Tropical Geometry and Biochemical Reaction Networks: A Mathematical Framework for Steady-State Topology

We present a mathematical framework connecting tropical geometry to the analysis of steady-state varieties in biochemical reaction networks under mass-action kinetics. The kinetic parameters of biological systems typically span many orders of magnitude, yet qualitative phenotypes remain robust to parametric uncertainty. We formalize this observation by studying the tropical limit-the piecewise-linear skeleton of the algebraic steady-state variety in logarithmic coordinates. Two primary results are proved. First, we establish convergence (in the Hausdor metric) of the logarithmically scaled positive real steady-state variety to a tropical polyhedral complex, under stated hypotheses on scale separation, coefficient genericity, and transversality (Theorem 2.10). Second, we prove a duality theorem (Theorem 2.11) establishing a bijection between maximal normal cones of Newton polytopes and dominant reaction subnetworks,and provide explicit conditions under which the tropical intersection lifts to a positive real steady state (Lemma 2.12). The framework is validated through five case studies of increasing complexity: a nonlinear metabolic branch point, Michaelis Menten enzyme kinetics, competitive inhibition, a cooperative binding switch, and a six-species upper glycolysis network with eight reactions. The glycolysis case study the first mid-sized demonstration of the method identifies five distinct metabolic regimes and four switching manifolds, with tropical predictions matching numerical steady states to within 2-6% for species deep in their dominant-balance regime, while species at regime boundaries show errors of 30-125%, consistent with the theory's predicted boundary limitations. We provide first-order Puiseux correction terms, quantitative finite-t error bounds, an explicit positivity criterion with a worked counterexample demonstrating its necessity, a genericity failure example with biological interpretation, measured runtimes benchmarking the algorithm against ODE integration, and a scalability discussion for genome-scale networks. A complete code repository accompanies this paper.