Chapter 6. Multi-locus coevolution, epistasis, and linkage disequilibrium
Biological Motivation
Obviously, more than a dsingle locus is involved. Here we develop a basic framework for studying two
locus systems introducing the concepts of epistasis, recombination, and linkage disequilibrium. After
studying how coevolution proceeds in a simple two locus system (motivated by???) we move on to
explore ??? INTRODUCE EPISTASIS AND LINKAGE DISEQUILIBRIUM
Key Questions:



What patterns of epistasis are likely to be generated by species interactions?
How do these patterns of epistasis influence the dynamics and outcome of coevolution?
What patterns of linkage disequilibrium do we expect to emerge in coevolving systems?
Building a 2-locus model of coevolution
Our goal is to develop the simplest possible model that captures the potentially important
consequences of the multi-locus gene-for-gene interactions for coevolution between X and X. Clearly,
the simplest starting point is to focus on only a single pair of loci and haploid sexual species. Within
haploid sexuals, recombination occurs in a transient diploid phase but selection occurs in the haploid
phase. Thus, we avoid the complexities of diploidy that we struggled with in the previous chapter. Of
course, ignoring diploidy also comes at the cost of reduced realism since both XX and XX are, indeed,
diploid species.
We imagine that rusts and flax’s run into each other at random, and that this has negative fitness
consequences for the flax and posoitive fitness consequences for the rust… Assuming random
encounters and that the probability of infection depends upon the two locus genotypes of flax and rust,
the fitness of the four possible Flax genotypes is given by:
𝑊𝑋,𝐴𝐵 = 1 − 𝑠𝑋 (𝑌𝐴𝐵 𝛼𝐴𝐵,𝐴𝐵 + 𝑌𝐴𝑏 𝛼𝐴𝐵,𝐴𝑏 + 𝑌𝑎𝐵 𝛼𝐴𝐵,𝑎𝐵 + 𝑌𝑎𝑏 𝛼𝐴𝐵,𝑎𝑏 )
