Chapter 7. Coevolution, recombination, and the red queen
Biological Motivation
A long-standing mystery in evolutionary biology is the origin and maintenance of sexual
reproduction (REFS). One of the key reasons sexual reproduction is so perplexing is that it should
generally impose a twofold cost associated with the production of male offspring (REFS). Thus, unless
sexual reproduction confers significant fitness advantages, it should rapidly be supplanted by clonal
reproduction. One such possible fitness advantage of sexual reproduction could be the production of
offspring carrying novel combinations of genes as a result of recombination. For the production of novel
combinations of genes to be beneficial, however, the environment must be continually changing;
otherwise, the primary impact of recombination would be to break apart favorable gene combinations
tuned by generations of natural selection (REFS). The importance of persistent environmental change
suggests that interactions with other species, and particularly with antagonistic species such as parasites
that continually evolve to better infect their hosts, could play an important role in generating selection
for sexual reproduction and increased rates of recombination (REFS). This idea, now formalized as the
Red Queen Hypothesis for the evolution of sex, suggests that coevolutionary interactions between hosts
and parasites favor the evolution of increased rates of recombination and sexual reproduction within
host populations (REFS).
Despite the popularity of the Red Queen hypothesis, empirical evidence supporting a role for
coevolution in the origin and maintenance of sexual reproduction and recombination comes from a
relatively small number of systems (REFS). Perhaps the best studied of these is the interaction between
the New Zealand mud snail, Potamopyrgus antipodarum, and its castrating trematode parasite,
Microphallus (Refs). Within New Zealand, these snails inhabit freshwater lakes. What makes this system
so cool is that the proportion of sexual snails differs among lakes as does the intensity of parasitism by
Microphallus (REFS). Long running studies of this interaction suggest that it may be coevolution that
favors the evolution of increased rates of sexual reproduction within lakes experiencing high rates of
parasitism (REFS). Our focus in this chapter will be to develop mathematical models that allow us to
explore the conditions under which coevolution between snail and trematode would be likely to
maintain high levels of sexual reproduction and recombination in lakes with significant parasite
pressure.
Key Questions:



Does coevolution between snail and trematode favor increased host recombination?
Does coevolution between snail and trematode cause changes in parasite recombination?
Does infection genetics matter?
Modeling the Red Queen
In order to understand how coevolution drives changes in rates of sexual reproduction or
recombination, we must develop a model that allows these quantities to evolve. One possibility is that
we could simply study coevolution between a parasite and a population of hosts with some asexual
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
individuals and some sexual individuals (REFS). We could then track the frequency of sexual individuals
over time to see if they increase or decrease. Although conceptually straightforward, implementing this
approach mathematically is more challenging than you might think, and deriving biologically insightful
results is more challenging still. An alternative possibility is to assume the entire host population
reproduces sexually, but that some host genotypes recombine at a greater rate than others (REFS). This
latter approach, while by no means mathematically simple, often allows more insightful biological
results to be derived. For this reason, we will pursue this latter strategy in this chapter, and work to
develop a mathematical model that allows the rate of recombination to evolve within coevolving host
and parasite populations.
The simplest possible model that can shed light on the conditions favoring the evolution of
increased rates of recombination is one in which there are three possible loci, A, B, and M. We will
assume that the first two loci, A and B, are involved in coevolutionary interactions between the species
just as was the case for the two locus models we studied earlier in Chapter X. In contrast, the third locus,
M, has no direct fitness consequences, instead determining only the rate of recombination between the
A and B loci (Figure 1). In general, we will assume that the M allele increases recombination rates and
that its effect on recombination rate is additive (Table 1). Even if we limit ourselves to studying only
haploid diallelic loci, we must now confront the challenges of studying a coevolutionary model with
eight possible genotypes in each species.
