Hierarchical structure of genetic distances: Effects of matrix size, spatial distribution and correlation structure among gene frequencies

Hierarchical structure of genetic distances:

Effects of matrix size, spatial distribution and correlation structure among gene frequencies

Flávia Melo Rodrigues and José Alexandre Felizola Diniz-Filho

Departamento de Biologia Geral, Instituto de Ciências Biológicas, Universidade Federal de Goiás, Caixa Postal 131, 74001-970 Goiânia, Goiás, Brasil. E-mail: [e-mail] Send correspondence to J.A.F.D.-F.

ABSTRACT

Geographic structure of genetic distances among local populations within species, based on allozyme data, has usually been evaluated by estimating genetic distances clustered with hierarchical algorithms, such as the unweighted pair-group method by arithmetic averages (UPGMA). The distortion produced in the clustering process is estimated by the cophenetic correlation coefficient. This hierarchical approach, however, can fail to produce an accurate representation of genetic distances among populations in a low dimensional space, especially when continuous (clinal) or reticulate patterns of variation exist. In the present study, we analyzed 50 genetic distance matrices from the literature, for animal taxa ranging from Platyhelminthes to Mammalia, in order to determine in which situations the UPGMA is useful to understand patterns of genetic variation among populations. The cophenetic correlation coefficients, derived from UPGMA based on three types of genetic distance coefficients, were correlated with other parameters of each matrix, including number of populations, loci, alleles, maximum geographic distance among populations, relative magnitude of the first eigenvalue of covariance matrix among alleles and logarithm of body size. Most cophenetic correlations were higher than 0.80, and the highest values appeared for Nei's and Rogers' genetic distances. The relationship between cophenetic correlation coefficients and the other parameters analyzed was defined by an "envelope space", forming triangles in which higher values of cophenetic correlations are found for higher values in the parameters, though low values do not necessarily correspond to high cophenetic correlations. We concluded that UPGMA is useful to describe genetic distances based on large distance matrices (both in terms of elevated number of populations or alleles), when dimensionality of the system is low (matrices with large first eigenvalues) or when local populations are separated by large geographical distances.

INTRODUCTION

Evolutionary theory predicts that genetic divergence between pairs of local populations should be correlated with geographic distances, as a result of stochastic processes such as migration and drift, and with environmental differences, as a result of natural selection (Dillon, 1984). Of course, interactions between these two effects are expected because geographical and environmental distances are usually correlated and because multi-locus data sets must be the result of many different evolutionary processes. Therefore, the analysis of geographic variation within species is important to understand evolutionary processes of genetic differentiation and speciation (Sokal, 1986a; Lessa, 1990; Sokal and Jacquez, 1991).

One of the most common approaches to evaluate genetic differentiation based on allozyme data among local populations and to determine its geographic structure, is the application of sequential, agglomerative, hierarchical and non-overlapping (SAHN) clustering algorithms on multivariate genetic distances among samples, though many other analytical methods are available today (Sokal and Oden, 1978a,b; Sokal, 1986a; Swofford and Olsen, 1990; Lessa, 1990). The SAHN clustering algorithm normally chosen is the unweighted pair-group method by arithmetic averages (UPGMA), based on many types of genetic distances (reviewed by Swofford and Olsen, 1990), although Nei's formulation (Nei, 1972, 1978) is the most commonly applied measurement. UPGMA clustering possesses many desirable properties, such as high stability and maximization of the cophenetic correlation coefficient (Farris, 1969; Rohlf and Sokal, 1981). UPGMA based on genetic distances have also been considered an efficient estimator of phylogenetic linkages, especially at low taxonomic levels (Nei et al., 1983; Sokal, 1986b; Swofford and Olsen, 1990).

