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FUNCTIONAL DIVERGENCE TOPIC 1: FFTNS and the shifting balance theory It is worth revisiting neutral theory before moving on to models of adaptive evolution. It was clear from the beginning (Kimura 1968; King and Jukes 1968) that neutral molecular evolution was an implausible mechanism for phenotypic evolution at the level of the organism. There is virtually no controversy concerning the inadequacy of neutral theory on this point. This topic introduces two of the more influential models for adaptive evolution. Neutral theory and models of adaptive evolution should not be viewed as disjoint entities; there is a role for both in explaining molecular evolution. A review of the notion of fitness Remember that FITNESS is a measure of an individual’s reproductive contribution to the future, and has two components: (i) an individual’s ability to survive and reproduce [VIABILITY]; and (ii) an individuals reproductive output [FECUNDITY]. We used the symbols WAA, WAa, and Waa to specify the relative fitness values of the AA, Aa, and aa genotypes. Symbolism for selection AA Aa p 02 2p0q0 WAA WAa Genotype Frequency Phenotype aa q 02 Waa By our definition the fitness ratio is: WAA : WAa : Waa and the suvival ratio of AA, Aa, and aa genotypes among the adults will be: p2WAA : 2pqWAa : q2Waa The above ratios are correct, but they are NOT frequencies because they no longer add up to 1! So, to build a model that included selection, we divided the individual fitness values by the grand total after selection. This normalized the terms so that the frequencies sum to 1. The grand total is a measure of the AVERAGE FITNESS ( of individuals in the population. = ) p2WAA + 2pqWAa + q2Waa So, we work with a normalized fitness of a genotype; i.e., fitness divided by the average fitness. We consider fitness a phenotype because it is an interaction between the genotype and the environment. It is important to remember that in the following models we use fitness as phenotype, not phenotypic characters of organisms. It is important not to confuse these, because in natural populations people often measure phenotypic traits that they ASSUME correlate with fitness. Fisher (1930) viewed fitness as the per unit time increase of individuals with different genotypes, and that is where we will begin. Fisher’s fundamental theorem of natural selection (FFTNS) R. A. Fisher (1930) developed a one-locus model of natural selection that implied an interesting property of population evolution. In words, his theorem states that the rate of increase in the average fitness of a population ( ) is equal to the genetic component of the variation in fitness. Fisher (1930) gave this model the humble name: “The fundamental theorem of natural selection”. Let’s take a look at the basis of Fisher’s theorem. Fisher started with the well known formula for the RESPONSE OF A POPULATION PHENOTYPE TO SELECTION (R). R = h2 × S h2: The proportion of total phenotypic variance that is predictably transmitted to next generation (i.e., additive genetic component of variance) S: SELECTION DIFFERENTIAL; the difference between the mean phenotype of those under selection and the mean phenotype of the population. The details of this way of treating selection are interesting, but beyond the scope of this course. The general equations for R (not shown) will work for any phenotype that affects fitness, and remember that Fisher was interested in fitness as a phenotype itself. Fisher wanted to build a model based on R that would answer the question of what happens to the fitness of a population over time. By substitution and some clever manipulation (not shown) Fisher obtained the following result: [This is FFTNS] The change in population fitness depends on just two parameters. : Va(W): The average fitness of the population Additive component of the total variation in fitness Biological implications of FFTNS FFTNS is a very simple formula, but as a model it has a number of noteworthy implications for adaptation as a process of microevolution. 1. Populations can not adapt without additive variance in fitness. The parameter Va(W) can be zero or positive. When it is zero, there can be no change in the mean fitness of the population; ie., when Va(W)=0, =0. 2. The rate of change in the mean fitness of the population ( ) depends on its mean fitness ( ). The more fit the population is to start with the slower the rate of change due to selection. 