Chapter 8. Coevolution, major genes, and spatial patterns of genetic variation Biological Motivation IN THIS CHAPTER WE EXPLORE WHEN COEVOLUTION GENERATES SPATIAL PATTERNS. Focused on very simple models that illustrate how we can include spatial structure START WITH MUTUALISM AND POINT OUT HOW WEIRD IT IS THAT THERE IS VARIATION WITHIN POPS Until now we have focused on interactions in single pops. We know however, that spatial structure matters… Let’s look at the snails and trematodes again… One way the importance of space has been discerened is by conducting a reciprocal cross infection experiment… These experiments have revealed very large levels of local adaptation. What is the underlying cause of these patterns? Key Questions: ο· ο· ο· Can spatial structure maintain genetic polymorphism within coevolving mutualistic populations? Can coevolution generate genetic differentiation among populations? How do coevolutionary selection and gene flow interact to shape rates of mutualistic interaction within and among populations? Building a model of spatially structured coevolution When we first developed a model of coevolution between M. truncatula and S. meliloti in Chapter 2, we assumed that the two species interacted within a single, isolated population. In order to study how spatial structure influences the maintenance of genetic variation within and between populations, we will obviously need to relax this assumption. Although it might be tempting to relax this assumption by expanding our model to include a large number of populations and complex patterns of gene flow, this would preclude the possibility of mathematical analysis. Instead, we will focus on developing the simplest possible model that allows us to understand spatially structured coevolution: a pair of coevolving populations connected by gene flow (Figure 1). Within each population, we will assume that M. truncatula and S. meliloti encounter one another at random and that their populations are sufficiently large that the impacts of random genetic drift can be ignored. We will also restrict ourselves to the case we studied previously in Chapter 2 where both of the interacting species are haploid and the interaction is mediated by a single diallelic locus in each species. With these assumptions, results we developed in Chapter 2 show that the frequency of the A allele within the M. truncatula population after a single round of coevolutionary selection is given by: Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm ′ ππ,π = ππ,π ππ,π΄,π Μ π,π π (1a) and the frequency of the B allele within the S. meliloti population is given by: ′ ππ,π = ππ,π ππ,π΅,π Μ π,π π (1b) where ππ,π is the frequency of the A allele in M. truncatula within population i, ππ,π is the frequency of the B allele in S. meliloti within population i, ππ,π΄,π is the fitness of the A allele in population i, ππ,π΅,π is Μ π,π is the mean fitness of M. truncatula in population i, and the fitness of the B allele in population i, π Μ π,π is the mean fitness of S. meliloti in population i. Our next challenge in building our model of π spatially structured coevolution is to figure out how gene flow between populations changes the frequencies of the A and B alleles. If we assume that individuals move between the two populations at random with a rate of mX in M. truncatula and mY in S. meliloti we can write down expressions for the allele frequencies within each population after gene flow occurs. The key to making progress is to focus on the proportion of each population that is made up of resident individuals (1-m) and the proportion of each population made up of immigrant individuals (m). Keeping this in mind, we can now calculate the allele frequency of each species within each population after gene flow occurs by calculating the weighted average allele frequency of residents and immigrants: ′′ ′ ′ (1 − ππ ) + ππ,π (ππ ) ππ,π = ππ,π (2a) ′′ ′ ′ (1 − ππ ) + ππ,π (ππ ) ππ,π = ππ,π (2b) where the double prime notation indicates the frequency of the allele after both selection and migration. In equations (2) the first term indicates the contribution to allele frequency within population i made by individuals that did not leave and the second term indicates the contribution made by individuals that arrived as immigrants from population j. Substituting (1) into (2) yields expressions for the allele frequencies of each species within each population in the next generation: ′′ ππ,π = ππ,π ππ,π΄,π (1 − Μ π,π π ππ ) + ππ,π ππ,π΄,π (ππ ) Μ π,π π (3a) ′′ ππ,π = ππ,π ππ,π΅,π (1 − Μ π,π π ππ ) + ππ,π ππ,π΅,π (ππ ) Μ π,π π (3b) Together, equations (3) provide a very general description of coevolution within two coupled habitat patches. To make further progress in understanding spatially structured coevolution, however, we will need to get a bit more specific about the relationship between species interactions and fitness. As we have seen in previous chapters, there are very many different assumptions we can make about the relationship between genotypes and fitness in coevolutionary models. Of central importance is what we assume about the relationship between genotypes and the outcome of encounters between individuals captured in the interaction matrix, α. As we discussed in Chapter 2, the genetic details surrounding the interactions between M. truncatula and S. meliloti are not entirely clear. Some evidence points to a GFG type model? While other points to a matching alleles type (REFS). To keep things consistent with Chapter 2, we will assume the interaction is mediated by a matching alleles type mechanism such that the interaction matrix takes the following form: 1 πΌ=( 0 0 ) 1 (4) where M. truncatula genotypes are in ROWS? and S. meliloti genotypes are in columns? and 1’s indicate an encounter