Conceptual analysis of methods applied to assessment of diversity within and distance between populations with asexual or mixed mode of reproduction

Types of diversity indices

The following provides a general analysis of relationships between different diversity indices and explains some properties of these indices. Application of these indices in a few abstract examples as well as actual data sets are included to illustrate how properties of the indices may affect results and interpretations of the analyses. For convenient reference, all coefficients and indices mentioned and discussed in the text are presented together in Appendix A. In the examples, we express molecular and virulence markers data in binary characters for presence/absence of DNA bands or virulence/avirulence of plant pathogens to differential hosts. Virulence markers used in analyses of genetic diversity in plant pathogens typically occur in polymorphic gene-for-gene relationships in which each major gene for resistance in the host plant is effective only if it is matched by a corresponding gene for avirulence in the pathogen. The resistance gene is ineffective if the pathogen is homozygous for the virulence gene at the corresponding locus. For diversity analyses, a virulent or avirulent reaction may be regarded as equivalent to the presence or absence of a band at a specific location in a DNA gel.

Designations and dissimilarity between individuals

Consider a sample from population P and two samples collected from two populations P₁ and P₂, which consist of the same number n of individuals tested on k differentiating factors and represented by binary patterns of 1 for positive response (virulence, presence of a band) and 0 for negative (avirulence, absence of a band). If the numbers of individuals in P₁ and P₂ are different, then two bootstrapped samples of the same size, n, from P₁ and P₂ could be considered. We denote by q_i, q_1i and q_2i the frequencies of appearance of 1 at the ith differentiating factor for populations P, P₁ and P₂, respectively. For example, if the differentiating factors comprise a typical set of differential host lines used in virulence tests, q_i would be the frequency of virulence in population P on the ith differential line. The frequencies of genotype r in populations P, P₁ and P₂ are denoted by p_r, p_1r and p_2r, respectively. We denote by ρ any coefficient of dissimilarity between two individuals x₁ and x₂, ρ(x₁, x₂), and we compare results from use of the commonly used dissimilarity measures: simple mismatch coefficient, m, Euclidean distance, e, Jaccard, j, and Dice, d, coefficients of dissimilarity (see Appendix A1). A recently developed measure of dissimilarity between diploid organisms tested with codominant molecular markers (Kosman & Leonard, 2005) could also be considered.

Classification of diversity indices

Measures of diversity within and distance between populations may be initially divided into three groups according to sources of information used for their calculation (Table 1). These are indices based on: (1) dissimilarity between individuals; (2) frequencies of appearance of differentiating factors (alleles or DNA bands); and (3) frequencies of individuals’ genotypes determined according to a set of differentiating factors. The first group, based on dissimilarity between individuals, includes the average difference within (ADW) and between (ADB) populations (McCain et al., 1992), Kosman's distance between (KB) and diversity within (KW) populations (Kosman, 1996; Manisterski et al., 2000; Schachtel & Kosman, 2002) and the Müller index (Mu) of diversity (Müller et al., 1996). The second group, based on factor (allele) frequencies, includes Nei's standard genetic distance (N) and Nei's minimum genetic distance (N_M) (Nei, 1972, 1978), mean character difference (MCD) (Sneath & Sokal, 1973; Long et al., 1998; Manisterski et al., 2000), Nei's coefficient of differentiation (G_ST) and Nei's measure of the average gene diversity per locus (H_S) (Nei, 1973), and Kosman's index (K) (Manisterski et al., 2000). The third group, based on genotype frequencies, includes Rogers distance (R) (Rogers, 1972), Simpson's diversity index (Si) (Simpson, 1949), Stoddart's index (St) (Stoddart, 1983; Stoddart & Taylor, 1988), Shannon's entropy (Sh) (Shannon & Weaver, 1949) and the Shannon evenness parameter (E) (Sheldon, 1969).

This three-way division of the indices is not absolute. For example, Manisterski et al. (2000) showed that Müller's mean dissimilarity Mu could be considered an index from the second group, because it may be expressed by the virulence frequencies. Moreover, the Müller index can be considered as the correction of Nei's measure of the average gene diversity per locus H_S for small samples (Kosman, 2003) due to the following equalities:

Thus, Nei's diversity H_S = (1 − 1/n)Mu, and it may be considered an index from the first group. In addition, we will further show that the Rogers distance and the Simpson diversity from the third group and the Nei genetic distance and Nei's minimum genetic distance from the second group may also be represented as indices based on dissimilarity between individuals (i.e. first group).

