The Annals of Applied Probability

Muller’s ratchet with compensatory mutations

P. Pfaffelhuber, P. R. Staab, and A. Wakolbinger

Full-text: Open access

Abstract

We consider an infinite-dimensional system of stochastic differential equations describing the evolution of type frequencies in a large population. The type of an individual is the number of deleterious mutations it carries, where fitness of individuals carrying $k$ mutations is decreased by $\alpha k$ for some $\alpha>0$. Along the individual lines of descent, new mutations accumulate at rate $\lambda$ per generation, and each of these mutations has a probability $\gamma$ per generation to disappear. While the case $\gamma=0$ is known as (the Fleming–Viot version of) Muller’s ratchet, the case $\gamma>0$ is associated with compensatory mutations in the biological literature. We show that the system has a unique weak solution. In the absence of random fluctuations in type frequencies (i.e., for the so-called infinite population limit) we obtain the solution in a closed form by analyzing a probabilistic particle system and show that for $\gamma>0$, the unique equilibrium state is the Poisson distribution with parameter $\lambda/(\gamma+\alpha)$.

Article information

Source
Ann. Appl. Probab. Volume 22, Number 5 (2012), 2108-2132.

Dates
First available in Project Euclid: 12 October 2012

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1350067996

Digital Object Identifier
doi:10.1214/11-AAP836

Mathematical Reviews number (MathSciNet)
MR3025691

Zentralblatt MATH identifier
1251.92035

Subjects
Primary: 92D15: Problems related to evolution 92D15: Problems related to evolution 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60H35: Computational methods for stochastic equations [See also 65C30] 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Muller’s ratchet selection back mutation Fleming–Viot process Girsanov transform

Citation

Pfaffelhuber, P.; Staab, P. R.; Wakolbinger, A. Muller’s ratchet with compensatory mutations. Ann. Appl. Probab. 22 (2012), no. 5, 2108--2132. doi:10.1214/11-AAP836. http://projecteuclid.org/euclid.aoap/1350067996.


Export citation

References

  • Antezana, M. A. and Hudson, R. R. (1997). Era reversibile! Point-mutations, the Ratchet, and the initial success of eukaryotic sex: A simulation study. Evolutionary Theory 11 209–235.
  • Audiffren, J. (2011). Ph.D. thesis. Univ. de Provence, Marseille.
  • Audiffren, J. and Pardoux, E. (2011). Muller’s ratchet clicks in finite time. Unpublished manuscript.
  • Chao, L. (1990). Fitness of RNA virus decreased by Muller’s ratchet. Nature 348 454–455.
  • Cuthbertson, C. (2007). Limits to the rate of adaptation. Ph.D. thesis, Univ. Oxford.
  • Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • Dawson, D. A. (1993). Measure-valued Markov processes. In École D’Été de Probabilités de Saint-Flour XXI—1991 (P. Hennequin, ed.). Lecture Notes in Math. 1541 1–260. Springer, Berlin.
  • Dawson, D. and Li, Z. (2010). Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 813–857.
  • 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 Mathematical Society Lecture Note Series 353 365–390. Cambridge Univ. Press, Cambridge.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Ethier, S. N. and Kurtz, T. G. (1993). Fleming–Viot processes in population genetics. SIAM J. Control Optim. 31 345–386.
  • Ethier, S. N. and Shiga, T. (2000). A Fleming–Viot process with unbounded selection. J. Math. Kyoto Univ. 40 337–361.
  • Gabriel, W., Lynch, M. and Bürger, R. (1993). Muller’s ratchet and mutational meltdowns. Evolution 47 1744–1757.
  • Gerrish, P. and Lenski, R. (1998). The fate of competing beneficial mutations in an asexual population. Genetica 102/103 127–144.
  • Gessler, D. D. (1995). The constraints of finite size in asexual populations and the rate of the ratchet. Genet. Res. 66 241–253.
  • Gordo, I. and Charlesworth, B. (2000). On the speed of Muller’s ratchet. Genetics 156 2137–2140.
  • Haigh, J. (1978). The accumulation of deleterious genes in a population—Muller’s ratchet. Theoret. Population Biol. 14 251–267.
  • Handel, A., Regoes, R. R. and Antia, R. (2006). The role of compensatory mutations in the emergence of drug resistance. PLoS Comput. Biol. 2 e137.
  • Higgs, P. G. 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.
  • Howe, D. K. and Denver, D. R. (2008). Muller’s ratchet and compensatory mutation in Caenorhabditis briggsae mitochondrial genome evolution. BMC Evol. Biol. 8 62–62.
  • Jain, K. (2008). Loss of least-loaded class in asexual populations due to drift and epistasis. Genetics 179 2125–2134.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • Loewe, L. (2006). Quantifying the genomic decay paradox due to Muller’s ratchet in human mitochondrial DNA. Genet. Res. 87 133–159.
  • Maia, L. P., Botelho, D. F. and Fontanari, J. F. (2003). Analytical solution of the evolution dynamics on a multiplicative-fitness landscape. J. Math. Biol. 47 453–456.
  • Maier, U., Bozarth, A., Funk, H., Zauner, S., Rensing, S., Schmitz-Linneweber, C., Börner, T. and Tillich, M. (2008). Complex chloroplast rna metabolism: Just debugging the genetic programme? BMC Biol. 6 36–36.
  • Maisnier-Patin, S. and Andersson, D. I. (2004). Adaptation to the deleterious effects of antimicrobial drug resistance mutations by compensatory evolution. Res. Microbiol. 155 360–369.
  • Maynard Smith, J. (1978). The Evolution of Sex. Cambridge Univ. Press, Cambridge.
  • Muller, H. J. (1964). The relation of recombination to mutational advance. Mutat. Res. 106 2–9.
  • Park, S. C. and Krug, J. (2007). Clonal interference in large populations. PNAS 104 18135–18140.
  • Poon, A. and Chao, L. (2005). The rate of compensatory mutation in the DNA bacteriophage phiX174. Genetics 170 989–999.
  • Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin.
  • Rouzine, I. M., Wakeley, J. and Coffin, J. M. (2003). The solitary wave of asexual evolution. Proc. Natl. Acad. Sci. USA 100 587–592.
  • Shiga, T. and Shimizu, A. (1980). Infinite-dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20 395–416.
  • Stephan, W., Chao, L. and Smale, J. (1993). The advance of Muller’s ratchet in a haploid asexual population: Approximate solutions based on diffusion theory. Genet. Res. 61 225–231.
  • Wagner, G. and Gabriel, W. (1990). What stops Muller’s ratchet in the absence of recombination? Evolution 44 715–731.
  • Waxman, D. and Loewe, L. (2010). A stochastic model for a single click of Muller’s ratchet. J. Theor. Biol. 264 1120–1132.
  • Yu, F., Etheridge, A. and Cuthbertson, C. (2010). Asymptotic behavior of the rate of adaptation. Ann. Appl. Probab. 20 978–1004.