## An exact steady state solution of Fisher’s geometric model and other models.(English)Zbl 1210.92030

Summary: Because nearly neutral substitutions are thought to contribute substantially to molecular evolution, and much of our insight about the workings of nearly neutral evolution relies on theory, solvable models of this process are of particular interest. I present an analytical method for solving models of nearly neutral evolution at steady state. The steady state solution applies to any constant fitness landscape under a dynamic of successive fixations, each of which occurs on the background of the population’s most recent common ancestor. Because this dynamic neglects the effects of polymorphism in the population beyond the mutant allele under consideration, the steady state solution provides a decent approximation of evolutionary dynamics when the population mutation rate is low ($$Nu \ll 1$$). To demonstrate the method, I apply it to two examples: R. A. Fisher’s geometric model (FGM) [The genetical theory of natural selection. Revised reprint of the 1930 original. (1999; Zbl 1033.92022); Oxford Univ. Press (1930; JFM 56.1106.13)], and a simple model of molecular evolution. Since recent papers have studied the steady state behavior of FGM under this dynamic, I analyze its behavior in detail and compare the results with previous work.

### MSC:

 92D15 Problems related to evolution 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)

### Citations:

JFM 56.1106.13; Zbl 1033.92022
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### References:

 [1] Andolfatto, P., Adaptive evolution of non-coding DNA in drosophila, Nature, 437, 1149-1152, (2005) [2] Barton, N.H., Coe, J., Banglawala, N., 2008. On the application of statistical physics to evolutionary biology (submitted for publication) [3] Berg, J.; Willman, S.; Lässig, M., Adaptive evolution of transcription factor binding sites, BMC evolutionary biology, 4, 42, (2004) [4] Boyko, A.R., Assessing the evolutionary impact of amino acid mutations in the human genome, Plos genetics, 4, e1000083, (2008) [5] Crow, J.F.; Kimura, M., An introduction to population genetics theory, (1970), Harper & Row New York · Zbl 0246.92003 [6] Ewens, W.J., Mathematical population genetics, (1979), Springer-Verlag Berlin · Zbl 0422.92011 [7] Eyre-Walker, A., The genomic rate of adaptive evolution, Trends in ecology and evolution, 21, 569-575, (2006) [8] Fay, J.C.; Wyckoff, G.J.; Wu, C., Testing the neutral theory of molecular evolution with data from drosophila, Nature, 415, 1024-1026, (2002) [9] Fisher, R.A., The genetical theory of natural selection, (1958), Dover New York · JFM 56.1106.13 [10] Gillespie, J.H., On ohta’s hypothesis: most amino acid substitutions are deleterious, Journal of molecular evolution, 40, 64-69, (1995) [11] Hartl, D.L.; Taubes, C.H., Compensatory nearly neutral mutations: selection without adaptation, Journal of theoretical biology, 182, 303-309, (1996) [12] Hartl, D.L.; Taubes, C.H., Towards a theory of evolutionary adaptation, Genetica, 103, 525-533, (1998) [13] Iwasa, Y., Free fitness that always increases in evolution, Journal of theoretical biology, 135, 265-282, (1988) [14] Ohta, T., Slightly deleterious mutant substitutions in evolution, Nature, 246, 96-98, (1973) [15] Orr, H.A., The population genetics of adaptation: the distribution of factors fixed during adaptive evolution, Evolution, 52, 935-949, (1998) [16] Poon, A.; Otto, S.P., Compensating for our load of mutations: freezing the meltdown of small populations, Evolution, 54, 1467-1479, (2000) [17] Sella, G.; Hirsh, A.E., The application of statistical physics to evolutionary biology, Proceedings of national Academy sciences USA, 102, 9541-9546, (2005) [18] Smith, N.G.C.; Eyre-Walker, A., Adaptive protein evolution in drosophila, Nature, 415, 1022-1024, (2002) [19] van Kampen, N.G., Stochastic processes in physics and chemistry, (1981), North-Holland Publishing Company Amsterdam · Zbl 0511.60038 [20] Wright, S, Evolution in Medelian populations, Genetics, 16, 97-159, (1931)
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