Evolutionary Accessibility of Modular Fitness Landscapes
A fitness landscape is a mapping from the space of genetic sequences, which is modeled here as a binary hypercube of dimension L, to the real numbers. We consider random models of fitness landscapes, where fitness values are assigned according to some probabilistic rule, and study the statistical properties of pathways to the global fitness maximum along which fitness increases monotonically. Such paths are important for evolution because they are the only ones that are accessible to an adapting population when mutations occur at a low rate. The focus of this work is on the block model introduced by A.S. Perelson and C.A. Macken (Proc. Natl. Acad. Sci. USA 92:9657, 1995) where the genome is decomposed into disjoint sets of loci (‘modules’) that contribute independently to fitness, and fitness values within blocks are assigned at random. We show that the number of accessible paths can be written as a product of the path numbers within the blocks, which provides a detailed analytic description of the path statistics. The block model can be viewed as a special case of Kauffman’s NK-model, and we compare the analytic results to simulations of the NK-model with different genetic architectures. We find that the mean number of accessible paths in the different versions of the model are quite similar, but the distribution of the path number is qualitatively different in the block model due to its multiplicative structure. A similar statement applies to the number of local fitness maxima in the NK-models, which has been studied extensively in previous works. The overall evolutionary accessibility of the landscape, as quantified by the probability to find at least one accessible path to the global maximum, is dramatically lowered by the modular structure.
KeywordsEvolution Fitness landscapes Adaptive walks Spin glasses
We acknowledge useful discussions with Peter Hegarty, Anders Martinsson, Johannes Neidhart, Stefan Nowak and Ivan Szendro, and support by DFG within SFB 680 and SPP 1590. J.K. takes this opportunity to thank Herbert Spohn for many years of guidance, encouragement and inspiration.
- 11.Orr, H.A.: The population genetics of adaptation: the adaptation of DNA sequences. Evolution 56, 1317–1330 (2002) Google Scholar
- 14.Weinreich, D.M., Watson, R.A., Chao, L.: Perspective: sign epistasis and genetic constraints on evolutionary trajectories. Evolution 59, 1165–1174 (2005) Google Scholar
- 21.Szendro, I.G., Schenk, M.F., Krug, J., de Visser, J.A.G.M.: Quantitative analyses of empirical fitness landscapes. J. Stat. Mech.: Theory Exp. P01005 (2013) Google Scholar
- 22.Klözer, A.: NK fitness landscapes. Diploma thesis, University of Cologne (2008) Google Scholar
- 25.Hegarty, P., Martinsson, A.: On the existence of accessible paths in various models of fitness landscapes, arXiv:1210.4798 (2012). To appear in Ann. Appl. Probab.
- 27.Berestycki, J., Brunet, É., Shi, Z.: How many evolutionary histories only increase fitness? Preprint. arXiv:1304.0246 (2013)
- 28.Roberts, M.I., Zhao, L.Z.: Increasing paths in trees. Preprint. arXiv:1305.0814 (2013)
- 32.Franke, J., Wergen, G., Krug, J.: Records and sequences of records from random variables with a linear drift. J. Stat. Mech.: Theor. Exp. P10013 (2010) Google Scholar
- 34.Kauffman, S.A.: The Origins of Order. Oxford University Press, London (1993) Google Scholar
- 46.Altenberg, L.: NK fitness landscapes. In: Bäck, T., Fogel, D.B., Michalewicz, Z. (eds.) Handbook of Evolutionary Computation. IOP Publishing, Bristol (1997) Google Scholar