Chapter 5. Coevolution in diploid, sexual systems Biological
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 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 N
X,RR
homozygous flax individuals carrying two copies of the resistant R gene, N
X,Rr heterozygous flax individuals carrying one copy of the resistant R gene and one copy of the susceptible r gene, and N
X,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 W
X,RR
, whereas for a heterozygous flax individual the survival probability is W
X,Rr
, and for a homozygous susceptible flax individual the probability is W
X,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
𝑁
′
𝑋,𝑅𝑅
= 𝑁
𝑋,𝑅𝑅
𝑊
𝑋,𝑅𝑅
= 𝑋
𝑅𝑅
𝑁
𝑋
𝑊
𝑋,𝑅𝑅 the number of heterozygous individuals is:
𝑁
′
𝑋,𝑅𝑟
= 𝑁
𝑋,𝑅𝑟
𝑊
𝑋,𝑅𝑟
= 𝑋
𝑅𝑟
𝑁
𝑋
𝑊
𝑋,𝑅𝑟
(1a)
(1b) the number of homozygous susceptible individuals is:
𝑁
′
𝑋,𝑟𝑟
= 𝑁
𝑋,𝑟𝑟
𝑊
𝑋,𝑟𝑟
= 𝑋 𝑟𝑟
𝑁
𝑋
𝑊
𝑋,𝑟𝑟
(1c) and the total number of individuals is:
𝑁 ′
𝑋
= 𝑋
𝑅𝑅
𝑁
𝑋
𝑊
𝑋,𝑅𝑅
+ 𝑋
𝑅𝑟
𝑁
𝑋
𝑊
𝑋,𝑅𝑟
+ 𝑋 𝑟𝑟
𝑁
𝑋
𝑊
𝑋,𝑟𝑟
= 𝑁
𝑋
𝑊
𝑋
(1d) where 𝑊
𝑋
is the average probability a flax individual survives to reproduce, or the population mean
fitness.
Since we now know how many flax individuals of each genotype will survive to reproduce, calculating genotype frequencies after selection is straightforward:
𝑋
′
𝑅𝑅
=
𝑁
′
𝑋,𝑅𝑅
𝑁
′
𝑋
=
𝑋
𝑅𝑅
𝑁
𝑋
𝑊
𝑋,𝑅𝑅
𝑁
𝑋
𝑊
𝑋
=
𝑋
𝑅𝑅
𝑊
𝑋,𝑅𝑅
𝑊
𝑋
𝑋
′
𝑅𝑟
=
𝑁
′
𝑋,𝑅𝑟
𝑁
′
𝑋
=
𝑋
𝑅𝑟
𝑁
𝑋
𝑊
𝑋,𝑅𝑟
𝑁
𝑋
𝑊
𝑋
=
𝑋
𝑅𝑟
𝑊
𝑋,𝑅𝑟
𝑊
𝑋
𝑋
′ 𝑟𝑟
=
𝑁
′
𝑋,𝑟𝑟
𝑁
′
𝑋
=
𝑋 𝑟𝑟
𝑁
𝑋
𝑊
𝑋,𝑟𝑟
𝑁
𝑋
𝑊
𝑋
=
𝑋 𝑟𝑟
𝑊
𝑋,𝑟𝑟
𝑊
𝑋
(2a)
(2b)
(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:
𝑋
′
𝑅𝑅
= 𝑝
2
𝑋
𝑊
𝑋,𝑅𝑅
𝑊
𝑋
𝑋
′
𝑅𝑟
=
2𝑝
𝑋
(1−𝑝
𝑋
)𝑊
𝑋,𝑅𝑟
𝑊
𝑋
(3a)
(3b)
𝑋
′ 𝑟𝑟
=
(1−𝑝
𝑋
) 2 𝑊
𝑋,𝑟𝑟
𝑊
𝑋
(3c) where 𝑊
𝑋
= 𝑝
2
𝑋
𝑊
𝑋,𝑅𝑅
+ 2𝑝
𝑋
(1 − 𝑝
𝑋
)𝑊
𝑋,𝑅𝑟 of the R allele can also be written as 𝑝
𝑋
+ (1 − 𝑝
𝑋
)
2
𝑊
𝑋,𝑟𝑟
. Finally, recognizing that the frequency
