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, and are given in the offspring
genotype columns of Table 1.
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
3
standpoint. Specifically, plugging away at equations (5) algebraically for a while (or even a great while)
allows them to be re-written as:
′′
′
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 (i.e., the 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 provided 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 substitute 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)
4
∆𝑋𝑎𝐵 =
̅ 𝑋 )𝑊
̅𝑋
𝑟𝑋 (𝑊𝑋,𝑎𝑏 𝑊𝑋,𝐴𝐵 𝑋𝑎𝑏 𝑋𝐴𝐵 −𝑊𝑋,𝑎𝐵 𝑊𝑋,𝐴𝑏 𝑋𝑎𝐵 𝑋𝐴𝑏 )+𝑋𝑎𝐵 (𝑊𝑋,𝑎𝐵 −𝑊
2
̅
𝑊𝑋
(8c)
∆𝑋𝑎𝑏 =
̅ 𝑋 )𝑊
̅𝑋
𝑟𝑋 (𝑊𝑋,𝑎𝐵 𝑊𝑋,𝐴𝑏 𝑋𝑎𝐵 𝑋𝐴𝑏 −𝑊𝑋,𝑎𝑏 𝑊𝑋,𝐴𝐵 𝑋𝑎𝑏 𝑋𝐴𝐵 )+𝑋𝑎𝑏 (𝑊𝑋,𝑎𝑏 −𝑊
̅ 𝑋2
𝑊
(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
5
(11c)
′′
𝑝𝑌,𝐴
=
𝑊𝑌,𝐴𝑏 𝑌𝐴𝑏 +𝑊𝑌,𝐴𝐵 𝑌𝐴𝐵
̅𝑌
𝑊
(12a)
′′
𝑝𝑌,𝐵
=
𝑊𝑌,𝑎𝐵 𝑌𝑎𝐵 +𝑊𝑌,𝐴𝐵 𝑌𝐴𝐵
̅𝑌
𝑊
(12b)
𝐷𝑌′′ =
(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′′ =
(15a)
_
𝑊𝑋
(15b)
_
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
(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)
6
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
(16c)
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. Our last move
is to re-write these recursions as difference equations by subtracting their values at the start of the
generation from (15-16):
_
∆𝑝𝑋,𝐴 =
𝑝𝑋,𝐴 (𝑝𝑋,𝐵 𝑊𝑋,𝐴𝐵 +𝑞𝑋,𝐵 𝑊𝑋,𝐴𝑏 −𝑊𝑋 )+(𝑊𝑋,𝐴𝐵 −𝑊𝑋,𝐴𝑏 )𝐷𝑋
(17a)
_
𝑊𝑋
_
∆𝑝𝑋,𝐵 =
∆𝐷𝑋 =
𝑝𝑋,𝐵 (𝑞𝑋,𝐴 𝑊𝑋,𝑎𝐵 +𝑝𝑋,𝐴 𝑊𝑋,𝐴𝐵 −𝑊𝑋 )+(𝑊𝑋,𝐴𝐵 −𝑊𝑋,𝑎𝐵 )𝐷𝑋
(17b)
_
𝑊𝑋
̅ 𝑋2
(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)−𝐷𝑋 𝑊
̅ 𝑋2
𝑊
(17c)
and,
∆𝑝𝑌,𝐴 =
̅ 𝑌 )+(𝑊𝑌,𝐴𝐵 −𝑊𝑌,𝐴𝑏 )𝐷𝑌
𝑝𝑌,𝐴 (𝑞𝑌,𝐵 𝑊𝑌,𝐴𝑏 +𝑝𝑌,𝐵 𝑊𝑌,𝐴𝐵 −𝑊
̅
𝑊𝑌
(18a)
∆𝑝𝑌,𝐵 =
̅ 𝑌 )+(𝑊𝑌,𝐴𝐵 −𝑊𝑌,𝑎𝐵 )𝐷𝑌
𝑝𝑌,𝐵 (𝑞𝑌,𝐴 𝑊𝑌,𝑎𝐵 +𝑝𝑌,𝐴 𝑊𝑌,𝐴𝐵 −𝑊
̅𝑌
𝑊
(18b)
∆𝐷𝑌 =
̅ 𝑌2
(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)−𝐷𝑌 𝑊
̅ 𝑌2
𝑊
(18c)
PHHHEEEEWWWWYYYY! We have done it. With the bulk of the tedious algebraic book-keeping behind
us, we can finally move on.
