Further comments on models of pathogen evolution in cultivar mixtures and the definition of pathogen fitness.

Yang and Ferrandino (12) make a number of thought-provoking points about Ostergard and Shaw (8) and Sun and Zeng (11) that we would like to comment on. We first clarify our concern about the estimation procedure proposed in Sun and Zeng (11) and then respond to some specific comments made in Yang and Ferrandino (12). Changes in frequencies of pathogen genotypes in cultivar mixtures or other heterogeneous host populations have been modeled for at least two reasons: (i) to predict theoretically how genotype distributions will evolve (1,6,7); and (ii) to give an expected pattern or null hypothesis with which to compare observed changes (5). Sun and Zeng (11) proposed a method to estimate relative fitnesses of pathogen genotypes growing on a heterogeneous host using observations of changes in frequencies of the genotypes on several mixtures of different composition. Sun and Zeng (11) suggested that their method could be used also to predict the effect of host composition on “race” frequencies. Their method was based on writing the change in frequencies of one genotype relative to a reference genotype as a linear combination of parameters, fijk, with the frequencies of the components as the coefficients (11). They described fijk as “the relative seasonal fitness of race i to race j on cultivar k.” Ostergard and Shaw (8) used a deliberately very simple situation to show that the relative changes in genotype frequencies in a mixture cannot, in general, be written as a linear function of the relative fitnesses on the pure components with the frequencies of the components as the coefficients. If the purpose of Sun and Zeng’s (11) method is to relate the observed changes of genotype frequencies in mixtures to the proportions of cultivars used in various experiments, the regression coefficients fijk cannot be interpreted as relative fitnesses on the components as Sun and Zeng (11) define them in their Equation 2. A simple example will clarify this. Consider two cultivars, A and B, and two pathogen genotypes, 1 and 2. Over a season, the genotypes produce offspring per original spore on isolated plots of each cultivar as implied in Table 1, lines 1 and 2. Thus, following Sun and Zeng (11), the relative fitnesses are to be referred to genotype 1, because it has the greatest absolute fitness on any cultivar in the system. The fitness of genotype 2 relative to genotype 1 on cultivar A is /4, and on cultivar 2, it is 0. Now suppose Sun and Zeng’s (11) procedure is followed in a very small experiment, using just a 0.5:0.5 mixture and a 0.8:0.2 mixture (Table 1, lines 3 and 4). For simplicity only, we assume there is no spread of inoculum between plants. We further assume that the initial inoculum is evenly spread over the mixture components, so that the initial density is the same on each cultivar. At the end of the season in the 0.5:0.5 mixture, twice as many spores of genotype 1 as of genotype 2 have been produced per initial spore, so the relative fitness of genotype 2 to genotype 1 is /2. In the 0.8:0.2 mixture this value is /3. Now, these figures can be inserted into the least squares equations suggested by Sun and Zeng (11), which will, in this case, have an exact solution since there are only two equations in two unknowns. Since there is no error, any estimation procedure should recover the original relative fitnesses. We denote estimated quantities by f ijk and the corresponding observed ones by fijk. After solving the equations, we obtain f 21A = /9 and f 21B = /9. But, in the construction of the example, f21A = /4 and f21B = 0! In conclusion, the procedure is not consistent. Furthermore, from experiments using different sets of mixtures, a very wide range of estimates, possibly including negative values, can be obtained even though there is no experimental error and the biology of the host-pathogen interaction is unchanged. In general, the regression coefficients will be strongly dependent on the exact set of mixture compositions used in the estimating equations, and the fit of the equations will not necessarily be good. We would next like to comment on the second part of Yang and Ferrandino’s (12) paper, in which they present definitions of fitness that they feel will better correspond to the intuitive notion. Given the confusion, this seems a good idea! (Yang and Ferrandino [12] use the phrase “population numbers” on each cultivar. We take this to mean population densities, since if actual population numbers are used, the cultivar density in effect enters twice into their Equation 6.) Yang and Ferrandino’s (12) Equation 9 is, as they point out, a restatement of the basic comparison of frequencies made in their Equation 1 (which is Equation 1 in Ostergard and Shaw [8] and Equation 2 in Sun and Zeng [11]). We would definitely agree that this ratio, Yang and Ferrandino’s (12) Equation 9, is a measurement of the relative fitness of two isolates on a given mixture or pure cultivar, and that this definition is fundamental. We would go further and say that it is basically the same definition used by both Ostergard and Shaw (8) and Sun and Zeng (11). However, the definition by itself does not allow measurements of relative fitness made on single cultivars to be related to measurements of relative fitness on mixtures. Sun and Zeng (11) proposed to predict changes in frequencies in single cultivars and arbitrary mixtures using experimental data on changes in frequencies in a restricted set of mixtures. Yang and Ferrandino (12) do not attempt this task. The derivation in Ostergard and Shaw (8) was not intended as a general method to make this connection nor as an introduction of a new set of definitions, as was made clear in that paper. We would emphasize that the aim in presenting Equations 5 to 9 in Ostergard and Shaw (8) was to set out the problems demonstrated above that we felt Sun and Zeng’s (11) procedure for estimating fitnesses on mixture components raised. We chose a drastically simplified (and therefore unrealistic) situation, since any suggested general method should at least be logically consistent in such a case. Yang and Ferrandino (12) point out that their Equation 6 differs in general from Ostergard and Shaw’s (8) Equation 6. The two equations are the same if the initial population densities for a genotype, Pik0, are the same on each component of Corresponding author: M. W. Shaw; E-mail address: M.W.Shaw@reading.ac.uk