Basic Population Genetics and One and Two locus models of Selection.

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

TIFF (LZW) decompressorare needed to see this picture.

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

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.










are needed to see this picture.Key: Internal equilibrium frequency. Stable equilibrium

Application: Stabilizing selection

A common model for stabilizing selection on a trait is to use a normal-type curve for trait fitness



As detailed in Example 5.6, we can use Wright’s formulato compute allele frequency change under this type ofselection



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



Consider average excess inrelative fitness for Ai



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







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)



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



Here, W(AA) is independent of freq(A) = p, andwe can use Wright’s formula to compute allelefrequency change.





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



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)



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





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.

QuickTime™ and aTIFF (Uncompressed) decompressor


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

QuickTime™ and aTIFF (Uncompressed) decompressor


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







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.



How much selection on a QTL given selectionon a trait?



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

Basic Population Genetics and One and Two locus models of Selection.

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