Analyzing the model
We can now transform the general two locus model described by difference equations (17-18)
into a specific model of coevolution between Flax and Flax-Rust by replacing the general values of
fitness W with their specific given by equations (X) and values of the interaction matrix appropriate for
our gene-for-gene model:
0
1
𝛼=[
1
1
1
0
1
1
1
1
0
1
1
1
]
1
0
Here, the interaction matrix depicts the outcome of the classical gene-for-gene model where host
resistance genes (A and B) are able to recognize parasite avirulence genes (a and b), but not parasite
virulence genes (A and B).
Even after working long and hard to simplify the resulting equations, however, I couldn’t get them to fit
on a single line of this page. As a general rule of thumb, if your equation doesn’t fit on a single line, you
aren’t going to learn much from it. So, what can we do? One option is to charge straight ahead and
7
simply rely on our computer to simulate coevolution by iterating our recursion equations for a large
number of parameter combinations. Although there is nothing wrong with this approach, we can
actually gain quite a bit of biological insight by using a bit more mathematical finesse, and developing
approximations that assume selection is not too strong and that recombination occurs with some
reasonable frequency. To be a bit more specific, one way to proceed is to pursue a Quasi-Linkage
Equilibrium (QLE) approximation (REFS).
Although a great deal of difficult math has gone into rigorous mathematical investigation when
and where the QLE approximation can be applied (REFS), our approach here will be more informal and, I
hope, more practical for those who simply want to learn something about biology rather than
mathematics. As a general rule of thumb, anytime selection is not too strong (less than a 1% difference
in fitness among genotypes) and recombination is of a larger magnitude than selection (if the fitness
difference among genotypes is 1%, recombination should be at least 0.1 or greater), linkage
disequilibrium will change much more rapidly than allele frequencies and will, in fact, approach a quasiequilibrium state where its value is small, and a function of the current allele frequencies within the
population. What this means to us is that if selection is weak (< 1%) and recombination is frequent
(>10%), linkage disequilibrium will be as small as selection (< 0.01). As a result, as long as we are willing
to tolerate some small amount of inaccuracy in our prediction, we can ignore all terms in our difference
equations that include things like s2, D2, and s*D because these terms will all be very small and quite
negligible. To be a bit more formal, if we are willing to assume recombination is frequent, and that
selection is weak and of some small order ε, linkage disequilibrium will also be weak and of order ε,
allowing us to ignore all terms of order ε2 and higher. Clearly, what this means is that our QLE
approximation will be more and more accurate as the difference in fitness among genotypes decreases,
because the terms we ignore become ever smaller in relation to the terms we keep.
Returning to the specific case of coevolution between Flax and Flax Rust, what we are going to
assume is that the fitness consequences of the interaction are relatively weak such that 𝑠𝑋 and 𝑠𝑌 are
both of small order ε, and that recombination within both species is relatively frequent (i.e., > ε). Our
next step in implementing our QLE approximation is to replace each of our difference equations with its
first order Taylor Series Expansion in ε. Using Mathematica, this is an incredibly trivial thing to do, and
yields the following approximate expressions for evolutionary change in the Flax:
∆𝑝𝑋,𝐴 ≈ 𝑠𝑋 𝑝𝑋,𝐴 𝑞𝑋,𝐴 𝑞𝑌𝐴 (1 − 𝑝𝑋𝐵 𝑞𝑌𝐵 )
(19a)
∆𝑝𝑋,𝐵 ≈ 𝑠𝑋 𝑝𝑋𝐵 𝑞𝑋𝐵 𝑞𝑌𝐵 (1 − 𝑝𝑋𝐴 𝑞𝑌𝐴 )
(19b)
∆𝐷𝑋 ≈ −𝑠𝑋 (1 − 𝑟𝑋 )𝑝XA 𝑞XA 𝑝XB 𝑞XB 𝑞YA 𝑞YB − 𝑟𝑋 𝐷𝑋
(19c)
and Flax Rust:
∆𝑝𝑌,𝐴 ≈ 𝑠𝑌 𝑝YA 𝑞YA 𝑝XA (1 − 𝑝XB 𝑞YB )
(20a)
∆𝑝𝑌,𝐵 ≈ 𝑠𝑌 𝑝YB 𝑞YB 𝑝XB (1 − 𝑝XA 𝑞YA )
(20b)
∆𝐷𝑌 ≈ 𝑠𝑌 (1 − 𝑟𝑌 )𝑝YA 𝑞YA 𝑝YB 𝑞YB 𝑝XA 𝑝XB − 𝑟𝑌 𝐷𝑌
(20c)
8
The beauty of the QLE approximation, and the primary reason for using it, (other than the fact that it
makes truly lovely equations) 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, even a
passing inspection of our approximation reveals 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 by simply setting (19c and 20c) equal to zero and solving for 𝐷𝑋 and 𝐷𝑌 :
̃𝑋 ≈ − 𝑠𝑋 (1−𝑟𝑋 )𝑝XA 𝑞XA 𝑝XB 𝑞XB 𝑞YA 𝑞YB
𝐷
𝑟
(21a)
̃𝑌 ≈ 𝑠𝑌 (1−𝑟𝑌 )𝑝YA 𝑞YA 𝑝YB 𝑞YB 𝑝XA 𝑝XB
𝐷
𝑟
(21b)
𝑋
𝑌
Remarkably, this shows that the sign of linkage disequilibrium should be different in Flax and Flax rust.
