Is the Genotype-Phenotype Map Modular?: A Statistical Approach Using Mouse Quantitative Trait Loci Data

Corresponding author: Günter P. Wagner, Department of Ecology and Evolutionary Biology, Yale University, POB 208106, New Haven, CT 06520-8106., gunter.wagner{at}yale.edu (E-mail)

Communicating editor: Z-B. ZENG

	ABSTRACT

TOP ABSTRACT THE UTILITY OF QTL... STATISTICS FOR MODULARITY RESULTS DISCUSSION LITERATURE CITED

Various theories about the evolution of complex characters make predictions about the statistical distribution of genetic effects on phenotypic characters, also called the genotype-phenotype map. With the advent of QTL technology, data about these distributions are becoming available. In this article, we propose simple tests for the prediction that functionally integrated characters have a modular genotype-phenotype map. The test is applied to QTL data on the mouse mandible. The results provide statistical support for the notion that the ascending ramus region of the mandible is modularized. A data set comprising the effects of QTL on a more extensive portion of the phenotype is required to determine if the alveolar region of the mandible is also modularized.

IT has been hypothesized that characters that collectively serve a common functional role should be genetically integrated and relatively independent from the rest of the body (OLSON and MILLER 1958 ; RIEDL 1978 ; CHEVERUD 1982 ; BONNER 1988 ; WAGNER and ALTENBERG 1996 ). The genetic organization of a complex of characters thus is expected to reflect functional relationships. This is also called the hypothesis of the "imitatory epigenotype" because it is predicted that the pattern of developmental constraints "imitates" the pattern of functional constraints (RIEDL 1978 ). Closely associated with the hypothesis of an imitatory epigenotype is the concept of a module (RAFF 1996 ; WAGNER and ALTENBERG 1996 ). A module is defined as a complex of phenotypic traits that is both tightly integrated by pleiotropic effects and relatively independent of the rest of the phenotype (WAGNER 1996 ). So defined, a module is recognized by the statistical distribution of mutation effects among phenotypic traits. The imitatory epigenotype hypothesis thus predicts that functional units of the phenotype should be genetically modular.

The study of CHEVERUD et al. 1997 addresses the issue of modularity with a study of the mouse mandible (Fig 1). A quantitative trait locus (QTL) study was performed to determine the effects of alleles at individual loci. For the QTL with more than two effects (21 in total), the authors determined that only a minority of 23% had effects on both the ascending ramus (traits 1–11 and 21) and the alveolar region (traits 10 and 12–20) of the mandible. Furthermore, most QTL with effects on more than two traits had effects significantly focused on either the ascending ramus (50%) or the alveolar region (27%). These results were interpreted as consistent with the theory that selection tends to produce modular structure in functionally distinct units. However, without analyzing these findings in a statistical framework for all the loci simultaneously, it is difficult to determine whether 27 and 50% of QTL biased to the ascending ramus or the alveolar region are significantly different than expected under a random model.

View larger version (33K): In this window In a new window Download PPT slide   Figure 1. Diagram of the mouse mandible. (A) Division demarcates the ascending ramus and alveolar region. (B) Traits used in the QTL study on the mouse mandible of CHEVERUD et al. 1997 : (1) coronoid height; (2) superior condylar length; (3) condylar width; (4) inferior condylar length; (5) condylar base length; (6) posterior angular height; (7) posterior angular length; (8) anterior angular length; (9) superior angular length; (10) posterior corpus length; (11) coronoid base length; (12) posterior-inferior basal length; (13) anterior-inferior basal length; (14) inferior incisor alveolus length; (15) incisor alveolus width; (16) superior incisor alveolus length; (17) anterior corpus height; (18) molar alveolus height; (19) superior molar alveolus length; (20) inferior molar alveolus length; (21) superior coronoid length.

In this study we propose an additional method for assessing the modularity of a portion of the phenotype using QTL data. We suggest two statistics that measure the two aspects of modularity: integration by pleiotropic effects and parcelation, i.e., fewer than expected pleiotropic couplings with the rest of the phenotype (WAGNER and ALTENBERG 1996 ). These statistics are used to test whether units of the phenotype are more modular than expected by chance. In this article, we apply the test to the same QTL reported in CHEVERUD et al. 1997 and find that the ascending ramus is more modular than expected by chance.

