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Genetical and Mathematical Properties of Similarity and Dissimilarity Coefficients Applied in Plant Breeding and Seed Bank Management

Institute of Plant Breeding, Seed Science, and Population Genetics, Univ. of Hohenheim, 70593 Stuttgart, Germany

^* Corresponding author (melchinger{at}uni-hohenheim.de)

	ABSTRACT

TOP ABSTRACT INTRODUCTION Nature of Molecular Marker... Distance and Euclidean... Genetic Dissimilarity... Genetic Dissimilarity... Relationships among... CONCLUSIONS REFERENCES

 
A proper choice of a dissimilarity measure is important in surveys investigating genetic relationships among germplasm with molecular marker data. The objective of our study was to examine 10 dissimilarity coefficients widely used in germplasm surveys, with special focus on applications in plant breeding and seed banks. In particular, we (i) investigated the genetical and mathematical properties of these coefficients, (ii) examined consequences of these properties for different areas of application in plant breeding and seed banks, and (iii) determined relationships between these 10 coefficients. The genetical and mathematical concepts of the coefficients were described in detail. A Procrustes analysis of a published data set consisting of seven CIMMYT maize populations demonstrated close affinity between Euclidean, Rogers', modified Rogers', and Cavalli-Sforza and Edwards' distance on one hand, and Nei's standard and Reynolds dissimilarity on the other hand. Our investigations show that genetical and mathematical properties of dissimilarity measures are of crucial importance when choosing a genetic dissimilarity coefficient for analyzing molecular marker data. The presented results assist experimenters to extract the maximum amount of information from genetic data and, thus, facilitate the interpretation of findings from molecular marker studies on a theoretically sound basis.

Abbreviations: AFLP, amplified fragment length polymorphism • OTU, operational taxonomic unit • SSR, simple sequence repeat

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Nature of Molecular Marker... Distance and Euclidean... Genetic Dissimilarity... Genetic Dissimilarity... Relationships among... CONCLUSIONS REFERENCES

 
QUANTIFYING THE DEGREE of dissimilarity among genera, species, subspecies, populations, and elite breeding materials is of primary concern in population genetics and plant breeding. Before 1970, measures of genetic dissimilarity between taxonomic units relied on pedigree analysis and morphological, physiological, or cytological markers, as well as biometric analyses of quantitative and qualitative traits, heterosis, or the segregation variance in crosses (Melchinger, 1999). During the following two decades, isozymes have successfully been employed in numerous taxonomic and evolutionary studies (Hamrick and Godt, 1990), but their use in other applications was hampered by the small number of polymorphic markers available.

Molecular markers, such as restriction fragment length polymorphisms (RFLPs), random amplified polymorphic DNA (RAPDs), amplified fragment length polymorphisms (AFLPs), simple sequence repeats (SSRs), and single nucleotide polymorphisms (SNPs), have meanwhile replaced isozymes and are heavily used for (i) detection of genetic relationships among different germplasm in seed banks and breeding programs (cf., Brummer, 1999), (ii) prediction of heterosis (cf., Melchinger, 1999), (iii) search for promising heterotic groups for hybrid breeding (cf., Reif et al., 2003), (iv) identification of duplicates in seed banks (cf., van Treuren et al., 2001), (v) assessment of the level of genetic diversity present in germplasm pools and its flux across time (cf., Dubreuil and Charcosset, 1998; Labate et al., 2003), and (vi) identification of essentially derived varieties in plant variety protection (cf., Smith et al., 1991; Lombard et al., 2000).

In these various applications, a proper choice of a similarity s or dissimilarity coefficient d = 1 – s (following the terminology of Gower, 1985) is important and depends on factors such as (i) the properties of the marker system employed, (ii) the genealogy of the germplasm, (iii) the operational taxonomic unit (OTU) (adopting the terminology of Sneath and Sokal, 1973) under consideration (e.g., lines, populations), (iv) the objectives of the study, and (v) necessary preconditions for subsequent multivariate analyses.

In a recent review, Mohammadi and Prasanna (2003) discussed the use of six coefficients d for the analysis of dichotomous molecular marker data, but ignored those coefficients based on allele frequencies, which are especially suitable for codominant marker data. Several authors (Goodman, 1972; Gower, 1985; Gower and Legendre, 1986) investigated the mathematical properties and relationships among various coefficients d. However, the above mentioned surveys disregarded coefficients, which are based on specific genetic models and, therefore, suitable for studies with seed bank or plant breeding materials.

