The biological fitness of microbes is largely determined by the rate with which they replicate their biomass composition. Mathematical models that maximize this balanced growth rate while accounting for mass conservation, reaction kinetics, and limits on dry mass per volume are inevitably non-linear. Here, we develop a general theory for such models, termed Growth Balance Analysis (GBA), which provides explicit expressions for protein concentrations, fluxes, and growth rates. These variables are functions of the concentrations of cellular components, for which we calculate marginal fitness costs and benefits that are related to metabolic control coefficients. At maximal growth rate, the net benefits of all concentrations are equal. Based solely on physicochemical constraints, GBA unveils fundamental quantitative principles of cellular resource allocation and growth; it accurately predicts the relationship between growth rates and ribosome concentrations in E. coli and yeast and between growth rate and dry mass density in E. coli.

Protein synthesis is the most expensive process in fast-growing bacteria. Experimentally observed growth rate dependencies of the translation machinery form the basis of powerful phenomenological growth laws; however, a quantitative theory on the basis of biochemical and biophysical constraints is lacking. Here, we show that the growth rate-dependence of the concentrations of ribosomes, tRNAs, mRNA, and elongation factors observed in Escherichia coli can be predicted accurately from a minimization of cellular costs in a mechanistic model of protein translation. The model is constrained only by the physicochemical properties of the molecules and has no adjustable parameters. The costs of individual components (made of protein and RNA parts) can be approximated through molecular masses, which correlate strongly with alternative cost measures such as the molecules’ carbon content or the requirement of energy or enzymes for their biosynthesis. Analogous cost minimization approaches may facilitate similar quantitative insights also for other cellular subsystems.

Much recent progress has been made to understand the impact of proteome allocation on bacterial growth; much less is known about the relationship between the abundances of the enzymes and their substrates, which jointly determine metabolic fluxes. Here, we report a correlation between the concentrations of enzymes and their substrates in Escherichia coli. We suggest this relationship to be a consequence of optimal resource allocation, subject to an overall constraint on the biomass density: For a cellular reaction network composed of effectively irreversible reactions, maximal reaction flux is achieved when the dry mass allocated to each substrate is equal to the dry mass of the unsaturated (or “free”) enzymes waiting to consume it. Calculations based on this optimality principle successfully predict the quantitative relationship between the observed enzyme and metabolite abundances, parameterized only by molecular masses and enzyme–substrate dissociation constants (Km). The corresponding organizing principle provides a fundamental rationale for cellular investment into different types of molecules, which may aid in the design of more efficient synthetic cellular systems.

The biological fitness of unicellular organisms is largely determined by their balanced growth rate, i.e., by the rate with which they replicate their biomass composition. Natural selection on this growth rate occurred under a set of physicochemical constraints, including mass conservation, reaction kinetics, and limits on dry mass per volume; mathematical models that maximize the balanced growth rate while accounting explicitly for these constraints are inevitably nonlinear and have been restricted to small, non-realistic systems. Here, we lay down a general theory of balanced growth states, providing explicit expressions for protein concentrations, fluxes, and the growth rate. These variables are functions of the concentrations of cellular components, for which we calculate marginal fitness costs and benefits that can be related to metabolic control coefficients. At maximal growth rate, the net benefits of all concentrations are equal. Based solely on physicochemical constraints, the growth balance analysis (GBA) framework introduced here unveils fundamental quantitative principles of cellular growth and leads to experimentally testable predictions.

The corresponding organizing principle -the minimization of the summed mass concentrations of solutes -may facilitate reducing the complexity of kinetic models and will contribute to the design of more efficient synthetic cellular systems.peer-reviewed)

9Protein synthesis is the most expensive process in fast-growing bacteria 1,2 . The economic 10 aspects of protein synthesis at the cellular level have been investigated by estimating 11 ribosome activity 3-5 and the expression of ribosomes 3,6 , tRNA 7-9 , mRNA 2 , and elongation 12 factors 10,11 . The observed growth-rate dependencies form the basis of powerful 13 phenomenological bacterial growth laws 5,12-16 ; however, a quantitative theory allowing us to 14 understand these phenomena on the basis of fundamental biophysical and biochemical 15principles is currently lacking. Here, we show that the observed growth-rate dependence of 16 the concentrations of ribosomes, tRNAs, mRNA, and elongation factors in Escherichia coli 17 can be predicted accurately by minimizing cellular costs in a detailed mathematical model of 18 protein translation; the mechanistic model is only constrained by the physicochemical 19properties of the molecules and requires no parameter fitting. We approximate the costs of 20 molecule species through their masses, justified by the observation that cellular dry mass 21 per volume is roughly constant across growth rates 17 and hence represents a limited 22resource. Our results also account quantitatively for observed RNA/protein ratios and 23 ribosome activities in E. coli across diverse growth conditions, including antibiotic stresses. 24Our prediction of active and free ribosome abundance facilitates an estimate of the 25 deactivated ribosome reserve 14,18,19 , which reaches almost 50% at the lowest growth rates. 26 We conclude that the growth rate dependent composition of E coli's protein synthesis 27 machinery is a consequence of natural selection for minimal total cost under 28 physicochemical constraints, a paradigm that might generally be applied to the analysis of 29 resource allocation in complex biological systems. 30

The physiology of biological cells evolved under physical and chemical constraints such as mass conservation, 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 such nonlinear systems, and we presented analytical conditions for optimal balanced growth in the special case that the active reactions are known. Here, we develop Growth Mechanics (GM) as a more general, succinct, and powerful analytical description of the growth optimization of GBA models, which we formulate in terms of a minimal number of dimensionless variables. GM uses Karush-Kuhn-Tucker (KKT) conditions in a Lagrangian formalism. It identifies fundamental principles of optimal resource allocation in GBA models of any size and complexity, including the analytical conditions that determine the set of active reactions at optimal growth. We 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. GM 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. GM thereby provides a general theoretical toolbox for the study of fundamental mathematical properties of balanced cellular 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.