leads to successful interaction and nodulation and 0’s indicate a failed interaction and an absence of nodulation. Also in keeping with Chapter 2, we will assume the interaction is a strict mutualism such that both species receive fitness benefits π π and π π when an encounter leads to a successful interaction (e.g., nodulation). With these assumptions, the expected fitness of M. truncatula A and a genotypes is: ππ,π΄,π = 1 + π π (πΌπ΄,π΅ ππ,π + πΌπ΄,π (1 − ππ,π )) = 1 + π π (ππ,π ) (5a) ππ,π,π = 1 + π π (πΌπ,π΅ ππ,π + πΌπ,π (1 − ππ,π )) = 1 + π π (1 − ππ,π ) (5b) and the expected fitness of S. meliloti B and b genotypes is: ππ,π΅,π = 1 + π π (πΌπ΄,π΅ ππ,π + πΌπ,π΅ (1 − ππ,π )) = 1 + π π (ππ,π ) (5c) ππ,π,π = 1 + π π (πΌπ΄,π ππ,π + πΌπ,π (1 − ππ,π )) = 1 + π π (1 − ππ,π ) (5b) where the quantity in parentheses is the probability of a successful interaction. Substituting (5) into (3) yields the following set of recursion equations that completely describes the process of coevolution in our two population system: ′′ ππ,π = (1−ππ )ππ,π (1+π π ππ,π ) 1+π π (ππ,π −(1−2ππ,π )ππ,π ) (1−ππ )ππ,π (1+π π ππ,π ) ′′ ππ,π = 1+π π (ππ,π −ππ,π (1−2ππ,π )) + ππ ππ,π (1+ππ,π π π ) 1+π π (ππ,π −(1−2ππ,π )ππ,π ) ππ ππ,π (1+ππ,π π π ) + 1+(π π,π −(1−2ππ,π )ππ,π )π π (6a) (6b) where the first term in each expression represents the post-selection frequency of resident individuals and the second term the post-selection frequency of immigrant individuals. Although coevolutionary recursions (6) completely describe the process of coevolution in our two population system, it probably comes as little surprise that without further simplification they are largely intractable. A first step in simplifying the problem is to employ a change of variables that allows us to more easily identify the conditions under which coevolution generates and maintains spatial genetic differentiation. Specifically, we will define the following new variables: πΜ π = πΜ π = ππ,1 +ππ,2 2 ππ,1 +ππ,2 2 (7a) (7b) πΏπ = ππ,1 − ππ,2 (7c) πΏπ = ππ,1 − ππ,2 (7d) where πΜ π and πΜ π are the average frequency of the A and B alleles across both populations for M. truncatula and S. meliloti respectively and πΏπ and πΏπ are the difference in the frequency of the A and B alleles across both populations for M. truncatula and S. meliloti respectively. Thus, the new variables πΏπ and πΏπ explicitly quantify the amount of spatial genetic differentiation that exists for each of the coevolving species. Using definitions (7), we can write down recursions for these new variables: πΜ π′′ = ′′ ′′ ππ,1 +ππ,2 2 ′′ ′′ πΏπ′′ = ππ,1 − ππ,2 πΜ π′′ = ′′ ′′ ππ,1 +ππ,2 2 ′′ ′′ πΏπ′′ = ππ,1 − ππ,2 (8a) (8b) (8c) (8d) and complete are change of variables by making the following substitutions on the right hand side of the expressions: ππ,1 = πΜ π + πΏπ /2, ππ,2 = πΜ π − πΏπ /2, ππ,1 = πΜ π + πΏπ /2, and ππ,2 = πΜ π − πΏπ /2. We now have a set of recursions for our new variables written entirely in terms of the new variables. Our next challenge is to learn something from these new expressions we have created. Even with the change of variables, the equations (X) describing spatially structured coevolution remain challenging to analyze. It can be done, but it isn’t easy and it isn’t terribly transparent. Instead, we will make one final set of assumptions that greatly simplifies our exploration of coevolution. Specifically, we will assume the fitness consequences of species interactions are relatively small such that π π and π π are both of small order π, and that rates of gene flow are also small such that ππ and ππ are also of small order π. With these assumptions, we can use Taylor Series to approximate equations (8), ignoring all terms of πͺ(π 2 ) and smaller. Implementing this approximation and changing from recursion equations to difference equations yields the following set of more manageable equations: 1 βπΜ π ≈ 4 π π ((1 − 2πΜ π )(πΏπ2 − 4πΜ π πΜ π ) + 2(1 − 2πΜ π )πΏπ πΏπ ) + πͺ(π 2 ) 1 2 βπΏπ ≈ (−π π (2(1 − 2πΜ π )(1 − 2πΜ π )πΏπ + (πΏπ2 − 4πΜ π πΜ π )πΏπ )) − 2ππ πΏπ + πͺ(π 2 ) 1 βπΜ π ≈ 4 π π ((1 − 2πΜ π )(πΏπ2 − 4πΜ π πΜ π ) + 2(1 − 2πΜ π )πΏπ πΏπ ) + πͺ(π 2 ) 1 βπΏπ ≈ 2 (−π π (2(1 − 2πΜ π )(1 − 2πΜ π )πΏπ + (πΏπ2 − 4πΜ π πΜ π )πΏπ )) − 2ππ πΏπ + πͺ(π 2 ) (9a) (9b) (9c) (9d) With our change of variables complete and our key assumptions implemented, we can finally move on to trying to learn something about coevolution in spatially structured environments. Analyzing the Model A logical first step in analyzing our model is to identify equilibria. Unfortunately, even after all of our clever changes and approximations it is not possible (for me anyway!) to find equilibrium solutions using the systematic approach we used previously. Instead, we need to approach the problem in a different way that relies on making educated guesses using a combination of inspection, simple algebra, and biological insight. The most obvious and (boring) equilibria we can identify using simple inspection are the four cases where both M. truncatula and S. meliloti are fixed for a particular allele in both populations such that spatial genetic structure is absent for both species (Table 1; Equilibria 1-4). We know these must be equilibria because absent genetic variation within a population there can be no response to selection and since both populations are fixed for the same allele, gene flow too, has no consequences for evolution. The fact that these are, indeed, equilibria can be verified by substituting our guesses for the equilibrium values of the variables into (7) and verifying that the right hand sides all equal zero. So far, the equilibria we have been able to identify have been fairly obvious and not very interesting because they preclude genetic variation within and among populations and thus cannot