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A Diffusion Approach to Approximating Preservation Probabilities for Gene Duplicates

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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.

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References

  1. Barbour A.D. (1972) The principle of the diffusion of arbitrary constants. J Appl Prob 9, 519–541

    Article  MATH  MathSciNet  Google Scholar 

  2. Barbour A.D. (1974) On a functional central limit theorem for Markov population processes. Adv Appl Prob 9, 21–39

    Article  MathSciNet  Google Scholar 

  3. 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

    Article  MATH  MathSciNet  Google Scholar 

  4. Bleistein N., Handelsman R.A. (1975) Asymptotic expansions of integrals. Rinehart and Winston Holt, New York

    MATH  Google Scholar 

  5. Buchholz, H. The confluent hypergeometric function, by H. Lichtblau & K. Wetzel. Springer translated from German Berlin Heidelberg New York: 1969

  6. Daley D.J., Kendall D.G. (1965) Stochastic rumours. J Inst Math Appl 1, 42–55

    Article  MathSciNet  Google Scholar 

  7. van Doorn E.A., Zeifman A.I. (2005) Birth-death processes with killing. Stat Prob Lett 72, 33–42

    Article  MATH  Google Scholar 

  8. 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

    Article  MATH  MathSciNet  Google Scholar 

  9. Ewens W.J. (2004) Mathematical Population Genetics. I. Theoretical Introduction, 2nd edi. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  10. Haldane J.B.S. (1932) The causes of evolution. Harper & Bros, New York

    Google Scholar 

  11. 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

    Article  MATH  Google Scholar 

  12. Karlin S., Tavaré S. (1982b) Linear birth and death processes with killing. J Appl Prob 19, 477–487

    Article  MATH  Google Scholar 

  13. Karlin S., Taylor H.M. (1981) A second course in stochastic processes. Academic: New York Press

  14. Kimura M. (1964) Diffusion models in population genetics. J Appl Prob 1, 177–232

    Article  MATH  Google Scholar 

  15. 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

    Article  MATH  Google Scholar 

  16. Kummer E.E. (1836) Über die hypergeometrische reihe F(a;b;x). J die Reine Angewandte Mathe 15(39–83): 127–172

    Article  MATH  Google Scholar 

  17. Lynch M., Conery J.C. (2000) The evolutionary fate and consequences of duplicate genes. Science 290, 1151–1154

    Article  Google Scholar 

  18. Lynch M., Force A. (2000) The probability of duplicate gene preservation by subfunctionalization. Genetics 154, 459–473

    Google Scholar 

  19. Lynch M., Katju V. (2004) The altered evolutionary trajectories of gene duplicates. Trends in Genetics. 20, 544–549

    Article  Google Scholar 

  20. Lynch M., O’Hely M., Walsh B., Force A. (2001) The probability of preservation of a newly arisen gene duplicate. Genetics 159, 1789–1804

    Google Scholar 

  21. Moran P.A.P. (1958) Random processes in genetics. Proc Cambridge Phil Soc 54, 60–71

    Article  MATH  Google Scholar 

  22. Ohno S. (1970) Evolution by gene duplication. Springer, Berlin Heidelberg, New York

    Google Scholar 

  23. 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

    Article  MATH  MathSciNet  Google Scholar 

  24. 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

    Article  Google Scholar 

  25. 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

    Google Scholar 

  26. Ward R., Durrett R. (2004) Subfunctionalization: How often does it occur? How long does it take. Theor Pop Biol 66, 93–100

    Article  MATH  Google Scholar 

  27. Watterson G.A. (1983) On the time for gene silencing at duplicate loci. Genetics 105, 745–766

    Google Scholar 

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Correspondence to Martin O’Hely.

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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

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  • DOI: https://doi.org/10.1007/s00285-006-0001-6

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