Hastings%20Genetics%20p.1-14

Hastings Genetics% - 5 are worth a K 5;‘ 3 Population Genetics The famous geneticist Theodosius Dobzhansky once wrote that nothing in biology

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Unformatted text preview: 5 are worth a K 5. ;‘ 3 Population Genetics The famous geneticist Theodosius Dobzhansky once wrote that nothing in biology makes sense except in the light of evolution. This is certainly true about many of the ecological questions of interest to us. Life history patterns, for example, are shaped by selection. The ages at which individuals have offspring, when they disperse, and even when they die are the results of selection. We need to develop some concepts about the genetic properties of populations to help us understand questions in evolutionary ecology. In turn, we will consider the issue of variation within a species —- a strictly genetic question — and then turn to questions at the interface between ecology and genetics. 3.1 Genetic questions What properties of populations can be classified as genetic in na- ture? Perhaps the most important aspect concerns variation among individuals. One of the primary questions of population genet— ics is to determine the reasons for the maintenance of polymer— pbz‘smsMwo or more different types Within a single population. Until the 19603, the variants studied by geneticists were almost all 42 3. Population Genetics What other examples of visible polymorphisms can you think of? The technique of gel electrophoresis is restricted to looking at soluble proteins that catalyze reactions. Visible polymorphisms: they could be detected by looking at the organism in question. One striking example of a visible polymorphism is the exis— tence of two sexes in most higher organisms. There is clearly a mechanism for the maintenance of the polymorphism. If there is only one male in a population and many females, the number of offspring for the male will clearly be higher then the average number of offspring for the females. Thus, any males produced will also have more offspring than any females produced. Conse— quently, the fraction of males in a population will tend to increase over time. This argument also works in reverse if there is only one female to start with and many males. In general, if a sex becomes rare it is then more ‘valuable’. There are a number of other classic cases of visible polymor- phism. One is the maintenance of dark and light forms in the moth Biston betularia and the change in the frequency of these forms following an increase in soot from industrial pollution — the case of industrial melam'sm. Another well-known example is the maintenance of the gene responsible for sickle-cell anemia in humans, a disease in which the red blood cells are deformed and less efficient in carrying oxygen. We will explore this case in the problems at the end of the chapter. A third example is the snail Cepaea nemomlis, whose numerous shell color and banding polymorphisms have been the subject of extensive study. Until the 19605 many geneticists thought genetic variants were relatively rare. Then, beginning with the classic work of Lewontin and Hubby (1966), the technique of gel electrophoresis emerged as a method to look for polymorphisms that did not have obvious macroscopic visual effects. Classic gel electrophoresis begins by placing soluble proteins in a gel made of starch. The gel is then subjected to an electric current, and different variants of the same protein may move at different rates, depending on the properties of the amino acids on the outside of the protein. Slices of the gel are treated with the substrates and cofactors catalyzed by a particular enzyme (protein) and a dye that reacts with the end product of the reaction. Proteins migrate under the influence of the electric field and appear as bands on the gel. Bands which travel different distances are different alleles. (Some amino acid FIGURE 3.1.Scl Each numberc moves in the g chemical that' which travel a respectively, h arbitrarily. substitutio tion rates of populal of extensi‘ The qur tion of the variation r tion (diffe is the maj phism. Ar of differei pled with the major During look for p Techniqur even the l electrophr direct exa 1995). Th responsib The trai fa Iation Genetics king at the __is the exis- " is clearly a 'e . If there is l the number , the average '5 produced , ced. Conse— 3‘ to increase , is only one i x becomes Le polymor- rrns in the icy of these pollution — l example is ‘x 11 anemia 7 deformed ‘_ - this case ple is the d banding dy. ‘ ‘. 