Chapter 7 3_30_15
Chapter 7. Coevolution, recombination, and the red queen
Biological Motivation
A long-standing mystery in evolutionary biology is the origin and maintenance of sexual
generally impose a twofold cost associated with the production of male offspring. 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
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 (Jaenike 1978; Hamilton 1980b). 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 (Jaenike 1978; Hamilton 1980a; Bell 1982; Salathe et al. 2008; Lively 2010).
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. Perhaps the best studied of these is the interaction between the
New Zealand mud snail, Potamopyrgus antipodarum, and its castrating trematode parasite,
2009; King et al. 2011). 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
coevolution that favors the evolution of increased rates of sexual reproduction within lakes experiencing
this chapter will be to develop mathematical models inspired by the coevolutionary interaction between snail and trematode and use them to better understand when coevolution favors increased rates of sexual reproduction and recombination.
Key Questions:
Does coevolution favor increased host recombination?
Does coevolution cause changes in parasite recombination?
Does the genetic basis of infection and resistance matter?
Modeling the Red Queen
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
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 would be to study coevolution between a parasite and a mixed population of hosts where some individuals
track the frequency of sexual host individuals over time to see if they increase or decrease. Although conceptually straightforward, implementing this approach is more mathematically challenging than you might think! An alternative possibility would be to study coevolution between a parasite and a sexual
host population where some host genotypes recombine at a greater rate than others (e.g., Bell and
Smith 1987; Peters and Lively 1999; Agrawal 2009). This latter approach, while by no means
mathematically simple, is often more amenable to mathematical analysis. For this reason, we will pursue this latter strategy in this chapter, recognizing that the assumptions of this approach differ substantively from those of the interaction between the snail P. antipodarum and its trematode parasite
microphallus with which we will motivate our theoretical development.
Now that we have decided to focus on the coevolution of recombination rates, we need to get a bit more specific about the genetic details of our model. Keeping with the overall theme of this book, we will do our best to build the simplest possible model that can possibly shed light on the biological problem we hope to study. In this case, this means we need a haploid model with three possible diallelic 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 6. 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).
Table 1. The effect of locus M on recombination rates between loci A and B in species i
Genotype
MM Mm
Recombination rate between A and B loci 𝑟
𝑀𝑀 𝑖,𝐴𝐵 𝑟
𝑀𝑚 𝑖,𝐴𝐵 mm 𝑟 𝑚𝑚 𝑖,𝐴𝐵
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:
2
𝑋 𝑖
′
=
𝑋 𝑖
𝑊
𝑋,𝑖
𝑊
𝑋
𝑌 𝑖
′
=
𝑌 𝑖
𝑊
𝑌,𝑖
𝑊
𝑌 where the population mean fitnesses, 𝑊
𝑋
and 𝑊
𝑌
, are given by:
̅
𝑋
= ∑
8 𝑖=1
𝑋 𝑖
𝑊
𝑋,𝑖
𝑊
𝑌
= ∑
8 𝑖=1
𝑌 𝑖
𝑊
𝑌,𝑖
(3a)
(3b)
(4a)
(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.
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 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:
𝑋 𝑖
′′
= ∑
8 𝑗=1
∑ 8 𝑘=1
𝑋 𝑗
′
𝑋
′ 𝑘
Π
𝑋,𝑗+𝑘→𝑖
𝑌 𝑖
′′
= ∑
8 𝑗=1
∑
8 𝑘=1
𝑌 𝑗
′
𝑌 𝑘
′
Π
𝑌,𝑗+𝑘→𝑖
(5a)
(5b) where the terms Π
𝑋,𝑗+𝑘→𝑖 and Π
𝑌,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 Π
𝑋,𝑗+𝑘→𝑖
terms and 64 different Π
𝑌,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 some sense out of these ridiculously complex expressions is the Quasi-Linkage Equilibrium (QLE) approximation we introduced in the previous chapter.
3
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. Specifically, for each species, we now need to follow three allele frequencies, three pairwise linkage disequilibria, and one three-way linkage disequilibrium. For the host, the recursions for these new variables can be written in terms of the recursions for the old variables using the following expressions: 𝑝
′′
𝑋,𝐴
= 𝑋
′′
𝐴𝐵𝑀
+ 𝑋
′′
𝐴𝐵𝑚
+ 𝑋
′′
𝐴𝑏𝑀
+𝑋
′′
𝐴𝑏𝑚 𝑝
′′
𝑋,𝐵
= 𝑋
′′
𝐴𝐵𝑀
+ 𝑋
′′
𝐴𝐵𝑚
+ 𝑋
′′ 𝑎𝐵𝑀
+𝑋
′′ 𝑎𝑏𝑚 𝑝
′′
𝑋,𝑀
= 𝑋
′′
𝐴𝐵𝑀
+ 𝑋
′′
𝐴𝑏𝑀
+ 𝑋
′′ 𝑎𝐵𝑀
+𝑋
′′ 𝑎𝑏𝑀
(6a)
(6b)
(6c)
𝐷 ′′
𝑋,𝐴𝐵
= (𝑋 ′′
𝐴𝐵𝑀
+ 𝑋 ′′
𝐴𝐵𝑚
) − 𝑝 ′′
𝑋,𝐴 𝑝 ′′
𝑋,𝐵
𝐷
′′
𝑋,𝐵𝑀
= (𝑋
′′
𝐴𝐵𝑀
+ 𝑋
′′ 𝑎𝐵𝑀
) − 𝑝
′′
𝑋,𝐵 𝑝
′′
𝑋,𝑀
𝐷 ′′
𝑋,𝐴𝑀
= (𝑋 ′′
𝐴𝐵𝑀
+ 𝑋
′′
𝐴𝑏𝑀
) − 𝑝 ′′
𝑋,𝐴 𝑝 ′′
𝑋,𝑀
𝐷
′′
𝑋,𝐴𝐵𝑀
= 𝑋
′′
𝐴𝐵𝑀
− 𝐷
′′
𝑋,𝐴𝐵 𝑝
′′
𝑋,𝑀
− 𝐷
′′
𝑋,𝐴𝑀 𝑝
′′
𝑋,𝐵
− 𝐷
′′
𝑋,𝐵𝑀 𝑝
′′
𝑋,𝐴
− 𝑝
𝑋,𝐴 𝑝
′′
𝑋,𝐵 𝑝
′′
𝑋,𝑀
(6d)
(6e)
(6f)
(6g) where 𝑝
𝑋,𝑖
is the allele frequency at locus i, 𝐷
𝑋,𝑖𝑗
is the pairwise linkage disequilibrium between loci i and j, 𝐷
𝑋,𝑖𝑗𝑘
is the three-way disequilibrium between loci i, j, and k. Expressions for the parasite population are identical once all incidences of X are replaced by Y. Unfortunately for us, we now have rather ugly recursions on our hands. Specifically, these recursions are written in terms of the new variables on their left hand sides and a mix of old and new variables on their right. This is entirely unacceptable and must be rectified immediately if we are to make further progress.
