Population Genetics I

Transcription

1 Poulation Genetics I The material resented here considers a single diloid genetic locus, ith to alleles A and a; their relative frequencies in the oulation ill be denoted as and q (ith q 1 ). Hardy-Weinberg equilibrium (revie) Requirements: 1. no selection 2. no mutation 3. no stochastic (random) effects, i.e. infinitely large oulation 4. no emigration or immigration 5. random mating If these conditions hold, allele frequencies ill not change and genotye frequencies ill be AA 2, Aa 2, aa q 2 (1) Effect of selection In the folloing selection is assumed to be constant over time, and density- and frequencyindeendent: each genotye has a constant fitness, regardless of the generation, the oulation density, and the relative frequencies of the genotyes. The main results ill be: 1. equilibria are: a) fixation of one allele (i.e. 1 or 0), or b) a mixture (olymorhism), if the heterozygote is either more fit than either homozygote (in hich case the olymorhism is stable), or less fit than either homozygote (in hich case the olymorhism is unstable). 2. the rate of evolution (change in mean fitness) is (aroximately) roortional to the genetic variance times the strength of selection (Fisher s fundamental theorem of natural selection ) 3. selection maximizes (aroximately) the mean fitness of the oulation 4. the rate of change in the relative frequency of a rare allele is sloer for a recessive allele than a dominant allele. 1 of 8

2 Selection model t relative frequency of A allele in current generation q t relative frequency of a allele in current generation AA relative survival of AA genotye Aa relative survival of Aa genotye aa relative survival of aa genotye Relative frequencies in the next generation: First, multily each genotye frequency (in the current generation) by the relative survival: AA: t 2 AA Aa: aa: 2 t q t Aa q t 2 aa But the total size of the oulation ill have changed, so these need to be rescaled to be relative frequencies, ith t+1 + q t+1 1. Average survival ill be t 2 t AA + 2 t q t Aa + q 2 t aa (2) The ne relative genotye frequencies ill be the frequencies given above divided by this average survival, i.e. t AA: t 2 AA /, etc. Since each individual has to coies of this locus, the frequency of the A allele in the next generation ill be (ith N being the oulation size): 2N frequency of AA + N frequency of Aa N t 1 frequency of AA 2 1 t AA t q t Aa t ( t AA + q t Aa ) ( frequency of Aa) 2 (3). 2 of 8

3 No define A;t as the mean fitness of A alleles, given the current t : A;t (robability a given A allele is in AA) x (fitness of AA) + (robability a given A allele is in Aa) x (fitness of Aa) (4) A;t t AA + q t Aa (5) So no the relative frequency of the A allele in the next generation can be exressed as t At ; t 1 (6) The ratio At ; can be interreted as the mean relative fitness of A alleles (in generation t), and is the groth rate of. The receding equation relating t+1 and t is analogous to the equation for geometric groth (N t+1 λn t ) excet that in this genetic equation the mean relative fitness (i.e. groth rate) is not constant: it deends on t. Change in relative frequencies Define as the change in from generation t to t+1, so t + 1 t t At ; t At ; t t --- ( At ; ) (7) Temorarily droing the t subscrits for reduce clutter, and substituting from above for A;t and, this gives t --- [( AA + q Aa ) ( 2 AA + 2 Aa + q 2 aa ) (8) Gathering together terms involving each of the s, and simlifying the s and qs gives. 3 of 8

4 --- [( 2 ) AA + ( q 2) Aa q 2 aa --- [ ( 1 ) AA + q( 1 2) Aa q 2 aa --- [ AA + q( 1 ) Aa q 2 aa --- [ q AA + qq ( ) Aa q 2 aa (9) A q from each of the summands in the brackets can be ulled out of the brackets: [ AA + ( q ) Aa q aa (10) A slight rearrangement gives the final exression: [ ( AA Aa ) + q ( Aa aa ) (11) Equilibrium The equilibrium frequency of the A allele, ˆ, ill be the value of at hich equals 0, i.e. there is no change in frequencies. From equation (11) above for, e can see that there are to trivial equilbria: if either 0 or 1 (so q 0), 0. This makes sense: if only one allele is resent in the oulation (and e are assuming there is no mutation), there can be no change in allele frequencies. If there is a non-trivial equilibrium ith both alleles resent, it ill be the value of at hich the term in brackets in equation (11) equals 0: ˆ ( AA Aa ) + ( 1 ˆ ) ( Aa aa ) 0 ˆ [( AA Aa ) ( Aa aa ) + ( Aa aa ) 0 ˆ [ AA + aa 2 Aa ( Aa aa ) ˆ Aa aa (12) Aa ( AA + aa ) For ˆ to be greater than 0, the numerator and denominator of the final exression in (12) must have the same sign (both ositive or both negative). For ˆ to be less than 1, the to terms in the ˆ Aa aa ( Aa AA ) + ( Aa aa ). 4 of 8

5 denominator must have the same sign, so that the absolute value of the denominator is greater than that of the numerator. Combining these conditions gives: 0 < ˆ < 1 requires either 1. Aa > AA and Aa > aa : the heterozygote has the greatest fitness or 2. Aa < AA and Aa < aa : the heterozygote has the loest fitness In ords: under the assumtions of this simle model, a olymorhism both alleles remaining in the oulation requires either heterozygote advantage or heterozygote disadvantage. (It ill be seen belo that the latter is unstable, but the former is stable.) Rate of change in and change in average fitness From equation (2) above, t 2 t AA + 2 t q t Aa + q 2 t aa. Exressing the qs in terms of (and again droing the t subscrits), this is 2 AA + 2( 1 ) Aa + ( 1 ) 2 aa 2 AA + 2 Aa 2 2 Aa + aa 2 aa + 2 aa (13) We can then take the derivative of this to get an exression for ho the average fitness of the oulation changes as the allele frequencies change: d d 2AA Aa Aa aa aa 2[ AA + Aa 2 Aa aa + aa 2[ ( AA Aa ) + ( 1 ) ( Aa aa ) (14) d No note that this exression for is 2 times the bracketed term in equation (11) for Δ. So d e get d d (15) To interret this result, note that the roduct is roortional to the variance in allele frequencies (from the binomial formula) and d measures the strength of selection: ho strongly fitness d varies ith allele frequency. Equation (15) then says that the rate of evolution is roortional to the roduct of genetic variability x strength of selection. 5 of 8

