Chapter 5. Coevolution in diploid, sexual systems
Biological Motivation
So far, when we have studied coevolution in a genetically explicit context, we have invariably
assumed that the interacting species were haploid. Although there are many important species
interactions occurring between haploid species (e.g., XX), interactions between diploid species are
abundant and equally important. For instance, the interactions between Wild Flax and Flax Rust and
Daphinia and Pastueria we studied in Chapters 2 and 4 actually involve species that are diploid despite
our earlier genetic assumptions. How then, might our previous predictions about coevolutionary
dynamics and outcomes change if we integrate the additional complexities of diploidy into our earlier
coevolutionary models? Our central goal in this chapter will be to answer this broad question by
evaluating when and where diploidy matters for the process of coevolution. Specifically, we will develop
and analyze diploid models and use them to revisit the three questions we posed in Chapter 2 when we
first explored coevolutionary interactions mediated by major genes in haploid systems.
Key Questions:



Does coevolution maintain genetic polymorphism?
Can coevolution explain variation in infectivity within populations?
Should coevolution cause infection rates to differ among populations?
Building a diploid model of coevolution
Let’s begin by returning to the interaction between wild Flax and Flax-Rust that we considered
previously. Although we have studied this system in some depth already, we have done so under the
assumption that both Flax and Rust are haploid — an assumption that we know to be false. Our goal
here is to develop a diploid model of this interaction and use it to predict the dynamics and outcomes of
coevolution. Fortunately, even though our species are now diploid, we can follow a sequence of steps
almost identical to those we employed in Chapter 2 to develop our coevolutionary model. The only
novel challenge we face is the need to study how Genotype frequencies change rather than only Allele
frequencies. Although this requires that we specify fitnesses for three genotypes rather than only two
alternative alleles, if we are willing to assume individuals mate at random and have very large
population sizes, this is really the only additional complexity diploidy brings to the table.
Specifically, let’s imagine that prior to natural selection there are NX,RR homozygous flax
individuals carrying two copies of the resistant R gene, NX,Rr heterozygous flax individuals carrying one
copy of the resistant R gene and one copy of the susceptible r gene, and NX,rr homozygous flax
individuals carrying two copies of the susceptible r gene. In addition, let’s assume that the probability a
homozygous resistant flax individual survives to reproduce is WX,RR, whereas for a heterozygous flax
individual the survival probability is WX,Rr, and for a homozygous susceptible flax individual the
probability is WX,rr. Just prior to reproduction, then, the number of homozygous resistant individuals is:
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
′
𝑁𝑋,𝑅𝑅
= 𝑁𝑋,𝑅𝑅 𝑊𝑋,𝑅𝑅 = 𝑋𝑅𝑅 𝑁𝑋 𝑊𝑋,𝑅𝑅
(1a)
the number of heterozygous individuals is:
′
𝑁𝑋,𝑅𝑟
= 𝑁𝑋,𝑅𝑟 𝑊𝑋,𝑅𝑟 = 𝑋𝑅𝑟 𝑁𝑋 𝑊𝑋,𝑅𝑟
(1b)
the number of homozygous susceptible individuals is:
′
𝑁𝑋,𝑟𝑟
= 𝑁𝑋,𝑟𝑟 𝑊𝑋,𝑟𝑟 = 𝑋𝑟𝑟 𝑁𝑋 𝑊𝑋,𝑟𝑟
(1c)
and the total number of individuals is:
̅𝑋
𝑁𝑋′ = 𝑋𝑅𝑅 𝑁𝑋 𝑊𝑋,𝑅𝑅 + 𝑋𝑅𝑟 𝑁𝑋 𝑊𝑋,𝑅𝑟 + 𝑋𝑟𝑟 𝑁𝑋 𝑊𝑋,𝑟𝑟 = 𝑁𝑋 𝑊
(1d)
̅𝑋 is the average probability a flax individual survives to reproduce, or the population mean
where 𝑊
fitness.