The key to moving forward is to identify expressions that relate genotype frequencies X and Y to allele frequencies and linkage disequilibria. Fortunately, general results presented in Kirkpatrick et al.
for the host population:
(7a) 𝑋
𝐴𝐵𝑀
= 𝑝
𝑋,𝐴 𝑝
𝑋,𝐵 𝑝
𝑋,𝑀
+ 𝐷
𝑋,𝐴𝐵 𝑝
𝑋,𝑀
+ 𝐷
𝑋,𝐴𝑀 𝑝
𝑋,𝐵
+ 𝐷
𝑋,𝐵𝑀 𝑝
𝑋,𝐴
+ 𝐷
𝑋,𝐴𝐵𝑀
𝑋
𝐴𝐵𝑚
= 𝑝
𝑋,𝐴 𝑝
𝑋,𝐵 𝑞
𝑋,𝑀
+ 𝐷
𝑋,𝐴𝐵 𝑞
𝑋,𝑀
− 𝐷
𝑋,𝐴𝑀 𝑝
𝑋,𝐵
− 𝐷
𝑋,𝐵𝑀 𝑝
𝑋,𝐴
− 𝐷
𝑋,𝐴𝐵𝑀
𝑋
𝐴𝑏𝑀
= 𝑝
𝑋,𝐴 𝑞
𝑋,𝐵 𝑝
𝑋,𝑀
− 𝐷
𝑋,𝐴𝐵 𝑝
𝑋,𝑀
+ 𝐷
𝑋,𝐴𝑀 𝑞
𝑋,𝐵
− 𝐷
𝑋,𝐵𝑀 𝑝
𝑋,𝐴
− 𝐷
𝑋,𝐴𝐵𝑀
𝑋
𝐴𝑏𝑚
= 𝑝
𝑋,𝐴 𝑞
𝑋,𝐵 𝑞
𝑋,𝑀
− 𝐷
𝑋,𝐴𝐵 𝑞
𝑋,𝑀
− 𝐷
𝑋,𝐴𝑀 𝑞
𝑋,𝐵
+ 𝐷
𝑋,𝐵𝑀 𝑝
𝑋,𝐴
+ 𝐷
𝑋,𝐴𝐵𝑀
𝑋 𝑎𝐵𝑀
= 𝑞
𝑋,𝐴 𝑝
𝑋,𝐵 𝑝
𝑋,𝑀
− 𝐷
𝑋,𝐴𝐵 𝑝
𝑋,𝑀
− 𝐷
𝑋,𝐴𝑀 𝑝
𝑋,𝐵
+ 𝐷
𝑋,𝐵𝑀 𝑞
𝑋,𝐴
− 𝐷
𝑋,𝐴𝐵𝑀
(7b)
(7c)
(7d)
(7e)
4
𝑋 𝑎𝐵𝑚
= 𝑞
𝑋,𝐴 𝑝
𝑋,𝐵 𝑞
𝑋,𝑀
− 𝐷
𝑋,𝐴𝐵 𝑞
𝑋,𝑀
+ 𝐷
𝑋,𝐴𝑀 𝑝
𝑋,𝐵
− 𝐷
𝑋,𝐵𝑀 𝑞
𝑋,𝐴
+ 𝐷
𝑋,𝐴𝐵𝑀
𝑋 𝑎𝑏𝑀
= 𝑞
𝑋,𝐴 𝑞
𝑋,𝐵 𝑝
𝑋,𝑀
+ 𝐷
𝑋,𝐴𝐵 𝑝
𝑋,𝑀
− 𝐷
𝑋,𝐴𝑀 𝑞
𝑋,𝐵
− 𝐷
𝑋,𝐵𝑀 𝑞
𝑋,𝐴
+ 𝐷
𝑋,𝐴𝐵𝑀
(7f)
(7g)
𝑋 𝑎𝑏𝑚
= 𝑞
𝑋,𝐴 𝑞
𝑋,𝐵 𝑞
𝑋,𝑀
+ 𝐷
𝑋,𝐴𝐵 𝑞
𝑋,𝑀
+ 𝐷
𝑋,𝐴𝑀 𝑞
𝑋,𝐵
+ 𝐷
𝑋,𝐵𝑀 𝑞
𝑋,𝐴
− 𝐷
𝑋,𝐴𝐵𝑀
(7h)
Here too, expressions for the parasite population are identical once all incidences of X are replaced by Y.
Now, we just need to substitute (7) into (6) to yield expressions for coevolutionary change in allele frequencies and linkage disequilibria within host and parasite populations.