(1a)
𝑊𝑋,𝐴𝑏 = 1 − 𝑠𝑋 (𝑌𝐴𝐵 𝛼𝐴𝑏,𝐴𝐵 + 𝑌𝐴𝑏 𝛼𝐴𝑏,𝐴𝑏 + 𝑌𝑎𝐵 𝛼𝐴𝑏,𝑎𝐵 + 𝑌𝑎𝑏 𝛼𝐴𝑏,𝑎𝑏 )
(1b)
𝑊𝑋,𝑎𝐵 = 1 − 𝑠𝑋 (𝑌𝐴𝐵 𝛼𝑎𝐵,𝐴𝐵 + 𝑌𝐴𝑏 𝛼𝑎𝐵,𝐴𝑏 + 𝑌𝑎𝐵 𝛼𝑎𝐵,𝑎𝐵 + 𝑌𝑎𝑏 𝛼𝑎𝐵,𝑎𝑏 )
(1c)
𝑊𝑋,𝑎𝑏 = 1 − 𝑠𝑋 (𝑌𝐴𝐵 𝛼𝑎𝑏,𝐴𝐵 + 𝑌𝐴𝑏 𝛼𝑎𝑏,𝐴𝑏 + 𝑌𝑎𝐵 𝛼𝑎𝑏,𝑎𝐵 + 𝑌𝑎𝑏 𝛼𝑎𝑏,𝑎𝑏 )
(1d)
Similarly, the fitness of the four possible Rust genotypes is given by:
𝑊𝑌,𝐴𝐵 = 1 − 𝑠𝑌 (1 − 𝑋𝐴𝐵 𝛼𝐴𝐵,𝐴𝐵 − 𝑋𝐴𝑏 𝛼𝐴𝑏,𝐴𝐵 − 𝑋𝑎𝐵 𝛼𝑎𝐵,𝐴𝐵 − 𝑋𝑎𝑏 𝛼𝑎𝑏,𝐴𝐵 )
(2a)
𝑊𝑌,𝐴𝑏 = 1 − 𝑠𝑌 (1 − 𝑋𝐴𝐵 𝛼𝐴𝐵,𝐴𝑏 − 𝑋𝐴𝑏 𝛼𝐴𝑏,𝐴𝑏 − 𝑋𝑎𝐵 𝛼𝑎𝐵,𝐴𝑏 − 𝑋𝑎𝑏 𝛼𝑎𝑏,𝐴𝑏 )
(2b)
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
𝑊𝑌,𝑎𝐵 = 1 − 𝑠𝑌 (1 − 𝑋𝐴𝐵 𝛼𝐴𝐵,𝑎𝐵 − 𝑋𝐴𝑏 𝛼𝐴𝑏,𝑎𝐵 − 𝑋𝑎𝐵 𝛼𝑎𝐵,𝑎𝐵 − 𝑋𝑎𝑏 𝛼𝑎𝑏,𝑎𝐵 )
(2c)
𝑊𝑌,𝑎𝑏 = 1 − 𝑠𝑌 (1 − 𝑋𝐴𝐵 𝛼𝐴𝐵,𝑎𝑏 − 𝑋𝐴𝑏 𝛼𝐴𝑏,𝑎𝑏 − 𝑋𝑎𝐵 𝛼𝑎𝐵,𝑎𝑏 − 𝑋𝑎𝑏 𝛼𝑎𝑏,𝑎𝑏 )
(2d)
Now, if we assume that the probability of survival to mating for the various Flax and Rust 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. For the Flax, this yields the following expressions:
′
𝑋𝐴𝐵
=
𝑋𝐴𝐵 𝑊𝑋,𝐴𝐵
̅𝑋
𝑊
(3a)
′
𝑋𝐴𝑏
=
𝑋𝐴𝑏 𝑊𝑋,𝐴𝑏
̅𝑋
𝑊
(3b)
′
𝑋𝑎𝐵
=
𝑋𝑎𝐵 𝑊𝑋,𝑎𝐵
̅𝑋
𝑊
(3c)
′
𝑋𝑎𝑏
=
𝑋𝑎𝑏 𝑊𝑋,𝑎𝑏
̅𝑋
𝑊
(3d)
̅𝑋 is the population mean fitness of species X and is given by:
where, as usual, the symbol 𝑊
̅𝑋 = 𝑋𝐴𝐵 𝑊𝑋,𝐴𝐵 + 𝑋𝐴𝑏 𝑊𝑋,𝐴𝑏 + 𝑋𝑎𝐵 𝑊𝑋,𝑎𝐵 + 𝑋𝑎𝑏 𝑊𝑋,𝑎𝑏
𝑊
(3e)
The same procedure can now be applied to the rust population to calculate the frequency of two-locus
genotypes there after selection but prior to mating:
′
𝑌𝐴𝐵
=
𝑌𝐴𝐵 𝑊𝑌,𝐴𝐵
̅𝑌
𝑊
(4a)
′
𝑌𝐴𝑏
=
𝑋𝐴𝑏 𝑊𝑌,𝐴𝑏
̅𝑌
𝑊
(4b)
′
𝑌𝑎𝐵
=
𝑌𝑎𝐵 𝑊𝑌,𝑎𝐵
̅𝑌
𝑊
(4c)
′
𝑌𝑎𝑏
=
𝑌𝑎𝑏 𝑊𝑌,𝑎𝑏
̅𝑌
𝑊
(4d)
̅𝑋 is the population mean fitness of species X and is given by:
where, as usual, the symbol 𝑊
̅𝑌 = 𝑌𝐴𝐵 𝑊𝑌,𝐴𝐵 + 𝑌𝐴𝑏 𝑊𝑌,𝐴𝑏 + 𝑌𝑎𝐵 𝑊𝑌,𝑎𝐵 + 𝑌𝑎𝑏 𝑊𝑌,𝑎𝑏
𝑊
(4e)
OK, so now we know what the frequencies of the various genotypes are just before mating ensues. How
can we now move forward to incorporate changes to genotype frequencies that accrue during the
process of mating?
