Inbreeding and heterozygosity

Concept

Inbreeding and heterozygosity

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Statement

The genetic cost of mating with relatives.

Why it matters

hardy-weinberg describes allele and genotype frequencies in an idealised, randomly mating population; inbreeding is the most common, biologically important violation of that random-mating assumption, and this result quantifies exactly how much heterozygosity is lost when mating occurs preferentially between relatives. It supplies the direct genetic basis for inbreeding depression and is central to conservation genetics of small populations, where mates are frequently, unavoidably related.

genetic-drift and gene-flow describe how allele frequencies themselves change across generations; inbreeding, by contrast, leaves allele frequencies essentially unchanged and instead redistributes them into different genotype proportions — a distinct, complementary way a population can depart from Hardy-Weinberg expectations.

Hypotheses

Allele frequencies \(p\) and \(q\) themselves are unchanged by inbreeding alone; only genotype (specifically heterozygote) frequencies are affected.If selection, mutation, or migration were simultaneously changing \(p\) and \(q\) across the same generations, the clean relationship derived below would need additional terms to separate inbreeding's effect from those other forces. The inbreeding coefficient \(F\) is defined as the probability that the two alleles at a locus in an individual are identical by descent — literal copies of one single ancestral allele — not merely identical in sequence by chance.Identity by descent, not identity by state, is specifically what \(F\) measures; two unrelated homozygotes carrying the same allele by chance contribute nothing to \(F\). In any real, finite population, without any known mating between relatives on record, background inbreeding still accumulates slowly across generations through genetic drift; \(F=0\) is a strict idealisation, exact only in an infinite, randomly mating population.

Proof

F = P(\text{two alleles at a locus are identical by descent})

This is the defining quantity of the whole result, following Hypothesis 2 directly. A

\text{Identity by descent} \ \Rightarrow\ \text{homozygosity: a fraction } F \text{ of individuals are forced homozygous purely by shared ancestry.}

Two literal copies of the same ancestral allele are, trivially, the same allele, so identity by descent at a locus necessarily produces a homozygous genotype there, independent of what Hardy-Weinberg alone would predict for that individual. A

\text{The remaining fraction } (1-F) \text{ mate as an ideal, random-mating (Hardy-Weinberg) population, giving expected heterozygosity } 2pq \text{ among that fraction.}

Only the identical-by-descent fraction is forced to homozygosity; the rest of the population is unaffected by inbreeding and follows ordinary Hardy-Weinberg proportions. A

H = (1-F)(2pq) = H_0(1-F), \qquad H_0=2pq

Combining Steps 2–3 gives total expected heterozygosity as the Hardy-Weinberg value \(H_0\), scaled down by the fraction of the population not forced homozygous by descent. A

F = \sum_{\text{paths}} \left(\tfrac12\right)^{n_1+n_2+1}(1+F_A)

\(F\) is computed directly from a pedigree by summing, over every path connecting the two parents through a shared ancestor \(A\), a term halving once per generational step back to \(A\) and forward again, where \(n_1,n_2\) count generations from each parent to \(A\) — the standard path-counting method (Worked examples). B

Result

H = 2pq(1-F); \qquad \text{genotypes: } AA=p^2+Fpq,\ \ Aa=2pq(1-F),\ \ aa=q^2+Fpq

Reading. Heterozygosity declines linearly with the inbreeding coefficient \(F\); the "missing" heterozygotes reappear as extra homozygotes of both types, split in proportion to the existing allele frequencies.

Scope. Applies at a single locus with allele frequencies otherwise fixed for the generation considered; requires no simultaneous selection, mutation, or migration acting on the same locus (Hypothesis 1) for the clean decomposition to hold exactly.

Corollaries & converses

Because allele frequencies are unchanged by inbreeding alone (Hypothesis 1), the "lost" heterozygotes represent redistributed, not removed, genetic variation — inbreeding does not directly delete alleles from the gene pool, though it does expose rare recessive alleles to selection far more often.
mutation-selection-balance's account of harmful recessive alleles persisting at low frequency depends on their being mostly sheltered in heterozygotes; inbreeding directly undermines that shelter, which is the proximate genetic cause of inbreeding depression wherever deleterious recessive alleles are present at appreciable frequency.
Converse: a locus showing markedly fewer heterozygotes than Hardy-Weinberg predicts, given accurately measured allele frequencies, is itself used to estimate \(F\) empirically, and hence to detect non-random mating, without requiring a complete known pedigree.

