Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T21:36:32.523Z Has data issue: false hasContentIssue false
  • English
  • Français

Time reversal of some stationary jump diffusion processes from population genetics

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Martin Hutzenthaler  and
Jesse Earl Taylor
Martin Hutzenthaler*
Affiliation:
Goethe-University Frankfurt
Jesse Earl Taylor*
Affiliation:
Arizona State University
*
∗ Current address: LMU Biozentrum, Grosshadern Str. 2, D-82152 Planegg-Martinsried, Germany. Email address: hutzenthaler@bio.lmu.de
∗∗ Postal address: School of Mathematical and Statistical Sciences, Arizona State University, PO Box 871804, Tempe, AZ 85287-1804, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We describe the processes obtained by time reversal of a class of stationary jump diffusion processes that model the dynamics of genetic variation in populations subject to repeated bottlenecks. Assuming that only one lineage survives each bottleneck, the forward process is a diffusion on [0,1] that jumps to the boundary before diffusing back into the interior. We show that the behavior of the time-reversed process depends on whether the boundaries are accessible to the diffusive motion of the forward process. If a boundary point is inaccessible to the forward diffusion then time reversal leads to a jump diffusion that jumps immediately into the interior whenever it arrives at that point. If, instead, a boundary point is accessible then the jumps off of that point are governed by a weighted local time of the time-reversed process.

Keywords

Time reversaljump diffusionlocal timecoalescentpopulation bottleneckselective sweep

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60J55: Local time and additive functionals 92D10: Genetics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 1147 - 1171
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research supported by the DFG in the Dutch German Bilateral Research Group Mathematics of Random Spatial Models from Physics and Biology (For 498).

References

Barton, N. H. and Etheridge, A. M. (2004). The effect of selection on genealogies. Genetics 166, 1115–1131.CrossRefGoogle ScholarPubMed
Barton, N. H., Etheridge, A. M. and Sturm, A. K. (2004). Coalescence in a random background. Ann. Appl. Prob. 14, 754–785.Google Scholar
Birkhoff, G. and Rota, G.-C. (1989). Ordinary Differential Equations, 4th edn. John Wiley, New York.Google Scholar
Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory (Pure Appl. Math. 29). Academic Press, New York.Google Scholar
Coop, G. and Griffiths, R. C. (2004). Ancestral inference on gene trees under selection. Theoret. Pop. Biol. 66, 219–232.CrossRefGoogle ScholarPubMed
Donnelly, P. and Kurtz, T. G. (1999). Genealogical processes for Fleming–Viot models with selection and recombination. Ann. Appl. Prob. 9, 1091–1148.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
Ewens, W. J. (2004). Mathematical Population Genetics. I, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Getoor, R. K. and Sharpe, M. J. (1981). Two results on dual excursions. In Seminar on Stochastic Processes, 1981 (Evanston, IL, 1981; Progress Prob. Statist. 1), Birkhäuser Boston, MA, pp. 31–52.CrossRefGoogle Scholar
Gillespie, J. H. (2000). Genetic drift in an infinite population: the pseudohitchhiking model. Genetics 155, 909–919.Google Scholar
Itô, K. and McKean, H. P. Jr. (1974). Diffusion Processes and Their Sample Paths. Springer, Berlin.Google Scholar
Kaplan, N. L., Darden, T. and Hudson, R. R. (1988). The coalescent process in models with selection. Genetics 120, 819–829.Google ScholarPubMed
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113), 2nd edn. Springer, New York.Google Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kim, Y. (2004). Effect of strong directional selection on weakly selected mutations at linked sites: implication for synonymous codon usage. Mol. Biol. Evol. 21, 286–294.Google Scholar
Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 27–43.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.Google Scholar
Krone, S. M. and Neuhauser, C. (1997). Ancestral processes with selection. Theoret. Pop. Biol. 51, 210–237.CrossRefGoogle ScholarPubMed
Mandl, P. (1968). Analytical Treatment of One-Dimensional Markov Processes. Springer, New York.Google Scholar
McKean, H. P. Jr. (1956). Elementary solutions for certain parabolic partial differential equations. Trans. Amer. Math. Soc. 82, 519–548.Google Scholar
Mitro, J. B. (1984). Exit systems for dual Markov processes. Z. Wahrscheinlichkeitsth. 66, 259–267.Google Scholar
Mitro, J. B. (1984). Time reversal depending on local time. Stoch. Process. Appl. 18, 171–177.CrossRefGoogle Scholar
Nelson, E. (1958). The adjoint Markoff process. Duke Math. J. 25, 671–690.Google Scholar
Neuhauser, C. and Krone, S. M. (1997). The genealogy of samples in models with selection. Genetics 145, 519–534.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in C, 2nd edn. Cambridge University Press.Google Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Cambridge University Press.Google Scholar
Rong, R. et al. (2007). Unique mutational patterns in the envelope alpha 2 amphipathic helix and acquisition of length in gp120 hypervariable domains are associated with resistance to autologous neutralization of subtype C human immunodeficiency virus type 1. J. Virol. 81, 5658–5668.Google Scholar
Rouzine, I. M. and Coffin, J. M. (1999). Search for the mechanism of genetic variation in the pro gene of human immunodeficiency virus. J. Virol. 73, 8167–8178.Google Scholar
Rubin, L. G. (1987). Bacterial colonization and infection resulting from multiplication of a single organism. Rev. Infect. Diseases 9, 488–493.CrossRefGoogle ScholarPubMed
Taylor, J. E. (2007). The common ancestor process for a Wright–Fisher diffusion. Electron. J. Prob. 12, 808–847.CrossRefGoogle Scholar
Yuste, E. et al. (1999). Drastic fitness loss in human immunodeficiency virus type 1 upon serial bottleneck events. J. Virol. 73, 2745–2751.Google Scholar