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Keywords:

  • breeding for resistance;
  • evolutionary stable strategy;
  • lesser developed countries;
  • model;
  • sustainable agriculture;
  • virus multiplication rate

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. The model and the evolutionary stable strategy
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information
  • 1
    Plant virus diseases severely constrain agricultural production world-wide, especially in less developed countries. The use of resistant cultivars may put a selection pressure on the plant virus to evolve towards more aggressive types. We used a modelling approach to show the impact of resistance, expressed through different mechanisms, on selection for plant virus strains with a higher multiplication rate. Begomoviruses (Geminiviridae: Begomovirus) inducing leaf curl diseases on tomato were used as key examples.
  • 2
    A model for the epidemiology of the plant–virus–vector system was combined with a model for the within-plant dynamics of the virus. Four types of resistance were defined, each expressing resistance through one or more model parameters. The evolutionary stable strategy (ESS) approach was used to study the effect of resistance on the evolution of within-plant virus multiplication rate.
  • 3
    Resistance expressed through reduced virus acquisition by the vector, and resistance expressed through reduced inoculation of the plant, do not put a selection pressure on the virus to evolve towards a higher multiplication rate.
  • 4
    Resistance expressed through reducing within-plant virus titre and symptom reducing resistance do put a selection pressure on the virus to evolve towards a higher multiplication rate.
  • 5
    Synthesis and applications. We have shown how to disentangle different types of plant resistance. We have also shown that each type of resistance puts a different type of selection pressure on the virus. Virologists and plant breeders can use these results to develop methods to characterize the combination of resistance mechanisms in the cultivars they breed and, on the basis of this, determine whether the cultivar will put a selection pressure on the virus to evolve more harmful types. This new approach to plant resistance against virus diseases has been made possible by combining ecological insight in virus–vector–plant interactions with ESS calculations.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. The model and the evolutionary stable strategy
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Plant viruses are a major threat to agricultural production, especially in less developed countries (Waterworth & Hadidi 1998; Rybicki & Pietersen 1999; Fereres, Thresh & Irwin 2000). Despite efforts to manage plant virus diseases, some viral disease problems are in fact continuing to emerge (Rybicki & Pietersen 1999; Varma & Malathi 2003). This is exemplified in particular by an escalation in disease epidemics caused by whitefly-transmitted geminiviruses (family Geminiviridae, genus Begomovirus). Several, seemingly new, diseases and many virus strains with altered pathogenicity have been reported over the past couple of decades (Brown 1990; Polston & Anderson 1997; Padidam, Sawyer & Farquet 1999; Mansoor et al. 2003a,b; Varma & Malathi 2003). The reason for the emergence of new strains is considered to be related to more rapid evolutionary changes of the virus brought about by increased vector and virus population sizes, more polyphagous vector populations and the potential for rapid genetic change in geminiviruses (Padidam, Sawyer & Farquet 1999; Varma & Malathi 2003; García-Arenal & McDonald 2003; García-Arenal, Fraile & Malpica 2003; Seal, van den Bosch & Jeger 2006).

Few direct means of control exist for most viral plant diseases. The available disease management options include the organization of agricultural practice, cultural control such as sanitation programmes, control of the vector population and use of host cultivars that support lower vector and virus populations. Vector population control has, however, often been difficult (Satapathy 1998; Perring, Gruenhagaen & Farrar 1999). Sanitation, in the form of roguing, and the use of resistant cultivars has in many cases been effective (Holt & Chancellor 1996; Holt, Colvin & Muniyappa 1999; Jeger et al. 2004). Much effort has been put into programmes to breed for resistance to begomoviruses and their whitefly vector Bemisia tabaci (Thresh, Otim-Nape & Lennings 1994; Bellotti & Arias 2001; Morales 2001; Lapidot & Friedmann 2002; Rubio et al. 2003). Any disease management effort will, however, put a selection pressure on the virus population to adapt to the new circumstances (Roossinck 1997). Given the rapid evolutionary changes in viruses it is no surprise that disease control that is initially effective is sometimes quickly rendered ineffective as a result of the adaptation of the virus (Roossinck 1997; Harrison 2002; McDonald & Linde 2002; García-Arenal & McDonald 2003; Mansoor et al. 2003b).