Indices of average differences

The indices of average difference within and between populations can be defined for any measure of dissimilarity between individuals ρ. We denote as ADW_ρ and ADB_ρ the average differences within and between populations with respect to measure of dissimilarity ρ:

(Eqn 1)

(x_i, x_1i and x_2i are individuals from populations P, P₁ and P₂, respectively). Average differences within and between populations calculated by the Jaccard (Adhikari et al., 1999) and the Dice (Table 3 in Kolmer & Liu, 2000) measures, the Euclidean distance and the simple mismatch coefficient (Table 2 in Kolmer & Liu, 2000) may characterize populations in qualitatively distinct ways, because the sums of dissimilarities between multiple pairs of individuals will differ when described by the different measures. For instance, examples can be constructed in which

where j(y₁,y₃) is the Jaccard dissimilarity between individuals y₁ and y₃, d(y₁,y₃) is the Dice dissimilarity between individuals y₁ and y₃, etc. (data not shown). For dominant markers in diploid or dikaryotic fungi, the simple mismatch dissimilarity measure is superior to either the Dice or Jaccard dissimilarity measures for analyses of genetic diversity within populations or genetic distance between populations of the same species. For dominant markers, both the Dice and Jaccard measures ignore similarities between individuals that share the absence of a dominant trait, whereas both shared presence or shared absence of the dominant trait are taken into account by the simple match coefficient (Kosman & Leonard, 2005).

The index of average difference within population ADW_ρ may be used as measure of diversity within a population which ranges from 0 to 1 if 0 ≤ ρ ≤ 1, and ADW_ρ(P) = 0 if and only if population P consists of individuals of identical genotype. By contrast, the index of average difference between populations ADB_ρ cannot be used as measure of distance between populations, because it generally distinguishes between two identical populations: ADB_ρ(P,P) = ADW_ρ(P) > 0 if P includes at least two individuals of different genotype. The distance of average differences between populations DAD_ρ could be proposed as a correction of ADB_ρ index:

(Eqn 2)

The problem is that the DAD_ρ index may sometimes take negative values for certain dissimilarity measures ρ (E. Kosman, unpublished). One can prove that the distance of average differences with respect to the simple mismatch coefficient DAD_m equals Nei's minimum genetic distance N_M (Nei, 1972). It is always nonnegative, DAD_m(P₁,P₂) ≥ 0, and P₁ = P₂ implies DAD_ρ(P₁,P₂) = 0. Therefore, DAD_m for the simple mismatch coefficient could be used as measure of distance between populations.

Assessments of distance between populations using among populations diversity corrected for diversities within populations were also considered in earlier studies (for Dice dissimilarity, Lynch, 1990; Lynch & Milligan, 1994). However, we did not find any thorough justification of the validity of the methods proposed. Therefore, we would not recommend application of the distance of average differences with respect to the Jaccard or Dice dissimilarity because there is no proof yet that values of DAD_j and DAD_d are always nonnegative.

One can show that the Nei genetic distance between two populations may be represented in the form

(Eqn 3)

This means that the Nei distance can be considered an index from the first group, and that it is a function of the simple mismatch dissimilarities between individuals. The first approximation of the Nei distance is the distance of average differences with respect to the simple mismatch coefficient:

N(P₁,P₂) ADB_m(P₁,P₂) − (ADW_m(P₁) + ADW_m(P₂))/2 =DAD_m(P₁,P₂). This approximation is more accurate when values of ADB_m(P₁,P₂), ADW_m(P₁) and ADW_m(P₂) are closer to zero (P₁ and P₂ are closely related and quite homogeneous populations).

As was proved in (Kosman, 2003), Nei's measure of the average gene diversity per locus H_S and the Müller diversity Mu within population P can be expressed by the index of average differences within population ADW_m with respect to the simple mismatch dissimilarity:

(Eqn 4)

Thus, the Nei diversity H_S and ADW_m index are mathematically identical, calculation of both estimators for a data set (for instance, Table 1 in Gale et al., 2002) is a tautology, and therefore similarity of conclusions from these two measures does not show robustness of the results.

