`Biological Motivation Key Questions Building a Model of XX
`Biological Motivation
Key Questions
Building a Model of XX-XX interactions
Analyzing the Model
Answers to Key Questions
New Questions Arising
Extensions and Generalizations
Chapter 3. Coevolution of major genes
Biological Motivation
VARIATION EXISTS IN INFECTIVITY WITHIN AND AMONG POPULATIONS. GENETIC
POLYMORPHISM IS OFTEN PRESENT. MOTIVATE WITH FLAX AND FLAX RUST.
Melampsora lini and Linum marginale
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 Model of Flax-Rust interactions
Studies of the interaction between flax and rust date back to Flor’s early work investigating the genetic basis of resistance to the rust fungus, Melampsora lini, in the domestic flax, Linum usitatissimum
(REFS). These early studies demonstrated that there was genetic specificity between flax and rust genotypes such that resistance and infectivity did not result solely from flax genotype or rust genotype alone, but rather their interaction (GxG). This work led to the development of the Gene-for-gene model
— one of the most widely used and applied models of coevolution to date. We now know that the basis of the gene-for-gene model is most likely the interaction between receptors encoded by plant R genes which can recognize and mount an immune defense against surface proteins encoded by pathogen Avr genes but not against surface proteins encoded by pathogen Vir genes (REFS). In contrast, plants carrying r genes fail to recognize and mount an immune response against either Avr or Vir pathogens
(Figure 1: INTRODUCE INTERACTION MATRIX ALPHA HERE!). This type of gene for gene model is thought to underpin the interaction between Wild flax, L. marginale, and flax rust, M. lini, introduced at the beginning of this chapter.
Although we now know it is a simplification of the real-world genetics underlying the interaction between L. marginale, and flax rust, M. lini, the simple single-locus, haploid, model illustrated in Figure 1 presents a logical point to begin exploring coevolution in this system. The key to moving forward is to translate the simple infection matrix shown in Figure 1 into a system of equations which describe coevolutionary change in rust and flax. Our starting point is to define variables corresponding to the frequency of the various alleles shown in Figure 1. Specifically, we will define the frequency of the R gene in the wild flax (species X) population as p
X
and the frequency of the Vir gene in the rust population
(species Y) as p
Y
. Because we are assuming each locus is diallelic, the frequencies of the alternate alleles are simply (1-p
Y
) and (1-p
Y
) in flax and rust, respectively. Obviously, what we need now is some way to predict how these allele frequencies will evolve in response to natural selection imposed by the interaction.
If we imagine for the time being that the only evolutionary force acting on allele frequencies is natural selection, we can study how allele frequencies change over evolutionary time by simply counting up numbers of the two alleles before and after selection. Specifically, let’s imagine that prior to natural selection there are N
X,R
flax individuals carrying the resistant R gene and N
X,r flax individuals carrying the susceptible r gene. In addition, let’s assume that the number of offspring produced by each resistant flax individual is W
X,R
and the number of offspring produced by each susceptible flax individual W
X,r
. If we make the additional assumption that the flax is an annual with no adult survivorship, the number of resistant individuals in the next generation is:
𝑁
′
𝑋,𝑅
= 𝑁
𝑋,𝑅
𝑊
𝑋,𝑅
= 𝑝
𝑋
𝑁
𝑋
𝑊
𝑋,𝑅
(1a) the number of susceptible individuals is:
𝑁
′
𝑋,𝑟
= 𝑁
𝑋,𝑟
𝑊
𝑋,𝑟
= (1 − 𝑝
𝑋
)𝑁
𝑋
𝑊
𝑋,𝑅
(1b) and the total number of individuals is:
𝑁
′
𝑋
= 𝑝
𝑋
𝑁
𝑋
𝑊
𝑋,𝑅
+ (1 − 𝑝
𝑋
)𝑁
𝑋
𝑊
𝑋,𝑅
= 𝑁
𝑋
𝑊
𝑋
(1c) where 𝑊
𝑋
is the average number of offspring produced by each flax individual, or the population mean
fitness.
