## An end to endless forms

Elhanan Borenstein and David C. Krakauer **An end to endless forms: Epistasis, phenotype distribution bias and non-uniform evolution.** PLoS Comp. Bio. **4**(10), 2008. pdf

The paper analyses a simple model of development: the space 2^n of binary vectors (genotypes) mapped to the space 2^k of binary vectors (phenotypes; k>=n) by a linear transformation coupled with a heaviside function. More precisely, a genotype *g* is mapped to its corresponding phenotype *p* by the formula

*p = H(D(g))*

where *D* is a *n*x*k* matrix whose entries belong to {-1,0,1}, and H(x) is zero when x<0 and 1 for x>=0.

The model recreates the well-known result of the RNA folding studies [1]: the development map is highly degenerate, i.e. there are many genotypes mapped to the same phenotype ,and the distribution of degeneracy levels follows a power law. However, unlike the RNA folding framework, this model considers phenotypes which are not images of any genotype. It is therefore possible to talk about the fraction of realised phenotypes (called *visible* phenotypes in the paper). Quite as could be expected, it turns out that this fraction is very low, even when measured against 2^n rather than 2^k. The authors vary various properties of their model, such as sparseness of D, but the results remain reasonably robust. The last part of the paper explores the dynamics of neutral evolution of such models, the main result being that increase in the size of D reveals (in absolute, not relative terms) more phenotypes, but instead of founding new islands of visible phenotypes, they seem to chart preexisting ones with more and more resolution.

This is a very well written, engaging and important paper. It validates the theoretical evo-devo work on RNA, but the setting used is more general and thus provides more general explanations of the causes and properties of the degeneracy of the genotype-phenotype mapping. It would be interesting to see an analysis of the neutral spaces of these models, or, more generally, what an evolutionary meaningful distance function of the development matrices induces on the morphospace.

[1] P. Schuster, W. Fontana, P.F. Stadler and I.L.Hofacker *From Structures to Shapes and Back: a case study in RNA secondary structures*. Proc. Biol. Sci. 255:279-284.

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