To successfully conduct molecular marker surveys with plant breeding and seed bank materials, a thorough knowledge of genetical and mathematical properties of coefficients d is of crucial importance. Therefore, the objective of our study was to examine 10 coefficients d widely used in germplasm surveys, with special focus on applications in plant breeding and seed banks. In particular, we (i) investigated the genetical and mathematical properties of these coefficients, (ii) examined consequences of these properties for different areas of application in plant breeding and seed banks, and (iii) determined relationships between these 10 coefficients.

	Nature of Molecular Marker Data

TOP ABSTRACT INTRODUCTION Nature of Molecular Marker... Distance and Euclidean... Genetic Dissimilarity... Genetic Dissimilarity... Relationships among... CONCLUSIONS REFERENCES

 
We suggest the term allelic informative if allele frequencies can be determined from the molecular marker data. Marker data are denoted as allelic noninformative if this is not feasible. For instance, SSR data of individual genotypes are allelic informative. The AFLP data are mostly allelic noninformative although Geerlings et al. (1999), Piepho and Koch (2000), and Jansen et al. (2001) described methods to estimate allele frequencies and, thus, score AFLP data as allelic informative in specific situations.

Provided that molecular marker data are allelic informative, the estimates of coefficients d between OTUs can be calculated from the difference in the allele frequencies (Table 1). For allelic noninformative molecular marker data, coefficients d based on absence or presence of observation of bands or signals must be applied (Table 2).

	Distance and Euclidean Properties

TOP ABSTRACT INTRODUCTION Nature of Molecular Marker... Distance and Euclidean... Genetic Dissimilarity... Genetic Dissimilarity... Relationships among... CONCLUSIONS REFERENCES

[1]

[2]

[3]

Some simple but important properties follow from this definition. All elements of a distance matrix with respect to a set of OTUs S must be defined and positive or zero. The matrix is symmetric and the triangle inequality (Eq. [3]) holds true for all triplets (i, j, k S). The latter means that the length of any side of a triangle constructed with the three elements (i, j, k S) is less than or equal to the sum of the lengths of the other two sides, with equality occurring only when the triangle degenerates to a line.

The coefficient d is Euclidean if n points P_i Rⁿ exist such that the Euclidean distance between P_i and P_j is d_ij for all i, j, M (Gower and Legendre, 1986). An illustration of the Euclidean property is given by these authors.

The Euclidean property is important because it is an explicit or implicit assumption of many multivariate analysis methods such as principal coordinate analysis, also known as classical multidimensional scaling, hierarchical cluster analysis, classification, hierarchical classification, and graph theory (Gower, 1985). However, if a coefficient d is not Euclidean, then there exists a constant b greater than some minimal value such that the matrix with the elements (d_ij + b) is Euclidean (Cox and Cox, 2000). The problem of finding such a constant b has been referred to for many years, Messick and Abelson (1956) being an early reference. Thus, the Euclidean property is desirable but the main criteria for the choice of a coefficient d are its genetical properties. Both the Euclidean and genetical properties will be investigated for the coefficients d (Tables 1 and 2).

	Genetic Dissimilarity Coefficients for Allelic Informative Marker Data

TOP ABSTRACT INTRODUCTION Nature of Molecular Marker... Distance and Euclidean... Genetic Dissimilarity... Genetic Dissimilarity... Relationships among... CONCLUSIONS REFERENCES

 
Euclidean Distance
The Euclidean distance is defined as:

[4]

where p_ij and q_ij are allele frequencies of the jth allele at the ith locus in the two OTUs under consideration, n_i is the number of alleles at the ith locus, and m refers to the number of loci. The d_E ranges from zero to , the limits being assumed when the two OTUs have identical allele frequencies or are fixed for different alleles. Thus, an obvious disadvantage is that d_E values from different studies cannot be compared directly because d_E depends on the number of marker loci assayed.

The d_E is appropriate if allelic informative marker data are available and the relationships between OTUs (populations or individuals) are investigated in combination with multivariate methods that require dissimilarities possessing the Euclidean property.

