Closed-Form Information Capacity of Canonical Signaling Models

TL;DR

This work introduces a unified Fisher Information framework to quantify the information capacity of signaling systems, deriving closed-form or easily computable asymptotic capacities $C^*_A$ for canonical models and revealing how input range, noise scaling, cascade length, and population diversity shape sensing limits. By expressing capacity through $C^*_A = \log_2\left((2\pi e)^{-l/2} V\right)$ with $V = \int \sqrt{|\mathrm{FI}(x)|}\,dx$, the authors connect geometric information notions to practical bounds, and provide explicit formulas for Binomial, Multinomial, Poisson, Gaussian, and Gamma outputs, including heterogeneous populations. The results show, for example, that capacity can scale as $C^*_N = C^*_A + \tfrac{l}{2}\log_2 N$ for large $N$, and that non-overlapping phenotypic diversification yields a population gain of $\tfrac{1}{2}\log_2(kN)$ bits, while information decays linearly with cascade length via a factor like $\left(\frac{\kappa}{\kappa+\gamma}\right)^{(m-1)/2}$. Beyond biology, the framework offers efficient benchmarks for engineered sensing systems and sensor networks, obviating costly mutual information estimation in many settings, and enabling principled design choices based on input range, noise structure, and heterogeneity.

Abstract

We employ a unified framework for computing the information capacity of biological signaling systems using Fisher Information. By deriving closed-form or easily computable information capacity formulas, we quantify how well different signaling models, including binomial, multinomial, Poisson, Gaussian, and Gamma distributions, can discriminate among input signals. These expressions clarify how key features such as signal range, noise scaling, pathway length, and receivers' diversity shape the theoretical limits of sensing. In particular, we show how signal-to-noise ratio and fold-change sensitivity arise naturally within the Fisher formalism, and how signal degradation in cascades imposes linear information loss. Our results provide intuitive, analytically grounded tools to benchmark and guide the analysis of real signaling systems, without requiring computationally expensive mutual information estimation. While motivated by cellular communication, the framework generalizes to any system where noisy input-output relationships constrain transmission fidelity, including synthetic biology, sensor networks, and engineered communication channels.

Figures (3)

• Figure 1: (A) The probability of a sensor being active as a function of the signal $x$, modeled as Hill function $h(x)=\frac{(x/H)^n}{1+(x/H)^n}$ with $H=1$, $n=2$. (B) The probability of a sensor in each of the four equiprobable ($\omega_i=1/k$) functional states (k=4) being active, $\omega_i h_i(x)$, as a function of the signal $x$ for a strongly overlapping sensitivity ranges of each functional state modeled as a Hill function with $H_1=0.5$, $H_2=1$, $H_3=2$, $H_4=4$. (C) Same as in (C) but for small overlaps between sensitivity ranges of each functional state: $H_1=0.5$, $H_2=2$, $H_3=8$, $H_4=32$. (D) Capacity $C_N$ as a function of $N$ in a scenario with one conformational state ($k=1$, Binomial output), as well as two and four conformational states (k=2 and k=4, Multinomial output). (E) Capacity $C_N$ in a scenario with two $(k=4)$ and four $(k=4)$ conformational states as a function of the ratio $R=H_{i+1} / H_{i}$, together with the limit of fully distinct responses for each functional state established by Eq. \ref{['eq:multinomial:capacity:limited']}.
• Figure 2: (A) Graphical representation of a chemical linear conversion pathway of length $m$, in which molecules appear in state $S1$ at rate $\kappa_0 = \beta_0+\kappa_0^{\max} \cdot h(x)$ that depends on the external signal $x$ through a monotonic function of choice, here Hill function $h(x)= (x/H)^n/(1+(x/H)^n)$, and then are converted to the next subsequent state at rate $\kappa$ or degrade at rate $\gamma$. All events are assumed to have exponential waiting time implying Poissonian distribution of molecule copy number in each state. (B) Information capacity as a function of cascade length for different values of the degradation rate $\gamma$. For computations, we assumed ${\color{black}\beta_0=0.5}$, $\kappa=1$, $\kappa_0^{\max}={\color{black}20}$, $H=1$, $n=2$.
• Figure 3: (A) Mean response of a cell in a scenario where all cells are governed by the same response parameters, and mean response is described by $h(x)=\mu_{min}+ (\mu_{max}-\mu_{min})\frac{(x/H)^n}{1+(x/H)^n}$ with $\mu_{min}=1$, $\mu_{max}=2$, $H=1$, $n=2$. (B) Mean responses of four, $k=4$, distinct phenotypes with strongly overlapping sensitivity ranges. Mean response is modeled as in scenario (A) with phenotypes having different values of $H$: $H_1=0.5$, $H_2=1$, $H_3=2$, $H_4=4$, and hence $R=2$. (C) As in (B) but for phenotypes with small overlapping sensitivity ranges: $H_1=0.5$, $H_2=2$, $H_3=8$, $H_4=32$, and hence $R=4$. (D) Capacity $C_N$ as a function of $N$ in a scenario with homogenous mean response, as in (A) with Gaussian and Gamma output, for $\sigma=\lambda \mu$ and $\lambda=0.1$. (E) Capacity $C^*_A$ in a scenario with four response phenotypes $(m=4)$ as a function of the ratio $R=H_{i+1} / H_{i}$, for Gaussian and Gamma output, $\sigma=\lambda \mu$ and $\lambda=0.1$, together with the limit of fully distinct responses established by Eq. \ref{['eq:C_N:k_phenotypes_Gamma_cells']} and Eq. \ref{['eq:C_N:k_phenotypes_Gamma_cells']}, plotted as dashed lines.