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Keywords:

  • genetic drift;
  • metapopulation;
  • natural selection;
  • resistance;
  • virulence

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Plants and their parasites co-evolve at key genes of interactions following the so-called gene-for-gene (GFG) relationship. Previous models of co-evolution assume (i) single infinitely large populations of hosts and parasites and (ii) costs of resistance and infectivity. The effects of three biologically realistic characteristics of plant and parasite populations on polymorphism maintenance at GFG loci were investigated. First, two components of the cost of resistance were disentangled: the cost of harbouring the resistance allele itself, and the cost of triggering resistance when encountering a parasite. Secondly, it was assumed that plants encounter parasites depending on fixed disease prevalence in time. Thirdly, finite sizes of host and parasite populations were introduced, assuming genetic drift and mutation. In a single population, statistical polymorphism in either host or parasite can be obtained in the finite population size model if there is no cost of harbouring the resistance allele and disease prevalence is low. On the other hand, long-term polymorphism can be maintained by heterogeneity in disease prevalence and costs of resistance in a spatially structured population with two demes linked by migration. More precisely, the trench warfare co-evolutionary dynamics occurs when assuming large host and parasite population sizes, and large differences between demes for disease prevalence or costs of triggering resistance. Moreover, the resistance allele does not need to harbour a fitness cost in itself for long-term stable polymorphism to occur in the co-evolutionary models. This observation may explain the lack of empirical evidence of high costs of carrying resistance alleles.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Host–parasite interactions are identified as important selective forces promoting genetic diversity in biological systems, and thus polymorphism at loci associated with host resistance and parasite virulence and infectivity is commonly found in natural populations (e.g. Laine, 2006, 2008; review in Tack et al., 2012). Two alternative scenarios have been proposed to explain the observed amount of polymorphism at loci associated with these antagonistic interactions (Holub, 2001; Woolhouse et al., 2002). In the ‘arms race’ model, there is unstable cycling of allele frequencies with recurrent fixation of alleles in both the host and the parasite, so polymorphism in virulence and resistance exists but is transient in time. Alternatively, in the ‘trench warfare’ model, the frequencies of resistance and virulence alleles are maintained at intermediate values (Stahl et al., 1999), generating a signature of balancing selection at these genes (e.g. Hörger et al., 2012). One of the main aims of current research is thus to determine the ecological, genetic and epidemiological factors that generate each of these two models of co-evolution.

The co-evolutionary scenarios have been studied using discrete time models based for example on gene-for-gene (GFG) interactions between hosts and parasites. GFG models are a common mechanism of infection in plant (Dodds & Rathjen, 2010) and invertebrate diseases (Wilfert & Jiggins, 2010). In the classic GFG model (Leonard, 1977), it is assumed that both the host and the parasite have one locus. The host has two alleles, for resistance and susceptibility, and the parasite has two alleles, for infectivity and non-infectivity. This current study uses the definition of infectivity as in the evolutionary biology literature (Gandon et al., 2002). Note that in the plant pathology literature and previous GFG models, infectivity is denoted as virulence or pathogenicity, and non-infectivity as avirulence (Leonard, 1977; Tellier & Brown, 2007). Infection occurs if the host is susceptible or if the parasite is infective. Otherwise, if the resistant hosts encounter non-infective parasites, a defence reaction prevents infection. Because disease is costly, hosts are selected to be resistant which in turn imposes a pressure for parasites to become virulent and overcome host defences. Co-evolutionary cycles emerge in such a model where the allelic frequencies of one species determine the fitness of the alleles in the other species (infectivity is favoured when resistance is common and resistance is favoured when infectivity is rare), a selective pressure known as indirect frequency dependence selection (iFDS, Tellier & Brown, 2007). In models where only iFDS occurs, the polymorphic equilibrium point and the cycles are unstable, driving alleles to fixation recurrently so the system behaves like an arms race scenario (Tellier & Brown, 2007). However, polymorphism can become stable in a GFG model if there is negative direct frequency dependence selection (ndFDS). This mechanism decreases the selective advantage of alleles that are in high frequency in the population, thus stabilizing polymorphism. Ecological and epidemiological factors such as polycyclic disease (Tellier & Brown, 2007), seed banks (Tellier & Brown, 2009), or spatial heterogeneity in the costs of infectivity (virulence) and resistance (Tellier & Brown, 2011) may generate ndFDS.

