conflict between alleles and modifiers in the evolution of genetic polymorphisms
DESCRIPTION
Conflict between alleles and modifiers in the evolution of genetic polymorphisms. Hans Metz. & Mathematical Institute, Leiden University. (formerly ADN ) IIASA. VEOLIA- Ecole Poly- technique. NCB naturalis. the tool. - PowerPoint PPT PresentationTRANSCRIPT
Conflict between alleles and modifiers in the evolution of genetic polymorphisms
(formerly ADN) IIASA
Hans Metz
QuickTime™ en eenTIFF (ongecomprimeerd)-decompressorzijn vereist om deze afbeelding weer te geven.VEOLIA-Ecole Poly-technique
QuickTime™ en eenTIFF (ongecomprimeerd)-decompressor
zijn vereist om deze afbeelding weer te geven.
&Mathematical Institute, Leiden University
QuickTime™ en eenTIFF (ongecomprimeerd)-decompressor
zijn vereist om deze afbeelding weer te geven.
NCB naturalis
the tool
(Assumptions: mutation limitation, mutations have small effect.)
the canonical equation of adaptive dynamics
X: value of trait vector predominant in the population Ne: effective population size, : mutation probability per birth C: mutational covariance matrix, s: invasion fitness, i.e., initial relative growth rate of a potential Y mutant population.
dXdt
= 2 Ne C ∂sX Y( )∂Y Y=X
⎡⎣⎢
⎤⎦⎥T
with Mendelian reproduction:
= 0evolutionary
stop
evolutionary constraints
phenotype
genotype
directional selection
coding region
regulatory regions
DNA
reading direction
Most phenotypic evolution is probably regulatory, and hence quantitative on the level of gene expressions.
Φ
the canonical equation of adaptive dynamics
The canonical equation is not dynamically sufficient as there is no need for C to stay constant.
Even if at the genotype level the covariance matrix stays constant, the non-linearity of the genotype to phenotype map Φ
will lead to a phenotypic C that changes with the genetic changes underlying the change in X.
additional (biologically unwaranted) assumption
symmetric phenotypic mutation distributions
saving grace?
I have reasons to expect that my final conclusions are independent of this symmetry assumption,
but I still have to do the hard calculations to check this.
I only showed (and use)the canonical equation for the case of
the canonical equation of adaptive dynamics
dXdt
= 2 Ne C ∂sX Y( )∂Y Y=X
⎡⎣⎢
⎤⎦⎥T
sX Y( ) ≈ln R0X Y( )⎡⎣ ⎤⎦
TrNe =
TrNTsse
2
dXdt
= 2 N
Tss e2 C
∂R0X Y( )∂Y Y=X
⎡⎣⎢
⎤⎦⎥T
R0 : average life-time offspring number
Ts : average age at death
: effective variance of life-time offspring numbers e2
of the residents of the residents
Tr : average age at reproduction
t
CE is derived via two subsequent limits
system size ∞ successful mutations / time 0
trait valuex
individual-basedsimulation
adaptive dynamicslimit
individual-based stochastic process
mutational step size 0
canonical equationlimit
branching
limit type:
t
this talk: evolution of genetic polymorphisms
system size ∞ successful mutations / time 0
trait valuex
individual-basedsimulation
adaptive dynamicslimit
individual-based stochastic process
canonical equationlimit
branching
limit type:mutational step size
0
the ecological theatre
Assumptions: but for genetic differences individuals are born equal,random mating, ecology converges to an equilibrium.
equilibria for general eco-genetic models
(1) setting the average life-time offspring number over the phenotypes equal to 1,
(2) calculating the genetic composition of the birth stream from equations similar to the classical (discrete time) population genetical ones,
with those life-time offspring numbers as fitnesses.
