Modeling induced resistance to plant disease using a dynamical systems approach

Standard epidemiological models describe that Susceptible plants (S) will become infected and develop Disease (D) after inoculation with a compatible pathogen under appropriate environmental conditions. These dynamic relationships can be affected by subtle changes to any one parameter and may result in a proportion of the plant population being able to exhibit Resistance (R) to infection. An example of this is the use of elicitors to promote an increase in plant basal resistance and so enable a proportion of formerly susceptible plants to express resistance i.e., a shift in the population from S to R. This phenomenon is termed induced resistance (IR). In this paper, a prototype mathematical model of IR is presented to describe the effects of a chemical elicitor compound, methyl jasmonate (MeJA), on the resistance of Pinus radiata seedlings to Diplodia pinea the causal agent of pine stem canker and tip dieback. Pine seedlings were sprayed with 0.1% MeJA at 27, 13, 6, or 3 days before inoculation with D. pinea using methods previously described by Gould et al. (2008). Disease assessments commenced at 1 week after inoculation and continued at 3–4 day intervals thereafter for 5 weeks. Disease development on the MeJA-treated seedlings was compared to that on a cohort of untreated plants. In this model system, the IR response is transient and it is modeled here using a forward-and-backward kinetics framework to describe the temporal nature of the phenomenon.

Because the expression of IR can only be detected after pathogen challenge, the model is formulated with the treated plants divided into two regimes: (1) pre-inoculation and (2) post-inoculation. The assumptions for the model's formulation can be summarized as follows. The plant population is divided into three compartments according to the above definitions where S + R + D = 1. At the time when plants are treated with an elicitor (t = 0), a proportion of the plant population will exhibit natural resistance (R_i). The induction period (t_p) describes the time interval between elicitor application and pathogen inoculation. Upon inoculation, a proportion of plants (D_i) will become infected immediately. This prototype IR model is based on the model by Jeger et al. (2009) and Xu et al. (2010). The model's equations for the treated plants are as follows:

Pre-inoculation: For 0 ≤ t < t_p

Post-inoculation: For t_p ≤ t ≤ T where we take [days⁻¹] as the elicitor effectiveness in the plants where [days⁻¹] is the maximum elicitor effect and L [days] is the time where this is at its peak, γ [days⁻¹] is the rate that resistant tissue becomes susceptible, and also β [days⁻¹] is the rate of disease development. The form for e(t) is chosen to reflect the temporal nature of IR with an initial increase in resistance which then decays over time. R_p is the degree of the resistance at the time of the pathogen inoculation obtained from Equation (1). Also, T is a finite (sufficiently large) time after the pathogen inoculation. The Equations (1)–(3) are based on the assumption that the rate of change of R and D are directly proportional to the amount of S available at a particular time (that is, S = 1 − R − D). For Equation (1), D is not in the equation because the pathogen is absent during this period. Pathogen is inoculated at time t_p, therefore the D term is occurs only in Equations (2) and (3). For the untreated plants, they share the same parameter values, especially β, R_i, D_i, since these untreated plants are characterized as the control group. The untreated plants will not have the e(t) term in their model equations and so are autonomous, that is, independent of time except implicitly. Therefore, the model's equations for the untreated plants are as follows: where α [days⁻¹] is the rate that untreated susceptible tissue becomes resistant.

The seven unknown parameters (α, β, γ, k, L, R_i, D_i) are determined by matching data to the model using computer-based algorithms. The following figure (Figure 1) was plotted using the optimal parameter values, and it illustrates the characteristics of the two compartments R and D for the Equations (1)–(5). The flows in the figure indicate the time-evolution of the system of the differential equations described above based on the different values of t_p and the untreated case. It shows that when t_p is between 3 and 6 days, the subsequent development of disease is less severe than with the other induction times and with the untreated 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 + D = 1. The model has three compartments (S, R, D), but the data sets have only two ([S + R], D). The Susceptible cohort is given by S = 1 − R − D and can be separated out by this model. The model has a line of equilibrium states in the R, D plane which are attracting, and predicts that disease development depends on the induction time t_p.

Figure 1

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