Up to this point we have been focused on the first step of our QLE approximation which required a change of variables from genotype frequencies to statistical moments (i.e., allele frequencies and linkage disequilibria). The second and final step in our QLE approximation involves identifying and eliminating terms in these expressions that should be negligible if selection is relatively weak and recombination relatively frequent. In the previous chapter we did this in a very straightforward and concrete way by first specifying the infection matrix and its relationship to genotypic fitness. A weakness of this approach is that it locks us into a particular infection matrix and model of fitness. Now that we are a bit more familiar with the QLE, we will work to do this in a more powerful and general way that does not require that we pin ourselves down to a particular infection matrix. The key to this more general approach is to re-write genotypic fitnesses in terms of the force of selection acting on individual loci ( 𝑎
𝐴
, 𝑎
𝐵
) and pairs of loci ( 𝑎
𝐴𝐵
):
𝑊 𝑖,𝐴𝐵
= 𝑊 𝑖
(1 + 𝑎 𝑖,𝐴
(1 − 𝑝 𝑖,𝐴
) + 𝑎 𝑖,𝐵
(1 − 𝑝 𝑖,𝐵
) + 𝑎 𝑖,𝐴𝐵
((1 − 𝑝 𝑖,𝐴
)(1 − 𝑝 𝑖,𝐵
) − 𝐷 𝑖,𝐴𝐵
)) (8a)
𝑊 𝑖,𝐴𝑏
= 𝑊 𝑖
(1 + 𝑎 𝑖,𝐴
(1 − 𝑝 𝑖,𝐴
) + 𝑎 𝑖,𝐵
(0 − 𝑝 𝑖,𝐵
) + 𝑎 𝑖,𝐴𝐵
((1 − 𝑝 𝑖,𝐴
)(0 − 𝑝 𝑖,𝐵
) − 𝐷 𝑖,𝐴𝐵
))
𝑊 𝑖,𝑎𝐵
̅ 𝑖
(1 + 𝑎 𝑖,𝐴
(0 − 𝑝 𝑖,𝐴
) + 𝑎 𝑖,𝐵
(1 − 𝑝 𝑖,𝐵
) + 𝑎 𝑖,𝐴𝐵
((0 − 𝑝 𝑖,𝐴
)(1 − 𝑝 𝑖,𝐵
) − 𝐷 𝑖,𝐴𝐵
))
(8b)
(8c)
𝑊 𝑖,𝑎𝑏
̅ 𝑖
(1 + 𝑎 𝑖,𝐴
(0 − 𝑝 𝑖,𝐴
) + 𝑎 𝑖,𝐵
(0 − 𝑝 𝑖,𝐵
) + 𝑎 𝑖,𝐴𝐵
((0 − 𝑝 𝑖,𝐴
)(0 − 𝑝 𝑖,𝐵
) − 𝐷 𝑖,𝐴𝐵
)) (8d) using formula (7) in Kirkpatrick et al. (2002). Re-writing fitness in this way yields recursions that depend on allele frequencies (p terms), statistical associations among loci (D terms), and the force of selection acting on individual loci and pairs of loci (a terms). At QLE, it is the a terms and the D terms that are weak and of 𝒪(𝜀)
We are now in a position to implement our QLE approximation by approximating our evolutionary recursions (now written in terms of p’s and D’s) using Taylor Series Expansions in which we ignore all terms of 𝒪(𝜀
2 ) and greater. Taking this approach reveals that the evolutionary change in the frequency of the M allele in species i is given by:
∆𝑝 𝑖,𝑀
≈ 0 + 𝒪(𝜀 2 ) (9)
5
demonstrating that, to leading order, recombination rates do not evolve in response to coevolution. The fact that our approximation yields no real prediction (whether recombination rates should increase or decrease) is frustrating and suggests we have been a bit too stringent with our QLE approximation.
Perhaps we should not have ignored so many terms! In order to explore this possibility, we can carry our
Taylor Series Expansion out to an additional order such that we keep terms of 𝒪(𝜀
2 ) and ignore only those terms that are 𝒪(𝜀
3
) or smaller. Taking this more precise approach, we find that the change in the frequency of the M allele is equal to:
∆𝑝 𝑖,𝑀
≈ 𝑎 𝑖,𝐴
̃ 𝑖,𝐴𝑀
+ 𝑎 𝑖,𝐵
̃ 𝑖,𝐵𝑀
+ 𝑎 𝑖,𝐴𝐵
̃ 𝑖,𝐴𝐵𝑀
+ 𝒪(𝜀
3 ) (10) where the quantities 𝐷 are the QLE values for the statistical associations among loci. Equation (10) reveals important insight into the forces driving evolutionary changes in the frequency of the M allele.
Specifically, the first two terms describe how the frequency of the M allele changes in response to selection acting on individual loci. If the M allele is statistically associated with the selectively favored allele at the A or B locus, these terms cause the frequency of the M allele to increase. If, in contrast, the
M allele is statistically associated with the selectively disfavored allele at the A or B locus, these terms cause the frequency of the M allele to decrease. Unlike the first two terms in (10) that quantify how the frequency of the M allele changes in response to selection at individual loci, the third term describes how its frequency changes in response to epistatic selection acting on the A and B loci. Specifically, if the
M allele is statistically associated with favorable combinations of A and B alleles, it will increase in frequency. If, instead, it is statistically associated with unfavorable combinations of A and B alleles, it will decrease in frequency.
