The equilibrium solution for mutation-selection balance can be characterized simply when all mutations have equivalent effects. Let Q denote the average number of deleterious mutations per gamete. Q is the product of the number of loci (L) and the frequency of the deleterious allele at each locus (q). While Q is not measured directly in most experimental studies, estimable measures of genetic load are simple functions of this variable. For example, numerous studies have estimated the ‘inbreeding load’, B, which is the slope of the regression of -Ln[fitness] onto f, the inbreeding coefficient of individuals (Morton et al., 1956; Charlesworth and Charlesworth, 1987). The average Ln[fitness] of outbred individuals is –2Qhs while the corresponding value for fully inbred (homozygous) individuals is –Qs. This implies that B = Qs (1 − 2 h), i.e. the inbreeding load is a linear function of Q.

A closed-form equation for Q can be obtained if the population is random mating (fully outcrossing) or reproduces exclusively by either self-fertilization or asexual means (this theory is reviewed in section 3 below). However, many hermaphroditic plants and animals have mixed mating systems (Vogler and Kalisz, 2001; Jarne et al., 2000). The basic mixed-mating model stipulates that progeny are produced by outcrossing with probability t, and by self-fertilization with probability 1− t. Kondrashov (1985), abbreviated K85, introduced an explicit multi-locus model for mutation-selection balance in mixed mating populations. This deterministic model retains the assumption that loci are unlinked, but still allows associations among loci which can be substantial with mixed mating (Lande et al., 1994).

Charlesworth and Charlesworth (1992) generalized K85 to allow selection on male gametes. In plants, there is substantial overlap between genes expressed in pollen grains and those expressed in the sprorophyte (Willing and Mascarenhas, 1984; Honys and Twell, 2004). While selection might also act on female gametes, pollen tube competition provides clear opportunity for selection on male gametes and this process has garnered most empirical attention (Mulcahy et al., 1996). Gametic selection reduces genetic load, and because dominance is irrelevant for haploid pollen genotypes, it alters the relationship between equilibrium load and mating system.

Let X denote the number of heterozygous loci per zygote and Y denote the number of loci that are homozygous for the deleterious allele. The equivalence of mutations in this deterministic model implies that allele frequencies are the same across loci (deleterious alleles are equally rare). If we then stipulate that loci are independent, it follows that X and Y should be Poisson distributed. This stipulation is correct if the population is either fully outcrossing (t = 1) or fully selfing (t = 0) and allows direct calculation of Q (Crow, 1970; Heller and Maynard Smith, 1978, 1979; Charlesworth et al., 1990a; Section 3). As noted by Charlesworth et al. (1991) however, the number of deleterious mutations per gamete will not be Poisson with mixed mating. The population of gametes that merge to form outbred zygotes is a mixture of contributions from outbred and inbred adults. The mean number of mutations per gamete will differ between these groups owing to differences in the effectiveness of selection on outbred vs. inbred genotypes. The consequence is a subtle form of linkage disequilibrium within the outbred zygote population and the variance in mutation count will slightly exceed the mean.

The other major complication associated with mixed mating is that individuals will vary in the extent to which they are inbred. A fraction t of zygotes are outbred, while the remainder are inbred to varying extents (f = ½, ¾, 7/8, etc.). This variation causes identity disequilibria, genotypic associations among loci even when these loci are unlinked (Haldane, 1949; Bennett and Binet, 1956). However, if we conceptually subdivide a population into ‘cohorts’ of individuals that share the same inbreeding coefficient (e.g. Campbell, 1986; Kelly, 1999a, b), variation in inbreeding history can be accommodated (see Figure 1). Identity disequilibrium is absent within cohorts, at least as long as the relevant loci are unlinked.

The Inbreeding History Model (IHM) is a set of recursions predicting changes in the joint distribution of X and Y across an infinite array of cohorts. Let M_X(i) and M_Y(i) denote the means of X and Y among zygotes in cohort i. Q is an average across cohorts:

where K_i is the fraction of zygotes in cohort i. The cohort proportions are determined by both mating system and selection. The outbred fraction is fixed (K₀ = t) while the inbred cohort fractions satisfy the following simple recursion:

where _i is the mean fitness within cohort i and is the overall mean fitness.

