ending at time $t$ at the same value $L\left(t\right)$ (red dot) yield different estimates of $L\left(t\right)$ because of their different average receptor occupancies $p_{{\tau }_{\mathrm{r}}}$ over the past ${\tau }_{\mathrm{r}}$. Figure and caption are adapted from Ref. [16].

Figure 2

Overview of the decomposition of the variance ${\sigma }_{{\widehat{p}}_{{\tau }_r}}^2$ in the estimate ${\widehat{p}}_{{\tau }_r}$ of the receptor occupancy $p_{{\tau }_{\mathrm{r}}}$ over the past integration ${\tau }_{\mathrm{r}}$ given that the current ligand concentration $L\left(t\right)=L$. The error in the estimate of $p_{{\tau }_{\mathrm{r}}}$ can be decomposed into the sampling error ${\sigma }_{{\widehat{p}}_{{\tau }_r}}^{2,\mathrm{samp}}$, denoted in blue, and the dynamical error ${\sigma }_{{\widehat{p}}_{{\tau }_r}}^{2,\mathrm{dyn}}$, denoted in green [see Eq. (9)]. The sampling error is a statistical error, arising from the finite cellular resources to sample the state of the receptor; see Eq. (22). The dynamical error is a systematic error that arises from the input dynamics and depends only on timescales; see Eq. (24). To arrive at this decomposition, ${\sigma }_{{\widehat{p}}_{{\tau }_r}}^2$ is split into a contribution from the stochasticity in the number of samples (“variance of the mean”) and one from the error for a fixed number of samples (“mean of the variance”); see Eq. (8). The first is given by Eq. (10). The latter consists of three contributions and is given by Eq. (12). The second of these three, the receptor covariance, contributes both to the sampling error [see Eq. (17)] and the dynamical error [see Eq. (23)].

Figure 3

Design of the optimal sensing system. (a) Sensing precision of the optimal system, as quantified via the mutual information $I(x^{*};L)$, as a function of the number of receptors $R_{\mathrm{T}}$ and the number of readout proteins $X_{\mathrm{T}}$, where $I(x^{*};L)$ has been optimized over the integration time ${\tau }_{\mathrm{r}}$ and the receptor occupancy $p$. The gray line shows one contour line of a constant total protein cost $C=R_{\mathrm{T}}+X_{\mathrm{T}}$. The black point on this line is the combination of the number of receptors $R_{\mathrm{T}}$ and readout proteins $X_{\mathrm{T}}$ that maximizes the sensing precision $I(x^{*};L)$ along this contour line of constant protein cost $C$. The black line shows the optimal design ($R_{\mathrm{T}}, X_{\mathrm{T}}$) that maximizes the sensing precision $(I(x^{*};L)$ for a given protein cost $C$, as a function of $C$. The red dashed line is the same parametric plot but then for a system that is constrained to obey the optimal allocation principle $X_{\mathrm{T}}=R_{\mathrm{T}}(1+{\tau }_{\mathrm{r}}/{\tau }_{\mathrm{c}})$. (b) The optimal integration time that maximizes the mutual information as shown in (a) as a function of the number of receptors $R_{\mathrm{T}}$ and readout proteins $X_{\mathrm{T}}$. (c) Sensing precision $I(x^{*};L)$ as a function of $R_{\mathrm{T}}$ and $X_{\mathrm{T}}$ for a system that obeys the resource allocation principle $X_{\mathrm{T}}=R_{\mathrm{T}}(1+{\tau }_{\mathrm{r}}/{\tau }_{\mathrm{c}})$, where $I(x^{*};L)$ has been optimized over the receptor occupancy $p$ (and the integration time ${\tau }_{\mathrm{r}}$ is set by the allocation principle). The red point shows the combination of $R_{\mathrm{T}}$ and $X_{\mathrm{T}}$ that maximizes $I(x^{*};L)$ along the shown contour line of constant protein cost $C=R_{\mathrm{T}}+X_{\mathrm{T}}$. The red line shows the parametric plot of $(R_{\mathrm{T}},X_{\mathrm{T}})$ that maximizes $I(x^{*};L)$ for a given $C$, as a function of $C$. For ease of comparison, this line is shown as a red dashed line in (a); similarly, the black solid line of panel (a), showing the optimal design $(R_{\mathrm{T}},X_{\mathrm{T}})$ as a function of protein cost $C$, is shown here as black dashed line. Since the minimal integration time ${\tau }_{\mathrm{r}}$ is zero, no system obeying the allocation principle $X_{\mathrm{T}}=R_{\mathrm{T}}(1+{\tau }_{\mathrm{r}}/{\tau }_{\mathrm{c}})$ exists for which $X_{\mathrm{T}}<R_{\mathrm{T}}$. The receptor correlation time is ${\tau }_{\mathrm{c}}/{\tau }_{\mathrm{L}}={10}^{-2}$, and the relative width of the signal distribution is ${\sigma }_L/\overline{L}={10}^{-2}$.