Kosman assignment based indices

The Kosman distance KB (Kosman, 1996) between two populations P₁ and P₂ of n individuals is defined as follows. To each individual from P₁ an individual from P₂ is matched so as to minimize the sum of dissimilarities between n corresponding pairs of individuals (optimal matching of individuals with similar genotypes). Finding the best matches is known as the ‘assignment problem’, and Kosman's distance is obtained by dividing the calculated minimum value of the sum of dissimilarities between matched pairs of individuals Ass_min(P₁,P₂) by the number n of matched pairs. If the numbers of individuals per population are not equal, two bootstrapped samples of the same size can be derived from the two populations for comparison. There are n! possible ways to match pairs of individuals between populations P₁ and P₂. When n is large, the number of possibilities increases very steeply. To achieve a workable solution, the ‘assignment problem’ algorithm provides an approach that eliminates a certain proportion of possibilities that have no chance of providing a minimum overall dissimilarity value. Even so, it is necessary to analyse great numbers of remaining possible combinations of matched pairs to find the combination that gives the minimum dissimilarity value.

The following example makes more tangible the method of calculation of Kosman distance.

Example 1 Consider samples of three individuals each from populations P₁ and P₂ tested on six differentials.

According to the simple mismatch coefficient m, dissimilarities between the individuals from these samples are represented by the following matrix:

In general, there are 6 (n! for n = 3) possibilities to match individuals from P₁ and P₂. The following two groups of matched pairs result in minimum of the sum of dissimilarities between corresponding pairs of individuals: A = {(a₁,b₃), (a₂,b₁),(a₃,b₂)} and B = {(a₁,b₂),(a₂,b₁),(a₃,b₃)}. Then Ass_min(P₁,P₂) = m(a₁,b₃) + m(a₂,b₁) + m(a₃,b₂) = m(a₁,b₂) + m(a₂,b₁) + m(a₃,b₃) = 8/6. The most similar pair of individuals (a₁,b₁) does not belong to any group of matched pairs, A and B. Moreover, pairs (a₁,b₃) from A and (a₃,b₃) from B are the most dissimilar possible pairs for individuals a₁ and a₃, respectively. Therefore, an algorithm for solving the ‘assignment problem’ requires analysing a great number of possible combinations of matched pairs, but not all of them (n!), and selecting at least one that gives the minimum overall dissimilarity value. Finally, the Kosman distance between populations P₁ and P₂ equals Ass_min(P₁,P₂)/n = 8/18 = 4/9.

The Kosman diversity KW (Kosman, 1996) within population P of n individuals is defined as follows. Individuals from P are matched to make up n pairs so as to maximize the sum of dissimilarities between the corresponding pairs (optimal matching of individuals with dissimilar genotypes). Finding such matches can be realized by the solution of the appropriate ‘assignment problem’, and Kosman's diversity is determined by dividing the obtained maximum value of the sum of dissimilarities between matched pairs of individuals Ass_max(P,P) by the number n of matched pairs.

Kosman's diversity within and distance between populations can be defined for different measures of dissimilarity between individuals. We will denote as KB_ρ and KW_ρ the Kosman diversities between and within populations with respect to dissimilarity ρ:

(Eqn 5)

Kosman's distance between populations KB_ρ is always nonnegative contrary to the distance of average differences DAD_ρ which may score negative values for some dissimilarity measures ρ. One can prove that if dissimilarity ρ between individuals is metric then the Kosman distance between populations KB_ρ is also metric (see Appendix B). This property of Kosman's distance may be important for some applications.

Let us show that the Rogers distance between two populations P₁ and P₂ and the Simpson diversity index can be represented as indices based on dissimilarity between individuals. Discrete measure of dissimilarity (discrete metrics) between individuals x₁ and x₂ is defined as follows

(Eqn 6)

Let the numbers of individuals of genotype r from populations P₁ and P₂ be n_1r and n_2r, respectively, and s is the total number of genotypes in both populations of n individuals. Then for a single differentiating factor the Kosman distance between populations P₁ and P₂ with respect to dissimilarity δ equals the Rogers distance between these populations:

(Eqn 7)

This means that with more than one differentiating factor the Kosman distance with respect to the simple mismatch coefficient KB_m(P₁,P₂) (Kosman, 1996) can be considered as a generalization of the Rogers distance, in which the measure of similarity between different genotypes is taken into account. One can prove that min{n_1r,n_2r} individuals from population P₂ are matched to the individuals of genotype r from population P₁ by the solution of the corresponding ‘assignment problem’Ass_min(P₁,P₂). Thus, the Kosman distance KB_m takes into account both the genotypic structure of populations and the measure of similarity between different genotypes.

