Weinreich, Daniel M., Suzanne Sindi, and Richard A. Watson. “Finding the boundary between evolutionary basins of attraction, and implications for Wright’s fitness landscape analogy.” Journal of Statistical Mechanics: Theory and Experiment 2013, no. 01 (2013): P01001.
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In 1932 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 to 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.