Mathematical properties of optimal fluxes in cellular reaction networks at balanced growth

The physiology of biological cells evolved under physical and chemical constraints, such as mass conservation across the network of biochemical reactions, nonlinear reaction kinetics, and limits on cell density. For unicellular organisms, the fitness that governs this evolution is mainly determined by the balanced cellular growth rate. We previously introduced growth balance analysis (GBA) as a general framework to model and analyze such nonlinear systems, revealing important analytical properties of optimal balanced growth states. It has been shown that at optimality, only a minimal subset of reactions can have nonzero flux. However, no general principles have been established to determine if a specific reaction is active at optimality. Here, we extend the GBA framework to study the optimality of each biochemical reaction, and we identify the mathematical conditions determining whether a reaction is active or not at optimal growth in a given environment. We reformulate the mathematical problem in terms of a minimal number of dimensionless variables and use the Karush-Kuhn-Tucker (KKT) conditions to identify fundamental principles of optimal resource allocation in GBA models of any size and complexity. Our approach helps to identify from first principles the economic values of biochemical reactions, expressed as marginal changes in cellular growth rate; these economic values can be related to the costs and benefits of proteome allocation into the reactions’ catalysts. Our formulation also generalizes the concepts of Metabolic Control Analysis to models of growing cells. We show how the extended GBA framework unifies and extends previous approaches of cellular modeling and analysis, putting forward a program to analyze cellular growth through the stationarity conditions of a Lagrangian function. GBA thereby provides a general theoretical toolbox for the study of fundamental mathematical properties of balanced cellular growth.Author summary: Mathematical models are an important tool to understand and predict the complex behavior of biological cells. This behavior is driven by nonlinear physical constraints that cannot be captured entirely in the prevalent modeling frameworks, which rely on simplified linear optimizations. The next generation of more realistic cell models will depend on an efficient mathematical formulation for the corresponding nonlinear optimization problem that facilitates the analytical study and numerical simulation of large models. Here, we present a succinct formulation for this nonlinear problem, and we derive the analytical properties of fluxes at optimal growth. We also show how these analytical properties can be understood in terms of economics and control theory, where they expose trade-offs related to the allocation of proteins.