This approach, however, has been criticized based on the fact that evolutionary processes do not always produce hierarchical patterns of geographic variation at the population level (Sokal et al., 1987; Lessa, 1990; Diniz-Filho, 1993). Hierarchical clustering can fail to produce an accurate representation of multivariate genetic distances in a low dimensional space, indicating a false hierarchy of populations, among which continuous (clinal) or reticulate patterns in fact exist (Lessa, 1990). One way to test these non-hierarchical effects is to estimate the cophenetic correlation coefficient (Sokal and Rohlf, 1962; Sneath and Sokal, 1973; Sokal, 1986b), which indicates the distortion produced by clustering in the original genetic distances. This coefficient is a matrix correlation between original genetic distances and a new distance matrix (the cophenetic matrix), derived directly from the UPGMA dendrogram (Sneath and Sokal, 1973). In low dimensional systems, clustering produces very small distortion in original distances among objects, in such a way that cophenetic correlation tends to one. This correlation coefficient cannot be tested in a statistical sense, even by matrix comparison techniques such as the Mantel test, because the null hypothesis of independence of matrices obviously does not hold (Sokal, 1979). This coefficient is also non-linearly related to the number of random interpolated gene frequency surfaces sampled in local populations, and so can be used as a measurement of hierarchy in multi-locus data matrices (Diniz-Filho, 1993).

The objective of this communication is to evaluate the variation in cophenetic correlations of UPGMA clustering based on distinct types of genetic distances, estimated for real data matrices obtained from the literature, to try to determine the situations in which these hierarchical models are adequate to describe spatial genetic structure among local populations.

METHODOLOGY

Fifty data matrices for allozyme data, obtained worldwide from natural animal populations, were examined in the literature (Table I). Groups analyzed ranged from Platyhelminthes to Mammalia (including humans). Geographic distances among local populations within species ranged from a few meters to thousands of kilometers.

Table I
- Animal populations from which genetic distance matrices were analyzed in this paper, with their taxonomic groups and sources from the literature.

Species	Taxonomic group	Source
Acridotheres tristis	Aves: Passeriformes	Baker and Moeed, 1987
Albinaria corrugata	Mollusca: Pulmonata	Schilthuizen and Lombaerts, 1994
Apis mellifera	Insecta: Hymenoptera	Del Lama et al., 1990
Apis mellifera	Insecta: Hymenoptera	Sheppard et al., 1991
Biomphalaria tenagophila	Mollusca: Basommatophora	Mascara and Morgante, 1991
Birgus latro	Crustacea: Decapoda	Lavery et al., 1995
Bombina bombina	Amphibia: Anura	Nürnberger et al., 1995
Bombina variegata	Amphibia: Anura	Nürnberger et al., 1995
Caledia captiva	Insecta: Orthoptera	Daly et al., 1981
Cervus elaphus	Mammalia: Artiodactyla	Hartl et al., 1991
Ceuthophilus gracilipes	Insecta: Orthoptera	Cockley et al., 1976
Cordylochernes scorpioides	Aracnidae: Pseudoscorpiones	Zeh and Zeh, 1994
Daphnia sp.	Crustacea: Cladocera	Hebert and Wilson, 1994
Drosophila buzzatii	Insecta: Diptera	Sokal et al., 1987
Drosophila melanogaster	Insecta: Diptera	Shamina and Parkash, 1994
Drosophila melanogaster	Insecta: Diptera	Jiang et al., 1989
Drosophila melanogaster	Insecta: Diptera	Long and Singh, 1995
Drosophila subobscura	Insecta: Diptera	Latorre et al., 1992
Drosophila tripunctata	Insecta: Diptera	Jaenike, 1989
Euphidryas editha	Insecta: Lepidoptera	Manly, 1985
Exoneura bicolor	Insecta: Hymenoptera	Blows and Schwarz, 1991
Fascioloides magna	Platyhelminthes: Echinostomida	Mulvey et al., 1991
Fringilla coelebs	Aves: Passeriformes	Baker, 1992
Fundulus heteroclitus	Pisces: Cyprinodontiformes	Ropson et al., 1990
Geomydoecus actuosi	Mammalia: Rodentia	Nadler et al., 1990
Heliconius erato	Insecta: Lepidoptera	Maurício-da Silva and Araújo, 1994
Homo sapiens (Yanomami)	Mammalia: Primates	Sokal et al., 1986
Hydroporus glabriusculus	Insecta: Coleoptera	Bilton, 1992
Lasioglossum zephyrum	Insecta: Hymenoptera	Crozier et al., 1987
Littorina mariae	Mollusca: Bivalvia	Totarenkov and Johannesson, 1994
Maniola jurtina	Insecta: Lepidoptera	Goulson, 1993
Mygalopsis marki	Insecta: Orthoptera	Dadour and Johnson, 1983
Nucella lapillus (Europe & USA)	Mollusca: Gastropoda	Day et al., 1993
Nucella lapillus (Norway)	Mollusca: Gastropoda	Day et al., 1993
Otiorhynchus alpicola	Insecta: Coleoptera	Mesaros and Tucic, 1995
Passer montanus	Aves: Passeriformes	Louis and Barlow, 1988
Podarcis wagleriana	Reptilia: Lacertidae	Capula, 1994
Rhinogobius brunneus	Pisces: Perciformes	Katto and Nishida, 1994
Rhytidopnera chalybaea	Hymenoptera: Formicidae	Ward, 1980
Rhytidoponera impressa	Insecta: Hymenoptera	Ward, 1980
Salmo truta	Pisces: Salmoniformes	Riffel et al., 1995
Salmo truta	Pisces: Salmoniformes	Karakousis and
		Triantaphyllidis, 1990
Sceloporus grammicus	Reptilia: Squamata	Sites-Jr. and Greenbaum, 1983
Telespiza cantans	Aves: Passeriformes	Fleischer et al., 1991
Thimallus articus	Pisces: Salmoniformes	Lynch and Vyse, 1979
Thomomys bottae	Mammalia: Rodentia	Patton and Yang, 1977
Thomomys sp.	Mammalia: Rodentia	Patton et al., 1972
Thomomys umbrinus	Mammalia: Rodentia	Hafner et al., 1987
Urocyon littoralis	Mammalia: Carnivora	Wayne et al., 1991
Warramaba virgo	Insecta: Orthoptera	Honeycutt and Wilkinson, 1989