We need to specify the joint distribution of X and Y within each cohort to determine how selection changes M_X(i) and M_Y(i). As noted above, linkage disequilibria will make mutation counts non-Poisson within the outbred cohort, and these associations persist in selfed progeny. However, the deviation from the Poisson (within cohorts) must usually be quite modest, and for this reason, we assume that the intra-cohort distribution of X and Y is nearly-Poisson:

where = [1 + ε₁λ_x (1 + λ_x ) + ε₂λ_y (1 + λ_y) + ε₃λ_x λ_y] is a normalization factor. The quantities λ_x, λ_y, ε₁, ε₂, and ε₃ are cohort specific parameters. Equation 3 converges on the bivariate Poisson as ε₁, ε₂, and ε₃ vanish and I will assume these quantities are small (and thus that the distribution of X and Y remains nearly Poisson within cohorts). Higgs (1994) used the univariate version of equation 3 to study the distribution of deleterious mutations per gamete with synergistic selection.

In principle, one could develop recursions for λ_x, λ_y, ε₁, ε₂, and ε₃ under the joint action of selection, mutation, and outcrossing/selfing. However, because the effects of mutation, outcrossing, and selfing can be described directly in terms of the distribution moments without regard to the underlying distribution, it is natural to seek recursions for the moments. To first order in ε₁, ε₂, and ε₃, the first and second central moments of this distribution are:

where V_X is the variance in X, V_Y is the variance in Y, and C_XY is the covariance of X and Y. I have suppressed the subscripts here, but these moments and the recursions outlined below apply distinctly to each cohort.

The fitness of a zygote with X heterozygous mutations and Y homozgotes is (1 − hs)^X (1 − s)^Y, where h is the dominance coefficient and s is the selection coefficient (Haldane 1927). K85 requires that mutations are not completely recessive (h > 0) and thus remain sufficiently rare so as to invariably appear as heterozygotes in outbred zygotes. I retain this assumption here. To first order in ε₁, ε₂, and ε₃, the predicted moments within cohorts after selection are

After mutation within each cohort, and , while the other moments remain unaltered: , , and . Here, U is the genomic deleterious mutation rate, a sum of locus specific (diploid) rates across all loci subject to mutation.

Pollen-level selection can occur during both outcrossing and selfing. The critical difference: with selfing, grains are competing against siblings. Let s_p denote the selective effect of a deleterious mutation in pollen grains. Allowing haploid selection (with the two alleles initially equally frequent) on male but not female gametes, a heterozygous locus in the parent will segregate to a homozygous mutant with probability (1 − s_p)/(4 − 2s_p), to a homozygous wild-type with probability 1/(4 − 2s_p), and to a heterozygote with probability ½. Noting that homozygous genotypes are directly transmitted to progeny and that the selfed progeny of cohort i constitute cohort i +1 in the next generation (Figure 1), we find that

The outbred cohort in the next generation is formed by random union of gametes recruited from adults across the entire population. Let Z denote the number of mutations per gamete prior to any gametic selection. Taking appropriate averages over cohorts, the mean (M_Z) and variance (V_Z) of Z are

Applying selection to the population of pollen grains and combining pollen grains with (unselected) female gametes:

And by assumption, M_{Y (0)} = V_{Y (0)} = C_{XY (0)} = 0. The rightmost term in equations 8 is based on the univariate version of equation 3.

Finally, we need to specify the mean fitness of each cohort. To first order in ε₁, ε₂, and ε₃,

which exactly matches iterative results from the generalized K85 with t = 1.0 (see also the Appendix of Charlesworth and Charlesworth, 1992).

Now consider a completely selfing population (t = 0). Heller and Maynard Smith (1978) showed that, in the absence of gametic selection, X and Y are independent Poisson random variables (see also Hopf et al., 1988). Thus, the respective means after zygotic selection are and . Mutation adds a Poisson distributed value (mean U) to X. Finally, segregation of alleles in heterozygotes will occur with the production of selfed progeny. If there is no gametic selection, the predicted means in the next generation of zygotes are

and

Solving ( and ) yields the following equilibria:

and Q = M_Y + ½ M_X. These predictions are also confirmed by iteration of K85 with s_p = 0 and t = 0.

Why do OC74 and K85 differ for complete selfing (Figure 3)? The OC74 recursions for a fully selfing population can be re-written in notation of this paper:

and

Solving, M_X = 2U/(1 + hs) and M_Y = U (1 − hs)/(2s(1 + hs)). The predicted number of heterozygotes is greater while homozygotes are fewer than predicted by equations 12.

