×

The rate of multi-step evolution in Moran and Wright-Fisher populations. (English) Zbl 1322.92047

Summary: Several groups have recently modeled evolutionary transitions from an ancestral allele to a beneficial allele separated by one or more intervening mutants. The beneficial allele can become fixed if a succession of intermediate mutants are fixed or alternatively if successive mutants arise while the previous intermediate mutant is still segregating. This latter process has been termed stochastic tunneling. Previous work has focused on the Moran model of population genetics. I use elementary methods of analyzing stochastic processes to derive the probability of tunneling in the limit of large population size for both Moran and Wright-Fisher populations. I also show how to efficiently obtain numerical results for finite populations. These results show that the probability of stochastic tunneling is twice as large under the Wright-Fisher model as it is under the Moran model.

MSC:

92D15 Problems related to evolution
92D10 Genetics and epigenetics
60E05 Probability distributions: general theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adler, F. R., Modeling the Dynamics of Life (1998), Brooks Cole: Brooks Cole Pacific Grove, CA
[2] Corless, R.; Gonnet, G.; Hare, D.; Jeffrey, D.; Knuth, D., On the Lambert \(W\) function, Advances in Computational Mathematics, 5, 329-359 (1996) · Zbl 0863.65008
[3] Crow, J. F.; Kimura, M., An Introduction to Population Genetics Theory (1970), Harper & Row: Harper & Row New York · Zbl 0246.92003
[4] Draghi, J. A.; Parsons, T. L.; Wagner, G. P.; Plotkin, J. B., Mutational robustness can facilitate adaptation, Nature, 463, 353-355 (2010)
[5] Durrett, R.; Schmidt, D., Waiting for two mutations: with applications to regulatory sequence evolution and the limits of Darwinian evolution, Genetics, 180, 1501-1509 (2008)
[6] Durrett, R.; Schmidt, D.; Schweinsberg, J., A waiting time problem arising from the study of multi-stage carcinogenesis, The Annals of Applied Probability, 19, 676-718 (2009) · Zbl 1219.92038
[7] Dwass, M., Total progeny in a branching process and a related random walk, Journal of Applied Probability, 6, 682-686 (1969) · Zbl 0192.54401
[8] Ewens, W., Conditional diffusion processes in population genetics, Theoretical Population Biology, 4, 21-30 (1973) · Zbl 0283.92016
[9] da Fonseca, C. M., On the eigenvalues of some tridiagonal matrices, Journal of Computational and Applied Mathematics, 200, 283-286 (2007) · Zbl 1119.15012
[10] Hermisson, J.; Pennings, P., Soft sweeps: molecular population genetics of adaptation from standing genetic variation, Genetics, 169, 2335-2352 (2005)
[11] Iwasa, Y.; Michor, F.; Komarova, N. L.; Nowak, M. A., Population genetics of tumor suppressor genes, Journal of Theoretical Biology, 233, 15-23 (2005) · Zbl 1442.92103
[12] Iwasa, Y.; Michor, F.; Nowak, M. A., Evolutionary dynamics of escape from biomedical intervention, Proceedings of the Royal Society B: Biological Sciences, 270, 2573 (2003)
[13] Iwasa, Y.; Michor, F.; Nowak, M. A., Stochastic tunnels in evolutionary dynamics, Genetics, 166, 1571-1579 (2004)
[14] Komarova, N.; Sengupta, A.; Nowak, M., Mutation-selection networks of cancer initiation: tumor suppressor genes and chromosomal instability, Journal of Theoretical Biology, 223, 433-450 (2003) · Zbl 1464.92067
[15] Kopp, M.; Hermisson, J., The genetic basis of phenotypic adaptation i: fixation of beneficial mutations in the moving optimum model, Genetics, 182, 233-249 (2009)
[16] Lynch, M., Scaling expectations for the time to establishment of complex adaptations, Proceedings of the National Academy of Sciences, 107, 16577-16582 (2010)
[17] Lynch, M.; Abegg, A., The rate of establishment of complex adaptations, Molecular Biology and Evolution, 27, 1404-1414 (2010)
[18] Moran, P., The Statistical Processes of Evolutionary Theory (1962), Clarendon Press: Clarendon Press Oxford · Zbl 0119.35901
[19] van Nimwegen, E.; Crutchfield, J. P.; Huynen, M., Neutral evolution of mutational robustness, Proceedings of the National Academy of Sciences, 96, 9716-9720 (1999)
[20] O’Fallon, B. D.; Adler, F. R.; Proulx, S. R., Quasi-species evolution in subdivided populations favours maximally deleterious mutations, Proceedings of the National Academy of Sciences, 274, 3159-3164 (2007)
[21] Proulx, S. R.; Adler, F. R., The standard of neutrality: still flapping in the breeze?, Journal of Evolutionary Biology, 23, 1339-1350 (2010)
[22] Proulx, S. R.; Phillips, P. C., The opportunity for canalization and the evolution of genetic networks, American Naturalist, 165, 147-162 (2005)
[23] Ross, S., A First Course in Probability (1988), Macmillan: Macmillan New York
[24] Schweinsberg, J., The waiting time for \(m\) mutations, Electronic Journal of Probability, 13, 52, 1442-1478 (2008) · Zbl 1191.60100
[25] Taylor, H. M.; Karlin, S., An Introduction to Stochastic Modeling (1984), Academic Press: Academic Press Orlando
[26] Tong, A.; Evangelista, M.; Parsons, A.; Xu, H.; Bader, G.; Page, N.; Robinson, M.; Raghibizadeh, S.; Hogue, C.; Bussey, H.; Andrews, B.; Tyers, M.; Boone, C., Systematic genetic analysis with ordered arrays of yeast deletion mutants, Science, 294, 2364-2368 (2001)
[27] Tong, A. H.Y.; Lesage, G., Global mapping of the yeast genetic interaction network, Science, 303, 808-813 (2004)
[28] Usmani, R. A., Inversion of Jacobi’s tridiagonal matrix, Computers & Mathematics with Applications, 27, 59-66 (1994) · Zbl 0797.15002
[29] de Visser, J. A.G. M.; Hermisson, J.; Wagner, G. P.; Meyers, L. A.; Bagheri, H. C.; Blanchard, J. L.; Chao, L.; Cheverud, J. M.; Elena, S. F.; Fontana, W.; Gibson, G.; Hansen, T. F.; Krakauer, D.; Lewontin, R. C.; Ofria, C.; Rice, S. H.; von Dassow, G.; Wagner, A.; Whitlock, M. C., Perspective: evolution and detection of genetic robustness, Evolution, 57, 1959-1972 (2003)
[30] Weinreich, D.; Chao, L., Rapid evolutionary escape by large populations from local fitness peaks is likely in nature, Evolution, 59, 1175-1182 (2005)
[31] Weissman, D. B.; Desai, M. M.; Fisher, D. S.; Feldman, M. W., The rate at which asexual populations cross fitness valleys, Theoretical Population Biology, 75, 286-300 (2009) · Zbl 1213.92051
[32] Wilke, C. O.; Wang, J. L.; Ofria, C.; Lenski, R. E.; Adami, C., Evolution of digital organisms at high mutation rates leads to survival of the flattest, Nature, 412, 331-333 (2001)
[33] Wright, S., 1932. The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: Jones D.F. (Ed.) The Sixth International Congress of Genetics, Brooklyn Botanic Garden, Menasha, WI.; Wright, S., 1932. The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: Jones D.F. (Ed.) The Sixth International Congress of Genetics, Brooklyn Botanic Garden, Menasha, WI.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.