Figure 4

The optimal resource allocation principle accurately describes the optimal system. The panels compare as a function of the protein cost $C=R_{\mathrm{T}}+X_{\mathrm{T}}$, the sensing precision $I(x^{*};L)$, and design $(R_{\mathrm{T}},X_{\mathrm{T}},{\tau }_{\mathrm{r}})$ of the globally optimal system (black lines), where $I(x^{*};L)$ has been optimized over $R_{\mathrm{T}},X_{\mathrm{T}},{\tau }_{\mathrm{r}}$, and $p$, to the precision and design of the optimal system that is constrained to obey the resource allocation principle $X_{\mathrm{T}}=R_{\mathrm{T}}(1+{\tau }_{\mathrm{r}}/{\tau }_{\mathrm{c}})$ (red lines); here $I(x^{*};L)$ is also optimized over $p$ and $R_{\mathrm{T}},X_{\mathrm{T}},{\tau }_{\mathrm{r}}$, but the allocation principle imposes a constraint that removes one degree of freedom between the latter three. See also Figs. 3 and 3. (a) The sensing precision $I(x^{*};L)$ as a function of the protein cost $C=R_{\mathrm{T}}+X_{\mathrm{T}}$ for the globally optimal system (black line) and the optimal system that obeys the resource allocation principle (red line). (b) The number of receptors $R_{\mathrm{T}}$ (solid line) and the number of readout proteins $X_{\mathrm{T}}$ (dashed line) as a function of protein cost for for the globally optimal system (black lines) and the optimal system that obeys the resource allocation principle (red lines). (c) The integration time ${\tau }_{\mathrm{r}}$ for the globally optimal system (black line) and the optimal system that obeys the resource allocation principle (red line). The receptor correlation time is ${\tau }_{\mathrm{c}}/{\tau }_{\mathrm{L}}={10}^{-2}$ and the relative width of the signal distribution is ${\sigma }_L/\overline{L}={10}^{-2}$.

Figure 5

The predictive power of the resource allocation principle improves further when the readout cost decreases or the input varies more rapidly, containing the integration time. The panels compare as a function of the protein cost $C=R_{\mathrm{T}}+c_X{X}_{\mathrm{T}}$ the sensing precision $I(x^{*};L)$ and design $(R_{\mathrm{T}},X_{\mathrm{T}},{\tau }_{\mathrm{r}})$ of the globally optimal system (black lines), where $I(x^{*};L)$ has been optimized over $R_{\mathrm{T}},X_{\mathrm{T}},{\tau }_{\mathrm{r}}$, and $p$, to the precision and design of the optimal system that is constrained to obey the resource allocation principle $X_{\mathrm{T}}=R_{\mathrm{T}}(1+{\tau }_{\mathrm{r}}/{\tau }_{\mathrm{c}})$ (red lines); here $I(x^{*};L)$ is also optimized over $p$ and $R_{\mathrm{T}},X_{\mathrm{T}},{\tau }_{\mathrm{r}}$, but the allocation principle imposes a constraint that removes one degree of freedom between the latter three. Top row (a–c) Relative cost of readout compared to receptor is decreased tenfold, $c_X=0.1$. Bottom row (d, e) Input timescale ${\tau }_{\mathrm{L}}$ is decreased tenfold compared to ${\tau }_{\mathrm{c}}$. (a, d) The sensing precision $I(x^{*};L)$ as a function of the protein cost $C=R_{\mathrm{T}}+X_{\mathrm{T}}$ for the globally optimal system (black line) and the optimal system that obeys the resource allocation principle (red line). (b, e) The number of receptors $R_{\mathrm{T}}$ (solid line) and the number of readout proteins $X_{\mathrm{T}}$ (dashed line) as a function of protein cost for for the globally optimal system (black lines) and the optimal system that obeys the resource allocation principle (red lines). (c, f) The integration time ${\tau }_{\mathrm{r}}$ for the globally optimal system (black line) and the optimal system that obeys the resource allocation principle (red line). The receptor correlation time is ${\tau }_{\mathrm{c}}/{\tau }_{\mathrm{L}}={10}^{-2}$, and the relative width of the signal distribution is ${\sigma }_L/\overline{L}={10}^{-2}$.