Journal of Mathematical Biology

, Volume 61, Issue 6, pp 877–898 | Cite as

Combinatorial vector fields and the valley structure of fitness landscapes

  • Bärbel M. R. StadlerEmail author
  • Peter F. Stadler
Open Access


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.


Fitness landscape Adaptive walk Barrier tree Combinatorial vector field 

Mathematics Subject Classification (2000)

05C20 90C27 68R10 



We thank Jürgen Jost for his suggestion to consider the relation of landscapes and combinatorial vector fields. This work was supported in part by a grant from the VolkswagenStiftung.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Bärbel M. R. Stadler
    • 1
    Email author
  • Peter F. Stadler
    • 1
    • 2
    • 3
    • 4
    • 5
  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Bioinformatics Group, Department of Computer Science, and Interdisciplinary Center for BioinformaticsUniversity of LeipzigLeipzigGermany
  3. 3.Fraunhofer Institute for Cell Therapy and ImmunologyLeipzigGermany
  4. 4.Institute for Theoretical ChemistryUniversity of ViennaViennaAustria
  5. 5.Santa Fe InstituteSanta FeUSA

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