	THE UTILITY OF QTL DATA FOR EXPLORING THE GENOTYPE-PHENOTYPE MAP

TOP ABSTRACT THE UTILITY OF QTL... STATISTICS FOR MODULARITY RESULTS DISCUSSION LITERATURE CITED

Hypotheses about the structure of the genotype-phenotype map are statements about the statistical distribution of mutation effects among phenotypic traits. Ideally, data to test these hypotheses should directly sample the distribution of the effects of individual mutations. Data on the distribution of individual mutation effects, however, is difficult to produce. QTL data provide a reasonable alternative for testing ideas about the structure of the genotype-phenotype map. One limitation when using QTL data, however, is that it reflects both the distribution of individual mutation effects as well as the fixation probabilities of the alleles. In this section, we discuss some of the problems of utilizing QTL data to analyze the structure of the genotype-phenotype map.

QTL are defined as genes influencing quantative traits (LIU 1998 ). Although identification of QTL has a long history (STURTEVANT 1913 ; SAX 1923 ; PENROSE 1935 ), only relatively recently, with advances in molecular marker technology, have "marker locus" QTL studies become possible at a larger scale (LANDER and BOT- STEIN 1989; KNOTT and HALEY 1992 ; ZENG 1993 ; ROUTMAN and CHEVERUD 1994 ). There exists some variation with respect to marker locus methodology, but a common strategy is to produce a F₂ or backcross population from two divergent homozygous lines and identify genetic markers associated with quantitative trait variation [see LYNCH and WALSH 1998 for a review of protocols and statistical techniques]. Markers identified in this manner are interpreted as being closely linked to loci with an effect on the given traits. Assuming a proper density of molecular markers along the chromosomes, the marker locus technique allows identification of a large number of QTL that may be located anywhere in the genome. The technique may be used to identify the set of genes causing heritable differences between lines and the individual pleiotropic effects of each gene (CHEVERUD et al. 1997 ).

The marker locus technique identifies loci where different alleles have become fixed in divergent lines. Whether a particular allele becomes fixed in a line depends on two probabilities: (1) the probability that a mutation with this effect will occur, and (2) the probability of fixation given that the mutation has occurred. The first probability is relevant for estimating the distribution of mutation effects. It is this statistical distribution that is predicted to be modular. The second probability is determined by the history of drift and selection effects in the study populations. Of the mutations that were segregating in the original base population and the mutations that occurred during selection, some are more likely to be fixed than others. For example, if a line used in a QTL study is produced by selecting for larger character values, mutations with large positive effects on the character are more likely to be fixed. The result is that marker locus QTL data are a "filtered" sample of the mutational distribution.

For the purposes of this study, we are interested in marker locus studies where multiple traits were measured and a large number of QTL were identified. In this study, we reanalyze the data set of CHEVERUD et al. 1997 that identified QTL with effects on the mouse mandible. The inbred lines used in this study were produced from lines selected for large and small body weight, respectively. Hence, the lines were selected for alleles with large overall effect on body size. As explained below, we therefore think that this data set is biased against the modularity hypothesis and we consider the test results reported here as conservative.

	STATISTICS FOR MODULARITY

TOP ABSTRACT THE UTILITY OF QTL... STATISTICS FOR MODULARITY RESULTS DISCUSSION LITERATURE CITED

Modularity of a set of traits can be defined by two attributes (WAGNER and ALTENBERG 1996 ): (1) a higher than average level of integration by pleiotropic effects of genes among the traits of a set, and (2) a higher than average level of independence from other trait sets. The first of these conditions reflects the degree of integration within a module and the second reflects the degree of parcelation of the phenotype into distinct units. We assess the modularity of a set of traits T₁ that is a proper subset of the total number of traits observed T₁ T. Ideally, the number of traits in the subset is much smaller than the total set of observables. We define two statistics for assessing the modularity of T₁, one to measure integration within the set and one to measure parcelation between trait sets. We call the total number of QTL that have effects on at least two traits n. Of these n QTL, we call m the number of QTL that affect at least one trait in T₁, m < n. Note that we only consider QTL with effects on at least two traits in T when calculating these statistics.