possibly explain the patterns of variability we observe in the mutualistic interaction between M. truncatula and S. meliloti. How could we go about identifying more interesting equilibria that allow for the maintenance of genetic variation? The first step we will take is to simply propose equilibria where the spatial average allele frequencies πΜ π and πΜ π are equal to ½. The next step is to solve for the values of πΏπ and πΏπ given our proposed values of πΜ π = 1/2 and πΜ π = 1/2, yielding five candidate equilibria (Table 1; Equilibria 5-9). Finally, we substitute these candidate equilibria back into the difference equations (7) and verify that they are indeed equilibria (they are!). We now have an interesting set of equilibria that may help us to shed light on spatial patterns of genetic variation in mutualistic interactions. iSAY MORE ABOUT THESE EQUILIBRIA. POINT OUT THAT ONLY THE FINAL FOUR ARE SPATIALLY VARIABLE Identifying equilibria that correspond to biologically interesting scenarios of spatial variation is an important step, but in order to proceed we need to evaluate when (or even if) these equilibria exist. Although finding existence conditions for equilibria sounds rather technical and dull, don’t despair, the results are almost certain to provide us with interesting insights into the biology of mutualism! Our general approach to identifying the conditions under which each of the spatially variable equilibria (Table 1; 6-9) exist is to find conditions on the parameters guaranteeing that πΏπ and πΏπ lie between -1 and 1. If these variables do not lie within this interval, the equilibrium cannot possibly exist because it predicts allele frequencies that are either less than zero or greater than 1. Solving these inequalities shows that the first two spatially variable equilibria (Table 1; 6-7) never exist for mutualistic interactions because πΏπ and πΏπ cannot both lie within the required interval {-1, 1} if π π and π π are both positive as required for a strictly mutualistic interaction. In contrast, solving the inequalities for the final two spatially variable equilibria shows that the equilibria can exist as long as the following condition holds: π π π π > 16ππ ππ (10) This existence condition provides us valuable insight into the conditions that allow spatial variation in genotype frequencies to be maintained within mutualistic interactions. Specifically, (8) reveals that the product of the fitness benefits accruing through mutualistic interaction (π π π π ) must exceed sixteen times the product of the rate of gene flow in the interacting species. Thus, even if the fitness benefits of mutualism to one species are weak, spatial variation can still be maintained if the fitness benefits to the interacting species are quite strong. Similarly, even if rates of gene flow in one of the species are quite high, spatial variation can still be maintained in both species if the rate of gene flow in the interacting species is quite low. This result is an interesting coevolutionary twist on classical single species results for the balance between spatially variable selection and gene flow (REFS). We now know that two of the spatially variable equilibria we have identified can exist and thus are worthy of further investigation. The obvious next step is to take a look at what these equilibria look like for various combinations of model parameters. Perhaps the most biologically intuitive way to explore these equilibria is to plot allele frequencies in both species and populations for a handful of representative parameter combinations (Figure 1). Taking this numerical approach to exploring the equilibria reveals several important points. First, there appear to be two possible spatial configurations for any particular combination parameters. Specifically, the frequencies of the A and B alleles can be larger in population 1 than in population 2 (Equilibrium #8; Figure 1, left hand panels) or they can both be smaller in population 1 than in population 2 (Equilibrium #8; Figure 1, right hand panels). These two possible equilibrium states are simply mirror images of one another. The second thing we can learn by plotting the spatially variable equilibrium frequencies is that, not surprisingly, the greater the overall fitness impacts of the mutualistic interaction, measured as the product π π π π , the larger the degree of spatial variation in allele frequencies (Figure 1). Similarly, the lower the overall rate of gene flow, measured as the product ππ ππ , the greater the spatial differentiation in allele frequencies (Figure 1). Finally, our equilibrium results give us a tool that allows us to predict which of the interacting species will show a greater degree of spatial genetic variation. Specifically, studying plots of equilibrium allele frequencies suggests that spatial differentiation will be greater in M. truncatula (Species X) anytime: π π π π > ππ ππ but greater in S. meliloti (Species Y) anytime: π π π π < ππ ππ This general prediction can be verified by solving for the conditions under which |πΏπ | > |πΏπ | and vice versa. Together, these analyses of the spatially variable equilibria give us a pretty good idea of how much spatial variation we should expect to see in coevolving mutualists if they have reached the spatially variable equilibria. The final step in our mathematical analysis is to evaluate the local stability of the various equilibria shown in Table 1. These analyses will help us to identify biological scenarios under which we expect coevolution between mutualists to lead to spatial variation in gene frequencies and genetic polymorphism within populations as observed for the interaction between M. truncatula and S. meliloti. As we learned previously in Chapter 2, the first step in performing a