Although equation (10) provides very general insights into the forces driving changes in frequency at the M locus, making more precise and concrete predictions requires that we solve for the
QLE values of the statistical associations among loci ( 𝐷 ). To accomplish this goal, we need to approximate our evolutionary recursions for the statistical associations (6d-g) using a Taylor Series
Expansion that ignores all terms of 𝒪(𝜀
2 ) and greater. Taking this approach and re-writing the recursions as difference equations yields the following expressions for evolutionary change in the statistical associations among loci at QLE:
∆𝐷 𝑖,𝐴𝑀
≈ −(𝑟
𝑀𝑚 𝑖,𝐴𝐵
+ 𝑟 𝑖,𝐵𝑀
)𝐷 𝑖,𝐴𝑀
+ 𝒪(𝜀
2 )
∆𝐷 𝑖,𝐵𝑀
≈ −𝑟 𝑖,𝐵𝑀
𝐷 𝑖,𝐵𝑀
+ 𝒪(𝜀 2 )
(11a)
(11b)
∆𝐷 𝑖,𝐴𝐵𝑀
≈ −𝛿 𝑖 𝑝 𝑖,𝑀 𝑞 𝑖,𝑀
(𝐷 𝑖,𝐴𝐵
+ 𝑎 𝑖,𝐴𝐵 𝑝 𝑖,𝐴 𝑞 𝑖,𝐴 𝑝 𝑖,𝐵 𝑞 𝑖,𝐵
) − (𝑟
𝑀𝑚 𝑖,𝐴𝐵
+ 𝑟 𝑖,𝐵𝑀
)𝐷 𝑖,𝐴𝐵𝑀
+ 𝒪(𝜀
2 ) (11c)
∆𝐷
𝑋,𝐴𝐵
≈ 𝑎 𝑖,𝐴𝐵 𝑝 𝑖,𝐴 𝑞 𝑖,𝐴 𝑝 𝑖,𝐵 𝑞 𝑖,𝐵
(1 − 𝑟̅ 𝑖𝐴𝐵
) − 𝛿 𝑖
𝐷 𝑖,𝐴𝐵𝑀
− 𝑟̅ 𝑖𝐴𝐵
𝐷 𝑖,𝐴𝐵
+ 𝒪(𝜀 2 ) (11d) where we have assumed interference prevents double recombination events, 𝑟̅ 𝑖𝐴𝐵
= 𝑝
2 𝑖𝑀 𝑟
𝑀𝑀 𝑖,𝐴𝐵
+ 2𝑝 𝑖,𝑀 𝑞 𝑖,𝑀 𝑟
𝑀𝑚 𝑖,𝐴𝐵 𝛿 𝑖
= 𝑟
𝑀𝑚 𝑖,𝐴𝐵
− 𝑟 𝑚𝑚 𝑖,𝐴𝐵
, and
+ 𝑞
2 𝑖𝑀 𝑟 𝑚𝑚 𝑖,𝐴𝐵
. Next, we find the equilibrium values of these statistical associations by setting the left hand sides of the equations to zero and simultaneously solving the system of equalities. The result is the following expressions for the QLE values of the statistical associations among loci:
6
̃ 𝑖,𝐴𝑀
≈ 0 + 𝒪(𝜀
2 )
̃ 𝑖,𝐵𝑀
≈ 0 + 𝒪(𝜀 2 )
̃ 𝑖,𝐴𝐵
≈ 𝑎 𝑖,𝐴𝐵 𝑝 𝑖,𝐴 𝑞 𝑖,𝐴 𝑝 𝑖,𝐵 𝑞 𝑖,𝐵
((𝑟
𝑀𝑚 𝑖,𝐴𝐵
+𝑟 𝑖,𝐵𝑀
)(1−𝑟̅ 𝑖𝐴𝐵
)+𝛿
2 𝑖 𝑝 𝑖,𝑀 𝑞 𝑖,𝑀
)
(𝑟
𝑀𝑚 𝑖,𝐴𝐵
+𝑟 𝑖,𝐵𝑀
)𝑟̅ 𝑖𝐴𝐵
−𝛿 𝑖
2 𝑝 𝑖,𝑀 𝑞 𝑖,𝑀
+ 𝒪(𝜀
2 )
̃ 𝑖,𝐴𝐵𝑀
≈ − 𝛿 𝑖 𝑎 𝑖,𝐴𝐵
(𝑟
𝑀𝑚 𝑖,𝐴𝐵 𝑝
+𝑟 𝑖,𝐴 𝑞 𝑖,𝐵𝑀 𝑖,𝐴 𝑝 𝑖,𝐵
)𝑟̅ 𝑖𝐴𝐵 𝑞 𝑖,𝐵
−𝛿 𝑖
2 𝑝 𝑖,𝑀 𝑞 𝑝 𝑖,𝑀 𝑞 𝑖,𝑀 𝑖,𝑀
+ 𝒪(𝜀 2 )
(12a)
(12b)
(12c)
(12d)
Equations (12) reveal that, to leading order, the only non-zero statistical associations are those between the two selected loci ( 𝐷 𝑖,𝐴𝐵
) , and among all three loci 𝐷 𝑖,𝐴𝐵𝑀
. These equations also reveal something very important: at QLE, the sign of the statistical association between the selected loci A and B ( 𝐷 𝑖,𝐴𝐵
) is always the same as the sign of epistatic selection acting on loci A and B ( 𝑎 𝑖,𝐴𝐵
) . This is important because it indicates that those combinations of alleles currently found in excess within the population are exactly those combinations currently favored by epistatic selection.
The final step in predicting evolutionary change in recombination rates is to substitute our QLE solutions for the statistical associations among loci (12) into our QLE expression for evolutionary change in the frequency of the M allele (10), yielding our final solution:
∆𝑝 𝑖,𝑀
≈ −𝛿 𝑖 𝑎
2 𝑖,𝐴𝐵 𝑝 𝑖,𝐴
(𝑟
𝑀𝑚 𝑖,𝐴𝐵 𝑞
+𝑟 𝑖,𝐴 𝑝 𝑖,𝐵𝑀 𝑖,𝐵 𝑞 𝑖,𝐵
)𝑟̅ 𝑖𝐴𝐵 𝑝 𝑖,𝑀
−𝛿 𝑖
2 𝑞 𝑖,𝑀 𝑝 𝑖,𝑀 𝑞 𝑖,𝑀
+ 𝒪(𝜀
3 ) (13)
If we take the time to look through the terms of this expression in detail, we see that selection and coevolution are captured (exclusively) by the term 𝑎 𝑖,𝐴𝐵
which measures the strength of epistatic selection acting on the A and B loci. Although the exact value of this term depends on our specific assumptions about the structure of the infection matrix, the fact that it is squared within (13) means that, at least from a qualitative standpoint, these coevolutionary details are irrelevant. No matter what model of coevolution we select, as long as selection is relatively weak and recombination relatively frequent, the frequency of the M allele will always decrease. Thus, equation (13) reveals that, contrary to the Red Queen hypothesis, reduced rates of recombination are always favored for any form of coevolutionary interaction. The reason increased rates of recombination are not favored is primarily because under QLE conditions, linkage disequilibrium is always of the same sign as epistatic selection, such that the predominant impact of recombination is to tear apart combinations of genes currently favored by selection.
Answers to key questions:
Does coevolution favor increased host recombination?
No. Instead, our results show that as long as coevolutionary selection is relatively weak and recombination relatively frequent, coevolution actually favors decreased rates of recombination within the host.
Does coevolution cause changes in parasite recombination?
7
Yes. Our results show that as long as coevolutionary selection is relatively weak and recombination relatively frequent, coevolution favors decreased rates of recombination within the parasite.
Does the genetic basis of infection and resistance matter?
No. Our results show that as long as coevolutionary selection is relatively weak and recombination relatively frequent, coevolution favors decreased rates of recombination irrespective of the genetic basis of infection and resistance.
