Journal of Theoretical Biology 347 (2014) 144–150

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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi

Modelling induced resistance to plant diseases Nurul S. Abdul Latif a,b, Graeme C. Wake b,n, Tony Reglinski c, Philip A.G. Elmer c a b c

Faculty of Agro Based Industry, Universiti Malaysia Kelantan, Jeli, Kelantan, Malaysia Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealand The New Zealand Institute for Plant and Food Research Limited, Hamilton, New Zealand

H I G H L I G H T S

 Determining a plausible dynamical system that describes the SIR compartments for disease.  Investigating control strategies that will minimise the onset of disease.  Using a case study of experimental data enables parameter values to be found.

art ic l e i nf o

a b s t r a c t

Article history: Received 1 June 2013 Received in revised form 25 November 2013 Accepted 20 December 2013 Available online 5 January 2014

Plant disease control has traditionally relied heavily on the use of agrochemicals despite their potentially negative impact on the environment. An alternative strategy is that of induced resistance (IR). However, while IR has proven effective in controlled environments, it has shown variable field efficacy, thus raising questions about its potential for disease management in a given crop. Mathematical modelling of IR assists researchers with understanding the dynamics of the phenomenon in a given plant cohort against a selected disease-causing pathogen. Here, a prototype mathematical model of IR promoted by a chemical elicitor is proposed and analysed. Standard epidemiological models describe that, under appropriate environmental conditions, Susceptible plants (S) may become Diseased (D) upon exposure to a compatible pathogen or are able to Resist the infection (R) via basal host defence mechanisms. The application of an elicitor enhances the basal defence response thereby affecting the relative proportion of plants in each of the S, R and D compartments. IR is a transient response and is modelled using reversible processes to describe the temporal evolution of the compartments. Over time, plants can move between these compartments. For example, a plant in the R-compartment can move into the S-compartment and can then become diseased. Once in the D-compartment, however, it is assumed that there is no recovery. The terms in the equations are identified using established principles governing disease transmission and this introduces parameters which are determined by matching data to the model using computer-based algorithms. These then give the best match of the model with experimental data. The model predicts the relative proportion of plants in each compartment and quantitatively estimates elicitor effectiveness. An illustrative case study will be given; however, the model is generic and will be applicable for a range of plant–pathogen–elicitor scenarios. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Elicitor Induced resistance Diplodia pinea Methyl jasmonate Dynamical system

1. Introduction Conventional crop production systems rely heavily on the use of synthetic agro-chemicals to manage plant disease. However, concerns about potentially negative effects of agro-chemicals on the environment have increased the demand for the development of biologically based disease control methods. One strategy that has gained more popularity in the last decade involves the application of inducing agents known as elicitors to stimulate the natural plant resistance response to pathogen attack; this

n

Corresponding author. E-mail address: [email protected] (G.C. Wake).

0022-5193/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2013.12.023

approach is referred to as induced resistance (IR). A range of naturally occurring elicitors have been identified and these include polysaccharides such as glucan and chitosan and also plant hormones such as salicylates and jasmonates. Elicitors are not antimicrobial per se but operate primarily by stimulating plant defences to enable a more rapid and intense resistance response to future pathogen attacks. The elicited responses are orchestrated via complex signalling cascades and typically there is a lag of several days between the elicitor application and the onset of IR. Some studies suggest that elicitors can induce a broad spectrum and long lasting resistance; however, more commonly it has been shown that IR is transient and lasts for a few weeks only. The efficacy of IR has been reported to range from 20 to 85% (Walters et al., 2005) and this variability is a function of the dynamic

N.S. Abdul Latif et al. / Journal of Theoretical Biology 347 (2014) 144–150

relationships between the plant, the pathogen and the environment. As we gain a greater understanding of these relationships so will we be better placed to develop smarter elicitor application strategies and so reduce the variability of IR in the field. This is essential for the practical implementation and acceptance of IR as a crop protection strategy in the future. In this paper, we describe a generic mathematical model that incorporates the elicitor effect to combat disease infection that was initially introduced in Abdul Latif et al., 2013 Numerical evaluation of this IR model provides a baseline to determine the optimum timing of the elicitor application. We extended this proposed model to include the effects of the following factors on disease development: (i) the timing of elicitor application, (ii) the multiple applications of elicitor, and (iii) post-disease elicitor treatment.

