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Copyright 0 1988 by the Genetics Society of America

Genetic Divergence in Mandible Form in Relation to Molecular Divergence in Inbred Mouse Strains

William R. Atchley,* Scott Newman+ and David E. Cowley* *Department of Genetics, North Carolina State University, Raleigh, North Carolina 27695, and TBiometry Section,

Waite Agricultural Research Institute, Glen Osmond, South Australia 5064

Manuscript received October 27, 1987 Revised copy accepted May 1 1, 1988

ABSTRACT Genetic divergence in the form of the mandible is examined in ten inbred strains of mice. Several

univariate and multivariate genetic distance estimates are given for the morphological data and these estimates are compared to measures of genealogical and molecular divergence. Highly significant divergence occurs among the ten strains in all 11 mandible traits considered individually and simultaneously. Genealogical relationship among strains is highly correlated with genetic divergence in single locus molecular traits. However, the concordance between genealogical relationship and multivariate genetic divergence in morphology is much more complex. Whether there is a significant correlation between morphological divergence and genealogy depends upon the method of analysis and the particular genetic distance statistic being employed.

B IOLOGISTS have a long tradition of relying upon morphological data to resolve complex ev-

olutionary problems. Indeed, many significant decisions about taxonomic classification, evolutionary relationships, and rates of divergence are based upon the degree of morphological resemblance. For example, in the absence of data on reproductive isolation, the working taxonomist often uses the level of morphological divergence to decide whether organisms are conspecific. Decisions about phylogenetic relationships among species and higher taxa are often based upon some quantitative measure of morphological similarity (SNEATH and SOKAL 1973) or upon whether taxa share “primitive” or “derived” morphological features (WILEY 198 1). Finally, the extent of morphological divergence between related taxa is often used to measure the rate at which evolutionary divergence has occurred and whether evolutionary change has occurred gradually or in spurts (STANLEY 1979; KI-

In spite of this heavy reliance on morphological data, the genetic aspects of evolutionary divergence in morphology remain poorly understood. BAILEY (1959), GREWAL (1962), HOI-SEN (1972), FESTING (1 973), WAYNE and O’BRIEN (1 986) and others have examined the relationship between time of separation and morphological divergence. However, objective estimates are lacking on how much actual genetic divergence is necessary to generate a given amount of morphological differentiation. In addition, it is unclear how variability in morphology relates to variability at other organizational levels. For example, many recent studies have quantified evolutionary divergence at the molecular level, but, when these mo-

MURA 1985).

Genetics 120 239-253 (September, 1988)

lecular results are compared to evolutionary changes at the morphological level, serious discordances have been revealed (e.g., KING and WILSON 1975; LEWIN 1985; LOWENSTEIN 1985; SIBLEY and AHLQUIST 1984; WILSON 1975). Such results suggest that, in spite of extensive reliance upon morphological variability to resolve complex evolutionary questions, we lack understanding of the genetic basis for evolutionary divergence in morphology.

Indeed, a significant number of fundamental questions about morphological divergence remain unan- swered. Several of these questions are examined in this paper including: (1) How much actual genetic divergence underlies a specific amount of morphological differentiation among taxa? (2) How much of the genetic information relating to variability in morphological traits is useful in deducing the phylogenetic history of the organisms? (3) What is the relationship between proximity of common ancestry for two taxa and the level of genetic divergence between them in complex morphological traits? (4) What is the relationship between genetic divergence at the single locus “molecular” level of organization and that observed for polygenically controlled morphological traits?

Answers to questions such as these have not been given in the evolutionary biology literature not because they are unimportant questions, but rather because they generally could not be obtained using conventional methods of analysis. However, FITCH and ATCHLEY (1985a) have shown that inbred strains of mice can be used to examine important questions about evolutionary theory and methodology at the molecular level. Moreover, inbred strains of mice can help resolve certain important questions at the mor-

240 W. R. Atchley, S. Newman and D. E. Cowley

phological level as well. Here and elsewhere (FITCH and ATCHLEY 1987), we utilize inbred strains of mice of known genealogy to examine several questions concerning the genetic basis for divergence in morphological form (=size and shape). In the present paper, we use the mouse mandible as a paradigm to examine the genetic aspects of divergence in morphological form. Herein, we (1) estimate the amount of polygenic divergence in mandible form that has occurred in 1 1 traits among 10 inbred strains, (2) examine several different estimators of univariate and multivariate genetic divergence for morphological traits, (3) determine whether estimates of genealogical relationships among strains of inbred mice parallel estimates of molecular and morphological divergence, (4) examine the concordance between several polygenic estimates of divergence as well as examine the concordance of divergence in morphology versus that found at the molecular level, and ( 5 ) examine the question of extent of genetic divergence as a function of time of separation.

MATERIALS AND METHODS

Rationale for using inbred strains: There are a number of attributes of inbred strains of mice which make them useful to resolve complex evolutionary problems. First, the genealogy of some strains is known with considerable accuracy because the times of separation and initial degrees of relatedness are known. As a result, the rate of genetic divergence among taxa can be estimated with reasonable accuracy. Second, divergence in these inbred mice has occurred primarily through random fixation of alleles at those loci which varied among the original founders. This fact permits specific genetic models, e.g., neutral evolution theory, to be used to test hypotheses about the rates of morphological evolution (LYNCH and HILL 1986). Third, there are extensive data available on many traits at varying levels of biological organization, e.g., molecular, morphological, physiological and reproductive. Fourth, inbred strains of varying degrees of relatedness can be crossed so that the genetic aspects of evolutionary divergence can be examined in detail. This permits the investigation of the genetic ar- chitecture underlying morphological change (e.g., number of loci involved, type of gene action involved, etc.).

Mouse strains examined: Ten inbred mouse strains are used in these analyses. Data on seven of the strains derive from the parental strains of a diallel analysis of morphological variability (S. NEWMAN and W. R. ATCHLEY, unpublished data). These seven strains and their sample sizes are as follows: A/J (21), BALB/cByJ (36), C57BL/6ByJ (38), C3H/HeJ (46), SEA/GnJ (51), SEC/lReJ (41) and SWR/J (40). For all strains, only Jackson laboratory stocks are used throughout which accounts for the ‘1” designation on the strain names. For these strains, extensive data are available for various skeletal traits (S. NEWMAN and W. R. ATCHLEY, unpublished data), life history and live body measurements, and growth curve statistics (our unpublished data).

In addition, limited experimental data are available for three additional inbred strains, ie., CBA/J (18), DBA/2J (17) and C58/J (7). These latter three strains were not part of the original diallel analysis and the sample sizes are smaller. However, inclusion of these data permits us to expand some of the analyses.

FOUNDER STOCK

GENERATIONS OF

STRAIN INBREEDING

PASTEUR SWR (l 25) FIGURE 1 .-Phylogenetic relationships among the ten strains of

mice examined in this study. The thin lines show periods of brother X sister mating, the box-like lines show periods where random breeding is known (boxes have smooth ends) or is assumed (boxes have jagged ends).

