Quantitative plant resistance in cultivar mixtures: wheat yellow rust as a modeling case study

Quantitative plant resistance in cultivar mixtures: wheat yellowrust as a modeling case study

Natalia Sapoukhina1,2,3, Sophie Paillard4, Franc�oise Dedryver4 and Claude de Vallavieille-Pope5

1INRA, UMR1345 Institut de Recherche en Horticulture et Semences – IRHS, SFR 4207, PRES UNAM, 42 rue Georges Morel, F-49071, Beaucouz�e Cedex, France; 2AgroCampus-Ouest,

UMR1345 Institut de Recherche en Horticulture et Semences – IRHS, F-49045, Angers, France; 3Universit�e d’Angers, UMR1345 Institut de Recherche en Horticulture et Semences – IRHS,

F-49045, Angers, France; 4INRA, UMR 1349, Institut de G�en�etique, Environnement et Protection des Plantes - IGEPP , BP 35327, F-35653, Le Rheu Cedex, France; 5INRA, UR1290

BIOGER-CPP, BP 01, F-78850, Thiverval-Grignon, France

Author for correspondence:Natalia Sapoukhina

Tel: +33 2 41 22 57 17Email: [email protected]

Received: 6 December 2012

Accepted: 17 June 2013

New Phytologist (2013) 200: 888–897doi: 10.1111/nph.12413

Key words: competition, diversifiedagroecosystems, Puccinia striiformis,quantitative/partial resistance,reaction–diffusion model, Triticum aestivum,wheat yellow (stripe) rust.

Summary

� Unlike qualitative plant resistance, which confers immunity to disease, quantitative resis-

tance confers only a reduction in disease severity and this can be nonspecific. Consequently,

the outcome of its deployment in cultivar mixtures is not easy to predict, as on the one hand it

may reduce the heterogeneity of the mixture, but on the other it may induce competition

between nonspecialized strains of the pathogen.� To clarify the principles for the successful use of quantitative plant resistance in disease

management, we built a parsimonious model describing the dynamics of competing pathogen

strains spreading through a mixture of cultivars carrying nonspecific quantitative resistance.� Using the parameterized model for a wheat–yellow rust system, we demonstrate that a

more effective use of quantitative resistance in mixtures involves reinforcing the effect of the

highly resistant cultivars rather than replacing them. We highlight the fact that the judicious

deployment of the quantitative resistance in two- or three-component mixtures makes it pos-

sible to reduce disease severity using only small proportions of the highly resistant cultivar.� Our results provide insights into the effects on pathogen dynamics of deploying quantita-

tive plant resistance, and can provide guidance for choosing appropriate associations of culti-

vars and optimizing diversification strategies.

Introduction

In plant epidemiology, quantitative or partial plant resistancethat reduces the disease severity rather than conferring immunityoffers an alternative to qualitative or total resistance (Singh et al.,2004), which does confer immunity to disease. Quantitativeresistance can be used to reduce the size of the pathogen popu-lation whilst also avoiding exerting high selective pressure(Zhan et al., 2002; Chen, 2005). For the same reason, cultivarscarrying quantitative resistance are potential components of low-selective pressure cultivar mixtures intended to control not onlythe spread of disease, but also the adaptive dynamics of the path-ogen population and its genetic diversity (Sommerhalder et al.,2011). However, so far, most of the theoretical and empiricalresearch on the use of cultivar mixtures for disease reduction hasfocused on diseases caused by biotrophic pathogens that interactwith their hosts on a gene-for-gene basis (Flor, 1956), which hasmade it possible to identify major resistance genes conferring cul-tivar immunity to disease. The presence of major resistance genesin a cultivar is still one of the most common criteria for selectingit as a mixture component, despite the fact that this can lead torapid pathogen evolution and the emergence of a super-virulentpathogen strain (Mundt, 2002).

The principles of the mixture theory were originally elaboratedfor a two-component mixture, consisting of one susceptible culti-var and one totally resistant cultivar (Leonard, 1969). Cultivarmixtures including qualitative resistance reduce the rate of diseasespread by eliminating large numbers of spores that are depositedon resistant cultivars, thus diluting the inoculum falling on thesusceptible hosts. In this way, the spatial heterogeneity of a culti-var mixture creates a physical barrier to disease spread (Garett &Mundt, 1999). Studies of the epidemiology and theory ofbiological invasions have contributed considerably to our under-standing of the impact of the spatial heterogeneity of a hostpopulation on the dynamics and rate of spread of disease. It hasbeen shown that there is an epidemic threshold, a minimumpercentage of suitable hosts, of c. 10–40%, which is required fordisease to spread (Collingham & Huntley, 2000; Otten et al.,2004; Dewhirst & Lutscher, 2009; Mundt et al., 2011). Thedevelopment of theoretical approaches to integrating spatial het-erogeneity into models has made it possible to demonstrate thatthe epidemic threshold depends on various spatial factors, such asthe degree of landscape fragmentation, the spatial arrangement ofthe mixture components, the spatial scale, and the dispersalcapacity of the disease (Kinezaki et al., 2010; Papa€ıx et al., 2011;Suzuki & Sasaki, 2011). Understanding the process of disease

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spread over a genetically and spatially heterogeneous cultivar mix-ture has made it possible to identify the key characteristics thatdetermine the effectiveness of a mixture: genotype unit area,dispersal gradient of pathogen spores, disease pressure, sowingdensity, and the number of components (Garett & Mundt,1999). In the case of wheat yellow (stripe) rust (caused byPuccinia striiformis f.sp. tritici), it has been shown that mixingcultivars carrying different major resistance genes slows thespread of epidemics, and that this effect can be enhanced by usingan appropriate proportion of a resistant cultivar. The 133 empiri-cal studies analyzed by Huang et al. (2012) show that 83% ofwheat cultivar mixtures produced disease intensities that werelower than the mean values found for pure stands. Overall, thereduction ranged from 30 to 50%, with an average of 28%.Empirical studies demonstrated that the reduction of rust severitydepended on characteristics that modify the degree of heteroge-neity of the mixture. For instance, Mundt et al. (1995) showedthat two-component mixtures produce a smaller mixture effectthan mixtures of larger numbers of components.

