*Proceedings of The National Academy of Sciences*, Harvard University researchers say they have demonstrated that simple changes in beak length and depth can explain the important morphological diversity of all beak shapes within the famous genus

*Geospiza*. Broadly, the work suggests that a few, simple mathematical rules may be responsible for complicated biological adaptations.

Using digitization techniques, researchers found that 14 distinct beak shapes, that at first glance look unrelated, could be categorized into three broader, group shapes. Despite the striking variety of sizes and shapes, mathematically, the beaks within a particular group only differ by their scales.

"It is not possible, however, to explain the full diversity of beak shapes of all Darwin's finches with only changes in beak length and depth," explains first author Otger Campàs. "By combining shear transformations (basically, what happens when you transform a square into a rhombus by shoving the sides toward one another), with changes in length and depth, we can then collapse all beak shapes onto a common shape."

*(Photo Credit: Otger Campàs and Michael Brenner, Harvard School of Engineering and Applied Sciences.)*

Using Micro-Computed Tomography (CT) scans on the heads for the different species in the genus Geospiza, researchers verified that the bone structure of the birds exhibits a similar scaling pattern as the beaks.

Thus, beak shape variation seems to be constrained by only three parameters: the depth of the length for the scaling transformation and the degree of shear.

"This is really significant because it means that adaptive changes in phenotype can be explained by modifications in a few simple parameters," adds co-author Ricardo Mallarino. "These results have encouraged us to try to find the remaining molecules responsible for causing these changes."

In fact, the mathematical findings also have a parallel genetic basis. The team explored the role of the two genes responsible for controlling beak shape variation. Bmp4 expression affects width and depth and Calmodulin expression relates to length. It turns out that the expression levels of the two genes, in particular Bmp4, are fundamentally related to the scaling transformations.

"We wanted to know how beaks changed on a fundamental level during evolution of Darwin's finches and how many unique beak shapes we need yet to explain using our developmental genetics approach," says Harvard evolutionary biologist Arhat Abzhanov. "Our joint study demonstrates that we understand the species-level variation really well where scaling transformations match up perfectly with expression and function of developmental genes which regulate precisely such type of change. Now we want to understand how novel beak shapes resulting from higher order transformations evolved in Darwin's finches and beyond."

The finding will help address an idea that Darwin raised nearly 175 years ago in the Voyage of the Beagle: "The most curious fact is the perfect gradation in the size of the beaks in the different species of Geospiza, from one as large as that of a hawfinch to that of a chaffinch, and even to that of a warbler … Seeing this gradation and diversity of structure in one small, intimately related group of birds, one might really fancy that from an original paucity of birds in this archipelago [Galapagos], one species had been taken and modified for different ends."

**Citation**: Campàs et al., 'Scaling and shear transformations capture beak shape variation in Darwin’s finches', PNAS, February 2010; doi: 10.1073/pnas.0911575107

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