2. Model development 2.1. Background and definitions Standard epidemiological models for plant disease development are based upon interactions between three critical components: the plant, the pathogen and the environment. Spread of disease is the result of migration of the pathogen. Dispersal is the movement of the pathogens0 dispersal units (e.g spores) from the place where they are formed to the plant tissue they infect. Some have referred this as the “conquest of space” (Zadoks and Schein, 1979). Only a few fungi can move actively from host to host and they belong to the wood rotting fungi where they form rhizomorphs (underground stems, up to 10 m to progress from one root to another). Active movement of fungi is the exception and passive movement is the rule. There are many mechanisms of dispersal such as leaf rubbing, rain splash, turbulent transfer (wind borne), water-borne, animal or vertebrate-borne, insect (invertebrateborne) and human-borne. Under disease conducive conditions susceptible (S) plants become infected and develop disease (D) when inoculated with a compatible pathogen. However, this is a dynamic relationship and a subtle change to any one parameter may result in a proportion of the plant population being able to exhibit resistance (R) to infection. In this study a plant defence elicitor was applied to susceptible plants in order to elevate their basal resistance and so enable a proportion of formerly susceptible plants to express IR i.e. a shift in the population from S to R. This prototype IR model is based on the model by Jeger et al. (2009) and Xu et al. (2010). Their models were developed for a generic biocontrol system. We extended these models to include IR as a specific biocontrol system and to determine the effectiveness of the elicitor used as the agent for the IR mechanism. There are no models in the literature which specifically describe the interactions among the plant, the pathogen, and the elicitor. Typically, inducible defences are triggered in response to pathogen attacks. Therefore, the model is formulated with the elicitor treated plants divided into two regimes: (i) pre-inoculation (before pathogen arrives) and (ii) postinoculation (after pathogen arrives). The assumptions for the model0 s formulation can be summarised as follows: 1. The plant population is divided into three compartments according to the above definitions where S þR þ D ¼ 1. 2. At the time when plants are treated with an elicitor ðt ¼ 0Þ, a proportion of the plant population will exhibit natural resistance ðRi Þ. 3. The induction period ðt p Þ describes the time interval between elicitor treatment and pathogen inoculation. Upon inoculation, a proportion of plants ðDi Þ will become infected immediately. This is shown schematically in Fig 1.

145

The model0 s equations for the treated plants are as follows: Pre-inoculation: For 0 o t ot p dR ¼ ðeðtÞ  γ RÞð1  RÞ; dt

Rð0Þ ¼ Ri

ð1Þ

Post-inoculation: For t p o t o T dR ¼ ðeðtÞ  γ RÞð1  R  DÞ; dt dD ¼ βDð1  R DÞ; dt

