2025-10-10 理化学研究所,東京科学大学
「大域的制約原理」の模式図:リービッヒの”段々”樽(たる)
<関連情報>
- https://www.riken.jp/press/2025/20251010_1/index.html
- https://www.pnas.org/doi/10.1073/pnas.2515031122
微生物の増殖法則に対する全体的制約原理 Global constraint principle for microbial growth laws
Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama
Proceedings of the National Academy of Sciences Published:October 3, 2025
DOI:https://doi.org/10.1073/pnas.2515031122
Significance
Cellular growth kinetics is a classical and fundamental problem in biology, traditionally described by Monod’s law. However, its universality has long remained uncertain. While the universal validity of this law itself is not argued, we prove that its key features—monotonicity and concavity in nutrient dependence—are universal consequences of intracellular resource allocation, providing a systematic theoretical basis for cellular growth kinetics. Unlike Monod’s model, which implies a local view centered on a single limiting reaction, our framework highlights global constraints on growth, thus generalizing the classical idea of Liebig’s law into a modern theory. By unifying these classical phenomenological laws of Monod and Liebig, our theory paves the way for a more comprehensive understanding of organismal growth.
Abstract
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.
