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**Titel:**

**
The topology of the possible:
Formal spaces underlying patterns of evolutionary change
**

**Author(s):**

Bärbel M. R. Stadler,
Peter Stadler,
Günter Wagner,
Walter Fontana

**Submitted to:**

*J. Theor. Biol.*

**Abstract:**

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 an unfamiliar structure, known as
a pretopology. 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 pretopology relative to which we can speak of
continuity in the genotype-phenotype map and in evolutionary
trajectories. We connect the factorization of a pretopology into a product
space with the notion of phenotypic character and derive a condition for
factorization. Based on anecdotal evidence from the RNA model, we
conjecture that this condition 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 is to offer a perspective, a
framework to implement this perspective, and a few results illustrating how
this framework can be put to work. The RNA case is used as an example
throughout the text.

**Keywords:**
Evolution, Genotype-Phenotype Map, Character, Pretopology

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