Rðt p Þ ¼ Rp

Dðt p Þ ¼ Di

ð2Þ ð3Þ

and we take the elicitor effectiveness in the plants as eðtÞ ¼

kt

ð4Þ

t 2 þ L2 1

where k=2Lðdays Þ is the maximum elicitor effect and L (days) is 1 the time where this is at its peak, γ ðdays Þ is the rate that 1 resistant tissue becomes susceptible and βðdays Þ is the rate at which the disease spreads. The elicitor effectiveness e(t) is assumed to be a time dependent function (see comments in the introduction). Other functional forms for e(t) were tried, and this one gave a best fit and matched the biological insight from experiments. Rp is the degree of the resistance at the time of the pathogen inoculation obtained from Eq. (1). Also, T is a finite (sufficiently large) time after the pathogen inoculation. Note that, Eqs. (1)–(3) are based on the assumption that the rate of change of R and D is directly proportional to the amount of S available at a particular time (that is, S ¼ 1  R  D) using the usual “mass-action-kinetics” (see Keener and Sneyd, 1998). This obviously applies in a standard way for Eqs. (3) and (6), however, it is less clear for the R compartment. The only mechanism that could cause this would be genetic in origin and this is beyond the scope of this study. The data sets (see Fig. 3) in our case study show that the steady state attained for D ( o 1) is not unique and is dependent on the initial point ðRi ; Di Þ which can be modelled by the inclusion of S ¼ 1  R  D (or 1  R) in the kinetics of the R compartment. For Eq. (1), D is not in the equation because the pathogen is absent during this period. Pathogen is inoculated at time tp, therefore the D term occurs only in Eqs. (2) and (3). The model0 s equation for the untreated plants is as follows: dR ¼  αRð1  R  DÞ; dt dD ¼ βDð1  R DÞ; dt

Rð0Þ ¼ Ri Dð0Þ ¼ Di

ð5Þ ð6Þ

where α (days  1) is the rate at which the untreated plants lose the resistance due to the pathogen attack. The untreated model shared the same parameters β, Ri , Di as the treated model as these untreated plants are characterised as the control group. This means there is no elicitor treatment for the control group.

Fig. 1. Here S, D, R are the proportion of plants in the respective compartments and the parameters γ, β and e(t) are the specific rates of movement between the linked compartments.

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Therefore, the untreated model will not have the e(t) term in the model0 s equations and so are autonomous, that is, independent of time except implicitly. Note that, the rate of R and D for the untreated plants is also directly proportional to the amount of S available this is shown schematically in Fig 2.

2.2. Model simulation The seven unknown parameters (α, β, γ, k, L, Ri , Di ) are determined by matching data to the model using MATLAB toolbox ‘fminsearch’. For illustration, this proposed model was fitted to the experimental data based on the effect of the elicitor, methyl jasmonate (MeJA), on the resistance of Pinus radiata seedlings to Diplodia pinea the causal agent of pine stem canker and tip dieback. The pine seedlings were divided into groups and sprayed with 0.1% MeJA at 35, 27, 21, 14, 13, 7, 6 or 3 days before inoculation with D. pinea using methods previously described in Gould et al. (2008). There was also an untreated control group for this experiment. Disease assessments for each treatment commenced at 1 week after inoculation and continued at 3–4 day intervals thereafter for 5 weeks. Here, the induction times are tp ¼ [3 6 7 13 14 21 27 35]. Our assumption is that for each induction time tp there will be different dynamics outcomes for the plant population, but share the same parameter values that apply for each tp. The ordinary least squares method was used to estimate the unknown parameter values. The model has three compartments (S, R, D), but the data sets have

Fig. 2. Here S, D, R are the proportion of plants in the respective compartments and the parameters are the specific rates of movement between the compartments.

