thomas lenormand - génétique des populations
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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
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