3. Fitness always increases. The parameter must be ≥ 0. Hence, the population can only increase in fitness, and once it goes up in fitness it can never go back down. a. This seems trivial, but it means that a population will always go to the top of the “nearest peak in fitness”. If there exists multiple fitness peaks, and to get to the global peak a population must go through a “fitness valley”, FFTNS predicts that the population will be unable to reach global fitness; it can only go to the top of the local fitness peak. b. Once at the top of a peak, selection will act to keep it at this point. Adaptive topography FFTNS led to the notion of an ADAPTIVE TOPOGRAPHY; a surface of mean fitness for a population where peaks represent the highest values of mean fitness, and valleys the lowest values of mean fitness. An adaptive topography is sometimes called an ADAPTIVE LANDSCAPE. Under FFTNS populations can only go “uphill”. The simplest case of an adaptive topography can be illustrated with a plot of population mean fitness for all possible frequencies of two alleles at a single locus (see plot below). Population fitness surface for two alleles at one locus under directional selection and overdominant selection WAA = 0.5 WAa = 1 Waa = 0.1 WAA = 0.1 WAa = 0.75 Waa = 1 Directional selection Over-dominant selection In both cases shown in the above figure, there was only one peak in fitness on the topography, so the population always goes to the peak. In a slightly more complex case with one locus and three alleles, the adaptive topography has multiple fitness peaks. In such a case the fate of the population depends on where its starts on the fitness landscape. To view the more complex fitness landscape we need a different method of graphically representing the population allele frequencies. The De Finetti diagram turns out to be very useful for this but it takes a little time to become familiar with its layout. To help familiarize you with it, I have provided a generalized version of a De Finetti diagram below. We assume three alleles whose frequencies sum to 1. The three vertices of the triangle represent the three points where a population is fixed for one of the three alleles. Any point in space within the triangle represents three allele frequencies; the frequency of the allele of interest is the perpendicular distances from the site opposite the involved vertex to the point in space. Example of a De Finetti diagram of three allele frequencies at a locus Allele 1 Indicates: Alleles 2 and 3 have freq =0 at this vertex Allele 1: freq = 0.45 Allele 2: freq = 0.25 Allele 3: freq = 0.60 0.6 0.25 Alleles 1 and 2 have freq =0 at this vertex 0.45 Allele 2 Allele 3 Alleles 1 and 3 have freq =0 at this vertex We are interested in the fitness topography, so we plot the contours of the three allele frequencies that yield the same population fitness. An example is shown below that is based on estimates of the fitness of certain human populations based on three alleles at the beta-globin locus. Adaptive topography [mean fitness surface] for all possible frequencies of three alleles at a single locus. Allele 1 Lowest mean fitness Valley on the fitness landscape Local peak in mean fitness Global peak in mean fitness Allele 2 Allele 3 In the above fitness topography there are two peaks; one is a local peak reflecting a population polymorphism for alleles 1 and 2, and the other is the globally maximum peak where allele 3 is fixed in the population. Note the yellow line. This is a FITNESS VALLEY, a region of locally minimum fitness that separates two peaks on the fitness topography. Natural selection cannot move a population across a valley in the adaptive topography. start here end here end here start here From FFTNS we see that a population cannot cross a fitness valley by natural selection, as it cannot go “down-hill”. This leads to an important notion: selection can actually prevent globally adaptive evolution! It also means that the results of natural selection are subject to historical effects; i.e., the end-point depends on the initial allele frequencies. Difficult assumptions of FFTNS 1. Constant fitness throughout time. Fitness of an allele can vary in different environments, and consequently as environments vary over time (season of year, year to year, etc.) so do the fitness levels. The fitness of an allele could be density dependent, which can vary in time as well. 2. Linkage equilibrium. Alleles at different loci can be in non-random association. Thus an allele can obey the theorem as it applies to another allele to which it is closely linked. The most dramatic effect is when a deleterious allele is tightly linked to a highly beneficial allele; the deleterious allele can increase in frequency due to the “hitchhiking effect”. If multiple loci affect fitness, then linkage and rate of recombination must be considered in a model that predicts their microevolutionary fate. 3. Fitness must be the phenotype. People measure traits in natural populations that they think correlate with fitness. In order to make prediction under FFTNS, one must be able to separate genetic