'nts were .I‘ Lewontin g emerged - e obvious It begins by gel is then the same 3. properties es of the A, ed by a ‘i the end r uence of Y. ds which ’ ino acid 3.1 Genetic questions FIGURE 3.1 . Schematic diagram illustrating gel electrophoresis as a tool forfinding genotypes. Each number corresponds to an individual. Afterthe application of an electric current, a protein moves in the gel, and then the substrates of a reaction catalyzed by that protein as well as a chemical that will react with the end product are placed on the gel. The alleles of the protein, which travel at different rates, show up as bands on the gel. Thus individuals 1 through 5, respectively, have the genotypes BC, BB, CC, AB, and AC. The names of the alleles are chosen arbitrarily. substitutions which do not result in proteins with different migra- tion rates under electrophoresis will not be detected.) The result of population surveys using gel electrophoresis was the discovery of extensive polymorphisms in almost all species examined. The question of interest to geneticists became the determina- tion of the forces responsible for the maintenance of the extensive variation revealed by electrophoresis. One possibility is that selec— tion (differential reproduction or survival of different genotypes) is the major force responsible for the maintenance of polymor— phism. Another possibility is that dnft (changes in the frequency of different genotypes caused by random sampling effects) cou- pled with mutation (the chance production of new alleles) are the major forces responsible for polymorphism. During the past decade the techniques used by geneticists to look for polymorphisms have become even more sophisticated. Techniques from molecular biology have been used to show that even the high estimates of levels of polymorphism revealed by gel electrophoresis were lower than those determined by direct or in- direct examination of DNA sequences (e.g., Begun and Aquadro, 1995). The major goal remains the determination of the forces responsible for the maintenance of polymorphism. The traits examined through electrophoresis or DNA sequence 43 Genetic techniques may help ecologists to determine rates of exchange of breeding individuals between different populations. 44 What other quantitative traits can you think of? 3. Population Genetics Frequency 0 50 1 00 1 50 200 Ratio of wing vein length x 100 FIGURE 3.2. Distribution of a quantitative trait, the ratio of wing vein lengths in Drosophi/a melanogaster (data from Scharloo et al., 1967). The frequency distribution is very roughly normal. techniques may not be the ones that are important to an ecolo— gist, in contrast to a trait such as the size of an individual. A very different kind of question emerges when focusing on quantita— tive characters — the kind that can be measured, such as height, weight, or the ratio of sizes of different body parts (Figure 3.2). Here the issue is why is every individual not the same — presum- ably optimal - size? Although some variation results from envi- ronmental factors, there is also underlying genetic variability in virtually all cases. A related issue is how fast such a quantitative trait evolves in response to selection. 3.2 Evolutionary and ecological questions There are several areas where the interface between ecology and genetics has produced fruitful insights. One involves the use of optimization concepts in ecology. Many ecologists studying the behavior of organisms assume that the behavior is ‘optimal’. The natural question is whether it is reasonable to expect that the outcome of is the evolz takes place theory — hc food items. For ecolc 0 Under traits ‘ 0 Why optim o If the envirr spon: evolv Other topi: elude coev and parasi the very di 3.3 Or. We will no to all thee complex, subject of :‘ that may 5 our under Hardy—ii At the tin factors re: understoo generatior ents. Greg heritance Lulation Genetics 3 ll ‘ in Drosophila is very roughly 7? an ecolo- al. A very I quantita- as height, igure 3.2). L' -— presum- m envi- ‘ ,. 'ability in antitative r is and e e use of ‘ ying the WI 1’. The that the 3.3 One-locus model without selection outcome of evolution is optimal behavior? One specific example is the evolution of life histories — when and how reproduction takes place. Another general area of study is optimal foraging theory — how a foraging animal searches for patchily distributed food items. For ecological questions such as these we may ask: 0 Under what circumstances will natural selection produce traits that are optimal? 0 Why are all organisms within the population not of the optimal type: why is variability maintained? 