Let n_r be the number of individuals of genotype r from population P of n individuals, and s is the total number of genotypes in this population. Then the average difference within population P with respect to dissimilarity δ equals the Simpson diversity index within this population:

(Eqn 8)

Thus, the Rogers distance and the Simpson diversity, which were originally defined as functions of frequencies of genotypes, can be considered as indices based on dissimilarities of individuals with respect to discrete metrics.

Genotype based indices

The Shannon entropy (Shannon & Weaver, 1949) measures diversity within population on the basis of frequencies of genotypes p_r, r = 1, 2, … , s. Its value ranges between 0 for a single genotype and lns when genotypes are evenly distributed (p_r = 1/s for all r). If the number s of genotypes increases then the Shannon entropy may tend to infinity. For convenience, it is possible to normalize this index in order to make it quantitatively comparable with other diversity measures, which vary between 0 and 1. There are two ways to normalize the Shannon entropy. The first one was proposed by Sheldon (1969) and implies division of the Shannon entropy by its maximum value ln s for a sample which comprises the same number s of genotypes. This is the so-called evenness parameter:

The second way implies division of the Shannon entropy by its maximum value ln n for a sample that consists of the same number n of individuals (in this case n is also the maximum possible number of genotypes for such a sample) (Goodwin et al., 1992; Andrivon & de Vallavieille-Pope, 1995). We will call this measure of diversity within populations the Shannon normalized index:

Both evenness and richness aspects of genotypic diversity are taken into account by this last index because it can be represented as the product of the evenness and richness parameters of the population as follows:

(Eqn 9)

where the richness parameter ln s/ln n reflects relative abundance of the sample. Hence, the Shannon normalized index is more informative than the evenness parameter, and therefore we would recommend to use this index for measuring genotypic diversity. We agree only in part with the conclusion of Grünwald et al. (2003) that scaling the Shannon entropy by sample size (ln n) should be avoided. This conclusion is right only in cases in which the maximum theoretically possible number of genotypes that can be detected by the technique used to assay for variation is less than the sample size. For example, if 1000 isolates are tested on eight binary differentiating factors (virulence/avirulence, presence/absence of DNA bands) then the maximum theoretically possible number of genotypes equals 2⁸ = 256. In this case (similar to simulation parameters considered by Grünwald et al. (2003): 1000 for sample size and 200 for number of genotypes) scaling by sample size may distort actual diversity assessments because the richness component ln s/ln n of the Shannon normalized index in equation 9 never reaches its maximum theoretically possible value 1 (ln s/ln n ≤ ln 256/ln 1000 0.803). However, a number of differentiating features used in regular diversity analyses of plant pathogen populations typically varies from 10 to 30 differential hosts for virulence data and from dozens to hundreds band positions for molecular markers, whereas a number of isolates tested (sample size) does not exceed 1000 and is usually measured in a few dozens or a couple of hundreds. Then the number of theoretically possible genotypes (1024 = 2¹⁰ at least) is considerably larger than sample size, which allows normalization of the Shannon entropy by logarithm of sample size.

Similarly the Stoddart diversity index , which ranges between 1 and s ≤ n, can be normalized in two different ways: dividing by s and by n, respectively. The first index St(P)/s can be considered as evenness parameter, whereas the second one takes into account both evenness and richness characteristics of populations because

Genotypic diversity (Shannon, Simpson and Stoddart) and distance (Rogers) indices usually overestimate actual diversity within and difference between populations. This bias becomes more significant for larger number of differentiating characters and can be explained as follows. In the case of k binary characters, the number of theoretically possible genotypes equals 2^k. The number of individuals tested, n, is limited and considerably less than 2^k even for relatively small values of k. Therefore, it is very likely that nearly all sampled individuals are of different genotypes (s ≈ n and p_r ≈ 1/n for all genotypes r = 1, 2, … , s). In such a case, both evenness and richness parameters are very close to their maximum values, which result in high score of genotypic diversity. Actual population diversity might be much lower because of possible similarity between overall allelic patterns of individuals. However, degree of similarity between individuals is not taken into account by indices based only on genotype frequencies. Similarly, the Rogers distance may overestimate actual difference between two populations because of the high probability that none or a very small number of individuals in the two populations will have identical genotypes.

