INTRODUCTION TO POPULATION GENETICS (2024)

INTRODUCTION TO POPULATION GENETICS

In this and the next few lectures we will be dealingwith population genetics which generally views evolution as changesin the genetic makeup of populations. This is a somewhat reductionist approach:if we could understand the combined action of the forces that change genefrequencies in populations, and then let this run over many generationswe might understand long term trends in evolution. Continuing debate: canthe processes of microevolution account for the patterns of macroevolution?Population genetics is an elegant set of mathematical models developedby largely by R. A. Fisher and J. B. S. Haldane in England and Sewall Wrightin the US. Continues to be developed by many mathematical, theoreticaland experimental biologists today (see J. Crow and M. Kimura Introductionto Population Genetics Theory).

In very simple terms, population genetics involvesanalyses of the interactions between predictable, "deterministic"evolutionary forces and unpredictable, random, "stochastic" forces.The deterministic forces are often referred to as "linear pressures"because they tend to push allele frequencies in one direction (up, downor towards the middle). Important forces of this nature are selection,mutation, gene flow, meiotic drive (unequal transmission of certainalleles [a form of selection]), nonrandom mating (also a form ofselection). The primary stochastic evolutionary force is genetic driftwhich is due to the random sampling of individuals (and genes) in smallpopulations. It is important to realize that the deterministic forces mayact together or against one another (e.g., selection may "try"to eliminate an allele that is pushed into the population by recurrentmutation). Moreover, deterministic forces may act with or against geneticdrift, to determine the frequencies of alleles and genotypes in populations(e.g., gene flow tends to hom*ogenize different populations while drifttends to make them different). Hence, the interaction of these forces iswhat we are really interested in (a later lecture), but since this canget very complex mathematically, we will start by analyzing one force ata time.

To begin we need to understand some simple populationgenetic "bookkeeping." Consider a locus with two alleles(alternative forms of the DNA sequence that "reside" at thatlocus, e.g., one from mother other from father). Now consider a populationof N individuals (N=population size); this means that there are2N alleles in the population. We can thus talk about genotypefrequencies and allele frequencies. In a population of N = 100individuals, if there are 25 AA, 50 Aa and 25 aa, then the genotype frequenciesare f(AA) = 0.25, f(Aa) = 0.50 and f(aa) = 0.25. If we count up the individualalleles there are 200 of them (because there are 100 diploid individuals).Hence to determine the frequency of the "A" allele we have tocount each individual "A" allele that is specified in each diploidgenotype. We get f(A) = (25+25+50) / 200 = 0.5. We generally refer to thefrequency of the "A" allele as f(A) = p; the frequency of the"a" allele is f(a) = q. Note that p = (1-q) because the sum ofthe allele frequencies must be 1.0. Common "language errors"in learning population genetics are to refer to the "p" allelewhen you really mean the "A" allele, or to say "the frequencyof the p allele" when you really mean: "...p, the frequency ofthe "A" allele..." Got it?? Good.

Since evolution is change in the genetic makeup ofa population over time, a general approach to modeling this is to determinethe allele and genotype frequencies in the next generation (p_t+1)that result from the action of a force on those frequencies in the currentgeneration (p_t). Thus :

Consider a simplistic life cycle where thegenotypes (a single locus way of referring to adults) produce gametes.These gametes mate to form new genotypes (=adults). See 5.1, pg.93 and 5.3, pg. 99. The relationship between allele frequencies (sometimescalled "gene" frequencies) and genotype frequencies is determinedby the Hardy Weinberg Theorem which defines the probabilities bywhich gametes will join to produce genotypes. Consider a coin toss: probabilityof a head = 0.5; of a tail = 0.5; prob. of two heads = 0.5x0.5 = 0.25;prob. of one head and one tail = 0.5x0.5 = 0.25, etc. Each coin is analogousto the type of allele you can get from one of your diploid parents; thetossing of two coins is analogous to the mating of two individuals to producefour possible genotypes (but heads,tails is the same as tails,heads). Nowconsider a roll of the dice. The probability of each face is 1/6, and isactually analogous to cases where more than two different alleles existin the population at a given locus. The probability of any combinationis 1/6 x 1/6 = 1/36. But recall that there can be more than one way toget many of the combinations (2,3 is the same as 3,2). The general expressionfor the number of genotypes that can be assembled from n different allelesis: [n(n+1)/2].

