Abstract
Stochastic models of population genetics are studied with special reference to the biological interest. Mathematical methods are described for treating some simple models and their modifications aimed at the problems of the molecular evolution. Unified theory for treating different quantities is extensively developed and applied to some typical problems of current interest in genetics. Mathematical methods for treating geographically structured populations are given. Approximation formulae and their accuracy are discussed. Some criteria are given for a structured population to behave almost like a panmictic population of the same total size. Some quantities are shown to be independent of the geographical structure and their dynamics are described.
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Literature
Arnold, L. 1973.Stochastic Differential Equations: Theory and Applications. New York: John Wiley & Sons.
Crow, J. F. and M. Kimura. 1970.An Introduction to Population Genetics Theory. New York: Harper and Row.
Dynkin, E. B. 1965.Markov Processes, Vols 1 and 2. Berlin: Springer-Verlag.
Ethier, S. N. and T. Nagylaki. 1980. “Diffusion Approximations of Markov Chains with Two Time Scales and Applications to Population Genetics.”Adv. appl. Prob. 12, 14–49.
Ewens, W. J. 1963. “The Diffusion Equation and a Pseudo-distribution in Genetics.Jl R. statist. Soc. (B) 25, 405–412.
— 1979.Mathematical Population Genetics. Berlin: Springer-Verlag.
Feller, W. 1951. “Diffusion Processes in Genetics.”Proc. 2nd Berkeley Symp. on Math. Stat. and Prob., pp. 227–246.
— 1954. “Diffusion Processes in One Dimension.”Trans Am. math. Soc. 77, 1–31.
Fisher, R. A. 1922. “On the Dominance Ratio”.Proc. R. Soc. Edinburgh 42, 321–341.
Itô, K. and H. P. McKean. 1965.Diffusion Processes and Their Sample Paths. Berlin: Springer-Verlag.
Itoh, Y. 1979. “Random Collision Process of Oriented Graph.” Institute of Statistical Mathematics (Japan), Research Memorandum No. 154, pp. 1–20.
Kimura, M. 1954. “Process Leading to Quasi-fixation of Genes in Natural Populations due to Random Fluctuation of Selection Intensities.Genetics 39, 280–295.
— 1955a. “Solution of a Process of Random Genetic Drift with a Continuous Model.”Proc. natn. Acad. Sci. U.S.A. 41, 144–150.
— 1955b. “Stochastic Processes and Distribution of Gene Frequencies under Natural Selections”.Cold Spring Harb. Symp. quant. Biol. 20, 33–53.
— 1962. “On the Probability of Fixation of Mutant Genes in a Population.”Genetics 47, 713–719.
— 1964. “Diffusion Models in Population Genetics.”J appl. Prob. 1, 177–232.
— 1968. “Evolutionary Rate at the Molecular Level”.Nature, Lond. 217, 624–626.
— 1980. “Average Time until Fixation of a Mutant Allele in a Finite Population under Mutation Pressure: Studies by Analytical, Numerical and Pseudo-sampling Methods.”Proc. natn Acad. Sci. U.S.A. 77, 522–526.
— and J. F. Crow. 1964. “The Number of Alleles that can be Maintained in a Finite Population.”Genetics 49, 725–728.
— and T. Ohta. 1969. “The Average Number of Generations until Fixation of an Individual Mutant Gene in a Finite Population.”Genetics 63, 701–709.
— and —. 1971. “Protein Polymorphism as a Phase of Molecular Evolution.”Nature, Lond. 229, 467–469.
Kolmogorov, A. 1935. “Deviations from Hardy’s Formula in Partial Isolation.”C. r. Acad. Sci. U.R.S.S. 3, 129–132.
Li, W.-H. 1973. “Total Number of Individuals Affected by a Single Deleterious Mutant in a Finite Population.”Am. J. Human. Genet. 241, 667–679.
— 1978. “Maintenance of Genetic Variability under the Joint Effect of Mutation, Selection and Random Drift.”Genetics 90, 349–382.
Maruyama, T. 1972. “The Average Number and the Variance of Generations at a Particular Gene Frequency in the Course of Fixation of a Mutant Gene in a Finite Population.”Genet. Res. 19, 109–113.
— 1974. “The Age of an Allele in a Finite Population.”Genet. Res.,23, 137–143.