Rationale for choosing diversity indices

What measures of diversity within and distance between populations are suitable for organisms with asexual and mixed mode of reproduction (for instance, plant pathogens)? The choices should take into account that populations with clonal reproduction may differ from each other in ways that sexual random mating populations do not.

Basic properties of diversity measures

Plant pathogen populations often exhibit considerable linkage disequilibrium because of asexual or mixed mode of reproduction. An optimal index for measuring diversity within such populations should satisfy several conditions (Groth & Roelfs, 1987). A population is more diverse and diversity index is higher if: (i) that population consists of a larger number of genotypes for a given number of individuals; (ii) it is characterized by a more even distribution of genotypes; (iii) the number of differences in virulence between genotypes within the population is larger.

The Shannon evenness parameter E satisfies the property (ii) only (Table 2). The first two properties are fulfilled by Simpson's Si, Stoddart's St and Shannon's entropy and normalized Sh diversity indices (Table 2), but they do not satisfy the third property because the virulence patterns are not taken into account by these indices. By contrast, the measure of average difference within populations ADW, the Müller index Mu and Nei's measure of the average gene diversity per locus H_S satisfy the property (iii) and do not meet the properties (i) and (ii) (Table 2) because genotypes are not considered by these indices. The Kosman diversity KW_m is a more complicated index. It takes into account the contribution of dissimilarity among individuals to the diversity within a population, not only the relative frequencies of different genotypes like Si, St and Sh indices. In addition, the KW_m index considers a population as a set of different genotypes with possibly associated virulences/avirulences and does not characterize a population by an independent set of virulence frequencies in contrast to ADW, Mu, H_S and K indices. The properties (i) and (ii) are not generally compelling by the Kosman diversity KW_m. The extent of dissimilarity among individuals contributes considerably to the diversity within a population. The following example demonstrates why Kosman's diversity KW_m is preferred over other indices in accounting for multiple aspects of diversity within a population.

Example 3 Consider samples of four individuals each from populations P₁ and P₂ tested on four differentials.

Populations P₁ and P₂ are considered as equally diverse by the following estimators: Si = 3/4, Sh = ln(4)/ln(4) = 1, ADW_m = H_S = 3/8, Mu = 1/2 are equal for both populations. By contrast, population P₂ is more diverse than P₁ according to Kosman's measure: KW_m(P₁) = 1/2 and KW_m(P₂) = 3/4.

This example demonstrates that the three properties (i), (ii) and (iii) of an optimal diversity index are not enough to distinguish between population diversities. The number of genotypes in each of two populations P₁ and P₂ is equal (four), the genotypes are evenly distributed within each population (genotype frequencies equal 1/4) and average dissimilarity between genotypes (individuals) equals ADW_m = 3/8 for both populations. Nevertheless, the Kosman KW_m index distinguishes between diversities of these populations. It could be explained by the absolute deviation of pairwise dissimilarities from the average dissimilarity:

(Eqn 10)

(x_i and x_j are individuals from population P of n individuals). In our case s(P₁) = 3/16 is less than s(P₂) = 1/4, and this is a reason why population P₂ can be considered as more diverse than population P₁. Therefore, it could be reasonable to supplement three properties of an optimal diversity index by adding the following fourth property. A population is more diverse and diversity index is higher if: (iv) deviation of pairwise differences between genotypes (individuals) from the average difference is larger.

The next example demonstrates that effects of conditions (i)–(iv) on diversity within populations may compensate each other if they counteract.

Example 4 Consider samples of four individuals each from populations P₁ and P₂ tested on four differentials.