Genetic dissimilarity between pairs of local populations was estimated for each data matrix using Nei's (1978) and Rogers (1972) distance coefficients and Cavalli-Sforza and Edwards' (1967) arc distance coefficient. Each of these distance matrices was subjected to UPGMA clustering, and cophenetic correlation (rH) was calculated to evaluate the magnitude of hierarchical effects on genetic distances (Sneath and Sokal, 1973).

The following parameters were also determined for each data matrix to try to establish in which situations a hierarchical structure of genetic distances can be detected: 1) number of local populations (NPOP); 2) number of loci (NLOC); 3) total number of alleles (NALL); 4) maximum geographic distance (in km) among populations (DMAX); 5) percentage of contribution of the first eigenvalue in relation to trace of the covariance matrix between pairs of alleles (EIGEN); 6) order of magnitude (in natural logarithms) of body size of species (SIZE). This last variable seems to be important since many biological processes at the population level associated with genetic differentiation, including local population density and growth and dispersion rates, are correlated with body size, even when measured over many orders of magnitude (Brown, 1995). All analyses were performed using Numerical Taxonomy and Multivariate Analysis System (NTSYS-PC), version 1.5 (Rohlf, 1989).

RESULTS

Frequency distributions of rH from UPGMA clustering based on the 50 genetic distance matrices using Nei's, Rogers' and Cavalli-Sforza and Edwards' coefficients were similar. Most values were above the normally acceptable critical values of 0.80 or 0.85 (Sokal, 1986b). Correlations between cophenetic coefficients based on the three types of genetic distances were very high (all r values larger than 0.760, with Nei's and Rogers' coefficients being most highly correlated; r = 0.943), indicating that when hierarchical effects are present, they are detected by all types of genetic distances.

Because of the non-linear and even non-monotonic relationships found between rH and the other parameters obtained from data matrices, correlation coefficients were useless to determine the contribution of these parameters to rH values. Multiple regressions also failed to explain rH by all predictors simultaneously, and the best R² estimate was 0.389, for rH values derived from Rogers' genetic distances, with significant angular coefficients for EIGEN and NALL. Results were very similar for rH values for Nei's and Rogers' analyses, and less similar for rH derived from arc distances. Statistical transformations of data also failed to improve these results. Despite these initial problems, visual inspection of scatterplots between pairs of variables permitted some conclusions about hierarchical structure among local populations. To conserve space, and considering the similarity between the three types of genetic distance matrices, only the scatterplots and analyses for rH from Nei's genetic distances are presented here.