For a physiologically structured population with all individuals born in the same physiolocal state, mating randomly with respect to genetic differences,
the equilibria can be calculated by
the eco-genetic model
Organism with a potentially polymorphic locus with two segregating alleles, leading to the phenotype vector , with .XG G ∈ aa, aA, AA{ }
X := Xaa XaA XAA( )T
Abbreviations: , etc. (and similar abbreviations later on). μG := μ (XG;E)
: expected per capita lifetime microgametic output times fertilisation propensity ( average number of kids fathered)
μ(XG;E)
: instantaneous ecological environmentE
l(XG ;E) : expected expected per capita lifetime macrogametic output (= average number of kids mothered)
the eco-genetic model C = classical discrete time model
baa =paaB,random union of gametes:
baA =(pAa + paA)B, bAA =pAAB.
Point equilibria: lqA = λ AA pAqA + 12 λ aA (pAqa + paqA ),
l := pAqAλ AA + (pAqa + paqA )λ aA + paqaλ aa = 1,
μpA = μ AA pAqA + 12 μ aA (pAqa + paqA ),
with
μ := pAqAμ AA + (pAqa + paqA )μ aA + paqaμ aa .
pa =1−pA
, : allelic frequencies in the micro- resp. macro-gametic outputs ( and )
pa , pA qa ,qA
qa =1−A
: total birth rate density (C: total population density, )B N
baa ,baA ,bAA : genotype birth rate densities (C: genotype densities, , etc)naa
naa =Ts,aabaa , etc.
example ecological feedback loop: E =Φ φ1(E, XG )nGG∑ , ... , φk(E, XG )nG
G∑⎛
⎝⎜⎞⎠⎟
the evolutionary play
Assumptions: no parental effects on gene expressions(mutation limitation, mutations have small effect)
long term evolution
Two modelsI. Evolution through allelic substitutions
Xa , XA : allelic trait vectorsXaA =Φ(Xa, XA),Φ : genotype to phenotype map: etc.
Abbreviations: etc.∂Φ(Xa , XA )∂Xa
=: ∂1Φ(Xa , XA ) =: ∂1ΦaA ,
II. Evolution through modifier substitutions b : original allele on generic modifier locus,
B: mutant, changing into X =Xaa
XaA
XAA
⎛
⎝⎜⎜
⎞
⎠⎟⎟ =
Xbb,aa
Xbb,aA
Xbb,AA
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟=: Xb
XbB,aa
XbB,aA
XbB,AA
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟=: XB.
smooth genotype to phenotype maps
IfModel I (allelic evolution)
Xa =XA + eZA
Xaa =XAA + e ∂1ΦAAZA + e ∂2ΦAAZA + O(e2 ) =XAA + 2 e ∂2ΦAAZA + O(e
2 )
Model II (modifier evolution)
XbB,aa
XbB,aA
XbB,AA
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟=
Xbb,aa
Xbb,aA
Xbb,AA
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟+ e
Zaa
ZaA
ZAA
⎛
⎝⎜⎜
⎞
⎠⎟⎟If
Xaa = XaA + e ∂2ΦaAZA + O(e2 ), XAa = XAA + e ∂2ΦAAZA + O(e
2 ),
then
XBB,aa
XBB,aA
XBB,AA
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟=
Xbb,aa
Xbb,aA
Xbb,AA
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟+ 2e
Zaa
ZaA
ZAA
⎛
⎝⎜⎜
⎞
⎠⎟⎟ +O(e
2 )then
Model I: phenotypic change in the CE limit
with dXa
dt = ψ aN pa2aCa
∂R0,XaXA(Xa )
∂Xa
⎛⎝⎜
⎞⎠⎟a=a
T
,dXA
dt = ψ AN pA2ACA
∂R0,XaXA(Xa )
∂Xa
⎛⎝⎜
⎞⎠⎟a=A
T
ψ a = 2 (Ts,aσ a2 ), ψ A = 2 (Ts,Aσ A
2 ),witha , θ A the mutation probabilities per allele per birth,
the mutational covariance matrices,Ca , CA
p a := 12 (pa + qa ), π A := 1
2 (pA + qA ).and
dXdt
=dXdt
⎛⎝⎜
⎞⎠⎟a
+dXdt
⎛⎝⎜
⎞⎠⎟A
=
2∂1ΦaadXa
dt
∂1ΦaAdXa
dt
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟
+ ∂2ΦaAdXA
dt
2∂2ΦAAdXA
dt
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟
0
0
Model I: phenotypic change in the CE limit
∂R0.Xa XA(Xa )
∂Xa
=∂R0, Xaa , XaA , XAA
(Xaa , XaA )∂Xaa
∂1Φaa +∂R0, Xaa , XaA , XAA
(Xaa , XaA )∂XaA
∂1ΦaA
Convention:
Differentiation is only with respect to the regular arguments, not the indices.
