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Basic Population Geneticsand One and Two locus
models of Selection
Allele and Genotype Frequencies
62pi = freq(A i) = freq(AiAi) +
1 X
i=j
freq(AiA j )
Given genotype frequencies, we can always compute allelefrequencies, e.g.,
The converse is not true: given allele frequencies we cannot uniquely determine the genotype frequencies
For n alleles, there are n(n+1)/2 genotypes
If we are willing to assume random mating,
freq(AiA j ) =Ωp2
i for i = j
2pipj for i6= j
Hardy-Weinberg proportions
Hardy-Weinberg• Prediction of genotype frequencies from allele freqs
• Allele frequencies remain unchanged over generations, provided:
• Infinite population size (no genetic drift)
• No mutation
• No selection
• No migration
• Under HW conditions, a single generation of randommating gives genotype frequencies in Hardy-Weinbergproportions, and they remain forever in these proportions
Gametes and Gamete Frequencies
freq(AABB) = freq(ABjfather) freq(ABjmother)
freq(AaBB) =freq(ABjfather)freq(aBjmother)+freq(aBjfather)freq(ABjmother)
When we consider two (or more) loci, we follow gametes
Under random mating, gametes combine at random, e.g.
Major complication: Even under HW conditions, gametefrequencies can change over time
AB
AB
ab
ab
abAB
AB
abIn the F1, 50% AB gametes50 % ab gametes
If A and B are unlinked, the F2 gamete frequencies are
AB 25% ab 25% Ab 25% aB 25%
Thus, even under HW conditions, gamete frequencies change
Linkage disequilibrium
freq(AB) = freq(A) freq(B)freq(ABC) = freq(A)freq(B)freq(C)
Random mating and recombination eventually changesgamete frequencies so that they are in linkage equilibrium (LE).Once in LE, gamete frequencies do not change (unless actedon by other forces)
At LE, alleles in gametes are independent of each other:
When linkage disequilibrium (LD) present, alleles are nolonger independent --- knowing that one allele is in the gamete provides information on alleles at other loci
freq(AB)6= freq(A) freq(B)The disequilibrium between alleles A and B is given by
DA B = freq(AB) ° freq(A)freq(B)
freq(AB) = freq(A) freq(B) + DAB
D(t) = D(0)(1 c)t°
The Decay of Linkage Disequilibrium
The frequency of the AB gamete is given by
LE valueDeparture from LEIf recombination frequency between the A and B loci
is c, the disequilibrium in generation t is
Initial LD valueNote that D(t) -> zero, although the approach can beslow when c is very small
Key PointsUnder HW changes in allele frequencies are permanent
Changes due to LD are transient, and decay away ontheir own
Implications for trait selection: Changes in allelefrequencies produce permanent change
Changes in genotype frequencies induced by selectionthat are out of HW decay back to HW (at new allelefrequencies) when selection stops.
Genetic Drift
Random sampling of 2N gametes to form the N individuals making up the next generation results inchanges in allele frequencies.
This process, originally explored by Wright and Fisher,is called Genetic Drift.
Suppose there are currently i copies of allele A,so that freq(A) = p = i/(2N)
That probability that, following a generation ofrandom sampling, the freq of A is j/(2N) is
This probability follows binominal sampling,
Pr(i copies ! j copies) =N!
(N ° j)!j!
µiN
∂ j µN ° i
N
∂N ° j- -
)- ( ()
Hence, if the current allele frequency is p, theexpected allele frequency in the next generationis also p, but with sampling variance p(1-p)/(2N)
Thus, with N is large, the changes in allele frequencyover any generate are expected to be rather small
However, the cumulative effects of generations ofsuch sampling are very considerable.
Eventually, any random allele will either be lostfrom the population or fixed.
If the allele has initial frequency p, then
Pr(Fixation) = p
Pr(loss) = 1- p
The expected time to fixation is on order of4N generations.
If selection is sufficiently weak, it can be overpoweredby drift.