local stability analysis is to create a Jacobian Matrix. The only difference is that we now have a system of four recursion equations, and as a consequence, our Jacobian is given by the following 4x4 matrix of partial derivatives: ππΜ π∗ ππΜ π ππΜ π∗ ππΜ π π½= ππΏπ∗ ππΜ π ππΏπ∗ [ππΜ π ππΜ π∗ ππΜ π ππΜ π∗ ππΜ π ππΏπ∗ ππΜ π ππΏπ∗ ππΜ π ππΜ π∗ π πΏπ ππΜ π∗ π πΏπ ππΏπ∗ π πΏπ ππΏπ∗ π πΏπ ππΜ π∗ π πΏπ ππΜ π∗ π πΏπ ππΏπ∗ π πΏπ ππΏπ∗ π πΏπ ] where π Μ ∗π = πΜ π + βπΜ π , πΏπ∗ = πΏπ + βπΏπ , πΜ ∗π = πΜ π + βπΜ π , and πΏπ∗ = πΏπ + βπΏπ . Next, we substitute in the equilibrium of interest and calculate the eigenvalues. As before, if the magnitude of the largest eigenvalues is greater than 1, the equilibrium is unstable; if it is less than one, it is stable. Taking this approach yields easily interpretable eigenvalues for the first five spatially homogenous equilibria (Table 2). What our local stability analyses reveal is that only those equilibria corresponding to the fixation of matching alleles in both populations (1&4) is locally stable. This is not surprising, really, because it is these scenarios of fixed matching that yield the highest reciprocal benefits of the mutualistic interaction, and maximize the population mean fitness of both interacting species. In contrast, those equilibria corresponding to fixed mismatching (2&3) are never stable. This, too, is not surprising since these equilibria minimize reciprocal benefits of interacting and also population mean fitness of both interacting species. Finally, the equilibrium characterized by equal levels of polymorphism in both populations and species (5), is never stable for mutualistic interactions. This result also makes good sense, intuitively, as one would expect small perturbations of this equilibrium to lead rapidly to the increase in frequency of matching alleles in both interacting species. Evaluating the stability of the spatially homogenous equilibria was relatively simple and yielded very clear insights. In contrast, determining when the spatially variable equilibria (8&9) are stable is not at all straightforward, and requires interpretation of complicated eigenvalues with expressions much too long to easily summarize in a table. Because I was unable to simplify these in any elegant way, we will need to proceed by evaluating them numerically/graphically. Although not elegant, these numerical analyses of the eigenvalues associated with the spatially variable equilibria (8&9) suggests that as long as these equilibria exist (defined by (X)) they are guaranteed to be locally stable as well (Figure 2). Thus, as long as π π π π > 16ππ ππ , the spatially variable equilibria exist, and are locally stable. The observation that four possible equilibria may all be simultaneously stable is a phenomenon we have not previously encountered in our previous investigations of coevolutionary interactions (TRUE?). What this means, is that if — for whatever reason — allele frequencies of M. truncatula and S. meliloti begin near one of these equilibria, they will remain there in perpetuity. Thus, if allele frequencies are initially spatially structured, conforming to either equilibrium 8 or 9, we would observe spatial genetic variation in this mutualistic interaction and also genetic polymorphism and variability in compatibility within populations. In contrast, if allele frequencies were initially spatially homogenous and matched the equilibria described by 1 or 4, we would observe no genetic variation across space and genetic polymorphism and variation in compatibility would be absent from individual populations. Obviously, only the former scenario is consistent with empirical observation. This raises an important question: just how differentiated do populations need to be initially for mutualistic coevolution to maintain spatial genetic variation? Unfortunately, answering this question mathematically is not straightforward. We can, however, simulate a handful of scenarios to gain some crude insight into the conditions that allow spatial variation to coevolve within mutualistic interactions. In order to simulate mutualistic coevolution, we can iterate the exact recursion equations (8) for particular combinations of parameters and initial conditions. Performing simulations across a range of initial conditions and strengths of mutualistic interaction reveal several important points. First, the more symmetrically differentiated the populations of the two mutualists are initially, the more likely it is that coevolution can drive further spatial genetic differentiation (Figures 3, 4). The reason for this is that, because of positive frequency dependence, the strength of spatially divergent natural selection increases with the degree of initial genetic differentiation. Thus, if the two mutualist species within one of the populations initially have high frequencies of the A and B alleles whereas the two mutualists within the other population have initially low frequencies of these alleles, spatially divergent selection is quite strong and maintains genetic differentiation even in the face of gene flow. Second, as the fitness consequences of the mutualistic interaction increase, the smaller the amount of initial spatial genetic differentiation required for coevolutionary differentiation to proceed (Figures 3, 4). Together, these simulations reveal that although mutualistic coevolution can drive the evolution of spatial genetic differentiation, it cannot do so de novo. Only if spatial genetic differentiation already exists does mutualistic coevolution lead to anything other than spatially homogenous allele frequencies (e.g., Figure 3c-f and Figure 4e-f). Later in this chapter we will grapple with the problem of how initial spatial genetic differentiation of this sort might, in