∂R0, Xa XA(XA )
∂XA
=∂R0, Xaa , XaA , XAA
(XAA , XaA )∂XAA
∂2ΦAA +∂R0, Xaa , XaA , XAA
(XAA , XaA )∂XaA
∂2ΦaA
notation
A ⊗B=
a11B L a1nBM M
aμ1B L aμnB
⎛
⎝⎜⎜
⎞
⎠⎟⎟
I the identity matrix of any required size
and
denotes the Kronecker product:⊗
Model I: phenotypic change in the CE limit
dXdt
= 2 0 0 00 1 1 00 0 0 2
⎛
⎝⎜⎜
⎞
⎠⎟⎟⊗I
∂1Φaa
∂1ΦaA
00
00
∂2ΦaA
∂2ΦAA
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
dXa
dtdXA
dt
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
in matrix notation:
dXa
dtdXA
dt
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=ψ allelicN 2allelic
%ψ a%apaCa 00 %ψ A
%ApACA
⎛⎝⎜
⎞⎠⎟
∂1ΦaaT
0∂1ΦaA
T
0
0∂2ΦaA
T
0∂2ΦAA
T
⎛⎝⎜
⎞⎠⎟
∂R0,Xaa , XaA, XAA(Xaa, XaA)
∂Xaa
⎡⎣⎢
⎤⎦⎥T
∂R0,Xaa , XaA, XAA(Xaa, XaA)
∂XaA
⎡⎣⎢
⎤⎦⎥T
∂R0,Xaa , XaA , XAA(XAA, XaA)
∂XaA
⎡⎣⎢
⎤⎦⎥T
∂R0,Xaa , XaA , XAA(XAA, XaA)
∂XAA
⎡⎣⎢
⎤⎦⎥T
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
and (the allelic coevolution equations)
withψ allelic = π aψ a + π Aψ A , allelic = π aθa + π AθA ,
%ψ a = ψ a ψ allelic , %ψ A = ψ A ψ allelic , %a = θa θallelic , %A = θA θallelic .
structurematrix
Model I: phenotypic change in the CE limit
dXdt
= ψ allelicN 2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I 2allelicYCallelicGallelic
combining the previous results gives:
Y :=
%ψ a%θaπ a 0 0 00 %ψ a
%θaπ a 0 00 0 %ψ A
%θAπ A 00 0 0 %ψ A
%θAπ A
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I
with
and
Callelic :=
∂1Φaa
∂1ΦaA
00
00
∂2ΦaA
∂2ΦAA
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Ca 00 CA
⎛⎝⎜
⎞⎠⎟
∂1ΦaaT
0∂1ΦaA
T
0
0∂2ΦaA
T
0∂2ΦAA
T
⎛⎝⎜
⎞⎠⎟.
Model I: phenotypic change in the CE limit
Gallelic =
∂R0,Xaa ,XaA , XAA(Xaa, XaA)
∂Xaa
⎡⎣⎢
⎤⎦⎥T
∂R0,Xaa ,XaA , XAA(Xaa, XaA)
∂XaA
⎡⎣⎢
⎤⎦⎥T
∂R0,Xaa , XaA , XAA(XAA, XaA)
∂XaA
⎡⎣⎢
⎤⎦⎥T
∂R0,Xaa , XaA , XAA(XAA, XaA)
∂XAA
⎡⎣⎢
⎤⎦⎥T
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
with
ba = 1−(pa −a)
laaμaA
4lμ−laAμaa
4lμ⎛⎝⎜
⎞⎠⎟, bA = 1−(pA−A)
lAAμaA
4lμ−laAμAA
4lμ⎛⎝⎜
⎞⎠⎟
Model I: phenotypic change in the CE limit
Gallelic =B−1 Gcoμ μ on
an explicit expression for the allelic (proxy) selection gradient:
B =
ba 0 0 00 ba 0 00 0 bA 00 0 0 bA
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I
with
=I
on the Hardy-Weinberg manifold (pA = qA) :
Model I: phenotypic change in the CE limit
with μaa = μ (Xaa;EXaa XaA XAA), etc.