Genealogical relationships: Genealogical relationships among these ten strains are given by STAATS (1 980, 198 1) and FESTINC (1 979) and are diagrammed in Figure 1. Where possible, the coefficient of kinship, f (CROW and KIMURA 1970), is used to quantify the relatedness of the mice that produced these inbred strains. The coeffkient of kinship (= coefficient of consanguinity) is the probability that two homologous genes drawn at random, one from each of two individuals will be identical by descent. Since divergence among inbred strains is primarily by genetic drift, measures of relationship such as f are useful as short-term genetic distance measures (REYNOLDS, WEIR and COCKERHAM 1983). T o provide continuity with previous results by FITCH and ATCHLEY ( 1 985a, b), we use a measure of relatedness, c, to describe genealogical affinities which is defined as (see Table 1 for complete list of symbols);

c.. = 1 -f. 9. (1)

This c is an expectation of relatedness given by the pedigree of a particular breeding design. As an expectation, the values of c have no variance; however, a given group of mice may deviate from the expected value of c due to random fixation of alleles with inbreeding. When c = 1, two strains are considered as unrelated.

C57BL/6 and C58 were derived in 1921 from a cross of a single male and two different females of the Lathrop stock of mice. Hence, C57BL and C58 were initially related as half-sibs and c = 0.875.

The A strain was derived from a cross of the Cold Spring Harbor and Bagg albino stocks. These latter stocks were probably random-bred or at least pen-bred prior to the cross which lead to the A strain. Subsequently, BALB/c was produced from inbreeding of the Bagg albino stock. While these strains are related, it is not possible from the historical data to compute an accurate estimate of c, however, it is probably close to unity.

Genetics of Morphological Divergence 24 1

TABLE 1

Summary of symbols

Symbol Definition

One minus the coeffkient of kinship for

Pairwise multivariate genetic distance be- strains i and j

tween strains i and j computed from principal components scores

Pairwise multivariate polygenic genetic distance between strains a and j based upon sex-corrected and log-transformed data. The pooled covariance matrix is used

Pairwise multivariate polygenic genetic distance between strains i and j based upon sex-corrected and log-transformed data. The within-group covariance matrix is used

Percentage difference between strains i and j for 97 molecular loci

Intraclass correlation coefficient for the ith trait

Pairwise difference between strains i and j divided by the within groups root mean square for strains i and j

Pairwise difference between strains i and j divided by the between groups variance component for strains i and j

Pairwise difference between strains i and j divided by the between-groups variance component for all ten strains considered simultaneously

Pairwise between-groups variance component between strains i and j divided by the total between-groups variance component for all ten strains considered simultaneously

DBA/2 was produced in 1909 by inbreeding a stock segregating for coat color (STAATS 1980; FESTING 1979; GREEN et al. 1985). Both C3H and CBA arose from a single cross in 1920 of the Bagg and the inbred DBA lines. Thus, C3H and CBA were initially related as full sibs where one parent mouse was from a strain which had been inbred for about 11 yr. For C3H and CBA, c = 0.375.

SEA was derived by crossing the inbred BALB/c and the P strain. The P strain had previously been derived from a cross involving the inbred BDP strain and another unknown stock (FESTING 1979) while the BDP strain was generated from the same strain used to produce the various DBA strains (STAATS 1980). SEC was produced from a cross involving BALB/c and the NB strain. A value of 0.5 for c is obtained among BALB/c, SEA and SEC since SEA and SEC have the completely inbred BALB/c genome in common.

For the SWR strain, the original mice were obtained from a dealer by the Pasteur Institute (Paris). Beginning in 1926, these mice were brother X sister mated to produce the SWR strain (LYNCH 1969).

E = 1 for any pairwise comparison among the C57, DBA and SWR strains. WAYNE and O’BRIEN (1986) have suggested that C57 and DBA were originally derived from the pen- or random-bred Lathrop colony. However, as in the case of A and BALB/c, c must be close to unity for C57 and DBA.

In many of the analyses reported later, we will contrast genetic divergence with genealogical divergence using pairs

of strains that are distantly related (cq = l.O), e.g., C57BL/ 6 and SWR, DBA and C57BL/6, SWR and DBA, and DBA and C58. These results are compared to strains that share some common ancestry, e.g., C57BL and C58, A and BALB/ c, CBA and C3H, SEC and BALB/c, SEA and BALB/c, and SEC and SEA.

Molecular data: A benchmark of genetic similarity among these inbred strains is provided by a set of 95 loci coding for protein and immunological phenotypes. These data, obtained from STAATS (1 980), RODERICK, STAATS and WOMACK (1 98 1) and FESTING (1 979), have been examined extensively by FITCH and ATCHLEY (1 985a, b) for ten strains of inbred mice, some of which are the same strains as analyzed here. These data will be referred to in this paper as “molecular” data for lack of a better descriptor. The degree of genetic distance between strains for these data (hereafter called “molecular” distance, m ) is given as the proportion of loci in any pair of strains where different alleles have become fixed (FITCH and ATCHLEY 1985a). With a molecular data set of this magnitude (1 0 taxa by 95 loci), missing data will occur, e.g., with less well-studied strains such as SEA and SEC. Therefore, the actual number of loci involved in each computation is given in parentheses in Table 2.

These molecular data were deliberately chosen by FITCH and ATCHLEY (1 985a) to represent loci which are polymorphic between inbred strains and which could be used to examine the phylogeny of these inbred lines. These data are also very useful for providing an alternative estimate of genetic similarity to compare with the polygenic data described later because the molecular data exhibit the following qualities: (1) they exhibit simple patterns of inheritance; (2) they are discrete traits and can usually be objectively scored; (3) there is little intercorrelation between molecular traits except that due to linkage; (4) they provide an extensive source of genetic information about genetic divergence from another level of biological organization which can be compared to genetic divergence in morphology; (5) these molecular data are not expected to be directly related to the loci controlling expression of the morphological changes in the mandible and therefore, they provide independent measures of genetic diversity among these inbred strains; and (6) these data have previously been shown to provide a very accurate estimate of the known genealogical relationships among these mouse strains (FITCH and ATCHLEY 1985a). As a result, these data are used to examine the concordance of patterns of genetic divergence at different levels of biological organization as well as to provide an accurate alternative measure of relatedness among these inbred strains.

In addition to those loci known to differ between strains, it is obvious that many additional loci exist in the mouse genome that exhibit no genetic differences between inbred strains. No attempt was made to include these invariant loci here since their only effect would be to lower proportion- ately the overall estimate of molecular divergence. How- ever, genetic distance should remain proportional between strains and the same conclusions reached here and elsewhere (e.g., FITCH and ATCHLEY 1985a) will result.

Husbandry: Ten males and ten females of each strain were obtained from The Jackson Laboratory and permitted to acclimatize to our laboratory for approximately 2 weeks prior to mating. Single pair matings between 6-week-old mice were carried out within each strain. Litters were standardized to 8 pups at birth and the pups were weaned at 21 days of age. The mated pairs were permitted to produce up to two litters. The male was removed approximately 7 days prior to parturition and returned to the female directly after


weaning. All progeny were sacrificed at 70 days of age. The carcasses were skinned, eviscerated and skeletonized by dermestid beetles prior to data recording.

Morphological traits analyzed: The mandible was chosen as a paradigm for the analyses of morphological divergence for several reasons. First, growth and morphogenesis of the mandible is well-understood (HALL 1978, 1982a, b; MOORE 1981; MOORE and LAVELLE 1974). Second, it is a complex morphogenetic structure whose various component parts have become highly integrated during development and evolution (HALL 1978; MOORE 1981). Third, the mandible has an important functional role throughout the organism’s ontogeny. Fourth, it has been widely studied from an evolutionary perspective at many different taxonomic levels (MOORE 1981; MOORE and LAVELLE 1974; BERRY, JACKOBSON and PETERS 1978; DAVIS 1983; FESTINC 1973). Fifth, the rodent mandible has been the subject of considerable genetic analysis (ATCHLEY 1983; ATCHLEY, RUTLEDGE and COWLEY 1981, 1982; ATCHLEY, PLUMMER and RISKA 1985a, b; BAILEY 1956, 1985, 1986; FESTINC 1973; HALL 1982a, b; LEAMY 1974; LOVELL and JOHNSON 1983; NONAKA and NAKATA 1984; and others); however, only the work of WAYNE and O’BRIEN (1 986) and FITCH and ATCHLEY (1987) relate specifically to the question of the extent of genetic divergence between taxa for morphological traits.