Since we now know how many flax individuals will be carrying each allele and the total number of flax individuals, calculating the new allele frequency of the resistant R allele in the next generation is straightforward: 𝑝
′
𝑋
=
𝑁
′
𝑋,𝑅
𝑁
′
𝑋
= 𝑝
𝑋
𝑁
𝑋
𝑊
𝑋,𝑅
𝑁
𝑋
𝑊
𝑋
= 𝑝
𝑋
𝑊
𝑋,𝑅
𝑊
𝑋
(2)
If we now subtract the frequency of the resistant R allele within the previous generation, 𝑝
𝑋
, from
Equation (2) we arrive at an expression for evolutionary change over a single generation:
∆𝑝
𝑋
= 𝑝 ′
𝑋
− 𝑝
𝑋
= 𝑝
𝑋
𝑊
𝑋,𝑅
𝑊
𝑋
− 𝑝
𝑋
= 𝑝
𝑋
(𝑊
𝑋,𝑅
−𝑊
𝑋
)
𝑊
𝑋
(3a)
An identical sequence of mathematical operations yields a parallel expression for the frequency of the virulent V gene in the rust:
∆𝑝
𝑌
= 𝑝
′
𝑌
− 𝑝
𝑌
= 𝑝
𝑌
𝑊
𝑌,𝑉
𝑊
𝑌
− 𝑝
𝑌
= 𝑝
𝑌
(𝑊
𝑌,𝑉
−𝑊
𝑌
)
𝑊
𝑌
(3b)
Where 𝑊
𝑌,𝑉
is the fitness of the rust Vir allele, and 𝑊
𝑌
is the population mean fitness of the rust population. 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 alleles.
How might the interaction between flax and rust influence the fitness of each organism? It seems clear from empirical studies that infection by rust reduces the fitness of wild flax (REFS). Similarly, it seems obvious that infection of a host should, at the very least, increase the fitness of the rust. How can we translate these simple ideas about the relationship between infection and fitness into mathematical expressions for fitness? An obvious possibility, and one that is widely employed in coevolutionary modeling, is to assume that each individual encounters one randomly selected individual of the interacting species within each generation. If this is the case, and species interactions are the only source of fitness variation, the expected fitness of a flax individual carrying the resistant R gene is:
𝑊
𝑋,𝑅
= 1 − 𝑠
𝑋
(𝛼
𝑅,𝑉 𝑝
𝑌
+ 𝛼
𝑅,𝑣
(1 − 𝑝
𝑌
)) (4a)
In equation (4), 𝑠
𝑋
measures the fitness reduction caused by rust infection and 𝛼 𝑖,𝑗
the probability that a host with allele i is infected by a pathogen carrying allele j. Biologically, the quantity in parentheses is the probability that an individual flax carrying the R allele is infected, and the parameter 𝑠
𝑋
the reduction in fitness an infected individual experiences. Following similar logic allows us to write a parallel expression for the fitness of the r allele in the flax population:
𝑊
𝑋,𝑟
= 1 − 𝑠
𝑋
(𝛼 𝑟,𝑉 𝑝
𝑌
+ 𝛼 𝑟,𝑣
(1 − 𝑝
𝑌
)) (4b)
Making identical assumptions but recognizing that infection should increase the fitness of individual rusts yields the following expressions for fitness of rust alleles:
𝑊
𝑌,𝑉
= 1 + 𝑠
𝑌
(𝛼
𝑅,𝑉 𝑝
𝑋
+ 𝛼 𝑟,𝑉
(1 − 𝑝
𝑋
)) (4c)
𝑊
𝑌,𝑣
= 1 + 𝑠
𝑌
(𝛼
𝑅,𝑣 𝑝
𝑋
+ 𝛼 𝑟,𝑣
(1 − 𝑝
𝑋
)) (4b) where 𝑠
𝑌
measures the fitness increase infection provides to an individual rust. With fitness functions for flax and rust alleles now specified, we can tackle the challenge of predicting coevolution between flax and rust.
Developing mathematical expressions for coevolutionary change requires only that we substitute fitness functions (4) into evolutionary recursions (3) and replace the general interaction matrix entries 𝛼 𝑖,𝑗
with the values appropriate for the gene-for-gene model shown in Figure 1. Making these substitutions and simplifying the resulting expressions algebraically yields the following expressions for the change in the frequency of flax R and rust Vir alleles:
∆𝑝
𝑋
= 𝑠
𝑋 𝑝
𝑋 𝑞
𝑋
(1−2𝑝
𝑌
𝑊
𝑋
)
∆𝑝
𝑌
= 𝑠
𝑌 𝑝
𝑌 𝑞
𝑌
(1−2𝑝
𝑋
)
𝑊
𝑌
(5a)
(5b) where 𝑞
𝑋
= (1 − 𝑝 respectively.