New Questions Arising:
Our analysis provides no support for the Red Queen hypothesis for the evolution of sexual reproduction. Instead, our results suggest that coevolution, whatever its form, actually selects for decreased rates of recombination in interacting species. These predictions, however, rest on the specific assumptions we relied on to derive our QLE approximation. Consequently, we are left with two very important unanswered questions:
Would our conclusions change if coevolutionary selection were strong?
Do our general conclusions hold if recombination is initially low or absent altogether?
In the next two sections, we will generalize our simple model in ways that allow us to answer these questions. Although it is possible to address these questions using more elaborate analyses than those
we used in this section (Barton 1995; Otto and Nuismer 2004), these analyses are beyond the
mathematical scope of this book. Instead we will take a more accessible, but less general, approach that relies on numerically iterating the exact recursions given by (5) for particular types of coevolutionary interaction.
Generalizations and Extensions
Generalization 1: Investigating strong selection
Our QLE results demonstrate that the Red Queen fails to cause increased rates of recombination to evolve when selection is weak. There is good reason to believe, however, that, selection may be quite strong in some coevolutionary interactions such as those between the castrating trematode
Microphallus and its snail host Potamopurgus antipodarum. For instance, frequencies of snail clones
change quite rapidly within lakes (Dybdahl and Lively 1998), local adaptation occurs over very small
change our conclusions about the Red Queen?
In order to tackle this question through simulation, we need to be more specific about fitness than we were in the first part of this chapter where we left things more general and implicit. Our approach will mirror that of the previous chapter and assume that host and parasite individuals encounter each other at random, and that when a host individual with genotype i encounters a parasite
8
individual with genotype j, an infection results with probability 𝛼 𝑖,𝑗
. If we assume that infection reduces the fitness of hosts by some amount 𝑠
𝑋
, the fitness of host genotype i is given by:
𝑊
𝑋,𝑖
= 1 − 𝑠
𝑋
∑
8 𝑗=1 𝛼 𝑖,𝑗
𝑌 𝑗
(14a)
Similarly, if we assume encountering a resistant host reduces the fitness of the parasite by some amount 𝑠
𝑌
, the fitness of parasite genotype i is given by:
𝑊
𝑌,𝑖
= 1 − 𝑠
𝑌
∑ 8 𝑗=1
(1 − 𝛼 𝑗,𝑖
) 𝑋 𝑗
(14b)
We now have expressions for the fitness of all host and parasite genotypes as a function of genotype frequencies and an infection matrix 𝛼 . Our next step will be to study how different assumptions about the mechanisms underlying infection and resistance as captured by 𝛼 influence the outcome of coevolution and whether or not increased rates of recombination ultimately evolve. We will explore three different forms of the infection matrix, 𝛼 , each of which we derived from mechanistic details of specific systems in the previous chapter.
Perhaps the most relevant infection matrix for the interaction between the snail, P.
antipodarum, and its trematode parasite Microphallus, is that which we derived previously for the interaction between the snail B glomerata and its schistosome parasite, XX:
0 1 1 1 𝛼 = [
1 0 1 1
1 1 0 1
]
1 1 1 0
(15)
WHERE ROWS ARE AND COLUMNS AREThis infection matrix arises because snails carrying a specific immune molecule (e.g., FREPS) are able to bind to and mount an immune response against only those parasites with a matching antigen (e.g., Mucin). If we now proceed to iterate the exact recursion equations (5) over generations, we can track coevolutionary changes in allele frequencies and statistical associations over time. We can also, of course, determine whether or not increased rates of recombination evolve in the interacting species as we increase the strength of coevolutionary selection and begin to violate the assumptions of our QLE approximation.
Simulating coevolution using infection matrix (15) shows that when selection is relatively weak, say a 5% fitness cost of infection or failure to infect, our QLE result performs very well, providing a very accurate forecast for the decrease in the frequency of the M allele in both host and parasite (Figure 2a).
If we increase the strength of selection further, say to a 15% fitness cost of infection or failure to infect, our QLE approximation is less accurate and begins to deviate from the observed frequency of the M allele over long evolutionary time scales. However, our QLE approximation remains qualitatively correct: recombination rates still decline in both of the interacting species (Figure 2b). Only when selection becomes quite strong, say 45% fitness cost of infection or failure to infect, do we see something new occur. Specifically, we now see that the frequency of the M allele increases over time in the host and parasite, indicating that under these conditions coevolution can favor the evolution of increased rates of recombination as predicted by the Red Queen Hypothesis (Figure 2c).
9
Next, we turn to another form of the infection matrix, 𝛼 , that can be derived under the assumption that resistance to infection depends on a quantitative traits in both of the interacting species such that:
1 3 4 ⁄ 𝛼 = [
⁄ 1
1
0
1 3 4
1 3 4
]
0 3 4 ⁄ 1
(16)
An infection matrix like (16) would be expected to occur if the probability of parasite infection increases with the similarity between host and parasite phenotype, as is the case for the warbler and cuckoo example we have studied previously in Chapters X and X. Next, we iterate exact recursions (5) assuming infection matrix (16) for various strengths of coevolutionary selection. Here too, we find that our QLE approximation performs very well when selection is weak (Figure 3a) and starts to break down as selection becomes stronger (Figure 3b). In contrast to the previous model of infection genetics, however, here we never see the frequency of the M allele increase. Instead, the overall prediction of our
QLE approximation appears to remain accurate even as selection becomes quite strong (Figure 3c).
Perhaps the reason for this is that the quantitative model of infection we study here generates epistasis of constant sign (positive in the host, negative in the parasite) whereas the model based on interactions between FREPS and Mucins causes epistasis to fluctuate wildly over time.
Finally, we consider the infection matrix, 𝛼 , that arises if we consider classical gene-for-gene interactions like those between Flax and Flax-Rust that we have considered in Chapters X, X, and X.
