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Keywords:

  • diversity indices;
  • population genetics

Abstract

  1. Top of page
  2. Abstract
  3. Example
  4. Acknowledgements
  5. References

It has been established that Nei's measure of the average genetic diversity per locus, HS, and the measure of average differences between isolates with respect to simple mismatch dissimilarity, are identical measures of diversity within populations. The Müller index of diversity can be considered as the correction of Nei's measure for small samples.

Various indices are used for analyses and comparisons of diversity within plant pathogen populations. These include Nei's measure of the average gene diversity per locus, HS (Nei, 1973); the measure of average differences (dissimilarities) within populations, ADW (McCain et al., 1992); and the Müller index, Mu (Müller et al., 1996). The Nei index is the most frequently used in applications, although the average dissimilarity between isolates is also employed quite often (Adhikari et al., 1999; Kolmer & Liu, 2000; Menzies et al., 2003). Sometimes these two indices are used together (Gale et al., 2002) to enhance the significance of conclusions. The objective of this short communication is to prove that the three above-mentioned indices are actually the same measure of diversity within populations in the case of binary data. This statement also holds true in the case of multiple states of multiallelic loci. The proof for multiallelic loci is similar to that for binary data, but is rather lengthy – it is not included in this letter, but is available from the author on request.

Consider a sample from population P, which consists of n individuals tested on k differentiating factors and represented by binary patterns. The frequency of appearance 1 at the sth differentiating factor for population P is denoted by qs. For example, if the differentiating factors represent a typical set of differential host lines used in virulence tests, qs would be the frequency of virulence in population P on the sth differential line.

The indices of average differences within populations depend on the measure of dissimilarity between isolates. Consider the simple mismatch coefficient of dissimilarity

  • image(1)

where xi and xj are isolates from population P(i, j = 1, … , n) and d(xi, xj) is the number of characters for which two isolates xi and xj respond differently. The index of average differences with respect to the simple mismatch coefficient m is defined as:

  • image(2)

The product of ADWm by the number of differentiating characters k is also a usable index (Table 1 in Gale et al., 2002):

  • image(3)

This is the average number of characters for which two arbitrary isolates from population P respond differently.

Müller's mean dissimilarity index Mu (Müller et al. (1996) was defined as the mean number of virulence loci differences between all pairs of different isolates in a sample. Its normalized version (dividing by the total number of loci) has the form:

  • image(4)

where 1 ≤ i < j ≤ n, that is, only one of the pairs of isolates (xi, xj) and (xj, xi) is considered, and [n(n − 1)]/2 is the total number of such pairs.

The Müller diversity Mu within population P can be expressed by the index of average differences within population ADWm with respect to the simple mismatch dissimilarity:

  • image(5)

Nei's measure of the average gene diversity per locus HS (Nei, 1973) is determined by the formula:

  • image(6)

where k is the total number of loci (differentiating factors), HSs = 1 − inline image − (1 − qs)2, and qs is the frequency of one of the two alleles at the sth diallelic locus (or virulence frequency, or band frequency, or frequency of appearance 1 at the sth differentiating factor).

It was mentioned by Manisterski et al. (2000) that the Müller index is a function of virulence frequencies, and the corresponding formula was presented. It will be demonstrated here that the measure of average differences with respect to the simple mismatch coefficient ADWm equals the Nei gene diversity parameter HS, and the Müller index Mu could be considered as the correction of Nei's measure HS for small samples.

Dissimilarity ds between two isolates xi and xj with regard to any character s can be measured as follows: ds(xi, xj) = 0 if xi and xj respond identically on s, and ds(xi, xj) = 1 if xi and xj respond differently on s. The dissimilarity d between these isolates equals to the following sum:

  • image(7)

Therefore:

  • image(8)

There are nqs and n(1 − qs) isolates with positive (1) and negative (0) responses, respectively, on the differential s. Then (nqs)2 pairs of isolates respond positively on s, [n(1 − qs)]2 pairs of isolates respond negatively on s, and n2  (nqs)2 [n(1 – qs)]2 = 2n2qs(1 − qs) pairs of isolates respond differently on s, where n2 is the total number of pairs. Thus:

  • image(9)

and the following equalities are fulfilled:

  • image(10)

This means that Nei's measure of the average gene diversity per locus, HS, and the index of average differences with respect to the simple mismatch coefficient, are identical measures of diversity within populations. In the case of diallelic loci (binary data) the maximum value of the HS and ADWm indices equals 0·5. The Müller index Mu could be considered as the correction of Nei's measure HS for small samples because:

  • image(11)

A more accurate unbiased estimate of HS for a small sample size is given by the formula:

  • image(12)

Example

  1. Top of page
  2. Abstract
  3. Example
  4. Acknowledgements
  5. References

Let population P consist of six isolates (n = 6) tested on eight differentials (k = 8):

s12345678
x111110000
x201111110
x311111000
x401111100
x511100000
x611000001
qs2/315/62/31/21/31/61/6
HSs4/905/184/91/24/95/185/18

The Nei index scores the following value of within-population diversity:

  • image(13)

The matrix of dissimilarities between isolates according to the simple mismatch coefficient m has the following form:

mx1x2x3x4x5x6
x101/21/83/81/83/8
x21/203/81/85/87/8
x31/83/801/41/41/2
x43/81/81/401/23/4
x51/85/81/41/201/4
x63/87/81/23/41/40

The index of average differences with respect to the simple mismatch coefficient equals the sum of all entries of the dissimilarity matrix divided by the total number of entries:

  • image(14)

and the Müller index equals the sum of all entries above the diagonal of the dissimilarity matrix divided by the number of entries above the diagonal:

  • image(15)

Thus the results obtained by Nei's measure of the average gene diversity per locus, HS, the measure of average differences within population, ADWm, and the Müller index are absolutely comparable. The simultaneous application of these measures is unnecessary.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Example
  4. Acknowledgements
  5. References

This study was partially supported by the Leiberman–Okinow foundation.

References

  1. Top of page
  2. Abstract
  3. Example
  4. Acknowledgements
  5. References
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  • Manisterski J, Eyal Z, Ben-Yehuda P, Kosman E, 2000. Comparative analysis of indices in the study of virulence diversity between and within populations of Puccinia recondita f. sp. tritici in Israel. Phytopathology 90, 6017.
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  • Nei M, 1978. Estimation of average heterozygosity and genetic distance from a small number of individuals. Genetics 89, 58390.