𝑋
) and 𝑞
𝑌
= (1 − 𝑝
𝑌
) are the frequencies of flax r and pathogen v alleles,
Analyzing the Model
Now that we have derived expressions for coevolutionary change in allele frequencies our goal is to understand what these equations can tell us about the dynamics and outcome of coevolution between Flax and Flax-rust. A logical starting point is to dissect equations (5) and try to understand their component pieces biologically. For instance, why does flax evolution take the particular form of equation (5a)? The simplest way to do this is by breaking the numerator into two distinct pieces: genetic variation ( 𝑝
𝑋 𝑞
𝑋
) and selection ( 𝑠
𝑋 𝑞
𝑌
). The term 𝑝
𝑋 𝑞
𝑋
quantifies genetic variability in the flax population and is maximized when the frequency of the two alternate alleles is equal. The term ( 𝑠
𝑋 𝑞
𝑌
) measures selection and increases with the fitness consequences of infection ( 𝑠
𝑋
) , and the frequency of the v allele
( 𝑞
𝑌
) in the rust population. The reason selection on the host increases with the frequency of the v allele in the rust is that it is this allele, and this allele only, which causes the two host alleles to have differential probabilities of infection (Figure 1). The reason rust evolution takes the form of equation
(5b) is equally simple and can be understood by again breaking the equation down into a component which measures genetic variation ( 𝑝
𝑌 𝑞
𝑌
) and a component which measures selection ( 𝑠
𝑌 𝑝
𝑋
). As was the case for flax, rust genetic variation is maximized when the frequencies of the two alternate alleles are equal. Selection on the rust population increases with the fitness benefits of infection ( 𝑠
𝑌
), and with the frequency of the, R, allele ( 𝑝
𝑋
) in the Flax population. Selection becomes more intense on the rust population as the frequency of the Flax R allele increases because it is this allele, and this allele only, which causes differential infection success for the alternate forms of the rust alleles (Figure 1).
Hopefully, the discussion in the preceding paragraph provided some insight into the reasons equations (5) look as they do from the biological perspective of the interaction between flax and rust. In addition to facilitating the biological interpretation of the equations, these insights allow us to make very general predictions about the dynamics of flax-rust coevolution without doing anything more than
“inspecting” equations (5). How does one go about drawing conclusions by implementing this mysterious notion of “inspecting” an equation? In the simplest terms, inspection proceeds by stepping through the pieces of the equation to form logical conclusions based on the properties of the component pieces. For instance, we might begin to draw conclusions about equations (5) by first recognizing that because the terms 𝑝
𝑋 𝑞
𝑋
and 𝑝
𝑌 𝑞
𝑌
are variances they must always be positive. In addition, the terms measuring selection 𝑠
𝑋 𝑞
𝑌 and 𝑠
𝑌 𝑝
𝑋
must also be positive because 𝑠
𝑋
and 𝑠
𝑌
are positive by definition, as are the allele frequencies 𝑝
𝑋
and 𝑞
𝑌
. Combining these simple observations with the fact that population mean fitness 𝑊 must also be positive (by definition), reveals that coevolution must always proceed in a very predictable way. Specifically, the frequency of the host R allele must increase until host genetic variance is exhausted or until the rust population becomes fixed for the virulent allele. This can be seen by noting that the sign of (5a) must always be positive and non-zero as long as both R and r alleles segregate ( 𝑝
𝑋 𝑞
𝑋
> 0 ) and the frequency of the Vir allele is greater than zero
( 𝑞
𝑌
> 0 ). In contrast, the frequency of the rust V allele must increase until rust genetic variation is exhausted, irrespective of flax allele frequency. This can be seen by recognizing that the sign of (5b) must be positive and non-zero as long as both V and v alleles segregate ( 𝑝
𝑌 𝑞
𝑌 is not absent ( 𝑝
𝑋
> 0 ) and the host R allele
> 0 ). Consequently, we expect coevolution to cause fixation of the pathogen V allele and increases in the frequency of the host R allele, but not necessarily fixation. Examples of actual coevolutionary dynamics predicted by (5) are shown in Figure 2 for various parameter combinations.