l =paqaλ aa + (paqA + pAqa )λ aA + pAqAλ AA = 1and
effect a mutation in the
a--allele A-allele
Gcommon =
pa +μaA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laa
2l∂Xaa
⎡⎣⎢
⎤⎦⎥T
+ a +laA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaa
2μ∂Xaa
⎡⎣⎢
⎤⎦⎥T
pA +μaa
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
+ A +laa
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
pa +μAA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
+ a +lAA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
pA +μaA
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂lAA
2l∂XAA
⎡⎣⎢
⎤⎦⎥T
+ A +laA
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μAA
2μ∂XAA
⎡⎣⎢
⎤⎦⎥T
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
Model I: phenotypic change in the CE limit
with μaa = μ (Xaa;EXaa XaA XAA), etc.
l =paqaλ aa + (paqA + pAqa )λ aA + pAqAλ AA = 1and
effect of the resulting phenotypic change in the
aa-homozygotes heterozygotes AA-homozygotes
Gcommon =
pa +μaA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laa
2l∂Xaa
⎡⎣⎢
⎤⎦⎥T
+ a +laA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaa
2μ∂Xaa
⎡⎣⎢
⎤⎦⎥T
pA +μaa
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
+ A +laa
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
pa +μAA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
+ a +lAA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
pA +μaA
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂lAA
2l∂XAA
⎡⎣⎢
⎤⎦⎥T
+ A +laA
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μAA
2μ∂XAA
⎡⎣⎢
⎤⎦⎥T
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
l =pa2λ aa + 2 pa pAλ aA + pA
2λ AA = 1
on the Hardy-Weinberg manifold (pA = qA)
summary of Model I (allelic trait substitution)
dXdt
= ψ allelicN
2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I 2allelicYCallelic B
−1 G coμ μ on
Gcommon =
pa +μaA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laa
2l∂Xaa
⎡⎣⎢
⎤⎦⎥T
+ a +laA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaa
2μ∂Xaa
⎡⎣⎢
⎤⎦⎥T
pA +μaa
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
+ A +laa
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
pa +μAA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
+ a +lAA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
pA +μaA
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂lAA
2l∂XAA
⎡⎣⎢
⎤⎦⎥T
+ A +laA
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μAA
2μ∂XAA
⎡⎣⎢
⎤⎦⎥T
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
on the Hardy-Weinberg manifold:
Model II: phenotypic change in the CE limit
dXdt
=2 ψ μ odifierN 2haplotψpe Cμ odifiersGμ odifier
ψ modifier = 2 (Ts,modifierσ modifier2 )with , the mutation probability per
haplotype per birth, the covariances of the mutational effects of modifiers.