To keep our model general for the time being, let’s not spend a lot of time thinking about the
underlying genetic mechanisms mediating coevolutionary interactions. Instead, let’s just forge ahead
and leave coevolution implicit, recognizing that its impact arises through modification of genotypic
fitnesses just as in the previous two chapters. Now, if we assume that the probability of survival to
mating for the various snail and trematode genotypes depends on these fitnesses, we can calculate the
frequency of each genotype after selection but prior to random mating. As before, we can calculate
these frequencies by multiplying the current frequency by its relative fitness. This yields the following
expression for snail and trematode genotype frequencies after selection:
𝑋𝑖′ =
𝑋𝑖 𝑊𝑋,𝑖
̅𝑋
𝑊
(3a)
𝑌𝑖′ =
𝑌𝑖 𝑊𝑌,𝑖
̅𝑌
𝑊
(3b)
̅𝑋 and 𝑊
̅𝑌 , are given by:
where the population mean fitnesses, 𝑊
̅𝑋 = ∑8𝑖=1 𝑋𝑖 𝑊𝑋,𝑖
𝑊
(4a)
̅𝑌 = ∑8𝑖=1 𝑌𝑖 𝑊𝑌,𝑖
𝑊
(4b)
We now know the frequency of each of the eight genotypes within both host and parasite after species
interactions have occurred. Next, we must predict how these genotype frequencies will change in
response to a single round of random mating and recombination.
2
Conceptually, integrating random mating and recombination is a simple matter. We need only
sum up the frequency with which all possible matings occur, and weight each possible mating the by the
probability that it produces a specific offspring genotype. Mathematically, this reasoning leads to the
following expressions for the frequency of genotype i after random mating and recombination in snail
and trematode:
𝑋𝑖′′ = ∑8𝑗=1 ∑8𝑘=1 𝑋𝑗′ 𝑋𝑘′ Π𝑋,𝑗+𝑘→𝑖
(5a)
𝑌𝑖′′ = ∑8𝑗=1 ∑8𝑘=1 𝑌𝑗′ 𝑌𝑘′ Π𝑌,𝑗+𝑘→𝑖
(5b)
where the terms Π𝑋,𝑗+𝑘→𝑖 and Π𝑌,𝑗+𝑘→𝑖 are the probabilities that parental genotypes j and k produce
offspring genotype i in snail and trematode, respectively. Although these equations appear quite simple,
they actually mask a huge amount of biological detail and tedious calculation. Specifically, there are 64
different Π𝑋,𝑗+𝑘→𝑖 terms and 64 different Π𝑌,𝑗+𝑘→𝑖 terms that must be worked out, each of which is a
function of the recombination rates between the A and B loci and the B and M loci. In the previous
chapter, we tackled this problem by creating a table depicting the frequency of offspring produced by
each possible mating. Although this was quite manageable with only a pair of loci, it becomes very
unwieldy for three loci because there are 64 possible types of matings, each of which can produce 8
possible offspring genotypes. As much fun as it would be to develop such a table, what would be the
point when it wouldn’t even fit on a single page? Instead, it is much more practical (and accurate) to
develop a simple computer algorithm that generates this table automatically. In the accompanying
Mathematica notebook, just such an algorithm is developed and used to perform the calculations
proscribed by equations (5).
At this point, we have a really huge algebraic mess on our hands. It is no understatement to say
that writing out the fully expanded versions of equations (5) would yield expressions of such length and
tedium that we would almost certainly cry blood. So what can we do? Perhaps the single most powerful
approach we can employ at this point to make sense out of these complex expressions is the QuasiLinkage Equilibrium (QLE) approximation we introduced in the previous chapter.
Analyzing the model
Although the general assumptions of our QLE approximation mirror those of the previous
chapter, the specific steps we must take to implement the approximation are now a bit more complex.
One significant change from the previous chapter is that in order to change variables from genotype
frequencies to statistical moments, we now need more than a pair of allele frequencies and a single
measure of linkage disequilibrium to fully describe the system. For each species, we now need to follow
three allele frequencies, three pairwise linkage disequilibria, and one three-way linkage disequilibrium.
