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MiS Preprint

Combinatorial Vector Fields and the Valley Structure of Fitness Landscapes

Bärbel M. R. Stadler and Peter F. Stadler


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.

Nov 3, 2009
Nov 10, 2009
Fitness landscape, Barrier tree, Adaptive Walk, Combinatorial Vector Field

Related publications

2010 Journal Open Access
Bärbel M. R. Stadler and Peter F. Stadler

Combinatorial vector fields and the valley structure of fitness landscapes

In: Journal of mathematical biology, 61 (2010) 6, pp. 877-898