## Posts Tagged ‘**theoretical evo-devo**’

## Waddington’s canalization revisited

Mark L. Siegal and Aviv Bergman **Waddington’s canalization revisited: Developmental stability and evolution.** PNAS **99**(16):10528-10532 PNAS page pdf

Siegal and Bergman build on the earlier work of A.Wagner (reviewed below), who showed that canalisation in (models of) gene networksÂ may evolve as a by-product of stabilising selection. Recall that in Wagner’s model, a regulatory gene network was represented as a matrix and the phenotype as the stable state of the deterministic, discrete-time dynamical process it encodes.

This setup is retained in the present paper but, crucially, situations where the network does not have (or rather: appears not to have) a stable state are considered as well. This allows the authors to decouple the effect of stabilising selection from that of selection for the existence of the steady state of the network. The result is that, perhaps surprisingly, canalisation can be accounted for by the latter mechanism alone, and therefore is an intrinsic property of stable complex networks regardless of whether their evolution is driven by natural selection.

Siegal’s and Bergman’s model has a number of parameters, most notably the interconnectedness of the network (defined as the number of non-zero entries in the matrix). It turns out that highly connected networks display low initial canalisation, but evolve it rapidly and to a greater extent than relatively sparse ones.

## Does evolutionary plasticity evolve?

Andreas Wagner **Does evolutionary plasticity evolve?** Evolution **50**(3), 1996. pdf

The focus is on epigenetic buffering of mutations, the phenomenon called here (perhaps unfortunately) evolutionary plasticity. With the help of a simple computational model of regulatory networks, Wagner shows that the plasticity can increase when the network’s stable state is put under stabilising selection. This is an indication that stabilising selection can alone explain the canalisation observed in real regulatory networks.

A regulatory network is modelled as a discrete-time dynamical system, which in turn is encoded as a real matrix. The matrix together with an initial state determines the steady state (if any), which is treated as a phenotype. Matrices “evolve” through recombination (swapping rows between pairs of different matrices), mutation (random alteration of entries) and stabilising selection (deviations from the target steady state are punished). Epigenetic stability of such networks was assessed before and after 400 rounds of evolution, and found to have increased significantly in the process. In addition, the evolved networks converge to their stable states much faster.

Apart from the valuable scientific findings, the paper is notable for the dilligence with which Wagner (now heading a successful lab in Zurich) sets up and carries out his experiments. For example, networks and their stable states are chosen *independently*; and stability is assessed with respect to the original mutation constructs *and an additional one*, which was not used during the simulated evolution. While this is perhaps no more than good practice, it is still good to see these measures taken.

## 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, butÂ 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.