= 𝑋
𝑅𝑅
+ 𝑋
𝑅𝑟
⁄ 2 allows the frequency of the resistant R allele to be calculate in the subsequent generation:
2
𝑝
′
𝑋
= 𝑝
2
𝑋
𝑊
𝑋,𝑅𝑅
+𝑝
𝑋
(1−𝑝
𝑋
)𝑊
𝑋,𝑅𝑅
𝑊
𝑋
(4)
If we now subtract the frequency of the resistant R allele within the previous generation, 𝑝
𝑋
, from
Equation (4) we arrive at an expression for evolutionary change over a single generation:
∆𝑝
𝑋
= 𝑝
′
𝑋
− 𝑝
𝑋
= 𝑝
2
𝑋
𝑊
𝑋,𝑅𝑅
+𝑝
𝑋
(1−𝑝
𝑋
)𝑊
𝑋,𝑅𝑟
𝑊
𝑋
− 𝑝
𝑋
= 𝑝
𝑋
(𝑝
𝑋
𝑊
𝑋,𝑅𝑅
+(1−𝑝
𝑋
)𝑊
𝑋,𝑅𝑟
−𝑊
𝑋
)
𝑊
𝑋
(5)
An identical sequence of mathematical operations (and assumptions) yields a parallel expression for the frequency of the virulent V gene in the rust:
∆𝑝
𝑌
= 𝑝
′
𝑌
− 𝑝
𝑌
= 𝑝
2
𝑌
𝑊
𝑌,𝑉𝑉
+𝑝
𝑌
(1−𝑝
𝑌
)𝑊
𝑌,𝑉𝑣
𝑊
𝑌
− 𝑝
𝑌
= 𝑝
𝑌
(𝑝
𝑌
𝑊
𝑌,𝑉𝑉
+(1−𝑝
𝑌
)𝑊
𝑌,𝑉𝑣
−𝑊
𝑌
)
𝑊
𝑌
(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…
𝑊
𝑋,𝑅𝑅
= 1 − 𝑠
𝑋
(𝑝
2
𝑌 𝛼
𝑅𝑅,𝑉𝑉
+ 2𝑝
𝑌
(1 − 𝑝
𝑌
)𝛼
𝑅𝑅,𝑉𝑣
+ (1 − 𝑝
𝑌
)
2 𝛼
𝑅𝑅,𝑣𝑣
)
𝑊
𝑋,𝑅𝑟
= 1 − 𝑠
𝑋
(𝑝
2
𝑌 𝛼
𝑅𝑟,𝑉𝑉
+ 2𝑝
𝑌
(1 − 𝑝
𝑌
)𝛼
𝑅𝑟,𝑉𝑣
+ (1 − 𝑝
𝑌
) 2 𝛼
𝑅𝑟,𝑣𝑣
)
𝑊
𝑋,𝑟𝑟
= 1 − 𝑠
𝑋
(𝑝
2
𝑌 𝛼 𝑟𝑟,𝑉𝑉
+ 2𝑝
𝑌
(1 − 𝑝
𝑌
)𝛼 𝑟𝑟,𝑉𝑣
+ (1 − 𝑝
𝑌
)
2 𝛼 𝑟𝑟,𝑣𝑣
)
(7a)
(7a)
(7a)
𝑊
𝑌,𝑉𝑉
= 1 − 𝑠
𝑌
(1 − 𝑝
2
𝑋 𝛼
𝑅𝑅,𝑉𝑉
− 2𝑝
𝑋
(1 − 𝑝
𝑋
)𝛼
𝑅𝑟,𝑉𝑉
− (1 − 𝑝
𝑋
)
2 𝛼 𝑟𝑟,𝑉𝑉
)
𝑊
𝑌,𝑉𝑣
= 1 − 𝑠
𝑌
(1 − 𝑝
2
𝑋 𝛼
𝑅𝑅,𝑉𝑣
− 2𝑝
𝑋
(1 − 𝑝
𝑋
)𝛼
𝑅𝑟,𝑉𝑣
− (1 − 𝑝
𝑋
)
2 𝛼 𝑟𝑟,𝑉𝑣
)
𝑊
𝑌,𝑣𝑣
= 1 − 𝑠
𝑌
(1 − 𝑝
2
𝑋 𝛼
𝑅𝑅,𝑣𝑣
− 2𝑝
𝑋
(1 − 𝑝
𝑋
)𝛼
𝑅𝑟,𝑣𝑣
− (1 − 𝑝
𝑋
)
2 𝛼 𝑟𝑟,𝑣𝑣
)
(8a)
(8a)
(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 𝛼 𝑖,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
𝛼 = [
1 1 1
0 0 1 ]
0 0 1
(9) 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:
∆𝑝
𝑋
= 𝑠
𝑋 𝑝
𝑋 𝑞
2
𝑋
(2𝑝
𝑌
𝑊
𝑋 𝑞
𝑌
+𝑞
2
𝑌
)
∆𝑝
𝑌
= 𝑠
𝑌 𝑝
2
𝑌 𝑞
𝑌
(𝑝
2
𝑋
+2𝑝
𝑋
𝑊
𝑌 𝑞
𝑋
) where 𝑞
𝑋
= (1 − 𝑝
𝑋
) and 𝑞
𝑌
= (1 − 𝑝
𝑌
) are the frequencies of flax r and pathogen v alleles, respectively. What can we learn from these relatively simple expressions?