None of the parameters analyzed displayed clear linear, or even monotonic, relationships with rH from Nei's genetic distances (Figure 1). The local populations, however, were usually restricted to an "envelope" triangular space, that can be interpreted in terms of mathematical or biological constraints in the relationship between pairs of variables (Brown, 1995). The most common pattern is that when the predictor increases, rH values increase in the mean and decrease in variance, such as can be seen for matrix dimensions (NPOP, NLOC, NALL) and for DMAX and EIGEN (Figures 1a-e). So, when predictors have small values (when few alleles or populations are analyzed, for example), rH can be small or large, but when these predictors have large values, rH will always tend to be close to 1.0. No clear relationship appears between rH and SIZE. The plots were also repeated using residuals from a regression of rH on NALL, with very similar results. So, values of rH are related to the predictors in a constraint space even after controlling for the dimensionality of the multivariate genetic distances.

Figure 1
- Relationships between cophenetic correlation (rH) and other parameters of genetic data matrices, including NPOP (a), NLOC (b), NALL (c), DMAX (d), EIGEN (e) and SIZE (f). For NLOC and DMAX, a significant power regression (P < 0.05) (broken line) was found.

We used the method of Blackburn et al. (1992) to statistically evaluate these patterns, which consists of fitting linear regressions within the borders of constraint space (upper and lower), using maximum and minimum values of y (in this case, the rH values) established for intervals of x (all other predictors). This strategy of fitting regression to borders of triangular regions has been continuously used in recent work on macroecological analysis (Brown, 1995). The results of these analyses applied to our genetic data (Table II) permit us to confirm the presence of triangular constraint spaces by testing the differences between slopes of the upper and lower regressions. For most predicted variables, the lower slope was significant at around the 10% level, and the upper slope was not significant at the same level. For NPOP, NALL, NLOC and DMAX the lower slope was at least 12 times larger than the upper slope, even for NPOP, for which both slopes were not significant at the 10% level. A power regression significantly improved the fit of lower border regressions for NLOC and DMAX (P < 0.05) (Figure 1b and 1d). Only the upper slope was significant for EIGEN, but this was the only variable for which a significant linear correlation with rH was found (r = 0.325; P < 0.05). None of the regressions was significant for SIZE. This method is strongly affected by the number and distribution of the points in the constraint space. Since there is a relative small number of points (macroecological studies usually work with hundreds of points - Brown, 1995), the number of class intervals in the predictors to establish the regressions is also small, in such a way that significant slopes are difficult to detect in our analyses. So, we considered acceptable significant regressions at the 10% level, because of the overall shape of constraint spaces and especially because of the difference between the upper and lower regression slopes. Changes in the number of classes in the predictors did not affect these results qualitatively.

Table II
- Regression analyses of the constraint lines of the relationship between cophenetic correlation coefficients and the predictors. n refers to the number of classes in the predictor variable, b is the regression coefficients of the upper and lower constraint lines, R² and F are the standard regression data for slopes and RT refers to a descriptive ratio between the two regression coefficients (lower/upper), in the module.

Predictor	Constraint line	n	b	R2	F	RT
NPOP	upper	3	-0.000192	0.030	0.030	16.922**
	lower	3	0.003249	0.906	9.642
NALL	upper	6	-0.000169	0.204	1.027	19.941**
	lower	6	0.003370	0.612	6.308*
NLOC	upper	5	-0.000475	0.070	0.226	14.737**
	lower	5	0.007000	0.666	5.976*
DMAX	upper	4	-0.000002	0.933	27.75*	12.500**
	lower	4	0.000025	0.979	93.46*
EIGEN	upper	5	0.001233	0.666	5.866*	1.435
	lower	5	0.001770	0.196	0.729
SIZE	upper	5	-0.001828	0.340	1.548	1.577
	lower	5	0.002883	0.185	0.680

