Generalized Topology in Biology and Chemistry

The current implementation of the Neo-Darwinian model of evolution typically assumes that the set of possible phenotypes is organized into a highly symmetric and regular space equipped with a notion of distance, for example, a Euclidean vector space. Recent computational work on a biophysical genotype-phenotype model based on the folding of RNA sequences into secondary structures suggests a rather different picture. If phenotypes are organized according to genetic accessibility, the resulting space lacks a metric and is formalized by a rather unfamiliar structure. Patterns of phenotypic evolution - such as punctuation, irreversibility, modularity - result naturally from the properties of this space. The classical framework, however, addresses these patterns by exclusively invoking natural selection on suitably imposed fitness landscapes.

We propose to extend the explanatory level for phenotypic evolution from fitness considerations alone to include the topological structure of phenotype space as induced by the genotype-phenotype map. We introduce the mathematical concepts and tools necessary to formalize the notion of accessibility in terms of a generalization of point set topology, relative to which we can speak of continuity in the genotype-phenotype map and in evolutionary trajectories. In the case of mutation as the only genetic operator we obtain pretopologies in a natural way. When recombination is considered, a further generalization becomes necessary.

We connect the factorization of these abstract spaces with the notion of phenotypic character. Based on anecdotal evidence from the RNA model, we conjecture that factorizability is not globally fulfilled, but rather confined to regions where the genotype-phenotype map is continuous. Equivalently, local regions of genotype space on which the map is discontinuous are associated with the loss of character autonomy. This is consistent with the importance of these regions for phenotypic innovation. The intention of the present paper work is to offer a perspective, a framework to implement this perspective, and a few results illustrating how this framework can be put to work.