THE OUTCOME OF THIS IS PERFECT INFECTIVITY (ADD PINFECT LINE TO FIGURE 2). With our mathematical analyses complete, we can begin to answer the questions we posed earlier in this chapter.
Answers to Key Questions
Does coevolution maintain genetic polymorphism?
Our analysis of gene-for-gene coevolution motivated by the interaction between wild flax and flax rust suggests that coevolution should not maintain genetic polymorphism and will instead erode polymorphism within both flax and flax rust. Specifically, equations (5) demonstrate that the frequency of the Flax R allele will increase until it reaches fixation or the Flax-rust becomes fixed for the Vir allele
(Figure 2). Even if the Rust Vir allele becomes fixed while Flax R and r still segregate, coevolution itself plays no role in the maintenance of this polymorphism and it would ultimately be eroded by the action of random genetic drift. For the rust, the erosion of polymorphism is even more aggressive, with the frequency of the Rust V allele increasing until it reaches fixation; there is no possibility of polymorphism for the rust, neutral or otherwise.
Can coevolution explain variation in infectivity within a population?
The results of our mathematical analysis suggest the ultimate outcome of coevolution is a population of hosts which is uniformly susceptible. Although transient variation in infectivity and resistance may be maintained for some period of time, ultimately the pathogen population will become fixed for the Vir allele. Because the Vir allele can infect all possible Flax genotypes, the result is uniform susceptibility in the host (Figure 2). Thus, our simple model gene-for-gene model suggests coevolution can maintain variation in infectivity within populations only over relatively short time-scales.
Should coevolution cause infection rates to differ among populations?
Although our model does not include any explicit spatial structure, it can be used to gain some insights into how coevolution might proceed within a very simple metapopulation where no gene flow occurs among populations. If all populations begin with the same allele frequencies in Flax and Flax-
Rust, and the values of the coefficients 𝑠
𝑋
and 𝑠
𝑌
are identical in all populations, equations (5) tell us that allele frequencies will be identical in all populations such that no spatial variation exists in rates of infection. If, however, initial allele frequencies or the coefficients 𝑠
𝑋
and 𝑠
𝑌
differ among populations, this is no longer the case, and we expect to find some populations with much higher rates of infection than others (FIGURE 3). However, given a sufficient amount of time, coevolution will ultimately cause the fixation of the Vir allele within all populations causing all populations of Flax to be entirely susceptible to the Rust. Thus, our model can explain transient, but not permanent, variation in infection rates among populations.
New Questions Arising:
Our simple model of gene-for-gene coevolution between Flax and Flax-Rust suggests that genetic polymorphism and variation in rates of infection within and among populations occur only transiently and are ultimately eroded by coevolution. Although thought provoking, the tenuous connection between this prediction and available empirical data immediately raises several important questions:
Do other forms of interaction matrix facilitate the maintenance of genetic polymorphism and variation in infectivity and resistance?
Can integrating costs of resistance and infectivity allow the gene-for-gene model to maintain genetic polymorphism and variation in infectivity and resistance?
Are other forms of ecological interaction, such as mutualism, more or less likely to maintain genetic polymorphism?
In the next three sections, we will develop generalizations of our simple model which allow us to answer these questions and gain further insight into the process of coevolution.
Generalizations
Generalization 1: Interactions with alternative interaction matrices
Although empirical evidence suggests the interaction between Flax and Flax-Rust is mediated by a gene-for-gene interaction matrix, alternative forms of the interaction matrix have been proposed and much debate has centered on which of the possible interaction matrices is “correct” (REFS). Almost without doubt, the most popular alternative to the gene-for-gene interaction matrix is the matching allele interaction matrix shown in Figure 4. Although empirical support for a Matching Alleles matrix has been mixed, it is built upon a sound mechanistic model of molecular interaction (REF) and has been implicated in some naturally occurring interactions such as those between Daphnia and Pasteuria ramosa
(Luijckx et al. 2013). How should the outcome and dynamics of coevolution differ for this
alternative model of interaction? In the remainder of this section, we will work to address this question by integrating a matching alleles matrix into the coevolutionary model we developed previously. We will then use this model to explore the capacity of the Matching Alleles model to maintain genetic polymorphism and variation in infection rates within and among populations.
