Can high risk fungicides be used in mixtures without selecting for fungicide resistance? 3 1 0 2 Alexey Mikaberidze, p e Bruce A. McDonald, S 0 Sebastian Bonhoeffer 2 ] E P . o i b - q [ 2 v 1 6 5 6 . affiliation: Institute of Integrative Biology, ETH Zurich 1 0 3 keywords: epidemiology, plant disease, mathematical model, host-pathogen interaction, 1 : fungicide resistance, population dynamics v i X r corresponding author: a Alexey Mikaberidze ([email protected]) Abstract Mikaberidze, A., McDonald, B. A., Bonhoeffer, S. 2013. Can high risk fungicides be used in mixtures without selecting for fungicide resistance? Phytopathology. Fungicide mixtures produced by the agrochemical industry often contain low-risk fungicides, to which fungal pathogens are fully sensitive, together with high-risk fungi- cides known to be prone to fungicide resistance. Can these mixtures provide adequate disease control while minimizing the risk for the development of resistance? We present a population dynamics model to address this question. We found that the fitness cost of resistance is a crucial parameter to determine the outcome of competition between the sensitive and resistant pathogen strains and to assess the usefulness of a mixture. If fitness costs are absent, then the use of the high-risk fungicide in a mixture selects for resistance and the fungicide eventually becomes nonfunctional. If there is a cost of resis- tance, then an optimal ratio of fungicides in the mixture can be found, at which selection for resistance is expected to vanish and the level of disease control can be optimized. Fungicide resistance is a prime example of adaptation of a population to an environ- mental change, also known as evolutionary rescue [6, 24]. While global climate change is expected to result in a loss of biodiversity in natural ecosystems, evolutionary rescue is seen as a mechanism that may mitigate this loss. In the context of crop protec- tion the point of view is quite the opposite: reducing adaptation of crop pathogens to chemical disease control would help stabilize food production. Better understanding of the adaptive process may help slow or prevent it. This requires a detailed quantitative understanding of the dynamics of infection and the factors driving the emergence and development of fungicide resistance [11]. Despite the global importance and urgency of fungicide resistance, this problem has received relatively little theoretical considera- tion (see [22, 21, 51, 42, 50, 37] and [11] for a comprehensive review) as compared, for example, to antibiotic resistance [40, 10, 34, 4]. In recent years, agrochemical companies have begun marketing mixtures that contain fungicides with a low-risk of developing resistance with fungicides that have a high-risk developing of resistance. In extreme cases the high-risk fungicide is no longer effective against some common pathogens because resistance has become widespread. For exam- ple, a large proportion of the European population of the important wheat pathogen Mycosphaerella graminicola (recently renamed Zymoseptoria tritici) [39, 41] is resistant to strobilurin fungicides [54]. A number of previous modeling studies addressed the effect of fungicide mixtures on selection for fungicide resistance (for example, [28, 52, 27, 49, 22, 23]). Different studies used different definitions of “independent action” (also called “additivity” or “zero interaction” in the literature) of fungicides in the mixtures [48] and reported somewhat different conclusions. One study [48] critically reviewed the outcomes of these earlier studies and attempted to clarify the consequences of using different definitions of “independent action”. Some studies found that alternations are preferable to mixtures 3 [28], while others found that mixtures are preferable to alternations [52]. A more recent study [23] addressed this question using a detailed population dynamics model and found that in all scenarios considered, mixtures to provided the longest effective life of fungicides as compared to alternations or concurrent use (when each field receives a single fungicide, but the fungicides applied differ between the fields). This study used the Bliss’ definition of “independent action” of the two fungicides [9] (also called Abbot’s formula in the fungicide literature [1]). Weaddressed thequestion ofwhether mixtures of low-riskandhighriskfungicides can provide adequate disease control while minimizing further selection for resistance using a simple population dynamics model of host-pathogen interaction based on a system of ordinary differential equations. We found that the fitness cost associated with resistance mutations is a crucial parameter, which governs the outcome of the competition between the sensitive and resistant pathogen strains. A single point mutation associated with fungicide resistance sometimes makes the pathogen completely insensitive to a fungicide, as is the case for the G143A mutation giving resistance to strobilurin fungicides in many fungal pathogens [15, 17]. In many other cases the resistance is partial, for example, resistance of Z. tritici and other fungi to azole fungicides [13, 57]. Therefore, we considered varying degrees of resistance in our model. In contrast to our study, resistance in [22] was assumed to bear no fitness costs for the pathogen. It was found that in the absence of fitness costs the use of fungicide mixtures delays the development of resistance [22]. This conclusion is in agreement with our results (see AppendixA.4). Here we focus on finding conditions under which the selection for the resistant pathogen strain is prevented by using fungicide mixtures. 