If we are willing to assume that both Flax and Rust mate at random and have quite large
population sizes, we can derive basic expressions for changes in genotype frequencies. The long and
2
tedious way to go about this is to first tabulate the frequency of offspring with various genotypes that
are produced by all possible combinations of parents (Table 1). RECOMBINATION! INTRODUCE IT HERE
Table 1. Genotype frequencies produced by random matings
Maternal|Paternal
genotypes
AB|AB
AB|Ab
AB|aB
AB|ab
Ab|AB
Ab|Ab
Ab|aB
Ab|ab
aB|AB
aB|Ab
aB|aB
aB|ab
ab|AB
ab|Ab
ab|aB
ab|ab
Frequency of mating
AB
𝑋𝐴𝐵 𝑋𝐴𝐵
𝑋𝐴𝐵 𝑋𝐴𝑏
𝑋𝐴𝐵 𝑋𝑎𝐵
𝑋𝐴𝐵 𝑋𝑎𝑏
𝑋𝐴𝐵 𝑋𝐴𝐵
𝑋𝐴𝐵 𝑋𝐴𝑏
𝑋𝐴𝐵 𝑋𝑎𝐵
𝑋𝐴𝐵 𝑋𝑎𝑏
𝑋𝐴𝐵 𝑋𝐴𝐵
𝑋𝐴𝐵 𝑋𝐴𝑏
𝑋𝐴𝐵 𝑋𝑎𝐵
𝑋𝐴𝐵 𝑋𝑎𝑏
𝑋𝐴𝐵 𝑋𝐴𝐵
𝑋𝐴𝐵 𝑋𝐴𝑏
𝑋𝐴𝐵 𝑋𝑎𝐵
𝑋𝐴𝐵 𝑋𝑎𝑏
1
1/2
1/2
(1 − 𝑟)/2
1/2
0
𝑟/2
0
1/2
𝑟/2
0
0
(1 − 𝑟)/2
0
0
0
Offspring genotype
Ab
aB
0
1/2
0
𝑟/2
1/2
1
(1 − 𝑟)/2
1/2
0
(1 − 𝑟)/2
0
0
𝑟/2
1/2
0
0
0
0
1/2
𝑟/2
0
0
(1 − 𝑟)/2
0
1/2
(1 − 𝑟)/2
1
1/2
𝑟/2
0
1/2
0
ab
0
0
0
(1 − 𝑟)/2
0
0
𝑟/2
1/2
0
𝑟/2
0
1/2
(1 − 𝑟)/2
1/2
1/2
1
What Table 1 provides us with is the raw material for calculating the frequency of the various genotypes
in the offspring generation. All we need to do now is sum up the entries in each column, weighting each
entry by the frequency with which the two relevant parental genotypes encounter one another at
random and mate. Mathematically, this amounts to evaluating the following expression for each of the
four possible offspring genotypes, i:
𝑋𝑖′′ = ∑4𝑗=1 ∑4𝑘=1 𝑋𝑗′ 𝑋𝑘′ Π𝑋,𝑗+𝑘→𝑖
(5a)
and the following expression for the four possible offspring genotype in Rust:
𝑌𝑖′′ = ∑4𝑗=1 ∑4𝑘=1 𝑌𝑗′ 𝑌𝑘′ Π𝑌,𝑗+𝑘→𝑖
(5b)
where Π𝑋,𝑗+𝑘→𝑖 and Π𝑌,𝑗+𝑘→𝑖 are the probability that two parents with genotypes j and k produce an
offspring of genotype i within the Flax and Rust populations, respectively.
Although equations (5) help to see, mechanistically speaking, how the genotype frequencies
within one generation are translated into those of the next through the process of segregation and
recombination, they are quite clunky and not terribly insightful. Fortunately, these equations can be
greatly simplified and re-expressed in a way that is much easier to implement from a practical
standpoint, and also much more biologically insightful. Specifically, plugging away algebraically allows
equations (5) to be re-written as:
3
′′
′
𝑋𝐴𝐵
= 𝑋𝐴𝐵
+ 𝑟𝑋 𝐷𝑋′
(6a)
′′
′
𝑋𝐴𝑏
= 𝑋𝐴𝑏
− 𝑟𝑋 𝐷𝑋′
(6b)
′′
′
𝑋𝑎𝐵
= 𝑋𝑎𝐵
− 𝑟𝑋 𝐷𝑋′
(6c)
′′
′
𝑋𝑎𝑏
= 𝑋𝑎𝑏
+ 𝑟𝑋 𝐷𝑋′
(6d)
in the Flax and as
′′
′
𝑌𝐴𝐵
= 𝑌𝐴𝐵
+ 𝑟𝑌 𝐷𝑌′
(7a)
′′
′
𝑌𝐴𝑏
= 𝑌𝐴𝑏
− 𝑟𝑌 𝐷𝑌′
(7b)
′′
′
𝑌𝑎𝐵
= 𝑌𝑎𝐵
− 𝑟𝑌 𝐷𝑌′
(7c)
′′
′
𝑌𝑎𝑏
= 𝑌𝑎𝑏
+ 𝑟𝑌 𝐷𝑌′
(7d)
in the rust. In these equations, DX and DY quantify linkage disequilibrium, a measure of the statistical
′
′
′
′
association or covariance between alleles at the A and B loci. Specifically, 𝐷𝑋′ = 𝑋𝐴𝐵
𝑋𝑎𝑏
− 𝑋𝐴𝑏
𝑋𝑎𝐵
and
′
′
′
′
′
𝐷𝑌 = 𝑌𝐴𝐵 𝑌𝑎𝑏 − 𝑌𝐴𝑏 𝑌𝑎𝐵 such that linkage disequilibrium is positive if there is an excess of AB and ab
genotypes within a population and negative if it is, instead, the Ab and aB genotypes that are in excess.