Specifically, linkage disequilibrium between resistance genes within the flax should always be negative
whereas linkage disequilibrium between virulence genes in the rust should always be positive. The
biological reason for this intriguing pattern is that Flax individuals receive a fitness benefit by carrying a
single resistance gene at either locus (A or B) whereas rust individuals must carry virulence alleles at
both loci (A and B) in order to evade detection and elimination by the host. Consequently, within the
Flax population, the quantity 𝑊𝑋,𝐴𝐵 𝑊𝑋,𝑎𝑏 − 𝑊𝑋,𝐴𝑏 𝑊𝑋,𝑎𝐵 is negative, indicating it experiences negative
epistasis. In contrast, within the Rust population the quantity 𝑊𝑌,𝐴𝐵 𝑊𝑌,𝑎𝑏 − 𝑊𝑌,𝐴𝑏 𝑊𝑌,𝑎𝐵 is positive,
indicating the rust population experiences positive epistasis.
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. Inspecting equations (19a-b, 20a-b) reveals
that as long as genetic variation exists at all loci, host resistance genes and parasite virulence genes will
increase in frequency (Figure 1). Only when the parasite fixes both virulence alleles, or the host has no
resistance alleles at either locus, does coevolution cease. This picture of coevolutionary dynamics is
remarkably similar to what we saw when we studied gene-for-gene coevolution in a haploid, single locus
model. Only when we study the relative rates of coevolution in the two species do we see the novel
twist that multiple loci and epistasis bring to the table. Specifically, if both Flax and Rust initially have
very low frequencies of resistance and virulence alleles, the Flax population will evolve resistance much
more rapidly than the rust can evolve to overcome it (Figure 1). The reason for this striking difference in
coevolutionary rates is, again, epistasis. Because the host realizes fitness benefits by having a resistance
allele at a single locus (because that is sufficient to recognize and clear the rust), selection is quite
effective at increasing the frequency of even a very rare resistance allele. In contrast, the parasite must
carry virulence alleles at both loci in order to avoid recognition by hosts with even only a single
resistance allele. Thus, if virulence alleles are initially very rare, rust individuals carrying virulence alleles
at both loci are incredibly rare, and selection has a very difficult time increasing the frequency of the
virulence alleles (Figure 1). Together with our earlier results about the sign of linkage disequilibrium, this
shows that it is epistasis that causes our two locus model to differ in interesting ways from the single
locus model we studied earlier in Chapter 2.
9
At this point, you should be wondering just how much we should trust our QLE approximation.