The morphological traits analyzed herein include 11 measurements from the mandible of 70-day-old mice. The dentary bones were separated at the mandibular symphysis and the right half placed on a glass microscope slide in a photographic enlarger. The image of the mandible was projected onto a digitizer connected to a microcomputer and 19 landmark points were recorded in x-y coordinate space. These landmark points are illustrated and described in detail by ATCHLEY, PLUMMER and RISKA (1985a). Mor- phological data recorded with a digitizer in this manner have very low measurement error and high repeatability. From these 19 points, 1 1 traits were chosen to represent the functional and morphogenetic aspects of the rodent mandible. These traits include: (1) posterior mandible length, (2) anterior mandible length, (3) height at the mandibular notch, (4) height at the incisor region, (5) height of the ascending ramus, (6) condyloid width, (7) condyloid height, (8) coronoid height, (9) coronoid area, (1 0) angular process length, and (1 1) tooth bearing area.

A description of morphogenesis of the mandible, the functional aspects of the various regions of the mandible, together with the heritabilities and genetic correlations of these traits from randombred ICR mice are described in detail by ATCHLEY, PLUMMER and RISKA (1985a, b). S. NEWMAN and W. R. ATCHLEY (unpublished data) report the genetic components of mandible form from a diallel analysis of seven of these ten strains.

Statistical analyses: With completely inbred organisms, morphological variability within strains is environmental in origin while the variability between strains is of genetic origin. The assumption that within strain variability is environmental in origin rests on the assumption that these strains are completely homozygous as a result of many generations of brother X sister mating. This assumption has been validated for polygenic traits by DEOL et al. (1960).

For the ith continuously varying trait, y,, within a large sample of inbred mice, the value of y in the jth mouse is

yli = pi + eli (2)

where pi is the parametric mean of the trait and e is an environmental component of variability. Within the popu-

lation, e is assumed to be normally distributed with zero mean and variance a:. If environmental effects are symmet- rical about a mean of zero, the mean of a large sample of inbred mice is a good estimator of the genotypic value of the polygenic trait for that strain.

The morphological data are adjusted for sex and parity and the residual values analyzed by the linear model

yjh = p + s, + ejh (3)

where yjk is an observation on the kth individual from the j th strain, I.C is the overall mean, sj is the random effect of the j th strain, and ejk is the residual error. si and t j h are both assumed to be normally distributed with a mean of zero and variance uf or u:,respectively.

Based on expression (3), the expected mean squares for the between strains variation (EM&) for 2 or more strains is

EMSb = u: + n,uf (4)

where u; is the within-groups variance (due to environmen- t a l causes), uf is the variance component between strains (variability due to genetic effects), and no is the average sample size. The intraclass correlation, t , is

t = s f / ( s f + s:) (5)

where sf is the sample estimate of uf. This is a useful statistic because it gives the proportion of variance between groups relative to the total variance (SOKAL and ROHLF 1981). Thus, in an analysis of variance of completely homozygous stocks, t for any morphological trait estimates the proportion of the variance in phenotypic divergence between groups which is of genetic origin. Both additive and nonadditive genetic effects may be included in the between group variance component.

Univariate polygenic distance measures: In order to analyze genetic divergence in continuously varying polygenic traits, an accurate measure of genetic distance must be obtained. In these results, the amount of genetic divergence in morphology between strains is estimated using several different univariate and multivariate distance estimates. These distance measures have different biological and statistical assumptions (GOODMAN 1972).

In the univariate case, genetic divergence is usually defined as the difference between strain means divided by a particular root mean square value in order to express genetic divergence in a standard metric. Several different procedures are suggested including:

1. u, = the difference in means between strains i and j divided by their within-groups root mean square. Recall that the within-strain component of variance reflects environmental variability so that u, expresses the between-strains differences on a common scale reflecting environmental variability.

2. u b = the difference between strains i and j divided by the square root of the between-groups variance component si for i and j. The between-groups variance component estimates the genetic component of variance so that u b

reports among-strain divergence on a common scale reflecting genetic variability for the two strains.

3. ut = the difference between strains i and j divided by the square root of the between groups variance component from an analysis of all ten strains considered simultaneously. This provides an estimate of the level of genetic divergence expressed on a scale of the total genetic divergence among all strains for the trait in question.

4. U%sb = the between-groups variance component, s f , from an analysis of variance of strains i and j , divided by the

Genetics of Morphological Divergence 243

square root of the between-groups variance component from an analysis of variance of all ten strains considered simultaneously. This statistic estimates the genetic variability between a pair of strains on the scale of the total genetic variance expressed among all strains in the analysis.

The values of u, for each individual trait and the average over all 1 1 traits are given to summarize univariate genetic divergence. Ub, ut and U%sb were calculated for each trait although these results are not presented here. These values were computed to assess the efficacy of uu to accurately describe univariate genetic divergence and reconstruct the genealogy of the inbred strains.

Multivariate polygenic genetic distances: A major problem with averaging univariate distances to estimate genetic distance is that it ignores the intercorrelations among traits. Ignoring intercorrelations among traits will cause the average univariate distance estimate to be biased (ATCHLEY 1980). The MAHALANOBIS (1936) generalized distance, d Z , which was developed to resolve the problem of intercorrelated traits, has the form

d? = (G - Zj)%0V" (G - Zj) (6) where zi is the centroid of the ith group and COV is the covariance matrix of the traits. The square root of this distance gives a multivariate analog to some of the univariate measures of divergence described earlier with the exception that the multivariate distance corrects for the within-groups covariance among traits (CAMPBELL and ATCHLEY 1981). This multivariate approach to a polygenic distance is described in more detail by Ca~uss r et al. (1 985).

The Mahalanobis distance is a statistical distance as op posed to many so-called "genetic" distances (e.g., the Nei or Rogers distances) which are based upon average frequencies of electrophoretic phenotypes. As noted in (6), statistical distances have their properties determined by vectors of means and the sample variance-covariance matrix (ATCHLEY et al. 1982). The numerical value of a statistical distance is a function of the probability density ellipses of the various samples because the distances reflect multivariate "standard deviation" units.

As shown graphically in ATCHLEY et al. (1982, Figure l), differences in the shape and orientation of the covariance ellipses for different taxa could considerably affect the distance estimate. Such differences in shape and orientation of ellipses result from the covariance matrices of the taxa being statistically heterogeneous. The standard statistical procedure for testing the homogeneity of covariance matrices is the likelihood ratio test (KSHIRSACER 1972; JOHNSON and WICHERN 1982; PERLMAN 1980). The most obvious result of heterogeneity in covariance structures is that the statistical distances may be asymmetrical, i.e., the distance from i to j may be different to that from j to i (ATCHLEY et al. 1982). This asymmetry generates problems when one attempts to (1) interpret the origin and evolutionary signifi- cance of heterogeneous covariances matrices and (2) analyze and interpret the resultant asymmetrical distances.

