Researchers have identified a previously unrecognised mathematical property that explains why one of evolutionary biology's most widely used statistical models can produce convincing but incorrect conclusions. They also offer a practical way to recognise and avoid this problem.
Why do some groups of organisms contain thousands of species while others have only a handful? Evolutionary biologists have spent decades trying to answer this question using mathematical models that estimate how biological traits and environmental factors influence the formation and extinction of species.
These models have become a cornerstone of modern biology and have been used in more than a thousand scientific studies. Yet the models carry a known weakness: they can sometimes lead scientists to the wrong conclusions. For years, no one fully understood why.
A puzzle hidden in mathematics
Several years ago, researchers found that many evolutionary models can generate exactly the same observations even when they rest on entirely different assumptions about evolutionary history. This meant that scientists could unknowingly reach different conclusions that were all equally consistent with the same data. Whether the same ambiguity also affected the more sophisticated models used to study how traits shape biodiversity remained unclear, because their mathematics was too complex to analyse directly.
From beetles to apples - and mathematics
Sergei Tarasov at the Finnish Museum of Natural History and Josef Uyeda at Virginia Tech approached the problem from a different angle. Their path to the solution started with a simple but unusual question: imagine three apples - one red, one light green and one dark green. Should the two green apples be grouped together, or treated as different colours? The researchers ran into the same classification puzzle while studying beetle anatomy. Searching for an answer led them to lumpability, a mathematical concept introduced in the 1960s that defines when different states of a Markov model can be safely grouped together without changing how a system behaves. Building on it, they unexpectedly discovered a new way of representing Markov models, one of the most fundamental classes of stochastic models used across science.
They showed that every discrete-state Markov model can be rewritten as an equivalent hidden-state model, a decomposition they call Hidden Expansion. Although the rewritten model looks larger, it is built from simple, identical mathematical components. This representation exposed previously hidden mathematical symmetries and turned an intractable problem into a solvable one.
"This mathematical property had gone unnoticed despite decades of research on Markov models," says Tarasov.
From mathematics back to biology
The new decomposition allowed the researchers to answer the question that had resisted mathematical analysis for years. They showed that the same hidden symmetries also affect the sophisticated models used to study biodiversity, and that the misleading conclusions these models sometimes produce are not isolated statistical mistakes. Instead, they stem from a deeper mathematical ambiguity built into the models themselves.
"For years, researchers could see the symptoms," says Tarasov. "Our work uncovered the underlying mathematical cause. Once we recognised that many fundamentally different evolutionary histories can produce exactly the same observations, it became clear why these methods can sometimes point to a convincing but ultimately incorrect biological explanation."
"A good concrete example comes from our own paper. We reanalysed a published dataset on stick insects (Phasmatodea) that asked whether the evolution of male weapons - leg protrusions used in fights over females - was associated with diversification," Tarasov describes.
"The original study found no evidence that these weapons affected diversification, and we agree with that conclusion. However, when we analysed the same data using a broader range of models, standard statistical methods favoured scenarios suggesting that the weapons did affect diversification. In other words, the same dataset can appear to support a compelling story about a trait driving diversification even when that story is false."
The new framework does not remove the ambiguity entirely, but shows where these misleading results can come from and clarifies the limits of what current methods can tell us.
"We don't claim to have solved the entire problem," says Tarasov. "But we now have a much clearer picture of the problem and where to look for solutions."
When biology inspires mathematics
Scientific discoveries often begin with advances in mathematics that later transform biology. This study followed the opposite path. A biological question about how to represent anatomical traits of beetles led to a new mathematical discovery, which in turn solved a long-standing problem in evolutionary biology.
"It's a wonderful example of biology and mathematics driving each other forward," says Tarasov.
Original article
- Tarasov, S., Uyeda, J. . Nat Commun 17, 9076 (2026).
