species interactions and coevolutiongomulki/mathbiol494/materials/ubm_nuismer_v... · 2011. 1....
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What is coevolution?
"Thus I can understand how a flower and a bee mightslowly become, either simultaneously or one after theother, modified and adapted to each other in the mostperfect manner, by the continued preservation of all theindividuals which presented slight deviations ofstructuremutually favourable to each other."— Charles Darwin, The Origin of Species
Coevolution: Reciprocal evolutionary change in interacting species (Janzen, 1980)
Species 1
Species 2
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My lab uses mathematical models to study coevolution
Spatially structured
species interactions
Structure of biological
communities
Genomestructure
Species 1
Species 2
Coevolution
Focus on this project today
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An example of spatially structured coevolution: toxic newts and resistant
snakes
Thamnophis sirtalis Taricha
+ =
• Predator-prey interaction
Butch Brodie
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Toxic newts
Taricha granulosa
• Newts contain Tetrodotoxin, a potent neurotoxin
• Some newts contain enough toxin to easily kill a human
• Toxin causes snakes to only “taste” the newts
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Resistant snakes
Thamnophis sirtalis
• Some snakes have evolved modified sodium channels
• These snakes are more resistant to tetrodotoxin
• Consequently, resistant snakes can eat toxic newts
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Toxic newts and resistant garter snakes(Hanifin et al. 2008, PLoS Biology)
Observation #1:
Newt toxicity and snake resistance Are spatially variable
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Toxic newts and resistant garter snakes(Hanifin et al. 2008, PLoS Biology)
Observation #2:
Newt toxicity and snake resistance are positively correlated
Predator resistance
Pre
y t
oxic
ity
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Summarizing the Data(Hanifin et al. 2008, PLoS Biology)
• Newt toxicity varies across space
• Snake resistance varies across space
• Toxicity and resistance are positively correlated
These observations have led to the development of a coevolutionary hypothesis
Predator resistance
Prey
tox
icit
y
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A coevolutionary hypothesis
Location 1: Strong coevolution
Time
*** We can test this coevolutionary hypothesis using mathematical models ***
Location 2: Weak coevolution
Resistance
&
Toxicity
Resistance
&
Toxicity
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Developing an appropriate model
• The data consists of toxicity and resistance measured in many populations
mmz
mmz
6.1
2.1
2
1
mmz
mmz
6.1
3.1
2
1
mmz
mmz
3.1
1.1
2
1
mmz
mmz
7.1
4.1
2
1
mmz
mmz
9.1
8.1
2
1
Our model must predict mean trait
values in replicate populations
2z
1z
1.0 1.2 1.4 1.6 1.8 2.01.0
1.2
1.4
1.6
1.8
2.0
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Let’s start by modeling one of these populations
If we assume that additive genetic variance is constant:
mmz
mmz
6.1
2.1
2
1
mmz
mmz
6.1
3.1
2
1
mmz
mmz
3.1
1.1
2
1
mmz
mmz
7.1
4.1
2
1
mmz
mmz
9.1
8.1
2
1
1
1
1
11
1
z
W
WGz
2
2
2
22
1
z
W
WGz
To predict (co)evolution we need to
calculate mean fitness
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Defining individual fitness
),()()( jiBiAiT zzWzWzW
Abiotic environment:
WA
Species interactions:
Phenotype, z
Optimal phenotype,
Probabilityof
consumption
1.0 0.5 0.0 0.5 1.0
0.2
0.4
0.6
0.8
1.0
Snake resistance - Newt toxicity
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Developing recursions for trait means
21212111 )}()(),()( dzdzzzzzWzWW BA
i
i
i
iiz
W
WGz
1
21211222 )}()(),()( dzdzzzzzWzWW BA
Assume weak selection
2
D2 OszGz iiiiiii
Incorporate genetic drift
Abiotic selection DriftBiotic selection
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Model predictictions for local coevolution
Weak selection on newtsStrong selection on snakes
Toxicity and Resistance
Equilibrium trait values depend on the strength of biotic selection
Strong selection on newtsWeak selection on snakes
Strong selection on newtsStrong selection on snakes
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But we need a model of MANY populations!
mmz
mmz
6.1
2.1
2
1
mmz
mmz
6.1
3.1
2
1
mmz
mmz
3.1
1.1
2
1
mmz
mmz
7.1
4.1
2
1
mmz
mmz
9.1
8.1
2
1
• Multiple populations
• Gene flow (island model)
Requires more equations
Empirical Data Minimal model
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Adding multiple populations and gene flow
2
1,D1,1,M1,1,1, )1(22 OZmzmszzszGz iiiiiiijiiiiii
2
2,D2,2,M2,2,2, )1(22 OZmzmszzszGz iiiiiiijiiiiii
•
•
•
2
,D,,M,,, )1(22 OZmzmszzszGz iiiniiininjininiiini
• In principle, we could then just solve this system of 2n equations
• In practice, this is impossible
This difficulty can be overcome by making a change of variables that reveals a tractable approximation
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What does our final approximation predict?
At equilibrium and assuming weak selection:
The spatial variability in toxicity or resistance is:
The correlation between toxicity and resistance is:
iiii
iz
GmN
Gi
22
ˆ 2
)(0ˆ 2 O
What is missing from these equations?
What does this tell us?
Does this provide support for the coevolutionary hypothesis?