A key insight illuminated by equations (6-7) is that the change in genotype frequencies that occurs in
response to random mating depends entirely on the rate of recombination. If no recombination occurs,
genotype frequencies within the offspring population remain identical to those within the parental
population. If, instead, recombination occurs, genotype frequencies in the offspring generation differ
from those in the parental generation by an amount proportional to linkage disequilibrium. Clearly,
then, recombination can influence coevolution only in cases where coevolutionary selection, or some
other evolutionary force, acts to create linkage disequilibrium within populations of interacting species.
We are now at a point where we have successfully described how genotype frequencies change
over the course of a single generation. To maintain some generality, let’s wait to subsititute in the
specific values for fitness corresponding to our GFG model, and simply express how genotype
frequencies change in terms of arbitrary fitness values, W. Specifically, subsitututing (3) into (6) and
changing from recursion equations to difference equations, yields the following expressions for the
change in host genotype frequencies that occurs over the course of a single generation:
∆𝑋𝐴𝐵 =
̅ 𝑋 )𝑊
̅𝑋
𝑟𝑋 (𝑊𝑋,𝑎𝐵 𝑊𝑋,𝐴𝑏 𝑋𝑎𝐵 𝑋𝐴𝑏 −𝑊𝑋,𝑎𝑏 𝑊𝑋,𝐴𝐵 𝑋𝑎𝑏 𝑋𝐴𝐵 )+𝑋𝐴𝐵 (𝑊𝑋,𝐴𝐵 −𝑊
2
̅
𝑊𝑋
(8a)
∆𝑋𝐴𝑏 =
̅ 𝑋 )𝑊
̅𝑋
𝑟𝑋 (𝑊𝑋,𝑎𝑏 𝑊𝑋,𝐴𝐵 𝑋𝑎𝑏 𝑋𝐴𝐵 −𝑊𝑋,𝑎𝐵 𝑊𝑋,𝐴𝑏 𝑋𝑎𝐵 𝑋𝐴𝑏 )+𝑋𝐴𝑏 (𝑊𝑋,𝐴𝑏 −𝑊
̅ 𝑋2
𝑊
(8b)
∆𝑋𝑎𝐵 =
̅ 𝑋 )𝑊
̅𝑋
𝑟𝑋 (𝑊𝑋,𝑎𝑏 𝑊𝑋,𝐴𝐵 𝑋𝑎𝑏 𝑋𝐴𝐵 −𝑊𝑋,𝑎𝐵 𝑊𝑋,𝐴𝑏 𝑋𝑎𝐵 𝑋𝐴𝑏 )+𝑋𝑎𝐵 (𝑊𝑋,𝑎𝐵 −𝑊
̅ 𝑋2
𝑊
(8c)
4
∆𝑋𝑎𝑏 =
̅ 𝑋 )𝑊
̅𝑋
𝑟𝑋 (𝑊𝑋,𝑎𝐵 𝑊𝑋,𝐴𝑏 𝑋𝑎𝐵 𝑋𝐴𝑏 −𝑊𝑋,𝑎𝑏 𝑊𝑋,𝐴𝐵 𝑋𝑎𝑏 𝑋𝐴𝐵 )+𝑋𝑎𝑏 (𝑊𝑋,𝑎𝑏 −𝑊
2
̅
𝑊𝑋
(8d)
Equations for the pathogen species, Y, are essentially identical and so are not shown. We are now to a
point where we could, if we wished, simply simulate the process of coevolution by plugging in the values
for fitness we derived previously for the GFG system (EQUSTIONS X) and iterating equations (X).