(10a)
(10b)
Analyzing the model
One potentially important difference is that, now, the rate of coevolution in the Flax depends on a term 𝑝
𝑋 𝑞
2
𝑋
rather than 𝑝
𝑋 𝑞
𝑋
as in the haploid case and the rate of coevolution in the Rust depends on a term 𝑝 2
𝑌 𝑞
𝑌
rather than 𝑝
𝑌 𝑞
𝑌
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 𝑝
2
𝑌
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?
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?
Although the pattern of dominance considered so far for the gene-for-gene interaction between
Flax and Flax-rust is commonly observed between plants and fungal pathogens (REFS), there are many other important interactions for which dominance is unknown. In these cases, it is worth considering alternative forms of dominance that might arise from what we know about the molecular mechanisms underpinning the interaction. For instance, the medically important interaction between the snail,
Biomphalaria glabrata and its schistosome parasite, Schistosoma mansoni, exhibits significant variability in outcome depending on the genotypes of the interacting individuals (REFS). Thus, the potential clearly exists for coevolution to drive changes in genotype frequencies within the interaction. However, unlike the interaction between Flax and Flax-Rust we considered previously, the pattern of dominance within this interaction is unclear. Instead, making progress modeling this interaction requires using what we know about the molecular mechanisms of infection and resistance to infer the structure of the alpha matrix from this information.
Recent infection experiments conducted between these species have identified the interaction between S. mansoni mucin molecules and B. glabrata FREP molecules as a candidate mechanism
glabrata to potentially clear an infection by encapsulating the parasite, its FREP molecules must be able to successfully bind to mucin molecules produced by the parasite. Thus, it appears that this interaction is, at least partially, mediated by a mechanism of targeted recognition where the host must match the
modeled as an inverse matching alleles model, and as long as both species are haploid, it is straightforward to write down an appropriate interaction matrix, α, as we did in Chapter 2 (DID WE DO
THAT?). If the interacting species are diploid (as is the case for the interaction between S. mansoni and
B. glabrata), however, a bit more thought must be given to the pattern of dominance and the structure of the interaction matrix, α, that results.
5
If we are willing to ignore some of the true genetic complexities underlying the production of
FREP and Mucin molecules and assume both proteins are encoded by a single genetic locus with two possible alleles (F and f in B. glabrata and M and m in S. mansoni) we can begin to predict the phenotype of individual snails and schistosomes and the outcome of interactions. If both FREPS and
Mucins are monomeric proteins, phenotypes of individual snails and schistosomes might be expected to be as depicted in Table 1.
