Is evolution doing linear algebra when it creates new species? An analysis by a team of evolutionary biologists and applied mathematicians indicates that may be the case, at least for one famous evolutionary feature: the beaks of Darwin's finches.
Charles Darwin was intrigued by the variety of finches on his visit to the Galápagos Islands in 1835. He noticed that the birds' beaks seemed fine‑tuned to their diet: those with small, pointed beaks tended to feast on insects, for example, while those with stout beaks ate vegetation. The find helped him formulate his theory of evolution by natural selection.
Wondering if there was some sort of mathematical pattern behind the adaptations, a team from Harvard University, including postdoc Otger Campàs and grad student Ricardo Mallarino, analyzed the beak shapes of 13 of Darwin's finches, including six species in the genus Geospiza and three in the genus Camarhynchus. Using carefully digitized profiles from specimens in Harvard's Museum of Comparative Zoology, the researchers then set out to see to what extent linear transformations‑the simplest mathematical functions that can act on geometric objects‑could "collapse" the two-dimensional curves representing the finches' upper beaks onto a common shape.
Mathematically, any two curves are related by some kind of transformation. But linearity imposes severe constraints. In two dimensions, it allows shapes to change only via scaling and shearing. Roughly speaking, a scaling transformation is one that stretches or squeezes two perpendicular axes but keeps them perpendicular, while a shear transformation shoves one axis toward the other, changing the angle between them.
The researchers found that the six Geospiza finches and their ancestor the Black-faced Grassquit matched up with scaling transformations alone. The other 6 specimens also matched up using scaling alone. The two groupings became one via shear, the team reports  online this week in the Proceedings of the National Academy of Sciences.
"We were quite shocked that they all collapsed so well," says study co-author Michael Brenner. "At the moment, this is just empirical," he adds. "We're trying to figure out what it means."
The answer, whatever it turns out to be, likely lies in the details of gene expression. Study co-author Arkhat Abzhanov and colleagues had already established that two proteins are largely responsible for beak shape in Geospiza‑‑calmodulin controls length, while bmp4 affects width and depth‑so "we were expecting that there would be some kind of correspondence between these beaks," Mallarino says. "But the fact that we got such clean levels of correspondence is indeed very surprising." Presumably the same or similar proteins are involved in the second group, but those studies are yet to be done.
Marc Kirschner, a systems biologist at Harvard Medical School in Boston, calls the work "a nice case of applied mathematics being added to qualitative biological observation." It may help explain how Darwin's finches‑and possibly other organisms as well‑were able to adapt so quickly to their environments: If successful changes in phenotype depend only on two or three parameters instead of thousands, he says, "it makes it much more feasible that you can get that much change in a relatively short time."
Charles Stevens, a molecular biologist at the Salk Institute in San Diego, California, who has studied the interplay between evolution and groups of mathematical transformations, agrees that the Harvard group is onto something. "The trick now is to find the conserved pattern formation rules for gene networks that have the property of giving rise to all these groups," he says. "When it's done, this will be a major advance in our understanding of evolutionary mechanisms."