principle, be generated. Answers to Key Questions: Can mutualistic coevolution maintain or generate genetic differentiation among populations? Our results suggest that mutualistic coevolution can maintain spatial genetic differentiation but cannot generate such differentiation de novo. Thus, if genotype frequencies of the mutualists M. truncatula and S. meliloti, are initially very similar across populations, mutualistic coevolution will simply reinforce this uniformity. If, in contrast, genotype frequencies are initially somewhat divergent among populations and the fitness consequences of mutualism are large relative to rates of gene flow, coevolution can enhance and maintain spatial genetic differentiation. Can spatial structure maintain genetic polymorphism within coevolving mutualistic populations? Sometimes. In those cases where coevolution maintains genetic differentiation among populations, gene flow will continually move alleles between populations and thus maintain genetic polymorphism. The amount of genetic polymorphism depends on the fitness consequences of mutualistic interactions relative to the rate of gene flow between populations. How do coevolutionary selection and gene flow interact to shape rates of mutualistic interaction within and among populations? Gene flow reduces rates of mutualistic interaction within populations when spatial heterogeneity is present. This is because individuals with incompatible alleles are continually introduced, thus reducing the average compatibility between species within populations. For the interaction between M. truncatula and S. meliloti this might lead to populations with variation in compatibility between genotypes. If, however, spatial heterogeneity is absent, gene flow has no consequence on rates of interaction because immigrants are genetically identical to residents. New Questions Arising: Our simple model of spatially structured mutualism between M. truncatula and S. meliloti suggests that coevolution can reinforce pre-existing spatial genetic differentiation but cannot cause initially homogenous populations to diverge from one another. Given that we observe spatial genetic variation within many coevolving interactions, this simple theoretical result raises several important questions: ο· Do similar results hold for other forms of ecological interaction? ο· How would our results change if the environment were heterogeneous? ο· What is the role of random genetic drift? In the next three sections, we will generalize our simple model in ways that allow us to answer these questions. Generalizations Generalization 1: Alternative forms of ecological interaction Our investigation of spatially structured mutualism suggested that coevolution can reinforce pre-existing spatial genetic structure but cannot cause initially genetically homogenous populations to diverge. To some extent, this result is intuitive and follows from the positive frequency dependence built into the model of mutualism we explored where individuals whose genotype matches the predominant genotype of the interacting species are most likely to interact successfully and accrue the benefits of the mutualism. But how might things differ for antagonistic interactions such as those between hosts and parasites or predators and prey that can generate negative frequency dependence? To investigate this scenario, we now turn our attention back to the interaction between the castrating trematode, Microphallus, and its snail host, P. antipodarum. As you might recall from the previous chapter, the interaction between Microphallus and P. antipodarum has been very well-studied within the context of the Red Queen Hypothesis for the evolution of sex. These studies have revealed significant genetic differentiation among lakes as well as substantial genetic polymorphism within lakes. In addition, studies of clone frequencies over time suggest cyclical dynamics (REFS), and are thus compatible with coevolutionary dynamics we expect from a matching alleles model. If we are willing to take some liberties with the genetics of the interacting species, and assume that both are haploid and that interactions involve only a single diallelic locus in each species, we can apply the same mathematical framework we used to study the spatially structured mutualism between M. truncatula and S. meliloti. In fact, as long as we remain comfortable with our key assumptions of weak selection and infrequent gene flow, we can even use the exact same equations! The only change we need to make is to recognize that the fitness consequences of the interaction are now negative for P. antipodarum. Thus, if we label P. antipodarum as species X and Microphallus as species Y, the fitness consequences of interacting for P. antipodarum must be negative such that π π < 0 and the fitness consequences of interacting for Microphallus must be positive such that π π > 0. With this small change in the ranges of these key parameters, we can repurpose our previous analyses of mutualistic interactions in a way that allows us to explore coevolution between P. antipodarum and Microphallus. Our first step will be to take a fresh look at the equilibria we identified previously as well as their local stability. A logical first step in our analysis of coevolution between P. antipodarum and Microphallus is to identify equilibria. Because the equations describing coevolution between these species are identical to those we used to study coevolution between M. truncatula and S. meliloti, the equilibria must also be the same (e.g., Table 1). Although we can be certain that the equilibria themselves remain identical, this need not be the case for the conditions under which the equilibria exist. The reason the existence conditions may no longer be the same as they