Chapter 8. Coevolution, major genes, and spatial patterns of genetic variation
Biological Motivation
IN THIS CHAPTER WE EXPLORE WHEN COEVOLUTION GENERATES SPATIAL PATTERNS. Focused on very
simple models that illustrate how we can include spatial structure
START WITH MUTUALISM AND POINT OUT HOW WEIRD IT IS THAT THERE IS VARIATION WITHIN POPS
Until now we have focused on interactions in single pops. We know however, that spatial structure
matters… Let’s look at the snails and trematodes again… One way the importance of space has been
discerened is by conducting a reciprocal cross infection experiment… These experiments have revealed
very large levels of local adaptation. What is the underlying cause of these patterns?
Key Questions:



Can spatial structure maintain genetic polymorphism within coevolving mutualistic populations?
Can coevolution generate genetic differentiation among populations?
How do coevolutionary selection and gene flow interact to shape rates of mutualistic interaction
within and among populations?
Building a model of spatially structured coevolution
The first decision we must make is exactly what type of spatial structure we want to explore. For
instance, should we focus on an extremely simple scenario such as a mainland island model where a
very large and evolutionarily static mainland repeatedly sends migrants to a small evolutionarily
dynamic island? Or should we instead focus on a more realistic and complex scenario where a large
number of evolutionarily dynamic populations share migrants with one another in a complex and largely
idiosyncratic pattern? Although the first approach is likely to be mathematically tractable, it is of limited
utility for studying coevolution because, unlike single species models exploring directional selection, we
have no clear a priori expectations for the evolutionary state of the mainland population. Although the
second approach avoids this problem, it comes at the cost of mathematical intractability; modeling the
scenario shown in Figure 1b would require at least XX equations! Rather than taking one of these two
extreme approaches, we will tackle what I think is the minimal model that can shed light on spatially
structured coevolution: two evolutionarily dynamic populations sharing migrants (Figure 1c).
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
as we begin to develop a model of spatially structured coevolution between M. truncatula and S.
meliloti,
we need to first decide just how we
In Chapter 2, we developed a model for coevolution between two interacting species mediated by a
single diallelic haploid locus. Our results revealed that, in general, the allele frequencies of the A and B
alleles after a single episode of coevolutionary selection are given by:
′
𝑝𝑋,𝑖
=
′
𝑝𝑌,𝑖
=
𝑝𝑋,𝑖 𝑊𝑋,𝐴,𝑖
̅ 𝑋,𝑖
𝑊
𝑝𝑌,𝑖 𝑊𝑌,𝐵,𝑖
̅ 𝑌,𝑖
𝑊
where 𝑝𝑋 is the frequency of the A allele in species X and 𝑝𝑌 is the frequency of the B allele in species Y.
If we let
𝑝𝑖′′ =
Now that we have expressions for the expected fitness of the various genotypes, we can use these to
predict how coevolution will change allele frequencies in the two species. Specifically, substituting (9)
into (3) and replacing the general interaction matrix entries 𝛼𝑖,𝑗 with the values appropriate for the
matching alleles model shown in Figure 1, yields the following expressions for coevolutionary change in
the frequency of M. truncatula and S. meliloti A and B alleles:
∆𝑝𝑋 =
−𝑠𝑋 𝑝𝑋 𝑞𝑋 (1−2𝑝𝑌 )
̅𝑋
𝑊
(10a)
∆𝑝𝑌 =
−𝑠𝑌 𝑝𝑌 𝑞𝑌 (1−2𝑝𝑋 )
̅𝑌
𝑊
(10b)
where 𝑞𝑋 = (1 − 𝑝𝑋 ) and 𝑞𝑌 = (1 − 𝑝𝑌 ). These expressions for evolutionary change are almost
identical to those describing antagonistic coevolution mediated by a matching alleles model (i.e., Eq. 6),
differing only in the sign of the equation predicting evolutionary change in the “host” individual — in
this case, the plant, M. truncatula.
Analyzing the Model
Answers to Key Questions:
2
Can spatial structure maintain genetic polymorphism within coevolving mutualistic populations?
Can coevolution generate genetic differentiation among populations?
How do coevolutionary selection and gene flow interact to shape rates of mutualistic interaction within
and among populations?
New Questions Arising:



Do similar results hold for other forms of ecological interaction?
How would our results change if the environment were heterogeneous?
What is the role of random genetic drift?
In the next three sections, we will generalize our simple model in ways that allow us to answer these
questions.
