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

Abstract

Determination of the age of an allele based on its population frequency is a well-studied problem in population genetics, for which a variety of approximations have been proposed. We present a new result that, surprisingly, allows the expectation and variance of allele age to be computed exactly (within machine precision) for any finite absorbing Markov chain model in a matter of seconds. This approach makes none of the classical assumptions (e.g., weak selection, reversibility, infinite sites), exploits modern sparse linear algebra techniques, integrates over all sample paths, and is rapidly computable for Wright-Fisher populations up to N e = 100,000. With this approach, we study the joint effect of recurrent mutation, dominance, and selection, and demonstrate new examples of “selective strolls” where the classical symmetry of allele age with respect to selection is violated by weakly selected alleles that are older than neutral alleles at the same frequency. We also show evidence for a strong age imbalance, where rare deleterious alleles are expected to be substantially older than advantageous alleles observed at the same frequency when population-scaled mutation rates are large. These results highlight the under-appreciated utility of computational methods for the direct analysis of Markov chain models in population genetics.

Haldane’s Probability of Mutant Survival is Not the Probability of Allele Establishment

“…However, it is interesting to note that mildly deleterious alleles take a significantly longer time to establish than do either neutral or strongly deleterious alleles. Similar effects have been observed in computations of expected allele age and times to absorption, and have been attributed to ‘stochastic slowdowns’ under weak selection in the presence of dominance (Mafessoni and Lachmann 2015; de Sanctis et al 2017) and mutation (de Sanctis et al 2017). Notably, we see these effects here even when the mutation rate is zero.…”

“…This reduced model is constructed by truncation of Q , and by setting the second column of R to . With the full transition matrix P ′ thus defined, where we call the corresponding transient-to-transient submatrix Q ′, and the transient-to-absorbing submatrix R ′, we can find the properties of interest by using N ′ = ( I − Q ′), as in equation S3. We can then use N ′ to derive properties such as probabilities and expected times using standard definitions (Krukov et al 2017; de Sanctis et al 2017). The matrix Q ′ (just as matrix Q ) can be based on any parameterization of the underlying model, including with arbitrary mutation, dominance, and selection. To integrate quantities of interest over the likely distribution of starting states c 0 , which can become important when the population mutation rate is not small, we integrate over each state according to the probability of mutation creating 1, 2, 3, … copies in a single generation, starting from zero mutant copies ( i.e.…”

“…, P 0,1 , P 0,2 , …). As in de Sanctis et al (2017) and de Koning and de Sanctis (2018), the summation can be truncated at x terms for x when P 0, x falls below some small threshold ( e.g. , 10 −7 ).…”

“…Here we derive establishment properties for arbitrary discretetime Wright-Fisher models. Detailed discussion of efficient computational solutions to the discrete-time model can be found in Krukov et al (2017); de Sanctis et al (2017); de Koning and de Sanctis (2018).…”

The Rate of Molecular Evolution When Mutation May Not Be Weak

“…S1). Although simple closed-form expressions for the component quantities are not available using diffusion theory, they can be easily calculated directly from the underlying Markov model using efficient computational techniques we recently described (De Sanctis et al, 2017;Krukov et al, 2016).…”