Since we now know how many flax individuals of each genotype will survive to reproduce,
calculating genotype frequencies after selection is straightforward:
′
𝑋𝑅𝑅
=
′
𝑋𝑅𝑟
=
′
𝑋𝑟𝑟
=
′
𝑁𝑋,𝑅𝑅
′
𝑁𝑋
′
𝑁𝑋,𝑅𝑟
′
𝑁𝑋
′
𝑁𝑋,𝑟𝑟
′
𝑁𝑋
=
𝑋𝑅𝑅 𝑁𝑋 𝑊𝑋,𝑅𝑅
̅𝑋
𝑁𝑋 𝑊
=
𝑋𝑅𝑟 𝑁𝑋 𝑊𝑋,𝑅𝑟
̅𝑋
𝑁𝑋 𝑊
=
𝑋𝑟𝑟 𝑁𝑋 𝑊𝑋,𝑟𝑟
̅𝑋
𝑁𝑋 𝑊
=
=
=
𝑋𝑅𝑅 𝑊𝑋,𝑅𝑅
̅𝑋
𝑊
(2a)
𝑋𝑅𝑟 𝑊𝑋,𝑅𝑟
̅𝑋
𝑊
(2b)
𝑋𝑟𝑟 𝑊𝑋,𝑟𝑟
̅𝑋
𝑊
(2c)
Next, we need to model what happens to these genotype frequencies when the surviving individuals
mate and reproduce. The good news is that if we are willing to assume individuals of both species mate
at random, undergo sexual reproduction, and that population sizes are very large, our model simplifies
substantially and we need only follow change in a single allele frequency to understand evolution within
the Flax population. The reason for this is that if mating is sexual and occurs at random, the genotype
frequencies X following mating but before selection are given by their Hardy Weinberg proportions such
that equations (2) can be re-written as:
′
𝑋𝑅𝑅
=
2
𝑝𝑋
𝑊𝑋,𝑅𝑅
̅𝑋
𝑊
(3a)
′
𝑋𝑅𝑟
=
2𝑝𝑋 (1−𝑝𝑋 )𝑊𝑋,𝑅𝑟
̅𝑋
𝑊
(3b)
′
𝑋𝑟𝑟
=
(1−𝑝𝑋 )2 𝑊𝑋,𝑟𝑟
̅𝑋
𝑊
(3c)
̅𝑋 = 𝑝𝑋2 𝑊𝑋,𝑅𝑅 + 2𝑝𝑋 (1 − 𝑝𝑋 )𝑊𝑋,𝑅𝑟 + (1 − 𝑝𝑋 )2 𝑊𝑋,𝑟𝑟 . Finally, recognizing that the frequency
where 𝑊
of the R allele can also be written as 𝑝𝑋 = 𝑋𝑅𝑅 + 𝑋𝑅𝑟 ⁄2 allows the frequency of the resistant R allele to
be calculate in the subsequent generation:
2
𝑝𝑋′ =
2
𝑝𝑋
𝑊𝑋,𝑅𝑅 +𝑝𝑋 (1−𝑝𝑋 )𝑊𝑋,𝑅𝑅
̅𝑋
𝑊
(4)
If we now subtract the frequency of the resistant R allele within the previous generation, 𝑝𝑋 , from
Equation (4) we arrive at an expression for evolutionary change over a single generation:
∆𝑝𝑋 = 𝑝𝑋′ − 𝑝𝑋 =
2
𝑝𝑋
𝑊𝑋,𝑅𝑅 +𝑝𝑋 (1−𝑝𝑋 )𝑊𝑋,𝑅𝑟
̅𝑋
𝑊
− 𝑝𝑋 =
̅ 𝑋)
𝑝𝑋 (𝑝𝑋 𝑊𝑋,𝑅𝑅 +(1−𝑝𝑋 )𝑊𝑋,𝑅𝑟 −𝑊
̅
𝑊𝑋
(5)
An identical sequence of mathematical operations (and assumptions) yields a parallel expression for the
frequency of the virulent V gene in the rust:
∆𝑝𝑌 = 𝑝𝑌′ − 𝑝𝑌 =
2
𝑝𝑌
𝑊𝑌,𝑉𝑉 +𝑝𝑌 (1−𝑝𝑌 )𝑊𝑌,𝑉𝑣
̅𝑌
𝑊
− 𝑝𝑌 =
̅ 𝑌)
𝑝𝑌 (𝑝𝑌 𝑊𝑌,𝑉𝑉 +(1−𝑝𝑌 )𝑊𝑌,𝑉𝑣 −𝑊
̅
𝑊𝑌
(6)
We now have general expressions for the change in the frequencies of relevant alleles in flax and rust.
For these expressions to be useful to us, however, we must specify the fitness of the various genotypes.