only two (½S þ R; D). The susceptible cohort is given by S ¼ 1  R D and can be separated out by this model. The solution curve is plotted as in Fig. 3 using the optimal parameter values as in Table 1, and shows the dynamics of each compartment corresponding to its induction time tp ¼3 days. In the figure the comparison between the treated and untreated model is shown. The calculated coefficient of determination for the whole system i.e. for all induction case tp is R2 ¼ 0:6598. The experimental data is available for D (and hence S þ R ¼ 1  D) only and is shown in the marked points. Similar graphs are available for all tp and are similar to those chosen. Fig. 4 illustrates the characteristics of the two compartments R and D for Eqs. (1)–(6) with phase plane. The flows in the figure indicate the time-evolution of the system of the differential equations described above based on the different values of tp and the untreated case. It shows that when tp is between 3 and 7 days, the subsequent development of disease is less severe compared with the other induction times and with the untreated control plants. That is, the resistance induced by the elicitor application is at its peak. The figure shows that the trajectories for each induction case will eventually approach that straight line R þD ¼ 1. The model has a line of equilibrium states in the R, D plane which are attracting, and shows that disease development depends on the induction time tp. Fig. 5 shows the proportion of diseased seedlings which were treated at the final time against the induction time tp. The IR model can determine the optimal induction time tp that gives the best outcome for disease control for the treated pine seedlings. This can be done by taking the point at the final time tf as a fixed variable e.g. let t f ¼ t p þ 35, and minimise the trajectories of DFinal ¼ Dðt ¼ t f Þ. For tp ¼0 days, it means the elicitor treatment and the pathogen inoculation were introduced at the same time. Experimental data for diseased seedlings measured at t p þ 35 days for each tp case is taken into account to compare with the model outcome. This numerical experiment shows that induced resistance was greatest when MeJA was applied to pine seedlings 6 days before pathogen inoculation. Although the error gaps between the model outcomes and the experimental data are noticeable, especially for the case tp ¼14 days, it can be attributed to the variability of the data sets or the experimental error. Also, note that we assumed for each induction time tp, the dynamics of each tp share the same parameter values (see Table 1).

Healhty (S + R)

Diseased (D)

1

0.5

0 0

10

20

1

0.5

0

30

0

10

Susceptible (S)

Resistant (R)

1

0.5

0

0

10

20

Time (days)

20

30

Time (days)

Time (days)

30

1

0.5

0

0

10

20

30

Time (days)

Fig. 3. The vertical dashed line (here at 3 days) is shown at the induction time t ¼ t p at which time the pathogen arrives and causes the disease to develop further. The blue lines (—-) are the treated model and the red lines (– –) are the untreated model. The data is shown in the marked points. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

N.S. Abdul Latif et al. / Journal of Theoretical Biology 347 (2014) 144–150

147

Table 1 Description of variables and parameters used in the IR model for pine seedlings with elicitor MeJA with values obtained from matching model simulating with data. Variable / Parameter Description R D S α β γ k L tp Ri Di te r

Value [Units]

Proportion of plant population being able to express resistance to infection (dimensionless) Proportion of plant population being infected and become diseased (dimensionless) Proportion of plant population which is susceptible (dimensionless) The rate at which untreated plants lose the resistance due to the pathogen attack 0.0908 (days  1) The rate at which the disease spreads 0.7379 (days  1) The rate of the resistant plant become susceptible 0.2801 (days  1) Determines the effectiveness of the elicitor 5.3749 (dimensionless) The time where the elicitor effectiveness is at the peak 10.7607 (days) The induction time of the pathogen i.e. the time interval between the elicitor application and the pathogen challenge (days) A proportion of the plant population exhibit natural resistance at the initial time t¼ 0 0.6118 (dimensionless) A proportion of the plant population which becomes infected immediately after the pathogen challenge 0.0168 (dimensionless) Time of the elicitor application (see later) (days) The sublinear effect of the elicitor % concentration (see later) 0.6650 (dimensionless)

1

0.8 tp = 3 days

0.8

t = 6 days p

tp = 7 days

0.7

t = 13 days

Diseased

p

0.6

tp = 14 days t = 21 days

0.5

p

Pathogen is introduced here

0.4

t = 27 days p

t = 35 days p

0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Resistant Fig. 4. Phase plane for different induction time tp. These trajectories of the R, D compartments are plotted in the feasible region ðR; D 4 0; R þ D o 1Þ. The lines schematically represent the values of the two compartments R and D as time passes for Eqs. (1)–(5) based on each induction case tp and the untreated case. When D ¼ 0, the straight lines illustrate the dynamics of the R compartment before the pathogen inoculation. These lines will have a discontinuity when the pathogen is introduced at time t ¼ t p , and show the state of the R compartment at that particular time. As can be seen in the figure, there is a jump in the D values of Di at t ¼ t p and the trajectories of the R, D compartments will continue to approach the straight line R þ D ¼ 1. For the untreated case, the dynamics of the R, D compartments will depend on the initial condition of the systems i.e. Ri and Di.

