Structure of autosynthetic models of balanced cell growth and numerical optimization of their growth rate

Abstract

Genome-scale reaction network models are available for many prokaryotic organisms. Yet, to predict the proteome and metabolome of the cell from them, additional information about (i) the nonlinear enzyme kinetics and (ii) the regulation of protein expression by metabolic signals is necessary. Knowledge about the latter could be sidestepped by assuming that expression regulation has evolved to achieve the protein composition that maximizes cellular growth rate. A general mathematical framework for optimizing the growth rate of models comprising an arbitrarily complex metabolic network and a relatively simple protein-synthesis network was recently formulated independently by two research groups [de Groot et al., PLoS Comput. Biol. 16, e1007559 (2020); Dourado & Lercher, Nature Commun. 11, 1226 (2020)]. Here, this formalism is further developed with particular focus on carrying out the optimization numerically. To this end, we identify the concentrations of the enzymes as the independent variables of the optimization problem and propose novel multiplicative updates for the iterative calculation of the dependent metabolite concentrations. The reduced gradient method, with analytical derivatives, is employed for the numerical optimization. Additionally, the roles of the dilution of the metabolite concentrations by growth and the commonly invoked constraint on the cell dry mass density are clarified. These developments should lay the basis for the practical optimization of large-scale kinetic models, thus formally connecting the physiological "macrostate" of the cell, characterized by its growth rate, to its "microstate", described by the cell proteome and metabolome. Author summaryAn evolving population of non-interacting, unicellular organisms in a constant environment will maximize its growth rate. By expressing the growth rate as a mathematical function of the cellular composition, it becomes possible to formulate an optimization problem whose solution yields the cell proteome and metabolome at the maximal growth rate. The formulation and solution of such an optimization problem has the potential to elucidate fundamental optimality principles in living cells and to enable the engineering of complex biological systems. Building on previous work, here we address the task of solving this optimization problem numerically. In the process, we elucidate the mathematical role of some common simplifying approximations. This allows us to organize many of the existing formulations of the optimization problem into a hierarchy, whose lower levels are reached by invoking these approximations.