Although this approach would surely provide us with some insights into the process of a coevolution, a
much more insightful and elegant approach is to first make a change of variables (Appendix 3) that
allows us to focus on allele frequencies and linkage disequilibrium rather than genotype frequencies. In
addition to facilitating biological interpretation and intuition, this change of variables simplifies our
model by reducing the number of variables we follow from four in equations (X) to three, which is the
actual number of degrees of freedom in the system.
In order to make the change of variables from genotype frequencies to allele frequencies and
linkage disequilibrium, we first need to clearly define the new variables. Specifically, we define allele
frequencies:
𝑝𝑋,𝐴 = 𝑋𝐴𝐵 + 𝑋𝐴𝑏
(9a)
𝑝𝑋,𝐵 = 𝑋𝐴𝐵 + 𝑋𝑎𝐵
(9b)
𝑝𝑌,𝐴 = 𝑌𝐴𝐵 + 𝑌𝐴𝑏
(9c)
𝑝𝑌,𝐵 = 𝑌𝐴𝐵 + 𝑌𝑎𝐵
(9d)
and linkage disequilibrium:
𝐷𝑋 = 𝑋𝐴𝐵 𝑋𝑎𝑏 − 𝑋𝐴𝑏 𝑋𝑎𝐵
(10a)
𝐷𝑌 = 𝑌𝐴𝐵 𝑌𝑎𝑏 − 𝑌𝐴𝑏 𝑌𝑎𝐵
(10b)
for both of the interacting species. The next step in our change of variables is to write down new
recursions that capture the way in which our new variables change over the course of a single
generation. The easiest way to do this is to just substitute the predicted values for the genotype
′′
′′
frequencies in the next generation (e.g., 𝑋𝐴𝐵
, 𝑋𝐴𝑏,
etc.) into expressions (9-10), yielding:
′′
𝑝𝑋,𝐴
=
𝑊𝑋,𝐴𝑏 𝑋𝐴𝑏 +𝑊𝑋,𝐴𝐵 𝑋𝐴𝐵
̅𝑋
𝑊
(11a)
′′
𝑝𝑋,𝐵
=
𝑊𝑋,𝑎𝐵 𝑋𝑎𝐵 +𝑊𝑋,𝐴𝐵 𝑋𝐴𝐵
̅𝑋
𝑊
(11b)
𝐷𝑋′′ =
(𝑊𝑋,𝑎𝐵 𝑊𝑋,𝐴𝑏 𝑋𝑎𝐵 𝑋𝐴𝑏 −𝑊𝑋,𝑎𝑏 𝑊𝑋,𝐴𝐵 𝑋𝑎𝑏 𝑋𝐴𝐵 )(𝑟𝑋 −1)
̅ 𝑋2
𝑊
′′
𝑝𝑌,𝐴
=
𝑊𝑌,𝐴𝑏 𝑌𝐴𝑏 +𝑊𝑌,𝐴𝐵 𝑌𝐴𝐵
̅𝑌
𝑊
(11c)
(12a)
5
′′
𝑝𝑌,𝐵
=
𝐷𝑌′′ =
𝑊𝑌,𝑎𝐵 𝑌𝑎𝐵 +𝑊𝑌,𝐴𝐵 𝑌𝐴𝐵
̅𝑌
𝑊
(12b)
(𝑊𝑌,𝑎𝐵 𝑊𝑌,𝐴𝑏 𝑌𝑎𝐵 𝑌𝐴𝑏 −𝑊𝑌,𝑎𝑏 𝑊𝑌,𝐴𝐵 𝑌𝑎𝑏 𝑌𝐴𝐵 )(𝑟𝑌 −1)
̅ 𝑌2
𝑊
(12c)
Obviously, we still have a bit of a problem! Our equations now contain a mix of old and new variables
which can never be a good thing. The way to move forward is to recognize that the genotype
frequencies appearing in the right hand sides of the equations can be re-written using definitions (9-10)
in the following way:
𝑋𝐴𝐵 = 𝑝𝑋,𝐴 𝑝𝑋,𝐵 + 𝐷𝑋
(13a)
𝑋𝐴𝑏 = 𝑝𝑋,𝐴 𝑞𝑋,𝐵 − 𝐷𝑋
(13b)
𝑋𝑎𝐵 = 𝑞𝑋,𝐴 𝑝𝑋,𝐵 − 𝐷𝑋
(13c)
𝑋𝑎𝑏 = 𝑞𝑋,𝐴 𝑞𝑋,𝐵 + 𝐷𝑋
(13d)
𝑌𝐴𝐵 = 𝑝𝑌,𝐴 𝑝𝑌,𝐵 + 𝐷𝑌
(14a)
𝑌𝐴𝑏 = 𝑝𝑌,𝐴 𝑞𝑌,𝐵 − 𝐷𝑌
(14b)
𝑌𝑎𝐵 = 𝑞𝑌,𝐴 𝑝𝑌,𝐵 − 𝐷𝑌
(14c)
𝑌𝑎𝑏 = 𝑞𝑌,𝐴 𝑞𝑌,𝐵 + 𝐷𝑌
(14d)
Substituting (13 and 14) into (11 and 12) and doing a bit of algebra allows us to finally complete our
change of variables and arrive at a set of equations expressed entirely in terms of the new variables.