3. Extensions to the IR model 3.1. Varying the elicitor concentration Next, this IR Model was extended to include the effect of different concentrations of the elicitor on its capability to induce resistance since some experimental studies have reported elicitor dose responses (Godard et al., 1999; Boughton et al., 2006; Gould et al., 2009; Vivas et al., 2012; Hindumathy, 2012). For model0 s illustrations, liquid suspensions of MeJA at concentrations of 0.0%, 0.023%, 0.025%, 0.1% or 0.4% were applied to P. radiata seedlings as a foliar spray. Each concentration was applied once and the seedlings were inoculated with D. pinea7 days after MeJA application. We assumed that the elicitor concentration is a sub-linear effect to the e(t) term. We choose this term as f ðaÞ ¼ ar eð1  aÞ where a is the scale for concentration and r ¼ 0.6650 is the sub-linear effect which determined by the least squares method. Note that the function f(a) is maximised at a ¼r. Therefore the effect of elicitor concentration to IR can be defined as r ð1  aÞ

concentration  eðtÞ ¼ a e



kt t 2 þ L2

:

The proportion of diseased seedlings at a fixed final time ( DFinal )

Untreated

0.9

Treated Model Untreated Model Experimental data

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

5

10

15

20

25

30

35

40

Induction time tp days Fig. 5. Long-term outcome ðt p þ 35Þ for diseased plants for different induction times tp.

The values of a as a concentration were scaled as a ¼0, 0.23, 0.25, 1, 4. If a ¼1, then f ð1Þ ¼ 1. The parameter values, including the physiological ones for the case study of pine seedlings, are this in line with those in Table 1 which we therefore use. Of course we are assuming the dynamics of the physiological response remains the same, expect for the elicitor effect when the dose is changed. For 0% concentration, the seedlings are assumed to behave as untreated seedlings. We say that the seedlings had a “water” treatment. From the model equations given above, this is true. If a ¼0, then the term above is equal to zero which implies the seedlings should be evaluated as the untreated model. Therefore, the model formulation will be: For 0% elicitor concentration assume as untreated case For pre-inoculation: 0 o t o t p dR ¼  αRð1  RÞ; dt

Rð0Þ ¼ Ri

ð7Þ

For post-inoculation: t p o t oT dR ¼  αRð1  R  DÞ; dt dD ¼ βDð1  R DÞ; dt

Rðt p Þ ¼ Rpunt

Dðt p Þ ¼ Di

ð8Þ

ð9Þ

Rpunt is obtained from (7) and is the value of Rðt p Þ for the untreated plants. For other elicitor concentrations

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0% MeJA concentration 1 1 0.8

Healthy

Diseased

0.8 0.6 0.4

0.6 0.4

0.2 0.2 0 0

10

20

30

0

40

0

10

30

40

1

Susceptible

Resistant

1 0.8 0.6 0.4 0.2 0

20

Time (days)

Time (days)

0.8 0.6 0.4 0.2

0

10

20

30

40

0

0

10

Time (days)

20

30

40

Time (days)

Fig. 6. Disease progression when tp ¼7 days for the 0% elicitor concentration (untreated plants) with marked points (⋄; ) giving the separate experimental values. The calculated SSE with these points are 0.3811.

For pre-inoculation: 0 ot o t p   dR kt ¼ ar eð1  aÞ 2  γ R ð1  RÞ; dt t þ L2

Rð0Þ ¼ Ri

For post-inoculation: t p o t o T   dR kt ¼ ar eð1  aÞ 2  γ R ð1  R  DÞ; 2 dt t þL dD ¼ βDð1  R  DÞ; dt