* - 0.05 < P < 0.10;

** - significant difference at P < 0.05 between upper and lower slopes by the F-test.

A relationship was also found between EIGEN and maximum geographic distance (DMAX) (Figure 2A). When distance was small, EIGEN ranged from 20% to almost 100%, but when distance increased, EIGEN seemed to stabilize around 60%. This pattern in large distances can be due to sampling errors, since a few matrices with populations situated more than 10,000 km apart were used in this study, and a triangular pattern, such as previously described, or even a random distribution, would be expected to appear. However, mapping cophenetic correlation rH on the bidimensional space formed by EIGEN and DMAX, using a distance weighted least-squares (DWLS) algorithm (Figure 2B), revealed that qualitatively, the hierarchical pattern results were independent of these possible sampling errors. There was a negative correlation between EIGEN and DMAX for the maximum cophenetic correlation coefficient (the darkest region on Figure 2B). This indicates that, for populations close in space, elevated levels of hierarchical structure will appear only when variables (allelic frequencies) are correlated and associated with a few dimensions of variation (indicated by an elevated percentage of EIGEN).

Figure 2
- (A) Relationship between first relative eigenvalue of covariance matrix among allele frequencies (EIGEN) and maximum geographic distance (DMAX). (B) Cophenetic values superimposed to this relationship, interpolated using distance weighted least squares (DWLS) algorithm.

The level of genetic integration in the data matrix necessary to produce a strong hierarchical structure, however, decreases when geographic distance between populations increases. According to the interpolated DWLS surface of rH, this pattern will be maintained even if a more intensive sampling of genetic data matrices would furnish more matrices with populations separated by more than 10,000 km and with an elevated level of genetic integration (large values of EIGEN).

DISCUSSION

The constraint space found for most of the predictor variables analyzed through cophenetic correlation is expected from previous work in numerical taxonomy, that found robust estimates of hierarchical structure when large numbers of characters and species were analyzed (Sneath and Sokal, 1973; Sokal and Rohlf, 1981). These authors found that special situations also produce hierarchy in low dimensions, especially the increasing differences in taxonomic levels. In our study, done at a single taxonomic level (local populations within species), this pattern occurred because increasing matrix dimensions increases the probability of sampling the real dimensionality of the system, which in turn is related to the biological processes leading to population differentiation. So, stability of multivariate structure can in fact be explained by genetic and microevolutionary processes acting on distinct geographic scales.

The results derived from the analyzes performed on hierarchical clustering are in accordance with standard population genetics theory. When a few populations and alleles are analyzed in small geographic ranges, stochastic processes of local divergence can frequently produce random patterns of variation, and in this case each allelic frequency follows a distinct direction of variation in multivariate genetic space. This results in a relatively low explanatory power of the first eigenvalue (sphericity in multivariate space) and, consequently, in a low level of hierarchical structure (see Figure 2B). When geographic distance is low, hierarchical structure will appear only when dimensions are correlated (indicated by large EIGEN percentage), suggesting that both deterministic (natural selection) and stochastic (migration and demic diffusion) processes are important to produce population differentiation on a small scale, resulting in common hierarchical spatial patterns, such as patches of similar selective values and/or steep clines (Sokal, 1986a; Sokal and Jacquez, 1991), or that loci are under linkage disequilibrium.

When spatial scale increases, hierarchical structure tends to be more clear, even when the relative explanatory power of the first eigenvalue is not very high. Since EIGEN in fact measures the relative dimensions of the hypervolume of local populations in the multivariate space of all alleles (Johnson and Wichern, 1992), stochastic divergence over large geographic ranges that increases overall sphericity levels in multivariate space (because alleles evolve independently) may also generate some hierarchical patterns. This occurs because, although genetic drift can fix distinct alleles in distinct loci for some populations, there will be higher gene flow rates among local populations, that are closer in geographic space, producing common spatial patterns such as regional patches, that can be explained by isolation-by-distance or other genetic differentiation stochastic processes conditioned by population structure (Sokal and Wartenberg, 1981; Sokal and Uytterschaut, 1987). Another explanation is that, even if a few dimensions (alleles) are submitted to common selective processes (producing relatively low EIGEN), the large geographic distance and, possibly, the long time since divergence among local populations increase genetic divergence in a spatially structured way.

