D d ttxt Jxt dxdx

We can also substitute the flux in allele frequency from equation 3.27 (using x to represent p and 1 - x to represent q):

2 2N

With only random sampling error acting to change allele frequency, (M(x)dt = 0), this rearranges to the diffusion equation for genetic drift:

The diffusion equation predicts the probability distribution of allele frequencies in many populations over time and some examples are given in Fig. 3.13. Compare Fig. 3.13 with Fig. 3.10 and it is apparent that the diffusion equation and the Markov chain model both make similar predictions for the outcome of genetic drift. A final point is that the term diffusion approximation implies that the diffusion equation makes some assumptions. Noteworthy assumptions are that the number of populations is very large, approaching infinity, and the allele frequency distribution is continuous so that the distribution of allele frequencies is a smooth curve (compare these assumptions with the allelic state distribution in

Figure 3.13 Probability densities of allele frequency for many replicate populations predicted using the diffusion equation. The initial allele frequency is 0.5 on the left and 0.1 on the right. Each curve represents the probability that a single population would have a given allele frequency after some interval of time has passed. The area under each curve is the proportion of alleles that are not fixed. Time is scaled in multiples of the effective population size, N. Both small and large populations have identically shaped distributions, although small populations reach fixation and loss in less time than large populations. The populations that have reached fixation or loss are not shown for each curve. For a color version of this image see Plate 3.13.

Allele frequency

Figure 3.13 Probability densities of allele frequency for many replicate populations predicted using the diffusion equation. The initial allele frequency is 0.5 on the left and 0.1 on the right. Each curve represents the probability that a single population would have a given allele frequency after some interval of time has passed. The area under each curve is the proportion of alleles that are not fixed. Time is scaled in multiples of the effective population size, N. Both small and large populations have identically shaped distributions, although small populations reach fixation and loss in less time than large populations. The populations that have reached fixation or loss are not shown for each curve. For a color version of this image see Plate 3.13.

Fig. 3.11a with its discrete allelic states and finite number of populations).

The diffusion equation has been used to arrive at a number of generalizations about genetic drift. A widely used set of generalizations is the average time to fixation for alleles that eventually fix in a population and the average time to loss for alleles that eventually are lost from a population:

where p is the initial allele frequency (Kimura & Ohta 1969a). (Note that the natural log of a number less than one is always negative: ln(1) = 0 and ln(x) ^ as x approaches zero, so that the average time will always be a positive number.) These two expressions can be combined to obtain the weighted average time that an allele segregates in a population (the allele is neither fixed nor lost):

segregate

The predictions from these equations quantify our intuition about the action of genetic drift (Fig. 3.14). Alleles close to fixation or loss do not take long to reach fixation or loss. Alternatively, an allele initially very close to fixation (or loss) would take a long time, about 4N generations if N is large, if it were to reach the opposite condition of being very close to loss (or fixation). Also, the closer an allele is to the initial frequency of 1/2, the longer it will segregate before reaching fixation or loss up to a maximum of about 2.8N generations. The curves for times to fixation or loss and segregation times have an identical shape

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