The recursions for these new variables are:
′′
′′
′′
′′
′′
𝑝𝑋,𝐴
= 𝑋𝐴𝐵𝑀
+ 𝑋𝐴𝐵𝑚
+ 𝑋𝐴𝑏𝑀
+𝑋𝐴𝑏𝑚
(6a)
′′
′′
′′
′′
′′
𝑝𝑋,𝐵
= 𝑋𝐴𝐵𝑀
+ 𝑋𝐴𝐵𝑚
+ 𝑋𝑎𝐵𝑀
+𝑋𝑎𝑏𝑚
(6b)
3
′′
′′
′′
′′
′′
𝑝𝑋,𝑀
= 𝑋𝐴𝐵𝑀
+ 𝑋𝐴𝑏𝑀
+ 𝑋𝑎𝐵𝑀
+𝑋𝑎𝑏𝑀
(6c)
′′
′′
′′ )
′′
′′
𝐷𝑋,𝐴𝐵
= (𝑋𝐴𝐵𝑀
+ 𝑋𝐴𝐵𝑚
− 𝑝𝑋,𝐴
𝑝𝑋,𝐵
(6d)
′′
′′
′′ )
′′
′′
𝐷𝑋,𝐵𝑀
= (𝑋𝐴𝐵𝑀
+ 𝑋𝑎𝐵𝑀
− 𝑝𝑋,𝐵
𝑝𝑋,𝑀
(6e)
′′ )
′′
′′
′′
′′
𝐷𝑋,𝐴𝑀
= (𝑋𝐴𝐵𝑀
+ 𝑋𝐴𝑏𝑀
− 𝑝𝑋,𝐴
𝑝𝑋,𝑀
(6f)
′′
′′
′′
′′
′′
′′
′′
′′
′′
′′
𝐷𝑋,𝐴𝐵𝑀
= 𝑋𝐴𝐵𝑀
− 𝐷𝑋,𝐴𝐵
𝑝𝑋,𝑀
− 𝐷𝑋,𝐴𝑀
𝑝𝑋,𝐵
− 𝐷𝑋,𝐵𝑀
𝑝𝑋,𝐴
− 𝑝𝑋,𝐴 𝑝𝑋,𝐵
𝑝𝑋,𝑀
(6g)
where blah blah blah. The next step, in our change of variables is to substitute all expressions involving
the old variables X on the right hand sides of these expressions with the definitions of these old
variables in terms of the new variables. To do this, we can use the following formulae:
𝑋𝐴𝐵𝑀 = 𝑝𝑋,𝐴 𝑝𝑋,𝐵 𝑝𝑋,𝑀 + 𝐷𝑋,𝐴𝐵 𝑝𝑋,𝑀 + 𝐷𝑋,𝐴𝑀 𝑝𝑋,𝐵 + 𝐷𝑋,𝐵𝑀 𝑝𝑋,𝐴 + 𝐷𝑋,𝐴𝐵𝑀
(7a)
𝑋𝐴𝐵𝑚 = 𝑝𝑋,𝐴 𝑝𝑋,𝐵 𝑞𝑋,𝑀 + 𝐷𝑋,𝐴𝐵 𝑞𝑋,𝑀 − 𝐷𝑋,𝐴𝑀 𝑝𝑋,𝐵 − 𝐷𝑋,𝐵𝑀 𝑝𝑋,𝐴 − 𝐷𝑋,𝐴𝐵𝑀
(7b)
𝑋𝐴𝑏𝑀 = 𝑝𝑋,𝐴 𝑞𝑋,𝐵 𝑝𝑋,𝑀 − 𝐷𝑋,𝐴𝐵 𝑝𝑋,𝑀 + 𝐷𝑋,𝐴𝑀 𝑞𝑋,𝐵 − 𝐷𝑋,𝐵𝑀 𝑝𝑋,𝐴 − 𝐷𝑋,𝐴𝐵𝑀
(7c)
𝑋𝐴𝑏𝑚 = 𝑝𝑋,𝐴 𝑞𝑋,𝐵 𝑞𝑋,𝑀 − 𝐷𝑋,𝐴𝐵 𝑞𝑋,𝑀 − 𝐷𝑋,𝐴𝑀 𝑞𝑋,𝐵 + 𝐷𝑋,𝐵𝑀 𝑝𝑋,𝐴 + 𝐷𝑋,𝐴𝐵𝑀
(7d)
𝑋𝑎𝐵𝑀 = 𝑞𝑋,𝐴 𝑝𝑋,𝐵 𝑝𝑋,𝑀 − 𝐷𝑋,𝐴𝐵 𝑝𝑋,𝑀 − 𝐷𝑋,𝐴𝑀 𝑝𝑋,𝐵 + 𝐷𝑋,𝐵𝑀 𝑞𝑋,𝐴 − 𝐷𝑋,𝐴𝐵𝑀
(7e)
𝑋𝑎𝐵𝑚 = 𝑞𝑋,𝐴 𝑝𝑋,𝐵 𝑞𝑋,𝑀 − 𝐷𝑋,𝐴𝐵 𝑞𝑋,𝑀 + 𝐷𝑋,𝐴𝑀 𝑝𝑋,𝐵 − 𝐷𝑋,𝐵𝑀 𝑞𝑋,𝐴 + 𝐷𝑋,𝐴𝐵𝑀
(7f)
𝑋𝑎𝑏𝑀 = 𝑞𝑋,𝐴 𝑞𝑋,𝐵 𝑝𝑋,𝑀 + 𝐷𝑋,𝐴𝐵 𝑝𝑋,𝑀 − 𝐷𝑋,𝐴𝑀 𝑞𝑋,𝐵 − 𝐷𝑋,𝐵𝑀 𝑞𝑋,𝐴 + 𝐷𝑋,𝐴𝐵𝑀
(7g)
𝑋𝑎𝑏𝑚 = 𝑞𝑋,𝐴 𝑞𝑋,𝐵 𝑞𝑋,𝑀 + 𝐷𝑋,𝐴𝐵 𝑞𝑋,𝑀 + 𝐷𝑋,𝐴𝑀 𝑞𝑋,𝐵 + 𝐷𝑋,𝐵𝑀 𝑞𝑋,𝐴 − 𝐷𝑋,𝐴𝐵𝑀
(7h)