Several multivariate distance statistics based upon the Mahalanobis distance are possible which reflect different interpretations of the origin of variability and covariability within and among-groups and how divergence in centroids might be corrected for this intertrait covariability. In the first instance, a typical Mahalanobis distance statistic was computed simultaneously for all nine strains of mice using the sex-corrected and log-transformed data. Since the within-group covariance matrix is the environmental covariance, the Mahalanobis distance (d,) in these analyses gives a multivariate genetic distance among strains analogous to uu. The likelihood ratio test is significant for these strains

(covariance matrices of the strains involved are statistically heterogeneous). As a result, the covariance matrices of each group are used to compute the Mahalanobis distance, d, as

d ; = (z, - z~)'COV~' (G - z,) + log, lC0Vjl (7)

COVj = the covariance matrix within group j . dg might be different numerically from dj depending upon whether the covariance matrix of group i or j is used. Because of this, we have reported the arithmetic average of the two defined as d = (dg + 4)/2. The extent to which heterogeneous covariance structures have affected other estimates of polygenic distance (e.g., WAYNE and O'BRIEN 1986) is unclear.

A second estimate of the multivariate genetic distance employs the pooled covariance over all groups to estimate the distance (=dp). Thus,

d j = ( ~ i - zJ'COV~' ( ~ i - z,) (8)

where COV, is the pooled covariance matrix. The latter method ignores any differences in environmental covariance structure between strains.

The biological rationale for using d, as opposed to dp is as follows: Since the covariance matrix within each group describes the environmental component between traits, using the pooled matrix assumes that the environmental impact is essentially the same for all strains. Under this assumption, heterogeneity of the covariance matrices for the various strains is simply the consequence of sampling variability from a common environmental covariance matrix for all taxa. Alternately, one might assume that the environ- ments which the mice experience during development have a different impact on the various genotypes, i . ~ . , there is a genotype (=strain) X environmental interaction. Thus, a likelihood ratio test evaluates the null hypothesis that the various within-group covariance matrices are homogeneous. Heterogeneous covariance matrices imply a different strain X environment interaction which is a genetic phenomenon that must be included in genetic distance estimates.

More rigorous discussions and solutions to the problem of heterogeneous covariances in multivariate analyses have been presented (.g., ANDERSON and BAHADUR 1962). How- ever, these discussions do not deal with the biological problems of resolving divergent sources of variability and how these divergent sources affect the genetically determined intercorrelations between traits. If the within-group covariances among traits are of environmental origin, they could have a negligible impact on the estimation of a multivariate genetic distance.

A potential criticism of using d, and dp to estimate multivariate genetic distance is that the genetic intercorrelations among traits has not been dealt with. Robust estimates of the sample mean in turn estimates the genetic mean of each trait with the genetic model described in Equations 2 and 3. Variation among individuals within inbred strains arises from environmental rather than genetic causes. Therefore, the genetic correlations among groups must be estimated from the correlations among strain means. The efficacy of the estimation procedure will be a function of the number of strains examined, the adequacy of the estimation of the mean within samples, and the proportionality of the genetic and phenotypic covariances.

Figure 2 gives a UPGMA phenogram of the genetic correlation structure among traits based on these nine inbred line means. The four clusters of traits reflect the patterns of genetic covariation among the traits as it relates to genetic differentiation among these taxa.

With this in mind, we have attempted to correct for genetic correlations among traits in estimating genetic dis-


Among-Groups Corre la t ion 0 0.25 0.50 0.75 1 I I I I

/rc = 0.73J POSTMANLEN RAMUSHIGH CONDYLWID TOOTHAREA ANTMANLEN NOTCHHIGH CONDYLLEN INCISHIGH CORONHIGH CORONSIZE ANGULARLEN

r

- -

FIGURE 2.-UPGMA phenogram of the genetic correlations among mandible traits. r, = cophenetic correlation coefficient.

tance distances among strains through use of principal components analysis. A principal components analysis trans- forms the original intercorrelated data into a new set of traits (principal component scores) which are statistically uncorrelated UOHNSON and WICHERN 1982). The genetic distance (do) was computed as the euclidean distance between pairs of taxa for the mean principal components score of each taxon. Only the scores associated with significant eigenvalues were used.

Matrix congruence: Statistical comparisons of the extent of association among the various distance measures are generally difficult to carry out because of the lack of inde- pendence among distance values involving common taxa. Dependence violates the assumptions of the null distributions of many statistical procedures, such as regression and analysis of variance.

For comparing distance matrices, the statistical problems associated with dependency can be overcome through the use of permutation tests (DIETZ 1983). Herein, the permutation test is used to determine the extent of association between the various genetic distance matrices or between the genetic distance matrices and a matrix of c values of genealogical relationship. For each matrix comparison, 5000 permutations were used to estimate the probability value (P) for Kendall’s 7 statistic.

Elucidation of the statistical association between small subsets from within given distance matrices is much more diffkult. For this reason, simple bivariate plots are used to show associations between morphological distance estimates and molecular distances (m) or genealogical relationship (c).

RESULTS

Molecular divergence: Table 2 gives the genetic divergence which has occurred at the single locus level among these ten inbred strains. These divergence values (m) reflect the proportion of the 95 loci in which different alleles have become fixed during the inbreeding process. T h e results of a UPGMA cluster analysis are given in Figure 3. These hierarchial relationships in Figure 3 are very similar to those found in Figure 1. FITCH and ATCHLEY (1 985a) found these molecular data gave a very close fit to the known genealogy of the strains.

An obvious question is whether m values for strains with c = 1.0 exhibit more divergence than closely related strains (if?., those with c < 1.0). T h e average value of m in Table 2 where cg = 1.0 is 0.40 while the

TABLE 2

Molecular divergence (m) among inbred mouse strains

Strain

Strain A/J BALB/c CBA C57BL C3H C58 DBA/Z SEA SEC SWR

A/J -

(90)

(89) (88)

(92) (91) (90)

BALB/c 0.24 -

CBA 0.45 0.47 -

C57BL 0.42 0.42 0.52 -

C3H 0.31 0.41 0.20 0.51 - (91) (90) (89) (92)

C58 0.48 0.48 0.44 0.27 0.46 - (83) (82) (81) (84) (83)

(91) (90) (89) (92) (91) (84)

(49) (47) (48) (49) (49) (45) (47)

(64) (63) (64) (65) (65) (59) (63) (45)

(78) (76) (75) (77) (76) (72) (76) (45) (58)

DBA/2 0.38 0.38 0.36 0.53 0.38 0.49 -

SEA 0.33 0.21 0.42 0.31 0.41 0.31 0.45 -

SEC 0.25 0.13 0.44 0.32 0.450.34 0.35 0.20 -

SWR 0.38 0.42 0.40 0.49 0.420.44 0.33 0.420.33 -

based upon 95 loci described in FITCH and ATCHLEY (1985a). Matrix elements represent percent difference between strains

Numbers in parentheses below the divergence value are the actual number of loci upon which the distance value is based.

Genetic Distance 0.5 0.4 0.3 0.2 0.1

I A m BALE

fr, = 0.831 - I E SEC SEA c57

CBA C58

C3H DBA SWR

I I

I FIGURE 3.-UPMGA phenogram of genetic divergence based

on m values (molecular distances, Table 2) for ten inbred mouse strains.

average value when c < 1.0 is 0.20. T h e relationship between m and c is shown in Figure 4. With the exception of the m values for BALB/c and SEC and for SEC with A, the plot suggests that the line asymp- totes at an m value of about 0.3. Such a curvilinear relationship is to be expected because c cannot exceed unity; however, taxa with a c value of 1 .O will continue to diverge in m to a point where pairs of taxa could theoretically differ at all loci.