Most empirical studies focusing on mixtures of susceptible andpartially resistant cultivars have analyzed the dynamics of splash-dispersed necrotrophs on cereals: Rynchosporium secalis on barley,Stagonospora nodorum on wheat, Bipolaris sorokiniana on wheat,and Mycosphaerella graminicola on wheat. They have shown thatmixtures including cultivars with partial resistance provide rela-tively low amounts of disease control. Overall, the results ofempirical studies of quantitative resistance deployment rangefrom not effective, as in wheat Cephalosporium stripe (Mundt,2002), M. graminicola (Cowger & Mundt, 2002), wheat eyespot(Mundt et al., 1995) and barley scald (Abbott et al., 2000), to sig-nificantly effective, as in wheat Septoria nodorum blotch (Jegeret al., 1981b), wheat yellow rust (Huang et al., 2011), wheat leaf(brown) rust (Mahmood et al., 1991), wheat Septoria triticiblotch (Mille et al., 2006), barley powdery mildew (Newton &Thomas, 1992), and potato late blight (Andrivon et al., 2003).To understand the key factors that determine the success ofdeploying quantitative plant resistance in mixtures, we need todistinguish between specific and nonspecific quantitative resis-tance. Specific resistance is effective against some pathogenstrains, and so can create more pronounced mixture heterogene-ity than nonspecific quantitative resistance, which can be infectedby all pathogen strains, but with different degrees of severity. Wewould expect the effectiveness of these mechanisms to depend onthe type and degree of quantitative resistance used in a mixture.For instance, an association of a nonspecific, moderately resistantcultivar with a susceptible one can result in little differencebetween the susceptibilities of the cultivars, and therefore lesspronounced heterogeneity of the mixture. The physical barrierand dilution effects can be dramatically reduced if cultivars carry-ing strong specific resistance are excluded from the mixture. Thedegree of heterogeneity in susceptibility of the components canexplain why mixtures are more effective in reducing specializedpathogens than nonspecialized ones (Xu & Ridout, 2000). How-ever, Jeger et al. (1981a), modeling the dynamics of a single path-ogen strain, showed that some differences in the susceptibilityof the components can create functional heterogeneity in the

mixture that is sufficient to control a nonspecialized pathogen.Although competition between pathogen strains may play animportant role in determining the effectiveness of mixtures,including quantitative resistance (Garett & Mundt, 1999;Lannou et al., 2005; Abang et al., 2006), it has not received muchattention in theoretical studies. In their two-dimensional stochas-tic spatial contact model of fungal pathogens in cultivar mixtures,Xu & Ridout (2000) considered the dynamics of specialized andnonspecialized pathogen strains, but not the competitionbetween them, as a host unit was assumed to be occupied by asingle pathogen strain. Moreover, this assumption generatedadditional spatial heterogeneity for pathogen strains that was notrelated to the degree of host resistance. There has been almostno attempt to adapt the mixture theory to the situation of hostdiversification, in which quantitative resistance leads to a contin-uum in host susceptibility and competitive interactions amongpathogen strains.

Our goal was to study the dynamics of competing pathogenstrains spreading over a cultivar mixture, the components ofwhich carry nonspecific quantitative resistance, and thereby toclarify the role of the quantitative plant resistance in disease man-agement involving the use of cultivar mixtures. Spatiotemporalmodels of the propagation of the airborne fungal diseases includethe system of differential equations representing continuous epi-demic dispersal by diffusion (Yang et al., 1991) or of integrodif-ference equations, where the dispersal kernel can represent bothshort- and long-distance disease dispersal (Skelsey et al., 2005).These models are spatially explicit, and so can be used to studypathogen spread over heterogeneous host populations. Moreover,being mechanistic, they can easily be parameterized and canmimic disease dynamics quite well (Yang et al., 1991). However,the abundance of parameters and processes in the available epide-miologically relevant models make them unsuitable for use forstudying the dynamics of diversified host–pathogen systems,where both the host and pathogen populations can be dividedinto a large number of classes with different properties. Here, weconstructed a parsimonious, spatially explicit, host–pathogenmodel describing pathogen spread over a genetically diversifiedhost population distributed over a two-dimensional landscape.The simplicity of our reaction–diffusion model means that it canbe used to describe the dynamics of a host–pathogen system inwhich the host and pathogen populations are divided intonumerous classes, depending on their susceptibility and infectionefficiency, respectively. This general model can be used to repre-sent the dynamics of both specialized and nonspecialized patho-gen strains. We used field data from wheat plots inoculated withyellow rust to study the effect of host diversity on the dynamicsof four competing, nonspecialized pathotypes of P. striiformisf.sp. tritici. First, we parameterized the model from field data forthe spread of wheat yellow rust over a homogeneous, susceptiblewheat cultivar (de Vallavieille-Pope & Goyeau, 1996; Finckhet al., 2000; de Vallavieille-Pope, 2004). Second, we used theparameter estimates obtained from the field studies to investigatethe effectiveness of two- and three-component random cultivarmixtures in which the degrees of susceptibility and proportionsof mixture components were varied. Finally, we discuss what

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constitutes the effective management of quantitative plantresistance.

