Dobzhansky-Muller Incompatibilities in Parapatry
Dobzhansky-Muller Incompatibilities
in Parapatry
Joachim Hermisson, Claudia Bank & Reinhard Bürger
Mathematics & Biology, University of Vienna
Dobzhansky-Muller Incompatibilities in Parapatry
Speciation due to genetic
incompatibilities: allopatry
secondary
contact
Hermann J. Muller
(1890 - 1967)
“DMI”
AB
Ab
Theodosius Dobzhansky
(1900-1975)
aB
a→A
allopatric
phase
b→B
ab
ancestral population
Dobzhansky-Muller Incompatibilities in Parapatry
Speciation due to genetic
incompatibilities: allopatry
secondary
contact
Hermann J. Muller
(1890 - 1967)
“DMI”
AB
Ab
Theodosius Dobzhansky
(1900-1975)
aB
a→A
allopatric
phase
b→B
ab
ancestral population
• theory by Orr,
Turelli, et al.
• standard allopatric
speciation model
Dobzhansky-Muller Incompatibilities in Parapatry
Speciation due to genetic
incompatibilities: parapatry
“DMI”
Hermann J. Muller
(1890 - 1967)
AB
Ab
a→A
parapatric
phase
migration
m
Theodosius Dobzhansky
(1900-1975)
aB
b→B
ab
ancestral population
Dobzhansky-Muller Incompatibilities in Parapatry
Origin and maintenance of
DMI‘s in parapatry
When does the mechanism work?
AB
Ab
a→A
migration
m
ab
¾ Limiting rate of gene-flow mmax
aB
b→B
Dobzhansky-Muller Incompatibilities in Parapatry
Origin and maintenance of
DMI‘s in parapatry
When does the mechanism work?
AB
Ab
a→A
migration
m
ab
aB
b→B
¾ Limiting rate of gene-flow mmax
• influence of ecological factors
- heterogeneous selection
• genetic factors
- strength of incompatibility
- recombination rate
• evolutionary history
- order of substitution events
Dobzhansky-Muller Incompatibilities in Parapatry
Model: 1. Haploid Continent and Island
• 2 loci, 2 alleles, recomb. r
A
a
r
aB, AB
ab, Ab
B
b
• unidirectional migration
a→A
m
island
aB
migration
m
ab
b→B
continent
Dobzhansky-Muller Incompatibilities in Parapatry
Model: 1. Haploid Continent and Island
• 2 loci, 2 alleles, recomb. r
A
a
r
aB, AB
ab, Ab
B
b
• unidirectional migration
a→A
m
• fitness scheme on island:
island
ecological or
extrinsic selection
aB
migration
m
ab
b→B
continent
incompatibility:
intrinsic selection
Dobzhansky-Muller Incompatibilities in Parapatry
Model: 2. Diploid Continent and Island
• 2 loci, 2 alleles, recomb. r
A
a
r
aB, AB
ab, Ab
B
b
• unidirectional migration
a→A
m
• fitness scheme on island:
island
aB
migration
m
ab
b→B
continent
4 incompatibility parameters
γ1: double heterozygote aAbB
γ2,γ3: hetero/ homo AAbB, aABB
γ4: double homozygote AABB
γ1 < γ2,γ3 < γ4
Dobzhansky-Muller Incompatibilities in Parapatry
Evolutionary Histories
“continent – island”
aB, AB
ab, Ab
a→A
island
aB
migration
m
ab
b→B
continent
Dobzhansky-Muller Incompatibilities in Parapatry
Evolutionary Histories
“island – continent ”
aB, AB
ab, Ab
a→A
island
aB
migration
m
ab
b→B
continent
“continent – island”
aB, AB
ab, Ab
a→A
island
aB
migration
m
ab
b→B
continent
Dobzhansky-Muller Incompatibilities in Parapatry
Evolutionary Histories
“island – continent ”
aB, AB
ab, Ab
a→A
island
aB
migration
m
ab
b→B
continent
“continent – continent”
aB, AB
ab, Ab
island
aB
migration
m
Ab
b→B
A→a
continent
“continent – island”
aB, AB
ab, Ab
a→A
island
aB
migration
m
b→B
continent
ab
“island – island”
aB, AB
ab, Ab
a→A
B→b
island
aB
migration
m
aB
continent
Dobzhansky-Muller Incompatibilities in Parapatry
Evolutionary dynamics on the island
island type Ab
recombinant AB
x3
x4
‫٭‬
x1
wildtype ab
continuous
time dynamics
?