Rðt p Þ ¼ Rp

Dðt p Þ ¼ Di

ð10Þ

ð11Þ

ð12Þ

Figs. 6 and 7 show the model simulation for some different elicitor concentrations and match reasonably well the experimental behaviour. The results in Fig. 8 demonstrate that the effect of elicitor MeJA on seedling health is dose dependent. For example, it shows that if the MeJA concentration used is between 0.02% and 0.1% it will reduce disease incidence by approximately 70% compared with the 0.0% “water” treatment. On the other hand, if lower concentration ðe:g: o0:02%Þ is used, the outcome will be the same as the untreated seedlings (i.e. no disease efficacy). The numerical computation demonstrated that induced resistance was greatest when MeJA was applied at a concentration of at  0:067% the effect of the elicitor treatment to induced resistance is at the highest value. Note that the optimal value obtained is close to the value which maximises the function f(a), in the elicitor effectiveness factor. This finding has important cost benefit implications as this “optimal” dose of MeJA would increase its cost effectiveness making it more attractive as a plant protection product. In a study by Gould et al. (2009), it was indicated that if higher elicitor concentration (4 0:1%) was used, there was evidence of phytotoxicity in the treated seedlings where the disease incidence was higher. 3.2. Effect of multiple elicitor applications Next, we introduced multiple elicitor applications to the proposed IR Model. The elicitors were applied before the pathogen challenge. The scenarios are described in Fig. 9.

Here, we assumed that the term f ðaÞ ¼ ar eð1  aÞ is true. Therefore, the equations for a multiple application will be: First application: for 0 o t o t e2 dR kt ¼ ðf ðaÞ 2  γ RÞð1  RÞ dt t þ L2

ð13Þ

Second application: for t e2 o t o t p dR ¼ dt

" f ðaÞ

! # kðt  t e2 Þ þ f ðaÞ γ R ð1  RÞ;  t 2 þ L2 ðt  t e2 Þ2 þ L2 kt

Rðt e2 Þ ¼ Re2 ð14Þ

Pathogen inoculation: t p o t o T dR ¼ dt

" f ðaÞ

! # kðt  t e2 Þ þ f ðaÞ γ R  t 2 þ L2 ðt  t e2 Þ2 þ L2 kt

 ð1  R DÞ; dD ¼ βDð1  R DÞ; dt

Rðt p Þ ¼ Rp Dðt p Þ ¼ Di

ð15Þ ð16Þ

Here we only consider two elicitor applications. The equations above are formulated the same way as the single application. The same concentration of elicitor is applied at each application time. Eq. (13) describes the first application and Eq. (14) the second application at time t e2 . Eqs. (15) and (16) demonstrate the pathogen inoculation to the seedlings. Fig. 10 shows the dynamics of these multiple elicitor applications for each S,R and D compartments. The marked points are the experimental data. For this illustration, with 0.1% MeJA was applied 14 and 7 days before the pathogen inoculation. That is t e1 ¼ 0, t e2 ¼ 7 and tp ¼14. It must be noted that no parameterisation was done for this model extension. The same parameter values as in Table 1 and the r value stated in Section 3.1 were used to plot Fig. 10. The illustrations demonstrate the model that have the same patterns observed in the experiment.

N.S. Abdul Latif et al. / Journal of Theoretical Biology 347 (2014) 144–150

149

0.1% MeJA concentration 1 1 0.8

Healhty

Diseased

0.8 0.6 0.4

0.6 0.4

0.2 0.2 0 0

10

20

30

0

40

0

10

1

1

0.8

0.8

0.6 0.4 0.2 0

20

30

40

Time (days)

Susceptible

Resistant

Time (days)

0.6 0.4 0.2

0

10

20

30

40

0

0

10

20

30

40

Time (days)

Time (days)

Fig. 7. Disease progression when tp ¼ 7 days and for the treated plants with higher elicitor concentration (0.1%). The marked points (⋄, , n) are the separate experimental outcomes and the calculated SSE with these points are 0.1080.