Description

The model

In this section we present a model of the focal expansion of a fun-gal disease propagated by airborne spores over a host populationspatially distributed in a two-dimensional domain Ω = [0, Lx]9[0, Ly] (m

2), Ω ⊂ R 2. The spatial scale of the domain, Ω, dependson the pattern of disease dispersal being studied. We assume thatthe pathogen and host populations are both genetically diverse,and include n and m distinct genotypes, respectively. We use areaction–diffusion model to describe the spread of a diverse path-ogen population over a heterogeneous host population:

@Pi@t

¼ PiXmj¼1

ei;jHj þ d@2Pi@x2

þ @2Pi@y2

� �;

@Hj

@t¼ rHj 1�Hj

K

� ��Hj

Xni¼1

ei;jPi

Eqn 1

where, Pi(x,t) is the density of the leaf area (m2 m�2) infected by

the ith pathogen strain, and Hj(x,t) is the density of the healthyleaf area (m2 m�2) of the jth host genotype, i = 1,…, n,j = 1,…,m. The terms ‘pathogen’ and ‘host’ are used below torefer to infected and healthy leaf areas, respectively. We considerthe dynamics of interacting populations during one cropping sea-son, from stem elongation until the ripening stage, that is, duringthe period when green vegetation is available. Time t is measuredin days (d). The parameter ei,j is the infection efficiency of the jthhost genotype by the ith pathogen genotype. All pathogen geno-types are assumed to have the same diffusion coefficient, d (m2

d�1). Similarly, the densities of healthy leaf area of all hostgenotypes are assumed to have the same growth rate, r (d�1), andcarrying capacity, K. The plausible assumption of logistic hostgrowth matches the dynamics of the host–pathogen interactionsat the beginning of the cropping season. However, the

continuous growth of healthy leaves also compensates for the lossof diseased tissues during the last stage of the epidemic. Conse-quently, while the density of healthy leaf area is maintainedbelow the K threshold, the densities of the infected leaf area,Pn

i¼1 Pi , and of the total leaf area,Pm

j¼1 Hj þPn

i¼1 Pi , canexceed the carrying capacity of the healthy leaf area, K (Support-ing Information, Notes S1). While theoretically this can happen,under field conditions it does not, as the growth of diseasedplants is slowed (M. Trottet, pers. comm.). As we use relativemeasurements for the assessment of the mixture efficiency,exceeding K by

Pmj¼1 Hj þ

Pni¼1 Pi does not affect either the

simulation results or the conclusions. Moreover, the density ofthe infected plant area is regulated naturally by the availability ofthe healthy leaf area and the short duration of the season,T = 150, and so we do not limit it by the carrying capacity(Fig. S1). However, in order to track host–pathogen dynamicsover a longer season, it would be necessary to include a capacityparameter that limits disease development during the last stage ofthe epidemic.

The definitions and units of variables and parameters are sum-marized in Table 1. We assume that Eqn 1 has a reflectingboundary condition:

@Pi@x

��x¼0;Lx

¼ @Pi@y

��y¼0;Ly

¼ 0; i ¼ 1; . . .; n;

meaning that the habitat boundary is completely impermeablefor the pathogen population, so that disease propagation is forcedto turn back when it reaches the boundary. For a detailed descrip-tion of the model behavior, see Notes S1. The resulting model,the methods, and numerical simulations listed in the followingsections were implemented in Borland Delphi V4.5.

Case study: wheat yellow rust system

In this study, six single seed descent (SSD) wheat (Triticumaestivum L.) lines, issued from the cross between two winterbread wheat cultivars, cv Renan (Yr17), the susceptible cv R�ecital(Yr6), and two parental lines, were assessed for adult-plant

Table 1 Definition and values of the parameters used to model wheat yellow rust dynamics, and results of the sensitivity analysis

Parameter Definition (units) Measured value Estimated value Elasticity, E

Hj (x,0) Initial density of healthy leaf area of the jth host genotype (m2m�2) 0.1 – � 0.23Pi (x,0) Initial density of the leaf area infected by the ith pathogen genotype (m2m�2) – 7.8E-5 (1.4E-5) � 0.03T Season duration (d) 150 – –Dx Sideways distance between two field points (m) 0.2 – –Dy Lengthways distance between two field points (m) 0.4 – –Lx Field width (m) 9 – –Ly Field length (m) 18 – –e Nominal infection efficiency of the pathogen on a susceptible wheat cultivar (d�1) – 0.296 (0.023) � 0.42d Diffusion coefficient (m2 d�1) – 0.029 (0.008) � 0.11K Carrying capacity of the density of the healthy leaf area of themth host genotype (m2m�2) – 2.7 (0.2) 0r Growth rate of host population (d�1) – 0.055 (0.015) 0

Estimated values are reported with standard errors between parentheses. The elasticity, E, of predicted disease severity to model parameters (SupportingInformation Notes S1). Parameters varied by � 5%.

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resistance in a 3 yr field experiment. The SSD lines carried differ-ent combinations of the specific resistance gene (Yr17) and thetwo quantitative trait loci (QTLs), QYr.inra-2BS (Q2) andQYr.inra-6B (Q3), issued from cv Renan, and a QTL QYr.inra-2AS1 (Q1) derived from cv R�ecital (Table 2). Q1 was located inwheat region 2AS, which is homologous to the Yr17 introgres-sion (Aegilops ventricosa segment) carried by cv Renan (Dedryveret al., 2009).