DMI = internal
stable equilibrium
x2
continental aB
Dobzhansky-Muller Incompatibilities in Parapatry
Evolutionary dynamics on the island
island type Ab
recombinant AB
x3
x4
‫٭‬
x1
wildtype ab
continuous
time dynamics
?
DMI = internal
stable equilibrium
x2
continental aB
m
Dobzhansky-Muller Incompatibilities in Parapatry
Evolutionary dynamics on the island
secondary contact
island type Ab
recombinant AB
x3
x4
island – continent or
continent – continent
continent – island
or island – island
continuous
time dynamics
‫٭‬
x1
wildtype ab
?
DMI = internal
stable equilibrium
x2
continental aB
m
Dobzhansky-Muller Incompatibilities in Parapatry
Evolutionary dynamics on the island
Analytical results
island type Ab
recombinant AB
x3
x4
• equilibria and stability analysis
• Lyapunov functions
• limiting cases: m → 0, r → 0, D → 0
and perturbation theory
& numerical analysis
‫٭‬
x1
wildtype ab
?
x2
continental aB
m
Dobzhansky-Muller Incompatibilities in Parapatry
Evolutionary dynamics on the island
Analytical results
island type Ab
recombinant AB
x3
x4
• equilibria and stability analysis
• Lyapunov functions
• limiting cases: m → 0, r → 0, D → 0
and perturbation theory
& numerical analysis
‫٭‬
x1
wildtype ab
?
x2
continental aB
m
Limiting migration rates mmax for stable DMI:
• haploid: almost everything analytical
• diploid: only limiting cases analytical, full numerical overview
¾ at most a single stable DMI, which can be globally or locally stable
Dobzhansky-Muller Incompatibilities in Parapatry
Results: Types of parapatric speciation
1. Neutral DMI: α = β = 0
¾ never for m > 0, unless r = 0
• flow: Ab → ab → aB
•
island type Ab
recombinant AB
x3
x4
‫٭‬
general proof for diploid ?
x1
wildtype ab
?
x2
continental aB
m
Dobzhansky-Muller Incompatibilities in Parapatry
Results: Types of parapatric speciation
1. Neutral DMI: α = β = 0
¾ never for m > 0, unless r = 0
• flow: Ab → ab → aB
•
recombinant AB
x3
x4
‫٭‬
general proof for diploid ?
2. Adaptive DMI [Schluter 2009]:
a) “Ecological speciation”
•
•
island type Ab
x1
wildtype ab
globally stable DMI
Schluter: for local adaptation (β < 0 < α)
?
x2
continental aB
m
Dobzhansky-Muller Incompatibilities in Parapatry
Results: Types of parapatric speciation
1. Neutral DMI: α = β = 0
¾ never for m > 0, unless r = 0
• flow: Ab → ab → aB
•
recombinant AB
x3
x4
‫٭‬
general proof for diploid ?
