The geometry of evolution

Biologists already know lots of reasons to encourage mathematicians to get hooked on biology.  There’s structural biology; population genetics; epidemiology; ecology; bioinformatics; computational neuroscience; and yes, systems biology, to name but a few.  But here is a new one.  How long have we been studying Darwin’s finches?  About 170 years.  In all that time, nobody has noticed that the shapes of beaks of the different species are related by simple geometric transformations.  The divergence of beak shapes that so productively tickled Darwin’s brain can now be traced to a combination of scaling and shearing events, for which some of the molecular mechanisms are known.

The paper describing these insights (Campàs et al. Scaling and shear transformations capture beak shape variation in Darwin’s finches. 2010. Proc Natl Acad  Sci U S A. 107 3356-60) comes from Mike Brenner and colleagues at the School of Engineering and Applied Sciences on Harvard’s Cambridge campus.   What they found was that if you look at the whole set of beaks from Darwin’s finches (13 shapes), you can reduce them to three basic groups.  Within each group, you can map one beak onto another by a simple scaling transformation (stretching).  But scaling transformations are not enough to map one group onto another.  For that, you need shear transformations; the shear is along the axis of beak depth.  Scaling and shear transformations all fall into the same class of mathematical transformations, called the affine group; these are the transformations that preserve lines as lines, with points along them still in the same order, and preserve parallel pairs of lines.   The ability to transform one beak onto another is not just a trivial result of beaks generally looking like beaks: Campàs et al. tried using affine transformations to get the Darwin finch beaks to map onto the beak of a distant relative, the African seedcracker, and couldn’t.

It is not new, of course, for biologists to consider how the shape of one species maps onto the shape of another.  One of the first to do this was D’Arcy Thompson, whose famous book On Growth and Form (1917) contained a chapter comparing the forms of different species, and asking how one could be mapped onto the other by transformations such as scaling.  The whole field of allometry studies relationships between shape and size in different species.  How is this study different?

There are two key differences.  The first is that the groupings derived from geometry are clearly related to phylogeny.  If you look at the phylogenetic tree, organized so that closely related finches are lined up next to each other, the groups of beaks that can be related by scaling transformations fall out as neat blocks.  This means that you can pinpoint exactly where in evolution the changes that result in shear transformations happened.  And that means, at least in principle (the genomes of Darwin’s finches haven’t been sequenced yet, though (trivia) the zebra finch genome has) that you can use genomics to look for what caused the transition.

The second is that the molecular mechanism of these changes in shape is at least partly understood.  Arhat Abzhanov, working with Cliff Tabin in the HMS Department of Genetics, produced two classic papers showing that changes in the expression of bone morphogenetic protein 4 (Bmp4) and calmodulin cause differences in beak morphology; these beak changes can be replicated by upregulating the relevant signal in chicks.  [Abzhanov is now a faculty member at the Department of Organismic and Evolutionary Biology at Harvard, and is a co-author on the Campàs et al. paper.]  Using extra-small CAT scans and in situ hybridizations for one of the geometrical groupings, Campàs et al. showed that the differences in beak scaling were fully reflected in the underlying bone morphology, and that expression of BMP4 in the mesenchyme at the tip of the developing beak in finch embryos correlated well with the eventual shape of the adult beak.

Remember that scaling only accounts for part of the evolutionary diversity seen in these finches, though.  The shearing transformation remains to be explained.  The authors speculate that the shears could result from changes earlier in embryogenesis, for example changes that alter the shape of parts of the skeleton and thus change the relationship between the top and bottom parts of the bill.  Now, for the first time, we have a handle on those changes.  And there may be many other situations in which this new geometrical approach will be able to help tease out the underlying structure of morphological changes across species.

Campas, O., Mallarino, R., Herrel, A., Abzhanov, A., & Brenner, M. (2010). Scaling and shear transformations capture beak shape variation in Darwin’s finches Proceedings of the National Academy of Sciences, 107 (8), 3356-3360 DOI: 10.1073/pnas.0911575107