The presence of QTLs in these different SSD lines was deter-mined using both phenotype (seedling tests and field assays) andmolecular marker data (Dedryver et al., 2009). QTLs weredetected by mapping a population of 194 (F6 and F7) SSD linesissued from the cross between cv Renan and cv R�ecital. The iden-tified adult plant resistance (APR) QTLs Q1, Q2, and Q3 werestudied here in the SSD lines L32, L138, L64, L110, and L22,and the resistant cv Renan (Table 2). Line 15 and cv R�ecital werehighly susceptible. The lines were evaluated during the period2006–2008. The experiments were conducted at the INRA-Ver-sailles (Ile de France) experimental station. Four pathotypes ofP. striiformis f.sp. tritici were chosen on the basis of their viru-lence phenotype to overcome all the specific resistance genes car-ried by cv Renan (Yr17), and two pathotypes had a virulence of6, which allowed them to overcome the specific resistance of cvR�ecital (Yr6). The pathotypes inoculated in the field plots duringthree consecutive years were V1,2,3,9,17,Sd; V1,2,3,4,6,9,17,Sd,Su; V1,2,3,4,6,9,17,Sd,Su; and V1,2,3,4,9,17,32,Sd,Su (de Vall-avieille-Pope et al., 2012). Two lines of each SSD line wereplanted with 30 seeds per line in a 1.2 m row at the end of Octo-ber, and arranged in a completely randomized block design withfour replications. Each field plot was located at least 700 m fromany other plot to prevent cross-contamination with the patho-types. A susceptible disease spreader line, consisting of three sus-ceptible cultivars, Victo, Slejpner (Yr9) and Audace (Yr17), wasplanted perpendicularly to the test lines at both ends of the testlines. Only one pathotype was inoculated in each plot. The fieldinoculation was done with cv Victo seedlings, cultivated in agrowth chamber to the two-leaf stage, and then artificially inocu-lated with P. striiformis f.sp. tritici isolates. In March, the inocu-lated seedlings were planted out just before sporulation (oneseedling every 2 m) in the spreader rows. Visual scoring of diseaseseverity was carried out three times during the growing season,using a 0–12 scale, with 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12corresponding to 0.3, 0.7, 2, 8, 12, 16, 24, 33, 50, 66, 82, and

100% sporulating leaf area, respectively. Disease severity wasassessed on five tillers of each of the four replicates for each leaf ofthe main stem of the plants at the time of rust appearance, corre-sponding to the growth stages DC32, DC39, and DC79, and adiseased surface area was calculated per plant at the last scoringdate.

Fig. 1 shows differential severity of the pathogen pathotypesaveraged over 3 yr (2006–2008). No specificity was foundbetween the QTL lines and the four P. striiformis f.sp. tritici iso-lates tested in field tests (data not shown). After assigning Eqn 1notations, P. striiformis f. sp. tritici pathotypes V1,2,3,9,17,Sd,V1,2,3,4,6,9,17,Sd,Su, V1,2,3,4,6,9,17,Sd,Su and V1,2,3,4,9,17,32,Sd,Su became P1, P2, P3, and P4, respectively. Among thehost genotypes, we distinguished resistant {L22, Renan}, moder-ately resistant {L110, L64}, moderately susceptible {L32, L138},and susceptible {L15} lines. Hereafter, we use ‘S’ for susceptible,‘M’ for moderately susceptible, and ‘R’ for resistant. Table 2summarizes the notations used for the genetic compositions ofthe lines and their susceptibility.

Parameter estimation

We parameterized the model using epidemiological data forwheat yellow rust. Most of the parameter values were derivedfrom the results of field experiments performed with wheat yel-low rust (de Vallavieille-Pope & Goyeau, 1996; Clevers et al.,2002; de Vallavieille-Pope, 2004). A local sensitivity analysis wasperformed to identify which parameters need to be estimatedmore accurately (Notes S1).

Measured parameters Based on experimental data on thedynamics of yellow rust severity in pure susceptible wheat stands(de Vallavieille-Pope & Goyeau, 1996; de Vallavieille-Pope,2004), and on the dynamics of the leaf area index for a wheatfield (Clevers et al., 2002), we fixed the values of some ofthe model parameters. We assumed the wheat field size to be

Table 2 Adult-plant resistance reaction to yellow rust of the single seeddescent wheat (Triticum aestivum) lines and the resistant parent cv Renan,characterized by the presence or absence of three quantitative trait loci(QTLs),QYr.inra-2ASI (Q1),QYr.inra-2BS (Q2) andQYr.inra-6B (Q3),evaluated under field conditions, with four Puccinia striiformis f.sp. triticipathotypes (S, highly susceptible; MS, moderately susceptible; MR, mod-erately resistant; R, highly resistant), according to Dedryver et al. (2009)

Lines L15 L138 L32 L64 L110 Renan L22

QTL None Q3 Q2 Q1Q2 Q1Q3 Q2Q3 Q1Q2Q3

Degree ofresistance

S MS MS MR MR R R

Fig. 1 Disease severity on the seven wheat (Triticum aestivum) linescarrying different combinations of three quantitative trait loci (QTLs),QYr.inra-2ASI,QYr.inra-2BS, andQYr.inra-6B (Table 2), and the parentcultivar, Renan, averaged over three cropping seasons (2006–2008) afterinoculating with four pathotypes of Puccinia striiformis f.sp. tritici:V1,2,3,9,17,Sd (P1); V1,2,3,4,6,9,17,Sd,Su (P2); V1,2,3,4,6,9,17,Sd,Su(P3); and V1,2,3,4,9,17,32,Sd,Su (P4). Field data were obtained at the endof an epidemic season.




Lx = 9 (m), Ly = 18 (m), which corresponds to 162 m2 of realcropping surface of a pure susceptible stand (de Vallavieille-Pope& Goyeau, 1996; de Vallavieille-Pope, 2004). According to thenumerical method used to solve the system (1), the field con-tained 2116 cells of Dx9Dy = 0.2 m9 0.4 m size. We assumedthe duration of the growing season, T, to be 150 d, and the initialdensity of healthy leaf area, H0, to be equal to 0.1 (Clevers et al.,2002). In other words, we assumed that the disease arrives at thebeginning of the growing season, when the healthy leaf area avail-able has reached 4% of its carrying capacity.

Model fitting We first estimated the values of the r and Kparameters using data for the temporal variation of the leaf areaindex for a wheat field (Clevers et al., 2002). By setting infectionrates equal to zero, ei,j = 0, i = 1,…, 4, j = 1,…, 7, we obtained apure wheat growth model. We fixed the value of H0 (Table 1),and assessed the values of r and K by nonlinear fitting. We fittedthe wheat growth model with M = 2 adjustable parametersq = (r, K) to N1 = 5 field measurements of the leaf area index,ðti ; yiÞ; i ¼ 1; . . .;N1. At a given time, ti, the model leaf areaindex was calculated as the space-averaged density of the host

population yðti ; qÞ ¼ 1LxLy

RLy0

RLx0

H ðx; tiÞd x. The maximum likeli-

hood estimate of the model parameters was obtained by minimiz-

ing the merit function as follows: F ðqÞ ¼ PNi¼1

�yi � yðti ; qÞ

�2,

N =N1, defined as the sum of squared residuals between theexperimental data and the corresponding model output. Theparameters of the model were then adjusted to minimizethe merit function, thus yielding the best-fitting parameters. Tominimize the merit function overM parameters, we used the sim-plex-simulated annealing approach for the global optimization,which combines the downhill simplex and simulated annealingalgorithms (Press et al., 1992).