The existence of fitness costs of the resistance and infectivity (virulence) alleles has been debated and discussed previously (see review in Brown & Tellier, 2011). There is a lack of evidence for the universality of such costs, in particular for resistance in plants (Bergelson et al., 2001), and it is usually assumed that they should be very small (on the order of few percent, but see Tian et al., 2003). Recent molecular biology studies suggest that defence mechanisms, measured as gene expression and disease resistance network activation, are triggered upon recognition of a parasite by non-specific basal defences (Tsuda et al., 2009; Katagiri & Tsuda, 2010). Such defences are sufficient to prevent infection by non-specific parasites. However, specialized parasites harbour effectors, potentially switching off these defences (Dodds & Rathjen, 2010). The model in the current study suggests that resistant plants, when encountering parasites, trigger a resistance reaction which is either (i) fully effective against non-infective parasites (for example via local cell death) or (ii) non-effective against infective parasites which harbour effectors circumventing such defences. It is suggested here that the fitness costs of harbouring a specific resistance allele with a very low (if any) metabolic cost (Bergelson et al., 2001), and the cost of triggering defences when encountering infective and non-infective parasites as a result of gene expression and activation of gene networks (Katagiri & Tsuda, 2010), should be distinguished.

A key assumption of previous models (Leonard, 1977; Tellier & Brown, 2007) is that the host encounters the parasite every generation (but see Gandon et al., 2002; Salathé et al., 2005; Tellier & Brown, 2009 for realistic epidemiological models). However, it is often the case in natural populations that potential hosts are never exposed to the parasite during their lifetime. For plant diseases, both epidemiological and environmental conditions strongly affect the incidence (i.e. whether the disease is present or not) and prevalence (the proportion of infected plants) of the parasite in a population over space and time (Laine, 2006, 2008; Soubeyrand et al., 2009). For instance, microclimatic conditions, climatic effect on the dormancy or on the latent periods of the parasite, and mechanisms of parasite dispersion are factors that can modify the frequency of parasite encounters in plant and invertebrate populations (e.g. Laine, 2006, 2008; Soubeyrand et al., 2009; review in Tack et al., 2012).

Spatiotemporal variation in the encounter rates with the parasite can change the intensity of selection for resistance traits in the host. When the host does not encounter the parasite, the effect of iFDS disappears because the fitness of the host is independent of the frequency of virulence in the parasite population. The importance of variation in the parasite prevalence on the dynamics of host–parasite interactions has been shown experimentally in studies of plant and bacterial populations (Laine, 2006, 2008; Vogwill et al., 2009). These studies have focused mainly on heterogeneity at a spatial scale, a very common feature of natural systems where species occur as separate demes connected through migration. In these scenarios, high variation in the parasite encounter rates has been observed even for neighbouring populations (Laine, 2006). These findings have been integrated into the theory of the ‘geographic mosaic of co-evolution’, which suggests that variation in ecological conditions across spatially structured populations can lead to differences in selection pressures between demes (Gomulkiewicz et al., 2000; Thompson, 2005). Demes in which there is strong reciprocal selection are usually referred to as ‘hotspots’ of co-evolution while demes with weak reciprocal selection are referred to as ‘coldspots’. The simple GFG model has already been studied in spatially structured populations where it was shown that differences in costs of resistance and infectivity between demes can promote stability of genetic polymorphism by introducing ndFDS through migration (Tellier & Brown, 2011). The current study extends the simple GFG model to include differences in parasite encounter rates between populations.

Two key elements were introduced in the GFG model. First, the cost of resistance from the simple GFG was separated into two components: a cost of harbouring the resistance allele, and a cost of triggering defences when encountering a parasite. Secondly, the assumption of permanent exposure to the parasite in the simple GFG model was relaxed to determine the effect of parasite encounter rates, or disease prevalence, on co-evolutionary dynamics. There were three objectives in this study: (i) analysis of the effect of disease prevalence on the allele frequency and stability of polymorphism in one population in a deterministic model, (ii) introduction of variation in parasite encounter rates in a spatially structured population with two demes in a deterministic model, to show that differences in disease prevalence or costs of resistance among demes may generate polymorphism stability via ndFDS, and (iii) finally, the production of more realistic models by introducing finite population size including genetic drift and mutation. Statistical polymorphism for either host or parasite alleles could be observed in a single population model at low prevalence. The study showed that the emergence of the trench warfare dynamics with long-term stable polymorphism may occur only for large host and parasite population sizes and large differences in disease prevalence between demes.

Methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Model with one deme

Hosts and parasites are haploid with discrete generations in time. The GFG model in this study assumes two alleles in the host, resistance (RES) and susceptibility (res), and in the parasite infectivity (INF) and non-infectivity (ninf). Each generation the host encounters the parasite with probability ρ, that is the disease prevalence. Infective parasites harbour a cost (of virulence) b. Non-infective parasites do not bear this cost, but if they encounter a resistant host they have a cost c of not being able to infect it ( 1, as in Tellier & Brown, 2007).

Being diseased has a cost s for the host (resistant or susceptible). Resistant hosts encountering parasites trigger a resistance reaction that incurs a cost of fitness u, irrespective of whether the defence is successful. When hosts do not encounter the parasite, which happens with probability 1 − ρ, resistant hosts have a cost u*. This corresponds to the basal cost of harbouring the allele for resistance. Thus, u* + ε, where ε ≥ 0 is interpreted as the cost of activating and expressing the defence genes. Note that the advantage of harbouring a resistance allele against a non-infective parasite, i.e. to trigger hypersensitive response and cell death preventing infection, is measured as the difference between the cost of being diseased (s) and u (usually u, as in Tellier & Brown, 2007). Parasites that do not encounter hosts do not survive, regardless of their genotypes. The fitnesses of hosts and parasites are summarized in Table 1. As a simplifying assumption, all the costs and the encounter rate are assumed to be constant through time (but see Mostowy & Engelstädter, 2011).

Table 1. Fitnesses of hosts and parasites for the gene-for-gene model in a single population
Disease prevalenceGenotypes (frequency)Fitness
HostParasiteHostParasite
  1. ρ: disease prevalence; RES, res: alleles for host resistance and susceptibility, respectively; ninf, INF: alleles for pathogen non-infectivity and infectivity, respectively; b: cost of virulence; c: cost of inability to infect RES host; u: cost of fitness when a resistance reaction is triggered; u*: cost of fitness to resistant hosts that do not encounter a pathogen (basal cost of resistance); s: cost of being diseased; g: generation; na: not applicable.

ρ RES (Rg)ninf (Ag)1 − u1 − c
INF (ag)(1 − u) (1 − s)1 − b
res (rg)ninf (Ag)1 − s1
INF (ag)1 − s1 − b
1 − ρRES (Rg) 1 − u*na
res (rg) 1na

At generation g, the frequency of non-infective (infective) alleles in the parasite population is Ag (ag) and the frequency of resistance alleles (susceptibility) in the host population is Rg (rg; Tellier & Brown, 2007). The recurrence equations are built assuming frequency-dependent disease transmission:

  • display math(1)
  • display math(2)

Note that the probability of encountering a host is assumed to be equal for infective and non-infective parasites, thus ρ does not introduce fitness differences between these two alleles. This system has four trivial equilibria where inline image, and inline image, which are (0,0), (0,1), (1,0) and (1,1). The non-trivial polymorphic equilibrium for this system is:

  • display math(3)
  • display math(4)

If u* and ρ = 1, the system of Eqns (1)-(4) collapses to the model studied in Tellier & Brown (2007) with an unstable equilibrium point.

Model with two demes

Spatial heterogeneity in parasite prevalence was introduced in a model with two demes linked by migration of both hosts and parasites. Selection occurs within each deme following the one-deme model above, and migration takes place after selection (similar results are obtained if migration precedes selection (Tellier & Brown, 2011). Each deme i (= 1, 2) is characterized by its own parameters for costs and parasite encounter rate as defined in the model above: ui, bi, si, ci, ρi, ui* (similarly εi = ui − ui*). In addition to costs (bi, si, ci), the disease prevalence (ρi), cost of resistance allele (ui*) and cost of expression of a resistance reaction (εi) were included. For simplicity it was assumed that migration for both the host and the parasite is symmetrical between the two demes, with the migration rates for the host and the parasite being denoted mH and mP, respectively. The system of equations is detailed in the Supporting Information S1 (Section 2).

