A hidden square-root boundary between growth rate and biomass yield

Although the theoretical value of biomass yield can be calculated from metabolic network stoichiometry, the growth rate is difficult to predict. Since the rate and yield can vary independently, no simple relationship has been discovered between these two variables. In this work, we analyzed the well-accepted enzyme kinetics and uncovered a hidden boundary for growth rate, which is determined by the square-root of three physiological parameters: biomass yield, the substrate turnover number, and the maximum synthesis rate of the turnover enzyme. Cells cannot grow faster than the square-root of the product of these parameters. This analysis is supported by experimental data and involves essentially no assumptions except (i) the cell is not undergoing a downshift transition, (ii) substrate uptake enzyme activity is proportional to its copy number. This simple boundary (not correlation) has escaped notice for many decades and suggests that the yield calculation does not predict the growth rate, but gives an upper limit for the growth rate. The relationship also explains how growth rate is affected by the yield and sheds lights on strain design for product formation.