Abstract
Consider a haploid population and, within its genome, a gene whose presence is vital for the survival of any individual. Each copy of this gene is subject to mutations which destroy its function. Suppose one member of the population somehow acquires a duplicate copy of the gene, where the duplicate is fully linked to the original gene’s locus. Preservation is said to occur if eventually the entire population consists of individuals descended from this one which initially carried the duplicate. The system is modelled by a finite state-space Markov process which in turn is approximated by a diffusion process, whence an explicit expression for the probability of preservation is derived. The event of preservation can be compared to the fixation of a selectively neutral gene variant initially present in a single individual, the probability of which is the reciprocal of the population size. For very weak mutation, this and the probability of preservation are equal, while as mutation becomes stronger, the preservation probability tends to double this reciprocal. This is in excellent agreement with simulation studies.
Similar content being viewed by others
References
Barbour A.D. (1972) The principle of the diffusion of arbitrary constants. J Appl Prob 9, 519–541
Barbour A.D. (1974) On a functional central limit theorem for Markov population processes. Adv Appl Prob 9, 21–39
Barbour A.D., Ethier S.N., Griffiths R.C. (2000) A transition function expansion for a diffusion model with selection. Ann Appl Prob 10, 123–162
Bleistein N., Handelsman R.A. (1975) Asymptotic expansions of integrals. Rinehart and Winston Holt, New York
Buchholz, H. The confluent hypergeometric function, by H. Lichtblau & K. Wetzel. Springer translated from German Berlin Heidelberg New York: 1969
Daley D.J., Kendall D.G. (1965) Stochastic rumours. J Inst Math Appl 1, 42–55
van Doorn E.A., Zeifman A.I. (2005) Birth-death processes with killing. Stat Prob Lett 72, 33–42
Ethier S.N., Nagylaki T. (1980) Diffusion approximations of Markov chains with two time scales and applications to population genetics. Adv Appl Prob 12, 14–49
Ewens W.J. (2004) Mathematical Population Genetics. I. Theoretical Introduction, 2nd edi. Springer, Berlin Heidelberg New York
Haldane J.B.S. (1932) The causes of evolution. Harper & Bros, New York
Karlin S., Tavaré S. (1982a) A diffusion process with killing: the time to formation of recurrent deleterious mutant genes. Stoc Pro Appl 13, 249–261
Karlin S., Tavaré S. (1982b) Linear birth and death processes with killing. J Appl Prob 19, 477–487
Karlin S., Taylor H.M. (1981) A second course in stochastic processes. Academic: New York Press
Kimura M. (1964) Diffusion models in population genetics. J Appl Prob 1, 177–232
Kimura M., King J.L. (1979) Fixation of a deleterious allele at one of two “duplicate” loci by mutation pressure and random drift. Proce Nati Acad Sci USA 76, 2858–2861
Kummer E.E. (1836) Über die hypergeometrische reihe F(a;b;x). J die Reine Angewandte Mathe 15(39–83): 127–172
Lynch M., Conery J.C. (2000) The evolutionary fate and consequences of duplicate genes. Science 290, 1151–1154
Lynch M., Force A. (2000) The probability of duplicate gene preservation by subfunctionalization. Genetics 154, 459–473
Lynch M., Katju V. (2004) The altered evolutionary trajectories of gene duplicates. Trends in Genetics. 20, 544–549
Lynch M., O’Hely M., Walsh B., Force A. (2001) The probability of preservation of a newly arisen gene duplicate. Genetics 159, 1789–1804
Moran P.A.P. (1958) Random processes in genetics. Proc Cambridge Phil Soc 54, 60–71
Ohno S. (1970) Evolution by gene duplication. Springer, Berlin Heidelberg, New York
Pollett P.K., Stewart D.E. (1994) An efficient procedure for computing quasi-stationary distributions of Markov chains with sparse transition structure. Adv Appl Prob 26, 68–79
Takahata N., Maruyama T. (1979) Polymorphism and loss of duplicate gene expression: a theoretical study with application to tetraploid fish. Proc Nati Acad Sci USA 76, 4521–4525
Wagner A. (2000) The role of population size, pleiotropy and fitness effects of mutations in the evolution of overlapping gene functions. Genetics 154, 1389–1401
Ward R., Durrett R. (2004) Subfunctionalization: How often does it occur? How long does it take. Theor Pop Biol 66, 93–100
Watterson G.A. (1983) On the time for gene silencing at duplicate loci. Genetics 105, 745–766
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
O’Hely, M. A Diffusion Approach to Approximating Preservation Probabilities for Gene Duplicates. J. Math. Biol. 53, 215–230 (2006). https://doi.org/10.1007/s00285-006-0001-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-006-0001-6