from environmental components of variance; this is not easy to do. 4. No genetic drift. FFTNS predicts an allele with any advantage will be fixed with a probability of 1. As we have seen with a beneficial allele in a finite population, if the product of Ne and s is less than 1, the probability of fixation is just 1/2 Ne . There are many natural conditions under which we expect that the assumptions of FFTNS will be violated [e.g., two-sex viability selection, fecundity selection, ecological complications, etc,]. Nevertheless FFTNS has been influential in providing a useful way of thinking about population evolution under natural selection The marginal fitness of an allele Before we can move on to a real dataset under FFTNS, we need to develop one more measure of fitness. In contrast to measuring the average fitness of the population, we also want to measure the average fitness of all the individuals in a population who bear a certain allele; this latter measure of fitness is called the MARGINAL FITNESS or AVERAGE AFFECT OF AN ALLELE. Comparison of the marginal fitness of an allele with the average fitness of the population tells us where an allele will increase, stay the same or decrease in a population. Consider a locus with two alleles: Allele A (freq = p) Allele a (freq = q) The marginal fitness of the a allele (Wa) is: Wa = q(Waa) + p(WAa) Now consider a locus with three alleles: Allele A (freq = p) Allele a (freq = q) Allele c (freq = r) The marginal fitness of the a allele (Wa) is: Wa = q(Waa) + p(WAa) + r(Wac) The value of this system is that we can extend it to any number of alleles at a locus and do a quick calculation to determine the influence of selection on an allele Wa Wa Wa - = 0; no change in frequency of a a allele increases in frequency < 0; the a allele decreases in frequency > 0; the The details of this relationship come from FFTNS; you are referred to population genetic texts such as Hartl and Clark for a more complete description. Adaptation in human populations: sickle cell haemoglobin One of the classic examples of natural selection is overdominant selection for the A and S alleles of the beta-chain of Haemoglobin in humans of West Africa. The reason for such selection appears to be a degree of resistance to the malaria parasite (P. falciparum), whose merozoite stage infects red blood cells. The three possible genotypes, their phenotypic effects, and their fitness values are shown below. Genotypes AA AS SS Blood Phenotype Normal 40% sickling of RBCs Sickle cell anaemia Mortality1 moderate Low very high Fitness1 0.89 1 0.2 1: Fitness and mortality are estimated as an average over 72 west African populations of humans. Data from Cavalli-Sforza and Bodmer (1971). Sickle-cell anaemia in SS heterozygote is so detrimental that fitness is very low in all environments. Heterozygotes, AS, experience about 40% of RBCs sickling during an episode. Sickling of a significant proportion of cells will disrupt the development of the parasite. Although this is not sufficient to kill all the parasites, it slows reproduction enough for immune system to mount an effective attack on the remaining parasites. Based on the data in the above table, the fitness topography is easily determined. Perhaps surprisingly, fitness of the population is maximized by a frequency of the S allele of just 0.12. Adaptive topography for A and S alleles in malarial environment As before, the fitness of the population can only go “uphill”. Marginal fitness calculations verify this result. Initial freq of S = 0.01: WS = 0.99 and = 0.89 WS - = 0.10; the S allele will increase Initial freq of S = 0.25: WS = 0.8 and = 0.88 WS - = -0.088; the S allele will decrease Peak in average fitness of population The equilibrium frequency of heterozygotes is 0.212, and the average fitness of the population ( ) is 0.91. This means that at equilibrium only 20% of the population is protected! This is because many people with the normal RBCs survive to reproduce (the average chance of survival to reproduction is 0.91). Some populations in West Africa have a third allele, C, that is relevant to malarial resistance. Individuals homozygous for the C allele have resistance to malaria but no severe anaemic affects. Let’s look at what happens to a population where the C allele occurs, and the S allele is not present. Genotypes AA Blood Phenotype Normal Fitness 0.89 Data from Cavalli-Sforza and Bodmer (1971). AC Normal 0.89 CC Resistant 1 Adaptive topography for A and C alleles in malarial environment Marginal fitness calculations verify the plot. Initial freq of C = 0.01: WC = 0.891 and = 0.890 WC - = 0.001; the C allele will increase [slowly] Initial freq of C = 0.25: WC = 0.917 and = 0.90 WC - Frequency of “C” allele = 0.02; the C allele will increase Peak in average fitness of