Assumptions of Hardy Weinberg: 1) diploidsexual population 2) infinite size, 3) random mating, 4)no selection, migration or mutation. This is a Null Model; obviouslysome of these assumptions will not hold in real biological situations.The theorem is useful for comparison to real-world situations where deviationsfrom expectation may point to the action of certain evolutionary forces(e.g., mutation selection, genetic drift, nonrandom mating, etc.). Usea Punnet square to determine genotype frequencies: f(AA) = p²,f(Aa) = 2pq, f(aa) = q² and p²+ 2pq + q² = 1 Learn this: One generation of randommating restores Hardy Weinberg equilibrium. H-W equilibriumis when the genotype frequencies are in the proportions expected basedon the allele frequencies as determined by the relation p² +2pq + q². This is derived more thoroughly in table 5.1, andaccompanying text, pg. 94.

Example: consider a sample of 100 individuals withthe following genotype frequencies:

Observed are different from expected, thus some forcemust be at work to change frequencies.

NATURAL SELECTION

Selection occurs because different genotypesexhibit differential survivorship and/or reproduction. If we consider acontinuously distributed trait (e.g., wing length, weight) with a stronggenetic basis, the response to selection can be characterized by wherein the distribution the "most fit" (greatest survivorship&reproduction)individuals lie. If after selection one extreme is most fit thisis directional selection; if the intermediate phenotypesare the most fit this is stabilizing selection; if both extremesare the most fit this is disruptive selection.

R. A. Fisher proposed a simple bookkeeping, or populationgenetics, approach for one locus with two alleles: we have AA, Aa and aain frequencies p², 2pq, q² . Define l_iias the genotype-specific probability of survivorship, mii as the genotype-specificfecundity. We build a model that will predict the frequencies of allelesthat will be put into the gamete pool given some starting frequenciesat the preceding zygote stage;

The gamete column is what determines the frequenciesof A and a that will be put into the gamete pool for mating to build thenext generation's genotypes. We can simplify by referring to the fitnessof a genotype as w_ii = m_ii l_ii . Thesefitness values will determine the contribution of that genotype to thenext generation. Thus the frequency of A allele in the next generationp_t+1 (sometimes referred to as p') would be the contributionsfrom those genotypes carrying the A allele divided by all alleles contributedby all genotypes:

p_t+1 = (w_AA p²+ w_Aa pq)/(w_AA p² + w_Aa 2pq+ w_aa q²). Or for thea allele,

q_t+1 = (w_aa q² +w_Aa pq)/(w_AA p² + w_Aa 2pq +w_aa q²). Note thatthe heterozygotes are not 2pq but pq because in each case they are onlybeing considered for the one allele in question. If we scale all wii'ssuch that the largest = 1.0 we refer to these as the relative fitnessesof the genotypes. A worked example where p = .4, q = .6 and w_AA= 1.0 w_Aa = 0.8 w_aa = 0.6:

Genotype frequencies are p² = 0.16, 2pq= 0.48, q² =0.36, thus:

p_t+1 = ((.16 x 1.0) + (.24 x .8))/((.16x 1.0) + (.48 x .8) + (.36 x .6)) = .463; so q = .537 and thus f(AA)_t+1= .215, f(Aa)_t+1 = .497 and f(aa)_t+1 = .288. Noteboth allele frequencies and genotype frequencies have changed (compareto what we saw with inbreeding). This can be continued with the new allelefrequencies and so on. When will the selection process stop? when Dp= 0, i.e., when p_t+1 = p_t . In some situations thiswill stop only when one allele is selected out of the population (p = 1.0).