For post-treatment: t e ot o T ! dR kðt  t e Þ ¼  γ R ð1  R DÞ; dt ðt  t e Þ2 þ L2

Varying elicitor concentration when inoculation time tp = 7 days

Diseased at the final time ( DFinal= D (tp+35) )

1

Model Experimental data

0.9

Rðt e Þ ¼ Re

ð19Þ

0.8

dD ¼ βDð1  R DÞ; dt

0.7

The pre-treatment scenario (Eqs. (17) and (18)) describes the seedlings after they have been challenged by the pathogen at t ¼0. We have assumed that they behave like the untreated seedlings as in Eqs. (1) and (2). Then, after some time te, the seedlings are treated with the elicitor. Therefore, the post-treatment formulations are given by (19) and (20) with eðtÞ ¼ kðt  t e Þ=ðt  t e Þ2 þ L2 . Values for Re and De are obtained from (17) and (18) respectively at time te. This model extension0 s dynamics are plotted in Fig. 11. The same parameter values in Table 1 are used to plot the graph. The model demonstrates that if the elicitor treatment is done after the pathogen challenge, these treated seedlings will be less severely affected compared to the untreated seedlings. This shows that the post-inoculation elicitor treatment assists seedlings recovery from being infected and becoming severely diseased. The model0 s equations (Eqs. (17)–(20)) show that the proportion of diseased plants is an increasing function of te. Thus, delaying the treatment will result in no significant difference in the treatment0 s outcome when compared to the untreated seedlings.

0.6 0.5 0.4 0.3 0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

a (concentration) Fig. 8. Final disease as a function of elicitor concentration. The marked points ðnÞ indicate the final experimental outcome. The fact that there are two points at a¼ 0.4 is because there are two different experiment used.

3.3. Elicitor treatment after the pathogen challenge The model is flexible enough to investigate different pathogen scenarios and in this next example, we investigated the effect of elicitor treatment applied after pathogen inoculation. Here is the model: Treated model For pre-treatment: 0 o t o t e dR ¼  αRð1 R  DÞ; dt dD ¼ βDð1  R  DÞ; dt

Rð0Þ ¼ Ri

Dð0Þ ¼ Di

ð17Þ

ð18Þ

Dðt e Þ ¼ De

ð20Þ

4. Conclusion The potential for using a plant-defence elicitor to induce resistance against plant disease is evaluated by use of a compartment model, which is in turn based on a SIR-type model, which here has the three compartments: Susceptible–Diseased–Resistant. This leads to a potentially powerful tool for quantifying the development of more potent elicitors and strategies for their application, both before and/or after pathogen attack and subsequent disease development. This enables the better management of disease progression which is critical for commercial operations

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Diseased

Fig. 9. Time-line for multiple elicitor applications (illustrative only).

parameters of course) be used for any combination of plant, elicitor, and pathogen. Of particular interest is the optimal induction time tp which leads to a minimisation of the long-term effect for elicitor application, either before or after (or both) the inoculation of the plants by the pathogen. This was determined for the pine seedling experiments described in this paper. Secondly, the effect of different concentration of elicitor and the frequency of elicitor application can be evaluated without further expensive experiments by simulated studies using this model. Accordingly we put forward this model as a significant tool for underpinning the understanding of disease control and the development of more effective elicitor agents for plant protection in the future.

Multiple application at 14 & 7 days before inoculation at 0.4% concentration 1 0.5 0 0

5

10

15

20

25

30

35

40

45

30

35

40

45

30

35

40

45

30

35

40

45

Healhty

Time (days) 1 0.5 0

0

5

10

15

20

25

Resistant

Time (days) 1 0.5 0

0

5

10

15

20

25

Susceptible

Time (days) 1 0.5 0

0

5

10

15

20

25

Acknowledgements

Time (days)

Fig. 10. Multiple application at 14 and 7 days before inoculation at 0.4% elicitor concentration. The marked points (n) are the experimental outcomes and the calculated SSE ¼0.0178.