Rogers' Distance
Rogers' distance (Rogers, 1972) is a modification of d_E and was developed assuming no knowledge about evolutionary forces diverging the OTUs under consideration:

[5]

The d_R is the average d_E across all loci standardized with the factor to restrict the values to the interval [0, 1]. It is one only if two OTUs are fixed for different alleles, but if one or both OTUs are not fixed and they have no alleles in common, d_R is not equal to one. d_R fulfills the distance properties (Nei et al., 1983), but it is not Euclidean. This follows from the identity d_R = 1 – s_SM for homozygous inbred lines and the fact that 1 – s_SM is not Euclidean (Gower and Legendre, 1986).

Assuming that (i) F₁ was the cross between two homozygous inbred lines P1 and P2 and (ii) O was an inbred offspring derived from the F₁ cross, Melchinger et al. (1991) showed that d_R fulfilled following two genetical properties:

[6]

[7]

The first property can be illustrated geometrically as three points F₁, P1, and P2 forming a line with F₁ lying in its center.

On the basis of these two properties, Melchinger et al. (1991) derived theoretical results that d_R estimates between two homozygous inbreds are linearly related to the coancestry coefficient (Malecot, 1948). Consequently, d_R is suitable for studying the relationship between the genetic dissimilarity of inbreds based on allelic informative marker data and the coefficient of coancestry (Malecot, 1948). This linear relationship is also desired in surveys (i) investigating the assembly and validation of core collections and the identification of duplicates in seed banks and (ii) uncovering pedigree relationships among OTUs as needed for the detection of essentially derived varieties in plant breeding.

Modified Rogers' Distance
Wright (1978)(p. 91) and Goodman and Stuber (1983) modified d_R by assigning each allele one dimension in the modified Rogers' distance:

[8]

Obviously, d_W = and as an Euclidean distance with values in [0, 1] it can be used for the same applications as recommended for d_E. Like d_R, d_W is not equal to one in the case of multiple alleles, even if the two OTUs have no allele in common.

Consider two populations, 1 and 2, in Hardy-Weinberg equilibrium and their hybrid population 1 x 2. On the basis of results of Falconer and Mackay (1996), and assuming biallelism and absence of epistasis, Melchinger (1999) derived the following relationship between the mean of these populations:

[9]

where H is the panmictic-midparent heterosis (Lamkey and Edwards, 1999), _i is the dominance effect at QTL i, and y_i is the difference in gene frequencies. Consequently, a linear relationship between H and d²_W is expected under the above conditions. Therefore, d²_W is especially suitable in studies based on allelic informative marker data for examining (i) the prediction of heterosis with genetic dissimilarities or (ii) the establishment of heterotic groups. Furthermore, d_W can be used for the same applications as suggested for d_E, owing to its Euclidean property.

Cavalli-Sforza and Edwards' Chord Distance
Cavalli-Sforza and Edwards (1967) developed a genetic distance to analyze blood group allele frequencies in human populations. In this coefficient, an OTU with allele frequencies p₁, p₂, ..., p_n is represented by the vector . Such a vector is always of unit length and, thus, the OTU is located on a surface of a hypersphere with a radius of one considering one locus. The distance between two OTUs is then directly proportional to the length of the chord connecting the points representing the OTUs. In particular, for two OTUs with no allele in common, d_CE is equal to one (Wright, 1978, p. 91). For multiple loci, the distances of all loci are combined by applying the Pythagorean theorem in many dimensions, so that the square of the distance between the OTUs is given by the sum of squared distances for each locus:

[10]

The d_CE ranges from zero to one even in the case of multiple alleles, which is an advantage over d_R and d_W. It can be shown that:

[11]

Thus, d_CE is similar to d_W except that it uses the square root of the allelic frequencies as coordinates and is consequently an Euclidean distance. The d_CE was developed based on Kimura's (1954) model of selective drift by assuming that (i) the mutation rate is small and (ii) variation in selection pressure is rapid and haphazard (no constant direction in allele frequency changes). It seems doubtful that seed bank and plant breeding materials have evolved according to this model because selection pressure is rather directed than rapid and haphazard. However, if allelic informative marker data are available and one can assume the selective drift model, then d_CE is a proper coefficient to investigate phylogenetic relationships among populations. Because d_CE is Euclidean, it can be used for the same tasks as proposed for d_E.

