Abstract
A quantitative genetic model of “random pleiotropy” is introduced as reference model for detecting the kind and degree of organization in quantitative genetic variation. In this model the genetic dispersion matrix takes the form of G = BB T, where B is a general, real, Gaussian random matrix. The eigenvalue density of the corresponding ensemble of random matrices (ℰG) is considered. The first two moments are derived for variance-covariance matrices G as well as for correlation matrices R, and an approximate expression of the density function is given. The eigenvalue distribution of all empirical correlation matrices deviates from that of a random pleiotropy model by a very large leading eigenvalue associated with a “size factor”. However the frequency-distribution of the remaining eigenvalues shows only minor deviations in mammalian skeletal data. A prevalence of intermediate eigenvalues in insect data may be caused by the inclusion of many functionally unrelated characters. Hence two kinds of deviations from random organization have been found: a “mammal like” and an “insect like” organization. It is concluded that functionally related characters are on the average more tightly correlated than by chance (= “mammal like” organization), while functionally unrelated characters appear to be less correlated than by random pleiotropy (“insect like” organization).
Similar content being viewed by others
References
Atchley, W. R., Rutledge, J. J., Cowley, D. E.: Genetic components of size and shape. II. Multivariate covariance patterns in the rat and mouse skull. Evolution 35, 1037–1055 (1981)
Bailey, D. W.: A comparison of genetic and environmental principal components of morphogenesis in mice. Growth 20, 63–74 (1956)
Berg, R. L.: The ecological significance of correlation pleiades. Evolution 14, 171–180 (1960)
Bulmer, M. G.: The mathematical theory of quantitative genetics. Oxford: Clarendon Press 1980
Bürger, R.: Constraints for the evolution of functionally coupled characters: A nonlinear analysis of a phenotypic model. Evolution (in press) (1984)
Cheverud, J. M.: Phenotypic, genetic, and environmental morphological integration in the cranium. Evolution 36, 499–516 (1982)
Cheverud, J. M., Rutledge, J. J., Atchley, W. R.: Quantitative genetics of development: Genetic correlations among agespecific trait values and the evolution of ontogeny. Evolution 37, 895–905 (1983)
Dammasch, I. E., Wagner G. P.: On the properties of randomly connected McCulloch-Pitts networks: Differences between input-constant and input-variant networks. Cybernetics and Systems 15, 91–117 (1984).
Gimelfarb, A.: Quantitative character dynamics: Gametic model. Theoret. Population Biology 22, 324–366 (1982)
Gould, S. J., Lewontin, R. C.: The spandrels of San Marco and the Panglossian paradigm: A critique of the adaptionist programe. Proc. R. Soc. London B204, 581–598 (1979)
Graf, U., Henning, H.-J., Stange, K.: Formeln und Tabellen der mathematischen Statistik. 2nd edition. Berlin-Heidelberg-New York: Springer 1966
Griffith, J. S.: Mathematical neurobiology. London and New York: Academic Press 1971
Hegmann, J. P., DeFries, J. C.: Are genetic correlations and environmental correlations correlated? Nature 226, 184–286 (1970)
Johnson, C.: Introduction to Natural Selection. Baltimore: Univ. Park Press 1976
Kendall, M. G., Stuart, A.: The advanced theory of statistics, vol. I–III. London: Charles Griffin & Co. 1958–1966
Kimura, M.: A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Natl. Acad. Sci. USA 54, 731–736 (1965)
Lande, R.: Quantitative genetic analysis of multivariate evolution, applied to brain: bodysize allometry. Evolution 33, 402–416 (1979)
Lande, R.: The genetic covariance between characters maintained by pleiotropic mutations. Genetics 94, 203–215 (1980)
Leamy, L.: Genetic and environmental correlations of morphometric traits in randombred house mice. Evolution 31, 357–369 (1977).
Leamy, L., Atchley, W.: Static and evolutionary allometry of osteometric traits in selected lines of rats. Evolution 38, 47–54 (1984)
Mayr, E.: How to carry out the adaptionist program? Amer. Nat. 121, 324–334 (1983)
Mehta, M. L.: Random Matrices and the statistical theory of energy levels. New York and London: Academic Press 1967
Olson, E. C., Miller, R. L.: Morphological Integration. Chicago: Univ. of Chicago Press 1958
Riedl, R.: A systems-analytical approach to macro-evolutionary phenomena. Q. Rev. Biol. 52, 351–370 (1977)
Rohlf, F. J., Sokal, R. R.: Comparative morphometrics by factor analysis in two species of Diptera. Z. Morphol. Tiere 72, 36–45 (1972)
Ruttner, F., Tassencourt, L., Louveax, J.: Biometrical statistical analysis of the geographical variability of Apis mellifera L. Apidologie 9, 363–381 (1978)
Sneath, P. H. A., Sokal, R. R.: Numerical Taxonomy. San Francisco: Freeman & Co. 1973
Sokal, R. R.: Variation and covariation of characters of Alate Pemphigus populi-transversus in eastern North America. Evolution 16, 227–245 (1962)
Srivastava, M. S., Khatri, C. G.: An introduction to multi-variate statistics. New York: North Holland 1979
Waddington, C.: The strategy of the genes. London: Allen and Unwin 1956
Wagner, G. P.: Coevolution of functionally constrained characters: Prerequisites for adaptive versatil-ity. Bio Systems 17, 51–55 (1984a)
Wagner, G. P.: Adaptively optimal genetic variation of quantitative characters: Theorems of existence and of the significance of morphological integration. Theoret. Population Biology (in press) (1984b)
Wright, S.: Evolution and the Genetics of Populations. Vol. I. Chicago-London: Univ. of Chicago Press 1968
Wuketits, F. M.: Grundriss der Evolutionstheorie. Darmstadt: Wiss. Buchges. 1982
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wagner, G.P. On the eigenvalue distribution of genetic and phenotypic dispersion matrices: Evidence for a nonrandom organization of quantitative character variation. J. Math. Biology 21, 77–95 (1984). https://doi.org/10.1007/BF00275224
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00275224