Quasi-independence, homology and the unity of type: a topological theory of characters.

In this paper Lewontin's notion of "quasi-independence" of characters is formalized as the assumption that a region of the phenotype space can be represented by a product space of orthogonal factors. In this picture each character corresponds to a factor of a region of the phenotype space. We consider any region of the phenotype space that has a given factorization as a "type", i.e. as a set of phenotypes that share the same set of phenotypic characters. Using the notion of local factorizations we develop a theory of character identity based on the continuation of common factors among different regions of the phenotype space. We also consider the topological constraints on evolutionary transitions among regions with different regional factorizations, i.e. for the evolution of new types or body plans. It is shown that direct transition between different "types" is only possible if the transitional forms have all the characters that the ancestral and the derived types have and are thus compatible with the factorization of both types. Transitional forms thus have to go over a "complexity hump" where they have more quasi-independent characters than either the ancestral as well as the derived type. The only logical, but biologically unlikely, alternative is a "hopeful monster" that transforms in a single step from the ancestral type to the derived type. Topological considerations also suggest a new factor that may contribute to the evolutionary stability of "types". It is shown that if the type is decomposable into factors which are vertex irregular (i.e. have states that are more or less preferred in a random walk), the region of phenotypes representing the type contains islands of strongly preferred states. In other words types have a statistical tendency of retaining evolutionary trajectories within their interior and thus add to the evolutionary persistence of types.