As with any approximation, whether it be the QLE, quantitative genetics, or adaptive dynamics, it is
important to evaluate robustness using simulations. In addition to allowing the generality of our
conclusions to be explored, performing simulations also helps serve as a safeguard against
straightforward algebraic errors. Although there are many ways to perform simulations, each of which
makes its own set of assumptions, for our purposes it is sufficient to simply iterate the general recursion
equations (X). Our goal is to push the limits of our QLE approximation to see when it “breaks”. As we
already saw in Figure 1, when𝑠𝑋 = 𝑠𝑌 = 0.01 and 𝑟𝑋 = 𝑟𝑌 = 0.1 the predictions of our QLE
approximation and exact simulations are more or less indistinguishable, even over 5,000 generations of
coevolution. But what about scenarios where the interaction between Flax and Rust has large fitness
consequences for the interacting species, perhaps something more along the lines of 𝑠𝑋 = 𝑠𝑌 = 0.1? In
such cases, Figure 2 shows that significant errors begin to creep into our QLE approximation, which over
sufficient amounts of time, lead to significant quantitative errors in our predictions for allele frequencies
and linkage disequilibria. That said, the amount of error accruing over any single generation remains
vanishingly small, and qualitative predictions such as the sign of linkage disequilibrium, remain entirely
robust. What can we conclude from this exercise about the robustness of the QLE approximation? The
answer is that it largely depends on the question you wish to address. If the goal is qualitative
prediction, the QLE approximation often works remarkably well even when its basic assumptions are
grossly violated. If, on the other hand, the goal is quantitative prediction, selection really does need to
be quite weak and recombination quite frequent for accuracy to be maintained over thousands of
generations.
Answers to key questions
What patterns of epistasis are likely to be generated by species interactions?
Our exploration of coevolution between Flax and its pathogen M. lini, revealed interesting
patterns of epistasis. Specifically, our QLE approximation revealed that the host plant, L. marginale,
experiences negative epistasis because under the assumptions of our gene-for-gene model, a single
resistance gene can be sufficient to recognize and clear the pathogen. In contrast, we found that
epistasis within the rust, M. lini, was positive, a pattern that emerges because host recognition can be
thwarted only by evading recognition at both loci.
How do these patterns of epistasis influence the dynamics and outcome of coevolution?
By studying coevolution using our QLE approximation and numerical simulation, we found that
epistasis plays an important role in the dynamics of coevolution while having no real impact on its
ultimate outcome. Specifically, because epistasis in the Flax is negative, significant fitness gains accrue
to individuals carrying a resistance allele at only a single locus. Thus, selection is quite effective at
increasing the frequency of resistance genes in the host. In contrast, because epistasis within the rust is
positive, significant fitness gains accrue only to individuals carrying virulence alleles at both loci.
Consequently, selection has a very hard time gaining traction on rare virulence alleles, thus slowing the
rate at which they increase in frequency within the rust population.
10
What patterns of linkage disequilibrium do we expect to emerge in coevolving systems?
In light of our observation that epistasis is negative within the Flax population but positive
within the Rust population, it is perhaps not surprising that we also observe differences in the sign of
linkage disequilibrium in the two species. Specifically, linkage disequilibrium within the Flax population
tends to be negative whereas within the rust population it tends to be positive. What this means is that
if we sample a Flax individual and find that it is carrying a resistance allele at one locus, it is less likely
than expected based on allele frequencies that it is also carrying a resistance allele at another locus. In
contrast, if we sample a rust individual and find that it is carrying a virulence allele at one locus, it is
more likely than we would expect based on allele frequencies that it is also carrying a virulence allele at
another locus.
New Questions Arising:
Our simple model of gene-for-gene coevolution between Flax and Flax-Rust suggests that …. Although
thought provoking, the tenuous connection between this prediction and available empirical data
immediately raises several important questions:

Are the patterns of epistasis and linkage disequilibrium we uncovered for our simple gene-forgene model also likely to occur in other species interactions?

If species interactions are mediated by quantitative traits, rather than molecular recognition,
should we still expect epistasis and linkage disequilibrium to matter?
In the next two sections, we will develop extensions of our two-locus model that will help us to answer
these questions and gain further insight into the process of coevolution in multi-locus systems.
Extensions
Extension 1: Evaluating the consistency of epistasis and disequilibrium
Just how general are the patterns of epistasis and linkage disequilibrium we observed in our
investigation of gene-for-gene coevolution between Wild flax and flax rust? Perhaps the easiest way to
answer this question is to jump right into developing and analyzing a model of coevolution for a very
different interaction. Because it is still fresh in our memory from the previous chapter, let’s use the
interaction between the snail XX and its schistosome parasite, XX, as our test case. As a quick refresher,
recent empirical studies have identified two molecules that have been hypothesized to interact and play
an important role in the outcome of an encounter between an individual snail and an individual
schistosome. Specifically, snails deploy FREP molecules that bind to specific mucin molecules produced
by the schistosome; when the FREP “matches” the mucin, the infection is cleared. In the previous
chapter, we developed interaction matrices describing this molecular interaction under various
assumptions, all of which involved only a single diploid locus. As many of you probably guessed, the
assumption that the structure of these molecules depends on only a single genetic locus is most likely
false (REFS). Given this information, let’s revisit this interaction and explore how it is likely to coevolve
given a fresh set of genetic assumptions. To keep things simple, we are going to forget all about diploidy
11
and simply focus in on a scenario where FREP and mucin are produced by only a pair of haploid, diallelic
loci in each species.
