Exact path-integral representation of the Wright-Fisher model with mutation and selection

The Wright-Fisher model describes a biological population containing a finite number of individuals. In this work we consider a Wright-Fisher model for an outcrossing population, where selection and mutation act at an unlinked locus. The selection acting has a general form, and the locus may have two or more alleles. We determine an exact representation of the time dependent transition probability of such a model in terms of a path integral. Path integrals were introduced in physics and mathematics, and have found numerous applications in different fields, where a probability distribution, or closely related object, is represented as a ‘sum’ of contributions over all paths or trajectories between two points. Path integrals provide alternative calculational routes to problems, and may be a source of new intuition and suggest new approximations. For the case of two alleles, we relate the exact Wright-Fisher path-integral result to the path-integral form of the transition density under the diffusion approximation. We determine properties of the Wright-Fisher transition probability for multiple alleles. We show how, in the absence of mutation, the Wright-Fisher transition probability incorporates phenomena such as fixation and loss.