Population genetics

Thomas LenormandCEFE – CNRS, Montpellier

The mathematics of frequency change

• An allele arise by mutation

• Its frequency change for several reasons

• Genome change ( = evolution occurs)

Mendel + DarwinA ‘microscopic view’ founding other approaches

G P P G

(1892-1964)

(1911-1998)

(1890-1962) (1889-1988)

(1920-2004)(1924-1994)(1916- ) (1929 - )

Levins 1966

Généralité

Précision Réalisme

Un corpus de modèlesTheories - sexe - speciation - mort - altruisme

Des Stats…bcp de stats

- Mutations- Enzymes- Microsat- Sequences- Genomes

Réalisme

distance x0

1

0 20 40 60

{R}

{E}

fréquence

Précison

Saccheri I J et al. PNAS 2008;105:16212-16217

1/(4πDeσ2)

Généralité

Probabilité de fixationHaldane 1927

……

0 1 2 3 4 5 … 0 1 2 3 4 5 …

P(1+s) P(1)

P: proba fixation1-P: proba perteXt : nb de copies à la génération t

)(2/)1(1

1!

1

1Pr1

22

)1(

)1(

1

soPsP

e

Pj

se

PjXP

Ps

j

jjs

j

jt

P ≈ 2s

Modifier theory

2nn

2nn

Selection diploïde11 + h s1 + s

Selection haploïde11 + s

Recombinaison rMutation

Selfing rate Mating preferences

Modifier theory

Migration m

Modifier theory

Suppose un locus qui modifie le caractère d’interet

Regarde changement de fréquence d’un mutant

Evolution à long terme du modifieur renseigne sur comment le caractère peut evoluer et dans quelles conditions

Un modifieur peut évoluer- par sélection directe (naturelle, sexuelle, de parentèle)- par sélection indirecte

On parle de modifieur « neutre » lorsqu’il n’y a que de la sélection indirecte

Construire un modèle: combien de locus au minimum?

2nn

Selection diploïde11 + h s1 + s

Selection haploïde11 + s

sexRecombinaison rMutation

Selfing rate Mating preferences

Exemples

Migration m

Plasticity

Reduction principle

If viability loci at stable polymorphic equilibriumTransmission evolves to be « perfect » (r = 0, = 0, m = 0, sex = 0)

Selected loci are at equilibrium. Selection coefficients are constant. There is random mating. Only one transmission parameter is considered at a time. Viability is sex-independent.

(Altenberg and Feldman 1987)

Modifier vs. optimality

Does modifier evolve to maximise mean fitness ?

Knowing what would be best for the pop does not say that evolution will lead there

Modifiers are tools for modelling

‘True’ genes have *always* pleiotropic effects

What is sex?

2nn

Why sex?

What is the benefit of recombination?

How to construct a model to measure this?

Main hypothesis: sex allows for recombination

Part I

Building the modelfrom scratch

Barton, N. H. 1995. A general model for the evolution of recombination. Genetical Research 65:123-144.

Key insights

Recombination > 1 locus polymorphic otherwise uninteresting

Simplest polymorphismmutation – haploid selection

Keep it simplesingle populationvery large number of individuals (neglect drift)

Step 1: genetic setting

Locus kLocus jrjk

W00 = 1W10 = 1+aj

W01 = 1+ak

W11 = 1+aj+ak+ajk

XjXk

01

01

Step 2: modifier

Locus kLocus jrjk

XjXk

01

01

Locus irij

Xi01

If Xi = 0 then the recombination rates are rij and rjk

If Xi = 1 then the recombination rates are rij+ij and rjk+jk

Only effects of the modifier

+ij +jk

Step 3: life cycle

2nn

Step 3: life cycle

2nn

selection

haploid viability selection

fair meiosis

Random mating

Step 4: variables

With 3 biallelic loci, there are 8 possible haploid genotypes

000 x1

001 x2

010011100101110 x7

111 x8

.

.