Identical formulae hold for the parasite as well…
Finally, we QLE the shit out of it.
ALTHOUGH I HAVE WRITTEN ALL OF THESE OUT, IT IS FAR EASIER TO UNDERSTAND THEM IN A GENERAL
SENSE BY CONSULTING THE WONDERFUL PAPER BY KJB WHICH PROVIDES GENERAL FORMULAE.
Although this change of variables should be mostly familiar after the previous chapter, we do encounter
a novel association in the three way. What does this mean?
OK, NOW WHAT WE HAVE IS A NICE TIDY SYSTEM OF EQUATIONS DESCRIBING HOW ALLELE
FREQUENCIES AND DISEQUILIBRIA CHANGE IN EACH OF OUR INTERACTING SPECIES. UNFORTUNATELY,
there is still a catch: these equations remain ten miles long. So, let’s now apply our QLE approximation
which should hold as long as recombination is relatively frequent relative to the strength of selection
4
(REF). IF WE DO WHAT WE DID IN THE LAST CHAPTER, Mechanistically, this means that we Taylor
Expand our expressions for evolutionary change ignoring all terms that are of order eps or greater which
in this case includes terms like s^2, D^2, and s*D and higher. IF WE DO THAT NOW WE FIND THAT:
∆𝑝𝑖,𝑀 ≈ 0 + 𝒪(𝜀 2 )
(8a)
WHY DOES OUR MODIFIER NOT CHANGE??? BLAH BLAH BLAH
if we go out to the next order we see that…
∆𝑝𝑖,𝑀 ≈ 𝑎𝑖,𝐴 𝐷𝑖,𝐴𝑀 + 𝑎𝑖,𝐵 𝐷𝑖,𝐵𝑀 + 𝑎𝑖,𝐴𝐵 𝐷𝑖,𝐴𝐵𝑀
(9)
where 𝑎𝐴 measures the strength of directional selection acting on locus A and is given by:
𝑎𝑖,𝐴 =