Morphological divergence: A single classification analysis of variance on each trait for the seven strains involved in the diallel analysis (C57BL/6, A, SEA, BALB/c, SEA, C3H and SWR) indicates highly significant differences among strains for all individual mandible traits. Table 3 provides the intraclass cor-


0.3 ' 1 0.1 0.2 0.3 0.4 0.5 0.6

Molecular Distance (m)

FIGURE 4.-Plot of c (=1 - coefficient of kinship) vs. m (molecular distance) for strains of known genealogical relationship.

TABLE 3

Intraclass correlation coefficients, t , for mandible traits

Trait t

1. Posterior mandible length 0.72 2. Anterior mandible length 0.72 3. Height at mandibular notch 0.62 4. Height at incisor region 0.80 5. Height of ascending ramus 0.72 6. Condyloid width 0.44 7. Condyloid height 0.67 8. Coronoid height 0.67 9. Coronoid area 0.52

10. Angular process length 0.73 11. Tooth-bearing area 0.50

Standard errors for the intraclass correlation range from 0.02 to 0.03.

relation coefficient ( t ) for each trait from these analyses. The proportion of morphological divergence among strains that is genetic in origin ( t ) ranges from 0.44 for condyloid width to 0.80 for height of the mandible at the incisor. The average value over all 11 traits is 0.65.

Multivariate analysis of variance and a canonical variate analysis were carried out on these seven strains for 1 1 traits. The MANOVA indicates highly significant differences among centroids (P < 0.001). The posterior identification matrix from the canonical analysis indicates that the seven taxa were discrete, ie., all mice were correctly identified as to inbred strain by the canonical variates.

Table 4 provides three separate pairwise multivariate polygenic distance estimators (do, d , and dp) between nine strains (seven taxa from the diallel analysis plus CBA and DBA/2) when all 1 1 traits are considered simultaneously. C58 was excluded from the initial computations of d , and dp due to the very small sample size and resultant singular covariance matrix. The likelihood ratio test UOHNSON and WICHERN

1982) indicates that the covariance matrices for these nine strains are heterogeneous.

Table 5 summarizes some results on the effect of different estimation procedures for COV on the value of the multivariate polygenic distance. For four strains (A, BALB, SEA and SEC), d was estimated based on two-strain and nine-strain analyses and for both within-group us. pooled covariance matrices. In all five analyses, the covariance matrices between pairs of strains are significantly heterogeneous as indicated by the likelihood ratio test. Therefore, Table 5 gives the distance estimates from the two sample analyses for the within-group case for d,, dji, and the arithmetic average of the two defined as (dij + dji)/2.

Several things are obvious from Table 5. First, different statistical distance estimates result depending upon whether the pooled covariance matrix in the two-strain versus nine-strain analyses is used. Second, as characterized by the two-strain analyses, the values for d , and dji may be quite different. Thus, for the distance between BALB and SEC or SEA, dq and dji are reasonably similar. However, for A and SEC or SEA, d , and dji are quite different. Third, even within the two-strain analyses, the average distance values based upon the within-groups covariance are different from the distance from the pooled covariance solution. In the case of BALB, SEC and SEA, the differences are small. However, for the relationships of A with SEA and SEC, the differences are more substan- tial.

Relationships between genetic divergence estimates and genealogy: A fundamental question for evolutionary biologists is whether genetic divergence in morphology parallels genealogical divergence. In- clusion of inbred strains C58, CBA and DBA/2 with the seven strains from the diallel analysis makes it possible to inquire whether genetic divergence in morphological traits for several sets of taxa parallels genealogical divergence. These results are given in Ta- ble 6. First, comparisons are made that describe the level of divergence among independent genealogical lineages, ie., among taxa that share no known common strains (c = 1.0). These comparisons include (a) C57BL and SWR, (b) C57 and DBA, (c) SWR and DBA, and (d) C58 and DBA. These four comparisons should document the greatest levels of genetic divergence among these taxa at the single locus and polygenic levels and this is usually the case. Average intraclass correlation values ( t ) among these distantly related strains range from 0.37 between DBA and C58 to 0.59 for C57 and SWR. The average univariate polygenic divergence (h) among these distantly related strains ranges from 1.36 SD between DBA and C58 to 2.21 SD between C57 and SWR.

The three multivariate polygenic distances (do, dp and d,) are given in Table 6. Further, the probability

246 W. R. Atchley, S . Newman and D. E. Cowley

TABLE 4

Multivariate genetic distance (d) between strains for mandible traits

Strain

Strain AIJ BALB CBA C57BL C3H DBA SEA SEC SWR

AIJ - 6.83 5.57 7.04 6.27 5.55 6.04 7.21 7.27 BALB 9.48 - 6.22 7.84 6.13 6.66 4.99 2.92 8.60 CBA 10.97 12.08 - 4.87 4.98 3.38 4.80 6.10 5.06 C57BL 10.78 14.79 8.54 - 6.24 6.76 7.29 8.72 5.97 C3H 6.84 9.38 7.63 6.68 - 6.77 5.33 6.07 5.44 DBA 10.84 8.47 5.06 8.79 7.95 - 5.80 5.91 6.02 SEA 7.38 5.43 7.47 7.76 6.35 5.87 - 5.38 6.8 1 SEC 10.25 3.53 10.36 10.26 7.90 5.88 5.66 - 7.82 SWR 10.33 12.36 7.51 5.77 6.18 7.33 6.67 8.65 - A/J BALB 3.66 CBA C57BL 3.78 2.12 2.43 C3H 3.1 1 2.12 2.75 2.78 - DBA 2.86 2.78 1.72 3.07 3.59 SEA 3.09 1.81 2.52 2.76 2.08 2.54 SEC 3.66 1.29 2.52 3.22 2.59 2.44 1.94 SWR 3.72 3.24 2.74 2.95 2.69 3.01 2.24 2.83

- -

3.32 2.41 - -

- -

- -

In the first matrix, values above the diagonal use the pooled covariance matrix (dp); values below the diagonal use within-groups covariances (&). When the within group covariance solution is used, distance values are (dq + 4) /2 . The second matrix gives values of do.

TABLE 5

Effect of different covariance matrices upon genetic distance estimates

Within Pooled

Taxon dg dr d dns d m

BALB-SEC 3.29 3.78 3.53 3.32 2.92 BALB-SEA 5.65 5.21 5.43 5.01 4.99 SEC-SEA 5.29 6.03 5.66 5.24 5.38 SEC-A 11.07 9.43 10.25 6.84 7.21 SEA-A 9.19 5.57 7.38 4.75 6.04

Distances based on within-group covariance matrices involve only two strains analyzed at a time and d is the arithmetic average. Distances using the p o o l e d covariance matrix involved either two strains (dp(p)) or nine strains (dH9)).

value from the likelihood ratio test for homogeneity of covariance matrices is included. In two comparisons involving C58, the test of homogeneity of covariance matrices could not be performed because, as ex- plained above, the covariance matrix for C58 is singular. In these instances, only do and dp are given. The distance values for d, and dp in Table 6 may differ from those given in Table 4 because those in Table 6 are based on only two rather than ten groups. As a result, the differences between strain means in Table 6 are adjusted by a more restrictive within-groups covariance matrix.