Selection
Single-locus selection
The basic building block is a single locus underselection. Think of this as a trait controlledby only a single gene
Individuals differ in fitness when they leavedifferent numbers of offspring
When the fitness of at least one genotypeis different from the others, selection occurs
Genotype AA Aa aa
Frequency(before selection)
p2 2p(1-p) (1-p)2
Fitness WAA WAa Waa
Frequency(after selection)
p2 WAA 2p(1-p) WAa (1-p)2Waa
One locus with two alleles
W W W
W
is the mean population fitness, the fitness of an randomindividual, e.g. = E[W]
Where = p2 WAA + 2p(1-p) WAa + (1-p)2WaaW
The new frequency p’ of A is just freq(AA after selection) + (1/2) freq(Aa after selection)
p0 =p2WA A +p(1 ° p)WAa
W= p
pWA A + (1 ° p)WAa
W
--
The fitness rankings determine the ultimate fateof an allele
If WXX > WXx > Wxx, allele X is fixed, x lost
If WXx > WXX, Wxx, selection maintains both X & x
Overdominant selection
0 1p
+
-
Δp0
+
-
Δp0
+
-
Δp0
+
-
Δp0
0 1p
0 1p0 1p
( ) ,A Directional Selection
WAA > WAa > Waa
( ) ,B Directional Selection
WAA < WAa < Waa
( ) ,C Overdominant Selection
WAA < WAa > Waa
( ) ,D Underdominant Selection
WAA > WAa < Waa
~p ~p
Class problemEquations 5.3c-5.3d in notes given approximateexpressions for the time to change from frequency p0 to p
Compute the time to change from frequency 0.1 to 0.9(i) Fitness are 1 : 1.01: 1.02(ii) Fitness are 1: 1:02: 1.02(iii)Fitness are 1: 1: 1.02
Wright’s formulaQuickTime™ and a
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Computes the change in allele frequency asas function of the change in mean fitness
Requires frequency-independence: Genotypefitnesses are independent of genotype frequencies,
d Wij / pi = 0
Note sign of change in p = sign of dW/dp
Application: Overdominant selection
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QuickTime™ and aTIFF (LZW) decompressor
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Application: Stabilizing selection
A common model for stabilizing selection on a trait is to use a normal-type curve for trait fitness
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As detailed in Example 5.6, we can use Wright’s formulato compute allele frequency change under this type ofselection
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This is selective underdominance! If p < 1/2, Δp <0and allele gets lost. If p > 1/2, Δp > 0 and allelefixed.
Hence, stabilizing selection on a trait controlled bymany loci removes variation!
p0i = pi
Wi
W; Wi =
nX
j=1
pjWi j ; W =nX
i=1
piWi
Multiple Alleles
Let pi = freq(Ai), Wij = fitness AiAj
Wi = marginal fitness of allele AiW = mean population fitess = E[Wi] = E[Wij]
If Wi > W, allele Ai increases in frequency
If a selective equilibrium exists, then Wi = W for all segregating alleles.
Fitness as the ultimate quantitative trait
Recall that the average excess of allele Ai is meantrait value in Ai carries minus the population mean
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Consider average excess inrelative fitness for Ai
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Allele frequency change is a function of the average excess of that allele
Allele frequency does not change when its averageexcess is zeroAt an equilibrium, all average excesses are zero.Hence, no variation in average excesses andthus no additive variation in fitness at equilibrium
Wright’s formula with multiple alleles
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Key: Note that the sign of dW/dpi does not determinesign( p). Thus an allele can change in a directionopposite to that favored by selection if the changeson the other alleles improve fitness more
Prelude to the multivariate breeder’s equation, R = G
General features with multiple allele selection
With Wij constant and random mating, mean fitnessalways increases
What about polymorphic equlibrium?
Require Wi = W1 for all i.
Kingman showed there are only either zero, one, orinfinitely many sets of equilibrium frequencies foran internal equilibrium
p1
p3p2
1
11
p1 + p2 + p3 = 1
cornerequilibrium
edgeequilibrium
internalequilibrium
Equilibrium behavior
Single internal equilibrium if W has exactly onepositive and at least one negative, eigenvaluee an eigenvalue of W if We = eOther equilibrium can potentially fall anywhereon the simplex of allele frequencies
Two-locus selection
When two (or more) loci are under selection, singlelocus theory no longer holds, because of linkagedisequilibrium (i.e., freq(ab) = freq(a)*freq(b)
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Consider the marginal fitness of the AA genotype
Note that this is a function of p = freq(A), q=freq(B),and D = freq(AB)-p*q. When D = 0 this reduces to
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Here, W(AA) is independent of freq(A) = p, andwe can use Wright’s formula to compute allelefrequency change.
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For two loci, follow gametes
The resulting recursion equations, even for thesimple two biallelic loci, do not have a generalsolution for their dynamics
When selection is strong and linkage (c) tight, results can be unpredictable
Mean fitness can decline under two-locus selection
At equilibrium
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If LD at equilibrium, the second term is nonzero andnot all gametes have the same marginal fitness. Notethat when c=0, this is just a 4 allele model, and allsegregating alleles have the same marginal fitness
If equilibrium LD is not zero, mean fitness is notat a local maximum. However, unless c is very small,it is usually close
In such cases, mean fitness decreases during thefinal approach to the equilirium (again, effectsusually small)
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Note complete additivity in the trait.
W(z) = 1 - s(z-2)2
Fitness function induces dominance and epistasis for a completely additive trait
At equilibrium, no additive variance in FITNESS -- stillcould have lots of additive variance in the trait.
Example 5.11. Even apparently simple models can havecomplex behavior.
FFT: Fisher’s Fundamental Theorem
What, in general can be said above the behavior ofmultilocus systems under selection?
Other than they are complex, no general statement!
Some rough rules arise under certain generalizations,such as weak selection -- weak selection on each individual locus, selection on the trait could be strong.