𝑊𝑖,𝑎𝑏 −𝑊𝑖,𝐴𝑏 +𝑝𝑖,𝐵 (𝑊𝑖,𝐴𝐵 +𝑊𝑖,𝑎𝑏 −𝑊𝑖,𝐴𝑏 −𝑊𝑖,𝑎𝐵 )
̅𝑖
𝑊
(10a)
𝑎𝐵 measures the strength of directional selection acting on locus B and is given by:
𝑎𝑖,𝐵 =
𝑊𝑖,𝑎𝐵 −𝑊𝑖,𝑎𝑏 +𝑝𝑖,𝐴 (𝑊𝑖,𝐴𝐵 +𝑊𝑖,𝑎𝑏 −𝑊𝑖,𝐴𝑏 −𝑊𝑖,𝑎𝐵 )
̅𝑖
𝑊
(10b)
and 𝑎𝐴𝐵 measures the strength of epistatic selection acting on locus A and B together and is given by:
𝑎𝑖,𝐴𝐵 =
(𝑊𝑖,𝐴𝐵 +𝑊𝑖,𝑎𝑏 −𝑊𝑖,𝐴𝑏 −𝑊𝑖,𝑎𝐵 )
̅𝑖
𝑊
(10c)
SO WHAT CAN WE LEARN FROM THIS EXPRESSion? 1. If the modifier is associated with selectively
favored alleles, it will increase. If the modifier is associated with a pair of alleles that has particularly
good fitness it will increase. To learn more, we need to solve for the values of D we expect at QLE.
NOW INTRODUCE QLE RECURSIONS FOR MOMENTS AND SOLVE FOR THEIR QLE VALUES…
If we assume interference prevents multiple recombination events, assumed the modifier has an
additive effect,
FOR SPECIES X:
𝑀𝑚
∆𝐷𝑖,𝐴𝑀 ≈ −(𝑟𝑖,𝐴𝐵
+ 𝑟𝑖,𝐵𝑀 )𝐷𝑖,𝐴𝑀
(13a)
∆𝐷𝑖,𝐵𝑀 ≈ −𝑟𝑖,𝐵𝑀 𝐷𝑖,𝐵𝑀
(13b)
𝑀𝑚
∆𝐷𝑖,𝐴𝐵𝑀 ≈ −𝛿𝑖 𝑝𝑖,𝑀 𝑞𝑖,𝑀 (𝐷𝑖,𝐴𝐵 + 𝑎𝑖,𝐴𝐵 𝑝𝑖,𝐴 𝑞𝑖,𝐴 𝑝𝑖,𝐵 𝑞𝑖,𝐵 ) − (𝑟𝑖,𝐴𝐵
+ 𝑟𝑖,𝐵𝑀 )𝐷𝑖,𝐴𝐵𝑀
(13c)
∆𝐷𝑋,𝐴𝐵 ≈ 𝑎𝑖,𝐴𝐵 𝑝𝑖,𝐴 𝑞𝑖,𝐴 𝑝𝑖,𝐵 𝑞𝑖,𝐵 (1 − 𝑟̅𝑖𝐴𝐵 ) − 𝛿𝑖 𝐷𝑖,𝐴𝐵𝑀 − 𝑟̅𝑖𝐴𝐵 𝐷𝑖,𝐴𝐵
(13d)
5
𝑀𝑚
𝑚𝑚
where and 𝛿𝑖 = 𝑟𝑖,𝐴𝐵
− 𝑟𝑖,𝐴𝐵
is the change in recombination caused by substituting a M for a m allele
2 𝑀𝑀
𝑀𝑚
2 𝑚𝑚
and 𝑟̅𝑖𝐴𝐵 = 𝑝𝑖𝑀
𝑟𝑖,𝐴𝐵 + 2𝑝𝑖,𝑀 𝑞𝑖,𝑀 𝑟𝑖,𝐴𝐵
+ 𝑞𝑖𝑀
𝑟𝑖,𝐴𝐵 is the average rate of recombination between the A
and B loci within the host population.