4 1. Theory and approaches We use a deterministic mathematical model of susceptible-infected dynamics (see Fig.1) dH = r (K H I I ) b([1 ε (C,r )]I +[1 ε (C,r )](1 ρ )I )H, (1) H s r s B s r B r r dt − − − − − − − dI s = b[1 ε (C,r )]HI µI , (2) s B s s dt − − dI r = b[1 ε (C,r )](1 ρ )HI µI . (3) r B r r r dt − − − The model has three compartments: healthy hosts H, hosts infected by a sensitive pathogen strain I , hosts infected by a resistant pathogen strain I ; and is similar to the s r models described in [11, 21]. The subscript “s” stands for the sensitive strain and the subscript “r” stands for the resistant strain. The quantities H, I and I , represent the s r total amount of the corresponding host tissue within one field, which could be leaves, stems or graintissue, depending onthespecific host-pathogeninteraction. Healthy hosts H grow with the rate r . Their growth is limited by the “carrying capacity” K, which H may imply limited space or nutrients. Furthermore, healthy hosts may be infected by the sensitive pathogen strain and transformed into infected hosts in the compartment I with the transmission rate b. This is a compound parameter given by the product s of the sporulation rate of the infected tissue and the probability that a spore causes new infection. Healthy hosts may also be infected by the resistant pathogen strain and transformed into infected hosts in the compartment I . In this case, resistant mutants r suffer a fitness cost ρ which affects their transmission rate such that it becomes equal to r b(1 ρ ). The corresponding terms in Eqs.(1)-(3) are proportional to the amount of the r − available healthy tissue H and to the amount of the infected tissue I or I . Infected host s r tissue loses its infectivity at a rate µ, where µ−1 is the characteristic infectious period. Since our description is deterministic we do not take into account the emergence 5 of new resistance mutations but assume that the resistant pathogen strain is already present in the population. Therefore, when “selection for resistance” is discussed below, we refer to the process of winning the competition by this existing resistant strain due to its higher fitness with respect to the sensitive strain in the presence of fungicide treatment. Emergence of new resistance mutations is a different problem, which goes beyond the scope of our study and requires stochastic simulation methods. We do not consider the possibility of double resistance in the model, but by preventing selection for single resistance as described here, one would also diminish the probability of the emergence of double resistance for both sexually and asexually reproducing pathogens (see AppendixA.7). We consider two fungicides A and B. Fungicide A is the high-risk fungicide, to which the resistant pathogen strain exhibits a variable degree of resistance. However, the sensitive strain is fully sensitive to fungicide A. Fungicide B is the low-risk fungicide, i.e. both pathogen strains are fully sensitive to it. We compare the effects of the fungicide A applied alone, fungicide B applied alone and the effect of their mixture in different proportions. We assume that the fungicides will decrease the pathogen transmission rate b [see the expression in square brackets in Eq.(2), Eq.(3)]. For example, application of a fungicide could result in production of spores that are deficient essential metabolic products such as ergosterol or β-tubulin. Consequently, these spores would likely have a lower success rate in causing new infections. Spores of sensitive strains of Z. tritici produced shorter germ tubes when exposed to azoles [33]. Spores that produce shorter germ tubes are less likely to find and penetrate stomata, hence are less likely to give rise to new infections. Protectant activity of fungicides will also reduce the transmission rate b [56, 46]. These studies [56, 46] also reported that fungicide application leads to a reduction in the number of spores produced. This outcome can be attributed to the fungicide decreasing 6 the sporulation rate and thus affecting b or decreasing the infectious period and thus affecting µ, or both of these effects. More detailed measurements are often needed to distinguish between these different effects. When only one fungicide applied, the reduction of the transmission rate is described by C A ε (C ) = k , (4) A A kA C +C A 50A for the fungicide A, and by C B ε (C ) = k , (5) B B kB C +C B 50B for the fungicide B. These functions grow with the fungicide doses C , C and saturate A B to values k , k , respectively, which are the maximum reductions in the transmission kA kB rate (or efficacies). This functional form was used before in the fungicide resistance literature [21, 19]. We also performed the analysis for the exponential fungicide action more common in plant pathology and obtained qualitatively similar results. The reason for choosing the function in Eq.(5) was that it made possible to obtain all the results analytically. The parameters C , C represent the fungicide dose at which half of the 50A 50B maximum effect is achieved. These parameters can always be made equal by rescaling the concentration axis for one of the fungicides. Hence, we set C = C = C . 