Integration statistic:
To measure integration of the set T₁ consisting of k traits, we consider only the m QTL with an effect on at least one trait in the set T₁. The integration statistic M_I is based on the total number of traits in T₁ affected by the m QTL compared to the maximum number of traits these QTL could affect in T₁, which is km:

The statistic is scaled to the unit interval [0, 1]. The extreme values M_I = 1 and M_I = 0 correspond to the maximum and minimum amounts of integration of the traits in T₁. For example, the most completely integrated case would be one where all m QTL with at least one effect on T₁ have effects on all k traits in T₁ such that the total number of effects is km. This corresponds to M_I = 1 (Fig 2A). At the other extreme, each QTL with at least one effect on T₁ affects only a single trait. In this case, each trait in T₁ changes independently as a result of an allele substitution. This corresponds to the lowest possible amount of integration such that M_I = 0 (Fig 2A).

View larger version (16K): In this window In a new window Download PPT slide   Figure 2. Examples for an intuitive understanding of the statistics MI and MP. T1 and T2 are two sets of traits (containing two traits each, k = 2). An arrow indicates an effect of a QTL on a trait tj. (A) An illustration of the integration statistic MI: the two QTL with at least one effect on traits in T1 have effects on both traits of T1. This is the maximally integrated case, hence M1(T1) = = 1. Each QTL with an effect on T2 has an effect on one in T2. The value of the integration statistic is therefore zero: MI(T2) = = 0. (B and C) Illustrations of the parcelation statistic MP: (B) For all the QTL in this sample, the expected number of effects is 1(Eji = pTiSj = 2 x 0.5 = 1.0). For QTL 1 and QTL 2 the observed values are 2 and 0 (Oj1 = 2 and Oj2 = 0, where j = 1 or 2). For QTL 3 and QTL 4, the observed values are also 2 and 0 (Oj1 = 0 and Oj2 = 2, where j = 3 or 4). Hence the value of MP = + + + + + + + = 8. This system is highly parceled and this is reflected in a large value of the MP statistic. (C) For all the QTL in this sample, the expected and observed values are 1.0. This example reflects the minimum amount of parcelation possible and hence MP = 0.

Parcelation statistic:
For measuring parcelation, we consider all n QTL in the sample with effects on more than one trait. Parcelation is defined as a relative lack of pleiotropic effects between two sets of traits. Parcelation is thus a relation between any two nonoverlapping sets of traits T₁ and T₂. These two sets can be a subset T₁ of all traits T and its complement, i.e., all the traits in T that are not in T₁. Pearson's ² is used to compare the observed number of traits affected by each QTL in T₁ and T₂ to the expected number based on the marginal distribution of effects among all QTL and traits. For each QTL j, the observed number of traits affected in T₁ and T₂ is symbolized by O_j1 and O_j2, and the expected number of traits affected is calculated as E_j1 = p₁S_j and E_j2 = p₂S_j. Where S_j is the total number of effects of QTL j, and p₁ and p₂ are the relative frequencies with which a trait in T₁ or T₂ is affected by any QTL in the total sample,

The parcelation statistic M_P therefore reads

Intuitively, this statistic measures the degree to which QTL preferentially affect traits in either T₁ or T₂. The null hypothesis is that the effects of a QTL are randomly distributed among the traits of the two trait sets. Rejecting the null hypothesis means that QTL effects tend to be clustered within each trait set. In genetic terms, this means that the QTL are "character specific," i.e., their effects tend to be limited to a set of traits. This was tested in CHEVERUD et al. 1997 for each individual QTL. The present statistic assesses this association between QTL and trait sets for the whole data set. The larger the value of this statistic, the less likely it is that a mutation will have an effect on both trait sets T₁ and T₂ (Fig 2B and Fig C).