For taxa sharing a common ancestor, several pairs of strains which exhibit varying degrees of relationship are available with these data (Figure 1). As a result five comparisons can be made: C57 and C58, SEC and BALB/c, SEA and BALB/c, SEC and SEA, and CBA and C3H. Generally speaking, these latter

five comparisons exhibit genetic distances which are smaller than the previous five comparisons (Table 6). For all the multivariate distances, the average values in instances where c = 1.0 is greater than where c < 1.0. For d,, the average distance is 6.97 where c = 1.0 while it is 5.56 when c < 1.0. For do, they are 3.17 and 1.74. In summary, the average values for strains of t , u,,,, m, do, d, and dp are greater when c = 1 .O as compared to when c < 1 .O.

Next, a permutation test was carried out to assess the congruence of the various genetic divergence estimates based upon morphology and a matrix C which contains the expected values of cg for all nine taxa. The Kendall’s tau statistic comparing C with matrices of pair-wise values of m, d,, dp and do were significant at P < 0.01 in each instance. These results suggest that all genetic distance measures are significantly correlated with genealogical divergence, ie., the more genealogical separation, the greater the genetic distance.

However, one might question the sensitivity of these permutation analyses involving all the data, since, for the entire nine taxa, the C matrix was composed of 32 values of unity and four values where c < 1.0. The very large number of comparisons where c = 1 .O may be greatly reducing the ability of the analysis to detect the actual level of association in those few instances where c < 1 .O.

To further explore this question, bivariate plots are given in Figure 5 of c and the various distance estimates for those taxa in Table 6 where genealogical relationships are known. This subset of four pairs of


TABLE 6

Pair-wise comparisons in morphological divergence between strains of known genealogical relationship

C57-SWR C=l

C57-DBA c=1

SWR-DBA C = l

DBA-C58 C = l

BALB/c-A Unknown

Trait t u, 1 4. t u, t u, t u, ~

1 0.54* 1.68 0.60* 1.74 0.85* 3.37 -0.11 0.0 1 0.79* 2.79 2 0.34* 1.1 1 0.71* 2.23 0.27* 0.90 0.79* 2.80 0.76* 2.50 3 0.72* 2.49 0.24* 0.85 0.48* 1.38 0.20 0.85 0.86* 3.48 4 0.80* 3.09 0.92* 4.83 0.35* 1.07 0.53* 1.56 0.38* 1.14 5 0.78* 2.88 0.04 0.01 0.84* 3.27 0.59* 1.77 0.25* 0.86

7 0.68* 2.23 0.03 0.39 0.64* 1.92 0.64* 1.95 0.88* 3.91

9 0.24* 0.90 0.31* 0.99 -0.03 0.14 0.21 0.86 0.71* 2.23 10 0.88* 4.18 0.44* 1.28 0.72* 2.30 0.21 0.85 0.lO-t 0.55 11 0.52* 1.62 0.11 0.57 0.30* 0.98 0.59* 1.77 0.18t 0.71

6 0.21* 0.82 0.06 0.46 0.01 0.33 0.47* 1.41 0.61* 1.81

8 0.82* 3.32 0.65* 1.97 0.46* 1.34 0.36t 1.16 0.33* 1.03

Mean 0.59 2.21 0.37 1.39 0.44 1.54 0.41 1.36 0.53 1.91

m 0.49 0.53 0.33 0.49 0.24 do 2.95 3.07 3.01 2.35 3.66 dw 5.70 8.79 6.42 - 9.48

5.30 6.72 P 0.001 0.033 0.475 - 0.001

n 38 40 38 17 40 17 17 7 36 21

SD 0.24 1.09 0.31 1.34 0.30 1.07 0.26 0.74 0.29 1.16

dP 5.23 7.80 6.42

C = 0.875 C = 0.5 C = 0.5 C = 0.5 C = 0.375 C57-C58 SEC-SEA BALB/c-SEA BALB/c-SEC CBA-C3H

t u, t u, t u, t u, t u,

1 2 3 4 5 6 7 8 9

10 11

Mean SD m do dw dP P n

0.64* 0.5 1 *

0.84* 0.52* 0.307 0.50* 0.17

-0.08 -0.01

-0.09

0.46*

0.34 0.3 1

0.27 1.56

2.98 -

- 38

1.95 1.50 0.09 3.33 1.53 1.01 1.48 0.76 0.12 0.40 1.37

1.23 0.93

7

~

0.47* 1.34 0.44* 1.28 0.54* 2.13 0.19* 0.72 0.05 0.34 0.61* 1.79 0.19* 0.72 0.70* 2.15 0.67* 2.03 0.20* 0.73 0.50* 1.42

0.42 1.33 0.22 0.64

0.20 1.94 5.66 5.24 0.001

51 41

0.7 1 * 0.04 0.73* 0.07t 0.69* 0.60* 0.71* 0.19* 0.00 0.77* 0.17*

0.42 0.32

0.21 1.16 5.43 5.01 0.00 1

36

2.23 0.37 2.34 0.44 2.1 1 1.75 2.24 0.73 0.20 2.61 0.68

1.43 0.94

41

0.30* 0.95 0.25* 0.86

-0.01 0.18 0.42* 1.21 0.52* 1.51

-0.02 0.1 1 0.75* 2.45 0.8 1 2.94 0.70* 2.19 0.47* 1.35 0.17* 0.69

0.40 1.31 0.29 0.9 1

0.13 1.29 3.53 3.32 0.054

36 51

-0.04 0.01 -0.04 0.05

0.lOt 0.55 -0.04 0.08

0.56* 1.62 0.04 0.39 0.187 0.73 0.54* 1.56

-0.04 0.07 0.04 0.42

-0.02 0.22

0.12 0.52 0.23 0.58

0.20 2.75 7.63 5.96 0.062

18 46

Trait codes are given in Table 3. t and * refers to significant differences between group means at P < 0.05 and P < 0.01, respectively. n = sample sizes for analyses in each strain. P = probability value associated with the likelihood ratio test for heterogeneity of covariance matrices.

taxa where c = 1.0 had an average value of m = 0.46 compared to an average m of 0.40 for all 32 comparisons in Table 2 where c = 1.0. There seems to be little evidence of a linear relationship between c and any of these estimators in this subset of nine comparisons. In the case of %, what appears to be a linear relationship with c is apparently the result of two extreme values (Figure 5a).

WAYNE and O’BRIEN (1986) examined the relationship between divergence in mandible form and both molecular distance and time of separation between

strains. They reported a product-moment correlation between the Mahalanobis distance for mandible traits and molecular data to be 0.24 and not different from zero. Further, they also found that morphological distance was not significantly correlated with time of separation.

Relationships among genetic distance estimates: Several important questions can be examined with these data. First, what is the relationship among the various estimates of genetic divergence for these pairs of inbred strains? Second, are the various genetic


0.8 1 a

0.6 1 m a

a

I

0.4 t 0.2 '

0.4 0.8 1.2 1.6 2 2.4 FIGURE B.-Plots of c with %, do, 4 W )

d, and dp. Table 1 describes notation and abbreviations. = ( 4

a m a

7 8 9

4 w )

estimates based upon morphological data providing equivalent estimates of genetic divergence? Third, are the divergence estimates based upon morphological and molecular data congruent?