1 1 1 1 𝛼 = [
0 1 0 1
0 0 1 1
]
0 0 0 1
(17)
We already know that for this simplified version of the gene-for-gene model without costs of resistance or virulence, the only outcome of coevolution is the fixation of virulence alleles. Thus, our goal here is to simply ask what happens to the frequency of the M allele as resistance and virulence alleles rapidly sweep through the host and parasite populations. Simulating coevolution for this scenario by iterating the exact recursions with infection matrix (17) allows us to explore the accuracy of our QLE approximation and determine if increased recombination rates can be favored. Just as we saw for the previous two cases, as long as selection is relatively weak (say 5%), our QLE approximation is very accurate (Figure 4a). Not surprisingly, we also observe that as the strength of selection increases, the accuracy of our QLE approximation decreases (Figure 4b,c). Just as we saw for the quantitative model of infection, however, our overall QLE prediction appears quite robust: coevolution favors decreased rates of recombination even when selection is quite strong. Perhaps the reason is again that, like the quantitative model, gene-for-gene coevolution generates epistasis of a constant sign. Although including costs of carrying resistance alleles in the host or infectivity alleles in the parasite could allow allele frequencies to cycle, only specific types of costs would create the potential for fluctuations in the sign of epistasis.
10
Generalization 2: Exploring weak recombination
In addition to assuming selection is relatively weak, our QLE approximation requires that recombination occurs with a frequency sufficient to prevent large amounts of linkage disequilibrium from accumulating. As a practical rule of thumb, recombination needs to be significantly greater than the strength of epistatic selection for the QLE approximation to perform well. This assumption clearly poses a problem if we hope to use our QLE predictions to understand the evolution of sex and recombination in interactions like those between the snail P. antipodarum and its trematode parasite,
Microphallus. The catch here is that in this system, host individuals are either clones, and thus do not recombine at all, or are obligately sexual, and thus presumably experience frequent recombination. To study such a system using the model developed in this chapter, we need to be able to predict what will happen to a host population that does not recombine at all when we introduce a novel M allele that allows some amount of recombination to occur. Although there are analytical approaches we could take to study such a scenario, they are beyond the scope of this book. Instead, we will take the more pragmatic approach used in the previous section to study strong selection, and rely on numerical iteration of the exact recursions (5). To keep things simple, we will study the impact of initially weak recombination by focusing on scenarios of weak selection. Thus, we will continue manipulating only one biological factor at a time.
Following the organization of the previous section, we begin our investigation by focusing on interactions mediated by the infection matrix represented by (15). As discussed in the previous section and in Chapter 6, an infection matrix of this form might be expected for interactions like those between the snail Biomphalaria glabrata and its schistosome parasite, Schistosoma mansoni, where specific immune molecules in the snail (e.g., FREPS) are able to bind to and mount an immune response against only those parasites with a matching antigen (e.g., Mucin). If we begin by studying coevolution where recombination between the A and B loci is initially infrequent ( 𝑟 𝑚𝑚 𝑖,𝐴𝐵
= 0.02
), we see that our QLE approximation remains quite accurate and the frequency of the M allele decreases in both host and parasite (Figure 5a). Decreasing the initial rate of recombination further ( 𝑟 𝑚𝑚 𝑖,𝐴𝐵
= 0.01
), reduces the quantitative accuracy of our QLE prediction but the frequency of the M allele continues to decrease within host and parasite populations (Figure 5b). Only when we investigate the extreme scenario where recombination is initially absent altogether ( 𝑟 𝑚𝑚 𝑖,𝐴𝐵
= 0.00
) do we see something new emerge from our simulations. Specifically, we now see that the M allele increases in frequency within both host and parasite populations (Figure 5c). Thus, it appears that even with weak selection (as we have assumed in these simulations) increased levels of recombination can be favored if recombination is, for all intents and purposes, initially absent from both host and parasite populations.
We now know that, at least in some cases, our QLE approximation can break down when initial rates of recombination are weak. In order to evaluate the extent to which this result is general, we now turn to the infection matrix, (16), that can be derived under the assumption that resistance to infection depends on the degree to which quantitative traits match in the interacting species (see Chapter 6 for a more detailed discussion). Here, our simulation results predict rates of recombination will decrease in the interacting species no matter how low the initial rate of recombination (Figure 6). Although
11
inconsistent with the Red Queen hypothesis, these results are quite interesting from the perspective of how genes involved in resistance/infection should be organized within the genome. Specifically, for interactions mediated by the matching of quantitative traits, like those between the nest parasitic cuckoo Cuculus canorus and its reed warbler host Acrocephalus scirpaceus, we might expect genes involved in egg coloration to evolve into close physical linkage such that recombination is infrequent and the pattern of linkage disequilibrium favored by (constant) patterns of epistasis can effectively be maintained.
Finally, we consider the infection matrix, (17), motivated by classical gene-for-gene interactions
like those thought to be involves in interactions between wild Flax and Flax Rust (Burdon 1994).
Simulations of this scenario show that, even when recombination is initially absent, coevolution does not favor the spread of the M allele (Figure 7). Instead, the frequency of the M allele decreases as long as genetic polymorphism remains at the A and B loci. Thus, at least in the absence of costs associated with resistance or infectivity, gene-for-gene coevolution appears to favor reductions in the rate of recombination rather than the increased rates of recombination predicted by the Red Queen.
Conclusions and Synthesis
The models we have studied in this chapter, (and many others, Lively ; Hamilton 1980b; Bell and
Maynard Smith 1987; Peters and Lively 1999; Agrawal 2006; Agrawal and Otto 2006; Gandon and Otto
2007; Salathe et al. 2008, 2009), demonstrate that coevolution between hosts and their parasites can
favor the evolution of increased rates of recombination as envisioned by the Red Queen Hypothesis. The models also show, however, that increasing recombination rate is not a universal response to antagonistic coevolution, instead revealing a much more capricious world where increased recombination is favored in only a subset of cases. Fortunately, the models go a long way to clarify when we should, or should not, entertain the Red Queen as a possible explanation for the evolution of sexual reproduction. Specifically, the models show that as long as epistatic and directional selection are weak, and a modest amount of recombination already exists, the Red Queen inevitably favors decreased recombination rates in both host and parasite. This result holds for any possible infection matrix, whether matching alleles, gene for gene, quantitative or otherwise. Only when coevolutionary selection becomes stronger or initial rates of recombination lower, can the Red Queen begin to drive increases in rates of recombination. Even then, only specific combinations of parameters and infection genetics lead to the evolution of increasing rates of recombination (Figures 2-7). Together, these results suggest that in order to evaluate the overall importance of the Red Queen, we must develop a better understanding of just how strong coevolutionary selection is likely to be in natural populations, as well as what forms of the infection matrix best characterize real-world interactions between hosts and their parasites. As we will see in later chapters, these are among the most pressing questions confronting coevolutionary biology today.