WHAT A MESS… WHICH FORM SHOULD I WRITE THE r’s IN? AS DELTA OR NOT? SHOULD I IGNORE
DOUBLE RECOMBO’s? I did above…
NOW WE JUST SOLVE FOR THE EQUILIBRIUM VALUES OF THESE MOMENTS…
HOST:
̃𝑖,𝐴𝑀 ≈ 0
𝐷
(15a)
̃𝑖,𝐵𝑀 ≈ 0
𝐷
(15b)
̃𝑖,𝐴𝐵 ≈
𝐷
𝑀𝑚
𝑎𝑖,𝐴𝐵 𝑝𝑖,𝐴 𝑞𝑖,𝐴 𝑝𝑖,𝐵 𝑞𝑖,𝐵 ((𝑟𝑖,𝐴𝐵
+𝑟𝑖,𝐵𝑀 )(1−𝑟̅𝑖𝐴𝐵 )+𝛿𝑖2 𝑝𝑖,𝑀 𝑞𝑖,𝑀 )
𝑀𝑚
+𝑟𝑖,𝐵𝑀 )𝑟̅𝑖𝐴𝐵 −𝛿𝑖2 𝑝𝑖,𝑀 𝑞𝑖,𝑀
(𝑟𝑖,𝐴𝐵
𝑖,𝐴𝐵 𝑝𝑖,𝐴 𝑞𝑖,𝐴 𝑝𝑖,𝐵 𝑞𝑖,𝐵 𝑝𝑖,𝑀 𝑞𝑖,𝑀
̃𝑖,𝐴𝐵𝑀 ≈ − 𝛿𝑖𝑎𝑀𝑚
𝐷
2
(15c)
(15d)
(𝑟𝑖,𝐴𝐵 +𝑟𝑖,𝐵𝑀 )𝑟̅𝑖𝐴𝐵 −𝛿𝑖 𝑝𝑖,𝑀 𝑞𝑖,𝑀
(16d)
AND DRUMROLL PLEASE… SUBSITUTE THESE D’S INTO OUR DELTA M’s:
2
∆𝑝𝑖,𝑀 ≈ −𝛿𝑖 𝑎𝑖,𝐴𝐵
𝑝𝑖,𝐴 𝑞𝑖,𝐴 𝑝𝑖,𝐵 𝑞𝑖,𝐵 𝑝𝑖,𝑀 𝑞𝑖,𝑀
𝑀𝑚
+𝑟𝑖,𝐵𝑀 )𝑟̅𝑖𝐴𝐵 −𝛿𝑖2 𝑝𝑖,𝑀 𝑞𝑖,𝑀
(𝑟𝑖,𝐴𝐵
+ 𝒪(𝜀 3 )
(17a)
Both of these quantities will always be negative indicating that modifiers increasing recombination rates
will always decrease in frequency whereas modifiers decreasing recombination rates will always
increase in frequency (Figure 1). Exactly the opposite of what we expect under the Red Queen. Looking a
bit deeper, we can see also that these expressions are virtually identical to what we found in the
previous section for a different model of infection genetics. The only difference is the specific value of a.
Because a is squared, however, we could continue trying alternative models of infection genetics until
the cows come home, and yet we would never find a different qualitative solution. As long as selection
is relatively weak, coevolution does not favor increased rates of recombination.
NO MATTER WHAT FORM OF COEVOLUTION YOU THROW AT THESE EXPRESSIONS, YOU WILL ALWAYS
GET A DECREASE IN R
Answers to key questions:
6
WaWha! No sex favored. The reason for this, is that linkage disequilibrium will always have the same
sign as epistasis. As such, increasing rates of recombination rips apart favorable genetic associations
and decreases fitness. Of course, the reason this occurs is because we have assumed selection is
relatively weak and recombination relatively frequent. Under such conditions, there are no time lags
between LD and S.
New Questions Arising:
What if infection genetics differs?
What happens when selection is strong and recombination weak?