were for the mutualistic interaction between M. truncatula and S. meliloti is that the parameter π π is now negative, capturing the reduction in fitness of P. antipodarum individuals unfortunate enough to become infected by the castrating trematode Microphallus. Thus, we must again identify the conditions under which each equilibrium exists by solving for parameter combinations that yield equilibrium values of host and parasite allele frequencies that lie between 0 and 1, inclusive. Taking this approach for each of the equilibria we identified previously shows that fewer equilibria exist when the interaction is between host and parasite rather than between mutualists (Table 3). In fact, the only equilibria that can now possibly exist are those where allele frequencies are spatially homogenous. This result suggests that if our model is to explain the spatial variation in genotype frequencies observed in the interaction between P. antipodarum and Microphallus, it must be through temporally fluctuating patterns rather than static equilibria. In order to investigate this possibility, we need to turn our attention from finding equilibria to studying their local stability. By now, the steps involved in analyzing local stability should be old hat: 1) generate a Jacobian matrix, 2) substitute the equilibrium of interest, and 3) calculate the eigenvalues of the Jacobian. In this case, our life is made easier by recognizing that, because the equations remain unchanged from our previous study of mutualism, the eigenvalues associated with each equilibrium must remain the same. Thus, the only challenge we face in our transition from mutualism to antagonism is to re-interpret the local stability of each equilibrium while recognizing that one of the species is now harmed by the interaction such that π π < 0 (Table 4). Taking this approach reveals that changing the sign of this single parameter has significant consequences for coevolution, destabilizing the matching equilibria (1&4), and making oscillatory dynamics near the polymorphic equilibrium (5) inevitable. These results strongly suggest the outcome of spatially structured coevolution between P. antipodarum and Microphallus will be oscillations in allele frequencies. The crucial question that remains unanswered by these analyses, however, is whether allele frequency oscillations will be out of phase in the two populations such that allele frequencies would vary across space as observed in the real world interaction between P. antipodarum and Microphallus? To answer these questions, we will need to carry our local stability analysis of the polymorphic equilibrium (5) a bit further. Up until now, we have interpreted the results of our local stability analyses by focusing on only the leading (largest) eigenvalue. If this leading eigenvalue had a magnitude greater than one, we deemed the system unstable; if it had a leading eigenvalue less than one, we deemed the system stable. Using this (entirely correct) interpretation, the results in table 4 show that the polymorphic equilibrium is unstable, and — because it has an imaginary component — also oscillatory. But does this imply that allele frequencies will oscillate over time but remain spatially homogenous? Or does it imply that allele frequencies will oscillate in a way that generates spatial heterogeneity? How can we tell? The key to answering these questions is to recognize that the Jacobian Matrix has a block form when we substitute in the polymorphic equilibrium: 1 π 2 π 0 0 1 0 0 0 0 (1 − 2ππ ) [ 0 0 1 π 2 π 1 π 2 π 1 1 π π π½= 2 (1 − 2ππ )] What this means is that the fate of perturbations to the average allele frequencies (πΜ π and πΜ π ) can be studied independently of perturbations to the spatial differences in allele frequencies (πΏπ and πΏπ ). Specifically, we can analyze the local stability of average allele frequencies by calculating the eigenvalues of the 2×2 matrix defined by the upper left block, and the local stability of the spatial difference in allele frequencies by calculating the eigenvalues of the 2×2 matrix defined by the lower right hand block. Taking this approach reveals the following pair of eigenvalues for average allele frequencies: 1 1 ± 2 √π π √π π (12a) and the following pair of eigenvalues for the spatial differences in allele frequencies: 1 1 − (ππ + ππ ) ± 2 √4(ππ − ππ )2 + π π π π (12b) By decomposing our Jacobian matrix in this way and analyzing the dynamics of each block separately, we gain two important insights. First, since the leading eigenvalue associated with the average allele frequencies (12a) is always imaginary and greater than one in magnitude, we know that average allele frequencies will oscillate around the polymorphic equilibrium with ever increasing amplitude. Second, since the leading eigenvalue associated with the spatial difference in allele frequency (12b) is guaranteed to have a magnitude less than one, we know that any small difference in allele frequencies across populations will ultimately decay. We are now at a point where we can bring our mathematical analyses together and use them to sketch a rough picture of the coevolutionary dynamics we expect for the spatially structured interaction between P. antipodarum and Microphallus. What this picture depicts is the inevitable emergence of spatially synchronized cycles that grow in amplitude over time. Even if allele frequencies initially differ among populations by some small amount, our mathematical results suggest this initial spatial heterogeneity will inevitably decay over time. Somewhat surprisingly, these results suggest that antagonistic interactions may be even less capable of generating and maintaining spatially heterogeneity in allele frequencies than are mutualistic interactions, at least under the