Understanding complex living systems requires identifying universal phenomenological laws that are independent of species and molecular-biological details. The Monod equation is a cornerstone of phenomenological microbial growth laws, depicting the growth rate as a saturating function of a single substrate. Because of its similarity in functional form to the Michaelis–Menten equation, cellular growth is often thought to be limited by a single reaction. However, cellular growth generically results from the coordination of thousands of metabolic reactions and diverse intracellular limited resources. Consequently, the mechanistic origins of the Monod equation remain controversial, and its extension has encountered limitations. Here, we propose the global constraint principle for cellular growth: As one nutrient becomes more available, other intracellular resources become limiting, driving transitions to distinct modes of resource allocation. Based on a general framework of constraint-based modeling and its dual formulations, we mathematically prove that, in general, microbial growth kinetics curves are monotonically increasing and concave with respect to nutrient availability. Numerical simulations using genome-scale Escherichia coli models with proteome allocation, molecular crowding, and membrane capacity constraints reproduce these features in a multiphasic manner. In contrast to the original Monod’s growth law, the global constraint principle also captures the dependence of microbial growth on the availability of multiple nutrients, by generalizing the Liebig’s law of the minimum, another phenomenological growth law for higher organisms, into a terraced landscape of diminishing returns. It thus integrates the classical phenomenological laws proposed by Monod and Liebig into a comprehensive theory of cellular growth.
Yamagishi et al. (2025) studied this question.