Table 1. Mapping between genotype and phenotype (monomeric proteins)
B. glabrata genotype
FF
Ff ff
FREP phenotype
F
1/2 F and ½ f f
S. mansoni genotype
MM
Mm mm
Mucin phenotype
M
1/2 M and ½ m m
If we now assume that once the schistosome enters its snail host it produces a single Mucin molecule which is confronted with a single snail FREP molecule, successful infection results only when this FREP and Mucin do not match. Otherwise, the snail recognizes and clears the infection. In such a scenario, the interaction matrix would take the following form:
0 1/2 1 𝛼 = [ 1/2 1/2 1/2 ]
1 1/2 0
(11) where columns are defined by B. glabrata genotypes {FF, Ff, ff}, rows are defined by S. mansoni genotypes {MM, Mm, mm}, and the entries indicate the probability of successful infection. What if, however, upon entering a snail each schistosome produces many Mucin molecules and the snail produces many FREP molecules? Perhaps in such a case, a host immune response is triggered and the infection cleared if only a single FREP matches a single Mucin such that infection occurs only when no
FREP and Mucin molecules match one another. As the number of random encounters between FREP and
Mucin molecules becomes very large, the interaction matrix approaches:
0 0 1 𝛼 = [ 0 0 0 ]
1 0 0
(12)
Obviously, the infection matrices (11-12) are extreme cases and reality may lie somewhere in between.
What is really important, however, is that anywhere between these two extreme cases an interesting and important pattern of dominance emerges for this interaction. Specifically, what we see is that as long as encounters among snail and schistosome genotypes occur at random and all genotypes are present, heterozygous snails are more resistant, on average, than homozygous snails. This represents a case of effective overdominance. Within the schistosome population, however, heterozygous individuals are less infective, on average, than homozygous schistosomes. This represents a case of effective
underdominance.
6
If, in contrast, both FREPS and Mucins are dimeric proteins, phenotypes of individual snails and schistosomes might look something more like what is shown in Table 2.
Table 2. Mapping between genotype and phenotype (dimeric proteins)
B. glabrata genotype
FF
Ff ff
FREP phenotype
FF
1/4 FF: ½ Ff: ¼ ff ff
S. mansoni genotype
MM
Mm mm
Mucin phenotype
MM
1/4 MM: ½ Mm: 1/4 mm mm
As before, we first consider what happens if snail and schistosome each produce only a single FREP and
Mucin molecule. In such a case, infection results only when this pair of FREP and Mucin molecules does not match, leading to the following interaction matrix:
0 3/4 1 𝛼 = [ 3/4 5/8 3/4 ]
1 3/4 0
(12) where columns are defined by B. glabrata genotypes {FF, Ff, ff} and rows are defined by S. mansoni genotypes {MM, Mm, mm}. Next, we consider what happens if both snail and schistosome produce a very large number of FREP and Mucin molecules. In such a case, infection results only when no matches occur between any of these FREP and Mucin molecules. In the limit of a very large number of random encounters between Freps and Mucins the interaction matrix then approaches:
0 0 1 𝛼 = [ 0 0 0 ]
1 0 0
(13)
In contrast to what we observed for monomeric proteins, we now see that the effective pattern of dominance that emerges depends on the details of immune defense. Specifically, if snail immune defense relies on a single encounter between FREP and Mucin molecules, interaction matrix (12) shows that heterozygous snails are more resistant than homozygous snails and heterozygous schistosomes are more infective than homozygous schistosomes. In such a case, then, we would observe overdominance for both infectivity and resistance. If, instead, a very large number of Mucin and FREP molecules encounter one another at random, interaction matrix (13) shows that heterozygous snails are more resistant than homozygous snails but that heterozygous schistosomes are less infective than homozygous schistosomes.
Comparing the interaction matrices we have derived for various possible scenarios shows that the patterns of dominance we expect for infectivity and resistance depend quite a bit on molecular details of immune molecule structure and the details of immune interactions (Table 3).
Table 3. Expected patterns of effective dominance
Mucin and FREP Number of FREP and Mucin Schistosome Snail Resistance
7
structure
Monomeric
Monomeric
Dimeric
Dimeric molecules produced
One
Many
One
Many
Infectivity
No dominance
Strongly
Underdominant
Weakly
Overdominant
Strongly
Underdominant
No Dominance
Strongly
Overdominant
Weakly
Overdominant
Strongly
Overdominant
Clearly, these results show that the simple pattern of dominance we explored for the interaction between wild Flax and Flax-Rust is not likely to be general across other types of coevolving systems. In the next section, we explore how the different possible patterns of dominance shown in Table 3 influence the dynamics and outcome of coevolution.
