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2.00^† 50 0.99 3.99 3.60 ∞^‡ 1.00 4.00 4.00

      ^* Retention of heterozygosity is approximately equal to 1 − 1/(2N), where N is the population size after the bottleneck. If a population crashed to 10 individuals, about 1 − ½(10) = 1 − 0.05 = 0.95 of the genetic variation of the original population would remain.

      ^† The formula for estimating how many alleles would remain after a bottleneck is E = m − ∑ _j (1 − p_j) ^2N _, where m is the number of alleles before the bottleneck, p is the frequency of the jth allele, and N is the population size after the bottleneck. From an original set of four alleles the remaining number would be

       4 − ∑ (1 − 0.94)20 + (1 − 0.02)20 + (1 − 0.02)20 + (1 − 0.02)20 =

       4 − ∑ 0.0620 + 0.9820 + 0.9820 + 0.9820 =

       4 − ∑ ~0 + 0.666 + 0.666 + 0.666 = 2

      ^‡ With a population of infinite size no genetic bottleneck occurs.

      

A genetic bottleneck is the outcome of a process known as random genetic drift, a process similar in concept to sampling error that you may be familiar with from introductory statistics. Genetic drift is an erratic change in gene frequencies that is likely to occur in small populations because each generation retains just a portion of the gene pool of the previous generation (Frankel and Soulé 1981 ; Hartl and Clark 1997). It is like sampling error because the genetic diversity “sampled” by the few survivors may not be representative of the population as a whole. Table 5.3 presents a formula for estimating the implications of random genetic drift for genetic diversity. The formula is identical to the one for estimating the loss of genetic variation in a bottleneck, with an exponent added to represent the number of generations that a population has continued to remain small. In other words, random genetic drift is the same thing as passing through a genetic bottleneck except that the bottleneck usually lasts multiple generations (compare column 2 in Table 5.3 with column 2 in Table 5.2).

Table 5.3 The proportion of genetic variation retained in small populations of constant size after 1, 5, 10, and 100 generations is approximately [1 − 1/(2N)] ^t , where N is the population size and t is the number of generations. For example, 0.95⁵ = 0.77.

      Based on Frankel and Soulé 1981

	Generations
Population size (N)	1	5	10	100
2	0.75	0.24	0.06	<<0.01
6	0.917	0.65	0.42	<<0.01
10	0.95	0.77	0.60	<0.01
20	0.975	0.88	0.78	0.08
50	0.99	0.95	0.90	0.36
100	0.995	0.975	0.95	0.60

We can see that although a population of 10 individuals may retain 95% of its genetic variation after one generation (or after one bottleneck), with random genetic drift for 10 generations only 60% of the variation is likely to be retained, and after 100 generations virtually all the original genetic variation would be lost. A similar pattern exists for the loss of alleles; after many generations of random genetic drift, small populations will usually retain only one allele for a given gene (Table 5.4). In the language of genetics, the gene will have been fixed for that allele. In sum, random genetic drift in a population that remains small for many generations is much more likely to lead to a loss of genetic diversity than is a single bottleneck from which a population recovers quickly.

Table 5.4 Expected number of alleles remaining after t generations for a population of six individuals with 2, 4, or 12 alleles for a gene, assuming equal frequency of each allele.

      Based on Frankel and Soulé 1981

	Number of alleles
Generations	m = 2	m = 4	m = 12
0	2.00	4.00	12.00
1	1.99	3.87	7.78
2	1.99	3.55	5.88
8	1.67	2.18	2.64
20	1.24	1.36	1.44
∞	1.00	1.00	1.00

      If drift erodes genetic diversity then will mutation simply replenish it? Probably not. The problem is a severe imbalance between the rates at which the two processes operate. A population bottleneck can deplete genetic diversity from a population during just a few generations if the bottleneck is narrow enough. In contrast, it has been estimated that 10⁵–10⁷ generations are required to regenerate allelic diversity for a single gene (Lande and Barrowclough 1987). The genetic machinery of a cell is remarkable at avoiding

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