Reynolds' Dissimilarity
Reynolds et al. (1983) used the coancestry coefficient (Malecot, 1948) as the basis for a measure of genetic dissimilarity for short-term evolution, when the divergence between populations with a common ancestral population may be regarded as being caused solely by drift:

[12]

where

[13]

For populations completely fixed at each locus (i.e., two homozygous inbred lines) is equal to one and d_RE is undefined. Thus, d_RE is neither a distance nor Euclidean. The d_RE was developed assuming that an ancestral population was split into several subpopulations of the same size, which subsequently diverged due to drift. In such a situation, d_RE is expected to increase linearly with the time since the populations diverged (Weir, 1996, p. 91), that is, d_RE t/2N, where N is the subpopulation size and t the time measured in generations after divergence of the two populations. Thus, if mutation and selection can be neglected, and drift is the major evolutionary force, then d_RE is an appropriate dissimilarity coefficient for investigating the phylogenetic relationships among populations based on allelic informative marker data.

A recent application of d_RE was described by Labate et al. (2003), who examined relationships among U.S. maize landraces with SSR markers and assumed that an ancestral population split into several subpopulations diverging mainly due to drift. Mutation is known to have only small effects on genetic diversity compared with other forces and, thus, can safely be ignored in short-term evolution scenarios. However, neglecting selection as an evolutionary force in plant breeding or in seed bank populations seems questionable in most instances.

Nei's Standard Genetic Dissimilarity
In contrast to d_CE and d_RE, where it is assumed that populations diverged due to random genetic drift, Nei (1972) suggested a dissimilarity coefficient based on mutation and drift, often referred to as Nei's standard dissimilarity. This measure is intended to estimate the average number of codon substitutions per locus and was defined as:

[14]

Nei (1978) extended d_N72 with a bias factor. If two OTUs differ in all alleles, d_N72 is not defined, because it becomes – ln 0. Thus, d_N72 is neither a distance nor Euclidean. d_N72 was developed based on the infinite-allele model (Kimura and Crow, 1964) assuming that an ancestral population split into various subpopulations, which diverged due to drift and mutation. If (i) the mutation-drift balance is maintained throughout the evolutionary process, (ii) selection is absent, and (iii) the dissimilarity is not very large, then d_N72 = 2vt, where v is the mutation rate per locus and generation and t is the time measured in generations after divergence of the two populations (Nei et al., 1983). Under the above conditions, d_N72 is suitable for investigating phylogenetic relationships among populations based on allelic informative marker data but otherwise, the same constrains apply as for d_RE.

Nei et al.'s (1983) Dissimilarity
Assuming the infinite allele model (Kimura and Crow, 1964), Nei et al. (1983) suggested in a simulation study a dissimilarity coefficient, which is quite efficient in recovering the true evolutionary tree when it is reconstructed from allele frequency data (Nei and Kumar, 2000):

[15]

which equals d²_CE. Nevertheless, the result of the simulation study depends heavily on the underlying evolutionary model of the simulation scenario. The d_N83 was not developed based on a specific genetic model and it is neither a distance nor Euclidean. Thus, application of d_N83 in surveys for detecting phylogenetic relationships among populations seems questionable. For homozygous inbred lines, d_N83 = d_R, and hence it could be used for the same applications as d_R.

	Genetic Dissimilarity Coefficients for Allelic Noninformative Marker Data

TOP ABSTRACT INTRODUCTION Nature of Molecular Marker... Distance and Euclidean... Genetic Dissimilarity... Genetic Dissimilarity... Relationships among... CONCLUSIONS REFERENCES

 
With allelic noninformative marker data and two OTUs under consideration, one can form a 2 by 2 table with entries v_ij (number of bands in common between both OTUs), w_ij (number of bands present in the ith OTU and absent in the jth OTU), x_ij (number of bands absent in the ith OTU and present in the jth OTU), and y_ij (number of bands absent from both OTUs).

The simple matching coefficient is one of the oldest similarity coefficients (Sneath and Sokal, 1973):

[16]

For homozygous inbred lines, d_SM = 1 – s_SM = d_R, and therefore can be used for the same applications as suggested for d_R.

Jaccard (1908) suggested the similarity coefficient:

[17]

The Dice coefficient (Dice, 1945) is defined as:

[18]

The dissimilarity d_D = 1 – s_D is also called the Nei–Li distance (Nei and Li, 1979) and is related to d_J = 1 – s_J by a monotonic function.