0 If the environment is Changing, or the organism is in a novel environment, one would like to know how fast is the re- sponse to selection. How quickly does the population evolve toward the optimum? Other topics at the interface between ecology and evolution in- clude coevolution, the joint evolution of predator and prey, host and parasitoid, host and pathogen, or plant and pollinator, and the very difficult issue of the process of speciation. 3.3 One-locus model without selection We will not be able to produce models that will guide our answers to all these questions. The models involved can become very complex, and many of the issues we have raised are still the subject of intense study. Instead, we will study very simple models that may suggest some simple answers. These models will guide our understanding of more complex cases. Hardy— Wemberg law At the time of Darwin, the mid-to—late nineteenth century, the factors responsible for the maintenance of variability were not understood. It was unclear why variability was not reduced each generation by the offspring looking like the ‘average’ of the par- ents. Gregor Mendel, who deduced the particulate nature of in- heritance through his extensive experiments with peas, provided 45 We will look later at comparisons between plants that reproduce many times and those that reproduce once and die. Response to changing environments is a question of immense current interest. 46 We start with the simplest case, a single locus with two alleles. We do” not claim that this is realistic. Think ofhow each assumption might be violated. As we formulate the model, consider how the model might have to be changed to accommodate changes in the assumptions. There are twice as many alleles as individuals because we are thinking of a diploid organism. 3. Population Genetics the mechanism — heritable genes —— that explained how variabil- ity could be maintained. Although Mendel was a contemporary of Darwin, his work was not well known until the early twen- tieth century. Also, in the early twentieth century, G.H. Hardy, a British mathematician known for his contributions to number theory and analysis, showed mathematically that in the absence of other forces variability is preserved, as we now demonstrate. Consider in a diploid organism a single locus with just two alleles, A and a. Thus there are three different possible genotypes in the population: AA, Aa, and am. We will make the following assumptions. 0 Mating is random among individuals in the population. 0 There is no selection: the probability of mating and survival is independent of the genotype. 0 Generations are nonoverlapping. c There is no immigration or emigration. o The population is so large that we can ignore stochastic effects and consider only the frequencies (fractions) of dif- ferent genotypes. 0 There are no mutations at this locus. What happens to variability in the population at this locus with these assumptions? Let pAA be the frequency of AA individuals, 1),,“ be the frequency of Ad individuals, and pm be the frequency of aa individuals. Let the frequency of A alleles be p. If there are N individuals and 2N alleles, then the number of A alleles is 2NpAA + NpAa. The frequency of A alleles is P=———-—-——=PAA+—- (3.1) The frequency of a alleles is 6] = 1 *‘P = 1% +Paa- (3.2) To determine what happens after one generation of mating, we consider all possible matings, their frequency, and the possi- ble offspring and their frequency. These are listed in Table 3.1. 3.3 One—locusr TABLE 3.1. Matin fraction of offspri by multiplying tl Summing l eration’, w Similar rea and Thus, after frequencie We drav First, after pletely de " ation Genetics variabil- ' temporary arly twen— .H. Hardy, to number :1 e absence ' - genomes following lation. ‘~ (1 survival stochastic of dif- .§ a locus with ' individuals, _ frequency fp. If there A alleles is (3.1) (3.2) 1‘ of mating, ' the possi- ‘Table 3.1. 3.3 One-locus model without selection 47 TABLE 3.1. Matings, frequencies, and offspring in a one-locus, two-allele model. The three 'offspring fraction’ columns are the fraction of offspring of the mating that are of the given genotype. The ’contribution to next generation' columns are obtained by multiplying the fraction of offspring of each genotype by the frequency of the mating. mating offspring fraction contribution to next generation mating frequency AA Aa aa AA Aa cm I!“ pi) “‘- 1/2 1/2 In“ AA x a“ nun 2mm -_ p2) n 1/2 1/2 In nun Summing up the entries in the column ‘contribution to next gen— eration’, we find the frequency of AA the in next generation: pAA =17)?