haplotype
Cmodifiers
γa = a
μaA
2μ+ pa
laA
2l−1+ bA, γA = A
μaA
2μ+ pA
laA
2l−1+ ba
G = γabA +γAba( )−1
γa
0 0 0
0 γa
0 0
0 0 γA
0
0 0 0 γA
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I
with Gmodifier =
100
010
010
001
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I G Gcoμ μ on
G = Π :=
pa
0 0 0
0 pa
0 0
0 0 pA
0
0 0 0 pA
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I
on the Hardy-Weinberg manifold:
summary: model comparison
dXdt
= ψ allelicN 2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I 2allelicYCallelic B
−1G coμ μ on
dXdt
= 2ψ μ odifierN 2haplotψpeCμ odifiers 100
010
010
001
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I G G coμ μ on
Model I (allelic substitutions):
Gcommon =
a +laA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaa
2μ∂Xaa
⎡⎣⎢
⎤⎦⎥T
+ pa +μaA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laa
2l∂Xaa
⎡⎣⎢
⎤⎦⎥T
A +laa
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
+ pA +μaa
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
a +lAA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
+ pa +μAA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
A +laA
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μAA
2μ∂XAA
⎡⎣⎢
⎤⎦⎥T
+ pA +μaA
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂lAA
2l∂XAA
⎡⎣⎢
⎤⎦⎥T
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
Model II (modifier substitutions):
summary: model comparisonModel I (allelic substitutions):
Gcommon =
a +laA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaa
2μ∂Xaa
⎡⎣⎢
⎤⎦⎥T
+ pa +μaA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laa
2l∂Xaa
⎡⎣⎢
⎤⎦⎥T
A +laa
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
+ pA +μaa
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
a +lAA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
+ pa +μAA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
A +laA
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μAA
2μ∂XAA
⎡⎣⎢
⎤⎦⎥T
+ pA +μaA
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂lAA
2l∂XAA
⎡⎣⎢
⎤⎦⎥T
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
Model II (modifier substitutions):
dXdt
= ψ allelicN 2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I 2allelicYCallelic B
−1G coμ μ on
dXdt
= ψ μ odifierN 2haplotψpeCμ odifiers 200
020
020
002
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I G G coμ μ on
summary: model comparison
Gcommon =
a +laA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaa
2μ∂Xaa
⎡⎣⎢
⎤⎦⎥T
+ pa +μaA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laa
2l∂Xaa
⎡⎣⎢
⎤⎦⎥T
A +laa
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
+ pA +μaa
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
a +lAA
2lA−pA( )
⎛⎝⎜
⎞⎠⎟
∂μaA
2μ∂XaA
⎡⎣⎢
⎤⎦⎥T
+ pa +μAA
2μpA−A( )
⎛⎝⎜
⎞⎠⎟
∂laA
2l∂XaA
⎡⎣⎢
⎤⎦⎥T
A +laA
2la −pa( )
⎛⎝⎜
⎞⎠⎟
∂μAA
2μ∂XAA
⎡⎣⎢
⎤⎦⎥T
+ pA +μaA
2μpa −a( )
⎛⎝⎜
⎞⎠⎟
∂lAA
2l∂XAA
⎡⎣⎢
⎤⎦⎥T
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
Model I (allelic substitutions):
Model II (modifier substitutions):
dXdt
= ψ allelicN 2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I 2allelicYCallelic B
−1G coμ μ on
on the Hardy-Weinberg manifold
dXdt
= ψ μ odifierN 2haplotψpeCμ odifiers 200
020
020
002
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I Π G coμ μ on
summary: model comparison on the Hardy-Weinberg manifold
2haplotψpeCμ odifiers 200
020
020
002
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I Π
2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I 2allelicYCallelic
summary: model comparison
2haplotψpeCμ odifiers 200
020
020
002
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I Π
2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I 2allelicYCallelic
on the Hardy-Weinberg manifold
summary: model comparison
2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I 2allelicYCallelic
Y :=
%ψ a%θaπ a 0 0 00 %ψ a
%θaπ a 0 00 0 %ψ A
%θAπ A 00 0 0 %ψ A
%θAπ A
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗ I = Π Ξ
on the Hardy-Weinberg manifold
2haplotψpeCμ odifiers 200
020
020
002
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I Π
summary: model comparison
Y :=
%ψ a%θaπ a 0 0 00 %ψ a
%θaπ a 0 00 0 %ψ A
%θAπ A 00 0 0 %ψ A
%θAπ A
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗ I = Π Ξ
A
B
on the Hardy-Weinberg manifold
2haplotψpeCμ odifiers 200
020
020
002
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I Π
2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I Π 2allelicXCallelic
summary: model comparisonModel I (allelic substitutions):
Model II (modifier substitutions):
dXdt
= ψ allelicN 2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I 2allelicYCallelic B
−1G coμ μ on
B =
ba 0 0 00 ba 0 00 0 bA 00 0 0 bA
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I
ba = 1−(pa −a)
laaμaA
4lμ−laAμaa
4lμ⎛⎝⎜
⎞⎠⎟, bA = 1−(pA−A)
lAAμaA
4lμ−laAμAA
4lμ⎛⎝⎜
⎞⎠⎟
G = γaβ A + γ Aβa( )−1
γ a 0 0 00 γ a 0 00 0 γ A 00 0 0 γ A
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
⊗ I
γa = a
μaA
2μ+ pa
laA
2l−1+ bA, γA = A
μaA
2μ+ pA
laA
2l−1+ ba
dXdt
= ψ μ odifierN 2haplotψpeCμ odifiers 200
020
020
002
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⊗I G G coμ μ on
in reality alleles and modifiers will both evolve
dXdt
= dXdt
⎛⎝⎜
⎞⎠⎟μ odifier
+dXdt
⎛⎝⎜
⎞⎠⎟a
+dXdt
⎛⎝⎜
⎞⎠⎟A
= dXdt
⎛⎝⎜
⎞⎠⎟μ odifier
+dXdt
⎛⎝⎜
⎞⎠⎟allelic
combining Models I and II:
evolutionary statics
genetical and developmental assumptions
In biological terms: there are no local developmental or physiological constraints.