Even if we do not yet know enough about the strength of coevolutionary selection in the wild or the structure of real world infection matrices to make grand statements about the overall importance of the Red Queen, we can use insights from the models studied here to evaluate the likelihood that the
Red Queen is operating within specific systems. An obvious place to start is by returning to the
12
interaction between the snail P. antipodarum and its trematode parasite Microphallus. This system is a strong candidate for the Red Queen for many reasons, including the strong negative fitness
consequences of infection (McArthur and Featherston 1976), strong patterns of genetic specificity
coevolutionary selection in this system or the structure of the infection matrix? Although I am unaware of any direct estimates of the strength of coevolutionary selection in this system, we can be virtually certain that it is very strong for several reasons. First, as we will see from models we will study in later chapters, the amount of local adaptation observed in this system is almost impossible to generate without strong coevolutionary selection. Second, remarkable empirical studies have demonstrated that it can take as little as three years for a clonal genotype to go from infrequent to frequent and back to
cycles of such frequency. In addition to evidence supporting strong coevolutionary selection in this system, we might expect the infection genetics underlying this interaction to be similar to those supporting increased recombination rates in Figure 1C that were derived from a detailed mechanistic understanding of immunity in the snail Biomphalaria glabrata to the schistosome parasite Schistosoma
mansoni. Taken together, the case for the Red Queen in this interaction is very strong.
13
Figure Legends
Figure 1. A schematic of the genome showing the arrangement of loci involved in coevolutionary interactions (A and B), the locus determining recombination rate (M), and the rate of recombination between each locus. Although the rate of recombination between the B and M locus (𝑟 𝑖,𝐵𝑀
) is fixed, the rate of recombination between the A and B loci (𝑟
𝕄 𝑖,𝐴𝐵
) depends on the diploid genotype of a mated pair at the M locus.
Figure 2. The coevolution of recombination rates across three different strengths of coevolutionary selection in the XX model. The left hand panels show the frequency of the M allele in the host (Black) and parasite (Grey) predicted by the QLE approximation (Dots) and exact numerical simulations (Solid).
The middle column shows the frequencies of the A and B alleles in the host (Black Solid and Black
Dashed) and in the parasite (Grey Solid and Grey Dashed). The right hand panels show linkage disequilibrium between the A and B alleles in the host (Black) and the parasite (Grey). In the first row, the fitness consequences of the interaction were set to 𝑠
𝑋
0.15, 𝑠
𝑌
= 0.15
, and the third row 𝑠
𝑋
= 0.45, 𝑠
𝑌
= 0.05, 𝑠
𝑌
= 0.05
, in the second row 𝑠
𝑋
=
= 0.45
. Remaining parameters were identical in all cases and set to:
0.08
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵 𝑟
= 0.06
𝑋,𝐵𝑀
.
= 0.1
, 𝑟
𝑀𝑀
𝑋,𝐴𝐵
= 0.1
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵
= 0.08
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.06
, 𝑟
𝑌,𝐵𝑀
= 0.1
, 𝑟
𝑀𝑀
𝑌,𝐴𝐵
= 0.1
, 𝑟
𝑀𝑚
𝑌,𝐴𝐵
=
Figure 3. The coevolution of recombination rates across three different strengths of coevolutionary selection in the XX model. The left hand panels show the frequency of the M allele in the host (Black) and parasite (Grey) predicted by the QLE approximation (Dots) and exact numerical simulations (Solid).
The middle column shows the frequencies of the A and B alleles in the host (Black Solid and Black
Dashed) and in the parasite (Grey Solid and Grey Dashed). The right hand panels show linkage disequilibrium between the A and B alleles in the host (Black) and the parasite (Grey). In the first row, the fitness consequences of the interaction were set to 𝑠
𝑋
0.15, 𝑠
𝑌
= 0.15
, and the third row 𝑠
𝑋
= 0.45, 𝑠
𝑌
= 0.05, 𝑠
𝑌
= 0.05
, in the second row 𝑠
𝑋
=
= 0.45
. Remaining parameters were identical in all cases and set to:
0.08
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵 𝑟
= 0.06
𝑋,𝐵𝑀
.
= 0.1
, 𝑟
𝑀𝑀
𝑋,𝐴𝐵
= 0.1
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵
= 0.08
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.06
, 𝑟
𝑌,𝐵𝑀
= 0.1
, 𝑟
𝑀𝑀
𝑌,𝐴𝐵
= 0.1
, 𝑟
𝑀𝑚
𝑌,𝐴𝐵
=
Figure 4. The coevolution of recombination rates across three different strengths of coevolutionary selection in the XX model. The left hand panels show the frequency of the M allele in the host (Black) and parasite (Grey) predicted by the QLE approximation (Dots) and exact numerical simulations (Solid).
The middle column shows the frequencies of the A and B alleles in the host (Black Solid and Black
Dashed) and in the parasite (Grey Solid and Grey Dashed). The right hand panels show linkage disequilibrium between the A and B alleles in the host (Black) and the parasite (Grey). In the first row, the fitness consequences of the interaction were set to 𝑠
𝑋
0.15, 𝑠
𝑌
= 0.15
, and the third row 𝑠
𝑋
= 0.45, 𝑠
𝑌
= 0.05, 𝑠
𝑌
= 0.05
, in the second row 𝑠
𝑋
=
= 0.45
. Remaining parameters were identical in all cases and set to:
0.08
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵 𝑟
= 0.06
𝑋,𝐵𝑀
.
