The transition density function of the Wright-Fisher diffusion describes the evolution of population-wide allele frequencies over time. This function has important practical applications in population genetics, but finding an explicit formula under a general diploid selection model has remained a difficult open problem. In this article, we develop a new computational method to tackle this classic problem. Specifically, our method explicitly finds the eigenvalues and eigenfunctions of the diffusion generator associated with the Wright-Fisher diffusion with recurrent mutation and arbitrary diploid selection, thus allowing one to obtain an accurate spectral representation of the transition density function. Simplicity is one of the appealing features of our approach. Although our derivation involves somewhat advanced mathematical concepts, the resulting algorithm is quite simple and efficient, only involving standard linear algebra. Furthermore, unlike previous approaches based on perturbation, which is applicable only when the population-scaled selection coefficient is small, our method is nonperturbative and is valid for a broad range of parameter values. As a by-product of our work, we obtain the rate of convergence to the stationary distribution under mutation-selection balance.
DIFFUSION processes, which are continuous-time Markov processes with almost surely continuous sample paths, have been successfully applied in various population genetic analyses in the past. Examples include finding the stationary distribution of allele frequencies and approximating fixation times and probabilities (see Karlin and Taylor 1981;Ewens 2004;Durrett 2008 for other applications of diffusion processes). This success is largely due to the fact that the diffusion approximation captures the key features of an evolutionary model while ignoring unimportant details, thereby arriving at a simpler process that facilitates computation. However, when a reasonably complex model of evolution is considered, one is faced with unwieldy equations even under the diffusion approximation. In particular, for Wright-Fisher diffusions with general diploid selection, finding an explicit analytic transition density function, which characterizes the evolution of population-wide allele frequencies over time, has remained a challenging open problem. The diffusion theory allows one to write down a partial differential equation (PDE) satisfied by the transition density, but solving the PDE analytically has proved to be difficult.The transition density has several practical applications, including the following: Recently, there has been growing interest in analyzing samples taken from the same or related populations at different time points. For example, such data arise from experimental evolution of model organisms in the laboratory (e.g., bacteria, see Lenski 2011), from viral/ phage populations (Shankarappa et al. 1999;Wichman et al. 1999), or from ancient DNA (Hummel et al. 2005); see also Bollback et al. (2008) and references therein. In part...