Generalization 2: How do these alternative patterns of dominance influence coevolution?
The last section revealed several different interaction matrices that could, in principle, emerge when we consider the molecular underpinnings of coevolution between the snail XX and its schistosome parasite XX. Each of these interaction matrices is quite different from the classical gene-for-gene model we studied in the first part of the chapter. Importantly, rather than the simple pattern of dominant resistance and recessive infectivity commonly associated with gene-for-gene systems (REFS), the interaction matrices we derived in the previous section are often characterized by overdominance and underdominance for resistance and infectivity. What are the consequences of these alternative types of dominance for coevolution between S. Mansoni and XX? In the remainder of this section, we will explore the dynamics and outcomes of coevolution predicted by the interaction matrices we derived based on various hypotheses about the molecular basis of the immune interaction between S. Mansoni and XX.
Our general approach to studying the dynamics and outcomes of coevolution will be to capitalize on much of the mathematical machinery we developed earlier in this chapter for the gene-forgene model. Really, this requires only a few small notational changes to equations (5-8) followed by substitution of the interaction matrix appropriate for the biological scenario in question (11-13). First, let’s clear up the notation a bit and try to make things a bit more consistent with the snail schistosome system. Specifically, rather than studying the R, r, V, and v alleles of the gene-for-gene system, we now focus on alleles F and f that confer alternative conformations of the FREP protein within the snail population and alleles M and m that produce two different conformations of the Mucin protein within the schistosome population. Using this new notation and maintaining our previous assumption of random encounters between individual snails and schistosomes, the fitness of snails with various FREP genotypes is:
𝑊
𝑋,𝐹𝐹
= 1 − 𝑠
𝑋
(𝑝
2
𝑌 𝛼
𝐹𝐹,𝑀𝑀
+ 2𝑝
𝑌
(1 − 𝑝
𝑌
)𝛼
𝐹𝐹,𝑀𝑚
+ (1 − 𝑝
𝑌
)
2 𝛼
𝐹𝐹,𝑚𝑚
)
𝑊
𝑋,𝐹𝑓
= 1 − 𝑠
𝑋
(𝑝
2
𝑌 𝛼
𝐹𝑓,𝑀𝑀
+ 2𝑝
𝑌
(1 − 𝑝
𝑌
)𝛼
𝐹𝑓,𝑀𝑚
+ (1 − 𝑝
𝑌
) 2 𝛼
𝐹𝑓,𝑚𝑚
)
(14a)
(14b)
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𝑊
𝑋,𝑓𝑓
= 1 − 𝑠
𝑋
(𝑝
2
𝑌 𝛼 𝑓𝑓,𝑀𝑀
+ 2𝑝
𝑌
(1 − 𝑝
𝑌
)𝛼 𝑓𝑓,𝑀𝑚
+ (1 − 𝑝
𝑌
)
2 𝛼 𝑓𝑓,𝑚𝑚
) (14c) and the fitness of schistosomes with various Mucin genotypes is:
𝑊
𝑌,𝑀𝑀
= 1 − 𝑠
𝑌
(1 − 𝑝
2
𝑋 𝛼
𝐹𝐹,𝑀𝑀
− 2𝑝
𝑋
(1 − 𝑝
𝑋
)𝛼
𝐹𝑓,𝑀𝑀
− (1 − 𝑝
𝑋
)
2 𝛼 𝑓𝑓,𝑀𝑀
)
𝑊
𝑌,𝑀𝑚
= 1 − 𝑠
𝑌
(1 − 𝑝
2
𝑋 𝛼
𝐹𝐹,𝑀𝑚
− 2𝑝
𝑋
(1 − 𝑝
𝑋
)𝛼
𝐹𝑓,𝑀𝑚
− (1 − 𝑝
𝑋
)
2 𝛼 𝑓𝑓,𝑀𝑚
)
𝑊
𝑌,𝑚𝑚
= 1 − 𝑠
𝑌
(1 − 𝑝
2
𝑋 𝛼
𝐹𝐹,𝑚𝑚
− 2𝑝
𝑋
(1 − 𝑝
𝑋
)𝛼
𝐹𝑓,𝑚𝑚
− (1 − 𝑝
𝑋
)
2 𝛼 𝑓𝑓𝑚𝑚
)
(15a)
(15b)
(15c) where 𝛼 𝑖,𝑗
is the probability of infection when a snail with FREP genotype i encounters a schistosome with Mucin genotype j.