So-called genetic constraints are rooted more deeply than in the physiology or developmental mechanics.Example: some phenotypes can only be realised by heterozygotes.
When there are developmental or physiological constraints, we can usually define a new coordinate system on any constraint manifold that the phenotypes run into, and proceed as in the case without constraints.
IF: There are no constraints whatsoever, that is, any combination of phenotypes may be realised by a mutant in its various heterozygotes.
(known in the literature as the “Ideal Free” assumption).
uniformly has full rank and uniformly has maximal rank.Cmodifiers Callelic
evolutionary stops
Evolutionary stops satisfy
I:
II:
2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I YCallelicB
−1G coμ μ on =0
100
010
010
001
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I G G coμ μ on = 0
& dCallelic
dt=0 ⎡
⎣⎢⎤⎦⎥
that is, Gcommon should lie in the null-space of
I:
respectively
II:
2 0 0 00 1 1 00 0 0 2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I YCallelicB
−1
100
010
010
001
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I G
evolutionary stops
Allelic evolution for model I:
dXa
dtdXA
dt
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=ψ allelic2N allelicY
Ca 00 CA
⎛⎝⎜
⎞⎠⎟
∂1ΦaaT
0∂1ΦaA
T
0
0∂2ΦaA
T
0∂2ΦAA
T
⎛⎝⎜
⎞⎠⎟Gallelic
Gallelic =B−1 Gcoμ μ on
Ca 00 CA
⎛⎝⎜
⎞⎠⎟
∂1ΦaaT
0∂1ΦaA
T
0
0∂2ΦaA
T
0∂2ΦAA
T
⎛⎝⎜
⎞⎠⎟B
−1 G coμ μ on =0
Hence at the stops:
CallelicB−1G coμ μ on =0
or equivalently,
when do the alleles and modifiers agree?
The alleles on the focal locus and the modifiers agree about a stop only if
Iand
II100
010
010
001
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⊗I G G coμ μ on =0
CallelicB−1 G coμ μ on =0
The seemingly simpler Gcommon = 0, amounts to 4n equations.
If the dimensions of phenotypic and allelic spaces are n resp. m, then I is a system of min{4n , 2m}, II a system of 3n equations.
Hence, generically there is never agreement.
Xaa , XaA , XAA( )Xa , XA( )
In the case of modifier evolution, these have to be satisfied by 3n, in the case of allelic evolution by min{2m , 3n} unknowns(since the act only through the ).
(When 2m > 4n, the alleles cannot even agree among themselves!)
exceptions to the generic case
We have already seen a case where the alleles and modifiers agree:
if pA = qA.
This can happen for two very different reasons:
1. When (HW)
(the standard assumption of population genetics).
2. Phenotype space can be decomposed (at least locally near the ESS) into a component that influences only l, and one that only influences μ (as is the case in organisms with separate sexes), and moreover the Ideal Free assumption applies. In that case at ESSes laa = laA = lAA = 1 and μaa = μaA = μA., Hence (HW) applies, and therefore pA = qA.
μ(XG;E) ∝ λ (XG;E)
inverse problem: find all the exceptions
Assumption: 4m ≥ n
In that case there is only agreement at evolutionary stops iff at those stops
Gcommon = 0.
inverse problem: find all the exceptions
If not (a), any individual-based restriction doing the same job implies (b).
