The information presented below was taken from: “Hartl, D. L., and A. G. Clark. 1989. Principles of Population Genetics. Sunderland, MA: Sinauer.”

F, or the inbreeding coefficient, defines (1) the probability that two alleles are identical by descent (i.e., autozygous), or (2) the proportional reduction in heterozygosity relative to a randomly mating, infinitely large population. The second definition of F is given below in mathematical form, where 2pq = predicted heterozygosity and H_obs = observed heterozygosity:

Eq. 1

Defining H_t to be the proportion of original heterozygosity remaining in generation t (i.e. H_obs/2pq):

Eq. 2

Note that in either equation 1 or 2, F defines the proportional reduction in heterozygosity relative to a randomly mating, infinitely large population.

As it turns out, the reduction of heterozygosity in small populations due to random drift results from an increase in the probability of resampling alleles in small populations (i.e., definition 1 of F above). Specifically, the probability of resampling alleles equals 1/(2N), which is approximately 0 for large populations, but increases as population size decreases. Similarly, the probability of not resampling alleles is 1-1/(2N). Altogether, the probability of resampling alleles in generation t equals:

Eq. 3

Equation 3 is derived as follows. The probability that two alleles will be identical by descent in generation t (F_t) is equal to the probability of sampling two alleles that are identical by descent (1/(2N) added to the product of the probability of sampling two alleles that are not identical by descent (1-1/[2N]) and the probability that two alleles were identical by descent in the previous generation (F_t-1).

For the case where F₀ = 0 (i.e., H_t = 1), we can rearrange equation 3 to give the probability that two alleles will be identical by descent as a function of generation time, t:

Eq. 4

Alternatively, we can use equation 2 to rearrange equation 3 with respect to heterozygosity (H_t):

Eq. 5

You probably remember this equation as the formula for calculating the reduction of heterozygosity in small populations. As such, we have shown that the reduction in heterozygosity for small populations is the result of an “inbreeding” type effect, or in other words, an increase in the probability of resampling alleles that are identical by descent.

As we are learning in class, non-random mating systems that include some degree of inbreeding (i.e., selfing in plants) also increase the probability of sampling alleles that are identical by descent. However, note that inbreeding influences F independent of population size.

Finally, we can use the equations above to see how F influences genotype frequencies. Overall, the probability that a randomly chosen individual will be AA is p²(1-F) (for the case where alleles are not identical by descent) + p(F) (for the case where alleles are identical by descent). Similarly, the probability that an individual will be Aa is 2pq(1-F) (because individuals that are heterozygous can’t be identical by descent), and the probability that an individual will be aa is q²(1-F) + q(F). Note that in the absence of inbreeding (F = 0), these probabilities are identical to those derived from Hardy-Weinberg.

SUMMARY TABLE

		Generation
		0	t	Infinity
Ft		0	1-(1-1/(2N))^t	1
Genotype frequency	AA	p^2	p^2(1-Ft)+p(Ft)	p
	Aa	2pq	2pq(1-Ft)	0
	aa	q^2	q^2(1-Ft)+q(Ft)	q
Allele frequency	A	p	p	p
	a	q	q	q