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

Abstract

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.

The complex relationship between microbial growth rate and yield and its implications for ecosystem processes

“…I propose that all of these observations are part of the same continuum (Figure 1). The overall shape of this curve is consistent with the “hidden square root boundary” of Wong et al (2009), in which the maximum growth rate is limited to the square root of the product of yield, substrate turnover number, and the maximum synthesis rate of the transporter or turnover enzyme. The positive relationship posited by Pirt pertains to conditions where growth rate is highly constrained by nutrient limitation or physiological stress, such as super or suboptimal pH (Koussémon et al, 2003), superoptimal temperatures (Monod, 1942), and extreme cases of energy limitation (near-starvation conditions).…”

“…A more tangible biochemical perspective is that at rapid growth rates where anabolism and catabolism become unbalanced, energy is dissipated through energy-spilling reactions involving futile cycles (Russell and Cook, 1995) or overflow metabolism (secretion of excess metabolites) (Carlson et al, 2007). Additionally, because of the high energetic cost of producing protein, higher rates of protein synthesis in fast-growing cells can lead to lower efficiency relative to a state with lower growth rate in which yield is maximized (Molenaar et al, 2009; Wong et al, 2009). For this reason, amino acid costs are minimized in highly expressed and secreted proteins (Akashi and Gojobori, 2002; Smith and Chapman, 2010).…”

Biophysical mechanisms that maintain biodiversity through trade-offs

“…A negative correlation between growth rate and yield has been observed but without the quadratic form 12 . Moreover, positive rate-yield correlations, the antithesis of a rate-yield trade-off, have been observed in other bacterial species 9 and, curiously, this relationship is also predicted to be constrained by a quadratic geometry 10 . We can reconcile these apparent contradictions by following a suggestion 9 that r-k relationships can be manipulated by varying the carbon richness of the environment in which bacteria grow (for clarity, we use k for carrying capacity where others use K).…”

“…Due to its importance to biodiversity theory, a search for trade-off data has become the subject of a burgeoning experimental literature using microbial populations in the laboratory 1,[5][6][7][8][9][10][11][12] . However, despite the significance of trade-off geometry to the theory and the longpostulated hypothesis that trade-offs arise from life-history constraints, we know of no successful derivation of trade-off geometry in living systems from fundamental physical or biochemical principles.…”

Phenomenological Model for Predicting the Catabolic Potential of an Arbitrary Nutrient

“…Biomass yield is a time independent quantity that cannot be directly associated to growth rate. However, because biomass yield gives an upper limit for growth rate [48], our model can be used as a baseline for researchers to explore and model growth rates.…”