Numerical simulations with finite population size

Host and parasite populations have sizes NH and NP, respectively. Allelic frequencies were calculated each generation using the recursion equations above, and genetic drift was introduced by binomial sampling of the finite host and parasite populations at each generation. Population size was assumed to be constant over time and NH = NP for simplicity. Mutations between allelic types were introduced from RES to res or INF to ninf alleles and vice versa (Kirby & Burdon, 1997; Salathé et al., 2005). Backward and forward mutations occur at a symmetric rate μ per generation following a Poisson distribution (Kirby & Burdon, 1997). Mutation (μ) and population size (N) determine the population mutation rate parameter θ (θ ∝ ), which defines the strength of genetic drift and the rate of appearance of new alleles and thus the speed of evolution (e.g. Wakeley, 2008). The characteristics of the co-evolutionary dynamics which are relevant for population genetics (see below) for various values of the population mutation parameter (θ) were compared, with N taking values of 1000, 5000 and 10 000 and the mutation rate being fixed to 1 × 10−5.

The models were analysed by simulating allele dynamics over 10 000 host generations. Using R codes, the percentage of time that host or parasite alleles are fixed (frequency of allele = 1) was measured. High mutation rates can avoid complete fixation of alleles even if the internal equilibrium is unstable (Kirby & Burdon, 1997; Salathé et al., 2005). For this reason, the percentage of time that alleles are near fixation or extinction was also measured, assuming a 5% threshold for allele frequency. The speed of co-evolution was measured by counting the total number of cycles for both the resistance and infectivity alleles over 10 000 generations. This computation is realized by fitting a smooth spline curve to the trajectory of the allele frequencies, using the ‘smooth.spline’ function from the R package stats (smoothing parameter = 0·15). Various sliding window sizes (from 10 to 100) and smoothing parameters (0·05–0·5) were tested to find the values used, which ensure robustness and accuracy of computations over a wide range of possible dynamics. The initial allele frequencies (a0 and R0) may affect the behaviour of the system as a result of the existence of unstable limit cycles (Leonard, 1994). However, as unstable limit cycles only exist in the absence of reverse mutation (Kirby & Burdon, 1997), the initial allele frequencies were not expected to affect the behaviour of the system here. In order to be cautious, all statistics are averages over 100 runs with varying initial frequencies sampled from a uniform distribution over the interval (0·01–0·5). The co-evolutionary dynamics were studied over a realistic range of parameters assuming low costs (u and < 0·1) and an intermediate cost of disease, = 0·4 (Tellier & Brown, 2007).

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Model with one deme

The non-trivial equilibrium point is unstable in the deterministic model (Eqns (1) and (2)), as is shown analytically in the Supporting Information S1 (Section 1). When assuming finite host and parasite population sizes, the equilibrium is confirmed to be unstable and recurrent fixation of alleles occurs (Fig. S1). High disease prevalence favours infective parasites as they select in return for resistant hosts, as indicated by the allele equilibrium frequencies (Eqns (3) and (4)). Simulations of the finite population size model demonstrate that the time near fixation (>0·95) of the infectivity allele increases with the encounter rate (Fig. 1), in contrast to the time of fixation of the susceptibility allele (Fig. 1). However, note that if resistance is favoured for higher values of ρ, the RES allele does not get fixed unless population sizes are very small (< 1000; data not shown). It follows that, as expected, higher parasite prevalence increases the speed of co-evolutionary cycles (Fig. S2). Co-evolution only occurs when the parasite encounters the host as iFDS is needed for allele frequency change.

image

Figure 1. Effect of the parasite encounter rate (ρ) on the fixation time of susceptibility (a) and infectivity (b) alleles. The mean and standard deviation are computed over 100 runs with varying initial allele frequencies. The other parameters are = 1000; μ = 10−5; u* = 0·05; = 0·1; = 0·4 and = 0·95.