Making identical notational changes to the equations predicting evolutionary change we derived earlier (5-6) shows that the change in the frequency of the F FREP allele that occurs over a single generation is:
∆𝑝
𝑋
= 𝑝
𝑋
(𝑝
𝑋
𝑊
𝑋,𝐹𝐹
+(1−𝑝
𝑋
)𝑊
𝑋,𝐹𝑓
𝑊
𝑋
−𝑊
𝑋
)
(16) where 𝑊
𝑋
= 𝑝
2
𝑋
𝑊
𝑋,𝐹𝐹
+ 2𝑝
𝑋
(1 − 𝑝
𝑋
)𝑊
𝑋,𝐹𝑓
+ (1 − 𝑝
𝑋
)
2
𝑊
𝑋,𝑓𝑓
, and the change in the frequency of the
M Mucin allele is:
∆𝑝
𝑌
= 𝑝
𝑌
(𝑝
𝑌
𝑊
𝑌,𝑀𝑀
+(1−𝑝
𝑌
)𝑊
𝑌,𝑀𝑚
𝑊
𝑌
−𝑊
𝑌
)
(17) where 𝑊
𝑌
= 𝑝
2
𝑌
𝑊
𝑌,𝑀𝑀
+ 2𝑝
𝑌
(1 − 𝑝
𝑌
)𝑊
𝑌,𝑀𝑚
+ (1 − 𝑝
𝑌
)
2
𝑊
𝑌,𝑚𝑚
. With these basic expressions in hand, we can now turn to our primary question: how are the dynamics and outcomes of coevolution influenced by alternative patterns of dominance?
Case 1. Production of a single monomeric Frep and Mucin
The first case we will tackle is the scenario where both snail and schistosome produce only a single, monomeric FREP and Mucin molecule, leading to the interaction matrix X we derived previously.
Substituting this interaction matrix into equations (16-17) and simplifying algebraically shows that coevolutionary change is described by:
∆𝑝
𝑋
=
1
2 𝑠
𝑋 𝑝
𝑋 𝑞
𝑋
(2𝑝
𝑌
−1)
𝑊
𝑋
∆𝑝
𝑌
=
1
2 𝑠
𝑌 𝑝
𝑌 𝑞
𝑌
(1−2𝑝
𝑋
)
𝑊
𝑌
(18a)
(18b) which, if you remember way back to Chapter 2 and our study of coevolution in haploid systems, look a lot like the equations (XX) describing coevolutionary change in a haploid matching alleles system! In order to explore whether this similarity extends to the dynamics and outcome of coevolution, we need to embark in a more formal analysis of (18). Our general approach will be identify equilibria and evaluate their local stability, a strategy we previously employed in Chapters 2 and X. As a quick
9
refresher, we can identify possible coevolutionary equilibria by setting equations (18) equal to zero, and then solving for the values of allele frequencies px and py that satisfy the equality. The result is the identification of five possible equilibria (Table 4).
