Information geometry of chemical reaction networks: Cramer-Rao bound and absolute sensitivity revisited

TL;DR

This work develops an information-geometric framework for CRNs by equipping concentration and chemical-potential spaces with Hessian metrics from strictly convex potentials and proving Legendre duality acts as an isometry, enabling a multivariate Cramer-Rao bound via metric comparisons. It then defines absolute sensitivity as a canonical projection operator onto the equilibrium manifold, derives a global tensor formulation $A=g_Y U^* g_H U$, and analyzes first-order corrections to the ideal, quasi-thermostatic geometry under non-ideal thermodynamics such as van der Waals-type interactions. The theoretical framework is applied to the core module of the IDHKP-IDH glyoxylate bypass, revealing that non-ideal thermodynamics can induce negative self-feedback and hypersensitivity—phenomena usually requiring nonlinear kinetics—while remaining tractable to analysis and computation. The results highlight how crowded cellular environments, with volume exclusion and intermolecular interactions, can shape CRN thermodynamics and sensitivities, offering a geometrically principled lens for understanding robustness, regulation, and potential regulatory design in biology. The combination of Hessian geometry, a global equilibrium-manifold perspective, and explicit first-order corrections provides a versatile toolkit for studying both ideal and non-ideal CRNs, with implications for thermodynamic efficiency, control, and nonequilibrium generalizations.

Abstract

Information geometry is based on classical Legendre duality but allows to incorporate additional structure such as algebraic constraints and Bregman divergence functions. It is naturally suited, and has been successfully used, to describe the thermodynamics of chemical reaction networks (CRNs) based on the Legendre duality between concentration and potential spaces, where algebraic constraints are enforced by the stoichiometry. In this article, the Riemannian geometrical aspects of the theory are explored. It is shown that duality between concentration and potential spaces and the natural parametrizations of equilibrium subspace are isometries, which leads to a multivariate Cramer-Rao bound through the comparison of two Riemannian metric tensors. In the subsequent part, the theory is applied to the recently introduced concept of absolute sensitivity. Using the Riemannian geometric tools, it is proven that the absolute sensitivity is a projection operator onto the tangent bundle of the equilibrium manifold. A linear algebraic characterization and explicit results on first order corrections to the thermodynamics of ideal solutions are provided. Finally, the theory is applied to the IDHKP-IDH glyoxylate bypass regulation system. The novelty of the theory is that it is applicable to CRNs with non-ideal thermodynamical behavior, which are prevalent in highly crowded cellular environments due to various interactions between the chemicals. Indeed, the analyzed example shows remarkable behavior ranging from hypersensitivity to negative-self regulations. These are effects which usually require strongly nonlinear reaction kinetics. However, here, they are obtained by tuning thermodynamical interactions providing a complementary, and physically well-founded, viewpoint on such phenomena.

Figures (3)

• Figure 1: Distributions of the absolute sensitivity $\alpha_4$ for randomly sampled concentration vectors in $\log x^i \in [-6,2]$, and random deformation tensor components $C_{ij} \in [-c_{\textrm{max}},c_{\textrm{max}}]$ for $i \geq j$ and $C_{ij} = C_{ji}$ for $i <j$. Each histogram consists of 100,000 data points.
• Figure 2: The histograms from Figure \ref{['fig:numerics']} from a different perspective.
• Figure 3: The histograms from Figure \ref{['fig:numerics']} from a different perspective.

Theorems & Definitions (41)

• Theorem
• Lemma
• Theorem
• Theorem
• Corollary
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.1