Examples: A priori Hardy Weinberg: .Ecological effect only through one sex: either or .Sex determining loci: for AA females and aA males:
μ ∝ λ∂l ∂x = 0 ∂μ ∂x = 0
μAA = 0, λ aA = 0.
The conditions for higher dimensional phenotype spaces are that after a diffeomorphism the space can be decomposed into components in which one or more of the above conditions hold true.
For one dimensional phenotype spaces the individual-based restrictions on the ecological model that robustly guarantee that Gcommon = 0 are that (a) at evolutionary stops (HW) holds true,
∂μaA ∂xaA =0 ⇒ ∂laA ∂xaA =0 ∂laA ∂xaA =0 ⇒ ∂μaA ∂xaA =0
∂μaa ∂xaa =0 & ∂laa ∂xaa =0 ∂μAA ∂xAA =0 & ∂lAA ∂xAA =0
or (b) in their neighbourhood:
(i) oror(ii) or
biological conclusions
When the focal alleles and modifiers fail to agree the result will be an evolutionary arms race
between the alleles and the rest of the genome.
Generically there is disagreement,
PredictionHermaphroditic species have a higher turn-over rate of their genome
than species with separate sexes.
This arms race can be interpreted as a tug of war between trait evolution and sex ratio evolution.
(Even though in all the usual models there is agreement !)
QuickTime™ en eenTIFF (LZW)-decompressorzijn vereist om deze afbeelding weer te geven.
Olof Leimar
with one biologically supported exception: the case where the sexes are separate.
The end
QuickTime™ and a decompressor
are needed to see this picture.
Carolien de Kovel
history
dXdt
= 12 N e C ∂sY X( )∂Y
Y=X
⎡
⎣⎢⎢
⎤
⎦⎥⎥
T
with Poisson # offspring
discrete generations
Assumptions still rather unbiological (corresponding to a Lotka- Volterra type ODE model): individuals reproduce clonally, have exponentially distributed lifetimes and give birth at constant rate from birth onwards
general life histories
Mendelian diploids
Michel Durinx & me
extensions (2008)
2 Ne
QuickTime™ en eenTIFF (LZW)-decompressor
zijn vereist om deze afbeelding weer te geven.
Ulf Dieckmann & Richard Law
basic ideas and first derivation (1996)
QuickTime™ en eenTIFF (ongecomprimeerd)-decompressorzijn vereist om deze afbeelding weer te geven.
Nicolas Champagnat & Sylvie Méléard
hard proofs (2003)
QuickTime™ en een-decompressorzijn vereist om deze afbeelding weer te geven.
so far only for community equilibria
non-rigorous
not yet publishednon-rigorous
hard proof for pure age dependence
Chi Tran(2006)
in reality alleles and modifiers will both evolve
dXdt
= dXdt
⎛⎝⎜
⎞⎠⎟μ odifier
+dXdt
⎛⎝⎜
⎞⎠⎟a
+dXdt
⎛⎝⎜
⎞⎠⎟A
= dXdt
⎛⎝⎜
⎞⎠⎟μ odifier
+dXdt
⎛⎝⎜
⎞⎠⎟allelic
in “reality”:
Generically in the genotype to phenotype map all three equations are incomplete dynamical descriptions as , and may still change as a result of the evolutionary process.
ΦY Callelic Cmodifier
and are constant when is linear and and resp. the are constant (two commonly made assumptions!). Otherwise constancy of and requires that changes in the various composing terms precisely compensate each other.
Callelic Cmodifier Φ Ca CA
CB
Cmodifier Callelic
rarely will be constant as and generically change with changes in X.Y p a p A
the canonical equation of adaptive dynamics
X: value of trait vector predominant in the population ne: effective population size, e: mutation probability per birth C: mutational covariance matrix, s: invasion fitness, i.e., initial relative growth rate of a potential Y mutant population.
dXdt
= 2 ne e C ∂sY X( )∂Y
Y=X
⎡
⎣⎢⎢
⎤
⎦⎥⎥
T