The Annals of Applied Probability

Asymptotic behavior of the rate of adaptation

Feng Yu, Alison Etheridge, and Charles Cuthbertson

Full-text: Open access

Abstract

We consider the accumulation of beneficial and deleterious mutations in large asexual populations. The rate of adaptation is affected by the total mutation rate, proportion of beneficial mutations and population size N. We show that regardless of mutation rates, as long as the proportion of beneficial mutations is strictly positive, the adaptation rate is at least $\mathcal{O}(\log^{1-\delta}N)$ where δ can be any small positive number, if the population size is sufficiently large. This shows that if the genome is modeled as continuous, there is no limit to natural selection, that is, the rate of adaptation grows in N without bound.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 3 (2010), 978-1004.

Dates
First available in Project Euclid: 18 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1276867304

Digital Object Identifier
doi:10.1214/09-AAP645

Mathematical Reviews number (MathSciNet)
MR2680555

Zentralblatt MATH identifier
1193.92079

Subjects
Primary: 92D15: Problems related to evolution
Secondary: 82C22: Interacting particle systems [See also 60K35] 60J05: Discrete-time Markov processes on general state spaces 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
Adaptation rate natural selection evolutionary biology Moran particle systems

Citation

Yu, Feng; Etheridge, Alison; Cuthbertson, Charles. Asymptotic behavior of the rate of adaptation. Ann. Appl. Probab. 20 (2010), no. 3, 978--1004. doi:10.1214/09-AAP645. https://projecteuclid.org/euclid.aoap/1276867304


Export citation

References

  • Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions. Dover, New York.
  • Barton, N. H. (1995). Linkage and the limits to natural selection. Genetics 140 821–841.
  • Barton, N. H. and Coe, J. B. (2009). An upper limit to the rate of adaptation. Preprint.
  • Brunet, E., Derrida, B., Mueller, A. H. and Munier, S. (2006). Noisy traveling waves: Effect of selection on genealogies. Europhys. Lett. 76 1–7.
  • Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996). On the Lambert W function. Adv. Comput. Math. 5 329–359.
  • Cuthbertson, C., Etheridge, A. M. and Yu, F. (2009). Fixation probability for competing selective sweeps. Preprint. Available at arXiv:0812.0104.
  • Desai, M. M. and Fisher, D. S. (2007). Beneficial mutation–selection balance and the effect of linkage on positive selection. Genetics 176 1759–1798.
  • Etheridge, A. M., Pfaffelhuber, P. and Wakolbinger, A. (2009). How often does the ratchet click? Facts, heuristics, asymptotics. In Trends in Stochastic Analysis. London Math. Soc. Lecture Note Ser. 353. Cambridge Univ. Press, New York.
  • Felsenstein, J. (1974). The evolutionary advantage of recombination. Genetics 78 737–756.
  • Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Clarendon Press, Oxford.
  • Gerrish, P. J. and Lenski, R. E. (1998). The fate of competing beneficial mutations in an asexual population. Genetica 102/103 127–144.
  • Gillespie, J. H. (1991). The Causes of Molecular Evolution. Oxford Univ. Press, Oxford, UK.
  • Gordo, I. and Charlesworth, B. (2000). On the speed of Muller’s ratchet. Genetics 156 2137–2140.
  • Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd ed. Clarendon Press, Oxford.
  • Haigh, J. (1978). The accumulation of deleterious genes in a population—Muller’s ratchet. Theoret. Population Biol. 14 251–267.
  • Haldane, J. B. S. (1927). A mathematical theory of natural and artificial selection, part v: Selection and mutation. Proceedings of the Cambridge Philosophical Society 23 834–844.
  • Hegreness, M., Shoresh, N., Hartl, D. and Kishony, R. (2006). An equivalence principle for the incorporation of favourable mutations in asexual populations. Science 311 1615–1617.
  • Higgs, P. and Woodcock, G. (1995). The accumulation of mutations in asexual populations, and the structure of genealogical trees in the presence of selection. J. Math. Biol. 33 677–702.
  • Hill, W. G. and Robertson, A. (1966). The effect of linkage on limits to artificial selection. Genetics Research 8 269–294.
  • Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • Muller, H. J. (1964). The relation of recombination and mutational advance. Mutat. Res. 106 2–9.
  • Norris, J. R. (1998). Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics 2. Cambridge Univ. Press, Cambridge.
  • Novella, I. S., Elena, S. F., Moya, A., Domingo, E. and Holland, J. J. (1995). Size of genetic bottlenecks leading to virus fitness loss is determined by mean initial population fitness. J. Virol. 69 2869–2872.
  • Novella, I. S., Elena, S. F., Moya, A., Domingo, E. and Holland, J. J. (1999). Exponential fitness gains of rna virus populations are limited by bottleneck effects. J. Virol. 73 1668–1671.
  • Orr, H. A. (2000). The rate of adaptation in asexuals. Genetics 155 961–968.
  • Protter, P. (2003). Stochastic Integration and Differential Equations. Applications of Mathematics (New York) 21. Springer, Berlin.
  • Rouzine, I., Brunet, E. and Wilke, C. O. (2008). The traveling-wave approach to asexual evolution: Muller’s ratchet and speed of adaptation. Theorectial Population Biology 73 24–46.
  • Rouzine, I., Wakeley, J. and Coffin, J. M. (2003). The solitary wave of asexual evolution. Proc. Nat. Acad. Sci. 100 587–592.
  • Stephan, W., Chao, L. and Smale, J. (1993). The advance of Muller’s ratchet in a haploid asexual population: Approximate solution based on diffusion theory. Genet. Res. 61 225–232.
  • Taylor, M. E. (1996). Partial Differential Equations: Basic Theory. Texts in Applied Mathematics 23. Springer, New York.
  • Wilke, C. O. (2004). The speed of adaptation in large asexual populations. Genetics 167 2045–2054.
  • Yu, F. and Etheridge, A. M. (2008). Rate of adaptation of large populations. In Evolutionary Biology from Concept to Application. Springer, Berlin.