.

genotypes frequencies

Locus i(modifier)

Locus j

Locus k

Step 4: variables

kk

jj

ii

pXE

pXE

pXE

)(

)(

)(

frequencies

8765

8

1

)()(

xxxx

xgXXEg

gii

000 x1

001 x2

010 x3

011 x4

100 x5

101 x6

110 x7

111 x8

Step 4: variables

kk

jj

ii

pXE

pXE

pXE

)(

)(

)(

frequencies

8743

8

1

)()(

xxxx

xgXXEg

gjj

000 x1

001 x2

010 x3

011 x4

100 x5

101 x6

110 x7

111 x8

Step 4: variables

ikki

jkkj

ijji

CXXCov

CXXCov

CXXCov

),(

),(

),(

Pairwise ‘associations’

ji

gjigji

jiijjiji

jjiiji

ppxx

ppxgXX

ppXEpXEpXXE

pXpXEXXCov

87

8

1

)(

)()(][

))((),( 000 x1

001 x2

010 x3

011 x4

100 x5

101 x6

110 x7

111 x8

(usually referred to as ‘linkage disequilibrium’)

Step 4: variables

ijkkji CXXXCov ),,( triplet ‘association’

kjiijkikjjki

kkjjiikji

pppCpCpCpx

pXpXpXEXXXCov

8

))()((),,(

000 x1

001 x2

010 x3

011 x4

100 x5

101 x6

110 x7

111 x8

Step 4: variables

000 x1

001 x2

010 x3

011 x4

100 x5

101 x6

110 x7

111 x8

Sum to 1

7 independent variables

pi

pj

pk

Cij

Cjk

Cik

Cijk

7 independent variables

Part II

Writing the equations

Exact recursions

Aim : computing variations of variables over one generation

2nn

selection(a)

(b)

(c)

Step 1: selection

8

1ggg

ggselection

g

xWW

xW

Wx

Step 2: Fertilization (random mating)

000 001 010 011 100 101 110 111

000 x12

001 x1 x2 x22

010 x1 x3 x2 x3 x32

011 … … … …

100 … … … … …

101 … … … … … …

110 … … … … … … …

111 … … … … … … x7 x8 x82

Male gametesfe

mal

e ga

met

es

Step 3: Meiosis

000 001 010 011 100 101 110 111

000 x12

001 x1 x2 x22

010 x1 x3 x2 x3 x32

011 … … … …

100 … … … … …

101 … … … … … …

110 … … … … … … …

111 … … … … … … x7 x8 x82

Male gametes

fem

ale

gam

etes

001010

001

010

000

011

(1-rjk)/2

(1-rjk)/2

rjk/2

rjk/2

Diploid individualproduces gametes

Step 3: Meiosis

000 001 010 011 100 101 110 111

000 x12

001 x1 x2 x22

010 x1 x3 x2 x3 x32

011 … … … …

100 … … … … …

101 … … … … … …

110 … … … … … … …

111 … … … … … … x7 x8 x82

Male gametes

fem

ale

gam

etes

101110

001

010

000

011

(1-rjk-)/2

(1-rjk-)/2

(rjk+)/2

Diploid individualproduces gametes

(rjk+)/2

Exact recursions

),,...,,( 8721' xxxxfx gg

Aftermatingmeiosismutationselection

System of 7 independent equationsg = 1…7

Changing variables

),,...,,( 8721' xxxxfx gg

equivalently

ijkikjkijkji CCCCppp ,,,,,,

recursions on

Part III

Analyzing the model

Methods

When does the allele at the modifier locus change in frequency?

The exact recursions give the answer, but they are not helpful

Dynamical system: equilibria?