2. Adaptive DMI [Schluter 2009]:
a) “Ecological speciation”
•
•
island type Ab
x1
wildtype ab
globally stable DMI
Schluter: for local adaptation (β < 0 < α)
?
x2
continental aB
m
b) “Mutation order speciation”
• Origin of DMI depends on mutation order: historical contingency
¾ locally stable DMI: evolves only if second substitution on continent
(island – continent, continent – continent, secondary contact)
• Schluter: for global adaptation (0 < α, β)
Dobzhansky-Muller Incompatibilities in Parapatry
Results: Fitness parameters (haploid)
D = 0 (linkage
equilibrium)
mmax
“local adaptation”
β<0<α
globally stable
β
0
locally stable
α /4
γ
α
mmax
α /4
“global adaptation”
0 < α, β
locally stable
0
α,β
γ
Dobzhansky-Muller Incompatibilities in Parapatry
Results: Fitness parameters (haploid)
D = 0 (linkage
equilibrium)
mmax
“local adaptation”
β<0<α
¾ heterogeneous
selection
β
globally stable
0
locally stable
α /4
γ
α
mmax
α /4
“global adaptation”
0 < α, β
¾ selection against
hybrids
locally stable
0
α,β
γ
Dobzhansky-Muller Incompatibilities in Parapatry
Results: Fitness parameters (haploid)
D = 0 (linkage
equilibrium)
mmax
“local adaptation”
β<0<α
¾ heterogeneous
selection
β
globally stable
0
locally stable
α /4
γ
α
mmax
α /4
“global adaptation”
0 < α, β
¾ selection against
hybrids
locally stable
0
α,β
γ
Dobzhansky-Muller Incompatibilities in Parapatry
Results: Recombination & linkage (haploid)
linkage
r = 0; 0.001; mmax
0.003; 0.1; 100
“local adaptation”
β<0<α
¾ heterogeneous
selection
β
0
α
0
α,β
γ
linkage
mmax
“global adaptation”
0 < α, β
¾ selection against
hybrids
γ
Dobzhansky-Muller Incompatibilities in Parapatry
Results: Diploid, recessive DMI
γ1 = 0;
γ2,3 = γ4 /2
mmax
“local adaptation”
β<0<α
¾ heterogeneous
selection
β
0
α
γ2,3
0
α,β
γ2,3
mmax
“global adaptation”
0 < α, β
¾ selection against
hybrids
Dobzhansky-Muller Incompatibilities in Parapatry
Results: Diploid, co-dominant DMI
γ1 = γ4 /4; mmax
γ2,3 = γ4 /2
+ selection against
double heterozygote F1
“local adaptation”
β<0<α
¾ heterogeneous
selection
β
0
α
γ2,3
0
α,β
γ2,3
mmax
“global adaptation”
0 < α, β
¾ selection against
hybrids
Dobzhansky-Muller Incompatibilities in Parapatry
Summary & Outlook
• No neutral DMI’s with gene-flow
Dobzhansky-Muller Incompatibilities in Parapatry
Summary & Outlook
• No neutral DMI’s with gene-flow
• Two mechanisms for stable DMI’s in parapatry:
Heterogeneous selection
- can be globally stable
- weak DMI favored
- tight linkage
Hybrid deficiency (intrinsic)
- mutation order important
- strong DMI favored
- loose linkage
Dobzhansky-Muller Incompatibilities in Parapatry
Summary & Outlook
• No neutral DMI’s with gene-flow
• Two mechanisms for stable DMI’s in parapatry:
Heterogeneous selection
- can be globally stable
- weak DMI favored
- tight linkage
Hybrid deficiency (intrinsic)
- mutation order important
- strong DMI favored
- loose linkage
• In diploids:
• costs for double heterozygotes AaBb (γ1) crucial
• γ1 >> 0: DMIs due to deficient hybrids with tight linkage
Dobzhansky-Muller Incompatibilities in Parapatry
Summary & Outlook
• No neutral DMI’s with gene-flow
• Two mechanisms for stable DMI’s in parapatry:
Heterogeneous selection
- can be globally stable
- weak DMI favored
- tight linkage
Hybrid deficiency (intrinsic)
- mutation order important
- strong DMI favored
- loose linkage
• In diploids:
• costs for double heterozygotes AaBb (γ1) crucial
• γ1 >> 0: DMIs due to deficient hybrids with tight linkage
• Outlook:
• two islands, X-linked DMI’s, genetic drift, …
Dobzhansky-Muller Incompatibilities in Parapatry
Thanks !
Claudia Bank
Reinhard Bürger