Extensions
Extension 1: Alternative infection matrix
1
1−𝜔
𝛼=[
1−𝜔
1 − 4𝜔
1−𝜔
1
1
1−𝜔
1−𝜔
1
1
1−𝜔
1 − 4𝜔
1−𝜔
]
1−𝜔
1
HOST:
̃𝑋,𝐴𝑀 ≈ 0
𝐷
(18a)
̃𝑋,𝐵𝑀 ≈ 0
𝐷
(18b)
̃𝑋,𝐴𝐵 ≈
𝐷
𝑀𝑚
2
𝑎𝑋,𝐴𝐵 𝑝𝑋,𝐴 𝑞𝑋,𝐴 𝑝𝑋,𝐵 𝑞𝑋,𝐵 ((𝑟𝑋,𝐴𝐵
+𝑟𝑋,𝐵𝑀 )(1−𝑟̅𝑋𝐴𝐵 )+𝛿𝑋
𝑝𝑋,𝑀 𝑞𝑋,𝑀 )
𝑀𝑚 +𝑟
2
(𝑟𝑋,𝐴𝐵
𝑋,𝐵𝑀 )𝑟̅𝑋𝐴𝐵 −𝛿𝑋 𝑝𝑋,𝑀 𝑞𝑋,𝑀
𝑋,𝐴𝐵 𝑝𝑋,𝐴 𝑞𝑋,𝐴 𝑝𝑋,𝐵 𝑞𝑋,𝐵 𝑝𝑋,𝑀 𝑞𝑋,𝑀
̃𝑋,𝐴𝐵𝑀 ≈ − 𝛿𝑋𝑎𝑀𝑚
𝐷
2
(18c)
(18d)
(𝑟𝑋,𝐴𝐵 +𝑟𝑋,𝐵𝑀 )𝑟̅𝑋𝐴𝐵 −𝛿𝑋 𝑝𝑋,𝑀 𝑞𝑋,𝑀
where:
𝑎𝑋,𝐴 = −𝜔𝑠𝑋 (2𝑝𝑌,𝐴 + 2𝑝𝑌,𝐵 − 2𝑝𝑋,𝐵 − 1)
(19a)
𝑎𝑋,𝐵 = −𝜔𝑠𝑋 (2𝑝𝑌,𝐴 + 2𝑝𝑌,𝐵 − 2𝑝𝑋,𝐴 − 1)
(19b)
𝑎𝑋,𝐴𝐵 = 2𝜔𝑠𝑋
(19c)
PARASITE:
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̃𝑌,𝐴𝑀 ≈ 0
𝐷
(20a)
̃𝑌,𝐵𝑀 ≈ 0
𝐷
(20b)
̃𝑌,𝐴𝐵 ≈
𝐷
𝑀𝑚
2
𝑎𝑌,𝐴𝐵 𝑝𝑌,𝐴 𝑞𝑌,𝐴 𝑝𝑌,𝐵 𝑞𝑌,𝐵 ((𝑟𝑌,𝐴𝐵
+𝑟𝑌,𝐵𝑀 )(1−𝑟̅𝑌𝐴𝐵 )+𝛿𝑌
𝑝𝑌,𝑀 𝑞𝑌,𝑀 )
𝑀𝑚 +𝑟
2
(𝑟𝑌,𝐴𝐵
𝑌,𝐵𝑀 )𝑟̅𝑌𝐴𝐵 −𝛿𝑌 𝑝𝑌,𝑀 𝑞𝑌,𝑀
𝑌,𝐴𝐵 𝑝𝑌,𝐴 𝑞𝑌,𝐴 𝑝𝑌,𝐵 𝑞𝑌,𝐵 𝑝𝑌,𝑀 𝑞𝑌,𝑀
̃𝑌,𝐴𝐵𝑀 ≈ 𝛿𝑌 𝑎𝑀𝑚
𝐷
+𝑟
−𝛿 2 𝑝
𝑞
(𝑟𝑌,𝐴𝐵
𝑌,𝐵𝑀 )𝑟̅𝑌𝐴𝐵
(20c)
(20d)
𝑌 𝑌,𝑀 𝑌,𝑀
where:
𝑎𝑌,𝐴 = 𝜔𝑠𝑌 (2𝑝𝑋,𝐴 + 2𝑝𝑋,𝐵 − 2𝑝𝑌,𝐵 − 1)
(21a)
𝑎𝑌,𝐵 = 𝜔𝑠𝑌 (2𝑝𝑋,𝐴 + 2𝑝𝑋,𝐵 − 2𝑝𝑌,𝐴 − 1)
(21b)
𝑎𝑌,𝐴𝐵 = −2𝜔𝑠𝑌
(21c)
AND DRUMROLL PLEASE… SUBSITUTE THESE D’S INTO OUR DELTA M’s:
2
∆𝑝𝑋,𝑀 ≈ −𝛿𝑋 𝑎𝑋,𝐴𝐵
2
∆𝑝𝑌,𝑀 ≈ −𝛿𝑌 𝑎𝑌,𝐴𝐵
𝑝𝑋,𝐴 𝑞𝑋,𝐴 𝑝𝑋,𝐵 𝑞𝑋,𝐵 𝑝𝑋,𝑀 𝑞𝑋,𝑀
𝑀𝑚 +𝑟
2
(𝑟𝑋,𝐴𝐵
𝑋,𝐵𝑀 )𝑟̅𝑋𝐴𝐵 −𝛿𝑋 𝑝𝑋,𝑀 𝑞𝑋,𝑀
𝑝𝑌,𝐴 𝑞𝑌,𝐴 𝑝𝑌,𝐵 𝑞𝑌,𝐵 𝑝𝑌,𝑀 𝑞𝑌,𝑀
𝑀𝑚 +𝑟
2
(𝑟𝑌,𝐴𝐵
𝑌,𝐵𝑀 )𝑟̅𝑌𝐴𝐵 −𝛿𝑌 𝑝𝑌,𝑀 𝑞𝑌,𝑀
+ 𝒪(𝜀 3 )
+ 𝒪(𝜀 3 )
(22a)
(22b)
Both of these quantities will always be negative indicating that modifiers increasing recombination rates
will always decrease in frequency whereas modifiers decreasing recombination rates will always
increase in frequency (Figure 2). Exactly the opposite of what we expect under the Red Queen. Looking a
bit deeper, we can see also that these expressions are virtually identical to what we found in the
previous section for a different model of infection genetics. The only difference is the specific value of a.
Because a is squared, however, we could continue trying alternative models of infection genetics until
