Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry

Abstract

The search spaces in combinatorial chemistry as well as the sequence spaces underlying (molecular) evolution are conventionally thought of as graphs. Recombination, however, implies a nongraphical structure of the combinatorial search spaces. These structures, and their implications for search process itself, are heretofore not well understood in general. In this contribution we review a very general formalism from point set topology and discuss its application to combinatorial search spaces, fitness landscapes, evolutionary trajectories, and artificial chemistries.

Analysis of a complete DNA–protein affinity landscape

“…With the absence of these data, investigations on landscape properties have been based predominantly on model systems such as spin glasses, NK-landscapes (Kauffman & Levin 1987) and perhaps the more biologically pertinent RNA models (Schuster et al 1994). Less associated (but equally applicable) with biological fitness landscapes are landscape studies considering combinatorial chemical space (Stadler & Stadler 2002), where rules associated with neighbourhood and fitness can be directly applied to guiding chemical synthesis. While mapping the entire sequence space of an average protein is intractable owing to the 'hyper-astronomical' number of variants (Voigt et al 2000), even limited sampling of the sequence space has previously been prohibitive because of the cost of DNA sequencing and (especially) of synthesis.…”

The Topology of Evolutionary Biology

“…Related approaches to defining saddle points are discussed in [25]. The topological approach to fitness landscapes is largely unexplored.…”

“…Population genetics theory typically assumes that the set of possible phenotypes is organized into a highly symmetric and regular space equipped with a notion of distance; most conveniently, a Euclidean vector space [1]. Computational studies using an explicit genotype-phenotype model based on the RNA folding, however, suggest a quite different picture [13,17,22]: If phenotypes are organized according to genetic accessibility, the resulting space lacks a metric and is conveniently formalized by an unfamiliar structure that generalizes topological spaces [23][24][25].…”

A prime factor theorem for a generalized direct product

“…[3,1]. To this end one defines the closure c(A) of a set of organisms A as the set of organisms that can be obtained from A by application of the genetic operators [7,6].…”