= 0.1
, 𝑟
𝑀𝑀
𝑋,𝐴𝐵
= 0.1
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵
= 0.08
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.06
, 𝑟
𝑌,𝐵𝑀
= 0.1
, 𝑟
𝑀𝑀
𝑌,𝐴𝐵
= 0.1
, 𝑟
𝑀𝑚
𝑌,𝐴𝐵
=
Figure 5. The coevolution of recombination rates across three different initial recombination rates. The left hand panels show the frequency of the M allele in the host (Black) and parasite (Grey) predicted by
14
the QLE approximation (Dots) and exact numerical simulations (Solid). The middle column shows the frequencies of the A and B alleles in the host (Black Solid and Black Dashed) and in the parasite (Grey
Solid and Grey Dashed). The right hand panels show linkage disequilibrium between the A and B alleles in the host (Black) and the parasite (Grey). In the first row, recombination rates were set to 𝑟
𝑋,𝐵𝑀
0.06
, 𝑟
𝑀𝑀
𝑋,𝐴𝐵
= 0.06
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵
= 0.04
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.02
, 𝑟
𝑌,𝐵𝑀
= 0.06
, 𝑟
𝑀𝑀
𝑌,𝐴𝐵
= 0.06
, 𝑟
𝑀𝑚
𝑌,𝐴𝐵
= 0.04
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵
=
= 0.02
, in the second row 𝑟
𝑋,𝐵𝑀 𝑟
𝑀𝑚
𝑌,𝐴𝐵
= 0.03
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵
= 0.05
= 0.02
, 𝑟
𝑀𝑀
𝑋,𝐴𝐵
= 0.05
, 𝑟
, and in the third row
𝑀𝑚
𝑋,𝐴𝐵 𝑟
= 0.03
𝑋,𝐵𝑀
, 𝑟
= 0.04
𝑚𝑚
𝑋,𝐴𝐵
, 𝑟
= 0.02
𝑀𝑀
𝑋,𝐴𝐵
,
= 0.04
𝑟
𝑌,𝐵𝑀
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵
= 0.05
, 𝑟
𝑀𝑀
𝑌,𝐴𝐵
= 0.02
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.05
,
= 0.00
, 𝑟
𝑌,𝐵𝑀
= 0.04
, 𝑟
𝑀𝑀
𝑌,𝐴𝐵 cases and set to 𝑠
𝑋
= 0.04
, 𝑟
= 0.05, 𝑠
𝑌
𝑀𝑚
𝑌,𝐴𝐵
= 0.02
= 0.05
.
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵
= 0.00
. Remaining parameters were identical in all
Figure 6. The coevolution of recombination rates across three different initial recombination rates. The left hand panels show the frequency of the M allele in the host (Black) and parasite (Grey) predicted by the QLE approximation (Dots) and exact numerical simulations (Solid). The middle column shows the frequencies of the A and B alleles in the host (Black Solid and Black Dashed) and in the parasite (Grey
Solid and Grey Dashed). The right hand panels show linkage disequilibrium between the A and B alleles in the host (Black) and the parasite (Grey). In the first row, recombination rates were set to 𝑟
𝑋,𝐵𝑀
=
0.06
, 𝑟 𝑀𝑀
𝑋,𝐴𝐵
= 0.06
, 𝑟 𝑀𝑚
𝑋,𝐴𝐵 in the second row 𝑟
𝑋,𝐵𝑀
= 0.04
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.05
, 𝑟
𝑀𝑀
𝑋,𝐴𝐵
= 0.02
, 𝑟
𝑌,𝐵𝑀
= 0.05
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵
= 0.06
, 𝑟 𝑀𝑀
𝑌,𝐴𝐵
= 0.03
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.06
, 𝑟 𝑀𝑚
𝑌,𝐴𝐵
= 0.02
, 𝑟
𝑌,𝐵𝑀
= 0.04
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵
= 0.05
, 𝑟
𝑀𝑀
𝑌,𝐴𝐵
= 0.02
= 0.05
,
, 𝑟
𝑀𝑚
𝑌,𝐴𝐵 𝑟
𝑌,𝐵𝑀
= 0.03
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵
= 0.04
, 𝑟
𝑀𝑀
𝑌,𝐴𝐵
= 0.02
, and in the third row 𝑟
𝑋,𝐵𝑀
= 0.04
, 𝑟
𝑀𝑚
𝑌,𝐴𝐵
= 0.02
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵
= 0.00
= 0.04
, 𝑟 𝑀𝑀
𝑋,𝐴𝐵
= 0.04
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵
= 0.02
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.00
. Remaining parameters were identical in all
, cases and set to 𝑠
𝑋
= 0.05, 𝑠
𝑌
= 0.05
.
Figure 7. The coevolution of recombination rates across three different initial recombination rates. The left hand panels show the frequency of the M allele in the host (Black) and parasite (Grey) predicted by the QLE approximation (Dots) and exact numerical simulations (Solid). The middle column shows the frequencies of the A and B alleles in the host (Black Solid and Black Dashed) and in the parasite (Grey
Solid and Grey Dashed). The right hand panels show linkage disequilibrium between the A and B alleles in the host (Black) and the parasite (Grey). In the first row, recombination rates were set to 𝑟
𝑋,𝐵𝑀
=
0.06
, 𝑟
𝑀𝑀
𝑋,𝐴𝐵
= 0.06
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵 in the second row 𝑟
𝑋,𝐵𝑀
= 0.04
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.05
, 𝑟 𝑀𝑀
𝑋,𝐴𝐵
= 0.02
, 𝑟
𝑌,𝐵𝑀
= 0.05
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵
= 0.06
, 𝑟
𝑀𝑀
𝑌,𝐴𝐵
= 0.03
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.06
, 𝑟
𝑀𝑚
𝑌,𝐴𝐵
= 0.02
, 𝑟
𝑌,𝐵𝑀
= 0.04
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵
= 0.05
, 𝑟 𝑀𝑀
𝑌,𝐴𝐵
= 0.02
= 0.05
,
, 𝑟
𝑀𝑚
𝑌,𝐴𝐵 𝑟
𝑌,𝐵𝑀
= 0.03
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵
= 0.04
, 𝑟 𝑀𝑀
𝑌,𝐴𝐵
= 0.02
, and in the third row 𝑟
𝑋,𝐵𝑀
= 0.04
, 𝑟
𝑀𝑚
𝑌,𝐴𝐵
= 0.02
, 𝑟 𝑚𝑚
𝑌,𝐴𝐵
= 0.00
= 0.04
, 𝑟
𝑀𝑀
𝑋,𝐴𝐵
= 0.04
, 𝑟
𝑀𝑚
𝑋,𝐴𝐵
= 0.02
, 𝑟 𝑚𝑚
𝑋,𝐴𝐵
= 0.00
. Remaining parameters were identical in all
, cases and set to 𝑠
𝑋
= 0.05, 𝑠
𝑌
= 0.05
.
15
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