We then fixed the values of three parameters, r, K,H0 (Table 1),and assessed the values of e, d, and P0, the initial density of theinfected leaf area. We fitted the complete Eqn 1, e 6¼ 0, withM = 3adjustable parameters q = (e, d, P0) to N2 = 8 observations of the

mean disease severity of the pure, homogeneous, susceptible plot,ðti ; yiÞ; i ¼ 1; . . .;N2 (Fig. 2). At a given time, ti, the model dis-ease severity was calculated as the percentage of the leaf area

infected over the total leaf area, yðti ; qÞ ¼ 100%RLy0

RLx0

Pðx; tiÞ=ðPðx; tiÞ þH ðx; tiÞÞd x. We obtained the best-fitting parametersby minimizing the merit function, F (q), withN =N2.

To assess the accuracy of the evaluated parameter values, weused residual bootstrapping (Efron & Tibshirani, 1993). To assessthe fit of the model, we calculated the coefficient of determinationas R2 ¼ 1�PN

i¼1

�yi � yðti ; qÞ

�2=PN

i¼1

�yi � �y

�2, where �y

denotes the mean of the observations andN 2 fN1; N2g: R2 esti-mates the fraction of the observed variability that is explained bythe model. The higher the value of R2, the better the fit of themodel, with R2 = 1 denoting a perfect fit. Since it is risky to use R2

alone as a model-fitting criterion for nonlinear models (Kv�alseth,1985), we supplemented it by the estimation of the concordancecoefficient, qc, measuring the degree of agreement between theobserved and predicted variables (Lin, 1989). The concordancecoefficient assesses the degree to which data pairs fall on the 45°line through the origin. It is estimated by

qc ¼2PNi¼1

yi � �yð Þ yðti ; qÞ � �yðqÞð ÞPNi¼1

yi � �yð Þ2 þ PNi¼1

yðti ; qÞ � �yðqÞð Þ2 þ N �y � �yðqÞð Þ2;

i ¼ 1; . . .;N ;N 2 N1; N2f g;

where yi are experimental data, y(ti, q) are the model outputs, and�y and �yðqÞ are their respective means. qc =� 1 if, and only if, thereadings are in perfect agreement or perfect reversal.

Differential pathogen infection efficiency, (ei,j)1 ≤ i ≤ n, 1 ≤ j ≤ m.Fig. 1 shows that four P. striiformis f.sp. tritici pathotypes pro-duced different average disease severity on different wheatlines. We split seven wheat lines into four groups according totheir degrees of severity: ‘R’, resistant including L22 and Re-nan lines; ‘MR’, moderately resistant, including L110 and L64lines; ‘MS’, moderately susceptible, including L32 and L138lines; and ‘S’, susceptible, including L15 line. We assumed thatthe differences in severity were caused by differing pathogeninfection rates, ei,j, i = 1,…, 4, j = 1,…, 4. The estimated valueof the pathogen infection efficiency, e = 0.296, on a suscepti-ble wheat cultivar is a nominal value that we used to assess theinfection rates, ei,j, of all the pathogen pathotypes in four linegroups. The value e = 0.296 resulted in 100% disease severity,while pathotype P2 on susceptible cultivar H1 led to 72% dis-ease severity at the end of the season. Thus, starting ate2,1 = 0.296, and decreasing the e2,1 value, while the othermodel parameters were fixed, we found e2,1 = 0.082, corre-sponding to 72% disease severity. Repeating this procedure forall the pathotypes and four line-groups, we obtained the matrix(ei,j)1 ≤ i ≤ 4, 1 ≤ j ≤ 4, summarized in Table 3. Differentiated val-ues of infection efficiencies determine the degree of susceptibil-ity or resistance of the interactions between pathogen and hostgenotypes.

Fig. 2 The dynamics of yellow rust severity on a susceptible wheat(Triticum aestivum) cultivar in a pure stand – observed (circles) (deVallavieille-Pope & Goyeau, 1996; de Vallavieille-Pope, 2004) andpredicted (solid line) by Eqn 1. Simulation results were obtained with theparameter values presented in Table 1.


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Note that the differing infection rates of the strains naturallyresulted in competition between them, since the healthy leaf areaavailable was limited (Notes S1).

Numerical simulations

Using biologically relevant parameters for the wheat–P. striiformisf.sp. tritici pathosystem (Table 1), we performed numerical experi-ments in order to derive effective mixture compositions. Thespatial arrangement of the mixture components was random andinvariable in the numerical simulations. The initial inoculum ofthe pathogen population was focal, consisted of four pathotypeswith equal density, and was located in the top-left corner of thefield, P1(10Dx,10Dy,0) = P2(10Dx,10Dy,0) = P3(10Dx,10Dy,0) =P4(10Dx,10Dy,0) = 7.8E-5. We varied the proportions of the com-ponents from 10 to 90% in six two-component mixtures (R/MR,R/MS, R/S,MR/MS,MS/S, andMR/S) and four three-componentmixtures (MS/MR/R, S/MS/R, S/MS/R, and S/MS/MR). For all378 combinations, we calculated the disease severity of a cultivarmixture at the end of the growing season, yðT Þ ¼100%

RLy0

RLx0

Pðx;T Þ=�Pðx;T Þ þH ðx;T Þ�d x, and the absolute

disease reduction (%) =Pk

i¼1 aiciðT Þ � yðT Þ, where ci(T) is theseverity in the ith mixture component grown in a pure stand, i = 1,…, k (where k is the number of mixture components), ai is fre-quency of ith component in the mixture, and y(T) is the severity ofthe mixture. Absolute disease reduction reflects the absolute gain inseverity decrease resulting from the use of a cultivar mixture in com-parison to the weighted mean of severity of components grown inpure stands. Negative values of absolute disease reduction indicate anegative mixture effect, whereas positive values indicate that themixture deployment was beneficial. Further, we define an effectivemixture as one displaying positive absolute disease reduction, and adisease severity, y (T ), of < 20% (the acceptable disease severity).