One such rule, widely abused, is Fisher’s Fundamentaltheorem
Karlin: “FFT is neither fundamental nor a theorem”
Fisher: “The rate of increase in fitness of anyorganism at any time is equal to its genetic variance in fitness at that time”
Classical Interpretation: Wbar = VarA(fitness)
This interpretation holds exactly only under restricted conditions, but is often a good approximate descriptor
Important corollary holds under very general conditions: in the absence of new variation from mutation or other sources such as migration, selection is expected to eventuallyremove all additive genetic variation in fitness
For example, approximately true under weak selection
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Additive variance in fitness is key. Consider a selectiveoverdominant locus, 1:1+s:1. Maximal total geneticvariance occurs at p =1/2, but heritability iszero at this value, and hence no response to selection
Generally, traits associated with fitness components(e.g., viability, # offspring) have lower h2 and alsomore non-additive variance.
0.80.70.60.50.40.30.20.10.0
.001
.01
.1
1
Heritability
Traits more closely associated (phenotypically correlated) with fitness had lower heritabilities in Collared flycatchers (Ficedula albicollis) on the island of Gotland in the Baltic sea, (Gustafsson 1986)
0.50.40.30.20.10.00.0
0.2
0.4
0.6
0.8
1.0
Heritability
Lack of such associated (fitness measure by seed production) for Plox ( Phlox drummondii) in Texas (Schwaegerle and Levin 1991).Again, phenotypic correlated used.
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Life history and morphological traits in the Scottish red deer ( Cervus elaphus).Circles denote life-history traits, squares morphological traits. Genetic correlationbetween trait and fitness (Kruuk et al. 2000)
Common is variance standarization, x’ = x/ x
Houle: Evolvability
Traits are generalized standardized to compare themwith others:Houle agreed that our interest is typically in theproportion of change -- e.g, animals are 5% larger.
On such a scale, a response (change in mean) of 0.1implies a 0.1 standard-deviation change in the mean.variance-standardization thus a function of standingvariation in the population.This uses mean standardization, x’ = x/
A 0.1 response on this scale means the trait improved by10%
Houle said that evolvability of a trait, A/ is a bettermeasure of evolutionary potential than h2 = A
2 / z2
Houle found that life history traits had HIGHERevolvabilities. They had more genetic variation, but alsomore environmental variance, resulting in lower h2
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Life history and morphological traits in the Scottish red deer ( Cervus elaphus).Circles denote life-history traits, squares morphological traits. Genetic correlationbetween trait and fitness (Kruuk et al. 2000)
CVA = Genetic Coefficient of variation:CVA2 = Var(A)/mean2
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Robertson’s secondary theorem and the breeder’s equation
FFT under weak selection gives some approximaterules about how populations evolve by followingchanges in fitness
We are usually much more interest in changesin trait values. What can we say here?
Alan Robertson proposed a ``secondary theorem”to Fisher’s to treat trait evolution,
Response = change in mean equals the additivegenetic covariance between trait and fitness(the covariance within an individual for thebreeding values of these two traits).
With weak selection, we can use population-geneticmodels based on Robertson’s idea to expressresponse in terms of gene frequency changes
If no dominance, =0, and R = h2SMore generally, when there is selectionto change the variance as well ( )
If selection only on mean and no skew, = -Sand we recover Breeders, R = h2S
However, if there is skew in the phenotype, andskew in the breeding values, then breeder’s equationdoes not exactly hold
Selection and DriftIf the strength of selection is weak relativeto the effects of drift, drift will overcomethe directional effects of selection.
Suppose genotypes AA : Aa : aa have fitnesses1 + 2s : 1 + s : 1
Kimura (1957) showed that the probability U(p) thatA is fixed given it starts with frequency p is
U(p) =1 ° exp(°4Nsp)1 ° exp(°4Ns)
Note if 4Ns >> 1, allele A has a very high probabilityof fixation
If 4Ns << -1 (i.e. allele is selected against), A hasessentially a zero probability of becoming fixed.
If 4N| s | << 1, the U(p) is essentially p, and henceThe allele behaves as if it selective neutral
An interesting case is when p = 1/(2N), i.e., the alleleis introduced as a single copy
Even if 4Ns >> 1, U is 2s. Hence, even a stronglyfavored allele introduced as a single copy isusually lost by drift.
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How much selection on a QTL given selectionon a trait?
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Having a specificallele shifts theoverall traitdistribution slightly
Resulting strength (andform) of selection on aQTL
Strength of selection on a QTL
Genotype A1A1 A1A2 A2A2
Contribution toCharacter
0 a 2a
Have to translate from the effects on a trait under selection to fitnesses on an underlying locus (or QTL)
Suppose the contributions to the trait are additive:
For a trait under selection (with intensity i) andphenotypic variance P
2, the induced fitnessesare additive with s = i (a / P )
Thus, drift overpowersselection on the QTL when 4Nejs j =
4Neja i jæP
<<1
More generally
Genotype A1A1 A1A2 A2A2
Contribution to trait
0 a(1+k) 2a
Fitness 1 1+s(1+h) 1+2s
¢q ' 2q(1 q)[1+h(1 ° 2q)]Change in allele frequency:
s = i (a / P )
Selection coefficients for a QTL
h = k