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It has been suggested that drift and mutation can maintain transient polymorphism for long periods of time and without costs, in contrast to predictions of the deterministic infinite population size models (Salathé et al., 2005). Although the non-trivial equilibrium is unstable, polymorphism in host populations can be maintained via allele frequency cycling by the effect of drift and mutation in a finite population size model with no cost of the resistance allele (u* = 0, = 10 000; Fig. 2). This occurs when the prevalence of the parasite is low (small values of ρ) and there is a non-zero cost of expressing resistance (ε and > 0; Fig. 2). In this case, selection in the host is absent when the parasite is not present (because u* = 0), and the strength of selection acting on the host is weak as a result of low disease prevalence. Finally, the case when both costs of resistance are zero (u* = 0) is explored. For low parasite prevalence (small values of ρ), polymorphism in the host population can be maintained over long periods of time (Fig. S3a) by balance between drift, mutation and weak selection for resistance. The strength of selection on the host depends on the encounter rate with the parasite, so that when ρ increases, selection for resistance becomes stronger before infectivity is fixed in the parasite population and fixation of the RES allele is more common (Fig. S3b). Note, however, that the absence of any cost of resistance in the host always leads to fixation of infectivity alleles in the parasite population (Fig. S3).

image

Figure 2. Statistical polymorphism for the resistance allele under low disease prevalence (ρ = 0·2) and no cost of harbouring the resistance allele (u* = 0) as a result of genetic drift and mutation. The frequency of the resistance allele is in blue, and of the infectivity allele in red. Other parameter values: = 10 000; μ = 10−5; = 0·1; = 0·4 and = 0·95.

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Model with two demes

Computation of the Jacobian matrix coefficients reveals that ndFDS can be generated by migration between demes which exhibit different disease prevalence but all other parameters being identical (Supporting Information S1, Section 3). However, due to the non-linearity of the system of equations and the number of parameters involved, it is not possible to extract a simple closed form condition for the stability of the non-trivial equilibrium. Based on stability analysis of other spatial models with heterogeneous selection (Nagylaki, 1992; Gavrilets & Michalakis, 2008; Tellier & Brown, 2011), it is expected that higher differences in parameter values between demes promote stability of the equilibrium point.

The conditions stabilizing genetic polymorphism were investigated by simulations in the finite population size model. Stable long-term polymorphism, i.e. trench warfare dynamics, can be obtained in both demes when they differ only in their disease prevalence (Fig. 3). Polymorphism stability was recorded over a wide parameter range by varying ρ1 between 0 and 1, and u1* between 0 and 0·1 in deme 1, while fixing these parameters in deme 2 at ρ2 = 0·5 and u2= 0·05. All other parameters are kept equal in both demes to = 0·1 (u1 u2 and b1 b2), = 0·4 (s1 s2) and = 0·95 (c1 c2). Migration rates play a significant role in determining the stability of the equilibrium by coupling the dynamics in the two demes (Gavrilets & Michalakis, 2008; Tellier & Brown, 2011). Thus, comparisons of co-evolutionary dynamics are conducted for three different migration rates (mP = mH = 0·01, 0·05, 0·1), and three different population sizes (= 1000, 5000 and 10 000; Fig. 4). Fixation of resistance and non-infective alleles was not observed, and thus the stability of polymorphism was investigated by monitoring the frequencies of infectivity and susceptibility alleles in deme 1. Because the demes are coupled by migration it is enough to evaluate if stability occurs in one deme to assess stability in both (Gavrilets & Michalakis, 2008; Tellier & Brown, 2011).

image

Figure 3. One simulation run with heterogeneity in disease prevalence among demes. Disease prevalence is (a) low in deme 1 (ρ1 = 0·4) and (b) high in deme 2 (ρ2 = 0·8). The blue square indicates the initial allele frequencies and the yellow square the equilibrium point. Values for migration rates are mH = mP = 0·01. Other parameters are set to identical values in both demes: = 10 000; μ = 10−5; u* = 0·01; = 0·1; = 0·4; = 0·95.

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image

Figure 4. Percentage of time that infectivity is fixed in the parasite population of deme 1 as a function of the differences in parasite prevalence values (ρ1 − ρ2) and costs of harbouring resistance (u1* − u2*) among demes for three population sizes: (a) = 1000; (b) = 5000; (c) = 10 000. The average over 100 simulations with varying initial frequencies is shown for each parameter combination. Migration rates for hosts and parasites (mH = mP) are low (= 0·01), intermediate (= 0·05) and high (= 0·1). The red and black dotted lines indicate when disease prevalence (red line) and the cost of harbouring the resistance allele (black line), respectively, do not differ between demes. Parameters in deme 2 are set to ρ2 = 0·5; u2* = 0·05. Other parameter values are identical in both demes: u1 = u2 = b1 b2 = 0·1; s1 = s2 = 0·4 and c1 = c2 = 0·95.