In contrast to the relative disease reduction (%)(�1� yðT Þ=Pk

i¼1 aiciðT Þ�� 100%) used in mixture theory asa measure of the mixture performance (Finckh & Mundt, 1992;Akanda & Mundt, 1996), we used the absolute disease reduction,as this seemed to fit better with the estimation of the efficiency ofa mixture, which has some components that can be partially resis-tant. Indeed, in this case, relative disease reduction can take highvalues with only a small gain in severity decrease, when theweighted mean severity of components grown in pure stands is

small. For example, the relative disease reduction for the mixtureof 80% R and 20% MR is c. 23%. This high mixture effect valuewas obtained with a low disease severity gain, resulting from6.5% weighted mean of severity in pure stands, and 5% mixtureseverity.

Results

Model fitting

The smooth line in Fig. 2 depicts the fit between Eqn 1 and theexperimental data of wheat yellow rust spread over the homoge-neous susceptible plot. The smooth curve generated by the modelfits the field data quite closely, and the values obtained for themerit function were sufficiently small: F(e, d, P0) = F(0.296,0.029, 7.8E–5) = 107.7, F(r, K) = F(0.055, 2.7) = 0.1 (Fig. 2).This is a qualitative indicator of the goodness of fit of themodel. When we quantitatively tested the goodness of fit forEqn 1 (df = 5, df = 3), we did not reject the null hypothesesthat the data were derived from the model, as R2 = 0.98 andqc ¼ f0:993; 0:986g. The bootstrap estimates of the errors inan estimated parameter set are given in Table 1.

Table 1 shows that, overall, the model dynamics are moder-ately sensitive to changes in infection efficiency, e, and initialdensity of the healthy leaf area, H0. The impact of the variationof the diffusion coefficient, d, and initial infected leaf area, P0,on the system dynamics was very low and negligible. Eqn 1 isinsensitive to both the host growth rate, r, and the host carryingcapacity, K.

Effective deployment of quantitative resistance

Two-component mixtures Fig. 3 shows that the absolute dis-ease reduction increased from negative values to positive ones formixture R/S. It remained negative for mixtures MR/S, MS/S,and positive for R/MR and R/MS. It fluctuates around zero forMR/MS. Like the disease severity, it declined, as the proportionof the resistant components, MS, MR, or R, increased in the mix-tures. The rate of increase or decline in absolute disease reductionor severity curve depends on mixture composition.

A susceptible line needs to be combined with > 63% of a resis-tant line (Fig. 3a) to make up an effective mixture. Using thisminimum allowable percentage of a resistant line reduced the dis-ease severity by 31% (Fig. 3b). Progressively increasing theproportion of a moderately susceptible line in a mixture with asusceptible one slowly reduced disease severity. Moving from 60to 80% of MS reduced the severity by only 10%, while the samechange in the MR line resulted in a 15% reduction. However,the progressive severity decline was not enough for either MR/Sor MS/S to obtain a positive absolute disease reduction. Anincrease in the proportion of the resistant line in a mixture with asusceptible line produced the fastest decrease in severity: reducingR from 80 to 60% reduced the severity from 34 to 14%.

The mixture of moderately resistant and resistant lines was themost effective, as disease severity was reduced to 20% or even lessby adding small proportions of the resistant line (Fig. 3b).

Table 3 Matrix of estimated differentiated infection efficiencies(ei,j)1 ≤ i ≤ 4, 1 ≤ j ≤ 4 of Puccinia striiformis f.sp. tritici pathogen genotypesP1, P2, P3, P4 on susceptible (S), moderately susceptible (MS), moderatelyresistant (MR), and resistant (R) lines

P1 P2 P3 P4

S 0.078 0.082 0.074 0.082MS 0.059 0.062 0.06 0.058MR 0.054 0.056 0.054 0.051R 0.044 0.046 0.044 0.043

The matrix corresponds to the disease severities reported in Fig. 1.




A mixture of resistant and moderately susceptible lines attainedthe acceptable degree of disease severity with 45% of a resistantline. In both mixtures, the severity decreased slowly as the pro-portion of R increased. The severity values of R/MS drew closerto those of R/MR as the proportion of R increased, and from60% of the resistant line, the severities of the two mixturesbecome almost identical and extremely low.

When the proportion of MR was increased, the absolute dis-ease reduction for the MR/MS mixture oscillated closely aroundzero. Disease severity declined slowly from 35% as the propor-tion of the moderately resistant line increased, and was 21% with90% of the MR line.

Only three mixtures satisfied our criterion for mixture effec-tiveness – a severity at the end of the growing season of < 20%and positive absolute disease reduction: R/S, R/MS, and R/MR,where the R variety had to exceed values of 71, 42, and 5%,respectively. The MS/S and MR/S mixtures could not reduce dis-ease severity to the acceptable value, while the MR/MS mixtureapproached 21% severity when the proportion of MR wasincreased.