To keep things grounded in biology, let’s switch from the interaction between Flax and Flax-Rust to that between Daphnia magna and its parasitic bacterium Pasteuria ramosa which at least some
simple, we will assume the D. Magna population has two possible alleles, A and a with frequencies p
X and (1- p
X
) and the P. Ramosa population also has two possible alleles, B and b with frequencies p
Y
and
(1- p
Y
). The outcome of interactions between individual D. Magna and individual P. Ramosa will be
determined by a matching alleles interaction matrix as shown in Figure 4. All we need to do now to develop mathematical expressions for coevolutionary change in this system is substitute fitness functions (4) into evolutionary recursions (3) and replace the general interaction matrix entries 𝛼 𝑖,𝑗
with the values appropriate for the matching allele model shown in Figure 4.
Making these substitutions and simplifying the resulting expressions algebraically yields the following expressions for coevolutionary change in the frequency of D. Magna and P. Ramosa A and B alleles:
∆𝑝
𝑋
= 𝑠
𝑋 𝑝
𝑋 𝑞
𝑋
(1−2𝑝 𝑦
)
𝑊
𝑋
∆𝑝
𝑌
= 𝑠
𝑌 𝑝
𝑌 𝑞
𝑌
(1−2𝑝
𝑋
)
𝑊
𝑌
(6a)
(6b) where 𝑞
𝑋
= (1 − 𝑝
𝑋
) and 𝑞
𝑌
= (1 − 𝑝
𝑌
) are the frequencies of D. Magna a and P. Ramosa b alleles, respectively. If we take some time and inspect equations (6) carefully we see that things are more complicated than for the case of gene-for-gene coevolution we studied previously. Specifically, equations (6) have terms of the form (1-2p) which were absent for gene-for-gene coevolution. These terms complicate things because their sign is positive when the allele frequency of the interacting species is less than 1/2, but negative when the allele frequency of the interacting species is greater than
1/2 (FIGURE 5). Although this extra complexity frustrates simple conclusions drawn from inspection, it arises because of the richer spectrum of coevolutionary dynamics possible within the matching alleles model.
Because simple inspection is now insufficient, we must implement a more formal mathematical analysis if we are to predict how D. Magna and P. Ramosa are likely to coevolve. A logical starting point for such an analysis is to identify coevolutionary equilibria and evaluate their local stability. Biologically, such an analysis will allow us to identify those combinations of allele frequencies in D. Magna and P.
ramosa where coevolution is likely to cause populations to spend considerable time; it will also allow us to predict coevolutionary dynamics near these combinations of allele frequencies. Formally, we can identify coevolutionary equilibria by setting equations (6) equal to zero and solving for the values of allele frequencies p
X
and p
Y
which satisfy the resulting equality. The resulting equilibria, and the average infection rate for each, are shown in Table 1. Looking through the equilibria in Table 1 shows that there are three different types of possible equilibria. The first type occurs when both D. Magna and P. Ramosa are fixed for matching alleles, either A and B (p
X
= p
Y
= 1) or a and b (p
X
= p
Y
= 0). At these equilibria, the pathogen and host populations are entirely compatible and all hosts are susceptible to all pathogens.
The second type of equilibrium occurs when D. Magna and P. Ramosa are fixed for mismatching alleles, either A and b (p
X
= 1, p
Y
= 0) or a and B (p
X
= 0, p
Y
= 1). At these equilibria, the pathogen and host populations are entirely incompatible and all hosts are resistant to all pathogens. The final type of equilibrium occurs when D. Magna and P. Ramosa populations are each composed of exactly ½ A and B alleles and ½ a and b alleles, respectively. At this equilibrium, hosts are resistant to some pathogens but not others, and half of all random encounters between host and pathogen lead to infection and the other half to resistance. Given this diversity of possible equilibria and their very different consequences
for the interacting species, it becomes imperative to identify when each of these different types of equilibria might coevolve.
Identifying when each of the equilibria in Table 1 might evolve requires that we perform a local
stability analysis. Such an analysis allows us to investigate how coevolution proceeds near each of the various equilibria. With this information in hand, we can make some educated predictions about how the overall process of coevolution is likely to proceed in the interaction between D. Magna and P.
