SpectralTDF: transition densities of diffusion processes with time-varying selection parameters, mutation rates and effective population sizes

Abstract

Motivation: In the Wright-Fisher diffusion, the transition density function describes the time evolution of the population-wide frequency of an allele. This function has several practical applications in population genetics and computing it for biologically realistic scenarios with selection and demography is an important problem. Results: We develop an efficient method for finding a spectral representation of the transition density function for a general model where the effective population size, selection coefficients and mutation parameters vary over time in a piecewise constant manner. Availability and implementation: The method, called SpectralTDF, is available at https://source forge.net/projects/spectraltdf/.

Allele Age Under Non-Classical Assumptions is Clarified by an Exact Computational Markov Chain Approach

“…For example, Steinruecken et al . 29 recently showed how the transition density function of biallelic Wright-Fisher diffusions 30 could be approximately computed, eliminating the need for a variety of simulations (although allele age has not been considered in this framework). Here we have shown that even the exact computational analysis of biallelic Markov models (including Wright-Fisher models) can be made efficient enough to often eliminate the need for either simulations or diffusion approximations in the first place.…”

The Effects of Population Size Histories on Estimates of Selection Coefficients from Time-Series Genetic Data

“… Steinrücken et al (2015) showed that if the allele frequency density at time s in epoch is given by the expansion where are the coefficients encoding the density at time s in the basis of the eigenfunctions , then at time t in epoch , the allele frequency density is given by , where the coefficients are given by where is the time of the terminating boundary of epoch , and In equation (B.7) , and are given by where is defined in equation (A.13) and for an arbitrary collection of distinct values . In practice, we take to be the Chebyshev nodes ( Steinrücken et al 2015 ). By repeated application of equation (B.6) , it follows that if the coefficients encode the density at time s in epoch , then the coefficients encoding the density …”

“…Instead, we must use a formula for the transition density across multiple epochs of different sizes. Steinrücken et al (2015) showed that if the allele frequency density at time s in epoch is given by the expansion where are the coefficients encoding the density at time s in the basis of the eigenfunctions , then at time t in epoch , the allele frequency density is given by , where the coefficients are given by where is the time of the terminating boundary of epoch , and …”

“…In equation (B.7) , and are given by where is defined in equation (A.13) and for an arbitrary collection of distinct values . In practice, we take to be the Chebyshev nodes ( Steinrücken et al 2015 ).…”

diplo-locus: A lightweight toolkit for inference and simulation of time-series genetic data under general diploid selection

“…One of the most intriguing forces that shape the dynamics of genetic variants is natural or artificial selection, as it reflects how the population adapts to its environment. The Wright-Fisher model and its diffusion approximations are commonly used to model the dynamics of allele frequencies over time, especially when selection is acting, and they have been applied to analyze temporal genetic data ( e.g ., Bollback et al, 2008; Malaspinas et al, 2012; Mathieson and McVean, 2013; Steinrücken et al, 2014; Steinrücken et al, 2016; He et al, 2020). Nevertheless, despite several existing approaches to infer selection parameters, most analyze the data exclusively under a model of additive (semi-dominant) selection, and are often applied to time-series at specific loci, rather than genome-wide.…”