Mutation–Selection Balance: Ancestry, Load, and Maximum Principle

Abstract

We analyze the equilibrium behavior of deterministic haploid mutation-selection models. To this end, both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is called the ancestral distribution, which turns out as a key for the study of mutation-selection balance. We find that the ancestral genotype frequencies determine the sensitivity of the equilibrium mean fitness to changes in the corresponding fitness values and discuss implications for the evolution of mutational robustness. We further show that the difference between the ancestral and the population mean fitness, termed mutational loss, provides a measure for the sensitivity of the equilibrium mean fitness to changes in the mutation rate. The interrelation of the loss and the mutation load is discussed. For a class of models in which the number of mutations in an individual is taken as the trait value, and fitness is a function of the trait, we use the ancestor formulation to derive a simple maximum principle, from which the mean and variance of fitness and the trait may be derived; the results are exact for a number of limiting cases, and otherwise yield approximations which are accurate for a wide range of parameters. These results are applied to threshold phenomena caused by the interplay of selection and mutation (known as error thresholds). They lead to a clarification of concepts, as well as criteria for the existence of error thresholds.

An asymptotic maximum principle for essentially linear evolution models

“…For the two-state model discussed above, x ∈ [0, 1] is simply the fraction of mutated sites relative to the reference sequence. (In population genetics, the infinite sites limit N → ∞ at constant i (and hence i/n → 0) is more familiar; for a discussion of how this relates to the limiting procedure here, see [26] and [5]). For models with a nucleotide alphabet, x ∈ [0, 1] 3 tells us at which fraction of the sites there is a replacement of the reference letter by one of the three other nucleotides (in a suitable encoding).…”

“…In this form, Q is the generator of the backward process on the stationary distribution as described in [30,Corollary 1] for general multitype branching processes, and used in [26] in the context of mutation-selection models. Loosely speaking, Q describes the Markov chain which results from picking individuals randomly from the stationary distribution h and following their lines of descent backward in time.…”

“…In [26], this high-dimensional maximization could be reduced to a scalar one for special choices of M and R.…”

“…However, in the limit as N → ∞, uniqueness may be lost, which is also reflected by the fact that the supremum in (25) may be assumed at more than one point. It is these degenerate situations where error thresholds may occur [26].…”

“…(See [2] or [32] for the general context of multitype branching processes, and [26] for the application to mutation-selection models.) Important first questions concern the asymptotic properties of the systems discussed.…”