## Co

- Neutral allele

20 40 60 80 100 120 140 Generation

Figure 5.3 The probability that a novel mutation is lost from a population due to Mendelian segregation. A neutral allele is eventually lost from the population while a beneficial mutation has a probability of about twice its selective advantage of fixation. The cumulative probability over time is described by ec(x-1) where x is the probability of loss in the generation before and c is the degree of selective advantage, if any. This expected probability assumes an infinitely large population that has a Poisson-distributed variance in family size.

the summation in equation 5.7 is really (1 + x +

infinity to give (e-2)(e1+x) = ex-1. When a mutant first appears in a population x = 0.)

Using this result shows that the probability of a new mutation being lost in two generations is e-06321 = 0.5315 or the probability of being lost in three generations is e-04685 = 0.6295. Based on this progression, Fig. 5.3 shows the probability that a new mutation is lost over the course of 140 generations. The conclusion from this graph is that a new mutation must eventually be lost from a population given enough time.

We can also ask what impact natural selection might have on this prediction that a new mutation will eventually be lost. Let's imagine that a new mutation is slightly beneficial instead of being neutral. Natural selection will then improve the chances that the new mutation is transmitted to the next generation, giving it a slight advantage over any of the other alleles in the population. Let c be the selective advantage of a new mutation so that a value of 1.0 would indicate neutrality and a value of 1.01 would mean a transmission advantage of 1%. The cumulative probability that an allele is lost at generation t is then

This version of the equation multiplies the exponent for the neutral case by the selective advantage of a beneficial allele. This makes very little difference to the probability that a mutant is lost if only a few generations elapse, but makes a larger difference after more generations have passed (Fig. 5.3). In general, the chance that a new beneficial mutation is not lost is approximately twice its selective advantage, still a very low probability for realistically small values of the selective advantage. However, as Fisher pointed out, if something like 250 independent beneficial mutations occur singly over time then there is a very small chance (0.9 8 250 = 0.00 64) that all of them would be lost during Mendelian segregation. This suggests that at least some beneficial mutations will be established in populations as mutations continue to be introduced.

The conclusion that a new neutral mutation must always be lost from a population seems at odds with the possibility of random fixation of a new mutation due to genetic drift. Fisher's method of modeling the fate of a new mutation makes the assumption that population size is very large. This assumption allows use of the expected values for the proportion of parental pairs for each family size under the Poisson distribution and the chance of an allele being lost for each family size, probabilities that should only be met in the limit of many parental pairs that span a wide range of family sizes. Finite numbers of parental pairs would likely not meet these expected values due to chance deviations from the expected value. The assumption of infinite population size is justified because it is used to reveal that particulate inheritance by itself can lead to loss of new mutations even in the complete absence of genetic drift. Next, we will take up the fate of a new mutation in the context of a finite population.

### Fate of a new mutation in a finite population

A second perspective on new mutations is to consider their fate as an allele in a finite population in the absence of natural selection. We can then employ the concepts and models of genetic drift developed in Chapter 3 to predict the frequency of new mutations over time in a population. The first critical observation is to recognize that the initial frequency of any new mutation is simply

P(mutant lost generation t) = e'

p0(new mutation) =

because a new mutation is present as a single-allele copy in a population of 2Ne allele copies. If the frequency of a new mutation is determined strictly by genetic drift, then each new mutation has a probability of of going to fixation and a probability of

1--of going to loss. This result makes intuitive

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