Mathematical models have been used to study the effects of plant, plant virus and vector characteristics on the development of disease epidemics (Chan & Jeger 1994; Holt & Chancellor 1996; Jeger et al. 1998; Madden, Jeger & van den Bosch 2000; Zhang, Holt & Colvin 2000a,b; Jeger, Dutmer & van den Bosch 2002; Escriu, Fraile & García-Arenal 2003). One of the significant shortcomings of these models is that most ignore the evolutionary dynamics of the plant virus, even though evolution is likely to have a major impact on disease control. The aim of this study was to develop models to study plant disease management that do take into account the evolutionary response of the plant virus to disease control measures. Our approach used the theory of evolutionary stable strategies (ESS), as introduced by Maynard Smith (1982), such that if a resident population engages in an evolutionary stable strategy it cannot be invaded by a population with a different strategy.

We restricted our study to one evolving virus trait, the multiplication rate of the virus within an infected cell. It should be stressed that our conclusions are only valid for the evolution of this trait. The within-cell multiplication rate varies widely between virus strains (Barker & Harrison 1986; Gray, Smith & Altman 1993; Jimenez-Martinez & Bosque-Perez 2004) and therefore was a useful starting point for our study. We used the within-cell virus multiplication to model the virus dynamics at the tissue level. Relationships between within-plant virus dynamics and population level parameters of acquisition, inoculation and roguing were used in accordance with data from experimental studies (Gill 1969; Foxe & Rochow 1975; Barker & Harrison 1986; Rubio et al. 2003; Jimenez-Martinez & Bosque-Perez 2004).

Although our model is of a generic nature, we used the specific system of whitefly-transmitted geminiviruses (begomoviruses) infecting tomato as our key example. Both experimental and modelling approaches have been used to study the epidemiology of tomato diseases (Holt, Colvin & Muniyappa 1999; Moriones & Navas-Castillo 2000). Holt, Colvin & Muniyappa (1999), studying tomato begomovirus diseases in India, reported that disease spread will occur for very low vector populations, and hence varietal resistance will be an important component of disease management. The breeding of begomovirus disease-resistant tomato cultivars is presently the subject of much research (Michelson, Zamir & Czosenk 1994; Vidavsky & Czosnek 1998; Lapidot et al. 2001; Pietersen & Smith 2002; Maruthi et al. 2003a,b; Gomez et al. 2004). These breeding programmes have been based on the introgression of resistance from accessions of wild Lycopersicon species (mainly L. peruvianum, L. chilense, L. pimpinellifolium and L. hirsutum) to cultivated tomato. Some of the types of resistance reported in this study can be found in the breeding lines of tomato breeding programmes, although it is not always easy to separate the various components of resistance.

The model and the evolutionary stable strategy

  1. Top of page
  2. Summary
  3. Introduction
  4. The model and the evolutionary stable strategy
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

virus-vector population model

The flow diagram Fig. 1a will be used to describe the model, which is a simplified version of the model by Jeger et al. (1998) and Madden, Jeger & van den Bosch (2000). In Appendix S1 (see the supplementary material) a step-by-step derivation of the model and a short description of its dynamics is given.

image

Figure 1. (a) Graphical representation of the virus-vector model. Boxes are the state variables: H, density of healthy plants; I, density of infectious plants; X, density of non-viruliferous vectors; Z, density of viruliferous vectors. (b) Schematic illustration of the within-plant virus dynamics. Boxes are state variables: U, density of uninfected cells; W, density of virions; V, density of infected cells.

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The plant population

The model describes the dynamics of the density (plants per units area) of healthy, H(t), and infected, I(t), plants through time. Healthy plants are planted at a constant rate, σ, reflecting a continuous cropping system. The productive lifetime of a plant is denoted by 1/h, implying that the number of healthy and infected plants removed per unit area per time unit equals hH(t) and hI(t), respectively. Infected plants have an additional removal rate, φ, which can either describe a death rate as a result of the virus infection or a roguing sanitation rate as a means of disease management. A healthy plant can become infected when a viruliferous vector inoculates the plant while feeding. To calculate the rate with which healthy plants become inoculated we have applied the commonly used mass action assumption, meaning that the rate is proportional to the density of healthy plants and viruliferous vectors, with a proportionality constant α. The parameter α will be called the inoculation rate parameter. We assume that an infected plant is infectious and a vector can acquire the virus during feeding. This implies that we omit a latent period in our model. We discuss the consequences of this assumption in the Discussion.