“ +PAAPAa +pfla/4 (3-3) = (1m tom/2)2 (3.4) = p2. (3.5) Similar reasoning leads to the conclusion that p1“, =pfia/4 +pAapaa +122“ (3.6) = (pita +paa/2)2 (3.7) = q2 (38) and pfiia = 1 _p2 _ qz Equation (3.10) iollowsfromthe = zpq. (3.10) observation thatl = p + q implies that 1-(p+q)Z-P"+21>q+q3- Thus, after one generation, no matter what the initial genotypic frequencies were, the genotypic frequencies are given by (3.13) sum to 1. PM =p2 (3.11) pAa = 21261 (3.12) pad = ‘12- (3.13) We draw several very important conclusions from this result. First, after one generation the genotype frequencies are com- pletely determined by the initial allele frequencies. This also Note that the frequencies in (3.1 1) through 48 3. Population Genetics Measuring selection directly can be very difficult because very small differences at the individual level in survivorship probability or reproduction can have large effects at the population level. means that we can express our models in terms of the single quantity, p, the frequency of A, which is a great simplification. Second, the allele frequencies and genotype frequencies remain constant from generation to generation after the first generation; genetic variability is not eliminated. Third, any deviation from the genotype frequencies (3.13), which are called the Hardy— Weinberg frequencia to honor the mathematicians who derived the equations, must result from violation of one of our assump- tions. Unfortunately, in practice it is very difficult to detect statisti- cally deviations from Hardy—Weinberg frequencies, because none of the available statistical tests are very sensitive. 3.4 One-locus model with selection At this point we can begin exploring the consequences of chang- ing any of the assumptions we have made in our initial model. Our initial model predicts that variability would be maintained, but does not provide any reason why a particular allele frequency should be found. Also, some variability will clearly be lost because of random sampling, so we need to look at forces that can truly maintain variability. We thus turn to an examination of the role of selection. In this context, we are also motivated by our interest in understanding the dynamics of traits of ecological interest. We thus add to the simple one—locus model the complicating factor of selection and ask: 0 How strong must selection be to produce an observed change in allele frequencies? We will use the answer to this question to consider the case of ‘industrial melanism’, the rapid increase in the frequency of dark forms of a forest moth following the presence of soot on trees. 0 Under what conditions will a polymorphism be maintained? We will examine the case of sickle—cell anemia in the prob- lems. Fitness To answer tion. The l hind offsp survival, ai on the ph viva] prob whereby ti Viability . In additior cumstance model inc change thi in fitnesse pendent o in the met The moi we can a Weinberg of mating is requiret relative at by the rel tions to fr ' Iation Genetics f the single I» plification. Cies remain generation; iation from the Hardy— ho derived ,ur assump- s-tect statisti- r‘cause none is of chang- ‘,__'tial model. ‘ aintained, r frequency > St because 3 t can truly / the role of V interest in rest. mplicating ‘ 3 observed n er to this “ nism’, the of a forest i 7! . intained? the prob— 3.4 One-locus model with selection Juveniles Reproduction Differential by random surVIval mating Adults FIGURE 3.3. Life cycle used in the model Fitness To answer these questions, we must talk about fitness and selec— tion. The fitness of an individual is its probability of leaving be- hind offspring as determined through differential reproduction, survival, and possibly representation among gametes, depending on the phenotype. We will temporarily equate fitness with sur- vival probability, or viability. By selection we mean the process whereby the more fit individuals are chosen. Viability selection In addition to the assumptions made under Hardy—Weinberg cir— cumstances, we will make further assumptions to obtain a simple model incorporating selection. We assume that viabilities do not change through time. We also ignore any frequency dependence in fitnesses; fitnesses of each genotype are assumed to be inde- pendent of the current makeup of the population. The two stages in the model are illustrated in Figure 3.3. The model is described in Table 3.2. At the initial juvenile stage we can assume that the frequencies are given by the Hardy— Weinberg proportions, because the juveniles resulted from a cycle of mating and we have already shown that only one generation is required to establish Hardy—Weinberg proportions. To find the relative adult proportions, we multiply the juvenile frequencies by the relative survival rates. To convert these relative propor— tions to frequencies, we divide by their sum, the (relative) mean 49 Can you think of cases in which frequency-dependent selection is likely? You can add the entries in the last row of Table 3.2 to see that they sum to i. 50 Both by guessing, and by following similar steps, find the equation for q’. 