Three-component mixtures Fig. 4 shows that the mixture ofmoderately susceptible, moderately resistant, and resistant lines

was the most effective, since a reduction of disease severity to 20–10% could be obtained over a broader range of proportions ofthe components. The higher the proportion of the MS line, thehigher the proportion of the R line had to be in the mixture inorder to preserve the effectiveness of the mixture (Fig. 4a). Themaximum allowable proportion of the MS cultivar was 50%, andthis could be combined with 30–40% of the R line and 20–10%of the MR line. To maintain disease control at 20% severity, low

(a)

(b)

Fig. 3 Absolute disease reduction (a) and disease severity (b) are plottedfor various percentages of the first mixture component (20–90%) in sixtwo-component mixtures, R/MR, R/MS, R/S, MR/MS, MS/S, MR/S,where ‘S’ denotes susceptible, ‘M’, moderately susceptible, and ‘R’,resistant. Solid curves correspond to percentages for which the absolutedisease reduction is positive. Absolute disease reduction was calculated asthe weighted mean severity (pure stands) – severity (cultivar mixture). Forinstance, to be effective, the mixture of resistant and susceptible linesshould contain < 37% of the susceptible component. In this case, themixture severity becomes < 30%, which corresponds to the weightedmean severity of the mixture components grown in pure stands.Simulation results were obtained with the parameter values presented inTables 1 and 3.

(a)

(b)

(c)

Fig. 4 Percentage of the three components of a cultivar mixture for whichthe absolute disease reduction is positive, and the disease severity is< 30%: light gray, component proportions for which the disease severitybelongs to a set of [0%; 10%]; mid-gray, proportions for which thedisease severity belongs to a set of [10%; 20%]; dark gray, proportions forwhich the disease severity belongs to a set of [20%; 30%]. Outside thedepicted areas, the target level of control cannot be attained, as either theabsolute disease reduction becomes negative or the disease severityexceeds 30%. (a) MS/MR/R mixture, %MR = 100% – (%R +%MS); (b)S/MR/R mixture, %MR = 100% – (%R +%S); (c) S/MS/R mixture, %MS = 100% – (%R +%S) (S, susceptible; M, moderately susceptible; R,resistant). Simulation results were obtained with the parameter valuespresented in Tables 1 and 3.


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NewPhytologist894

proportions of the R line should be offset by low proportions ofthe MS line and high proportions of the MR line. To attain a dis-ease severity of 10%, we would have to increase the proportion ofthe R cultivar to at least 40%. The advantage of the MS/MR/Rcombination is that it consists of a well-balanced mixture: 30%of MS, 40% of MR and 30% of R lines can reduce the severity ofdisease by 17%.

The use of the susceptible line in three-component mixturesgreatly reduces the range of component proportions at which thedisease control can be effective (Fig. 4b,c). To reduce diseaseseverity to 20%, the proportion of the susceptible line should notexceed 20% in mixtures with moderately resistant, or moderatelysusceptible and resistant lines (Fig. 4b,c). The higher the propor-tion of the susceptible line in a mixture, the higher the propor-tion of the resistant or moderately resistant line that has to bedeployed to maintain effective disease control. To obtain 20%disease severity, an amount of only 10% of the S cultivar shouldbe offset by including at least 30% of R and 60% of MR lines,while including 20% of the S cultivar requires the inclusion of atleast 60% of the resistant line. Moderately resistant and resistantlines complement each other in reducing the negative impact ofan S cultivar. In comparison with the MS/MR/R mixture, withwhich 20% severity can be obtained with well-balanced cultivarproportions, the S/MR/R mixture can provide the same resultwith 10% of S, 60% of MR, and 30% of R.

Fig. 4(a,c) shows that a moderately susceptible line should bedeployed in combination with moderately resistant and resistantlines. A high proportion of 50% of a moderately susceptible linecan be successfully offset by 30% of resistant and 20% of moder-ately resistant lines, and results in 20% disease severity. The com-bination of a moderately susceptible line with a susceptible onerequires the deployment of > 50% of a resistant line to reducedisease severity to 20% (Fig. 4c).

Discussion

We built a parsimonious host–pathogen model describing thedynamics of competing pathogen strains spreading over a geneti-cally diversified host population distributed in a two-dimensionalenvironment. Applying our model to the wheat–yellow rust patho-system, we identified the conditions under which cultivar mixturesthat divide the pathogen population into nonspecialized patho-types can result in both an acceptable degree of disease severity anda positive absolute disease reduction. In particular, we concludethat the best way to use moderately resistant and moderately sus-ceptible cultivars is to associate them with a highly resistant culti-var, rather than with a susceptible one. Moreover, our results revealthat the effective deployment of quantitative resistance makes itpossible to reduce disease density using small proportions of thehighly resistant cultivar. The significance of our findings is thatthey demonstrate that the role of quantitative plant resistance incultivar mixtures is to reinforce the effect of the highly resistantcultivars rather than being a substitute for them.

The model is parameterized to approximate to yellow rust spa-tial dynamics, but our work is generic in nature, and the model isrelevant to any plant–pest system where the pest population has a

diffusive pattern of dispersal. According to the test of goodnessof fit, our model accurately captures the dynamics of the develop-ment of yellow rust epidemic. The results of the local sensitivityanalysis suggest that the infection efficiencies of pathogen strainsand the initial density of healthy leaf area must be estimatedaccurately to provide a reliable model output. The result yieldedby the model agrees with previous theoretical findings demon-strating that disease spread can be slowed markedly when at least60% of the landscape area becomes unsuitable for the disease(Collingham & Huntley, 2000; Otten et al., 2004; Dewhirst &Lutscher, 2009). The fact that our theoretical findings are consis-tent with empirical results showing that rust dynamics can becontrolled by a mixture consisting of one-third of a susceptiblecultivar and two-thirds of a resistant cultivar (de Vallavieille-Pope & Goyeau, 1996; de Vallavieille-Pope, 2004) is further evi-dence of the credibility of the model. Moreover, the model con-firms empirical findings reporting that the severity-reducingeffect of a resistant cultivar mixed with a moderately susceptiblecultivar on wheat yellow rust was greater than that obtainedwhen it was mixed with the same proportion of a susceptible cul-tivar (Aslam & Fischbeck, 1993; Huang et al., 2011). Thus, wecan conclude that the model is biologically relevant, and that itcan be used for educational and research purposes to illustratethe principles of mixture theory and generate hypotheses aboutthe deployment of resistant cultivars. In the present formulation,the model is valid for locations with short growing seasons, suchas cereal crops grown in a temperate climate. It has been shownthat the length of the growing season can influence the effects ofcultivar mixtures on disease dynamics (Garrett et al., 2009).Incorporating the carrying capacity of the pathogen strains couldprovide a model applicable to locations with longer seasons. Weintend to complete the validation of the model by carrying outexperiments to test the performance of the suggested deploymentstrategies of wheat cultivars carrying nonspecific quantitativeresistance. This will increase confidence in the model for furtherpractical application in designing cultivar mixtures includingquantitatively resistant cultivars to provide sustainable wheat rustmanagement.

