Wright-Fisher dynamics on adaptive landscape

Adaptive landscape, proposed by Sewall Wright, has provided a conceptual framework to describe dynamical behaviours. However, it is still a challenge to explicitly construct such a landscape, and apply it to quantify interesting evolutionary processes. This is particularly true for neutral evolution. In this work, the authors study one-dimensional Wright Fisher process, and analytically obtain an adaptive landscape as a potential function. They provide the complete characterisation for dynamical behaviours of all possible mutation rates under the influence of mutation and random drift. This same analysis has been applied to situations with additive selection and random drift for all possible selection rates. The critical state dividing the basins of two stable states is directly obtained by the landscape. In addition, the landscape is able to handle situations with pure random drift, which would be non-normalisable for its stationary distribution. The nature of non-normalisation is from the singularity of adaptive landscape. In addition, they propose a new type of neutral evolution. It has the same probability for all possible states. The new type of neutral evolution describes the non-neutral alleles with 0%. They take the equal effect of mutation and random drift as an example.