population In the case of the C allele above, the population will go to fixation, with 100% of the population protected (as compared with only 20% protection under AS over dominance). An interesting phenomenon is observed when we consider a population where both the S and C alleles are present. The fitness values for West Africa are as follows. Genotypes AA AS SS AC SC CC Frequency p2 2pq q2 2pr 2qr r2 Fitness 0.89 1 .2 .89 .71 1.31 Mortality1 moderate low Very high moderate moderate low Anaemia none some severe none some none 1: Fitness and mortality are estimated as an average over 72 west African populations of humans. Data from Cavalli-Sforza and Bodmer (1971). Consider the case where the population is at equilibrium for AS, and the C allele occurs by mutation. To investigate evolution in this case we can compare the marginal fitness of the S and A alleles to the average fitness of a population with both alleles present. Frequency of A = 0.879 = p Frequency of S = 0.120 = q Frequency of C = 0.001 = r The marginal fitness of the S allele is: WS = q(WSS) + p(WAS) + r(WSC) WS = 0.12(0.2) + 0.879(1) + 0.001(0.71) WS = 0.903 The marginal fitness of the C allele is: WC = r(WCC) + p(WAC) + q(WSC) WC = 0.001(1.31) + 0.879(0.89) + 0.12(0.71) WC = 0.869 The average fitness of the population is computed in the usual way. = 0.903 The fate of the S allele: WS - = 0; no change in the population The fate of the C allele: WC - = -0.03; the C allele cannot invade! What if both the S and C allele are rare? Frequency of A = 0.879 = p Frequency of S = 0.05 = q Frequency of C = 0.01 = r WS = 0.957 WC = 0.885 = 0.898 WS WC - = 0.06; The S allele invades! = -0.01; the C allele cannot invade! The result is that natural selection will push this population to a strategy where only 20% are protected from malaria, despite an optimal solution of 100% protection (i.e., fixing the C allele) being available. We can understand this result if we take a look at the adaptive topography for these three alleles: Our example from above comes from the three allele haemoglobin case. S allele In this region the C allele only needs to be a littler more than 10% The C allele has to start with a high frequency in order for it to invade C allele A allele The fitness valley prevents the population from achieving the globally optimum state when the initial frequency of the C allele is low. Natural selection, on its own, simply cannot cause the population to cross this region. Once it begins going up the local peak, in the direction of a stable AS polymorphism, it cannot go back (FFTNS). Let’s check it out by testing the marginal fitness of the C allele when it is in a higher frequency in the population. Frequency of A = 0.64 = p Frequency of S = 0.25 = q Frequency of C = 0.11 = r WS = 0.768 WC = 0.891 = 0.877 WS - = -0.10; The S allele is a gonner! WC - = 0.01; the C invades! The reason for the difference between the C and S alleles is due to the effectiveness of selection on heterozygotes. When an allele is rare it is most often found in the heterozygote. In the case of the AS heterozygote (fitness = 1), the phenotype is beneficial; in the case of the AC (fitness = 0.89) and SC (fitness = 0.71) heterozygote the phenotype is deleterious. Because most of the C alleles are found in the heterozygous genotypes when it is rare, it cannot invade a population that also has S alleles. We have seen that the C allele can invade when its allele frequency exceeds 10%. This represents the far side of a fitness valley. Once the C allele gets across this valley, FFTNS take over and the population moves to the global peak, where 100% have CC genotype. We noted earlier that natural selection cannot get the population across the valley on its own. Two other population genetic forces can provide a means of getting the population across: 1. Strong drift effect. A large change in frequencies due to a bottleneck or a founder event could get the frequency of the C allele high enough in one generation that it comes under the influence of the global peak. 2. Inbreeding: Inbreeding changes genotype frequencies, not allele frequencies. The adaptive topography changes because dominance effects are reduced (homozygosity is increased); this favours the C allele and disfavours the S allele. 