Figure 6 provides bivariate plots of m with u,, do, d, and dp. There appears to be an association between m and do, d,, and dp, but not between m and t~,. The permutation procedure for Kendall's rank correlation for m us. do is 0.56 (P < 0.05) while the correlation between m and d, = 0.48 ( P = 0.09). The more general question, however, is whether the genetic distance data for molecular and morphological traits are correlated over all 36 pairwise comparisons (36 as opposed to 45 comparisons are used since C58 is not included because d, cannot be estimated with C58).

The permutation test (DIETZ 1983) is used to compare the matrix of m values (Table 2) and the matrices of do, d, and d, values (Table 4). The null hypothesis is the elements of the two matrices are independent, ie., divergence at the molecular level is random with respect to genetic divergence in mandible form. The alternative hypothesis is that they are correlated. Ken- dall's T statistic associated with this test has an estimated probability of P = 0.04 for m and d,, but P = 0.39 for m and dp and P = 0.38 for m with do.

Thus, for d, and m, the permutation test would reject the null hypothesis that divergence at the molecular level over these nine strains is random with respect to genetic divergence in mandible morphology. For m and both do and dp , one would conclude that divergence at the molecular level, as described by m, is random with respect to divergence at the morphological level as reported by do and dp. The

o l 0.6

0 a a

o.8 t o . 6 ~ 0.4 0.2 3 4 5 4 P ) 6 7 8

significant association between d, and m is in disagree- ment with the results of WAYNE and O'BRIEN (1 986) who suggested a nonsignificant correlation between genetic distance at the molecular level and morphological divergence in mandible form.

Hierarchial analysis: Figure 7 provides a graphical depiction from UPGMA cluster analyses of the genetic interrelationships for dp and dw (Table 4) while Figure 3 gives the clustering for m. Figure 8 shows the clustering for the do values. It is obvious that the three genetic distances give different estimates of the hierarchial relationships among these nine inbred strains. As noted here and elsewhere (FITCH and ATCHLEY 1985a), the m values provide an accurate estimate of the known genealogical relationships. The d, values give a tree which is accurate in some instances (the BALB, SEC and SEA cluster, depiction of CBA and DBA as related). However, in other instances, e.g., CBA and C3H, SWR and C57, etc., the hierarchial relationships are in error.

The tree for do (Figure 8) also accurately describes the genealogical relationships of most taxa with the clustering of BALB/c, SEC and SEA, together with the clustering of C3H and DBA. However, this cluster analysis still does not correctly assign C3H. To test the robustness of the cluster analysis of do, data for C58 were added and the analyses recomputed. As depicted in Figure 1, C58 should cluster with C57 which is the case (Figure 8).

A confusing aspect of these results is the lack of agreement between the results from the matrix permutation procedure and from the cluster analyses. These results suggest a lack of proportionality be-

m

Genetics of Morpbologicai Divergence

rn

249

Oe6 r 0.5

0.4 .

0.3 '

o.2} 0

0

0

0

0

0

0 0.1 I I t I

0.4 0.8 1.2 1.6 2 2.4

m 0.6

0.5

a.4

0.3

0.2

0.1

0

0

0 0 0

t t 1

4 5 6 7 8 9

0.6

0.5

0.4

0.3

l a

0 0.1 I

1 1.5 2 2.5 3

4 0 )

m 0.6

0.5

0.4

0.3

0.2

0.1

0

0

FIGURE 6.-Plots of m with G, d , d, and de Table 1 describes notation and abbreviations.

Genetic Distance 10 8 6 4 2

I - r I r

A BALE SEC SEA CEIA DEJA c57 SWR C3H

7.5 6.0 4.5 3.0 1.5 I I

I A

(r, a O . ~ O / C ~ L;G SWR c57 BALE SEC SEA

FIGURE 7.-UPGMA phenograms of dp and d, for the nine strains of inbred mice. Table 1 describes notation and abbreviations.

tween the matrices of m and both do and dp values. However, in spite of this lack of proportionality, many

0

0

a

0

0

0

of the hierarchical relationships among these taxa are preserved.

Effect of missing molecular data: Some inbred strains had considerable missing data for the molecular loci (e.g., SEA and SEC). Therefore, three data sets containing 65, 40 and 30 loci selected at random were generated and the values of m recomputed. While this is not intended to be a very robust analysis, the correlation among the m values among pairs of strains for these and the 95 locus data set is high. Clustering algorithms, such as the UPGMA algorithm (SNEATH and SOKAL 1973) give the same groups of taxa for all four data sets. Thus, missing molecular data does not seem to have a very pronounced effect on the results.

Rate of divergence: The rate of divergence (=rate of allele fixation) among inbred strains is a function of the level of inbreeding, the initial levels of heterozygosity in the founding mice, and the mutation rate. To determine the rate of divergence between pairs of strains at the molecular and morphological levels, m, d,, dp and d, were divided by the average number of


Genetic Distance

3.6 3.0 2.4 1.8 1.2 I I I

3.5 3.0 2.5 2.0 1.5

" (re = 0.811

BAL0 SEC SEA C3H c57 E C58 C0A DBA SWR

FIGURE 8.-UPGMA phenograms for do for nine inbred strains of mice (a) or ten inbred strains (b). Table 1 describes notation and abbreviations.

I - -

I - 1

TABLE 7

Rate of genetic divergence (=rate of gene fixation) between pairs of inbred mouse strains

Rate of divergence

Taxa Generations m dp d m do

SEA-BALB/c 121 0.0017 0.041 0.045 0.033 SEC-BALB/c 120 0.0011 0.028 0.029 0.022 SEC-SEA 120.5 0.0017 0.043 0.047 0.033 CBA-C3H 162.5 0.0012 0.037 0.047 0.039 C57BL/C58 150.5 0.0018 0.020 0.027

Number of generations refers to the average number of generations of inbreeding as given by STAATS (1981). Rate of divergence is the genetic distance value divided by the average number of generations.

generations the two strains had been separated since founding of the strains. The latter is defined here as the number of generations of inbreeding (given in Figure 1). Rates are given in Table 7 for various combinations of BALB/c, CBA, C3H, C57BL/6, SEA, SEC and SWR.

The best comparisons involve strains BALB/c, SEA and SEC since it is suggested that they may be approximately equally interrelated ( c = 0.5) and have almost identical times of separation. As seen in Table 7, the rate of divergence (=rate of allele fixation) per generation at the molecular level is 0.001 7 allele changes per generation for SEA-BALB/c and SEC- SEA, while it is 0.001 1 for SEC-BALB/c. In the polygenic morphological data, the rate of divergence is about 0.028-0.043 multivariate SD units per gen-

eration using dp, 0.029-0.047 with d,, and 0.022- 0.039 with do. The rate of divergence in individual mandible traits varies widely as might be expected because of differing levels of heterozygosity for these traits in the initial populations.

For C58 and C57BL/6, the founding mice were less related (c = 0.875). However, the rates of divergence per generation are similar to those given above, i e . , 0.0018 for m, 0.020 for dp and 0.027 for do. With C3H and CBA, the original founders were related as full sibs and one parent was completely inbred. In this case, c = 0.375. The rate of divergence was 0.0012 per generation for m and from 0.037 to 0.047 for the various polygenic distance estimates.

DISCUSSION

In the analyses reported here, variability in the mandible among inbred strains of mice is used as a paradigm for examining some genetic aspects of variability in morphological form. In these results, the extent of genetic divergence at a large number of generally independent molecular loci as well as the differentiation in morphological form is described for taxa of known genealogical affinities. The design of these analyses permits comparison of the extent of correspondence in genetic divergence at the polygenic level with genetic divergence in an extensive set of identifiable loci. Further, we have compared several different measures of polygenic distance.

