Log-Linear Reaction Quotient Dynamics

Abstract

Chemical reaction networks in living cells maintain precise control over thousands of metabolites despite operating far from equilibrium under constant perturbations. While mass action kinetics accurately describe the underlying dynamics, the resulting nonlinear differential equations are difficult to analyze and control, particularly for large networks. We propose a simplified model where reaction quotients (the ratios that measure how far reactions are from equilibrium) evolve exponentially toward their equilibrium values when viewed on a logarithmic scale. This principle leads to linear dynamics in log-space, providing several key advantages: analytical solutions exist for arbitrary network topologies, thermodynamic constraints are automatically satisfied through the relationship between reaction quotients and Gibbs free energy, conservation laws decouple from reaction quotient dynamics simplifying both analysis and control design, and external energy sources couple linearly to the dynamics, unifying diverse biological regulatory mechanisms.