3. Population Genetics TABLE 3.2. The one-locus, two—allele model with selection. — — Juvenile relative survival rates wAA relative adult frequencies 2 adult frequencies prAA/E fitness: E = 1?sz + quwAa + qzwaa. (5.14) We then use these results to determine the frequency of A alleles in adults, which is the same as the frequency of A alleles in the juveniles of the next generation, p’ . From the last row of Table 3.2 we see that p’ = p14,, wig/2 (3.15) = prAA/w + (1/2)2pqua/E (3.16) = PCPwAA: qua) (5.17) w = ‘13, (3.18) w where in the last line we have defined the mean fitness of allele A to be WA = PwAA + qua. (3.19) Does this make sense? If we know that an individual has a single A allele, then with probability p its other allele is A, while with probability q its other allele is 61. Thus the mean relative fitness of an individual carrying an A allele is prA + qua. From equation (3.18) we see that allele A increases in frequency if the mean fitness of individuals carrying allele A is greater than the mean fitness of the population. Before using this model to examine some of the questions posed earlier, we phrase the model in terms of a different set of parameters that simplify the algebra and make our conclusions 3.4 One—locus more app: At this po parameter and that Equilibria We are n( posed ear] The notio the text. A constant ti Let Ap l to general Denote th change in so set Ap if tion Genetics M q wad waa/ w ‘1/ (3.14) 6fA alleles les in the 4: Table 3.2 (3.15) (3.16) (3.17) (3.18) l of allele (3.19) a single hile with f fitness of equation the mean :the mean 4 uestions 'l Serent set so nclusions 3.4 One-locus model with selection 51 more apparent. Let wAA = 1 — s (3.20) 10/151 = wad = 1 — t. (3.22) At this point, the sign of s and t is arbitrary. With this choice of Canyougiveaheuristicdiscussionofthe ' 7 parameters, we see that mean'”9°f””d" Remember to usethe fact that p + q = 1. wA=p(1—s)+q=p+q—ps=1—ps (3.23) wa=P+q(1—t)=p+q—qt=1—qt (3.24) and that Remember that 1=(p+q)2=p2+2pq+qz. w =p2(1 — s) + 2M + 42(1 — t) (3.25) =p2 + M + 612 —p25 — qzt (3.26) = 1 —pzs — qzt (3.27) =p(1 —ps) + q(1 -— qt). (3.28) Equilibria We are now ready to begin to answer the first of the questions posed earlier: when is a polymorphism maintained at equilibrium? The notion of an equilibrium is central to much of the rest of the text. An equilibrium is a value for a variable that can remain constant through time. Let Ap be the change in p, the frequency of A, from generation to generation. Then Compare the development hereto the graphical approach used in Figures 3.4 AP =p/ _p = p% _p. through3Jbelow. Denote the equilibrium value of p by At an equilibrium, the change in allele frequencies from generation to generation is zero, so set Ap = 0: _ [31,014 _ A 0 — —ZU_ p (3.30) 13(1 —fis) A = _-————-——-A ""‘ u 1 1_pzs_é2t p (33) 15(1 —135) —[3(1 —j)zs — qzt) A = e —p. (3.32) We get this pair because if the right-hand side of (3.34) is zero, one or the other of the factors must be. We are trying to divide by the factor 1 — p. 3. Population Genetics Box 3.1. Finding the equilibrium ofa discrete time model with a single variable. To find the equilibrium of the discrete time model p’ =f(p) write the change in p, 8p, as 6p =f(p) —p. Set Sp to be zero, and find the equilibrium, [3, as the solution of the resulting equation: 0 =f(13) -I3. We continue by setting the numerator of (3.32) to zero: 0 =13(1 —13$) —13(1 #325 — 812!) (3.33) =fi[1 —fis — (1 ~fazs — 21%)]. (3.34) Thus either .5 = 0 (3.35) 01‘ 1 —13s — (1 —132s — 8ft) = 0. (3.36) It makes sense that [a = 0 is an equilibrium, because if there are no A alleles in one generation and there is no mutation or immigration (as we have assumed), there will be no A alleles the next generation. Similar reasoning suggests that f) = 1 must be an equilibrium as well and thus 1 — 15 must be a factor of (3.36). We therefore notice that (using & = 1 — fi) 0 = 1 —135 — (1 12325 — 3ft) (3.37) = 1 —135 — 1 +1325 + (1 —;3)% (3.38) = — 135 +325 + (1 —13)2t (3.39) = (1 —13)[#135 + (1 — 13):] = 0. (3.40) We conclude that f) = 0, or 15 = 1 or that 4» + (1 — 13): = 0. (3.41) 3.4 One-locus Solving fo We finally Note that equilibriur which cle both posit tbe hetero. Stability of We have three equi frequencic But, what that is not on appro: times in tl We first happens i number oi rate or a s at the cas equation 1 We are that if p = approxim; ’ ‘rpulation Genetics 3 - 1'0: (3.33) (3.34) (3.35) (3.36) 7-4 use if there " mutation or A alleles the ._ 1 must be an of (3.36). We (3. 