′′
𝑝𝑋,𝐴
=
𝑝𝑋,𝐴 (𝑞𝑋,𝐵 𝑊𝑋,𝐴𝑏 +𝑝𝑋,𝐵 𝑊𝑋,𝐴𝐵 )+(𝑊𝑋,𝐴𝐵 −𝑊𝑋,𝐴𝑏 )𝐷𝑋
′′
𝑝𝑋,𝐵
=
𝑝𝑋,𝐵 (𝑞𝑋,𝐴 𝑊𝑋,𝑎𝐵 +𝑝𝑋,𝐴 𝑊𝑋,𝐴𝐵 )+(𝑊𝑋,𝐴𝐵 −𝑊𝑋,𝑎𝐵 )𝐷𝑋
𝐷𝑋′′ =
_
𝑊𝑋
_
𝑊𝑋
(𝑊𝑋,𝑎𝐵 𝑊𝑋,𝐴𝑏 (𝑞𝑋,𝐴 𝑝𝑋,𝐵 −𝐷𝑋 )(𝑝𝑋,𝐴 𝑞𝑋,𝐵 −𝐷𝑋 )−𝑊𝑋,𝑎𝑏 𝑊𝑋,𝐴𝐵 (𝑞𝑋,𝐴 𝑞𝑋,𝐵 +𝐷𝑋 )(𝑝𝑋,𝐴 𝑝𝑋,𝐵 +𝐷𝑋 ))(𝑟𝑋 −1)
̅ 𝑋2
𝑊
(15a)
(15b)
(15c)
and,
′′
𝑝𝑌,𝐴
=
𝑝𝑌,𝐴 (𝑞𝑌,𝐵 𝑊𝑌,𝐴𝑏 +𝑝𝑌,𝐵 𝑊𝑌,𝐴𝐵 )+(𝑊𝑌,𝐴𝐵 −𝑊𝑌,𝐴𝑏 )𝐷𝑌
̅𝑌
𝑊
(16a)
′′
𝑝𝑌,𝐵
=
𝑝𝑌,𝐵 (𝑞𝑌,𝐴 𝑊𝑌,𝑎𝐵 +𝑝𝑌,𝐴 𝑊𝑌,𝐴𝐵 )+(𝑊𝑌,𝐴𝐵 −𝑊𝑌,𝑎𝐵 )𝐷𝑌
̅𝑌
𝑊
(16b)
𝐷𝑌′′ =
(𝑊𝑌,𝑎𝐵 𝑊𝑌,𝐴𝑏 (𝑞𝑌,𝐴 𝑝𝑌,𝐵 −𝐷𝑌 )(𝑝𝑌,𝐴 𝑞𝑌,𝐵 −𝐷𝑌 )−𝑊𝑌,𝑎𝑏 𝑊𝑌,𝐴𝐵 (𝑞𝑌,𝐴 𝑞𝑌,𝐵 +𝐷𝑌 )(𝑝𝑌,𝐴 𝑝𝑌,𝐵 +𝐷𝑌 ))(𝑟𝑌 −1)
̅ 𝑌2
𝑊
6
(16c)
PHHHEEEEWWWWYYYY! We have done it. We now have a set of equations describing how allele
frequencies and linkage disequilibrium evolve in response to natural selection and random mating over
the course of a single generation. With the bulk of the mathematical tedium behind us, we can now
move on to use these equations to gain insights into the process of coevolution between rust and flax.
