The geometry of evolution

July 21, 2010 § 3 Comments

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

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§ 3 Responses to The geometry of evolution

  • Marc Kirschner says:

    While you are thinking about evolution and mathematics, you may want to check out a paper from Uri Alon’s group in PLoS Computational biology (2008): Parter et al., Facilitated variation: how evolution learns from past environments to generalize new environments. They deal with the question of the internalization of mechanisms that have been used in the past that can now be readily used in new environments. This kind of thinking may explain why it is so easy to generate new shapes of bird beaks by tweaking a very small number (in the case of Darwin Finches, just two) parameters. The Parter et al. mathematical paper is a computer model and hence not mapped directly to a biological problem like bird beak evolution but it has the strength that it is not just a post hoc explanation. It is a model that could have given very different results using other principles. The ability to compare results based on different assumptions is easy in mathematics but in biology if the principles are general and deeply embedded in all biological systems it may be hard to find counter examples. Both this and the Campas et al. paper show that modeling and mathematical analysis of phenotypic variation (as opposed to genotypic variation commonly studied in population genetic models) has an important place in evolutionary theory.

  • […] Ward at It Takes 30: The geometry of evolution. Becky describes how our understanding of Darwin’s finches, 170 years in the making, is still […]

  • Carol Morton says:

    For pictures of Abzhanov on the Galapagos Islands and the famous beaks, see this story about the classic papers and the origins of the research question. http://focus.hms.harvard.edu/2006/090106/genetics.shtml

    Yesterday, Abzhanov updated attendees of the medical genetics short course at Jackson Laboratory.

    Beak shape is established by two distinct development modules — prenasal cartilage and premaxillary bone, respectively. The timing and strength of exposure to calmodulin (to set length) and to bone morphogenetic protein 4 (to sculpt width and depth) exert their influence early on the cartilage. In the latest paper, scaling (but not shearing) correlated perfectly with Bmp4 activity on the cartilage.

    “The changes are not random,” he said. “There is a strict mathematical logic, which probably reflects the physics and chemistry by which they occur.” His group is seeking the molecular correlates of the shearing.

    In other work on the second module, Abzhanov has found, other genes more directly mold the premaxillary bone development at a later stage. This bone is now better known as the upper beak, but it likely began its evolutionary journey as the snout section in the upper jaw of the dinosaur.

    This growing knowledge of the potential mechanisms of cranial skeletal development may help people. Abzhanov is working with cranio-facial surgeons at Children’s Hospital Boston and Massachusetts General Hospital to understand similar mechanisms in human cells and mouse models to aid diagnosis and treatment methods for babies born with cranio-facial abnormalities that can affect normal development. See his website for more information.

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