conditions of weak selection we have considered here. This result is uncomfortable because it is not at all consistent with what is actually observed in empirical studies of the interaction between P. antipodarum and Microphallus, as well as in many other well-studied antagonistic interactions (REFS). Where could our model have gone wrong? One obvious possibility is our assumption that the fitness consequences of interactions are weak. Although tackling strong selection mathematically is challenging, simulating coevolution under conditions of strong selection is really pretty easy. All we need to do is write a simple loop that iterates the exact recursion equations (6), over and over again for specific parameter combinations and initial conditions. Taking this approach for a smattering of parameter values and initial conditions suggests that our mathematical results are pretty darn robust: in virtually every case simulated, spatial heterogeneity in allele frequencies ultimately decays over time (Figure 5). At the same time, however, simulations can only ever investigate specific parameter combinations, so the generality of this result is unclear. It may well be that specific combinations of parameters exist that do maintain spatial heterogeneity in allele frequencies (E.G., see Gav and Mik). What seems clear, however, is that our model lends little support to the idea that coevolution between hosts and parasites can generate significant levels of spatial genetic differentiation, even with strong selection. What we will see in the next two sections, however, is that by relaxing our assumptions that the fitness consequences of interactions are constant across space and that population sizes are infinite, creates substantially more scope for coevolution to drive genetic differentiation among populations. Generalization 2: Heterogeneous environments and selection mosaics When we first started writing down expressions describing the fitness of M. truncatula and S. meliloti at the beginning of this chapter (i.e., equations 5), we made a critical assumptions that might not have been entirely obvious at the time. Specifically, we assumed that the fitness consequences of the interaction, π π and π π , do not vary across space. Biologically, this means that a successful interaction between M. truncatula and S. meliloti has exactly the same fitness consequences in different geographic regions. Although that might sound reasonable at first glance, imagine what would happen if the amount of available nitrogen in the soil differed between geographic locations, perhaps due to more severe run-off from agricultural fields in some populations than in others. Under such conditions the benefits of interacting with the nitrogen fixing rhizobium, S. meliloti, may be significantly reduced for populations of M. truncatula living in nitrogen rich populations such that the value of the parameter π π is reduced. We now know that such selection mosaics are quite common (REFS), and can have important consequences for coevolution (REFS). Our goal here is to integrate selection mosaics into our model of spatially structured coevolution, and ask if they can facilitate the emergence of spatial differentiation among coevolving populations. The first step in our exploration of selection mosaics is to decide how we can integrate spatial variation in the fitness consequences of an interaction into our fitness expressions. The simplest way to do this is to modify our original expressions for fitness (5) to allow the values of the parameters π π and π π to vary in the two populations, yielding the following equations for fitness in the coevolving species: ππ,π΄,π = 1 + π π,π (ππ,π ) (13a) ππ,π,π = 1 + π π,π (1 − ππ,π ) (13b) ππ,π΅,π = 1 + π π,π (ππ,π ) (13c) ππ,π,π = 1 + π π,π (1 − ππ,π ) (13d) where the subscript i indicates population. Next, we can follow the same sequence of mathematical steps we used earlier in this chapter to explore coevolution between S. meliloti and M. truncatula until we arrive at expressions like (8) describing the allele frequency of both species in each of the populations. At this point, however, we need to diverge from the path we took previously if we are going to develop useful mathematical results. The reason is that in the presence of selection mosaics, even our previous assumptions of weak fitness consequences and weak gene flow fail to simplify the problem sufficiently for any substantive mathematical progress to be made. Because our previous approximation based on assumptions of weak selection and weak gene flow failed to sufficiently simplify the problem, we need to identify a fresh approach. One idea might be to flip our worldview and assume that rather than being infrequent, gene flow is actually quite common. Superficially, this seems like a dumb idea; how could assuming a parameter is large possibly simplify the problem more than assuming it is small? To see why this superficially counter-intuitive approximation might work, we need to focus our attention on the amount of genetic differentiation among populations we expect to see. Specifically, as the rate of gene flow increases, we expect the level of genetic differentiation among populations to be reduced. In fact, if selection is relatively weak (of small order π), and gene flow is substantial, we expect the difference in allele frequencies between populations (πΏπ and πΏπ terms) to also be small and of order π (REFS). What this means, is that if we are willing to assume gene flow is relatively strong and selection is relatively weak, we can approximate the dynamics of spatially structured coevolution by using a Taylor Series to approximate equations (8), ignoring all terms of πͺ(π 2 ). Taking this approach yields the following expressions for the