Ramosa. Formally, analyzing the local stability of the equilibria in Table 1 requires that we calculate the
Jacobian matrix for the system of equations (6):
𝐽 =
[
𝜕∆𝑝
𝑋
𝜕𝑝
𝑋
𝜕∆𝑝
𝑌
𝜕𝑝
𝑋
𝜕∆𝑝
𝜕𝑝
𝑌
𝑋
𝜕𝑝
𝑌
𝜕∆𝑝
𝑌
] and then calculate the eigenvalues of this Jacobian matrix for the various equilibrium values of p
X
and p
Y
.
The full set of equilibria and their associated eigenvalues are shown in Table 2 and derived step by step in the accompanying Mathematica notebook. For a complete and general discussion of local stability analyses, the wonderful book by Otto and Day is highly recommended (REFERENCE).
We now have mathematical expressions for coevolutionary equilibria and their associated eigenvalues. How can we use this information to learn something practical about coevolution between
D. Magna and P. Ramosa? We can begin by inspecting the eigenvalues associated with the matching equilibria shown in the first two rows of Table 2. Because we defined the fitness costs of the interaction s
X
and s
Y
to be on the interval [0 and 1], it is easy to see that the leading (largest) eigenvalue for these matching equilibria will always be a positive real number. Thus, these matching equilibria are locally unstable and host and pathogen allele frequencies will coevolve away from perfect matching. We can next inspect the eigenvalues associated with the mismatching equilibria shown in the third and fourth rows of Table 2. Here too, it is easy to see that the leading eigenvalue will always be positive indicating that the mismatching equilibria will also be locally unstable such that host and pathogen coevolve away from perfect mismatching. With a little thought, these results make perfect sense from a biological standpoint. Basically, the matching equilibria represent a scenario where the pathogen has “won” and can infect all hosts. Of course, under such conditions, any new host with a novel allele unrecognizable to the monomorphic pathogen population will rapidly invade, driving the host population away from this unfavorable equilibrium. Similarly, the mismatching equilibria represent a scenario where the host population has “won” and can resist all pathogens. Here too, it is clear that the alternative (and currently absent) pathogen allele, if introduced, would rapidly invade and increase in frequency because it can infect all available hosts. Together, these local stability analyses of the matching and mismatching equilibria suggest that neither the host nor pathogen can “win” when the interaction is mediated by a matching alleles mechanism as might be the case for D. Magna and P. Ramosa. If neither species can win, what do we expect the dynamics and outcome of coevolution to be? To answer this question, we need to analyze the local stability of the final equilibrium.
Unlike the first two types of equilibria, the final equilibrium is characterized by eigenvalues which are purely imaginary (because the denominator is a square root of a negative number). When the leading eigenvalue is purely imaginary, allele frequencies will oscillate with amplitude that increases over time. Thus, near this final equilibrium, where both species are composed of an equitable mix of both types of alleles, we expect allele frequencies to cycle in both species resulting in fluctuating rates of infection over time (Figure 6). This simple result has motivated and informed much work on the Red
Queen Theory for the evolution of sex, and much work on parasite local adaptation (REFS).
Because our system is characterized by two variables (p
X
and p
Y
) each possible equilibrium is associated with two eigenvalues, with the local stability of the system determined by the largest (REFERENCE TO
OTTO AND DAY).
Generalization 2: Integrating costs of infectivity and resistance
The previous section revealed that an alternative mechanism of infection and resistance — the matching alleles model — was very conducive to maintaining genetic variation and thus appears quite consistent with much empirical data. At the same time, however, the gene-for gene model seems quite well supported in some sustems suggesting there may perhaps be a way for this model to be reconciled with empirical data. One possibility, and one which receives at least some empirical support, is that resistance and virulence are associated with costs which promote the maintenance of genetic variation.
Generalization 3: Alternative forms of ecological interaction
Generalization #1: Alternative infection matrices
Sadly, equation X is significantly more complicated than equations (5) and it is not possible to reach any meanginful conclusions by simply inspecting the equations. Instead, we must reach into our mathematical toolbox and apply more formal approaches to gain biological insight. A first step is to identify Equilibria.
Luijckx, P., H. Fienberg, D. Duneau, and D. Ebert. 2013. A matching-allele model explains host resistance to parasites. Current biology : CB 23:1085-1088.
Luijckx, P., H. Fienberg, D. Duneau, and D. Ebert. 2013. A matching-allele model explains host resistance to parasites. Current biology : CB 23:1085-1088.