Analyzing the model
With our general model in hand, we can return to flax and flax rust and begin to answer the
questions we posed at the beginning of the chapter. In order to do so effectively, however, we are going
to need to take one final mathematical leap. Specifically, we are going to introduce the idea of a quasilinkage equilibrium (QLE) approximation that will allow us to better understand coevolution both here,
and in later chapters. Although frequently misunderstood, this approximation assumes only that
selection is relatively weak and that recombination is relatively frequent. Often, the idea of weak
selection causes a knee jerk reaction that the approximation does not allow us to understand cases of
“real world” coevolution. In my opinion, this knee jerk reaction is generally misguided. When we say
“weak selection” in the context of a QLE approximation, we mean as a rule of thumb that selection
should be less than 5% per generation. Although this qualifies as weak mathematically, it certainly is
quiet strong, and not often observed, within natural populations. The second misconception about the
QLE is that it assume linkage disequilibrium is zero. Again, this is untrue.
∆𝑝𝑋,𝐴 ≈ 𝑠𝑋 𝑝𝑋,𝐴 𝑞𝑋,𝐴 𝑞𝑌𝐴 (1 − 𝑝𝑋𝐵 𝑞𝑌𝐵 )
(17a)
∆𝑝𝑋,𝐵 ≈ 𝑠𝑋 𝑝𝑋𝐵 𝑞𝑋𝐵 𝑞𝑌𝐵 (1 − 𝑝𝑋𝐴 𝑞𝑌𝐴 )
(17b)
∆𝐷𝑋 ≈ −𝑠𝑋 (1 − 𝑟𝑋 )𝑝XA 𝑞XA 𝑝XB 𝑞XB 𝑞YA 𝑞YB − 𝑟𝑋 𝐷𝑋
(17c)
∆𝑝𝑌,𝐴 ≈ 𝑠𝑌 𝑝YA 𝑞YA 𝑝XA (1 − 𝑝XB 𝑞YB )
(18a)
∆𝑝𝑌,𝐵 ≈ 𝑠𝑌 𝑝YB 𝑞YB 𝑝XB (1 − 𝑝XA 𝑞YA )
(18b)
∆𝐷𝑌 ≈ 𝑠𝑌 (1 − 𝑟𝑌 )𝑝YA 𝑞YA 𝑝YB 𝑞YB 𝑝XA 𝑝XB − 𝑟𝑌 𝐷𝑌
(18c)
Now that is a pretty set of equations! The beauty of the QLE approximation, and the primary reason for
using it, is that it allows us to “see” things about the biology of a system that we might otherwise spend
hours upon hours simulating and still never pick up on. For instance, here we can immediately see that
coevolutionary change in allele frequencies is independent of linkage disequilibrium. As a result, we can
solve for the quasi-equilibrium values of linkage disequilibrium very easily by simply setting (17c and
18c) equal to zero and solving for D and D:
̃𝑋 ≈ − 𝑠𝑋 (1−𝑟𝑋 )𝑝XA 𝑞XA 𝑝XB 𝑞XB 𝑞YA 𝑞YB
𝐷
𝑟
(19a)
𝑋
7
̃𝑌 ≈ 𝑠𝑌 (1−𝑟𝑌 )𝑝YA 𝑞YA 𝑝YB 𝑞YB 𝑝XA 𝑝XB
𝐷
𝑟
(19b)
𝑌
A quick inspection of (19) shows that the sign of linkage disequilribium should be different in Flax and
Flax rust. Specifically, linkage disequilibrium between resistance genes within the flax should always be
negative whereas linkage disequilbrium between virulence genes in the rust should always be positive.
Why does this consistent pattern arise? MUST HAVE SOMETHING TO DO WITH THE FACT THAT THE
HOST NEEDS ONLY ONE R GENE TO “WIN” WHEREAS THE PATHOGEN NEEDS TWO V GENES TO “WIN”.
SIGN OF EPISTASIS SHOULD BE DIFFERENT IN THE TWO SPECIES. IS IT? Yup…
Our QLE approximation has already unearthed a valuable insight about our expectations for the form of
epistasis and sign of linkage disequilibrium that we expect to emerge from GFG coevolution. Can we
push our QLE approximation further to learn about the dynamics and outcomes of coevolution? The
place to start is with an analysis of allele frequency change.
Answers to key questions
New Questions Arising:
Extensions
Extension 1: Snails and schistosomes
Extension 2: Quantitative traits
Conclusions and Synthesis
Like dominance before, epistasis plays an important role in the dynamics and outcome of coevolution.
Yet, here too, we know so very little about actual patterns of epistasis within real systems it is almost
shocking; certainly humbling. This, along with dominance, is the frontier of coevolutionary genetics.
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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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