A more detailed explanation

In the absence of natural selection, mean fitness would be expected to decline exponentially as inbreeding increases, where inbreeding is measured using Wright's inbreeding coefficient F^[6] (the reason why decline is exponential on F instead of linear is just that fitness is usually considered a multiplicative trait). The rate at which fitness declines as F increases (the inbreeding depression rate δ) depends on the frequencies and deleterious effects of the alleles present in the population before inbreeding.

The above coefficient F is the standard measure of inbreeding, and gives the probability that, at any given neutral locus, an individual has inherited two copies of a same gene of a common ancestor (i.e. the probability of being homozygous "by descent"). In simple conditions, F can be easily computed in terms of population size or of genealogical information. F is often denoted using lowercase (f), but should not be confused with the coancestry coefficient.

However, the above prediction for the fitness decline rarely applies, since it was derived assuming no selection, and fitness is precisely the target trait of natural selection. Thus, Wright's inbreeding coefficient F for neutral loci does not apply to deleterious alleles, unless inbreeding increases so fast that the change in gene frequency is governed just by random sampling (i.e., by genetic drift).

Therefore, the decline of fitness should be predicted using, instead of the standard inbreeding coefficient F, a "purged inbreeding coefficient" (g) that gives the probability of being homozygous by descent for (partially) recessive deleterious alleles, taking into account how their frequency is reduced by purging.^[4] Due to purging, fitness declines at the same rate δ than in the absence of selection, but as a function of g instead of F.

This purged inbreeding coefficient g can also be computed, to a good approximation, using simple expressions in terms of the population size or of the genealogy of individuals (see BOX 1). However this requires some information on the magnitude of the deleterious effects that are hidden in the heterozygous condition but become expressed in homozygosis. The larger this magnitude, denoted purging coefficient d, the more efficient is purging.

An interesting property of purging is that, during inbreeding, while F increases approaching a final value F = 1, g can approach a much smaller final value. Hence, it is not just that purging slows the fitness decline, but also that it reduces the overall fitness loss produced by inbreeding in the long term. This is illustrated in BOX 2 for the extreme case of inbreeding depression caused by recessive lethals, which are alleles that cause death before reproduction but only when they occur in homozygosis. Purging is less effective against mildly deleterious alleles than against lethal ones but, in general, the slower is the increase of inbreeding F, the smaller becomes the final value of the purged inbreeding coefficient g and, therefore, the final reduction in fitness. This implies that, if inbreeding progresses slowly enough, no relevant inbreeding depression is expected in the long term. This results in the fitness of a small population, that has been a small population for a long time, can be the same as a large population with more genetic diversity. In conservation genetics, it would be very useful to ascertain the maximum rate of increase of inbreeding that allows for such efficient purging.

Predictive equations when inbreeding is due to small population size

Consider a large non-inbred population with mean fitness W. Then, the size of the population reduces to a new smaller value N (in fact, the effective population size should be used here), leading to a progressive increase of inbreeding.

Then inbreeding depression occurs at a rate δ, due to (partially) recessive deleterious alleles that were present at low frequencies at different loci. This means that, in the absence of selection, the expected value for mean fitness after t generations of inbreeding, would be:

${\displaystyle W_{t}=We^{-\delta F_{t}}}$

where ${\displaystyle F_{t}}$ is the population mean for Wright's inbreeding coefficient after t generations of inbreeding.^[6]

However, since selection operates upon fitness, mean fitness should be predicted taking into account both inbreeding and purging, as

${\displaystyle W_{t}=We^{-\delta g_{t}}}$

In the above equation, ${\displaystyle g_{t}}$ is the average "purged inbreeding coefficient" after t generations of inbreeding.^[4] It depends upon the "purging coefficient" d, which represents the deleterious effects that are hidden in heterozygosis but exposed in homozygosis.

The average "purged inbreeding coefficient" can be approximated using the recurrent expression

${\displaystyle g_{t}=\left[\left(1-{\frac {1}{2N}}\right)g_{t-1}+{\frac {1}{2N}}\right]\left[1-2dF_{t-1}\right]}$

There are also predictive equations to be used with genealogical information.

The example of inbreeding depression due to recessive lethals

Average population fitness (W) against number of generations (t) of maintenance with population size N=10 when, in the original population, individuals carried on the average one rare recessive lethal. Evolution of W expected only from inbreeding (red line) or from inbreeding and purging (blue line).

As an example of genetic purging, consider a large population where there are recessive lethal alleles segregating at very low frequency in many loci, so that each individual carries on the average one of these alleles. Although about 63% of the individuals carry at least one of these lethal alleles, almost none carry two copies of the same lethal. Therefore, since lethals are considered completely recessive (i.e., they are harmless in heterozygosis), they cause almost no deaths. Now assume that population size reduces to a small value (say N=10), and remains that small for many generations. As inbreeding increases, the probability of being homozygous for one (or more) of these lethal alleles also increases, causing fitness to decline. However, as those lethals begin to occur in homozygosis, natural selection begins purging them. The figure to the right gives the expected decline of fitness against the number of generations, taking into account just the increase in inbreeding F (red line), or both inbreeding and purging (blue line, computed using the purged inbreeding coefficient g). This example shows that purging can be very efficient in preventing inbreeding depression. However, for non-lethal deleterious alleles, the efficiency of purging would be smaller, and it can require larger populations to overcome genetic drift.

Genome renewal

The yeast Saccharomyces cerevisiae and Saccharomyces paradoxus have a life cycle that alternates between long periods of asexual reproduction as a diploid, ending in meiosis that is usually immediately followed selfing, with only rare outcrossing.^[7] Recessive deleterious mutations accumulate during the diploid expansion phase, and are purged during selfing: this purging has been termed "genome renewal".^[8][9]