A team of researchers, including a scientist from the Earth-Life Science Institute (ELSI) at the Institute of Science Tokyo in Japan, has identified a new principle in biology that mathematically explains why the growth of living organisms slows down when nutrients become plentiful. This well-known phenomenon is referred to as the “law of diminishing returns.”

How organisms grow in response to changing nutrient conditions has long been one of biology’s central questions. Across all forms of life – from microbes to plants and animals – growth depends on access to nutrients, energy, and the internal machinery of cells. Although scientists have studied how these factors affect growth, most research has focused on individual nutrients or specific biochemical pathways. What has remained unclear is how all these interconnected processes within a cell work together to control growth when resources are limited.

A Global Principle That Unites Living Systems

To explore this mystery, ELSI’s Specially Appointed Associate Professor Tetsuhiro S. Hatakeyama and RIKEN Special Postdoctoral Researcher Jumpei F. Yamagishi discovered a new unifying concept that describes how all living cells manage growth under resource constraints. Their work introduces what they call the global constraint principle for microbial growth – a framework that could reshape how scientists understand biological systems.

Since the 1940s, microbiologists have relied on the “Monod equation” to describe how microbes grow. This model shows that growth rates increase with added nutrients until they level off. However, the Monod equation assumes that only one nutrient or biochemical reaction limits growth at a time. In reality, cells perform thousands of simultaneous chemical processes that must share finite resources.

A Network of Constraints Inside Every Cell

According to Hatakeyama and Yamagishi, the traditional model captures only a small part of what’s happening. Instead of a single bottleneck, cellular growth is shaped by a complex network of limitations that interact to slow growth as nutrients accumulate. The global constraint principle explains that when one limiting factor – such as a nutrient – is alleviated, other constraints like enzyme production, cell volume, or membrane space begin to take over.

Using a technique known as “constraint-based modeling,” the team simulated how cells distribute and manage internal resources. Their results showed that while each additional nutrient helps microbes grow, its benefit gradually decreases – each one contributes less than the last.

“The shape of growth curves emerges directly from the physics of resource allocation inside cells, rather than depending on any particular biochemical reaction,” explains Hatakeyama.

Uniting Classic Laws of Biology

This new principle brings together two of biology’s foundational growth laws: the Monod equation and Liebig’s law of the minimum. Liebig’s law states that a plant’s growth is limited by whichever nutrient is scarcest (for example, nitrogen or phosphorus). Even if all other nutrients are plentiful, the plant can only grow as much as the least available one allows.

By merging these two concepts, the researchers created what they call a “terraced barrel” model. In this model, new limiting factors appear in stages as nutrient availability increases. This explains why organisms – from single-celled microbes to complex plants – experience diminishing growth returns even when conditions seem ideal, as each new stage reveals a fresh constraint.

Hatakeyama compares this to an updated version of Liebig’s famous barrel analogy, in which a plant’s growth is limited by its shortest stave, representing the scarcest resource. “In our model, the barrel staves spread out in steps,” he says, “each step representing a new limiting factor that becomes active as the cell grows faster.”

To test their hypothesis, the researchers built large-scale computer models of Escherichia coli bacteria. These models incorporated details about how cells use proteins, how crowded they are inside, and the physical limits of their membranes. The simulations accurately predicted the observed slowing of growth as nutrients were added and showed how oxygen and nitrogen levels affected the results. Laboratory experiments confirmed that the model’s predictions matched real biological behavior.

Toward Universal Laws of Life’s Growth

The discovery offers a new way to understand how life grows, without the need to model every molecule or reaction in detail. The global constraint principle provides a framework that unifies many aspects of biology. “Our work lays the groundwork for universal laws of growth,” says 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.”

This principle could have far-reaching applications. It may lead to more efficient microbial production in biotechnology, improved crop yields through better nutrient management, and stronger models for predicting how ecosystems respond to climate change. Future research may explore how this principle applies to different types of organisms and how multiple nutrients interact to influence growth. By bridging cellular biology with ecological theory, this study moves science closer to a universal framework for understanding life’s growth limits.

Earth-Life Science Institute (ELSI) is one of Japan’s prominent World Premiere International (WPI) research centers. It aims to drive breakthroughs in interdisciplinary science by attracting top researchers from around the world to collaborate on challenging scientific problems. ELSI’s mission focuses on studying the origin and co-evolution of Earth and life.

The Institute of Science Tokyo (Science Tokyo) was established on October 1, 2024, through the merger of Tokyo Medical and Dental University (TMDU) and Tokyo Institute of Technology (Tokyo Tech). Its mission is “Advancing science and human wellbeing to create value for and with society.”

Japan’s World Premier International Research Center Initiative (WPI), launched in 2007 by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), supports a network of elite research centers that operate with a high degree of independence and global collaboration. The program is managed by the Japan Society for the Promotion of Science (JSPS).

RIKEN, Japan’s largest research institute for basic and applied science, produces over 2,500 papers each year in leading journals across physics, chemistry, biology, engineering, and medicine. Known for its interdisciplinary and international approach, RIKEN has earned a worldwide reputation for scientific excellence.