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Trench warfare co-evolutionary dynamics are more likely when increasing the population size, as stable polymorphism is maintained over a larger parameter range for = 10 000 (Fig. 4c) compared to = 5000 (Fig. 4b) and = 1000 (Fig. 4a). This means that for some parameter combinations, ndFDS is generated in the deterministic model, but genetic drift drags allele frequencies away from the deterministic equilibrium point, generating fixation of alleles. Trench warfare dynamics are also more likely at intermediate migration rates (here = 0·01 and 0·05). Smaller migration rates (below 0·01) generated the opposite pattern, namely that populations become independent and fixation of alleles prevails, as in the model with one deme (Fig. 4).

Differences in disease prevalence between demes can stabilize polymorphism and prevent allele fixation even when u* is equal between the two demes (Fig. 3, and red dotted line for which u1= u2* in Fig. 4). On the other hand, when ρ is identical between demes, differences in u* between demes (u1* − u2*) stabilize polymorphism but in a smaller parameter range when compared to differences in disease prevalence (black dotted line in plots in Fig. 4). Stability of long-term polymorphism is favoured when ρ1< ρ2* and u1> u2* (lower right corner of plots in Fig. 4). In this region of the parameter space, selection pressures in the host are opposite for both demes: susceptibility is favoured in deme 1 because disease prevalence is low and the cost of harbouring the resistance allele is high, while in deme 2 resistance is favoured. Migration between deme 1 (coldspot) and deme 2 (hotspot) thus has the potential of preventing allele fixation in each deme, therefore stabilizing genetic polymorphism. Additionally, migration from demes with high disease prevalence increases the speed of co-evolution (measured as number of cycles per unit of time) in demes with low prevalence (Fig. S2). Indeed, analytical results for the equilibrium frequency predict a larger effect of migration for demes with lower ρ (Supporting Information S1, Section 3). Differences in prevalence between demes stabilize polymorphism because the strength of selection for both infectivity and resistance alleles increases with disease prevalence (Fig. 1), and migration couples the different dynamics, thus generating ndFDS. The deme with lower parasite prevalence is a coldspot – where co-evolution occurs at a lower speed and iFDS is weak – whereas the deme with higher parasite prevalence is a hotspot. Finally, differences in disease prevalence in both demes are also sufficient to stabilize genetic polymorphism even in the absence of costs of harbouring the resistance allele (u* = u1* = u2* = 0, but ε > 0; Fig. S5).

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Most previous models in host–parasite co-evolution assume that hosts are constantly exposed to the parasite, although there is much empirical evidence showing that this is not the case in natural populations of plants and invertebrates (Laine, 2006, 2008; Soubeyrand et al., 2009; Tack et al., 2012). Parasite survival, dispersal and transmission are highly dependent on environmental conditions, and thus it is not rare that hosts never encounter a parasite during their lifetime. When hosts are not exposed to parasites, there is no indirect FDS. Epidemiological models with ecological feedback assume that transmission of the parasite between host generations is density-dependent (Gandon et al., 2008; Boots et al., 2009), and thus parasite prevalence has a large effect on the co-evolutionary dynamics (Jeger, 1997; Gandon et al., 2008; Boots et al., 2009; Tellier & Brown, 2009).

Disease prevalence in a single population

For simplicity, this current study assumed a constant and fixed prevalence of disease over time which does not generate ndFDS in contrast to classic epidemiological models (Gandon et al., 2008; Boots et al., 2009; Tellier & Brown, 2009). The occurrence of an arms race scenario with recurrent fixation of alleles in both host and parasite populations is predicted. High disease prevalence selects for infective parasites by favouring resistance, while low prevalence of the parasite selects for non-infective parasites by favouring susceptibility (Fig. 2). These results are consistent with empirical studies in plant populations where resistance allele frequencies are positively correlated with the rate of parasite encounter (Laine, 2006, 2008).

