Decoding cell signaling via optimal transport and information theory

TL;DR

This analysis establishes geometric fidelity as a fundamental, yet previously unrecognized, dimension of signaling fidelity, which provides a quantitative, experimentally accessible framework for dissecting natural networks and designing task-specific synthetic circuits.

Abstract

A central challenge in cellular signal processing is understanding how biochemical networks perform reliably despite molecular noise. Traditionally, mutual information has been widely used to quantify signaling fidelity, capturing how well outputs discriminate distinct input states. However, it fails to capture whether the output also faithfully mirrors the statistical structure of the input, a property crucial in processes like morphogen patterning, dose-dependent signaling, and cellular communication. To address this gap, we introduce the 2-Wasserstein distance from optimal transport theory, which provides a geometric basis for comparing input and output distributions. In our proposed framework, we define mutual information as informational fidelity and the inverse of the 2-Wasserstein distance as geometric fidelity. Applying this dual-fidelity framework to canonical regulatory motifs under Gaussian channel approximation reveals a topology-dependent trade-off: coherent feed-forward loops can achieve high performance in both dimensions, whereas feedback architectures typically sacrifice informational fidelity to enhance geometric fidelity. We demonstrate that theoretical predictions of feedback regulation are well supported by experimental data from tumor necrosis factor signaling. Our results demonstrate that maximizing information alone is not always advantageous and that reliable signaling arises from balancing information transmission with the geometric aspects of signaling. Thus, our analysis establishes geometric fidelity as a fundamental, yet previously unrecognized, dimension of signaling fidelity. It also provides a quantitative, experimentally accessible framework for dissecting natural networks and designing task-specific synthetic circuits.

Figures (9)

• Figure 1: Schematics of signaling systems and the fidelity representations. Illustrative examples of signaling pathways: a morphogen signaling, b neuronal signaling, c intracellular signaling, d cell-to-cell communication, e bet-hedging, and f homeostatic control. g A general scheme of signal transduction. Here, the input $X$ acts as an internal representation of the external signals. The input distribution $P_X(x)$ is processed into an output $Z$ with distribution $P_Z(z)$. h Informational fidelity is defined as MI between $X$ and $Z$, $I(X;Z)$. The MI is shown in terms of the overlap of Shannon entropies. i Geometric fidelity is defined as the inverse of the 2-WD, $W(X,Z)^{-1}$. The 2-WD $W(X,Z)$ is introduced via an optimal transport problem.
• Figure 2: Schematic illustrations of informational and geometric fidelity.a Informational fidelity characterizes state-wise mapping between input and output distributions. This mapping resolves states accurately but sacrifices distributional correspondence. b Geometric fidelity accounts for shape, scale (variability) and mean-wise mapping of input distribution to output. While this geometric fidelity cannot measure the precise discrimination of input states in the output, it does measure their overall distributional correspondence. c A balance between these two fidelities governs cellular decision-making. The table summarizes the various combinations of these two fidelities and their interpretation in signaling.
• Figure 3: Schematics of gene regulatory motifs and governing dynamical equation. a Simple cascade: the input $X$ activates $Y$, which subsequently activates $Z$. b Coherent type-1 feed-forward loop: $X$ activates both $Y$ and $Z$, and $Y$ activates $Z$. c Incoherent type-1 feed-forward loop: $X$ activates both $Y$ and $Z$, but $Y$ represses $Z$. d Positive feedback loop: $X$ activates $Y$, which again activates $Z$, and $Z$ activates $Y$ forming the feedback. e Double negative feedback loop: $X$ activates $Y$, which represses $Z$, and $Z$ represses $Y$. f Negative feedback loop: $X$ activates $Y$, which again activates $Z$, but $Z$ represses $Y$. g Natural degradation with rate $g_M$ ($M \in \{Y, Z \}$) is illustrated. h The stochastic model for the gene regulatory motifs is shown. Here, $\bm{\xi}(t)$ denotes the Gaussian white noise satisfying $\overline{\xi_M(t)}=0$ and $\overline{\xi_M(t)\xi_{M'}(t')}=\delta_{M{M'}}\delta(t-t')$ with $M \in \{ Y,Z \}$ and $M' \in \{ Y,Z \}$. The overbar $\overline{\cdots}$ represents the ensemble average.
• Figure 4: Binding affinity parameters $\theta_X$ and $\theta_Z$.a Schematic illustration of $\theta_X$ and $\theta_Z$ across different network motifs. b Definition of the binding affinity parameters. $\theta_X$ and $\theta_Z$ represent the effective binding affinities of $X$ and $Z$, respectively, to the promoter of gene $Y$.
• Figure 5: Trade-off between informational and geometric fidelities across network motifs. The relationships between the two fidelities obtained from the dual-fidelity optimization framework for six canonical motifs: a Simple cascade, b Coherent type-1 feed-forward loop, c Incoherent type-1 feed-forward loop, d Positive feedback loop, e Double negative feedback loop, and f Negative feedback loop. Each curve represents how the operating regime shifts as BAP and $\lambda$ are varied. The model parameters used to generate these plots are mentioned in SI text.