## Abstract

Some genetic phenomena originate as mutations that are initially advantageous but decline in fitness until they become distinctly deleterious. Here I give the condition for a mutation–selection balance to form and describe some of the properties of the resulting equilibrium population. A characterization is also given of the fixation probabilities for such mutations.

MUTATIONS that change the normal genetic system of an organism may well have an initial selective advantage, but an advantage that deteriorates with time and later transforms into a disadvantage. Polyploids lose chromosomes and become unbalanced, asexuals miss out on the advantage of recombination, mutants that spend less energy on repair find themselves loaded with bad mutations, and so on. How can such mutations be studied and their evolutionary effects understood?

Haldane (1927) showed that an organism that suffers regular mutations with fixed deleterious effects evolves toward a stable mutation–selection balance. Wright and Dobzhansky (1946) introduced the study of nonfixed fitnesses and considered the effects of frequency-dependent fitness values, while Kimura and Ohta (1970) studied advantageous mutations (inversions) that gradually lose their fitness advantage. Here I present results for the population genetics of positive mutations that with time become truly deleterious.

The inheritance system for the considered organism is haploid, but the model is relevant also for the formation of reproductively separated clones of asexuals and polyploids at any ploidy level. Assumptions are kept at a minimum to find general conclusions for the considered class of mutations. Some additional derivations, examples, tables, and figures are given as supporting information.

It is my hope that these results will inspire further study of this type of contradictory but relevant mutations, their expected evolutionary behavior, and how they can be empirically recognized in nature.

## General model

Consider an infinitely large population of haploids. In each generation the standard type, *A*_{0}, changes to the mutant type, *A*_{1}, with probability μ (0 < μ < *A*_{0} is 1 and of *A*_{1} 1 + *s*, where *s* is strictly greater than 0. In general, the mutant type *t* generations after production is denoted *A _{t}* and its fitness is (1 +

*s*)

*f*. To make this a model of deteriorating mutants, it is assumed that

_{t}*f*≥

_{t}*f*for

_{t+1}*t*≥ 1. All

*f*-values are positive and we define

*f*

_{1}as 1.

In a particular generation, let the standard type (*A*_{0}) have a frequency of *x*_{0}, and let mutants of age-class *t* (*A _{t}*) have a frequency of

*x*. The recursion equations describing the relationship between these frequencies can then be written

_{t}If this dynamic system goes to an equilibrium state (*i.e.*, a state where *x _{t}*′ =

*x*for

_{t}*t*≥ 0), then the first of the equations tells us immediately that

*x*-values denote equilibrium frequencies).

The equilibrium frequencies are valid if and only if *A* takes a positive, limited value, and the necessary and sufficient condition for this is that there is a positive number *T* such that*A* will not converge to a limited value and there will be no stable equilibrium state; when condition 6 is fulfilled, on the other hand, the equilibrium frequencies given by 4 are always valid.) The fitness of mutants in age-class *t* is (1 + *s*)*f _{t}*, so another way to express this equilibrium condition is to say that the fitness of the mutant should be smaller than 1 – μ within a limited number of generations after it has been produced. Note that this condition is independent of how strong the initial fitness advantage is and for how long the favorable phase lasts.

The mutational load is, as seen from (3) above, equal to μ, just as in the standard mutation–selection balance for haploids. The average degree of selection against the mutant in the equilibrium population (denoted σ) can be shown to be (see supporting information, section S1, File S1)

The equilibrium distribution of mutant age-classes can be characterized as follows: In “the young mutant phase,” lasting for as long as *f _{t} >* (1 – μ)/(1 +

*s*), subsequent age-classes increase in relative frequency. Then, when

*f*(1 – μ)/(1 +

_{t}=*s*), the frequencies of the age-classes remain unchanged between generations. Finally, in “the old mutant phase,” for which

*f*(1 – μ)/(1 +

_{t}<*s*), the age-classes decline in frequency toward zero. The first two of these phases may be long or short or even missing, but they must be finite in length and they do not intercalate.

Since fitness is 1 – μ during the second mutant phase (if it exists), it follows that for all reasonably smooth distributions of *f*-values the most common mutant age-class(es) will have a fitness that is close to—in many situations indistinguishable from—the fitness of the standard, nonmutant type.