The vector population

The model describes the density of healthy vectors, X(t), and viruliferous vectors, Z(t), through time. We assume that total vector density, P, is constant and that thus X(t) +Z(t) = P. This assumption implies that the population birth rate, b(t), equals the population death rate that is calculated from ωX(t) + ωZ(t), where ω is the probability per time unit for a vector to die. Healthy vectors can acquire the virus when feeding on an infectious plant. Using similar assumptions as for the plant inoculation rate, the rate with which healthy vectors become viruliferous is calculated from δH(t)Z(t), where δ will be called the acquisition rate parameter. After acquisition of the virus vectors stay on average 1/τ time units viruliferous and then return to the healthy state.

The basic reproductive number

A key quantity in our analysis will be the basic reproductive number, R0, which for this model is given by:

  • image( eqn 1)

where ?=σ/h is the density of plants in the absence of the disease. This expression has a clear biological interpretation. Consider a disease-free crop. In this situation all plants are healthy, implying that ?=σ/h and also X=P. Now, consider the situation where one viruliferous vector flies into the crop. This viruliferous vector will infect αH plants per time unit. A viruliferous vector remains alive and viruliferous for, on average, 1/(ω + τ) time units (i.e. it either dies or loses the virus). Therefore, the total number of plants one viruliferous vector infects before it reaches the end of its life time or viruliferous period equals the first part on the left-hand side of equation 1. Next we consider the infected plants. Each infected plant will be visited by healthy vectors that then can become viruliferous themselves. The number of vectors that become viruliferous per time unit per infectious plant equals δP. An infectious plant remains infectious for, on average, 1/(h + φ) time units (i.e. either the plant is harvested or is rogued/removed). Therefore, the total number of vectors per infectious plant that acquire the virus before the plant is removed as a result of harvesting or roguing equals the second part on the left-hand side of equation 1. The product of the two terms thus has the interpretation of the number of viruliferous vectors resulting from one viruliferous vector, with infected plants as an intermediate step, when the viruliferous vector is surrounded by an entirely healthy plant population and an entirely non-viruliferous vector population. It is clear that if this number of new ‘cases’ per case is larger than unity an epidemic will develop, whereas if it is smaller than unity no epidemic will develop. The basic reproductive number, R0, thus is the threshold for epidemic development in a virus-vector system.

within-plant tissue virus dynamics

The flow diagram Fig. 1b will be used to describe the model. In Appendix S2 (see the supplementary material) a step-by-step derivation of the model is given and its dynamics are described. The model for the within-plant tissue virus dynamics describes the density of healthy cells, U(t), the density of infected cells, V(t), and the density of virions, W(t), in the plant. Virion density, W, will in the model (at least in steady state) be proportional to the virus titre, as used by many virologists.

Healthy tissue is assumed to produce cells at a constant rate λ. Each cell has a constant probability, d, per time unit to die. A healthy cell in the tissue can become infected if a virion enters the cell. Using the mass action assumption once more, we calculate the rate at which healthy cells become infected from βU(t)W(t), where β will be called the contact rate parameter. When a cell is infected the virion multiplies (through disassembly, multiplication of virus templates and subsequent reassembly; processes not explicitly covered in this paper) and produces new virions at a rate k per infected cell per time unit. An infected cells dies with a probability a per time unit. Finally, virions leave the system at a rate u, as a result of degradation of the virion or the virion moving into a plant tissue where no healthy and susceptible cells are available.

This model was introduced by Nowak & Bangham (1996) and Bonhoeffer et al. (1997) to describe animal viruses but in fact the model assumptions may be better suited to describe plant viruses. The amount of virions in an animal cell builds up till the cell bursts and releases the virus particles. In the actual model equations, however, virions are continuously produced and released by infected cells, as is the case for many of the plant viruses under consideration.