Equations 13 imply that all new mutations appear first as heterozygotes. In contrast, equations 11 imply that a fraction of new mutations will appear in their first zygote as homozgyotes. If deleterious mutations are at least partially recessive (h < 0.5), OC74 predicts a higher equilibrium load because new mutations are partially shielded from selection during their first full generation. This rather subtle effect can be quantitatively important when the average lifespan of mutations is short. For the purpose of comparisons, it is useful to re-derive the single locus model using the K85 life cycle. For arbitrary t, the equilibria are:

and

where Ω = s(1 − t) + 2t. When Q from equations 14 is plotted onto Figure 2, the trajectory is consistently lower than OC74, matching K85 at both extremes (t = 0 and 1).

Interestingly, gametic selection changes the mean of Y, but not of X. However, their joint distribution remains Poisson when t = 0. Applying the same methods, when s_p > 0, equation 11b becomes

Noting that and solving,

The IHM also provides insight about the process of multi-locus, mutation-selection balance. Consider selective interference, which is responsible for the large difference between single and multi-locus predictions in Figure 2. This effect is often associated with deleterious mutations of large effect because Lande et al. (1994) focused on embryonic lethals in their study of interference. Their parameter sets were clearly motivated by empirical studies, but it is worth noting that selective interference is not a consequence of high selection coefficients. Instead, the key property of mutations is their recessivity. Consider a population with U = 0.5, h = 0.02, and t = 0.7. If mutations are lethal (s = 1), the IHM predicts Q = 11.98. This is about seven times greater than the single locus prediction (OC74). However, with the same parameters except s = 0.1, the IHM predicts Q = 121.13 which is nine times greater than predicted by OC74. Thus, selective interference becomes (proportionally) more important as s declines.

Scofield and Schultz (2006) hypothesize that selective interference is responsible for the rarity of highly selfing, long-lived plants. It is thus important to determine the region of parameter space over which this process has substantial effects. Lethals are known to be highly recessive (h = 0.01–0.02), but it is unclear that they occur at a sufficient rate to generate the mutational load necessary for interference. The estimated lethal mutation rate is only a few percent per generation in Drosophila (Simmons and Crow, 1977). While the rate could be somewhat higher in long-lived plants (Klekowski and Godfrey, 1989), it seems likely that most inbreeding depression is due to the aggregate effects of mutations with smaller effects. While it is commonly stated that mildly deleterious mutations are likely to be ‘nearly additive’, estimates for the average dominance coefficient of such alleles are typically in the range of 0.05 to 0.3 (Johnston and Schoen, 1995; Willis, 1999; Charlesworth and Hughes, 2000). The IHM predicts substantial interference, at least in the lower part of this range (h = 0.05 – 0.15), if U ≥ 1.

Additional information is provided by other dynamical variables of the IHM, specifically the cohort proportions (K_i) and cohort specific mutation counts. Figure 6 illustrates cohort proportions before and after selection for a case in which selective interference is important. Selection dramatically alters the proportions, and as a consequence, the outbred cohort produces the great majority of individuals in the next generation. The increased efficiency of selection against deleterious mutations within inbred cohorts becomes inconsequential because these cohorts make a very limited contribution to the next generation (Ganders, 1972; Lande et al., 1994). In this case, it is the aggregate effect of many mutations and not the exposure of a single recessive lethal that limits the contribution of inbred cohorts.

Now consider the oscillatory approach to equilibrium of Figure 5. These fluctuations are not a simple consequence of the large immediate change in outcrossing rate (from t = 1 to t = 0.5). If t makes a more dramatic change, say from 1 to 0.2, K85 predicts a rapid approach to the new equilibrium Q. The sustained oscillation result from the complex evolution of the inter-locus associations that are most pronounced at intermediate outcrossing rates. The cohort proportions and the cohort specific moments of X and Y evolve in predictable ways over the course of a cycle. At the ebb of a cycle (when Q is below the equilibrium), M_X and M_Y are relatively low within the first few cohorts. Because the strength of selection is roughly proportional to these means (equations 4), Q increases. As the means increase, selection becomes more efficient which should subsequently reduce the rate of increase in Q. However, over most of the upward part of the cycle, V_X/M_X and V_Y/M_Y actually decline. This limits selection and allows Q to climb substantially above the equilibrium. At the crest of the cycle, V_X/M_X and V_Y/M_Y start to increase, but M_X and M_Y are sufficiently high that selection overtakes mutation and these means begin to decrease. However, V_X and V_Y continue to increase (or decline only slightly) over this first stage of downward cycle. V_X/M_X and V_Y/M_Y (within the first few cohorts) reach their maxima in the midst of the down cycle, and because selection becomes most efficient substantially after the crest, this drives Q below the eventual equilibrium. The cycling is essentially caused by a time-lag in adjustments of cohort specific means and variances.