TBI 02-04-015
A Mathematical Theory of Characters
## 02-04-015

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Abstract:

Quasi-Independence, Homology and the Unity of Type:

A Topological Theory of Characters

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G.P. Wagner, P.F. Stadler

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 continuity 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.
**Keywords:**
Quasi-Independence,
Characters,
Homology,
Evolutionary Innovation,
Body Plans,
Generalized Topology,
Product Spaces

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