Analysis involve assumptions

Method 1 : stability analysis

Method 2 : separation of time scales

Taylor Series

302

220

00 )(2

)()()()(

00

xxOdx

fdxx

dx

dfxxxfxf

xxxx

constant approximation

linear approximation

quadratic approximation

Notation: Series[f(x),{x, x0, 2}]

(like in Mathematica)

Assumptionsr >> a

pi changes at a even slower rate

because indirect selection

Associations change at a faster rate

because r is larger

pi

pj

pk

Cij

Cjk

Cik

Cijk

pj and pk change slowly

because a is small

fast changing variables

slow changing variables

Separation of time scales = treat associations as constantsalso known as « Quasi Linkage Equilibrium » (QLE)

Assumptions

aj , ak >> ajk

weak epistasisbecause more interesting (see later)

<< r weak modifier effectsinvestigate only evolution by the accumulation of small mutations

W00 = 1W10 = 1+aj

W01 = 1+ak

W11 = 1+aj+ak+ajk

W00 = 1W10 = 1+aj

W01 = 1+ak

W11 = (1+aj)(1+ak)+ejk

ajk=ejk+ajak

QLECjk

(association between thetwo selected loci)

1,0,,Series ' aCC jkjk

jkC

3)()1( aOrCrpqe jkjkjkjkjk

0 jkC 3)()1(

aOr

rpqeC

jk

jkjkjkQLEjk

Association is generated by epistasis between the lociRecombination REDUCES associations

QLE (leading orders)

4)(

aOrr

pqeC

jkijk

ijkjkjkQLEijk

5)()1(

aOrrr

rpqeaC

jkijkij

ijijkjkkjkQLEij

5)()1(

aOrrr

rpqeaC

jkijkik

ikijkjkjjkQLEik

3)()1(

aOr

rpqeC

jk

jkjkjkQLEjk

treat them as constant of knownorder

Modifier

6)(aOCaCaCap QLEijkjk

QLEikk

QLEijji

ikijkj rr

aa11

1

6)(

aOeerr

pqjkjk

ijkjk

ijkjk

where

Final Result

Sign of pi

ejk

pi

0

More recombination evolves if < ejk < 0

Convergence state

-20 -15 -10 -5

0.1

0.2

0.3

0.4

0.5

rij = 0.1 0.2 0.50.4

0

kj

jk

aa

e

*jkr

outsidehypothesismade

by the accumulation of modifier of small effect

(stabilityanalysis)

Part IV

Interpreting the model

Back to equations

ijkjkikkijji CaCaCap

Effect on the mean fitness of offspring

Effect onthe variancein fitness of

offspring

Effect on variance

W

Cjk > 0

W

Cjk < 0

VarianceResponse to selection ejk

Cjk

0

+ +

- -

Modifier becomes associated to beneficial alleles (Cij, Cik)and hitchhikes with them

IF Cjk < 0

Effect on mean

W11 + W00 – W01 – W10 = ajk

when ajk >0 extreme genotypes are fitter on average, it’s worthrecombining if it producesmore of them (Cjk < 0)

ejk

Cjk

-ajak

+

+-

-

W

Cjk > 0

W

Cjk < 0

extreme intermediate

Direction of selection

ejk

-ajak

Cjk

0

both effectspositive

‘effect on mean’ dominates

both effectsnegative

‘effect on variance’ dominates

Direction of selection

ejk

-ajak

Cjk

0

Cjk is only generated by ejk in this model

Recombination favored for weak negative epistasis

Part V

Relate to other models

Fluctuating Epistasis (Barton 1995, Peters & Lively 1999)

ejk

Cjk

0

Environmental heterogeneity(Lenormand & Otto 2000)

ejk

Cjk

0Cov(aj,ak)>0

Cov(aj,ak)<0

Hill-Robertson effect(originally Fisher 1930, Muller 1932)

ejk

Cjk

0

Interference among selected loci generate negative Cjk

Take home messages

Recombination…is neutral in absence of associations

can increase or decrease variance

may increase variation but variation needs not be favourable

can change both mean and variance in fitness

frequency change depends on both the sign of genes associations and epistasis

Further?

• When to include stochasticity?

• Extending modifier theory to genome…

• Relationship to other approaches