Molecular conclusions: The molecular data indicate a relationship between the degree of genetic divergence and degree of genealogical affinity. In view of these conclusions, we might inquire how closely the fit of c with m is in other studies. Unfor- tunately, very few analyses are available for other species. The relationship between indices of relationship and genetic similarity has been reported by Cox et al. (1985a) for 43 hard red winter wheat cultivars while Cox et al. (1 985b) described similar analyses for 115 soybean cultivars. In the case of the wheat cultivars, a rank correlation of 0.27 (P < 0.01) was found between the indices of relationship and gliadin polyacrylamide gel electrophoresis patterns. They suggested that a combination of selection and genetic drift was responsible for reducing the correlation between these two measures of relationship. In the soybean data, the authors compared genetic data for 20 loci with a coefficient of parentage and found a mod- est but significant correlation between the two sets of data. Both the soybean and wheat analyses had smaller levels of association between molecular and genealogical data than those described here for the inbred mouse data.

Polygenic conclusions: With regard to the polygenic data, these analyses also demonstrate that significant morphometric divergence has occurred even

Genetics of Morphological Divergence 25 1

among genealogically closely related strains of mice. These results indicate that much of the morphological divergence observed in the total sample of ten inbred strains arises from genetic causes. In many individual traits, the genetic component of morphological divergence can be very high, indeed, as much as 80% in some traits. When averaged over all 11 mandibular traits, the proportion of the total morphological variability of genetic origin was over 50%.

Does genetic divergence at the morphological level parallel phylogenetic (or genealogical in this case) divergence? Among pairs of taxa in which the geneal- ogies are known with considerable certainty, we have shown that genetic divergence in mandible morphology parallels genealogical divergence only in certain instances. None of the multivariate analyses provided evidence of a statistically significant relationship between morphology and genealogy as measured by c. Thus, these conclusions on a more diverse group of mouse strains do not support the findings of FESTING (1 973) who indicated that multivariate divergence in mandible morphology parallels the number of generations of separation in several sublines of the C57BL mouse.

Assessing the rate of polygenic divergence as a function of number of generations of inbreeding, comparisons among five strains (BALB/c, C57BL, C58, SEA and SEC) indicated that the rate of multivariate polygenic divergence was 0.020 to 0.047 genetic standard deviation units per generation. This value seems quite large. However, in their discussions about phenotypic evolution under neutral theory, LYNCH and HILL (1 986) have shown how this rapid change can be brought about. These authors indicate that while brother X sister mating within inbred strains minimized the within-line variance, the divergence among inbred strains is maximized and is independent of population size.

Congruence in patterns of genetic divergence across levels of organization: A topic of considerable interest to evolutionary biologists in recent years has been the extent of correspondence between estimates of genetic similarity obtained from different levels of biological organization, e.g., chromosomal us. electrophoretic, molecular us. morphological. Many of these analyses could be construed as sampling various components of the genome. Thus, an assumption that is important to test rigorously in phylogenetic analyses is whether different subsamples of the genome will produce comparable estimates of genetic similarity. Asked more specifically “what is the relationship between genetic divergence at the single locus molecular level and that at the polygenic morphological level?”

In several instances in the results being reported here, there is a reasonably close agreement in the patterns of genetic divergence at different levels of

organization. Further, when the molecular and multivariate distance matrices were compared by a permutation test in all nine strains without reference to genealogical affinity, there was a statistically significant association between the values of m and those for d,, (but not for do and dp). Thus, there is a correspondence in the patterns of genetic divergence between molecular and both univariate and multivariate morphological divergence.

However, it is obvious from these results that the level of statistical association is greater between genetic divergence at the molecular and multivariate morphological levels than it is between multivariate morphological divergence and genealogical affinity. There are several possible explanations for this. First, as noted above, the assumed expectations of c may not accurately reflect the actual degree of genealogical relationship. However, opposing this explanation is the high level of association between c and m.

Second, it may be that too small a sample of morphological traits is examined and the 11 mandible traits analyzed here are too small a sample to accurately describe genealogical relationships. Indeed, ROGERS and HARPENDINC (1 983) suggest that for data such as these, that one locus is approximately equal to one morphological trait in terms of genetic information. Therefore, the agreement between morphological and genealogical divergence should improve if more morphological data are added. WAYNE and O’BRIEN (1986) suggest that “extension of morphological measures to other skeletal characters which are independently regulated might improve the correlation between morphological and molecular varia- tions.” However, no significant improvement occurs in the relationship between morphological divergence and genealogy when additional morphological traits from other body regions are included (W. R. ATCH- LEY, unpublished data). In fact, the concordance often gets worse.

What is a proper polygenic distance measure? One striking feature about these results is the considerable heterogeneity among the various estimates of polygenic distance, depending on how the parameters are estimated. If estimates of polygenic distance are to be a significant component of discussions about multivariate evolutionary divergence, then this topic warrants some thought.

Considerable attention has been paid in the past to simple genetic distances based on electrophoretic and related data. However, the way one formulates genetic distances for simple, discrete molecular traits is quite different from the methods needed for complex, polygenic and continuously varying traits. The latter require use of statistical distances which are concep- tually and computationally more complex (ATCHLEY et al. 1982; REYMENT, BLACKITH and CAMPBELL


1984). Specifically, algorithms for computing statistical distance usually include not only a vector of differences in trait means but also a variance-covariance matrix for the various traits within each taxon. It is the latter feature that complicates the issue owing to problems with estimation.

As is evident from these results, a distance estimate for polygenic traits must account for the following complexities: (1) Morphological variability arises from both genetic and environmental causes. (2) The genetic component has both additive and nonadditive fractions. (3) Continuously varying morphological traits are often intercorrelated, the extent of which can be represented in the variance-covariance matrix. (4) Heterogeneity in the variance-covariance matrices among taxa may affect the distance estimates in complex ways. (5) Heterogeneity in genetic covariance may reflect important biological processes. That is, the loci underlying morphological variability may change considerably during ontogeny and, as a result, both the genetic covariance among traits and the genetic similarity among taxa may also change (ATCH- LEY 1984). Differences in heterogeneity of the covariance structure arise from differences in genetic variance, genetic covariance or both. (6) Differential response to environmental heterogeneity within a generation may produce changes in the genetic covariance structure as a result of a genotype X environment interaction.

Thus, the search for an accurate multivariate measure of polygenic divergence is complicated by the extent of heterogeneity in the covariance matrices among the various taxa, whether the heterogeneity stems from differences in variances (inflation) or covariances (orientation), and the type of covariance matrix actually used to produce the distance estimate. The solution to these and related problems is a complicated issue. Attempts such as ours, CAMUSSI et al. (1 985) and others address several of these problems; however, more detailed theoretical and experimental work is required before these problems can be re- solved.

We are indebted to DONALD W. BAILEY, JAMFS M. CHEVERUD, E. J. EISEN, WALTER M. FITCH, MAJOR M. GOODMAN, CATHY LAURIE, LARRY LEAMY, BRUCE S. WEIR and two anonymous review- ers for their critical comments on various drafts of this manuscript. We were afforded technical assistance from ALISON PLUMMER and BRUCE RISKA. This research was supported by National Science Foundation grants DEB-8109904, BSR-8507855 and BSR- 8605518 to WRA. Paper No. 10792 of the Journal Series of the North Carolina Agricultural Research Service, Raleigh, North Car- olina 27695-7601.

LITERATURE CITED

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Communicating editor: E. THOMPSON

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