Application and test distribution:
To illustrate how these statistics are utilized, consider a study where n QTL have been identified that have effects on at least two of N observed phenotypic traits. Let us assume that k of these traits describe a complex T₁ that performs some common function. The information about the functional significance of T₁ must be available from some independent source, such as a study of the functional morphology of the traits. We want to test whether T₁ is more modular than expected for a random set of traits. To do this, we first determine the values of the statistics for T₁ and then the cumulative mass function (cmf) of M_I and M_P with respect to all the possible subsets of k traits taken from the total N traits. If both the values of M_I and M_P are greater than the critical value (using = 0.05 for this study), the null hypothesis is rejected; i.e., the modularity of the set is greater than expected for a randomly chosen set of traits.

The "shape" of the cmf for the statistics M_I and M_P will depend on the distribution of pleiotropic effects among the QTL. Optimally, the value of M_I and M_P for all possible sets of k traits should be determined to calculate the cmfs exactly. This is the procedure adopted in our study. However, if the values of the E_ij are generally >5 and if no or only a few QTL have a number of effects >k or N - k, then the sampling distribution M_P will approximately follow a ² distribution with 2n d.f.

We do not in general expect the marginal distributions of the statistics M_I and M_P to be independent. If they are not independent, the probability of rejecting the null hypothesis for both statistics for a random set of k traits is somewhere between 0.0025 (for independent M_I and M_P) and 0.05 (for a correlation of 1.0). Regardless of the correlation structure, we still consider a functional unit to be significantly modular if we can reject the null hypothesis for both statistics. It is expected that the correlation of the statistics will not be useful for comparing the power of the test for different data sets as the distribution associated with an alternative hypothesis (how traits of the unit T₁ are assigned) will not necessarily be comparable between data sets.

It should be noted that for cases where we are assessing the integration and parcelation of more than one unit of a single data set, the units should not have many traits in common. If two units have many traits in common, we would expect similar results (rejection or nonrejection) for both units. We must therefore be cautious when assessing hierarchical and overlapping sets of traits for their degree of integration or parcelation. In this case, the two units of the mandible have only a single trait in common.

	RESULTS

TOP ABSTRACT THE UTILITY OF QTL... STATISTICS FOR MODULARITY RESULTS DISCUSSION LITERATURE CITED

For this study, we assessed the modularity of functional sets of the mandible of the mouse Mus musculus. A QTL study performed by CHEVERUD et al. 1997 discovered 41 QTL, of which 37 QTL affect at least 2 of the 21 traits of the mandible (Fig 1). A summary of the amount of pleiotropy in this data set (the frequency of QTL effecting z traits) is provided in Fig 3. The average number of traits affected by a QTL is 4.0, and the maximum is 13 traits. In this study, we assessed two functional sets of traits defined in CHEVERUD 2000 , the ascending ramus (traits 1–11 and 21; k = 12) and the alveolar region (traits 10 and 12–20; k = 10; Fig 2). All the major mandibular muscles attach to the ascending ramus. The shape of the ramus is critical for proper motion of the chewing and biting apparatus and therefore is considered to have been under strong selection pressure (ATCHLEY and HALL 1991 ). The alveolar region is the location of tooth attachment. The form of this region is critical for proper positioning of the teeth (ATCHLEY et al. 1992 ). The results discussed below are summarized in Table 1.

View larger version (37K): In this window In a new window Download PPT slide   Figure 3. The amount of pleiotropy in the QTL data of CHEVERUD et al. 1997 . x-axis, number of traits affected by an individual QTL; y-axis, number of QTL with effects on x traits. Note that the QTL with an effect on only a single trait were not considered in this study.

Ascending ramus:
The ascending ramus is described by k = 12 of the 21 traits in this study. These traits are affected by m = 32 QTL of the 37 that have effects on at least 2 traits. The maximum number of traits these QTL could affect is 12 x 32 = 384. The observed number of effects on ascending ramus traits is 98 and thus the integration statistic is M_I = 0.188. The cmf of the integration statistics for k = 12 yields a P value of 0.0102, which is significant at the 2% level. This value reflects the probability that this or greater levels of integration would have been observed if the traits were selected at random.