These patterns may be also produced by sampling effects on data matrices. The wider range of cophenetic coefficients found in a low number of populations, alleles and maximum geographic distance may be explained if the error in estimating genetic distances is large relative to the variability among local populations, reducing the possibility of producing hierarchical structure. This statistical explanation is, in fact, linked to the action of stochastic processes of genetic divergence at small geographic distances.

The results permit us to establish some strategies to use hierarchical clustering applied to genetic data. They suggest that UPGMA clustering is better applied to large matrices, both in terms of number of alleles and local populations, especially when sampled over large geographic ranges. When small matrices are available, UPGMA would still work only if the dimensionality of the system is low, concentrated on a few principal axes, or when analyses are done on a large geographic scale. In practice, it is recommended that cophenetic correlation must always be computed to check for the validity of representation of genetic structure by hierarchical clustering. Although this analysis was specifically conducted for UPGMA clustering based on genetic distances for allozymes, it is very probable that other hierarchical methods, including cladistic ones, based upon other types of genetic data (for example, divergence in DNA or mtDNA sequences), may be equally affected. When all of these methods are in adequate, alternative data analysis strategies, such as ordination techniques and spatial autocorrelation methods (Lessa, 1990; Sokal and Jacquez, 1991), would be a better alternative to describe genetic structure and investigate evolutionary processes at the population level.

ACKNOWLEDGMENTS

The authors thank L.M. Bini, M.I.B. Pignata, A.S.G. Coelho, and two anonymous referees for discussions and critical reading of the manuscript. Financial support was furnished by many research fellowships from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). Computations were performed in a microcomputer acquired with grants given by the Pró-Reitoria de Pesquisa e Pós-Graduação (PRPPG) and FUNAPE, Universidade Federal de Goiás.

RESUMO

A estrutura geográfica das distâncias genéticas entre populações locais dentro das espécies, baseada em dados de aloenzimas, tem sido usualmente avaliada com algorítimos hierárquicos, como o método não ponderado por médias aritiméticas (UPGMA). A distorção produzida no processo de agrupamento é estimada por um coeficiente de correlação cofenética. Estas abordagens hierárquicas, entretanto, podem falhar em produzir uma representação acurada das distâncias genéticas entre populações em espaços de pequena dimensão, especialmente quando padrões contínuos (clinais) ou reticulares de variação existem. Neste trabalho, nós analisamos 50 matrizes de distância genética entre populações animais da literatura, variando de platelmintos a mamíferos, para avaliar em que situações o UPGMA é útil para entender os padrões de variação genética entre as populações. Os coeficientes de correlação cofenética, derivados do UPGMA baseados em três tipos de coeficientes de distância genética, foram correlacionados com outros parâmetros de cada matriz, incluindo número de populações, loci, alelos, distância máxima geográfica entre as populações, magnitude relativa do primeiro autovalor da matriz de covariância entre os alelos e o logaritmo do tamanho do corpo. As correlações cofenéticas obtidas foram freqüentemente maiores que 0,80 e os valores mais elevados apareceram para as distâncias genéticas de Nei e Rogers. A relação entre o coeficiente de correlação cofenética e os outros parâmetros analisados foi definida para um "espaço de restrição", formando triângulos em que altos valores de correlação cofenética são observados para altos valores nos parâmetros, mas pequenos valores nestes não necessariamente correspondem a correlações cofenéticas elevadas. Isto permite concluir que o UPGMA é mais bem utilizado para descrever as distâncias genéticas baseadas em matrizes grandes (tanto em termos de números elevados de populações quanto alelos), quando dimensionalmente o sistema é pequeno (matrizes com elevados primeiros autovalores) ou quando as populações locais são separadas por grandes distâncias geográficas.

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(Received April 3, 1997)

Publication Dates

• Publication in this collection
  06 Jan 1999
• Date of issue
  June 1998

History

• Received
  03 Apr 1997