We discuss two quantities here that will be used in the rest of the paper. Nowak & Bangham (1996) have shown that the densities of uninfected cells, infected cells and virions develop towards a stable steady-state density. In steady state the ratio of infected to healthy cells, V*/U*, is given by:

  • image( eqn 2)

For large within-plant multiplication rates, the steady-state virion density is approximately:

  • image( eqn 3)

Note that equation 3 defines the relation between virus multiplication rate, k, and within-plant virus titre.

The evolving trait

As mentioned in the Introduction we restrict our attention to the evolution of the within-plant virus multiplication rate k because this is a virus trait that is known to vary widely between virus strains (Barker & Harrison 1986; Gray, Smith & Altman 1993; Jimenez-Martinez & Bosque-Perez 2004). All our conclusions will therefore only relate to the evolution of the virus multiplication rate. In the discussion section we will discuss possible trade-offs, biological attainable maximum values for k and speculate about its effects on the results.

relation between within-plant dynamics and population dynamic parameters

Here, we couple the within-plant virus dynamics with the epidemiological model, guided by published experimental research in this area.

The probability of acquiring the virus during feeding increases with virus titre and levels off at unity for very high virus titres (Gill 1969; Foxe & Rochow 1975; Barker & Harrison 1986; Jimenez-Martinez & Bosque-Perez 2004). We model this relation (Fig. 2a) as:

image

Figure 2. Relation between within-plant tissue dynamics and population dynamic parameters. (a) The acquisition rate parameter δ and inoculation rate parameter α as a function of the virus titre, W. (b) The roguing rate parameter φ as a function of the ratio of infected to healthy cells in the plant tissue, U/V.

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  • image( eqn 4)

where ø is the shape parameter and A=ψλ/au.

We assume that the inoculation rate parameter, α, is a constant, independent of the virus titre in the plant where the vector acquires the virus (Fig. 2a). However, we will show in the Discussion that this assumption can be relaxed without a change in the results.

Plants with a high ratio of infected to healthy cells will have a larger probability of dying as a result of the disease. Moreover, symptom severity is related to virus titre (Rubio et al. 2003) and thus symptom severity will be higher in plants with a high V*/U* ratio, which makes them more prone to roguing sanitation. The infected plant mortality rate, φ, will thus increase with V*/U*. Because of a lack of quantitative information on this aspect, we take a linear relation, Fig. 2b, giving:

  • image( eqn 5)

where B=ηβλ/a2u.

cultivar resistance

Resistance of cultivars to virus infection can be expressed through various model parameters. We summarize the knowledge of resistance of tomato to tomato yellow leaf curl virus (TYLCV) and a few other begomoviruses, and base our four types of resistance on this.

Gomez et al. (2004) developed four tomato lines introgressed from L. chilense (named LD3, LD4, LD5 and LD6). They showed that in these lines virus accumulation was very low (0·09–1·00 ng plant−1 60 days after inoculation) and no symptoms developed. These breeding lines are a clear example of the type of resistance we will name ‘virus titre-reducing resistance’. Gomez et al. (2004) compared these lines with the commercial F1 hybrids ARO 8479 and HA 3108. These hybrids developed high virus titres (more than 1000 ng viral DNA plant−1 60 days after inoculation). Symptom severity was very low in these hybrids and they represent clear examples of the type of resistance we describe as ‘symptom-reducing resistance’. Vidavsky & Czosnek (1998) found similar results with plants from accessions LA1777 and LA386 of L. hirsutum. Their BC1F4 line (denominated 902) does not show any virus accumulation even upon extensive whitefly mediated inoculation. Their BC1F4 line (denominated 908) did support virus accumulation but no symptoms developed, another example of symptom-reducing resistance. Various other authors have evaluated resistant cultivars and found examples of virus titre reduction and symptom-reducing resistance, often in combination (Michelson, Zamir & Czosenk 1994; Pietersen & Smith 2002; Maruthi et al. 2003a,b; Rubio et al. 2003).