Table 4. Equilibria and their local stability for the case of a single monomeric Frep and Mucin 𝑝
𝑋
0
0
1
1 𝑝
𝑌
1/2
0
1
0
1
1/2
Eigenvalues
− 𝑠
𝑋
2
, 𝑠
𝑌
2(1 − 𝑠
𝑌
) 𝑠
𝑋
2(1 − 𝑠
𝑋
)
, − 𝑠
𝑌
2 𝑠
𝑋
2(1 − 𝑠
𝑋
)
, − 𝑠
𝑌
2
− 𝑠
𝑋
2
, 𝑠
𝑌
2(1 − 𝑠
𝑌
)
√𝑠
𝑋
√𝑠
𝑌
(𝑠
𝑌
− 2)
2(2 − 𝑠
𝑌
)√2 − 𝑠
𝑋
, −
√𝑠
𝑋
√𝑠
𝑌
(𝑠
𝑌
− 2)
2(2 − 𝑠
𝑌
)√2 − 𝑠
𝑋
Stability
Unstable
Unstable
Unstable
Unstable
Cycles with increasing amplitude
If we next analyze the local stability of these equilibria just as we have before (generate a Jacobian
Matrix, calculate the eigenvalues, evaluate eigenvalues at the equilibrium of interest), we find that none of these equilibria are locally stable (Table 4). Instead, just as we found for the haploid matching alleles model, all equilibria are locally unstable with the exception of the internal polymorphic equilibrium where allele frequencies in both species are ½. This polymorphic equilibrium is characterized by eigenvalues that are purely imaginary, suggesting that we should expect coevolutionary cycles to occur around this equilibrium. Because this is a discrete time system, we would expect these cycles to increase in amplitude over time because the magnitude of the leading eigenvalue is greater than zero (See
Appendix X; Local stability). Numerical iteration of equations (18) confirms this expectation and shows that frequencies of alternative FREP and Mucin alleles cycle over time, with the amplitude increasing at a rate proportional to the fitness consequences of infection s
X
and s
Y
(Figure 2).
Case 2. Production of many monomeric Freps and Mucins
Our analysis in the preceding section showed that one possible outcome of coevolution between Schistosome and snail is cycling of alternative FREP and Mucin alleles. This analysis assumed that snail and schistosome produce only a single FREP and Mucin. What if, instead, individuals produce a large number of these immune molecules leading to the genotype specific probabilities of infection given by interaction matrix (12)? Substituting this interaction matrix into equations (16-17) and simplifying algebraically shows that coevolutionary change is described by:
∆𝑝
𝑋
=
1
2 𝑠
𝑋 𝑝
𝑋 𝑞
𝑋
(𝑝
𝑋
(1−2𝑝
𝑌 𝑞
𝑌
𝑊
𝑋
)−𝑝
2
𝑌
)
(19a)
10
∆𝑝
𝑌
=
1
2 𝑠
𝑌 𝑝
𝑌 𝑞
𝑌
(𝑝
𝑌
(1−2𝑝
𝑋 𝑞
𝑋
)−𝑝
2
𝑋
𝑊
𝑌
)
. (19b)
As was the case in the previous section, these equations are too complex for us to gather much information through a simple inspection of their terms. Instead, we must employ the more formal approach of finding their equilibria and evaluating the local stability of these equilibria. Without belaboring the details, performing the standard analyses reveals five unique equilibria with the eigenvalues, and associated stability conditions summarized in Table 5.
Table 5. Equilibria and their local stability for the case of many monomeric Freps and Mucins 𝑝
𝑋 𝑝
𝑌
0
0
1
0
1
0
Eigenvalues
0, 0 𝑠
𝑋
1 − 𝑠
𝑋
, −𝑠
𝑌 𝑠
𝑋
1 − 𝑠
𝑋
, −𝑠
𝑌
0, 0
Stability
Neutral
Unstable
Unstable
1 1 Neutral
1/2 1/2
4(𝑠
𝑌
− 𝑠
𝑋
) + 3𝑠
𝑌 𝑠
𝑋
(𝑠
𝑋
− 8)(7𝑠
𝑌
− √𝐾
− 8)
,
4(𝑠
𝑌
− 𝑠
𝑋
) + 3𝑠
𝑌 𝑠
𝑋
(𝑠
𝑋
− 8)(7𝑠
𝑌
+ √𝐾
Stable or unstable, cycles possible
− 8)
Where the compound parameter 𝐾 = 4𝑠 2
𝑋
− 56𝑠
𝑋 𝑠
𝑌
+ (4 − 3(−16 + 𝑠
𝑋
)𝑠
𝑋
)𝑠 2
𝑌
A quick look at the equilibria and stability conditions reveals that coevolution is likely to proceed quite differently now than it did when we had assumed each individual produced only a single FREP and
Mucin molecule. Specifically, we now see that the two equilibria corresponding to the fixation of matching FREP and Mucin molecules are neutrally stable rather than unstable. Also, if we investigate the eigenvalues associated with the polymorphic equilibrium numerically (Figure X), we see that although we still expect dynamics to be cyclical, this equilibrium can now be stable in a small range of parameter space (if selection on the snail is stronger than on the schistosome). Much of the reason for this difference is the shift in the pattern of dominance within the interaction matrix (X). Specifically, interaction matrix X shows that resistance in the snail population is expected to be strongly overdominant whereas infectivity in the schistosome population is expected to be weakly underdominant. This has two important consequences. First, there will be a stronger tendency for genetic variation to be mainted in host than in parasite. Second, when the parasite nears fixation of either form of mucin, it tends to get “stuck” in an evolutionary sense. This is because changing allele frequencies to the alternative form requires passing through an adaptive valley defined by low fitness heterozygous individuals.
