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Figure 5.4 The frequencies over time of new mutations that each have an initial frequency of In this example, one new mutation is introduced into the population every 30 generations and Ne = 10. All of the mutations except one (solid line) go to loss within a few generations. The one allele that does go to fixation takes a relatively long time to do so compared with the time to loss. At the start of the simulation the ancestral allele has a frequency of 1 (not shown). When a new mutation reaches fixation, the original ancestral allele is lost and the new mutation becomes the ancestral allele.

These predictions for the frequency and fate of new neutral mutations under genetic drift suggest that at least some genetic variation is maintained in populations simply due to the random allele-frequency walk that new mutations take before reaching either fixation or loss. If the population shown in Fig. 5.4 were observed at a single point in time, it is possible that it would be polymorphic since a new mutation happened to be somewhere between fixation and loss. Observing many such loci at one point in time, it would be very likely that at least some of them would be polymorphic. This observation forms the basis of the neutral theory of molecular evolution, the hypothesis that genetic variation in populations is caused by genetic drift, covered in Chapter 8.

Geometric model of mutations fixed by natural selection

The third perspective on the fate of new mutations will focus on beneficial mutations, looking first at mutations fixed by natural selection alone and then at mutations fixed by the combined processes of natural selection and genetic drift. In addition to considering how new mutations are lost during segregation, in 1930, Fisher (see Fisher 1999 variorum edition) constructed another model of the fate of mutations that are acted on by the process of natural selection. As discussed earlier in the chapter, mutations may have a range of effects on fitness as well as on any phenotype with variation that has a genetic basis. The model Fisher developed sought to determine the range or distribution of the effect sizes (the amount of change in the phenotype caused by each mutation) of the beneficial mutations that are fixed by natural selection over time. Are the mutations that are fixed by natural selection all of large effect or all of small effect, or do they have effect sizes that fall into some type of distribution? It is quite likely that you are aware of the generalizations of this model without being aware of where these conclusions came from or what assumptions are involved. The generalization is that beneficial mutations have small effects: we do not expect to see beneficial changes taking place in single big leaps. This view of evolution is called micromutationalism, a concept that has been profoundly influential in evolutionary biology and population genetics (see Orr 1998a and references therein). The model that leads to this conclusion is called the geometric model of mutation and is developed in this section.

Micromutationalism The view that beneficial mutations fixed by the process of natural selection have small effects and therefore that the process of adaptation is marked by gradual genetic change.

Fisher imagined a situation where the values of two phenotypes determined the survival and reproduction, or fitness, of an individual organism (fitness is defined rigorously in Chapter 6). An example of two phenotypes might be the number of leaves and the size of leaves to achieve the maximum light capture for photosynthesis for a species of plant. However, the exact nature of the phenotypes is not important in the model as long as they contribute to the fitness of individuals. The critical point to understand is that phenotypic values closer to the maximum fitness value (Fisher called this the "optimum") are favored by natural selection, causing genotypes conferring higher fitness phenotypes to increase in frequency and fix in a population over time. Figure 5.5 shows the model. The values of the two traits are represented by two axes and the optimum fitness value for the combination of the two traits is at the center, the point labeled O for optimum.

Let's say an individual has values of the two phenotypes that put it at point A on the phenotypic axes, some distance r from the optimum fitness. All the points on a circle of radius r centered at the optimum have the same fitness as the fitness of the individual at point A (the dashed circle in Fig. 5.5 a). Next, imagine that random mutations can occur to one allele of the genotype of the individual at point A. If the effects of mutations are random, then a mutation could move the phenotype in any direction from point A, and these moves could be of any distance large or small away from point A. Some mutations would move the phenotype a short distance whereas others cause a long-distance move; some mutations move the phenotype toward the optimum whereas others move the phenotype away from the optimum.

From this graphical model, can we determine what types of mutational change are likely to be fixed by natural selection and contribute to adaptive change? One conclusion from the model is that mutations with a very large effect on phenotype (change in phenotype greater than 2r) cannot get the phenotype any closer to the optimum even if they are in the right direction. Since mutations of very large effect always move the phenotype further away from the

Maximum fitness or O

Value of * phenotype 2

Maximum fitness or O

Phenotype of individual A

Circle of fitnesses equal to fitness of individual A

Value of phenotype 1

Phenotype of individual A

Circle of fitnesses equal to fitness of individual A

Value of phenotype 1

Mutation of larger effect

Mutation of smaller effect

Mutation of larger effect

Figure 5.5 R.A. Fisher's geometric model of mutations fixed by natural selection. (a) Axes for two hypothetical phenotypes that determine fitness with maximum fitness when both phenotypes have the values at the center point marked with the black dot. An individual (or the mean phenotype of a population) with a phenotypic value is some distance from the maximum fitness. The dashed circle shows a perimeter of equal fitness around the point of maximum fitness. Although only two phenotypes define fitness in this example, the dashed circle of equal fitness would be a sphere with three phenotypes and an n-dimensional hyperspace if n phenotypes contribute to fitness. (b) Two mutations with smaller or larger phenotypic effects. The phenotypic effect of the mutations could be in any direction around the current phenotype (circles with radius m). Mutations with smaller effects have a better chance of moving the phenotype toward the maximum fitness (more of the area of the mutation-effect circle is to the left of the dashed line of equal fitness).

optimum fitness outside the dashed circle, these will never be fixed by natural selection.

What about mutations with smaller effects? Figure 5.5b shows two situations where the pheno-typic effect of a mutation is smaller (change in phenotype less than 2 r). On the right there is a mutation with a larger effect on the phenotype and on the left a mutation with a smaller effect on the phenotype. Both of these mutations could be in any direction, specified by the circle around A with a radius m to indicate the magnitude of the mutational effect. Notice that as the mutation gets larger in effect, less of the circle describing the effect on the pheno-type falls inside of the dashed circle that describes the current fitness of the individual at point A. As the phenotypic effect of a mutation approaches zero (m ^ 0), its effect circle will approach being half inside the arc of current fitness and half outside the arc of current fitness. Said the other way, as the phenotypic effect of a mutation gets larger and larger its effect circle encompasses more and more area outside the arc of current fitness. As mutations increase in their phenotypic effect they have an increasing probability of being in a direction that will make fitness worse rather than better. Therefore, natural selection should fix more mutations of small effect than of large effect since smaller mutations have a greater probability moving the phenotypic value toward the optimum. Mutations with almost zero effect have close to a 1/2 chance of being favorable while large mutations have a diminishing chance of being favorable.

This can be described in an equation:

P(mutation improves fitness) = 1

v 20

where m is the radius of the phenotypic effect of a mutation and r is the distance to the optimum from the current phenotypic value. As m goes to zero, the probability that a mutation moves the phenotype closer to the optimum approaches 1/2. For mutations of increasing effect, there is a diminishing probability that they improve fitness. When m is equal to twice the value of r there is no chance that the mutation will improve fitness: a mutation of effect 2r could just reposition A on the opposite side of the equal fitness circle around the optimum at best.

Fisher also reasoned that the fitness of organisms depends on many independent traits since the pheno-types of organisms must meet many requirements for successful growth, feeding, avoidance of predation, mating, and so forth. He therefore assumed that the dashed circle of equal fitness shown in Fig. 5.5 for illustration was really better represented by a space of many dimensions. In n dimensions, the measure of whether or not a mutation is large or small relative to the distance to the point of maximum fitness (r)

2r yfn tion 5.12. The main point is that increasing phenotypic dimensions cause the probability that a mutation improves fitness to decline more rapidly as its pheno-typic effect gets larger. The top panel of Fig. 5.6 plots is gauged by

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