It is remarkable how little research has been done on resistance expressed through inoculation and acquisition. A notable exception is the work of Lapidot et al. (2001). They measured acquisition and transmission by whiteflies for several resistant lines, and measured amounts of TYLCV DNA in inoculated plants and viruliferous whiteflies. Their analysis, however, did not allow a lower acquisition rate as a result of a lower plant virus titre to be distinguished from pure acquisition resistance. In addition to mechanisms operating on a biochemical level, plant architecture can also lead to inoculation and/or acquisition resistance. For example, high densities of leaf trichomes can hinder the movement of whiteflies and cause a reduction of virus acquisition and/or inoculation (Snyder & Carter 1985; Muigai et al. 2002).

On the basis of the literature survey summarized above we introduce four resistance parameters, ɛ, such that for ɛ= 0 the cultivar shows no resistance and ɛ= 1 implies the full extent of the resistance. Note that we do not propose here that a resistant cultivar will have one type of resistance only but we separate the mechanisms to assess their contribution to the evolution of the virus.

Inoculation resistance

Inoculation resistance relates to a reduction of the inoculation parameter, α, in the virus-vector population model. Introducing the resistance parameter (Fig. 3b), we write:

image

Figure 3. Definition of the four types of resistance studied in this paper. (a) Acquisition resistance. The acquisition rate parameter δ as function of the virus titre. The arrow depicts the effect of this type of resistance. (b) Inoculation resistance. The inoculation rate parameter as function of the virus titre. The arrow indicates the effect of this type of resistance. (c) Virus titre-reducing resistance. The acquisition rate parameter δ as a function of the virus titre. The arrow indicates the effect of this type of resistance. (d) Virus titre reducing resistance. The roguing rate parameter φ as function of the ratio of infected to healthy cells in the plant tissue. The arrow indicates the effect of this type of resistance. (e) Symptom-reducing resistance. The roguing rate parameter φ as function of the ratio of infected to healthy cells. The arrow indicates the effect of this type of resistance.

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  • image( eqn 6)

where inline image is a constant.

Acquisition resistance

Acquisition resistance relates to a reduction of the acquisition rate parameter, δ, in the virus-vector population model. Introducing this resistance parameter (Fig. 3a) using equation 4, we write:

  • image( eqn 7)

We stress here that this type of resistance does not give a smaller acquisition rate because of lower within-plant virus titre, but that for equal virus titres the virus is acquired less easily from the resistant cultivar by the vector.

Virus titre-reducing resistance

The mechanisms by which the virus titre is reduced are largely unknown. We therefore model this type of resistance in the simplest way by assuming that the steady-state virion density, W*, is reduced by a factor 1 − ɛW. Substituting this assumption into equations 2 and 3 and subsequently in equations 4 and 5 we find (Fig. 3c,d):

  • image( eqn 8)
Symptom-reducing resistance

A symptom-reducing cultivar is defined as a cultivar showing less symptoms at a similar virus titre. This implies that acquisition and inoculation remain the same but the mortality/roguing rate, φ, is reduced by a factor 1 − ɛφ because roguing rates and plant mortality as a result of the virus infection will be smaller. We use equation 5 (Fig. 3e), and write:

  • image( eqn 9)

Note that for this type of resistance we deliberately did not use the word tolerance, as this word has been defined in different and incompatible ways.

In the analysis of the evolution of the virus we use the most general form involving all four types of resistance and write:

  • image( eqn 10)

evolutionary stable strategy

We calculate the ESS for the virus multiplication rate, kESS (Maynard Smith 1982), such that if a resident virus population engages in this ESS it cannot be invaded by a virus with a different multiplication rate. For models of the type used here and the type of relation between k and the population level parameters, the ESS value of k is that value of k that maximizes the basic reproduction number equation 1 (Claessen & de Roos 1995; Van Baalen & Sabelis 1995; Dieckmann et al. 2002). Here we will simply use this result for our analysis, a summary of which can be found in Appendix S3 (see the supplementary material). Note, however, that although the basic reproduction number is maximized in the present model, this does not apply more generally (Dieckmann et al. 2002).