Case 3. Production of a single dimeric Frep and Mucin
Based on our analyses of the previous two sections, it seems that the specific assumptions we make about the details of the mechanisms underpinning snail immunity to the shistosome XX have
11
important consequences for coevolution. To see if this trend holds, we need to explore what happens when snail and schistosome produce a single dimeric FREP or Mucin molecule. In such a case, the genotype specific probabilities of infection are given by interaction matrix (13). Substituting this interaction matrix into equations (16-17) and simplifying algebraically shows that coevolutionary change is described by:
∆𝑝
𝑋
=
1
4 𝑠
𝑋 𝑝
𝑋 𝑞
𝑋
((7−3𝑝
𝑌
)𝑝
𝑌
+𝑝
𝑋
(2−6𝑝
𝑌 𝑞
𝑌
)−3)
𝑊
𝑋
∆𝑝
𝑌
=
1
4 𝑠
𝑌 𝑝
𝑌 𝑞
𝑌
(2𝑝
𝑌
+2𝑝
𝑋
(2+𝑞
𝑋
(1−2𝑝
𝑌
))−3)
𝑊
𝑌 which are, to be quite honest, rather inelegant equations.
(20a)
(20b)
Table 6. Equilibria and their local stability for the case of a single dimeric Frep and Mucin 𝑝
𝑋 𝑝
𝑌
Eigenvalues Stability
0
0
1
1
0
1
0
1
−
3𝑠
𝑋
4
,
3𝑠
𝑌
4 ( 1 − 𝑠
𝑌
) 𝑠
𝑋
4(1 − 𝑠
𝑋
)
, − 𝑠
𝑌
4 𝑠
𝑋
4(1 − 𝑠
𝑋
)
, − 𝑠
𝑌
4
−
3𝑠
𝑋
4
,
3𝑠
𝑌
4(1 − 𝑠
𝑌
)
Unstable
Unstable
Unstable
Unstable
1/2 1/2
8(𝑠
𝑋
− 2𝑠
(21𝑠
𝑋
𝑌
) + 7𝑠
𝑋 𝑠
𝑌
− 32)(7𝑠
𝑌
− 2√𝐾
− 16)
,
8(𝑠
𝑋
− 2𝑠
𝑌
(21𝑠
𝑋
) + 7𝑠
𝑋
− 32)(7𝑠
𝑌 𝑠
𝑌
+ 2√𝐾
− 16)
Stable or unstable, cycles possible
Where the compound parameter 𝐾 = 16𝑠 2
𝑋
+ 56𝑠
𝑋
(−72 + 47𝑠
𝑋
)𝑠
𝑌
+ (64 + 7(240 − 161𝑠
𝑋
)𝑠
𝑋
)𝑠
𝑌
2
Case 4. Production of many dimeric Freps and Mucins
Same as previous case #X
Conclusions and Synthesis
The diversity of outcomes is striking with only slight differences in our assumptions about underlying mechanisms. Diversity is maintained or diversity is eroded. These are major and fundamental questions that remain quite open within the field. These observations also underscore the importance of GENETICALLY EXPLICIT MODELS IN MANY CASES.
12
ONE OF THE GREATEST STANDING CHALLENGES IN COEVOLUTIONARY GENETICS IS UNDERSTANDING
PATTERNS OF DOMINANCE.
XXXXXXXXXXXXXs such that individuals with genotype AA make only FREP molecules with phenotype
“A”, heterozygous Aa individuals make FREP molecules with phenotype “A” and also FREP molecules with phenotype “A”, and individuals with genotype aa make phenotype “a”. If
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 1 𝛼 = [ 0 0 1 ]
0 0 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.”
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
13
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
14
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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