Of the total 160 effects of the QTL, 98 affected traits of the ascending ramus. The probability that a trait affected by a QTL is in this region (p₁) is therefore 0.6125. Given this probability and the individual effects of the QTL, the parcelation statistic is calculated to be 77.650. This corresponds to a P value of 0.0003. We conclude that the ascending ramus is both more integrated and more parceled than expected by chance. It thus fulfills the criteria for a modularized unit (of the part of the phenotype that is described by the set of observed traits).

Alveolar region:
Of the 21 traits observed, k = 10 are ascribed to the alveolar region. These traits are affected by m = 28 QTL with at least 2 effects. The maximum number of effects on the alveolar region is 280. The number of effects on the alveolar region traits is 71, which leads to an integration statistic of M_I = 0.171. This value is similar to that obtained for the ascending ramus. The cmf of the statistic, however, reveals that this value is not significantly different from a random sample of traits (P = 0.0968) at the 5% level.

Of the total 160 effects of the QTL, 71 affected the alveolus such that p₁, the a priori probability of an alveolar trait affected by a QTL, is 0.444. Given this probability and the individual effects of the QTL, the parcelation statistic is 68.688. This corresponds to a P value of <10^-4. The alveolar region is thus a part of the phenotype that is well parceled from the rest of the mandible but not more integrated than expected by chance. The alveolar region can thus be seen as a collection of quasi-independent traits rather than an integrated character complex.

	DISCUSSION

TOP ABSTRACT THE UTILITY OF QTL... STATISTICS FOR MODULARITY RESULTS DISCUSSION LITERATURE CITED

There are two aspects to the test proposed in this article. On the one hand, it is a test for the nonrandomness of the distribution of QTL effects on a subset of traits. As such, this test does not address which evolutionary forces may have caused the nonrandom pattern. If, however, the test is performed on a set of traits that a priori have been classified as describing a functional unit, it is in fact an indirect test of the hypothesis that natural selection has shaped the genotype-phenotype map to modularize functionally integrated sets of characteristics. A rejection of the null hypothesis is considered as evidence in support of the hypothesis of the imitatory epigenotype (RIEDL 1978 ). In drawing this conclusion, it is of critical importance that the classification of the traits into functional units is done on the basis of independent evidence (see SCHWENK 2000 for a review of the concept of a functional unit). This is the case in this study, where the ascending ramus and alveolar region of the mandible have been identified as functional units independent of the genetic structure (ATCHLEY and HALL 1991 ; ATCHLEY et al. 1992 ).

For the data set of CHEVERUD et al. 1997 , the ascending ramus represents a well-modularized portion of the phenotype. For the alveolar region, rejection of the parcelation null hypothesis indicates that mutation effects are significantly parceled with respect to this region. Although the integration null hypothesis could not be rejected for the alveolar region, the P value is only 0.0968 (Table 1). Given the significant test results for the ascending ramus, the results for the alveolar region (significant result for parcelation and nonsignificant result for integration) are somewhat expected. Since the ascending ramus and alveolar region together comprise all 21 traits of the mandible, the parcelation statistic is measuring the degree to which QTL affect either the ramus or the alveolar region. Hence, if we observe a significant result for one region we are likely to observe a significant result for the other. Similarly, since the ascending ramus is highly integrated, the alveolar region would need to have a level of integration close to the ramus to produce a significant result as well. That is, both units may be integrated but the effect will only be detected if the integration of the units is comparable. A better test would consider more traits such that together the ascending ramus and alveolar regions would contain far fewer than the total number of traits. If this is the case, different levels of integration may still produce a significant result in the integration test and we might expect different results for the parcelation test. To determine if both of these units reflect a modular pattern, we would therefore need a QTL data set comprising more traits. Our study does at least confirm that the ascending ramus is a genetically modular unit of the phenotype (CHEVERUD et al. 1997 ).