The vector of virulence frequencies is the same for the both samples, and therefore, all diversity indices based just on virulence frequencies do not distinguish between diversities within populations P₁ and P₂. On the other hand, according to the Simpson and the Shannon normalized indices, which are based just on genotype frequencies, population P₁ is considered as more diverse than population P₂ (Si(P₁) = 3/4, Si(P₂) = 1/2) and Sh(P₁) = ln(4)/ln(4) = 1, Sh(P₂) = ln(2)/ln(4) = 1/2) because of the larger number of genotypes in P₁ (four) compared with P₂ (two) and even distribution of genotypes in both populations. This means that property (i) was the crucial one for making decision because condition (ii) is satisfied for the both populations P₁ and P₂, and properties (iii) and (iv) are irrelevant for Simpson's and Shannon's indices.

The Kosman diversity KW_m equals 1 for both populations (i.e. P₁ and P₂ are considered as equally diverse). This could be explained as follows. The average differences between isolates are equal for the both populations, ADW_m = 1/2, whereas the absolute deviation (Eqn 10) is less for population P₁, s(P₁) = 1/4 and s(P₂) = 1/2. Thus, populations P₁ and P₂ are of similar characteristics with respect to the conditions (ii) and (iii), but characterizations of these populations by properties (i) and (iv) are of opposite tendency. The greater number of different genotypes in population P₁ increases its level of diversity compared with P₂. However, the measure of variance between isolates is less for population P₁. Therefore, the former effect is compensated by the latter one, which leads to the supposed equal diversity within these populations.

Correlation of results from different types of indices

Different measures of diversity within and distance between populations were applied for analysis of Israeli populations of wheat leaf rust, caused by the fungus Puccinia triticina (Manisterski et al., 2000; J. Manisterski, unpublished data). The sexual stage of P. triticina occurs on alternate hosts in the genus Thallictrum, which do not occur in Israel but do occur in the Iberian peninsula and in Eastern Europe. Thus, P. triticina reproduces only asexually within Israel, but some level of migration of P. triticina into Israel may occur from sexually reproducing populations in Europe. Dozens of isolates were collected annually during 1993–2002 and tested on a set of 15 differentials, and the corresponding virulence patterns were arranged in two-way data tables of 0s (avirulence) and 1s (virulence). Diversities within 10 annual collections of leaf rust isolates and distances between them were calculated using the KOIND package (Schachtel & Kosman, 2002). The results are presented in the following examples 5 and 6.

Example 5 Comparison of different measures of diversity within populations.

Diversity within 10 populations of leaf rust isolates was estimated by all indices mentioned in Appendix A, A2. The corresponding values for all possible pairs of diversity indices were plotted one vs. another using the mxplot program of NTSYSpc package, version 2.1 (Exeter Software, Setauket, NY, USA). The results for three diversity measures of different nature (Kosman's diversity within population, KW, Nei's measure of the average gene diversity per locus, H_S, and the Shannon normalized index of diversity, Sh) are presented in Fig. 1. Rather strong correlation (r = 0.943) was observed between values of the Kosman and Nei diversity indices (Fig. 1a). Note, however, that the rank order of least diverse to most diverse was not exactly the same for the Kosman and Nei diversity indices. Results from the Shannon's index were not correlated with those from either the Kosman (Fig. 1b) or Nei (Fig. 1c) diversity indices. Similar results were obtained when the Kosman diversity within population (KW) was compared with another Kosman's index (K), which is based only on virulence frequencies like the Nei (H_S) index, and with the Simpson (Si) and Stoddart (St) indices, which are based only on genotype frequencies as is the Shannon normalized index (Sh). Weak correlation was observed between an index based only on virulence frequencies (K or H_S) and an index based only on genotype frequencies (Sh, Si or St). Conversely, the indices of the same nature were strongly correlated. Thus, different measures can result in qualitatively unlike descriptions of diversity within populations, especially if they are of different nature. As expected for genotypic diversity measures, the Shannon normalized index (Sh) is biased towards high values (Fig. 1b,c).

Example 6 Comparison of different measures of distance between populations.