Generalizations
Generalization 1: Alternative forms of ecological interaction
Generalization 2: Heterogeneous environments and selection mosaics
Generalization 3: Random genetic drift
Conclusions and Synthesis
3
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Figure Legends
Figure 1. Population dynamics predicted by the Lotka-Volterra model for three different rust death
rates, d. In all cases, the dashed line indicates rust population size and the solid line indicates flax
population size. Parameters held constant across panels were 𝛼 = .1, 𝛽 = .03, 𝜉 = .01, 𝑟 = .1,
and 𝑘 = 100. Rust death rates were: A) 𝑑 = .005 B) 𝑑 = .081, and C) 𝑑 = .1.
Figure 2. Population sizes and allele frequencies for the gene-for-gene model over time for three
different rust death rates, d. In all cases, dashed lines indicate rust population size or frequency of the
virulent allele and solid lines indicate flax population size or frequency of the resistant allele. Parameters
held constant across panels were 𝛽 = .003, 𝜉 = .001, 𝑟 = .1, and 𝑘 = 100. Rust death rates were: A)
𝑑 = .005 B) 𝑑 = .081, and C) 𝑑 = .1.
Figure 3. Population sizes and allele frequencies for the gene-for-gene model with costs of resistance
and virulence over time for three different costs of virulence. In all cases, dashed lines indicate rust
population size or frequency of the virulent allele and solid lines indicate flax population size or
frequency of the resistant allele. Parameters held constant across panels were 𝛽 = .003, 𝜉 = .001,
𝑟𝐴 = .05, 𝑟𝑎 = .09, 𝑑𝑏 = .01 and 𝑘 = 100. Death rates of the virulent rust allele were: A) 𝑑𝐵 = .005 B)
𝑑𝐵 = .081, and C) 𝑑𝐵 = .1.
Figure 4. Population dynamics and allele frequencies for the matching alleles model over time for three
different P. ramosa death rates, d. In all cases, dashed lines indicate P. ramosa population size or
frequency of the B allele and solid lines indicate D. magna population size or frequency of the A allele.
Parameters held constant across panels were 𝛽 = .003, 𝜉 = .001, 𝑟 = .1, and 𝑘 = 100. Rust death
rates were: A) 𝑑 = .005 B) 𝑑 = .081, and C) 𝑑 = .1.
Figure 5. Population sizes and phenotypes over time for the phenotype matching model and two
different cuckoo death rates, d. In all cases, dashed lines indicate cuckoo population size or egg color
phenotype and solid lines indicate warbler population size or egg color phenotype. Parameters held
constant across panels were 𝛽 = .006, 𝜉 = .005, 𝑟 = .05, 𝑘 = 100, and the mutation rates of the two
species 𝜇𝑋 = .02 and 𝜇𝑌 = .02. Rust death rates were: A) 𝑑 = .06 and B) 𝑑 = .03.
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Table 1. Summary of stability conditions and ecological dynamics
Condition
Implications for stability
Equilibrium (3c) unstable;
𝑑 > 𝑘𝛼𝜉
Equilibrium (3b) stable
Equilibrium (3c) stable;
𝑑 < 𝑘𝛼𝜉
Equilibrium (3b) unstable
Equilibrium
(3c) stable and
4𝑘𝛼𝜉
oscillatory;
𝑑 < 𝑘𝛼𝜉 (
)
𝑟 + 4𝑘𝛼𝜉
Equilibrium (3b) unstable
7
Biological consequences
Extinction of Rust. Flax at
carrying capacity
Coexistence of Flax and Rust
Coexistence of Flax and Rust.
Transient cycles likely.
Table 2. Equilibria corresponding to parasite extinction. Because we assume carrying the virulent allele is
costly, 𝑑𝐵 > 𝑑𝑏 .
Equilibrium
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 0, 𝑝𝑉 = 0
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 0, 𝑝𝑉 = 1
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 1, 𝑝𝑉 = 0
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 1, 𝑝𝑉 = 1
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 1 − (𝑑𝑏 ⁄𝑘𝜉 ), 𝑝𝑉 = 0
Conditions for instability
𝑑𝑏 < 𝑘𝜉
Always unstable
𝑑𝐵 − 𝑑𝑏 < 𝑘𝜉
𝑘𝜉 + 𝑑𝑏 < 𝑑𝐵 𝑜𝑟 𝑑𝐵 < 𝑘𝜉
𝑑𝐵 < 𝑘𝜉
8