A limitation of our tests is that they only take into account whether a QTL has a significant effect on a trait or not. Ideally one would like to take into account the magnitude of the effects to assess the degree of integration and/or parcelation. The problem, though, with including the magnitude of the effects is that most of the effects are small and thus the confidence intervals are relatively large. This implies that there is little actual information added to the analysis if the measured effect is taken into account other than the fact that the effect is greater than zero.

As explained previously, every QTL data set represents a filtered sample of mutations that may occur. That is, a QTL data set not only reflects the probability of a mutation occurring but also the fixation probability of a mutation during the preparation of the lines used in the QTL study. Since different QTL study conditions may produce different probabilities of fixation, it is possible that the tests may produce a significant result for one set of lines but may not with another set of lines. Consider, for instance, a functional complex that is modular. A group of mutations that occur in this system are likely to reflect this modular structure. However, a QTL study protocol may end up fixing a filtered sample of these mutations that no longer have a modular distribution. Because QTL study conditions could also have the opposite effect, making a nonmodular unit appear modular, it is important to test data from other QTL studies using different inbred lines before drawing a final conclusion concerning the modularity of a structure. Since multiple QTL data sets do not presently exist for the mouse mandible, we can only interpret the present results as consistent with the hypothesis that the ascending ramus is a modular unit.

The genetic structure of the mouse mandible has been studied in the past with quantitative genetic methods (BAILEY 1956 ; ATCHLEY et al. 1985 , ATCHLEY et al. 1992 ). BAILEY 1956 presented a principle component analysis of the genetic correlations matrix and found a general size factor and one factor with contrasting loadings on alveolar region and ascending ramus measurements. This result is consistent with our finding that these two parts of the mandible are modular. Bailey's result, however, may in part be a measurement artifact since all the measurements in his study have one landmark in common, located at the junction between the ascending ramus and the corpus of the mandible. This can cause spurious correlations proportional to the angle among the distances measured. ATCHLEY et al. 1985 presented a cluster analysis of genetic correlations among mandible traits, which showed two clusters of ascending ramus and alveolar measurements in addition to some independently varying traits. A reanalysis of the same data in ATCHLEY et al. 1992 did not reveal any clear pattern of modularity (see Figure 9 in ATCHLEY et al. 1992 ).

The comparison of genetic correlations and QTL data on the genetic architecture of characters is complicated by two factors. At the one hand, the genetic correlations reflect not only the pattern of mutational effects but also the frequencies of segregating alleles in the population in which the genetic correlations were estimated. Genetic correlations are as much a property of the population composition as they are a property of the genetic architecture of the traits. Furthermore, a low genetic correlation can indicate one of two things: it could be caused by a relative lack of pleiotropic effects among the traits or a pattern of positive and negative pleiotropic effects that cancel each other. There is no way to distinguish between these two scenarios by analyzing genetic correlations. Our results and that reported in CHEVERUD et al. 1997 show that the independent variation between the two major regions of the mandible is due to a relative lack of pleiotropic effects rather than due to antagonistic pleiotropic effects.

It should be noted that although the results from the QTL analysis are consistent with the hypothesis that natural selection may result in modular structures, results from this test alone are not sufficient to actually infer the action of natural selection in creating a modular pattern. To more directly investigate the possible role of natural selection in causing the association between functional and genetic integration, it is necessary to analyze data on the evolution of the genotype-phenotype map and the evolution of character function. However, with analyses of the sort described in this study, we can now begin to examine the architecture of genotype-phenotype maps and provide a foundation for such studies.

	FOOTNOTES

¹ Present address: Department of Biological Science, Florida State University, Tallahassee, FL 32306.

	ACKNOWLEDGMENTS

The authors thank Junhyong Kim and David Houle for critically reading the manuscript. We further thank Homayoun Bagheri, Chi-Hua Chiu, Ashley Carter, Christian Pazmandi, Scott Rifkin, Max Shpak, and Terri Williams for stimulating discussions on the topic of this article. The support by National Science Foundation grants IBN-9905403, DEB-9419992, and DEB-9726433 is gratefully acknowledged.

Manuscript received November 5, 1999; Accepted for publication May 24, 2000.

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