Our general approach to specifying fitness will be identical to that which we took when we
explored haploid models in Chapter 2. Specifically, we will assume each individual encounters one other
randomly selected individual over its lifetime. For Flax individuals, an encounter with a rust that leads to
a successful infection reduces individual fitness by an amount s, and for a Rust individual, an encounter
with a Flax that fails to produce a successful infection reduces individual fitness by some amount s.
NOTE THAT THIS IS SUBTLY DIFFERENT THAN OUR ASSUMPTIONS OF YORE…
𝑊𝑋,𝑅𝑅 = 1 − 𝑠𝑋 (𝑝𝑌2 𝛼𝑅𝑅,𝑉𝑉 + 2𝑝𝑌 (1 − 𝑝𝑌 )𝛼𝑅𝑅,𝑉𝑣 + (1 − 𝑝𝑌 )2 𝛼𝑅𝑅,𝑣𝑣 )
(7a)
𝑊𝑋,𝑅𝑟 = 1 − 𝑠𝑋 (𝑝𝑌2 𝛼𝑅𝑟,𝑉𝑉 + 2𝑝𝑌 (1 − 𝑝𝑌 )𝛼𝑅𝑟,𝑉𝑣 + (1 − 𝑝𝑌 )2 𝛼𝑅𝑟,𝑣𝑣 )
(7a)
𝑊𝑋,𝑟𝑟 = 1 − 𝑠𝑋 (𝑝𝑌2 𝛼𝑟𝑟,𝑉𝑉 + 2𝑝𝑌 (1 − 𝑝𝑌 )𝛼𝑟𝑟,𝑉𝑣 + (1 − 𝑝𝑌 )2 𝛼𝑟𝑟,𝑣𝑣 )
(7a)
𝑊𝑌,𝑉𝑉 = 1 − 𝑠𝑌 (1 − 𝑝𝑋2 𝛼𝑅𝑅,𝑉𝑉 − 2𝑝𝑋 (1 − 𝑝𝑋 )𝛼𝑅𝑟,𝑉𝑉 − (1 − 𝑝𝑋 )2 𝛼𝑟𝑟,𝑉𝑉 )
(8a)
𝑊𝑌,𝑉𝑣 = 1 − 𝑠𝑌 (1 − 𝑝𝑋2 𝛼𝑅𝑅,𝑉𝑣 − 2𝑝𝑋 (1 − 𝑝𝑋 )𝛼𝑅𝑟,𝑉𝑣 − (1 − 𝑝𝑋 )2 𝛼𝑟𝑟,𝑉𝑣 )
(8a)
𝑊𝑌,𝑣𝑣 = 1 − 𝑠𝑌 (1 − 𝑝𝑋2 𝛼𝑅𝑅,𝑣𝑣 − 2𝑝𝑋 (1 − 𝑝𝑋 )𝛼𝑅𝑟,𝑣𝑣 − (1 − 𝑝𝑋 )2 𝛼𝑟𝑟,𝑣𝑣 )
(8a)
Developing mathematical expressions for coevolutionary change requires only that we
substitute fitness functions (7-8) into evolutionary recursions (5-6) and replace the general interaction
matrix entries 𝛼𝑖,𝑗 with values appropriate for the interaction between Flax and Flax-Rust. But what are
the appropriate entries now that we are dealing with a diploid system and must specify values of 𝛼 for
the 9 different combinations of Flax and Rust genotypes that can possibly encounter one another?
MORE TEXT ON DOMINANCE!
3
1 1
𝛼 = [0 0
0 0
1
1]
1
where columns are defined by the Flax genotypes {RR, Rr, rr} and rows are defined by Rust genotypes
{VV, Vv, vv}. DOMINANCE
Making these substitutions and simplifying yields the following expressions for the change in
frequency of flax R and rust Vir alleles:
∆𝑝𝑋 =
2
2
𝑠𝑋 𝑝𝑋 𝑞𝑋
(2𝑝𝑌 𝑞𝑌 +𝑞𝑌
)
̅𝑋
𝑊
(5a)
∆𝑝𝑌 =
2
2
𝑠𝑌 𝑝𝑌
𝑞𝑌 (𝑝𝑋
+2𝑝𝑋 𝑞𝑋 )
̅𝑌
𝑊
(5b)
where 𝑞𝑋 = (1 − 𝑝𝑋 ) and 𝑞𝑌 = (1 − 𝑝𝑌 ) are the frequencies of flax r and pathogen v alleles,
respectively. What can we learn from these relatively simple expressions?
