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Lecture 18:

One and two locus models of

selection

Bruce Walsh lecture notes

Synbreed course

version 8 July 2013

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Single-locus selection

The basic building block of the theory of selection

response (from population-genetic standpoint) is a

single locus under selection. Think of this as a trait

controlled by only a single gene.

Individuals differ in fitness when they leave

different numbers of offspring.

When the fitness of at least one genotype

is different from the others, selection occurs.

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(1-p)2Waa2p(1-p) WAap2 WAAFrequency

(after selection)

WaaWAaWAAFitness

(1-p)22p(1-p)p2Frequency(before selection)

aaAaAAGenotype

One locus with two alleles

W W W

W

is the mean population fitness, the fitness of an random

individual, e.g. = E[W]

Where = p2 WAA + 2p(1-p) WAa + (1-p)2WaaW

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The new frequency p’ of A is just

freq(AA after selection) + (1/2) freq(Aa after selection)

p′ =p2WAA + p(1− p)WAa

W= p

pWAA + (1− p)WAaW

The fitness rankings determine the ultimate fate

of an allele

If WXX > WXx > Wxx, allele X is fixed, x lost

If WXx > WXX, Wxx, selection maintains both X & x

Overdominant selection

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Class problem: Required time for allele

frequency change

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

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Wright’s formula

Computes the change in allele frequency as

as function of the change in mean fitness

Requires frequency-independence: Genotype

fitnesses are independent of genotype frequencies,

dWij / dpi = 0

Note sign of change in p = sign of dW/dp

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Application: Overdominant selection

Key: Internal equilibrium frequency.

Stable equilibrium

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Application: Stabilizing selection

This is selective underdominance! If p < 1/2, !p < 0

and allele gets lost. If p > 1/2, !p > 0 and allele

fixed.

Hence, stabilizing selection on a trait controlled by

many loci removes variation!

A common model for stabilizing selection

on a trait is to use a normal-type

curve for trait fitness

As detailed in WL Example 5.6, we can use Wright’s formula

to compute allele frequency change under this type of

selection

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p′i = piWi

W, Wi =

n∑

j=1

pjWij , W =n∑

i=1

piWi

Multiple Alleles

Let pi = freq(Ai), Wij = fitness AiAj

Wi = marginal fitness of allele Ai

W = mean population fitness = E[Wi] = E[Wij]

If Wi > W, allele Ai increases in frequency

If a selective equilibrium exists, then Wi = W

for all segregating alleles.

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Fitness as the ultimate quantitative trait

Recall that the average excess of allele Ai is mean

trait value in Ai carries minus the population mean

Consider average excess in

relative fitness for Ai

Allele frequency change is a function

of the average excess of that allele

Allele frequency does not change when its average

excess is zero

At an equilibrium, all average excesses are zero.

Hence, no variation in average excesses and

thus no additive variation in fitness at equilibrium

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Wright’s formula with multiple alleles

Key: Note that the sign of dW/dpi does not determine

sign(!p). Thus an allele can change in a direction

opposite to that favored by selection if the changes

on the other alleles improve fitness to a greater extent

Prelude to the multivariate breeder’s equation, R = G"

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General features with multiple

allele selection

With Wij constant and random mating, mean fitness

always increases

What about polymorphic equilibrium?

Require Wi = W1 for all i.

Kingman showed there are only either zero, one, or

infinitely many sets of equilibrium frequencies for

an internal equilibrium (all alleles are segregating).

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p1

p3p2

1

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p1 + p2 + p3 = 1

corner

equilibrium

edge

equilibrium

internal

equilibrium

Equilibrium behavior

Other equilibrium can potentially fall anywhere

on the simplex of allele frequencies

Single internal equilibrium if W has exactly one

positive and at least one negative, eigenvalue

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Two-locus selection

When two (or more) loci are under selection, single

locus theory no longer holds, because of linkage

disequilibrium, D = freq(ab) - freq(a)*freq(b)

Consider the marginal fitness of the AA genotype

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Two-locus selection

Here, W(AA) is independent of freq(A) = p, and

we can use Wright’s formula to compute allele

frequency change. Can’t do this when D not zero!

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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For two loci, must follow gamete frequencies

The resulting recursion equations, even for the simple

two biallelic loci, do not have a general solution for

their dynamics (4th degree polynomials)

When selection is strong and linkage (c) tight,

results can be unpredictable

Mean fitness can decline under two-locus selection

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At equilibrium

If LD at equilibrium, the second term is nonzero and

not all gametes have the same marginal fitness. Note

that when c = 0, this is just a 4 allele model, and all

segregating alleles have the same marginal fitness

If equilibrium LD is not zero, mean fitness is not

at a local maximum. However, unless c is very small,

it is usually close

In such cases, mean fitness decreases during the final approach

to the equilibrium (again, effects usually 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 -- still

could have lots of additive variance in the trait.