Fails without

Drop the identity-by-descent definition of \(F\) (Hypothesis 2), and instead simply compare a population's observed homozygote frequency to Hardy-Weinberg expectation: any such deviation could then be wrongly attributed to inbreeding when it might equally arise from population subdivision (the Wahlund effect), selection against heterozygotes, or genotyping error — \(F\) specifically isolates the identity-by-descent cause.
Drop the unchanged-allele-frequency assumption (Hypothesis 1): if selection or migration is simultaneously altering \(p\) and \(q\) across the generations being compared, \(H=H_0(1-F)\) no longer isolates the effect of inbreeding alone, since part of the observed change in heterozygosity would then be due to changing allele frequencies rather than to \(F\).

Common errors

Treating \(F\) as measuring identity by state (the same allele by chance) rather than identity by descent (a literal shared ancestral copy).
Assuming inbreeding directly changes allele frequencies; at a single generation, absent associated selection, it changes only genotype — specifically heterozygote — frequencies.
Assuming \(F=0\) exactly is achievable in any real, finite population; genetic drift causes some background relatedness to accumulate even with no known common ancestor on record (Hypotheses, t3).
Confusing inbreeding depression, a fitness cost, with inbreeding itself, \(F\), a purely genetic and statistical quantity describing parental relatedness; \(F\) is one possible cause of inbreeding depression, not a synonym for it.

Discussion

Sewall Wright, one of the three founders of population genetics alongside R.A. Fisher and J.B.S. Haldane, formalised the inbreeding coefficient and the path-counting method for computing it from a pedigree during the early twentieth century, as part of his broader development of quantitative population genetics.

By the path-counting formula, full-sibling or parent-offspring mating (a single common-ancestor path of two generational steps) gives \(F=1/4\), while first-cousin mating (two paths through two shared grandparents) gives \(F=1/16\), both assuming the shared ancestors are themselves non-inbred — standard textbook values illustrating how quickly \(F\) falls as relatedness becomes more distant.

Common misconception: that inbreeding is inherently and always harmful. Its direct genetic effect — raising homozygosity — is fitness-neutral in itself, and becomes harmful specifically in a population carrying deleterious recessive alleles at appreciable frequency for that raised homozygosity to expose.

Worked examples

\text{Full-sibling mating: two paths, one through each shared parent, each of length } n_1=n_2=1, \text{ giving } (\tfrac12)^{1+1+1}=\tfrac18 \text{ per path.}

Summing the two paths (assuming the shared parents are themselves non-inbred) gives \(F = \tfrac18+\tfrac18=\tfrac14\), the standard textbook value for full-sib mating. A

\text{With } p=q=0.5,\ H_0=2pq=0.5,\ F=0.25:\ H = 0.5(1-0.25)=0.375

One generation of full-sib mating alone produces a 25% relative decline in heterozygosity compared with the Hardy-Weinberg expectation, purely from the identity-by-descent term. A

F_{\text{full-sib}}=0.25 \ \Rightarrow\ 25\%\text{ relative loss of heterozygosity at any locus with } p=q=0.5

Reading. Even a single generation of close-relative mating produces a substantial, directly computable drop in heterozygosity.

Scope. The same path-counting and \(H=H_0(1-F)\) calculation applies to any pedigree relationship once its paths are enumerated.

Problems

Compute \(F\) for first-cousin mating, which shares two grandparents through two paths, each of length \(n_1=n_2=2\).
Solution
Each path contributes \((\tfrac12)^{2+2+1}=(\tfrac12)^5=\tfrac{1}{32}\); summing the two paths through the two shared grandparents gives \(F=\tfrac{1}{32}+\tfrac{1}{32}=\tfrac{1}{16}\), the standard textbook value for first cousins.
Given \(p=0.3\), \(q=0.7\), \(F=0.1\), compute the expected genotype frequencies \(AA\), \(Aa\), and \(aa\).
Solution
\(AA=p^2+Fpq=0.09+0.1(0.21)=0.111\); \(Aa=2pq(1-F)=0.42(0.9)=0.378\); \(aa=q^2+Fpq=0.49+0.021=0.511\); these sum to \(1.000\), as required.
Explain why a rare recessive lethal allele becomes more consequential in a population practicing regular inbreeding, even though its allele frequency \(q\) is unchanged.
Solution
Inbreeding raises the homozygote frequency \(q^2+Fpq\) above the Hardy-Weinberg value \(q^2\) alone (Result), so a larger fraction of individuals now express the recessive phenotype at the same allele frequency — the allele is no longer as effectively sheltered from selection within heterozygotes (Corollaries).