the cows come home, and yet we would never find a different qualitative solution. As long as selection
is relatively weak, coevolution does not favor increased rates of recombination.
Generalization 2: Strong selection and weak initial recombination
Although this can be tackled analytically by making further assumptions about the strength of epistasis
relative to directional selection (e.g., barton, Otto/Nuismer) and maintaining additional terms, these
approaches are outside the scope of the book. Instead, we will take a less general, but much more
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straightforward approach that relies on simulating the coevolutionary process through numerical
iteration of the exact recursion equations (X).
Conclusions and Synthesis
For the Red Queen to favor more sex, selection must be strong. No matter what the form of genetic
interaction, weak selection just won’t cut it. Interestingly, in the interaction between snails and
trematodes, cycles look fast (insert figure). In fact, they look fast like strong selection fast. Thus, this
system might be one where the red queen is in operation. However, there is little evidence to suggest
coevolution is always so strong. Thus, it seems unlikely that the red queen is a general explanation for
the evolution of sex.
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References
Figure Legends
Dybdahl, M. F., C. E. Jenkins, and S. L. Nuismer. 2014. Identifying the Molecular Basis of Host-Parasite
Coevolution: Merging Models and Mechanisms. AMERICAN NATURALIST 184:1-13.
Mitta, G., C. M. Adema, B. Gourbal, E. S. Loker, and A. Theron. 2012. Compatibility polymorphism in
snail/schistosome interactions: From field to theory to molecular mechanisms. Developmental
and Comparative Immunology 37:1-8.
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