In contrast to s_SM, both s_J and s_D do not involve negative matches (y_ij). For instance, if the probability of nonamplification of bands is high and absence of bands in both OTUs cannot be interpreted as a common characteristic, it is appropriate to apply coefficients s excluding negative matches (s_J and s_D).

In contrast to 1 – s, is for all three presented coefficients a distance and Euclidean (Gower and Legendre, 1986). Thus, they could be used to examine relationships among OTUs based on allelic noninformative marker data in combination with multivariate methods, the explicit or implicit assumption of which is a dissimilarity coefficient with the Euclidean property (Gower, 1985).

	Relationships among Dissimilarity and Similarity Coefficients

TOP ABSTRACT INTRODUCTION Nature of Molecular Marker... Distance and Euclidean... Genetic Dissimilarity... Genetic Dissimilarity... Relationships among... CONCLUSIONS REFERENCES

 
If (i) band absence or presence can be interpreted as two alleles of one locus and (ii) the OTUs under consideration are homozygous inbreds, then the following relationships exist between the s and d coefficients:

[19]

Gower (1975) proposed a method of comparing different multivariate analyses of the same data set, also known as Procrustes analysis (Cox and Cox, 2000). We used this approach to illustrate the differences between the dissimilarity coefficients based on allele frequency differences (Table 1).

The Procrustes analysis is based on the pairwise comparison between two sets of dissimilarities, d_ij and d^*_ij (i, j = 1, 2, ..., n), among the same sample of n OTUs. Rather than concentrating on the distances themselves, geometric points P_i (i = 1, ..., n) of the n OTUs are constructed to give rise to all the interdistances, d_ij. The coordinates of these points were obtained with Kruskal's nonmetric multidimensional scaling (Cox and Cox, 2000). Kruskal's nonmetric multidimensional scaling is a technique to represent OTUs in a reduced space while preserving the distance relationships among them with high fidelity. It is not limited to Euclidean distance matrices and can produce ordinations of objects from any dissimilarity matrix. Similarly, the coordinates of the points P_i(i = 1, ..., n) are found for the dissimilarities d^*_ij by applying again Kruskal's nonmetric multidimensional scaling. The two configurations are then matched for best fit by means of translation, rotation, and reflection. The criterion of best fit adopted is the minimization of the residual sum of squares R² = ²_i=1d_E², where d_E is the Euclidean distance between corresponding points P_i and P^*_i.

We compared the seven coefficients d based on allele frequency differences (Table 1) of a published data set of seven tropical CIMMYT maize populations (Reif et al., 2003) by subjecting them pairwise to the Procrustes analysis. The resulting R² matrix (Table 3) was then used as input for Kruskal's nonmetric multidimensional scaling (Fig. 1) . The same analyses were also performed with other data sets and yielded similar results (data not shown). All analyses were performed with Version 2 of the Plabsim software (Frisch et al., 2000), which is implemented as an extension to the statistical software R (Ihaka and Gentleman, 1996).

View larger version (12K): [in this window] [in a new window]   Fig. 1. First two principal axes (PA1 and PA2) of Kruskal's nonmetric multidimensional scaling for comparison of Euclidean (dE), Rogers' (dR), modified Rogers' (dW) and Cavalli-Sforza and Edwards' distance (dCE), and Nei's (1972) (dN72), Nei et al.'s (1983) (dN83), and Reynolds' (dRE) dissimilarity coefficient based on a Procrustes analysis with a published data set of seven CIMMYT maize populations (Reif et al., 2003).

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION Nature of Molecular Marker... Distance and Euclidean... Genetic Dissimilarity... Genetic Dissimilarity... Relationships among... CONCLUSIONS REFERENCES

 
Our investigations show that genetical (Table 4) and mathematical (Tables 1 and 2) properties of dissimilarity measures are of crucial importance when choosing a genetic dissimilarity coefficient for analyzing molecular marker data. The presented results can assist experimenters in the choice of dissimilarity measures that allow the extraction of the maximum amount of information from genetic data for given objectives. Thus, they facilitate the interpretation of findings from molecular marker studies on a theoretically sound basis.

	REFERENCES

TOP ABSTRACT INTRODUCTION Nature of Molecular Marker... Distance and Euclidean... Genetic Dissimilarity... Genetic Dissimilarity... Relationships among... CONCLUSIONS REFERENCES

 

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