An Analytical Theory of Cellular Growth

The biological fitness of non-interacting unicellular organisms in constant environments is given by their balanced growth rate, i.e., by the rate with which they replicate their biomass composition. Evolutionary optimization of this growth rate occurred under a set of physicochemical constraints, including mass conservation, reaction kinetics, and limits on dry mass per volume (cellular capacity). Mathematical models that account explicitly for these constraints are inevitably nonlinear, and their optimization has been restricted to small, non-realistic cell models. Here, we show that states of maximal balanced growth are elementary flux modes of a related flux balance problem, i.e., deactivating any active reaction makes steady-state growth impossible. For any balanced growth state that corresponds to an elementary flux mode of an arbitrarily sized model, we provide explicit expressions for individual protein concentrations, fluxes, and growth rate; all variables are uniquely determined by the concentrations of metabolites and total protein. We provide explicit and intuitively interpretable expressions for the marginal fitness costs and benefits of individual concentrations. At optimal balanced growth, the marginal net benefits of each metabolite concentration and of total protein concentration equal the marginal benefit of the cellular capacity. Based solely on physicochemical constraints, our work unveils fundamental quantitative principles of balanced cellular growth, quantifies the effect of cellular capacity on fitness, and leads to experimentally testable predictions.