

Preprint 65/2009
Combinatorial Vector Fields and the Valley Structure of Fitness Landscapes
Bärbel M. R. Stadler and Peter F. Stadler
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Submission date: 03. Nov. 2009
Pages: 21
published in: Journal of mathematical biology, 61 (2010) 6, p. 877-898
DOI number (of the published article): 10.1007/s00285-010-0326-z
Bibtex
Keywords and phrases: Fitness landscape, Barrier tree, Adaptive Walk, Combinatorial Vector Field
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Abstract:
Adaptive (downhill) walks are a computationally convenient way of analyzing
the geometric structure of fitness landscapes. Their inherently stochastic nature has limited
their mathematical analysis, however. Here we develop a framework that interprets adaptive
walks as deterministic trajectories in combinatorial vector fields and in return associate these
combinatorial vector fields with weights that measure their steepness across the landscape.
We show that the combinatorial vector fields and their weights have a product structure
that is governed by the neutrality of the landscape. This product structure makes practical
computations feasible. The framework presented here also provides an alternative, and
mathematically more convenient, way of defining notions of valleys, saddle points, and barriers
in landscape. As an application, we propose a refined approximation for transition rates
between macrostates that are associated with the valleys of the landscape.