Distance between all possible pairs of 10 populations of leaf rust isolates was estimated by all indices mentioned in Appendix A, A3. The values for all possible pairs of distance indices were plotted one versus another using the mxcomp program of the NTSYSpc package, version 2.1 (Exeter Software), which also provided the corresponding values of Mantel test correlation between different indices. The results for three distance measures of different nature (Kosman's distance between populations, KB, Nei's genetic distance, N, and the Rogers distance, R) are presented in Fig. 2. Rather strong correlation (r = 0.951) was observed between values of the Kosman and Nei distances (Fig. 2a). Note, however, that in some cases the rank order for pairs of populations according to distance between them differed considerably between the Kosman and Nei distances. For example, one pair of populations was considered more distant than 23 other pairs by the Nei index, but more distant than only nine other pairs by the Kosman index. A discrepancy of this magnitude could produce a significant distortion in cluster analyses for the populations. Rather strong correlations also were obtained when the Kosman distance (KB) was compared with the mean character difference (MCD) and Nei's coefficient of differentiation (G_ST), which are based only on virulence frequencies as is the Nei distance (N). Results from Rogers’ distance (R), which is based only on genotype frequencies, were only weakly correlated with those from either the Kosman (r = 0.712) or the Nei (r = 0.619) distances. Weak correlation also was observed between Rogers’ distance (R) and the group of indices that are based only on virulence frequencies (MCD, N and G_ST). By contrast, the indices of the same nature (MCD, N and G_ST) were strongly correlated. Thus, different measures can result in qualitatively unlike descriptions of distance between populations, especially if they are of different nature. As expected for genotypic indices, the Rogers distance (R) is biased towards high values (Fig. 2b,c).

Genetic diversity within and between populations is a function of gene and genotypic structure of populations. Consequently, variance of any genetic population parameter can partly be explained by variability of the corresponding gene and genotypic parameters. The last two are not independent, and it seems that gene structure of populations may have a much stronger impact on genetic estimators than genotypic one has. Therefore, high level of correlation observed between genetic and gene parameters in Examples 5 and 6 (KW vs H_S diversity (r = 0.943) and KB vs N distance (r = 0.951), respectively) was expected. This means that about 90% (r²) of the variance of genetic diversity within and between the annual populations of wheat leaf rust can be explained by the variation in corresponding gene estimators. If the P. triticina populations in Israel reproduced sexually with random mating, the Kosman and Nei diversities and distances would be even more highly correlated, because the Nei indices assumes linkage equilibrium which typically results from random mating. In asexual populations, however, lack of genetic recombination causes even selectively neutral genes to increase or decrease in frequency in response to average differences in relative fitness of the genotypes in which the neutral alleles happen to occur. Thus, in asexual populations small enough to be affected by genetic drift or in asexual populations exposed to fluctuating environmental conditions from generation to generation, lack of genetic recombination is likely to result in at least some genetic disequilibria even between selectively neutral alleles at different loci.

Example 6 demonstrated that despite strong correlation between distance measures of different types the rank order for pairs of populations according to distance between them may differ considerably. The following example shows that a similar effect may occur even for distance measures of the same type.

Example 7 Comparison of distance measures based on allele frequencies.

Zhang et al. (2005) used the sequence-related amplified polymorphism (SRAP) technique to determine genetic diversity and population structure of Apiosporina morbosa on Prunus spp. Resemblance of the pathogen populations collected from 15 geographic locations was assessed on the basis of allele frequencies at 58 SRAP loci (two alleles corresponding to presence and absence of bands were considered at each putative locus). The Rogers modified genetic distance coefficient (Wright, 1978) and the Nei genetic identity (Nei, 1972) were calculated for all possible pairwise comparisons among 15 populations (Table 3 in Zhang et al., 2005). We calculated the Nei standard genetic distance (N) between the populations as a function of Nei identity, and the values of N were plotted versus the corresponding values of Rogers modified genetic distance (RG) as in Example 6 (Fig. 3). Despite relatively strong correlation (r = 0.897) between the indices, in some cases the rank order for pairs of populations according to distance between them differed considerably between the Nei and Rogers modified genetic distances. For example, among total 105 pairs of populations one pair was considered more distant than 93 other pairs by the Rogers genetic distances, but more distant than only 40 other pairs by the Nei index. By contrast, another pair of populations is more distant than only 11 other pairs by the Rogers genetic distances, but more distant than 34 other pairs by the Nei index. A discrepancy of this magnitude could produce a significant distortion in cluster analyses for the populations.

Note

Software for calculating Kosman's diversity indices and some other population parameters is free and available from the first author.