Scientists uncover a basic principle that shows how higher nutrient levels change the pace of cell growth, revealing a universal rule that applies to microbial growth.
A research group that includes a scientist from the Earth-Life Science Institute (ELSI) at Institute of Science Tokyo, Japan, has uncovered a new biological principle that mathematically describes why growth slows down as nutrient levels rise, a pattern known as “the law of diminishing returns.”
For decades, biologists have tried to understand how living organisms grow under different nutrient conditions. Microbes, plants, and animals all depend on nutrients, energy, and cellular machinery, yet most studies examine only single nutrients or isolated biochemical reactions. This leaves a larger issue unresolved: how do many interacting cellular processes work together to control growth when resources are limited?
Introducing the global constraint principle for microbial growth
To answer this question, ELSI’s Specially Appointed Associate Professor Tetsuhiro S. Hatakeyama and RIKEN Special Postdoctoral Researcher Jumpei F. Yamagishi identified a shared principle that governs how cells adjust their growth under resource constraints. Their work presents the global constraint principle for microbial growth, a framework that may reshape how scientists analyze complex biological systems.
For nearly eighty years, microbiologists have used the “Monod equation,” developed in the 1940s, to describe how microbial growth responds to increasing nutrients. The equation predicts that growth rises with nutrient availability and eventually levels off. But it is based on the assumption that only a single nutrient or reaction limits growth. In reality, cells rely on thousands of chemical processes that draw from the same limited pool of resources.
The researchers argue that the Monod equation reflects only one part of a much broader system. Instead of a single limiting factor, growth is influenced by many constraints acting simultaneously. This produces the same flattening of growth curves but for different underlying reasons.
The global constraint principle shows that when one nutrient becomes plentiful, other factors such as enzyme levels, available cell volume, or membrane capacity begin to restrict growth. Using “constraint-based modeling,” which simulates how cells allocate their internal resources, the team demonstrated that additional nutrients always promote growth, but each extra nutrient provides a smaller boost than the one before it.
Uniting classic laws of biology through a new model
“The shape of growth curves emerges directly from the physics of resource allocation inside cells, rather than depending on any particular biochemical reaction,” says Hatakeyama.
This new principle unites two classic biological laws: the Monod’s equation, which describes microbial growth, and the Liebig’s law of the minimum, which states that a plant’s growth is limited by whichever nutrient is in shortest supply, such as nitrogen or phosphorus. In other words, even if a plant has plenty of most nutrients, it can only grow as much as the scarcest nutrient allows.
By combining these concepts, the researchers created a “terraced barrel” model. In this model, different limiting factors take effect sequentially as nutrients increase. This explains why both microbes and higher organisms show diminishing returns and growth slows down even when more nutrients are added, because a new limiting factor becomes dominant.
Hatakeyama likens his theory to an updated version of Liebig’s barrel, where a plant can only grow as much as its shortest stave (i.e., its most limited nutrient) allows. “In our model, the barrel staves spread out in steps,” he explains, “each step representing a new limiting factor that becomes active as the cell grows faster.”
Testing the principle with E. coli and predicting real-world growth
To test their theory, the team used large-scale computer models of Escherichia coli, which include how the cells utilise proteins, how they are spatially packed, and the capacities of their membranes. The simulations showed the predicted slowing of growth as more nutrients were added and revealed how oxygen or nitrogen levels affect growth patterns. The results agreed well with lab experiments, confirming the model’s accuracy.
The discovery provides a fresh perspective for looking at growth across all forms of life. Combining different principles, the global constraint principle explains complex biological behaviors without needing to model every single molecule in detail. “Our work lays the groundwork for universal laws of growth,” remarks Yamagishi. “By understanding the limits that apply to all living systems, we can better predict how cells, ecosystems, and even entire biospheres respond to changing environments.”
The significance of the research goes beyond basic biology. It may help improve microbial production in industry, increase crop yields by pinpointing limiting nutrients, and guide predictions of ecosystem responses under changing climates.
Future studies could help explore how the principle applies to different organisms and the way multiple nutrients are used together. By connecting microbial biology with ecological theory, this study takes a major step towards a universal foundation for understanding the limits of life’s growth.
Reference: “Global constraint principle for microbial growth laws” by Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama, 3 October 2025, Proceedings of the National Academy of Sciences.
DOI: 10.1073/pnas.2515031122
J.F.Y. is supported by the RIKEN Research Fund for Special Postdoctoral Researcher (project code: 202501094096) and the Masason Foundation. T.S.H. is supported by Japan Society for the Promotion of Science KAKENHI Grant No. JP21K15048.
Never miss a breakthrough: Join the SciTechDaily newsletter.
Follow us on Google and Google News.
1 Comment
thanks for this