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. The final resu Sub 1 and 2 into 3 and 4, then sub 3 and 4 into 6
and 7.
_
_
𝑟𝑋 (𝑊XaB 𝑊XAb 𝑋aB 𝑋Ab − 𝑊Xab 𝑊XAB 𝑋ab 𝑋AB ) + 𝑋AB (𝑊XAB − 𝑊𝑋 )𝑊𝑋
_
𝑊𝑋2
between X and X Clearly we must compromise.
Analyzing the model
Introduce the QLE approximation
4
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.
5
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.
6