6 The result in equation (15) also tells us that (under the assumtions of this simle model) natural selection ill maximize mean fitness. To see this, note that at the non-trivial equilibrium, d ith both and q greater than 0, 0 requires that 0 ; this in turn imlies that is at d a maximum (actually sloe of 0 could also indicate a minimum, but this is not the case here). Dominance So far selection as defined simly by the three genotye fitnesses, AA, Aa and aa, hich ere left entirely unsecified. To exlore the effects of dominance, e can secify these three fitnesses using to arameters, one to reresent the difference in fitness beteen the to homozygotes and the second to reresent the degree of dominance, i.e. the fitness of the heterozygote. Secifically, let AA 1 Aa 1 ( h s) aa 1 s Let s assume s is ositive, so that the a allele is deleterious (aa has loer fitness than AA). The magnitude of s measures this deleterious effect. The arameter h, together ith s, determines the fitness of the heterozygote. If h 0, the heterozygote has fitness 1, the same as the AA homozygote: the A allele is comletely dominant. Conversely if h 1, the fitness of the heterozygote is the same as that of the aa homozygote (1-s): the a allele is comletely dominant. If 0 < h < 1, the heterozygote s fitness is somehere beteen those of the homozygotes: there is incomlete dominance. If h ½ exactly, the alleles have additive effects: the heterozygote fitness is the average of the to homozygotes fitnesses. If h < 0, the heterozygote s fitness is greater than 1, and thus greater than that of the AA homozygote; this is called overdominance. Similarly, if h > 1, the heterozygote has loer fitness than the aa homozygote (and of course also the AA homozygote); this is underdominance. Substituting these exressions for the genotye fitnesses into the exression for the change in allele frequency (equation 11) gives: [ ( AA Aa ) + q ( Aa aa ) [ ( 1 ( 1 hs) ) + q (( 1 hs) ( 1 s) ) [ hs + qs( 1 h) s [ h + q( 1 h) (16). 6 of 8

7 The quantity in this equation outside the brackets is never negative, and equals 0 only if either only one allele is resent ( 0 or 1) or there is no selection (s 0). The sign of therefore deends on the sign of the term in brackets in equation (1). Several situations are ossible, deending on the value of h (i.e. on the degree and nature of dominance): 1. If h < 0, i.e. overdominance or heterozygote advantage: When is large, the h term ill dominate and the bracketed term ill be negative: ill decrease. Conversely, hen is small, the q(1-h) term ill dominate and the bracketed term ill be ositive: ill increase. This constitutes stabilizing selection for a olymorhism (both alleles resent, i.e. 0 < < 1): if A alleles are rare ( small), selection increases their frequency (increases ), and if A alleles are at very high frequency ( large), selection reduces them (reduces ). 2. If h > 1, i.e. underdominance or heterozygote disadvantage: When is large, the h term ill dominate and the bracketed term ill be ositive: ill increase. Conversely, hen is small, the q(1-h) term ill dominate and the bracketed term ill be negative: ill decrease. This constitutes disrutive selection: there is a olymorhic equilibrium, but it is unstable. If is above the equilibrium, selection ill cause it increase further, until the A allele becomes fixed in the oulation; conversely, if is belo the equilibrium, selection ill cause it to decrease further and eventually the a allele ill become fixed. 3. If 0 h 1: [ h + q( 1 h) > 0, so > 0 for all values of (other than 0 or 1). In this case, reresenting the entire sectrum from comlete dominance of A to comlete dominance of a, there is directional selection for A: alays increases, and ill go to fixation ( 1). a) If h is 0, or very small (A is dominant): [ h + q( 1 h) q so Δ 2 s ( 1 ) 2 s In this case, as shon by the red curve in the figure, the resonse to selection is fastest is largest hen is smallish, i.e. A is rare. b) If h is 1, or nearly so (a is dominant): [ h + q( 1 h) so Δ 2 qs ( 1 )s In this case (black curve), the resonse to selection is fastest hen is largish, i.e. A is common.. 7 of 8

8 The rate of evolution in these latter cases, ith 0 h 1, is fastest hen the rare allele is the dominant one. The rare allele ill occur mostly in heterozygotes. If it is dominant, its fitness effect still ill be exressed even in the heterozygotes, and selection can act on it (for it, if advantageous, against it if deleterious). If, though, the rare allele is recessive, those heterozygotes and thus the rare recessive allele also ill have similar fitness to the common allele. With little variation in fitness, evolution ill be slo. In essence, a rare recessive allele hides from selection because its deleterious effect occurs only in the very rare homozygote recessives. Putting these effects another ay: If the advantageous allele is dominant it ill increase quickly from rarity (since it is exressed in the heterozygotes even hile it is rare), but if the advantageous allele is common, elimination of the deleterious recessive allele ill be slo (since it rarely is exressed). Conversely, if the advantageous allele is recessive, its increase from rarity ill be slo but once it becomes common, it ill go to fixation (the deleterious allele ill be eliminated) raidly.. 8 of 8