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. The model and the evolutionary stable strategy
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

evolutionary stable strategy

We find that the ESS value of k, kESS, is:

  • image( eqn 11)

This expression shows that the ESS is independent of the resistance parameters ɛα and ɛδ. This means that resistant cultivars with inoculation or acquisition resistance do not put a selection pressure on the virus to evolve a different multiplication rate, k. Virus titre-reducing resistance, ɛW, and symptom-reducing resistance, ɛφ, clearly do put a selection pressure on the virus to increase its multiplication rate.

introduction of the resistant cultivar

When the virus has not yet responded to the resistant cultivar, the virus multiplication rate is the default value. The virus titre in this situation, as a function of the resistance parameter ɛ is shown in Figs 4–6a and the density of healthy plants in this situation is given in Figs 4–6b.

image

Figure 4. The effect of inoculation resistance on virus multiplication rate, graph (c), within-plant virus titre, graphs (a) and (d), and the density of healthy plants, graphs (b) and (e). Default parameter values used are α= 0·008, σ= 0·003, h= 0·003, P= 50, ω= 0·12, φ= 0·1, θ= 0·016, A= 3 × 10−5, B= 0·01.

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image

Figure 5. The effect of virus titre-reducing resistance on virus multiplication rate, graph (c), within-plant virus titre, graphs (a) and (d), and the density of healthy plants, graphs (b) and (e). Default parameter values used are α= 0·008, σ= 0·003, h= 0·003, P= 50, ω= 0·12, φ= 0·1, θ= 0·016, A= 3 × 10−5, B= 0·01.

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image

Figure 6. The effect of symptom-reducing resistance on virus multiplication rate, graph (c), within-plant virus titre, graphs (a) and (d), and the density of healthy plants, graphs (b) and (e). Default parameter values used are α= 0·008, σ= 0·003, h= 0·003, P= 50, ω= 0·12, φ= 0·1, θ= 0·016, A= 3 × 10−5, B= 0·01.

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Inoculation resistance

For values of the resistant parameter, ɛα, close to unity the virus dies out and all plants are healthy (Fig. 4b).

Acquisition resistance

The results for acquisition resistant cultivars differ only in minor quantitative details from those of the inoculation resistance shown (data not shown).

Virus titre-reducing resistance

Virus titre decreases with increasing resistance (Fig. 5a). The density of healthy plants increases with increasing resistance. For very small values of the resistance parameter the virus is effectively removed (Fig. 5b).

Symptom-reducing resistance

Symptom-reducing resistance, as modelled, does not affect the virus titre (Fig. 6a). The density of healthy plants slightly decreases with increasing symptom-reducing resistance (Fig. 6b) because of the fact that symptom-reducing plants have a smaller mortality/roguing rate (equation 9) and infected plants remain longer in the system. This implies that for a tolerant cultivar to be effective this decrease in the density of healthy plants has to be outweighed by the gain, in terms of additional yield, compared with a non-tolerant cultivar, because of the symptom-reducing resistance of the crop to the disease.

evolution of the virus

The virus can respond to the resistant cultivar by evolving to a new evolutionary stable virus multiplication rate, kESS (Figs 4–6c–e).

Inoculation resistance

As the introduction of inoculation resistance (Fig. 4) does not select for virus strains with higher multiplication rates, the ESS value of k is equal for any resistance parameter, ɛα, value (Fig. 4c). Correspondingly, neither the virus titre nor the density of healthy plants change compared with the situation prior to evolution.

Acquisition resistance

The results are very similar to those for inoculation resistance and the same conclusions hold (data not shown).

Virus titre-reducing resistance

After the virus has fully evolved to the ESS, the multiplication rate, kESS, has increased (Fig. 5). The more resistant the cultivar is (the larger ɛW) the larger the ESS multiplication rate. As a consequence the virus titre evolves to the value it had before the resistant crop was introduced. The density of healthy plants decreases and in steady state it has reached the value it had before introduction of the resistant cultivar.

Symptom-reducing resistance

As with the virus titre-reducing cultivar, the introduction of a symptom-reducing resistant (Fig. 6) cultivar puts an evolutionary pressure on the virus to increase its within-cell multiplication rate, k. The ESS value of k is larger for more resistant cultivars (smaller ɛφ). Once the ESS is approached the virus titre in the plant has increased. The density of healthy plants has not changed much, although it is slightly reduced compared with the situation prior to the introduction of the new host cultivar.