— 1977.Lecture Notes in Biomathematics 17. Stochastic Problems in Population Genetics. Berlin: Springer-Verlag.
— 1980. “On an Overdominant Model of Population Genetics.”Adv. appl. Prob. 12, 274–275.
— 1981. “Stochastic Problems in Population Genetics: Applications of Itô’s Stochastic Integrals.” InSto chastic Nonlinear Systems, Eds. L. Arnold and R. Lefever, pp. 154–161. Berlin: Springer-Verlag.
Maruyama, T. and P. A. Fuerst. “Analyses of the Age of Genes and the First Arrival Times in a Finite Population.”Genetics. In press.
— and Kimura, M. 1971. “Some Methods for treating Continuous Stochastic Processes in Population Genetics.”Jap. J. Genet. 46, 407–410.
— and — 1975. “Moments for Sum of an Arbitrary Function of Gene Frequency along a Stochastic Path of Gene Frequency Change.”Proc. natn Acad. sci. U.S.A. 72, 1602–1604.
— and M. Nei. 1981. “Genetic Variability maintained by Mutation and Overdominant Selection in Finite Populations.”Genetics 98, 491–459.
— and N. Takahata. 1981. “Numericl Studies of the Frequency Trajectories in the Process of Fixation of Null Genes at Duplicated Loci.Heredity 46, 49–57.
McShane, E. J. 1974.Stochastic Calculus and Stochastic Models. New York: Academic Press.
Nagasawa, M. 1964. “Time Reversions of Markov Processes.”Nagoya math. J. 24, 177–204.
— and Maruyama, T. 1979. “An Application of Time Reversal of Markov Processes to a Problem of Population Genetics.”Adv. appl. Prob. 11, 457–478.
Nagylaki, T. 1974. “The Moments of Stochastic Integrals and the Distribution of Sojourn Times.”Proc. natn Acad. Sci. U.S.A. 71, 746–749.
Nei, M. 1975.Molecular Population Genetics and Evolution. New York: North-Holland/American Elsevier.
Pederson, D. G. 1973. “Note: An Approximation Method of Sampling a Multinomial Population.”Biometrics 29, 814–821.
Robertson, A. 1964. “The Effect of Non-random Mating within Inbred Lines on the Rate of Inbreeding.”Genet. Res. 5, 164–167.
Rumelin, W. 1980. “Numerical Treatment of Stochastic Differential Equations. Report No. 12, Forschungsschwerpunkt Dynamische Systeme, Universität Bremen.
Skorokhod, A. V., 1965.Studies in the Theory of Random Processes. Reading, MA: Addison Wesley.
Slatkin, M. 1981. “Fixation Probability and Fixation Times in a Subdivided Population.”Evolution 35, 477–488.
Watanabe S. 1971. “On the Stochastic Differential Equations for Multidimensional Diffusion Process with Boundary Conditions”.J. Math. Kyoto Univ. 11, 169–180.
Watterson, G. A. 1977. “Heterosis or Neutrality?”Genetics 85, 789–814.
Wong, E. and M. Zakai. 1965. “On the Convergence of Ordinary Integrals to Stochastic Integrals.”Ann. math. Statist. 36, 1560–1564.
Wright, S. 1931. “Evolution in Mendelian Populations.”Genetics 16, 97–159.
— 1945. “Differential Equations of the Distribution of Gene Frequencies.”Proc. natn Acad. Sci. U.S.A. 31, 382–389.
Wright, S. 1948. “Genetics of Populations.” InEncyclopedia Britannica, Vol. 10, 111, 111A-D, 112.
— 1949. “Adaptation and Selection.” InGenetics, Paleontology, and Evolution, Ed. G. G. Simpson, G. L. Jepsen and E. Mayr, pp. 365–389. Princeton, NJ: Princeton University Press.
— 1969.Evolution and Genetics of Populations, Vol. 2. The Theory of Gene Frequencies. Chicago, IL: University of Chicago Press.
— 1970. “Tandom Drift and the Shifting Balance Theory or Evolution.” InMathematical Topics in Population Genetics, Ed. K. Kojima, pp. 1–31. Berlin: Springer-Verlag.
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Maruyama, T. Stochastic theory of population genetics. Bltn Mathcal Biology 45, 521–554 (1983). https://doi.org/10.1007/BF02459586
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DOI: https://doi.org/10.1007/BF02459586