A research team including a scientist of Earth-Life Science Institute (ELSI) at Institute of Science Tokyo, Japan, has identified a novel principle in biology that mathematically explains why the growth of organisms slows as nutrients become more abundant—a phenomenon known as “the law of diminishing returns.”
Understanding how living organisms grow under various nutritional environments has long been a central question in biology. Across microbes, plants, and animals, growth is shaped by the availability of nutrients, energy, and cellular machinery. While extensive research has explored these limitations, most studies focus only on individual nutrients or specific biochemical reactions, leaving a broader question unanswered: how do complex, interconnected cellular processes collectively regulate growth under constrained conditions?
To address this, a research team consisting of ELSI’s Specially Appointed Associate Professor Tetsuhiro S. Hatakeyama and RIKEN Special Postdoctoral Researcher Jumpei F. Yamagishi has discovered a unifying principle that explains how all living cells regulate growth when resources are limited. Their study introduces the global constraint principle for microbial growth, a concept that could transform how scientists approach the study of biological systems.
For nearly eight decades, researchers have relied on the “Monod equation” in microbiology, formulated in the 1940s, to describe microbial growth. According to the Monod equation, growth rates increase with an increase in nutrients before reaching a stable growth. However, the model assumes that only one nutrient or biochemical reaction restricts microbial growth. In fact, cells carry out thousands of interacting chemical processes, all competing for the same limited resources.
According to the team, the Monod equation captures only part of the picture. Rather than a single bottleneck, cellular growth is shaped by a network of constraints acting together, resulting in the familiar flattening of growth rates, though caused by an entirely different reason. The global constraint principle explains the fact that when one nutrient becomes more abundant, other factors such as enzyme availability, cell volume, or membrane capacity, begin to limit growth. Using a method called “constraint-based modeling” that models how cells manage their resources, the team showed that adding more nutrients always helps microbes grow, but each additional nutrient has a lower effect on growth than the previous one.
“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.”
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
Jumpei F. Yamagishi1,2* and Tetsuhiro S. Hatakeyama3*, Global constraint principle for microbial growth laws, Proceedings of the National Academy of Sciences of the United States of America (PNAS), DOI: 10.1073/pnas.2515031122
Center for Biosystems Dynamics Research, RIKEN, Kobe 650-0047, Japan
Universal Biology Institute, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan
Earth-Life Science Institute, Institute of Future Science, Institute of Science Tokyo, Tokyo 152-8550, Japan
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Earth-Life Science Institute (ELSI) is one of Japan’s ambitious World Premiere International research centers, whose aim is to achieve progress in broadly inter-disciplinary scientific areas by inspiring the world’s greatest minds to come to Japan and collaborate on the most challenging scientific problems. ELSI’s primary aim is to address the origin and co-evolution of the Earth and life.
Institute of Science Tokyo (Science Tokyo) was established on October 1, 2024, following the merger between Tokyo Medical and Dental University (TMDU) and Tokyo Institute of Technology (Tokyo Tech), with the mission of “Advancing science and human wellbeing to create value for and with society.”
World Premier International Research Center Initiative (WPI) was launched in 2007 by Japan's Ministry of Education, Culture, Sports, Science and Technology (MEXT) to foster globally visible research centers boasting the highest standards and outstanding research environments. Numbering more than a dozen and operating at institutions throughout the country, these centers are given a high degree of autonomy, allowing them to engage in innovative modes of management and research. The program is administered by the Japan Society for the Promotion of Science (JSPS).
RIKEN is Japan's largest research institute for basic and applied research. Over 2500 papers by RIKEN researchers are published every year in leading scientific and technology journals covering a broad spectrum of disciplines including physics, chemistry, biology, engineering, and medical science. RIKEN's research environment and strong emphasis on interdisciplinary collaboration and globalization has earned a worldwide reputation for scientific excellence.
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