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. The model and the evolutionary stable strategy
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

When a resistant cultivar is introduced, the virus is not yet evolved to be adapted to the new situation. Our findings (Figs 4–6a,b) correspond with those of other modelling studies (Holt & Chancellor 1996; Holt, Colvin & Muniyappa 1999; Jeger et al. 2004). The effect of symptom-reducing resistance on the dynamics of a viral plant disease has not previously been modelled. The density of healthy plants slightly decreases with increasing levels of symptom-reducing resistance, which is understandable as the mortality/roguing rate decreases with increasing symptom-reducing resistance in the crop. We conclude that results obtained when the virus is allowed to evolve are thus not a result of a model specification that contradicts other models.

When the virus has had time to adapt to the use of host resistance, two broad categories of virus responses to resistance can be distinguished. The first type, including (i) inoculation resistance and (ii) acquisition resistance, does not put a selection pressure on the virus to evolve towards a higher virus multiplication rate. The second type, including (iii) virus titre-reducing and (iv) symptom-reducing resistance, does put a selection pressure on the virus to evolve towards higher virus multiplication rates.

The cultivars from this second group are not durable in the sense that they put selection on the virus that might reduce the effectiveness of the resistance. Whether symptom-reducing resistant cultivars and virus titre-reducing cultivars have a contribution to make is, however, dependent not only on this selection pressure but also on additional factors. The symptom-reducing resistant cultivar might still have a positive effect on yield even when the virus has evolved the new ESS simply because the symptom-reducing resistance provides a sufficient amount of additional damage reduction. Furthermore it could be that the ESS virus multiplication rate is outside the attainable range for the virus under consideration. Finally the time it takes the virus to evolve towards the new ESS virus multiplication rate may be so long to make it of little importance.

We are thus not proposing that symptom-reducing and virus titre-reducing cultivars do not have a contribution to make to the management of virus diseases, but rather that there are potential problems that need to be considered to maximize their durability and minimize their effects on viral populations. The breeding of inoculation resistant and acquisition resistant cultivars does not pose the same potential problems.

The model used in this paper is simple; several model extensions and other evolving virus traits have to be investigated before general conclusions can be drawn. Our model, however, does allow for several extensions without any change in the conclusions.

  • 1
    Figures 4–6 were constructed using the parameter values given in the figure legends. We have done the same calculations for a wide variety of parameter values around the values used (plus and minus 100%) and found that although quantitative differences occur, the qualitative trends do not change.
  • 2
    The plant–virus–vector population model does not include a latent period for the plant nor a latent period for the virus in the vector. These model extensions are discussed by Jeger et al. (1998) and Madden, Jeger & van den Bosch (2000). It is easily seen that the ESS value of the virus multiplication rate, k, is still the value maximizing the basic reproductive number, implying that the trends in Figs 4–6 do not change when introducing these model extensions.
  • 3
    We assumed that the inoculation rate, α, is independent of virus titre and, because virus titre is related to virus multiplication rate, equation 3, to k. This is probably the least supported assumption about the relation between virus titre and population model parameters we have used. It is, however, possible to relax this assumption without any change in the results and conclusions. One might argue that the inoculation rate is related to virus titre as:
  • image

and introducing the resistance parameters we have:

  • image

Substituting this expression into the basic reproductive number and calculating the ESS value of the virus multiplication rate k, we find that kESS can be calculated from:

  • image

This equation can only be solved numerically. We have calculated Figs 4–6 using this expression and found no qualitative differences, although quantitative differences do occur.

We have thus investigated a number of models. In future, special attention needs to be given to other virus traits that are under selection pressure. Furthermore, assumptions on model structure need to be investigated for their effect on model outcome. For example, we use a mass action assumption for the transmission of viruses between plants and virions between cells. This assumption has been criticized (Cuddington & Beisner 2005) for its widespread use and research is needed either to justify its use in the present context, or the consequences of relaxing this assumption on model output must be quantified.