## One-step fitness drop model

So far, the assumptions of the model have been very general. Let us now consider a specific example, built on the idea that newly formed mutations have a high fitness but that this drops drastically after a specified time (a second example based on the idea of a continuous deterioration of fitness is found in section S4, File S1). Or in more formal terms: Let the mutant type retain fitness 1 + *s* until and including generation *T* – 1, when fitness falls to 1 – *z* for all consecutive generations (*z* > μ). In addition, assume that the mutation rate is very small and that (1 + *s*)^{T}^{−1} can be approximated by 1 + (*T* – 1)*s*. Then the average strength of selection against the mutant type becomes*s*, as long as *s* is small relative to *z*. Indeed, if the positive effect of the new mutation is not very great and stays fixed for many generations before fitness plunges to a strongly deleterious state, then the mean selection coefficient (9) becomes close to *T*^{−1}. In this situation, the time structure of the model rather than the relationship between *s* and *z* determines the properties of the equilibrium population. A numerical example of this effect is given as supporting information (see section S2, File S1; and Table S1).

A key assumption in the general model, and also in this investigated special case, is that all changes in mutant fitness occur deterministically in time. The importance of this assumption is shown by the following model extension: Let the time to fitness drop, *T*, be a random variable with a geometric distribution. Denote the mean of the distribution *E(T)*; this value then equals the inverse of the probability for fitness to drop per generation.

Now the condition for the population to move to a mutation–selection balance becomes*s*, and the earlier derived condition for stability that is independent of the quantitative relationship between these parameters and requires only that the drop occurs in finite time. The reason for this discrepancy is that the rate with which mutants drop to their deleterious state in the stochastic case must be large enough to keep pace with their constantly growing number, while no such condition is necessary in the deterministic situation, where *all* positive mutations—irrespective of their number—become deleterious at a specific moment in time. For μ << *s* << 1 condition (10) becomes *E(T)* < *s*^{−1}, which supports this interpretation.

## On probabilities of fixation and fitness estimates

So far the analysis has been made under the assumption of an infinite population size. The important new factor that a finite population size brings to our investigation is the possibility that a mutation of the considered kind may go to fixation. Using simulations I have studied the probability of this occurring. The results can be summarized as follows (see also Table S2 and Table S3): Barring very rare events, a mutation of the considered kind will go to fixation only if it manages to do so before its fitness falls below the normal fitness value. The fixation probability of a new positive mutation is approximately 2*s* (Haldane 1927, based on Fisher 1922), with the mean time to fixation being strongly related to the inverse strength of selection, *s*^{−1}. Thus, mutations of the kind studied here are most likely to become fixed if their initial advantage is strong and this advantage lasts for a sufficiently long time. As expected, in the one-step fitness drop example this implies that the size of *s* becomes much more important for the probability of fixation than the size of *z* (see section S5, File S1; and Figure S1).

In general, positive but deteriorating mutations will be difficult to recognize as such in nature. Above is shown that the most common class of mutants will have fitness close to normal. In addition, the fixation simulations demonstrated that the mean fitness of mutants that ultimately become lost due to genetic drift is not only close to normal but often greater than normal (see section S5, File S1).

However, from expression (8) above is seen that the mean coefficient of selection against mutants can be estimated from the relative frequency of new mutations among mutant newborn (among *N* newborn, μ*N* are new mutants among a total number of (μ/σ)*N* mutant newborn; from this follows that their proportion is σ). Thus, given that parental relationships can be established, this method may function as a way to comprehend more complex fitness relationships for recurrent mutations.

## Acknowledgments

I acknowledge encouragement and help from Torbjörn Säll and Freddy B. Christiansen. Thanks also go to Sofia Adolfsson, who developed an earlier version of the program used for the fixation simulations, and the reviewers. The work has been supported in parts by the Nilsson-Ehle Fund and the Trygger Foundation.

## Footnotes

*Communicating editor: L. M. Wahl*

- Received March 25, 2012.
- Accepted May 11, 2012.

- Copyright © 2012 by the Genetics Society of America

Available freely online through the author-supported open access option.