The assumption of constant disease prevalence in time is to some extent an unrealistic simplification. However, if epidemiological dynamics are well documented within years for annual plant species, especially crops (Campbell & Madden, 1990), the relevance of such models for between-years dynamics is less clear (Soubeyrand et al., 2009). In fact, disease transmission from one year to the next in annual plants depends strongly on the environmental conditions and the availability of inoculum (resting stages and over-wintering structures) and less on surviving infected hosts for many plant or invertebrate parasites (Campbell & Madden, 1990; Soubeyrand et al., 2009). This may be particularly the case for hemibiotrophic and necrotrophic parasites. However, for biotrophic parasites, the inoculum and disease prevalence at the start of a yearly epidemic may depend on environmental conditions as well as surviving infected hosts. The strong influence of environmental conditions on between-year disease dynamics strongly contrasts with assumptions of classic epidemiological models in the animal or human literature (Gandon et al., 2008; Boots et al., 2009). The studied GFG model would be improved by integrating epidemiological (Jeger, 1997; Tellier & Brown, 2009) and fixed environmental effects to specify the disease prevalence between host generations. Studying the transmission of parasite inoculum between host generations and the effect of the cost parameters on over-wintering populations (Montarry et al., 2010) are thus of crucial importance for modelling the co-evolutionary dynamics.

Statistical polymorphism for resistance in a single population

Costs of the resistance allele and expressing resistance have a similar role as in the simple GFG model (Leonard, 1977; Tellier & Brown, 2007). At least one cost (u* or ε) is necessary to generate co-evolutionary dynamics, i.e. iFDS, but is not sufficient for stable polymorphism to occur. The values of these costs determine the equilibrium frequency of parasite alleles (Leonard, 1977). Genetic polymorphism can be maintained in the absence of ndFDS only under weak selection for resistance and large host population size with genetic drift (Fig. 2 and Fig. S3). This effect, known as ‘statistical polymorphism’, was shown in multilocus GFG systems without costs in a single population (Salathé et al., 2005) and in spatial models (Gavrilets & Michalakis, 2008). In the model in the current study, statistical polymorphism occurs in a single population with one GFG locus, when assuming low parasite prevalence and a cost of harbouring the resistance allele being zero (u*). In this scenario, selection does not occur on the host in the absence of the parasite, as there is no cost of carrying the resistance allele, so that susceptible and resistant hosts have the same fitness. Because disease prevalence is low, the effect of drift and mutation (i.e. neutral evolution) can be higher or equal to the strength of selection due to co-evolution (Fig. 2). However, note that under such assumptions, statistical polymorphism occurs only in the host population while the infective allele is fixed or nearly fixed most of the time in the parasite population (as in Salathé et al., 2005). Therefore, these dynamics are not defined as trench warfare (Stahl et al., 1999).

Spatial heterogeneity for disease prevalence

Spatial heterogeneity is an ecological reality and an important mechanism for stabilizing genetic polymorphism in host–parasite interactions, via the occurrence of ndFDS through migration (Gavrilets & Michalakis, 2008; Tellier & Brown, 2011). Stable polymorphism can occur in a heterogeneous spatially structured population in both deterministic and finite population size models. However, fixation of alleles could occur for finite population size for parameter ranges where the equilibrium is stable in the deterministic model, especially for small population sizes (Fig. 4). An alternative scenario for stable polymorphism to occur is suggested here that does not require differences in the costs of resistance (u) or infectivity (b) between populations. If they exist, these costs are potentially challenging to measure in empirical studies as discussed previously (Bergelson et al., 2001; Brown & Tellier, 2011). Large heterogeneity in parasite prevalence introduces ndFDS at intermediate migration between demes, generating long-term stable polymorphism and a trench warfare dynamics (red dotted line in plots in Fig. 4). The results of this study imply that if the costs of resistance, infectivity or disease are unknown in natural systems, documenting large heterogeneity in disease prevalence, for example in relation to environmental variables, may be sufficient to explain the observed polymorphism at various spatial scales (Laine, 2008; Laine & Tellier, 2008; Soubeyrand et al., 2009).