As Rubio et al. (2003) state, the lack of good methodologies for the evaluation of components of plant resistance is a limiting factor to the breeding for resistance. Further, the lack of consistent terminology related to resistance may be hindering the dissociation of resistance components in breeding programmes (Lapidot & Friedmann 2002). We hope that our definitions, as expressed in equations 6–9, help to categorize resistance components and develop methods to distinguish these.

In breeding programmes visual assessment of symptoms is often used to select plants for further breeding. This method will inevitably cause a bias towards the breeding of symptom-reducing or titre-reducing resistant lines. Plants expressing inoculation resistance or acquisition resistance will not be recognized because they will develop symptoms once infected. Further, in the evaluation of the usefulness of cultivars it is necessary to know which mixtures of resistance mechanisms are operating in the cultivar and their extent. Such information is a prerequisite to evaluating whether the evolution of the virus towards a higher virus multiplication rate will render the resistance of the cultivar ineffective.

Rubio et al. (2003) provide a major step towards a methodology to analyse the various types of resistance operating. They concentrated on unravelling the virus titre reduction and symptom-reducing resistance mechanisms. In combination with the methods used by Lapidot et al. (2001), which estimate acquisition and inoculation resistance, it should be possible to develop a methodology to separate these four types of resistance.

Besides the use of the model in distinguishing different types of resistance, our work can also help to develop hypotheses that can be tested experimentally. From the work presented in this paper we hypothesize that:

  • 1
    cultivars expressing resistance through virus titre reduction or through symptom reduction can eventually lead to the evolution of virus strains that build up higher virus titres in the plant compared with the cultivar they are derived from;
  • 2
    cultivars that reduce acquisition or inoculation, such as cultivars with leaf trichomes, etc., will not lead to the evolution of virus strains that build up a higher virus titre in the plant.

The key message from our work is that resistance to viral plant diseases can be expressed through several mechanisms. Each of these resistance mechanisms exerts its own characteristic selection pressure on the virus. There has been inadequate recognition of this by virologists and by breeders for virus resistance, but is, as we have shown in this paper, of key importance to the evolutionary response of the virus to the resistant cultivar. Methods need to be developed by practitioners to distinguish the different types of plant resistance, and to determine the combination of resistance mechanisms operating in a cultivar. Breeding for virus resistance can then be directed to deliver cultivars that put a minimal selection pressure on the virus to evolve more harmful strains. This paper provides the first step in disentangling the resistance components and their implications for virus evolution.

The ecological and epidemiological interactions included in the model allowed us to translate the mechanisms at the level of the within-plant virus dynamics to the epidemiological level. By so doing we have been able to quantify how the four types of resistance select or do not select virus strains with higher within-plant multiplication rates. The use of ecological theory and evolutionary stable strategy calculations has thus resulted in insight into the practical aspects that virologists and plant breeders will need to consider to improve programmes aimed at breeding for resistance. At present, few researchers determine the parameters through which resistance is expressed, and no adequate test of the hypotheses above is available. We hope that this paper inspires researchers to obtain experimental data with which to test the hypotheses.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. The model and the evolutionary stable strategy
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

This project is funded by the Crop Protection Programme (CPP) of the UK Department for International Development (DFID). Rothamsted Research receives support from the Biotechnology and Biological Research Council (BBSRC) of the UK. The authors are grateful for the interesting discussions during this research with the help of Richard Gibson, Francisco Morales, Pamela Anderson, John Colvin, James Legg and other participants in the Tropical Whitefly Initiative.

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  7. Acknowledgements
  8. References
  9. Supporting Information
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Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. The model and the evolutionary stable strategy
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Appendix S1. THE VIRUS-VECTOR POPULATION MODEL The plant population

Appendix S2. THE WITHIN-PLANT VIRUS DYNAMICS

Appendix S3. EVOLUTIONARY STABLE STRATEGY AND CONTINUOUSLY STABLE STRATEGY

FilenameFormatSizeDescription
JPE1159_AppendixS1.doc47KSupporting info item
JPE1159_AppendixS2.doc27KSupporting info item
JPE1159_AppendixS3.doc21KSupporting info item

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