Cellular Signaling Networks Function as Generalized Wiener-Kolmogorov Filters to Suppress Noise

Cellular signaling involves the transmission of environmental information through cascades of stochastic biochemical reactions, inevitably introducing noise that compromises signal fidelity. Each stage of the cascade often takes the form of a kinase-phosphatase push-pull network, a basic unit of signaling pathways whose malfunction is linked with a host of cancers. We show that this ubiquitous enzymatic network motif effectively behaves as a Wiener-Kolmogorov optimal noise filter. Using concepts from umbral calculus, we generalize the linear Wiener-Kolmogorov theory, originally introduced in the context of communication and control engineering, to take nonlinear signal transduction and discrete molecule populations into account. This allows us to derive rigorous constraints for efficient noise reduction in this biochemical system. Our mathematical formalism yields bounds on filter performance in cases important to cellular function—such as ultrasensitive response to stimuli. We highlight features of the system relevant for optimizing filter efficiency, encoded in a single, measurable, dimensionless parameter. Our theory, which describes noise control in a large class of signal transduction networks, is also useful both for the design of synthetic biochemical signaling pathways and the manipulation of pathways through experimental probes such as oscillatory input.

Popular Summary

Cells process and transmit information about their environment through complex systems of chemical reactions known as signaling cascades. Accurate transmission of information is crucial for normal function; defects in signaling cascades result in a variety of cancers. As in many designed communications systems, these biological circuits must cope with noise that inevitably corrupts the signal and reduces its fidelity. Efficiently filtering noise and reconstructing the best estimate of the input have been key problems in engineering. A significant advance in this area was a mathematical framework developed by Wiener and Kolmogorov during World War II, originally inspired by the need to filter noise in the targeting of anti-aircraft systems.

Remarkably, the same mathematical ideas help us understand the noise-filtering strategies in biological systems. We show that cells effectively implement the Wiener-Kolmogorov solution within signaling cascades, through a series of enzymatic reactions that activate and deactivate substrates. We generalize the Wiener-Kolmogorov approach, removing the inherent linearity of the framework to deal with some of the challenges that arise in the biological context, including signaling through discrete changes in molecular populations, and the highly nonlinear relation between input and output. We verify our analytical results using Monte Carlo simulations and provide mathematically rigorous bounds on the performance of biochemical noise filters, thus identifying the mechanism by which cells optimize signal fidelity.

Beyond illuminating naturally evolved noise-reduction strategies, our approach should be useful in the design of synthetic cellular signaling circuits for biomedical applications. We envision that our results may be applicable to a large class of biological signaling networks involved in cellular response to external stimuli.

Figure 1

Schematic of a signaling cascade. (a) A signaling pathway involving cascades of kinase phosphorylation, activated by a receptor embedded in the cell membrane that responds to extracellular ligands. (b) A close-up of one enzymatic push-pull loop within the cascade. Kinase ($K$) phosphorylates the substrate ($S$), converting it to active form ($S^{*}$), while phosphatase ($P$) reverts it to the original form through dephosphorylation. $S_K$ and $S_P^{*}$ represent the substrate in complex with the kinase and phosphatase, respectively. The rate parameters labeling the reaction arrows are described in the text. The input $I=K+S_K$ and output $O=S^{*}+S_P^{*}$. (c) A minimal signaling circuit, involving an input species $I$ and output species $O$, related by the production rate function $R(I)$.

Figure 2

Optimal noise reduction in the minimal signaling circuit [Fig. 1]. (a) Numerical optimization results for the Hill production function $R(I)$ that minimizes relative error $E$ between input and output, with each color corresponding to different values of the parameter $\tilde{\mathrm{\Lambda }}={10}^2–{10}^4$ (see text for other parameters). The input probability distribution $\mathcal{P}(I)$ is superimposed in black (the height scale is arbitrary). (b) For each value of $\tilde{\mathrm{\Lambda }}$ from (a), circles show the minimal $E$. The lower bound $E_{\text{opt}}$ [Eq. (9)] is drawn as a blue curve. (c) Analogous to (b), but showing the ratio ${\gamma }_O/{\gamma }_I$ at which the minimum $E$ is achieved. The blue curve shows the WK prediction for this ratio, ${\gamma }_O/{\gamma }_I=\sqrt{1+\tilde{\mathrm{\Lambda }}}$.

