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Finding the boundary between evolutionary basins of attraction, and implications for Wright’s fitness landscape analogy

Finding the boundary between evolutionary basins of attraction, and implications for Wright’s fitness landscape analogy
Finding the boundary between evolutionary basins of attraction, and implications for Wright’s fitness landscape analogy
In 1932 Sewall Wright introduced the notion of the fitness landscape. By analogy with a physical landscape, whose gradient predicts a rolling marble’s spatial trajectory, the contours of the fitness landscape are meant to predict an evolving population’s genetic trajectory. Wright’s chief interest was in the possibility that mutational interactions might frustrate natural selection, giving rise to multiple maxima on the fitness landscape. Here we study a dynamical system over the state space defined by allele frequencies and linkage disequilibria between alleles. We first analytically locate the saddle between basins of attraction in infinite-sized populations evolving under the influence of selection and recombination for the simplest two-locus case. We further show numerically that the boundary between basins is approximately linear with respect linkage disequilibrium, though not allele frequency. We also employ this framework to develop novel perspectives on two venerable results for single-peaked fitness landscapes. Finally we sought the potential function whose contours would predict evolutionary trajectories through this state space. Importantly not every dynamical system can be described by a potential function, and the present problem is provably one such case. Thus in the parlance of Wright’s analogy, in locating the floor of the fitness valley we have lost the landscape, and this conclusion is not limited to our choice of parameterization, nor of problem. This result motivates us to carefully review the formal implications and requirements of this widely used analogy.
1742-5468
523-536
Weinreich, Daniel M.
a5522cf4-5a31-4e13-9934-e3529fdd4d6f
Sindi, Suzanne
0b3cd87d-a5f0-44d5-b1ee-74aafda82d0c
Watson, Richard A.
ce199dfc-d5d4-4edf-bd7b-f9e224c96c75
Weinreich, Daniel M.
a5522cf4-5a31-4e13-9934-e3529fdd4d6f
Sindi, Suzanne
0b3cd87d-a5f0-44d5-b1ee-74aafda82d0c
Watson, Richard A.
ce199dfc-d5d4-4edf-bd7b-f9e224c96c75

Weinreich, Daniel M., Sindi, Suzanne and Watson, Richard A. (2012) Finding the boundary between evolutionary basins of attraction, and implications for Wright’s fitness landscape analogy [in special issue: Statistical Mechanics and the Dynamics of Evolution] Journal of Statistical Mechanics Theory and Experiment, pp. 523-536.

Record type: Article

Abstract

In 1932 Sewall Wright introduced the notion of the fitness landscape. By analogy with a physical landscape, whose gradient predicts a rolling marble’s spatial trajectory, the contours of the fitness landscape are meant to predict an evolving population’s genetic trajectory. Wright’s chief interest was in the possibility that mutational interactions might frustrate natural selection, giving rise to multiple maxima on the fitness landscape. Here we study a dynamical system over the state space defined by allele frequencies and linkage disequilibria between alleles. We first analytically locate the saddle between basins of attraction in infinite-sized populations evolving under the influence of selection and recombination for the simplest two-locus case. We further show numerically that the boundary between basins is approximately linear with respect linkage disequilibrium, though not allele frequency. We also employ this framework to develop novel perspectives on two venerable results for single-peaked fitness landscapes. Finally we sought the potential function whose contours would predict evolutionary trajectories through this state space. Importantly not every dynamical system can be described by a potential function, and the present problem is provably one such case. Thus in the parlance of Wright’s analogy, in locating the floor of the fitness valley we have lost the landscape, and this conclusion is not limited to our choice of parameterization, nor of problem. This result motivates us to carefully review the formal implications and requirements of this widely used analogy.

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Accepted/In Press date: March 2012
Organisations: Agents, Interactions & Complexity

Identifiers

Local EPrints ID: 336423
URI: http://eprints.soton.ac.uk/id/eprint/336423
ISSN: 1742-5468
PURE UUID: f727f03f-306f-4775-9dd3-dc186889134f

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Date deposited: 26 Mar 2012 10:45
Last modified: 18 Jul 2017 06:08

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Contributors

Author: Daniel M. Weinreich
Author: Suzanne Sindi

University divisions

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