WL Example 5.11. Even apparently simple models can have

complex behavior.

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FFT: Fisher’s Fundamental Theorem

What, in general can be said above the behavior of

multilocus 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 Fundamental

theorem

Karlin: “FFT is neither fundamental nor a theorem”

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Fisher: “The rate of increase in fitness of any

organism 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

eventually remove 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 selective

overdominant locus, 1:1+s:1. Maximal total genetic

variance occurs at p = 1/2, but heritability is

zero at this value, and hence no response to selection

Generally, traits associated with fitness components

(e.g., viability, # offspring) have lower h2 and also

more non-additive variance.

1.00.80.60.40.20.0

0.00

0.05

0.10

0.15

0.20

0.25

Allele Frequency p

Var/

s2

Additive

Dominance

Total

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0.80.70.60.50.40.30.20.10.0

.001

.01

.1

1

Heritability

Pro

po

rtio

n o

f fi

tnes

s ex

pla

ined

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)

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0.50.40.30.20.10.0

0.0

0.2

0.4

0.6

0.8

1.0

Heritability

Co

rrel

ati

on

wit

h f

ruit

pro

du

ctio

n

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 correlation

between trait and fitness (Kruuk et al. 2000)

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Common is variance standarization, x’ = x/#x

Houle: Evolvability

Traits are generally standardized to compare them

with others:

Houle agreed that our interest is typically in the

proportion of change -- e.g, animals are 5% larger.

With variance-standardization, a response of 0.1 implies

a 0.1 standard-deviation change in the mean.

Variance-standardization thus a function of standing

variation in the population.

Evolvability uses mean standardization, x’ = x/µ

A 0.1 response on this scale means the trait improved by

10%

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Houle: Evolvability

Houle argued that evolvability of a trait, CVA = #A/ µ

(mean-standardization) is a better measure of evolutionary

potential than h2 = #A2 / #z2 (variance-standardization)

Houle found that life history traits had HIGHER

evolvabilities = significant potential for large

proportional (percentage) change in the mean

They had more genetic variation, but also

more 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 correlation

between trait and fitness (Kruuk et al. 2000)

CVA = Genetic

Coefficient of variation:

CVA2 = Var(A)/mean2

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How much selection on a QTL given selection

on a trait?

Having a specific

allele shifts the

overall trait

distribution slightly

Resulting strength (and

form) of selection on a

QTL

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Strength of selection on a QTL

2a a 0Contribution toCharacter

A2A2A1A2A1A1Genotype

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) and

phenotypic variance #P2, the induced fitnesses

are additive with s = i (a /#P )

Thus, drift overpowers

selection on the QTL when4Ne |s | =

4Ne| ai |σP

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More generally

1+2s1+s(1+h)1Fitness

2aa(1+k)0Contribution to trait

A2A2A1A2A1A1Genotype

∆q " 2q(1 q)[1 + h(1 − 2q)]Change in allele frequency:

s = i (a /#P )

Selection coefficients for a QTL

h = k

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Class problem: How quickly do allele

frequencies at a QTL change?

2a a 0Contribution toCharacter

A2A2A1A2A1A1Genotype

s = i (a /#P )

(1) Suppose a/# = 0.5

Suppose i = 2 (strong TRAIT selection). How

long for a rare QTL (p0 = 0.05) to reach 50%?

(2) Suppose a/# = 0.005

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General selection response• Two locus theory in the general setting is

very complex!

• What can we say about k-locus selection?

• FFT under weak selection gives someapproximate rules about how populationsevolve by following changes in fitness

• We are usually much more interested inchanges in trait values. What can we sayhere?

• Robertson’s secondary theorem

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Robertson’s secondary theorem and the

breeder’s equation

Alan Robertson proposed a “secondary theorem”

to Fisher’s to treat trait evolution,

Response = change in mean equals the additive

genetic covariance between trait and fitness

(the covariance within an individual for the

breeding values of these two traits).

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Much more on FFT, Robertson’s theorem, set within

the Price-equation framework (no covered here) in WL

Chapter 6

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Robertson’s secondary theorem and the

breeder’s equation

If no dominance, $ = 0, and R = h2S

More generally (but no dominance), when

there is selection to change the variance as well ($# )

If selection only on mean and no skew (E[%3] = 0),

$# = -S, and we recover breeder’s equation, R = h2S