Chapter 9. Coevolution, polygenic traits, and spatial patterns of phenotypic variation
Biological Motivation
In the previous chapter, we explored how coevolution proceeds in spatially structured
environments where gene flow connects isolated populations inhabiting potentially heterogeneous
environments. Our primary focus was to understand how gene flow and coevolution interact to shape
patterns of genetic polymorphism and genetic differentiation among and within populations when the
outcome of interactions depends on a single genetic locus in each species. Although these results
informed our understanding of a wide range of naturally occurring species interactions, such as those
between M truncatula and S. meliloti, where a small number of genes are thought to be involved, they
cannot effectively address the broad swath of species interactions that are mediated by quantitative
traits. For instance, the well-studied interactions between wild parsnip XX and parsnip webworm XX are
mediated by concentrations of toxic furanocoumarins in the plant and levels of detoxifying enzyme
activity in the insect (REFS). Other well-studied interactions that appear to be mediated by quantitative
traits include those between the garter snake predator, Thamnophis sirtalis, and its toxic prey, T.
granulosa (REFS), and those between the Japanese Camellia and its seed predator XX (REFS), to name
only a few.
Over the past ten to twenty years, landscape level studies of interactions mediated by
quantitative traits have revealed a wide range of interesting phenotypic patterns. For instance, in the
interaction between the toxic newt, T. granulosa and its garter snake predator, T. sirtalis, average levels
of newt toxicity vary widely across the western United States as do average levels of snake resistance
(REFS). Within some regions, newt toxicity and snake resistance are wildly exaggerated, with a single
newt carrying enough tetrodotoxin to kill a Tyranosaurus Rex and local snakes possessing such extreme
resistance that they can actually manage to ingest these crawling toxic waste dumps (GET REAL FACTS
FROM BUTCH; REFS). Overall, when snake and newt populations are studied together across the
western Unites States, the pattern that emerges is one where newt toxicity and snake resistance vary in
concert, with geographic regions with elevated toxicity generally associated with elevated levels of
resistance (REFS). Comparable patterns have been observed in a wide range of interactions (REFS),
suggesting that similar evolutionary processes may be driving the evolution of quantitative traits
mediating these well-studied species interactions. The central goal of this chapter will be to explore the
potential for coevolution to generate these commonly observed phenotypic patterns. In order to
achieve this goal, and address the key questions that follow, we need to integrate spatial structure into
the quantitative genetic models of coevolution we first developed in Chapter 3.
Key Questions:



Can coevolution generate phenotypic variation across space?
Is there reason to expect exaggerated trait values within regions of intense coevolution?
Does coevolution lead to well-matched phenotypes in interacting species?
Building a Model of Snake-Newt Interactions
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
In Chapter 2, we developed a model of coevolution between two species mediated by a
quantitative trait and a mechanism of phenotype differences. This model assumes the probability of a
“successful” interaction occurring depends on the extent to which the phenotype of one species
exceeds that of the other (Chapter 2, Figure X). In the context of interactions between T. sirtalis and T.
granulosa, a “successful” interaction is one where the snake successfully consumes the newt, with the
probability of consumption depending on the quantity of tetrodotoxin in the newt (𝑥̅ ) and the resistance
to tetrodotoxin in the snake (𝑦̅). Being consumed reduces newt fitness (by an amount 𝑠𝑋 ) and increases
snake fitness (by an amount 𝑠𝑌 ). In addition to selection imposed by the interaction, an extension of this
basic model accounted for constraints on toxicity and resistance in the form of stabilizing selection
toward optimal trait values 𝜃𝑋 and 𝜃𝑌 . These assumptions, along with the classical quantitative genetic
assumptions of weak selection, fixed additive genetic variance, and Gaussian phenotype distributions
allowed us to derive expressions for the population mean phenotypes of the interacting species in the
next generation:
𝑥̅ ′ ≈ 𝑥̅ + 𝑆𝑋 𝐺𝑋 + 2𝛾𝑋 𝐺𝑋 (𝜃𝑋 − 𝑥̅ )
(1a)
𝑦̅ ′ ≈ 𝑦̅ + 𝑆𝑌 𝐺𝑌 + 2𝛾𝑌 𝐺𝑌 (𝜃𝑌 − 𝑦̅)
(1b)
where 𝑆𝑋 = 𝛼
𝑠𝑋
2(2+𝑠𝑋 )
and 𝑆𝑌 = 𝛼
𝑠𝑌
2(2−𝑠𝑌 )
, and XXX and XXX.
In order to use this model to study the spatial patterns that characterize many interactions
mediated by quantitative traits, we need to integrate multiple populations and gene flow between
them. CAUTION ABOUT THE PERILS OF SPATIAL STRUCTURE, GENE FLOW, AND BIMODALITY
𝑥̅𝑖′ ≈ 𝑥̅𝑖 + 𝑆𝑋,𝑖 𝐺𝑋 + 2𝛾𝑋 𝐺𝑋 (𝜃𝑋 − 𝑥̅𝑖 )
(2a)
𝑦̅𝑖′ ≈ 𝑦̅𝑖 + 𝑆𝑌 𝐺𝑌 + 2𝛾𝑌 𝐺𝑌 (𝜃𝑌 − 𝑦̅𝑖 )
(2b)
Our basic approach will be like that in the previous chapter. We will explore one of the simplest
possible spatial scenarios: a pair of populations connected by gene flow at rates X and X.
𝑥̅𝑖′′ ≈ (1 − 𝑚𝑋 )𝑥̅𝑖′ + 𝑚𝑋 𝑥̅𝑗′
(3a)
𝑦̅𝑖′′ ≈ (1 − 𝑚𝑌 )𝑦̅𝑖′ + 𝑚𝑌 𝑦̅𝑗′
(3b)
Analyzing the Model
We will then clarify and focus our unbderstanding by yusing the change of variables analogous
to that we also used in the previous chapter
∆𝜇𝑋 = 𝐺𝑋 (𝑆𝑋̅ − 2𝛾𝑋 (𝜇𝑋 − 𝜃𝑋 ))
(4a)
2
∆𝛿𝑋 = −2𝑚𝑋 𝛿𝑋 + 𝐺𝑋 (1 − 2𝑚𝑋 )(𝛿𝑆𝑋 − 2𝛾𝑋 𝛿𝑋 )
(4b)
∆𝜇𝑌 = 𝐺𝑌 (𝑆𝑌̅ − 2𝛾(𝜇𝑌 − 𝜃𝑌 ))
(4a)
∆𝛿𝑌 = −2𝑚𝑌 𝛿𝑌 + 𝐺𝑌 (1 − 2𝑚𝑌 )(𝛿𝑆𝑌 − 2𝛾𝑌 𝛿𝑌 )
(4b)
where XXX… Interestingly, (3) shows that the coevolutionary dynamics of average phenotypes are
decoupled from the coevolutionary dynamics of spatial structure. We can capitalize on this
independence to first solve for the coevolutionary dynamics oif trait means, which is identical to what
we did in Chapter 3 and has time depdnent solution:
Next, we can easily see that spatial structure must decay over time because XX. Thus, ultimately, we
expect coevolution between snake and newt to lead to an equilibrium where:
𝑆̅
𝜇̂ 𝑋 = 𝜃𝑋 + 2𝛾𝑋
(5a)
𝑋
(1−2𝑚𝑋 )
𝛿𝑆 𝐺𝑋
𝛿̂𝑋 = 2𝛾 𝐺 𝑋(1−2𝑚
(5b)
𝑋 )+2𝑚𝑋
𝑋 𝑋
𝑆̅
𝜇̂ 𝑌 = 𝜃𝑌 + 2𝛾𝑌
(5c)
𝑌
𝛿̂𝑌 =
𝛿𝑆𝑌 𝐺𝑌 (1−2𝑚𝑌 )
(5d)
2𝛾𝑌 𝐺𝑌 (1−2𝑚𝑌 )+2𝑚𝑌
Clearly, then, we expect no difference between spatially structured and unstructured interactions.
Taken together, our results suggest that c
Answers to Key Questions:
Can coevolution generate phenotypic variation across space?
Potentially. Our results show that in a homogenous environment, coevolution alone cannot
cause population mean phenotypes to diverge among populations. However, when species interactions
impose different fitness consequences in different geographic regions, population mean phenotypes
escalate to different degrees. The result of this differential escalation is spatial variation in mean
phenotype.
Is there reason to expect exaggerated trait values within regions of intense coevolution?
Yes. Regions where both species experience strong fitness consequences from interacting, and
thus intense coevolution, should be characterized by exaggerated trait values. The extent to which trait
3
values are exaggerated in these coevolutionary “hot spots” depends on the balance between
coevolutionary selection and stabilizing selection.
Does coevolution lead to well-matched phenotypes in interacting species?
Potentially. If both species experience stronger than average selection from species interactions
in the same geographic locations and weaker than average selection from species interactions in the
same geographic locations, the population mean phenotypes of the two species will be positively
correlated and well-matched. Put differently, if the fitness consequences of species interactions are
positively correlated, then so too will population mean phenotypes. If, however, the fitness
consequences of interactions are uncorrelated or negatively correlated, the population mean
phenotypes of the interacting species will also be uncorrelated or even systemically mismatched.
New Questions Arising:
Our simple model of XXX cannot explain anything we observe suggesting we have errored on
the side of too simple.

Can forces other than coevolution produce spatially variable and potentially correlated
phenotypes?

Are some functional forms of interaction more likely to produce spatial variation and correlation
than others?

Should we expect mutualistic interactions to produce similar patterns of spatial phenotypic
variation and trait correlation?
In the next three sections, we will develop generalizations of our simple model which allow us to answer
these questions and gain further insight into the process of coevolution.
Extensions
Extension 1: Can forces other than coevolution produce spatially variable and potentially correlated
phenotypes?
WARNING: EXPLAIN DANGERS OF DISTANT THETAS AND APPROXIMATIONS…
Although …
Extension 2: Are some functional forms of interaction more likely to produce spatial variation and
correlation than others?
4
Extension 3: Should we expect mutualistic interactions to produce similar patterns of spatial phenotypic
variation and trait correlation?
Although interactions …
Conclusions and Synthesis
5
Tables
Figure Legends
References
6