Spatial heterogeneity for costs

Previous studies have considered heterogeneity in space only for the cost for the host of being diseased (Nuismer, 2006; Gavrilets & Michalakis, 2008) and presence of coldspots and hotspots of co-evolution (Gomulkiewicz et al., 2000). Variation in costs of infectivity, resistance or costs of disease among demes generates ndFDS and stable polymorphism in deterministic and stable long-term polymorphism in finite population size models (Tellier & Brown, 2011). In the model in the current study, greater variation between demes of u enhances stability of polymorphism. The cost of the resistance allele itself (u*) is not necessary for polymorphism to be stable if ε > 0 in both demes (Fig. S5). This observation may explain in part the lack of fitness measures of the cost of harbouring a resistance allele (Brown & Tellier, 2011). Indeed, this cost may be very small, and thus not detectable, if a cost of triggering resistance exists (ε). Hosts regularly encounter pathogens and herbivores (even in the glasshouse environment), which may activate gene expression and gene networks often during the lifetime of a plant. As a consequence, measuring such a cost of gene or gene network activation may be empirically more challenging (Katagiri & Tsuda, 2010). It is suggested that variation in costs of triggering resistance gene networks may vary depending on the biotic and abiotic environment and be heterogeneous in space with potential larger values than the cost of harbouring the resistance allele. If both u* and ρ differ between populations, stability of genetic polymorphism can be achieved in a larger area of the parameter space compared to the case when only one of the two parameters varies (Fig. 4). This suggests that spatial variation in both disease prevalence and other costs may contribute to further stabilize polymorphism in natural populations. Interestingly, in demes with very low or non-existent disease (coldspots in this model), the costs of resistance (u*) may be expected to evolve towards very small or zero values, although this cost may be higher in other demes with high disease prevalence (hotspots in our model).

Speed of co-evolution

The speed of co-evolution, measured as the period of allele frequency cycles, in host–parasite systems may allow contrasting evolutionary scenarios in natural populations to be distinguished (Gandon et al., 2008). The results of the current study suggest that disease prevalence modifies these expectations. In fact, higher prevalence of the parasite speeds up co-evolutionary cycles even when all the other parameters are constant (Fig. S2), due to the effect of iFDS, which is stronger for higher values of ρ. Because variation in parasite prevalence is common in the wild, parasite encounter rates are likely to have an important effect on the speed of evolution observed in natural populations. Additionally, the results show that migration from demes with high prevalence of the parasite increases the speed of evolution in demes with low prevalence (Fig. S4). This is consistent with previous theoretical (Gomulkiewicz et al., 2000) and experimental studies in bacterial populations (Vogwill et al., 2009).

The current study has shown that accounting for parasite prevalence in the simple GFG model can generate (i) transient genetic polymorphism in a single population by ‘statistical polymorphism’ and (ii) long-term stable polymorphism in spatially structured populations with heterogeneous disease prevalence. Future empirical studies in natural populations are necessary to quantify selection in the host population in the absence of parasites, the conditions generating spatiotemporal variation in disease prevalence, but also the cost of triggering plant defences in different environmental conditions.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

AT acknowledges support from DFG grant HU 1776/1 to Stephan Hutter and the BMBF Synbreed AgroCluster ‘Genomics of domestication in plants and animals’. WS was funded by DFG grants HU 1776/1 and STE 325/14. SM was funded by the European Union through the Erasmus Mundus Master Program in Evolutionary Biology.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information
FilenameFormatSizeDescription
ppa12131-sup-0001-SuppInfoS1.pdfapplication/PDF530KSupporting Information S1. Effect of disease prevalence and spatial heterogeneity on polymorphism maintenance in host–parasite interactions
ppa12131-sup-0002-FigS1.pdfapplication/PDf86KFigure S1. Phase plot of the frequencies of infectivity and resistance for the model with one deme.
ppa12131-sup-0003-FigS2.pdfapplication/PDF67KFigure S2. Number of co-evolutionary cycles as a function of the parasite encounter rate in the host (a) and in the parasite (b).
ppa12131-sup-0004-FigS3.pdfapplication/PDF140KFigure S3. Statistical polymorphism for the resistance allele when there is no cost of resistance (u* = 0) for low disease prevalence (a) and high prevalence (b).
ppa12131-sup-0005-FigS4.pdfapplication/PDF149KFigure S4. Allele frequencies over time for a single deme with low disease prevalence (a), and for the same deme linked by migration with a deme with high disease prevalence (b).
ppa12131-sup-0006-FigS5.pdfapplication/PDF148KFigure S5. One simulation run with heterogeneity in disease prevalence among demes when u = 0 in both demes..
ppa12131-sup-0007-figuresdataS1.docxWord document11K 

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