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The next step in the model is to calculate the equilib^ rium condition q, or the allelic frequency when there is no change in allelic frequency from one generation to the next that is, when q (equation 20.3) is equal to zero: q Thus, pn Then, substituting (1 qn 1
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And, since p q 1,
Mutational Equilibrium
Mutation affects the Hardy-Weinberg equilibrium by changing one allele to another and thus changing allelic and genotypic frequencies. Consider a simple model in which two alleles, A and a, exist. A mutates to a at a rate of (mu), and a mutates back to A at a rate of (nu): A 7 a
(20.7)
We can see from equations 20.6 and 20.7 that an equilibrium of allelic frequencies does exist. Also, the equilib^ rium value of allele a (q ) is directly proportional to the relative size of , the rate of forward mutation toward a. ^ If , the equilibrium frequency of the a allele (q ) will be 0.5. As gets larger, the equilibrium value shifts toward higher frequencies of the a allele.
If pn is the frequency of A in generation n and qn is the frequency of a in generation n, then the new frequency of a, qn 1, is the old frequency of a plus the addition of a alleles from forward mutation and the loss of a alleles by back mutation. That is, qn
Stability of Mutational Equilibrium
Having demonstrated that allelic frequencies can reach an equilibrium due to mutation, we can ask whether the mutational equilibrium is stable. A stable equilibrium is
(20.1)
Tamarin: Principles of Genetics, Seventh Edition
IV. Quantitative and Evolutionary Genetics
20. Population Genetics: Process that Change Allelic Frequencies
The McGraw Hill Companies, 2001
Twenty
Population Genetics: Processes That Change Allelic Frequencies
one that returns to the original equilibrium point after being perturbed. An unstable equilibrium is one that will not return after being perturbed but, rather, continues to move away from the equilibrium point. As we mentioned in the last chapter, the Hardy-Weinberg equilibrium is a neutral equilibrium: It remains at the allelic frequency it moved to when perturbed. Stable, unstable, and neutral equilibrium points can be visualized as marbles in the bottom of a concave surface (stable), on the top of a convex surface (unstable), or on a level plane (neutral; g. 20.1). Although more sophisticated mathematical formulas exist for determining whether an equilibrium is stable, unstable, or neutral, we will use graphical analysis for this purpose. Figure 20.2 introduces the process of graphical analysis, which provides an understanding of the dynamics of an event or process by representing the event in graphical form. In gure 20.2, we have graphed equation 20.3, the q equation of mutational dynamics.The ordinate, or y-axis, is q, the change in allelic frequency.The abscissa, or x-axis, is q, or allelic frequency.The diagonal line is the q equation, the relationship between q and q. Note that q can be positive (q is increasing) or negative (q is decreasing), whereas q is always positive (0 1.0). Graphical analysis can provide insights into the dynamics of many processes in population genetics. The diagonal line in gure 20.2 crosses the q 0 ^ line at the equilibrium value (q ) of 0.167. This line also shows us the changes in allelic frequency that occur in a population not at the equilibrium point. We will look
at two examples of populations under the in uence of mutation pressure, but not at equilibrium: one at q 0.1 (below equilibrium) and one at q 0.9 (above equilibrium). If we substitute q 0.1 into equation 20.3, we get a q value of 4 10 6. If we substitute q 0.9 into the equation, we get a q value of 4.4 10 5. In other words, when the population is below equilibrium, q increases ( q 4 10 6 ); if the population is above equilibrium, q decreases ( q 4.4 10 5). We can read these same conclusions directly from the graph in gure 20.2. We can see that the mutational equilibrium is a stable one. Any population whose allelic frequency is not at the equilibrium value tends to return to that equilibrium value. A shortcoming of this model is that it provides no obvious information revealing the time frame for reaching equilibrium. To derive the equations needed to determine this parameter is beyond our scope. (We could use computer simulation or integrate equation 20.3 with respect to time.) In a large population, any great change in allelic frequency caused by mutation pressure alone takes an extremely long time. Most mutation rates are on
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