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.
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Acknowledgments
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.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Stadler, B.M.R., Stadler, P.F. Combinatorial vector fields and the valley structure of fitness landscapes. J. Math. Biol. 61, 877–898 (2010). https://doi.org/10.1007/s00285-010-0326-z
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DOI: https://doi.org/10.1007/s00285-010-0326-z