change in average allele frequencies and the difference in allele frequencies: βπΜ π ≈ π Μ π πΜ π πΜ π (1 − 2πΜ π ) + πͺ(π 2 ) (14a) βπΏπ ≈ πΏπ π πΜ π πΜ π (1 − 2πΜ π ) − 2ππ πΏπ + πͺ(π 2 ) (14b) βπΜ π ≈ π π πΜ π πΜ π (1 − 2πΜ π ) + πͺ(π 2 ) (14c) βπΏπ ≈ πΏπ π πΜ π πΜ π (1 − 2πΜ π ) − 2ππ πΏπ + πͺ(π 2 ) (14d) where the terms πΏπ π = π π,1 − π π,2 and πΏπ π = π π,1 − π π,2 measure the intensity of the selection mosaic acting on each species. A remarkable fact about our new approximation (14) is that the spatial genetic differentiation and average allele frequencies are now uncoupled! Just as we saw in Chapter 6 when we explored the QLE approximation, we have a situation where we expect one of the variables, in this case the degree of spatial genetic differentiation (πΏπ and πΏπ ), to approach a quasi-equilibrium state where its value depends only on that of the other variable. To see this more clearly, we can solve for the conditions under which (14b) and (14d) are equal to zero, yielding the following Quasi Panmixia approximation for levels of genetic differentiation between populations: (1−2πΜ π ) πΏπ πΜ π πΜ π πΏΜπ ≈ − π 2π π (1−2πΜ π ) πΏπ πΜ π πΜ π πΏΜπ ≈ − π 2π π (15a) (15b) A quick inspection of equations (15) reveals a very important result: as long as average allele frequencies do not become stuck at zero, one, or one half, the presence of a selection mosaic guarantees that allele frequencies will differ across populations. Thus, we need only identify those conditions where average allele frequencies do not approach and remain at these values in order to nail down conditions guaranteeing we see the type of genetic variation among populations observed in many coevolving systems. The logical way to identify these conditions is to turn again to an analysis of equilibria and their local stability. Because the dynamics of average allele frequencies are decoupled from differences in allele frequencies, we can identify these conditions by analyzing the equilibria and stability defined by the pair of equations (14a and 14c) in isolation. Following the usual steps reveals that there are five possible equilibria, none of which are ever stable for antagonistic interactions like those between P. antipodarum and Microphallus (Table 5). However, one of the equilibria (where average allele frequencies are near ½ in both species), is associated with imaginary eigenvalues, suggesting that average allele frequencies should oscillate around ½ in both species. GIVEN THAT, WHAT SHOULD WE SEE? IT APPEARS WE SHOULD SEE FLUCTUATING LEVELS OF SPATIAL DIFFERENTIATION. WHEN AVERAGE ALLELE FREQUENCIES ARE FAR FROM ½ DIFFERENTIATION IS LARGE, WHEN IT IS NEAR ½ small, on the other side of ½ the sign fluctuates. 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A subset of equilibria for spatially structured mutualism Equilibrium πΜ π πΜ π πΏπ πΏπ Existence 1 0 0 0 0 Always 2 0 1 0 0 Always 3 1 0 0 0 Always 4 1 1 0 0 Always 5 ½ ½ 0 0 Always ½ ½ −8ππ ππ π π + (8ππ2 + π π2 )π π + 4√πΎπ √ π π2 π π −8ππ ππ π π + (8ππ2 + π π2 )π π + 4√πΎπ √ π π2 π π Never ½ ½ −8ππ ππ π π + (8ππ2 + π π2 )π π + 4√πΎπ √ π π2 π π −8ππ ππ π π + (8ππ2 + π π2 )π π + 4√πΎπ −√ π π2 π π Never ½ ½ −8ππ ππ π π + (8ππ2 + π π2 )π π − 4√πΎπ −√ π π2 π π −8ππ ππ π π + (8ππ2 + π π2 )π π − 4√πΎπ √ π π2 π π π π π π > 16ππ ππ −8ππ ππ π π + (8ππ2 + π π2 )π π − 4√πΎπ −√ π π2 π π −8ππ ππ π π + (8ππ2 + π π2 )π π − 4√πΎπ −√ π π2 π π π π π π > 16ππ ππ 6 7 8 9 ½ ½ πΎπ = ππ2 (4ππ2 π π2 − 8ππ ππ π π π π + (4ππ2 + π π2 )π π2 ) and πΎπ = ππ2 (4ππ2 π π2 − 8ππ ππ π π π π + (4ππ2 + π π2 )π π2 ) Table 2. Spatially homogenous equilibria and associated eigenvalues Equilibrium πΜ π πΜ π πΏπ πΏπ Eigenvalues Stability 1 0 0 0 0 1 − 2ππ − π π , 1 − π π , 1 − 2ππ − π π , 1 − π π Stable 2 0 1 0 0 1 + π π , 1 − 2ππ + π π , 1 + π π , 1 − 2ππ + π π Unstable 3 1 0 0 0 1 + π π , 1 − 2ππ + π π , 1 + π π , 1 − 2ππ + π π Unstable 4 1 1 0 0 1 − 2ππ − π π , 1 − π π , 1 − 2ππ − π π , 1 − π π Stable 0 1 1 1 − √π π √π π , 1 + √π π √π π , 2 2 1 1 1 − ππ − ππ − √πΆ, 1 − ππ − ππ + √πΆ 2 2 Unstable 5 ½ ½ πΆ = 4(ππ − ππ )2 + π π π π 0 Table 3. A subset of equilibria for spatially structured antagonism Equilibrium πΜ π πΜ π πΏπ πΏπ Existence 1 0 0 0 0 Always 2 0 1 0 0 Always 3 1 0 0 0 Always 4 1 1 0 0 Always 5 ½ ½ 0 0 Always 6 ½ ½ −8ππ ππ π π + (8ππ2 + π π2 )π π + 4√πΎπ √ π π2 π π −8ππ ππ π π + (8ππ2 + π π2 )π π + 4√πΎπ √ π π2 π π Never 7 ½ ½ −8ππ ππ π π + (8ππ2 + π π2 )π π + 4√πΎπ √ π π2 π π −8ππ ππ π π + (8ππ2 + π π2 )π π + 4√πΎπ −√ π π2 π π Never 8 ½ ½ −8ππ ππ π π + (8ππ2 + π π2 )π π − 4√πΎπ −√ π π2 π π −8ππ ππ π π + (8ππ2 + π π2 )π π − 4√πΎπ √ π π2 π π Never 9 ½ ½ −8ππ ππ π π + (8ππ2 + π π2 )π π − 4√πΎπ −√ π π2 π π −8ππ ππ π π + (8ππ2 + π π2 )π π − 4√πΎπ −√ π π2 π π Never πΎπ = ππ2 (4ππ2 π π2 − 8ππ ππ π π π π + (4ππ2 + π π2 )π π2 ) and πΎπ = ππ2 (4ππ2 π π2 − 8ππ ππ π π π π + (4ππ2 + π π2 )π π2 ) Table 4. Spatially homogenous equilibria and associated eigenvalues Equilibrium πΜ π πΜ π πΏπ πΏπ Eigenvalues Stability 1 0 0 0 0 1 − 2ππ − π π , 1 − π π , 1 − 2ππ − π π , 1 − π π Unstable 2 0 1 0 0 1 + π π , 1 − 2ππ + π π , 1 + π π , 1 − 2ππ + π π Unstable 3 1 0 0 0 1 + π π , 1 − 2ππ + π π , 1 + π π , 1 − 2ππ + π π Unstable 4 1 1 0 0 1 − 2ππ − π π , 1 − π π , 1 − 2ππ − π π , 1 − π π Unstable 0 1 1 1 − √π π √π π , 1 + √π π √π π , 2 2 1 1 1 − ππ − ππ − √πΆ, 1 − ππ − ππ + √πΆ 2 2 Unstable and Oscillatory 5 ½ ½ πΆ = 4(ππ − ππ )2 + π π π π 0 Table 5. Spatially homogenous equilibria and associated eigenvalues Equilibrium πΜ π πΜ π πΏπ πΏπ Eigenvalues Stability for host-parasite 1 0 0 0 0 1 − π Μ π , 1 − π Μ π Unstable 2 0 1 0 0 1 + π Μ π , 1 + π Μ π Unstable 3 1 0 0 0 1 + π Μ π , 1 + π Μ π Unstable 4 1 1 0 0 1 − π Μ π , 1 − π Μ π Unstable 5 ½ ½ 0 0 1 1 1 − √π Μ π √π Μ π , 1 + √π Μ π √π Μ π , 2 2 Unstable and Oscillatory