The particular assumption we are going to make to adapt this interaction to the two-locus
modeling framework we developed earlier in this chapter is that each genotype makes a FREP or mucin
molecule with a unique conformation. If the genotypes of snail and shistosome match, the host FREP
molecule binds to the parasite mucin molecule and the infection is cleared. If, in contrast, the genotypes
of the two species do not match, the snail FREP fails to bind to the schistosome mucin and the infection
succeeds. Together, these assumptions lead to the following interaction matrix describing the
probability that a schistosome genotype evades recognition and successfully infects a host genotype:
0
1
𝛼=[
1
1
1
0
1
1
1
1
0
1
1
1
]
1
0
where snail genotypes are in columns {AB, Ab, aB, ab} and schistosome genotypes are in rows {AB, Ab,
aB, ab}.
∆𝑝𝑋,𝐴 ≈ −𝑠𝑋 𝑝XA 𝑞XA (1 − 2𝑝YA )(𝑞YB − 𝑝XB (1 − 2𝑝YB ))
∆𝑝𝑋,𝐵 ≈ −𝑠𝑋 𝑝XB 𝑞XB (1 − 2𝑝YB )(𝑞YA − 𝑝XA (1 − 2𝑝YA ))
∆𝐷𝑋 ≈ 𝑠𝑋 𝑝XA 𝑞XA 𝑝XB 𝑞XB (1 − 2𝑝YA )(1 − 2𝑝YB )(1 − 𝑟𝑋 ) − 𝑟𝑋 𝐷𝑋
∆𝑝𝑌,𝐴 ≈ 𝑠𝑌 𝑝YA 𝑞YA (1 − 2𝑝XA )(𝑞YB − 𝑝XB (1 − 2𝑝YB ))
∆𝑝𝑌,𝐵 ≈ 𝑠𝑌 𝑝YB 𝑞YB (1 − 2𝑝XB )(𝑞YA − 𝑝XA (1 − 2𝑝YA ))
∆𝐷𝑌 ≈ −𝑠𝑌 𝑝YA 𝑞YA 𝑝YB 𝑞YB (1 − 2𝑝XA )(1 − 2𝑝XB )(1 − 𝑟𝑌 ) − 𝑟𝑌 𝐷𝑌
̃𝑋 ≈
𝐷
𝑠𝑋 𝑝XA 𝑞XA 𝑝XB 𝑞XB (1 − 2𝑝YA )(1 − 2𝑝YB )(1 − 𝑟𝑋 )
𝑟𝑋
̃𝑌 ≈ −
𝐷
𝑠𝑌 𝑝YA 𝑞YA 𝑝YB 𝑞YB (1 − 2𝑝XA )(1 − 2𝑝XB )(1 − 𝑟𝑌 )
𝑟𝑌
These equations reveal that the sign of LD fluctuates over time in response to fluctuating epistasis
We now know that drawing conclusions about the sign of linkage disequilribium is much less
certain than it was for the gene-for-gene model. What about allele frequencies? Here too, drawing
definitive conclusions is much more challenging and requires that we make use of a more formal
approach than the simple method of inspection that sufficed for the GFG model. Specifically, we can
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solve for the equilibria and identify their local stability. Solving equations (x) set to zero yields a set of
30!!! equilibria. HOW TO HANDLE THIS????
What we learn from stability… Epistasis matters!!! Why should this equilibrium not be downright
unstable? The answer lies in epistasis. Although the host desperately wants to escape, it can’t because
only two mutations or two rare alleles will do it any good. So its fucked by epistasis!
Extension 2: Evaluating the importance of epistasis and disequilibrium for quantitative traits
HERE THERE ARE SO MANY POINTS TO MAKE… ASSUMPTIONS OF LANDE LAND, MAPPING BETWEEN
FITNESS FUNCTIONS AND EPISTASIS… ADDITIEV TRAITS VS ADDITIVE FITNESSES. MAPPING BETWEEN LD
AND VG, ETC…
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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