50A 50B 50 We next determine the effect of a mixture of two fungicides according to the Loewe’s definitionofadditivity (ornon-interaction) [7](anequivalent graphicprocedureisknown as the Wadley method in the fungicide literature [35]). It is based on the notion that a compoundcannotinteractpharmacologicallywithitself. Ashammixtureofacompound Awithitselfcanbecreatedanditseffectusedasareferencepointforassessingofwhether the components of a real mixture interact pharmacologically. When the two compounds A and B have the same effect as the sham mixture of the compound A with itself, they 7 are said to have no interaction (or an additive interaction). In this case, the isobologram equation C /C +C /C = 1 (6) A Ai B Bi holds (see Sec. VA of [7] for the derivation). Here, C and C are the doses of the A B compounds A and B, respectively, when applied in the mixture; C is the isoeffective Ai dose of the compound A, that is the dose at which compound Aalone has thesame effect asthemixture; andC istheisoeffectivedoseofthecompoundB.IfthemixtureofAand Bi B has a larger effect than the zero-interactive sham mixture, then C /C +C /C < 1 A Ai B Bi and the two compounds are said to interact synergistically. On the contrary, when the mixture of A and B has a smaller effect than the zero-interactive sham mixture, C /C +C /C > 1 and the two compounds interact antagonistically. A Ai B Bi Using the dose-response dependencies of each fungicide when applied alone, Eq.(5) and Eq.(6), we derive the dose-response function for the combined effect of the two fungicides on the sensitive pathogen strain in the case of no pharmacological interaction (see Sec. VIB of [7] for the derivation): k C +k C kA A kB B ε (C ,C ) = . (7) s A B C +C +C A B 50 Similarly, we determine the combined effect of the two fungicides on the resistant pathogen strain still without pharmacological interaction: k αC +k C kA A kB B ε (C ,C ) = , (8) r A B αC +C +C A B 50 where we introduced α, the degree of sensitivity of the resistant strain to the fungicide A (the high-risk fungicide). At α = 0 the pathogen is fully resistant to fungicide A and the effect of the mixture ε (C ,C ) in Eq.(8) does not depend on its dose C , while at r A B A α = 1 the pathogen is fully sensitive to fungicide A. 8 The expression in Eq.(7) and Eq.(8) are only valid in the range of fungicide con- centrations, over which isoeffective concentrations can be determined for both fungi- cides. Here, the isoeffective concentration is the concentration of a fungicide applied alone that has the same effect as the mixture. This requirement means that we are only able to consider the effect of the mixture at a sufficiently low total concentration: C = C +C < k C /(k k )/(1 r ). A B kB 50 kA kB B − − Next, we introduce deviations from the additive pharmacological interaction. There are several ways to do this, usually by adding an interaction term to the isobologram equation [18]. We chose a specific form of the interaction term, which is proportional to the square root of the product of the concentrations of the two compounds [Eq.(28) in [18]]. Assuming k = k = k , this form allows for a simple analytical expression for kA kB k the effect of the combination on the sensitive strain C ε (C,r ) = k , (9) s B k C +C /γ 50 s and on the resistant strain C ε (C,r ) = k . (10) r B k C +C /γ 50 r Here C = C +C , where C is the dose of the fungicide A and C is the dose of the A B A B fungicide B, r = C /C is the proportion of the fungicide B in the mixture and B B γ = 1+u r (1 r ), (11) s B B − p γ = α(1 r )+r +u αr (1 r ) (12) r B B B B − − p are the parameters which modify C due to pharmacological interaction and partial 50 9 resistance. Eqs.(9), (10) are obtained from Eq.(6) with an interaction term added and the dose-response functions of each fungicide when applied alone, Eq.(5). The degree of pharmacological interaction is characterized by the parameter u. At u = 0 the fungicides do not interact and Eqs(9), (10) are the same as Eqs(7), (8). The case when u > 0 represents synergy: the interaction term proportional to u in Eq.(9) and Eq.(10) is positive and it reduces the value of C , meaning that the same effect can be achieved 50 at a lower dose than at u = 0. The case when u < 0 corresponds to antagonism (see AppendixA.1). Note, that the interaction term is proportional to r (1 r ). This B B − p functional formguaranteesthat itvanishes, whenever onlyoneofthecompounds isused, i.e. r = 0 or r = 1. B B In order to make clear the questions we ask and the assumptions we make, we consider the dynamics of the frequency of the resistant strain p(t) = I (t)/[I (t)+I (t)]. The r r s rate of its change is obtained from Eqs.(1)-(3) dp = s(t)p(1 p), (13) dt − where s = b[(1 ε (C,r ))(1 ρ ) (1 ε (C,r ))]H(t) (14) r B r s B − − − − is the selection coefficient [a similar expression was found in [19]]. Here ε (C,r ) and s B ε (C,r ) are given by Eq.(9) and Eq.(10). If s > 0, then the resistant strain is favored r B by selection and will eventually dominate the pathogen population (p 1 at t → → ). Alternatively, if s < 0, then the sensitive strain is selected and will dominate the ∞ population (p 0 at t ). → → ∞ The focus of this paper is to investigate the parameter range over which s < 0, i.e. the sensitive strain is favored by selection. Mathematically this corresponds to finding the range of stability of the equilibrium (fixed) point of the system Eqs.(1)-(3), 10