Excerpt

A recurrent topic in evolutionary biology is the problem of the role played by contingency versus the existence of convergence. In other words, we can ask if the repertoire of our biosphere’s complex life forms is one in many or if there are instead strong constraints to the space of the possible. In the first scenario, the vagaries of fluctuating environments, along with the combinatorial possibilities of genomes and gene networks, would allow for an astronomic number of choices. In the latter, fundamental constraints affecting how morphologies can arise from developmental programs would lead to a different, but largely familiar, alternative biosphere. Both alternative paths can create complexity, and a legitimate question is: How can we define cartography of such complexity in terms of a space of the possible? If such a goal was reachable, what would such space look like? What evolutionary lessons could be learned from it? If evolution can freely explore the universe of possible forms over millions of years and under very different environmental conditions, we would expect that this space would have been more or less homogeneously visited (i.e., filled by different possible forms). Is that the case?

In 1966 paleontologist David Raup published a seminal paper that introduced the concept of a space of possible forms, using shell coiling as a case study. The choice made sense for two reasons. First, the shells of a large variety of invertebrates are coiled, and we have an accurate fossil record of them (shells are well preserved). Second, coiling allows us to exploit well-defined mathematical expressions that fully capture the main geometric features. Although a potentially large number of parameters could in principle be required to describe all possible shell morphologies, Raup was able to find three algebraically independent main numbers that successfully allowed him to visualize all theoretical shells within a three-dimensional cube. This in itself was a considerable tour de force that previous studies had tried to address, with limited success. Four dimensions were the smallest number variables chosen by different authors, but the choices were plagued by the problem of lack of orthogonality: it was difficult to choose truly independent dimensions. Raup’s clever solution involved the use of the so-called logarithmic model from which three parameters (W, D, T ) could be numerically estimated from direct measurements. In a nutshell, these parameters capture the impact of three different geometric transformations (including translations and expansions) on the final geometry. In this way, Raup’s work takes advantage of an earlier suggestion by D’Arcy Thompson (1942) that the many forms adopted by some structures in nature are in fact variations of a basic, simple model. Among these simple forms, the so-called logarithmic spiral was used as a minimal representation of coiled shells. This is a self-similar curve that several scholars used as a baseline to further mathematical extensions to account for the full three-dimensional problem.

Bibliography

Alberch, P. 1989. “The Logic of Monsters: Evidence for Internal Constraint in Development and Evolution.” Geobios 22:21–57. https://doi.org/10.1016/S0016-6995(89)80006-3.

Avena-Koenigsberger, A., J. Goñi, R. Solé, and O. Sporns. 2015. “Network Morphospace.” Journal of the Royal Society Interface 12 (103): 20140881. https://doi.org/10.1098/rsif.2014.0881.

Erwin, D. H. 2017. “The Topology of Evolutionary Novelty and Innovation in Macroevolution.” Philosophical Transactions of the Royal Society B: Biological Sciences 372 (1735). https://doi.org/10.1098/rstb.2016.0422.

McGhee, G. R. 2006. The Geometry of Evolution: Adaptive Landscapes and Theoretical Morphospaces. Cambridge, UK: Cambridge University Press.

Niklas, K. J., and V. Kerchner. 1984. “Mechanical and Photosynthetic Constraints on the Evolution of Plant Shape.” Paleobiology 10 (1): 79–101.

Ollé-Vila, A., S. Duran-Nebreda, N. Conde-Pueyo, R. Montañez, and R. Solé. 2016. “A Morphospace for Synthetic Organs and Organoids: The Possible and the Actual.” Integrative Biology 8 (4): 485–503. https://doi.org/10.1039/C5IB00324E.

Thompson, D. W. 1942. On Growth and Form. Cambridge, UK: Cambridge University Press.

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