Figure 3

(a) ${\gamma }_O$ (solid curve) and ${\gamma }_I\sqrt{1+\mathrm{\Lambda }}$ (dashed curves), based on the mapping in Eq. (11), for $\mathrm{\Lambda }=100$. The $\overline{P}$ value at the intersection of the solid and dashed curves, where the WK optimality conditions are fulfilled, is indicated by a diamond. (b) Gain $G$ (solid line, CL theory; circles, KMC simulations) for the same $\mathrm{\Lambda }$ value as in (a), versus the WK optimal value for $G$ (dashed line) given by Eq. (5). (c) Same as (b), but showing error $E$ versus the WK optimal prediction $E_{\mathrm{WK}}$ (dashed line) from Eq. (6). (d)–(f) Sample trajectories for the scaled input $G\delta I(t)$ (blue curve) and the output $\delta O(t)$ (purple curve) from KMC simulations of the push-pull loop, for $\mathrm{\Lambda }=100$ ($\overline{S}=8\times {10}^4$) and three different values of $\overline{P}$. These $\overline{P}$ values are marked by red triangles under (c) and correspond to cases where, relative to the input, the output is too smooth (d), optimal (e), and too noisy (f). (g)–(i) Power spectral densities of the scaled input $G\delta I(t)$ (blue curve) and the output $\delta O(t)$ (purple curve) for the three cases shown in (d)–(f).

Figure 4

Conditions for WK optimality as the enzymatic push-pull loop parameters are varied. (a) The solid blue curve shows the relation between mean phosphatase population $\overline{P}$ and the characteristic frequency scale ${\gamma }_K$ over which the active kinase input signal varies. This is at WK optimality [Eq. (5)], using the mapping of Eq. (11) and the parameter values ${\kappa }_b$, ${\rho }_b$, ${\kappa }_u$, ${\rho }_u$, ${\kappa }_r$, ${\rho }_r$ listed in the text after Eq. (10). The shaded region between the dashed curves shows the 68% confidence interval for achieving WK optimality, resulting from randomly perturbing all the parameter values so that they can be up to tenfold smaller or larger. (b) Analogous to (a), but showing the relation between $\overline{P}$ and substrate population $\overline{S}$ at WK optimality.

Figure 5

(a) Sample KMC simulation trajectories for $I(t)$ (solid blue) and $O(t)$ (solid purple) in a $\mathrm{\Lambda }=10$ system driven by an oscillatory upstream flux $F(t)$ (see text for parameters). Dashed lines are local means $I_{\text{loc}}(t)$ and $O_{\text{loc}}(t)$. (b) For trajectories in (a), the deviations from local means, $\delta I_{\text{loc}}(t)$ (blue) and $\delta O_{\text{loc}}(t)$ (purple). (c)–(e) Results calculated from KMC simulations for a system with $\mathrm{\Lambda }=10$ and oscillatory $F(t)$ at varying driving periods $T$. ${\gamma }_I^{-1}$ is marked by a vertical dashed line. (c) The minimum errors $E$ (circles) and $E_{\text{loc}}$ (squares). $E_{\mathrm{WK}}$ is marked by a horizontal dashed line. (d) The minimum local coefficient of variation ${\mathrm{CV}}_{\text{loc}}=\sqrt{⟨(\delta O_{\text{loc}}/{\overline{O}}_{\text{loc}}{)}^2⟩}$. (e) The mean phosphatase population values $\overline{P}$ at which the minima shown in (c) and (d) are achieved ($E$, circles; $E_{\text{loc}}$, squares; ${\mathrm{CV}}_{\text{loc}}$, crosses). The $\overline{P}$ value for WK optimality is marked by a horizontal dashed line.