3. Large Migration event of individual carrying the C allele in their genome. Wright’s shifting balance model of adaptive evolution The notion of a complex adaptive topography, with both adaptive peaks and valleys, is actually due to Sewall Wright, who proposed the shifting balance theory (SBT) of evolution. Wright (1932) was motivated by a particular problem: “The problem as I see it is that of a mechanism by which species may continually find its way from lower to higher peaks…” Wright (1932, 1977) proposed the idea of an adaptive topography as a conceptual model of the relationship between the mean fitness of a population and the frequencies of the involved genotypes. Wright’s solution was a model of adaptation he called the SHIFTING BALANCE THEORY; a model that proposes how a population can move from one locally stable peak in mean fitness to another, and in the process cross an adaptive valley, by means of genetic drift and population selection. In SBT genetic drift provides the mechanism for a population to “drift” about the local peak it finds itself on. You can think of genetic drift as allowing a population to explore the local surface of the adaptive topography; this is not possible under FFTNS, as that model is always kept on the peak by selection. On rare occasions, with a strong enough drift effect; a population can get across a valley. Before we get to the details of SBT, we must review its underlying assumptions: Assumption 1: A large amount of multi-locus polymorphism is maintained in an equilibrium state. This assumption is conditional on two premises: (i) such variation is relevant to fitness (i.e., generally not neutral; after all, this is a model of adaptation); and (ii) fitness variation is relevant to “minor factors”. Minor factors refer to small fitness consequences [small s], mutation selection equilibrium points [low frequencies], recurrent mutations, etc. Assumption 2: A single genes has effects on multiple phenotypic characters. This is called PLEIOTROPY. Assumption 3: A complex adaptive landscape with multiple fitness peaks and valleys. Assumption 4: Multiple, partially isolated populations. This increases the chance that an allele will get across a fitness saddle and get fixed in one of the local populations. Let’s take a look at a single population under assumptions 1, 2, and 3, and with a very large effective size (Ne). In this case drift does not move them around much. The figures below illustrate how such a population will appear on an adaptive topography. A small portion of a hypothetical adaptive topography. The lines are fitness contours This portion has two peaks (+) and two valleys (-). The landscape is a multidimensional field of allele frequencies, represented in 2 dimensions for easy visualization. Mass selection under uniform conditions. The population (shaded) is held tightly to the peak This case: 1. effective selection (Nes very large) 2. low mutation (Neµ lower) 3. drift is NOT moving the popn (Ne very large) Mutation pressure pushes the population out in all directions relative to the case above This case: 1. less effective selection (Nes still large) 2. more mutation (Neµ higher) 3. drift is NOT moving the popn (Ne very large) Now let’s add in the effects of genetic drift by substantially reducing the effective size of the population. This population has a very small effective size. It has very high levels of population inbreeding and moves more erratically over the landscape as movement is purely by drift. Such a population will be small with little variation. Over time, this population will fix too many deleterious mutations, leading to degeneration and extinction. This population has an intermediate effective size. It also movers over the landscape but stays in the vicinity of the peak. This population is in a perpetual state of “SHIFTING BALANCE” of altered gene frequencies and selection coefficients. On very rare occasions such a population could cross a “saddle” on the adaptive topography. Now let’s put a structured species on the landscape. This represents a very large population that is subdivided into numerous local populations. The local populations are connected to each other via individuals that move among populations and reproduce (i.e., gene flow). Each local population is part of a continually shifting balance. The complete model of SBT is specified by three phases of evolution: 1. Phase of genetic drift: Random drift moves the population, and recurrent mutation spreads it out. Selection acts to push the population back up towards the peak. Migration among local populations prevents them from evolving independently of each other. So alleles lost via drift in some populations can be regained by mutation and migration between populations. This results in a continually shifting balance that allows “exploration” of the local topography. 