1

References

Untreated Model

0.9

Treated Model

0.8

Pathogen is inoculated at t = 0

Diseased (D)

0.7 0.6 0.5 0.4 0.3

te = 6 days : the seedlings

te = 2 days : the seedlings

were treated with the elicitor

0.2

were treated with the elicitor

0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

The authors wish to thank J.T. Taylor for the experimental data collection and Ministry of Higher Education of Malaysia (KPTBS851110105416) for funding to support this study.

0.8

0.9

1

Resistant (R) Fig. 11. For this illustration, the seedlings were challenged by the pathogen at time t¼ 0 day, and then 2 days later or 6 days later (i.e. te ¼ 2 or te ¼6) the seedlings were treated with the elicitor. The blue curve illustrating the treated seedlings and the red curve corresponds to the untreated seedlings. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

(Gent et al., 2011). Further the model enables the likely long-term effect of the disease. The attracting steady states are in this model dependent on the initial conditions and not, as is usually the case, asymptotically stable. The case study used here was pine seedling (Pinus radiata) inoculated by the causal agent of terminal crook (Diplodia pinea) and was used to develop and validate the model0 s predictive capability. The model is generic and can (with different

Abdul Latif, N.S., Wake, G.C., Reglinski, T., Elmer, P.A.G., Taylor, J.T., 2013. Modelling induced resistance to plant disease using a dynamical system approach. Front. Plant Sci. 4, 1–3. Boughton, A.J., Hoover, K., Felton, G.W., 2006. Impact of chemical elicitor applications on greenhouse tomato plants and population growth of the green aphid, Myzus persicae. Entomol. Exp. Appl. 120, 175–188. Gent, D.H., De Wolf, E., Pethybridge, S.J., 2011. Perceptions of risk, risk aversion, and barriers to adoption of decision support systems and integrated pest management: an introduction. Phytopathology 101, 640–643. Godard, J.-F., Ziadi, S., Monot, C., Le Corre, D., 1999. Siluè, Benzothiadiazole (bth) induces resistance in cauliflower (brassica oleracea var botrytis) to downy mildew of crucifers caused by Peronospora parasitica. Crop Prot. 18, 397–405. Gould, N., Reglinski, T., Spiers, M., Taylor, J.T., 2008. Physiological trade-offs associated with methyl jasmonate-induced resistance in Pinus radiata. Can. J. Forest Res. 38, 677–684. Gould, N., Reglinski, T., Northcott, G.L., Spiers, M., Taylor, J.T., 2009. Physiological and biochemical responses in Pinus radiata seedlings associated with methyl jasmonate-induced resistance to Diplodia pinea. Physiol. Mol. Plant Pathol. 74, 121–128. Hindumathy, C.K., 2012. The defense activator from yeast for rapid induction of resistance in susceptible pearl millet hybrid against downy mildew disease. Int. J. Agric. Sci. 4, 196–201. Jeger, M.J., Jefferies, P., Elad, Y., Xu, X.M., 2009. A generic theoretical model for biological control of foliar plant disease. J. Theor. Biol. 256, 201–214. Keener, J., Sneyd, J., 1998. Mathematical Physiology. Springer-Verlag Inc, New York, United States of America. Vivas, M., Martín, J.A., Gil, L., Solla, A., 2012. Evaluating methyl jasmonate for induction of resistance to Fusarium oxysporum, F. circinatum and Ophiostoma novo-ulmi. Forest Syst. 21, 289–299. Walters, D., Walsh, D., Newton, A., Lyon, G., 2005. Induced resistance for plant disease control: maximizing the efficacy of resistance elicitors. Phytopathology 95, 1368–1373. Xu, X.M., Salama, N., Jeffries, P., Jeger, M.J., 2010. Numerical studies of biocontrol efficacies of foliar plant pathogens in relation to the characteristics of a biocontrol agent. Phytopathology 100, 814–821. Zadoks, J.C., Schein, R.D., 1979. Epidemiology and Plant Disease Management. Oxford University Press, New York, USA.

Modelling induced resistance to plant diseases.

Plant disease control has traditionally relied heavily on the use of agrochemicals despite their potentially negative impact on the environment. An al...
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