Despite the fact that the model does not account for the pro-cesses by which the pathogen adapts to quantitative resistance, itdoes allow us to derive evolutionarily stable strategies, if we usethe following theoretical criterion: in order to slow the process ofpathogen adaptation, the proportion of a resistant cultivar in amixture with a susceptible cultivar should be either < 30% or> 70%, as intermediate proportions lead to the rapid emergenceof super-virulent rust strains (van den Bosch & Gilligan, 2003;Bourget, 2013; Bourget et al., 2013). According to this criterion,our model shows that two-component mixtures consisting of aresistant cultivar and a susceptible, moderately susceptible, ormoderately resistant cultivar can be evolutionarily stable, if theproportion of the resistant cultivar exceeds 70%. Linking thecharacteristics of the plant resistance to epidemic dynamics in atheoretical model, Fabre et al. (2012) have shown that low crop-ping ratios of a resistant plant can prolong the resistance durabil-ity. We therefore also highlight the advantages of three-component mixtures, such as MS/MR/R and S/MS/R, as they




can provide evolutionarily stable disease control with low propor-tions of the resistant cultivar.

Modeling the dynamics of a nonspecialized pathogen in a cul-tivar mixture, Jeger et al. (1981a) showed that its dynamics canbe retarded if the components of the mixture create sufficientlyheterogeneous susceptibility. Our model allowed us to extendthis finding by determining what amounts of component resis-tance need to be combined to create a mixture possessing thisdegree of functional heterogeneity. According to our results, theheterogeneity created by a mixture of highly resistant and moder-ately resistant/susceptible cultivars is functional, as it can reducepathogen density to the target low value even with an intermedi-ate proportion of the highly resistant cultivar. Moreover, moder-ately resistant/susceptible cultivars can be used to reduce theproportion of the highly resistant cultivar required in mixtures,and thus to reduce the selective pressure on the pathogen popula-tion. They can be substituted for a susceptible line, but not for ahighly resistant cultivar. Our results reinforce the view that theproportions of the different components of the mixture cancompensate for weak heterogeneity in the susceptibility of thecomponents, or reinforce the effect of strong heterogeneity (vanden Bosch, 1993). The model showed that the higher the propor-tion of the highly resistant cultivar present in a mixture with amoderately susceptible cultivar, the less severe the yellow rust,which is consistent with the recent empirical results reported byHuang et al. (2011).

Our model shows that three-component mixtures includingdiffering amounts of resistance make it possible to design abroader range of effective control strategies. Finckh et al. (2000)argued that mixture performance depends on the mean amountof resistance of all the components of the mixture, rather thanon the number of components. However, empirical studiesdemonstrate that four-way mixtures are more effective thantwo-way mixtures in reducing severity of septoria tritici blotch(Mille et al., 2006). Furthermore, in the case of Rhynchosporiumsecalis, Newton et al. (1997) showed that increasing the numberof barley cultivars increased the mixture effect. Cultivarmixtures in practical use tend to include up to five cultivarscarrying qualitative or quantitative resistance. Our work showsthat successful disease control can be achieved with a three-component mixture, as they can create well-balanced diversityand functional heterogeneity in component susceptibility. Fur-thermore, in mixtures including combinations of a highly resis-tant line with a moderately resistant one, the proportion of theresistant line can be reduced to 10–30% without any loss ofmixture effectiveness. Our results suggest that it is worthincreasing the number of components in the mixture when asusceptible cultivar is unavoidable, or when we need to reducethe proportion of the resistant cultivar in order to obtain morelasting control.

Recently, cultivar mixtures have been promoted by Chinesegovernment agencies and researchers to control yellow rust inmountainous areas where P. striiformis can oversummer, but theyare not yet being used on extensive areas, such as those that wouldbe required for controlling rice blast (Huang et al., 2011). Morethan 30% of the wheat acreage in Washington State was planted

with mixtures of two or three cultivars, and this has reduced theyield losses caused by stripe rust (Kolmer et al., 2009). Despitethe high potential value of cultivar mixtures in disease manage-ment, the current standards of uniformity, marketing restrictions,and processing quality impede their wide use. In addition to tech-nical barriers, there is a dearth of the mathematical models thatcould help in selecting component cultivars and in tailoring culti-var mixtures to local pathogen populations. Here, we demon-strate that a parsimonious, spatially explicit model of a reaction–diffusion type could fill this gap. Our study contributes to thedevelopment of a general framework of mixture theory that couldallow us to design diversified agroecosystems that exert low selec-tive pressure on the pathogen populations. The model can beused in multidisease control by intercropping as well. It providestheoretical support for future experimental research intended todevelop disease control strategies based on diversified host popu-lations.

Acknowledgements

The authors are grateful to Laurent G�erard and Marc Leconte fortheir technical assistance, and to Christophe Montagnier and histeam (INRA Experimental Unit, Thiverval-Grignon) for fieldplot management. The research was funded by the ENDUREproject (2008–2010) under the 6th European FrameworkProgramme.

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Supporting Information

Additional supporting information may be found in the onlineversion of this article.

Fig. S1 Dynamics of densities of healthy, H(t), and infected, P(t),leaf area.

Notes S1 Technical information related to model analysis.

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Quantitative plant resistance in cultivar mixtures: wheat yellow rust as a modeling case study