2. Phase of Mass selection: Occasionally a population will cross a saddle in the adaptive topography. Once across, selection becomes a strong force to move the population close to the “new peak”. This transition is expected to occur rapidly, until a new shifting balance is reached. conditions of phase1. Thus this phase ends with a return to the 3. Phase of inter-population selection: Once a transition to a new peak is made, gene flow among populations is a force for moving the population. If there is sufficient excess reproduction coming from the population on the new peak, the entire set of local populations could be shift to an equilibrium surrounding the new peak. This effect is based on the assumption that a population that comes under the control of a higher fitness peak produced more offspring than populations under the control of relatively lower fitness peaks. Greater success and reproduction of individuals with equilibrium allele frequencies determined by the peak in the lower right quadrant means that there is net change in the allele frequencies of all the populations in that direction. The direction of the change to a new equilibrium is indicated by the arrow. Difficulties with SBT SBT sounds great; however, the theory has been controversial since the very beginning. R. A. Fisher, one of the founders of the discipline of population genetics, immediately criticized the theory on two points: (i) species never get stuck on peaks, as there are always paths that are going up; and (ii) the effective population sizes in nature were too large for drift to move them around in the required amount. These two points are not conclusive. Specific cases appear consistent with the notion of valleys between peaks (e.g., C allele in human haemoglobin). Moreover population structures and effective population sizes can vary considerable over macroevolution time periods; so SBT needs only for the appropriate conditions to be met at some point in time. Recently, more focused criticisms have been levelled at SBT: 1. Low migration rates are required for exploration and transition during phases 1 and 2, yet higher migration rates are necessary to complete phase 3. 2. Population structures typical of natural populations seem to be too small for phase 1 3. Group selection is a weak force for evolution, and hence unlikely to result in a shift in equilibrium. Because it is weak, an extremely high amount of migration is required among sub-populations for inter-population selection to work as suggested in phase 3. 4. Alternatives seem more likely. Alternatives to genetic drift as the force for movement on the topography Wright (1932) initially proposed several alternatives: 1. Change in environment changes the landscape. Essentially, the peak moves out from under you, or a new “ridge” appears in the fitness landscape. Remember industrial melanism; dramatic changes in the fitness topography clearly can result from changes in the environment. 2. Mutation. In this case a mutation occurs that allows a population to evolve to a higher fitness without going through a deleterious intermediate stage. Perhaps a population has a rare recessive allele that would be beneficial but the heterozygotes have severely reduced fitness. Such a population might need to “wait” for a compensatory mutation to occur. We know that compensatory mutations can be very strongly selected (e.g., the stem structures of RNA molecules) 3. Change in the strength of selection. In this case the structure of the adaptive topography stays the same but the relative height of the peaks, or depth of valleys, changes. A flattening of the topography would allow a population to move under FFTNS. These alternatives require a population to “wait” in time for such events to occur. The criticisms in the previous section point out the conditions for SBT that might not be optimal in natural populations. However, the alternative models require populations to “wait” for changes in the landscape or for certain mutations; why not expect that populations must also “wait” to work on those occasions when structure, or levels of gene flow, are appropriate for SBT? It is worth comparing SBT with Neutral theory one more time. SBT concerns events that are extremely rare as compared with events modeled under neutral theory. The rarity of such events means that the chance of observing data from populations that reflect evolution under SBT is very low. Hence, SBT does not provide us with many predictions we can test directly. Alternatively, most natural populations are likely to exhibit some data relevant to tests of neutral evolution. Some have suggested that SBT can neither be proven nor disproved (e.g., Whitlock 2002). Lastly, we cannot directly observe adaptive valleys, peaks, saddles etc, so we can never be completely sure if a population ever really did cross a valley, as opposed to following FFTNS to a fitness peak. Despite these difficulties, SBT has been important in evolution. It has provided us with a useful conceptual model, and encouraged further development of the theories of genetic drift, group selection, and complex adaptive topographies. SBT and the evolution of heterostyly [I will try to get it into the lecture in the future; no time this year]