Analyzing the model
One potentially important difference is that, now, the rate of coevolution in the Flax depends on
a term 𝑝𝑋 𝑞𝑋2 rather than 𝑝𝑋 𝑞𝑋 as in the haploid case and the rate of coevolution in the Rust depends on
a term 𝑝𝑌2 𝑞𝑌 rather than 𝑝𝑌 𝑞𝑌 as we saw in the haploid case. The reason for this difference is that the
rate of evolution in diploid systems depends on dominance. For the Flax population, resistance to the
Rust is dominant (see matrix alpha), and is expressed in both heterozygotes and homozygotes. As a
consequence, selection can “see” even very rare resistant alleles in the Flax population and effectively
increase them in frequency. In contrast, the ability to overcome Flax resistance within the Rust
population is conferred by a virulence allele which is recessive (see matrix alpha), and expressed only in
homozygous individuals. Thus, only rust individuals homozygous for this virulent allele can overcome
Flax resistance and these homozygous genotypes will be very, very rare when the virulent allele first
arises within the population (hence the 𝑝𝑌2 term). Putting all this together suggests that when both
resistant and virulent alleles are initially rare, the Flax population should be able to evolve resistance
more rapidly than the Rust population can evolve to overcome it (Figure 1).
Answers to key questions



Genetic variation is never maintained
Ultimately variation in infectivity will be lost
infectivity should, ultimately be constant across populations
SO… QUALITATIVELY, NOTHING DIFFERS FROM THE HAPLOID CASE… What about quantitatively?
New Questions Arising:
4
One potentially important difference is that, now, the Flax population should be able to evolve
resistance more rapidly than the Rust population can evolve to overcome it (Figure 1). Although
equations (5) show that this advantage experience by the Flax population is transient, it could have
important consequences if we were to simultaneously consider demography (REFS?), or if we allowed
for the repeated origin of novel resistance and virulence alleles (REFS). SUGGESTS DOMINANCE AND
PLOIDY COULD HAVE INTERESTING CONSEQUENCES FOR COEVOLUTION raising several important
questions:



What other patterns of dominance are possible in coevolving systems?
How do these alternative patterns of dominance influence coevolution?
How do diploidy and mode of reproduction interact??? ie. CLONALITY AND DIPLOIDY OR…
MIXED PLOIDY SYSTEMS… FOR INSTANCE DAPHNIA AND PASTUERIA
In the next two sections, we will generalize our simple model in ways that allow us to answer these
questions.
Generalizations
Generalization 1: What other patterns of dominance are possible in coevolving systems?
The interaction between Biomphalaria glabrata and Schistosoma mansoni, (Mitta et al. 2012)
FREPS and Mucins. The host FREP must match the parasite mucin in order for the host to recognize and
potentially clear the infection. What pattern of dominance might arise from a molecular mechanism like
this? Unfortunately, without knowing how these proteins are constructed Assuming the FREP and Mucin
are the product of a single locus, it is logical manifestWe might imagine, then, that if both Mucins and
FREPS are monomeric proteins, the outcome would be XX and XX. However, if they were dimeric, more
outcomes become possible…
1 1
𝛼 = [0 0
0 0
1
1]
1
FREPS AND MUCINS!
HERE WE EXPLORE SYSTEM X DOMINANCE
“Unlike other trematodes, the schistosomes are dioecious, i.e., the sexes are separate. The two
sexes display a strong degree of sexual dimorphism, and the male is considerably larger than the
female. The male surrounds the female and encloses her within his gynacophoric canal for the entire
adult lives of the worms, where they reproduce sexually.”
5
Biomphalaria glabrata are simultaneous hermaphrodites but apparently diploid…
Generalization 2: How do these alternative patterns of dominance influence coevolution?
HERE WE EXPLORE COEVOLUTION BETWEEN S
Generalization 3: MANY SYSTEMS HAVE MIXED PLOIDY… HOW CAN WE HANDLE THAT? How do
diploidy and mode of reproduction interact??? ie. CLONALITY AND DIPLOIDY OR… MIXED PLOIDY
SYSTEMS… FOR INSTANCE DAPHNIA AND PASTUERIA
Conclusions and Synthesis
6
References
Figure Legends
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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