Now we can consider various regimes of selection(s = selection coefficient, (1-s) is fitness):

Substitute the fitnesses (w_ii) in conditionI above into the expression Dp= p_t+1 - p_t and prove for yourself that the equationson page 101 (eqn. 5.5) is related to the expression for p_t+1shown above. First three are directional in that selection stops only whenallele is eliminated. In I the elimination process slows down because asq becomes small the a alleles are usually in heterozygote state and thereis no phenotypic variance. In II selection is slow at first because withq small most genotypes are AA so there is low phenotypic variance; as selectioneliminates A alleles q increases and the frequency of the favored genotype(aa) increases so selection accelerates. III is like the worked examplerun to fixation/loss. IV is known as balancing selection due tooverdominance (heterozygotes are "more" than either hom*ozygote).Both alleles maintained in population by selection. This is an exampleof a polymorphic equilibrium (fixation/loss is also an equilibriumcondition but it is not polymorphic). The frequencies of the alleles atequilibrium will be:

p_equil = t/(s + t); q_equil= s/(s+t).

Classic example = sickle cell anemia. A=normalallele; S=sickle allele. S should be eliminated because sickle cell anemialowers fitness. S is maintained where malarial agent (Plasmodium falciparum)exists because AS heterozygotes are resistant to malaria. Note that S alleleis very low frequency where there is no malaria (the selective coefficientof S is different because the environment is different). See figure5.8, pg. 120; table 5.9, pg. 119.

Another way that genetic variation can be maintainedis through multiple niche polymorphism (polymorphism maintainedby environmental heterogeneity in selection coefficients). If differentgenotypes are favored in different niches, patches or habitats, both allelescan be maintained.

Heterozygotes will have the highest averagefitness although they are not the most fit in either habitat (see figure5.12, pg. 124). The same dynamics would apply to temporal heterogeneity(spring and fall; winter and summer) assuming that selection did not eliminateone allele during the first period of selection. Classic example of temporalheterogeneity: third chromosome inversions of Drosophila pseudoobscurastudied by T. Dobzhansky. Different chromosomal arrangements ("Standard"and "Chiricahua") show reciprocal frequency changes during theyear.

Yet another way to maintain variation by selectionis through frequency dependent selection.

If an allele's fitness is not constant but increasesas it gets rare this will drive the allele back to higher frequency. Seefigure 5.9, pg. 121. Example: allele may give a new or distinct phenotypethat predators ignore because they search for food using a "searchimage" (e.g., I like the green ones).

Most (by no means all) evolutionary biologists believethat selection plays a major role in shaping organic diversity, but itis often difficult to "see" selection. One reason is that selectioncoefficients can be quite small (1-s ~1) so the response to selection issmall. When selection coefficients are large Dpcan be large, but the problem here is that with directional selection fixationis reached in a few generations and we still can't "see" selectionunless we are lucky enough to catch a population in the middle of the periodof rapid change.

What affects the rate of change under selection?Recall that Dp = p_t+1- p_t

Dp = [(w_AAp2 + w_Aa pq)/(w_AA p2 + w_Aa 2pq + w_aaq2)] - p . With some simple algebra we can rearrange this

It states that the rate of evolution is proportionalto the genetic variance of the population. In the above example wehave not explicitly defined the fitnesses wiis or the dominance relationshipsand these can have a major effect on Dpas written above.

Another important observation for looking at thisDp equation and pluggingin some values is that selection always increases the mean fitness ofthe population. For example with p=0.4, q=0.6 and w_AA=1,w_Aa=0.8 and w_aa=0.6, the mean fitness (w'bar') =0.76. After one generation of selection p' = 0.463 and q' = 0.537. Recalculatingw'bar' we get wbar_t+1 = 0.78, which is greater than 0.76. Whenwill this process stop? At fixation (or equilibrium with overdominance).

This treatment of the algebra of natural selectionillustrates what selection alone can do to allele and genotype frequencies.In the next lectures we will consider other evolutionary forces (mutationgene flow, genetic drift), how they act alone, and eventually, how theyinteract with each of the other evolutionary forces.