Abstract
Methods of functional analysis are applied to provide an exact mathematical analysis of Kimura's continuum-of-alleles model. By an approximate analysis, Kimura obtained the result that the equilibrium distribution of allelic effects determining a quantitative character is Gaussian if fitness decreases quadratically from the optimum and if production of new mutants follows a Gaussian density. Lande extended this model considerably and proposed that high levels of genetic variation can be maintained by mutation even when there is strong stabilizing selection. This hypothesis has been questioned recently by Turelli, who published analyses and computer simulations of some multiallele models, approximating the continuum-of-alleles model, and reviewed relevant data. He found that the Kimura and Lande predictions overestimate the amount of equilibrium variance considerably if selection is not extremely weak or mutation rate not extremely high. The present analysis provides the first proof that in Kimura's model an equilibrium in fact exists and, moreover, that it is globally stable. Finally, using methods from quantum mechanics, estimates of the exact equilibrium variance are derived which are in best accordance with Turelli's results. This shows that continuum-of-alleles models may be excellent approximations to multiallele models, if analysed appropriately.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Feichtinger, H. G.: Compactness in translation invariant Banach spaces of distributions and compact multipliers. J. Math. Anal. Appl. 102, 289–327 (1984)
Fleming, W. H.: Equilibrium distributions of continuous polygenic traits. SIAM J. Appl. Math. 36, 148–168 (1979)
Greiner, G.: A typical Perron-Frobenius theorem with applications to an age-dependent population equation. In: Infinite-dimensional systems. Kappel, F., Schappacher, W. (eds.) Lect. Notes Math. 1076. Berlin Heidelberg New York: Springer 1984
Henry, D.: Geometric theory of semilinear parabolic equations. Lect. Notes Math. 840, Berlin Heidelberg New York: Springer 1981
Kimura, M.: A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Natl. Acad. Sci. USA 54, 731–736 (1965)
Kimura, M.: The neutral theory of molecular evolution. New York: Cambridge Univ. Press 1983
Kingman, J. F. C.: On the properties of bilinear models for the balance between genetic mutation and selection. Math. Proc. Camb. Phil. Soc. 81, 443–453 (1977)
Kingman, J. F. C.: A simple model for the balance between selection and mutation. J. Appl. Prob. 15, 1–12 (1979)
Lande, R.: The maintenance of genetic variability by mutation in a polygenic character with linked loci. Genet. Res., Camb. 26, 221–235 (1976)
Latter, B. D. H.: Selection in finite populations with multiple alleles. II. Centripetal selection, mutation and isoallelic variation. Genetics 66, 165–186 (1970)
Lynch, M.: The selective value of alleles underlying polygenic traits. Genetics 108, 1021–1033 (1984)
Magnus, W., Oberhettinger, F., Soni, R. P.: Formulas and theorems for the special functions of mathematical physics, 3rd edn. Berlin Heidelberg New York: Springer 1966
Moran, P. A. P.: Global stability of genetic systems governed by mutation and selection. Math. Proc. Camb. Phil. Soc. 80, 331–336 (1977). II. ibid. Math. Proc. Camb. Phil. Soc. 81, 435–441 (1976)
Nagylaki, T.: Selection on a quantitative character. In Chakravarti, A. (ed.) Human population genetics: the Pittsburgh symposium. New York: Van Nostrand 1984
Ohta, T., Kimura, M.: Theoretical analysis of electro-phoretically detectable alleles: models of very slightly deleterious mutations. Am. Naturalist 109, 137–145 (1975)
Reed, M., Simon, B.: Methods of modern mathematical physics. I: Functional analysis (revised and enlarged edn.) New York San Francisco London: Academic Press, 1980
Reed, M., Simon, B.: Methods of modern mathematical physics. II: Fourier analysis, self adjointness. New York San Francisco London: Academic Press, 1975
Reed, M., Simon, B.: Methods of modern mathematical physics. IV: Analysis of operators. New York San Francisco London: Acadamic Press, 1978.
Rothe, F.: Global solutions of reaction-diffusion systems. Lect. Notes Math. 1072. Berlin Heidelberg New York: Springer 1984
Smoller, J.: Shock waves and reaction-diffusion equations: Berlin Heidelberg New York: Springer 1983
Turelli, M.: Heritable genetic variation via mutation-selection balance: Lerch's Zera meets the abdominal bristle. Theor. Pop. Biol. 25, 138–193 (1984)
Turelli, M.: Effects of pleiotropy on predictions concerning mutation-selection balance for polygenic traits. Genetics 111, 165–195 (1985)
Webb, G. F.: A semigroup proof of the Sharpe-Lotka theorem. In: Kappel, F., Schappacher, W. (ed.) Infinite-dimensional systems. Lect. Notes Math. 1076, Berlin Heidelberg New York: Springer 1984
Webb, G. F.: Theory of nonlinear age-dependent population dynamics. New York: Marcel Dekker 1985
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bürger, R. On the maintenance of genetic variation: global analysis of Kimura's continuum-of-alleles model. J. Math. Biol. 24, 341–351 (1986). https://doi.org/10.1007/BF00275642
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00275642