37) (3.38) (3.39) (3.40) (3.41) 3.4 One—locus model with selection Solving for f), we find: .135 _‘fit + t = 0 (3.42) t =[Js +1’3t. (3.43) We finally conclude that 13 = L (3.44) s+t‘ Note that for (3.44) to represent a polymorphic equilibrium (an equilibrium with both alleles present) we must have 0 < f) < 1, which clearly requires that either 5 and t are both negative or both positive. Yhus a polymorphic equilibrium is possible only if the heterozygote is the most fit or the least fit genotype. Stability of equilibria We have discovered that the one-locus model can have up to three equilibria. We now know what the outcome is if the allele frequencies are exactly zero, or exactly one, or exactly t/(s + t). But, what happens if the population starts at an allele frequency that is not one of the equilibria? We will use an argument, based on approximations near an equilibrium, that we repeat several times in this text. We first start near the 13 = 0 equilibrium. We first look at what happens if we start with no A alleles and then introduce a small number of A alleles, as would happen if there was a low mutation rate or a small number of immigrants with A. We thus are looking at the case where p is very small. We begin by writing out the equation for p’ in terms of p only, eliminating q. / = p(1 _p5) p ——1 _p25 _ qzt (3.45) _ p —p2s ————1 _p25 _ (1 _p)2t (3.46) _ 2 = p p S (5.47) 1—p25—t+2pt—p2t' We are ready to use our assumption that p is very small. Notice that ifp = 10—2, then p2 = 10—4, so ifp is very small we can approximate the numerator by [2 because the other term will be 53 Thus this model can have either two or three possible equilibria. What does (3.44) imply aboutf; ifs > 0 and t < 0? We begin by going through a series of steps that in factjustify the simpler procedure we will use in general. The equilibrium [3 = 0 corresponds to a population with noA alleles. 54 3. P0pulation Genetics We are ignoring the special case in which t is precisely 1. Give a definition of an unstable equilibrium. We are still justifying, and not simply describing, the procedure we will use in practice to find stabiliw. much smaller. Similarly, in the denominator the term 1 —t will be much larger than any of the other terms, all of which contain p. Thus, we approximate (3.47) as L 1 _ t. (3.48) p’ % Do you recognize this equation? It is identical to the equation for discrete time exponential growth which we studied earlier. Thus, we identify two distinct cases: 1 o If t > 0, then 1—; > 1 so p increases when it is small. 0 If t < 0 then % < 1, so p decreases when it is small. Does this conclusion make sense? If p is small and A is rare, there will be so few AA individuals that we can ignore them. If the heterozygote Aa is more fit than the homozygote aa, we would expect that A would increase in frequency. A more fit heterozygote corresponds to t > 0. We can use a similar argument to show that if q is small, or p is near 1, then: 0 If s > 0, then q increases (p decreases). o If s < 0, then q decreases (p increases). We have just determined when each of the equilibria f) = 0 and f) = 1 is stable. An equilibrium is stable if a population starting near the equilibrium approaches the equilibrium. The notion of stability is very important, because we expect to find populations near stable equilibria and do not expect to find populations near unstable equilibria because only stable equilibria are approached as time increases. A simpler way of computing stability We would not want to go through all these steps every time we want to compute stability. Luckily, there is a different way of view- ing the calculation we have just done. Think of p’ as a function ofp, FTP)- If we wish to approximate 19’ near the equilibrium [3 = 1, we can use a Taylor series (Box 3.2). The ‘variable’ in the Taylor series 3.4 One-locu When equilil: will be p We knov just this ( approxir. We subs ...
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This note was uploaded on 02/09/2011 for the course EEP 115 taught by Professor Waynem.getz during the Fall '10 term at University of California, Berkeley.

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Hastings Genetics% - 5 are worth a K 5;‘ 3 Population Genetics The famous geneticist Theodosius Dobzhansky once wrote that nothing in biology

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