Bulletin of Mathematical Biology

, Volume 62, Issue 5, pp 799–848 | Cite as

Metastable evolutionary dynamics: Crossing fitness barriers or escaping via neutral paths?

  • Erik van Nimwegen
  • James P. Crutchfield


We analytically study the dynamics of evolving populations that exhibit metastability on the level of phenotype or fitness. In constant selective environments, such metastable behavior is caused by two qualitatively different mechanisms. On the one hand, populations may become pinned at a local fitness optimum, being separated from higher-fitness genotypes by a fitness barrier of low-fitness genotypes. On the other hand, the population may only be metastable on the level of phenotype or fitness while, at the same time, diffusing over neutral networks of selectively neutral genotypes. Metastability occurs in this case because the population is separated from higher-fitness genotypes by an entropy barrier: the population must explore large portions of these neutral networks before it discovers a rare connection to fitter phenotypes.

We derive analytical expressions for the barrier crossing times in both the fitness barrier and entropy barrier regime. In contrast with ‘landscape’ evolutionary models, we show that the waiting times to reach higher fitness depend strongly on the width of a fitness barrier and much less on its height. The analysis further shows that crossing entropy barriers is faster by orders of magnitude than fitness barrier crossing. Thus, when populations are trapped in a metastable phenotypic state, they are most likely to escape by crossing an entropy barrier, along a neutral path in genotype space. If no such escape route along a neutral path exists, a population is most likely to cross a fitness barrier where the barrier is narrowest, rather than where the barrier is shallowest.


Barrier Height Peak Individual Error Threshold Neutral Network Barrier Width 
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  1. Adami, C. (1995). Self-organized criticality in living systems. Phys. Lett. A. 203, 29–32.CrossRefGoogle Scholar
  2. Barnett, L. (1998). Ruggedness and neutrality: the NKp family of fitness landscapes, in ALIFE VI, Available at:
  3. Bergman, A. and M. W. Feldman (1996). Question marks about the period of punctuation. Technical Report, Santa Fe Institute Working paper 96-02-006.Google Scholar
  4. Christiansen, F. B., S. P. Otto, A. Bergman and M. W. Feldman (1998). Waiting with and without recombination: The time to production of a double mutant. Theor. Pop. Biol. 53, 199–215.CrossRefGoogle Scholar
  5. Crutchfield, J. P. and M. Mitchell (1995). The evolution of emergent computation. Proc. Natl. Acad. Sci. USA 92, 10742–10746.Google Scholar
  6. Crutchfield, J. P. and E. van Nimwegen (2000). The evolutionary unfolding of complexity, in Evolution as Computation, Lecture Notes in Computer Science, L. F. Landweber, E. Winfree, R. Lipton and S. Freeland (Eds), New York: Springer-Verlag, Santa Fe Institute Working Paper 99-02-015; adap-org/9903001.Google Scholar
  7. Derrida, B. and L. Peliti (1991). Evolution in a flat fitness landscape. Bull. Math. Biol. 53, 355–382.CrossRefGoogle Scholar
  8. Eigen, M. (1971). Self-organization of matter and the evolution of biological macromolecules. Naturwissen. 58, 465–523.CrossRefGoogle Scholar
  9. Eigen, M., J. McCaskill and P. Schuster (1989). The molecular quasispecies. Adv. Chem. Phys. 75, 149–263.Google Scholar
  10. Elena, S. F., V. S. Cooper and R. E. Lenski (1996). Punctuated evolution caused by selection of rare beneficial mutations. Science 272, 1802–1804.Google Scholar
  11. Ewens, W. J. (1979). Mathematical Population Genetics, volume 9 of Biomathematics, New York: Springer-Verlag.Google Scholar
  12. Fontana, W. and P. Schuster (1998). Continuity in evolution: On the nature of transitions. Science 280, 1451–1455.CrossRefGoogle Scholar
  13. Frauenfelder, H. (Ed.) (1997). Landscape Paradigms in Physics and Biology. Concepts, Structures and Dynamics (Papers originating from the 16th Annual International Conference of the Center for Nonlinear Studies. Los Alamos, NM, USA, 13–17 May 1996), Amsterdam: Elsevier Science, Published as a special issue of Physica D 107, 2–4 (1997).Google Scholar
  14. Gardiner, C. W. (1985). Handbook of Stochastic Methods, New York: Springer-Verlag.Google Scholar
  15. Gavrilets, S. (1997). Evolution and speciation on holey adaptive landscapes. Trends Ecol. Evol. 12, 307–312.CrossRefGoogle Scholar
  16. Gavrilets, S. (1999). A dynamical theory of speciation on holey adaptive landscapes. Am. Naturalist 154, 1–22.CrossRefGoogle Scholar
  17. Gavrilets, S. and J. Gravner (1997). Percolation on the fitness hypercube and the evolution of reproductive isolation. J. Theor. Biol. 184, 51–64.CrossRefGoogle Scholar
  18. Gould, S. J. and N. Eldredge (1977). Punctuated equilibria: The tempo and mode of evolution reconsidered. Paleobiology 3, 115–251.Google Scholar
  19. Harris, T. E. (1989). The Theory of Branching Processes, New York: Dover publications.Google Scholar
  20. Huynen, M., P. F. Stadler and W. Fontana (1996). Smoothness within ruggedness: The role of neutrality in adaptation. Proc. Natl. Acad. Sci. USA 93, 397–401.CrossRefGoogle Scholar
  21. Kauffman, S. A. and S. Levin (1987). Towards a general theory of adaptive walks in rugged fitness landscapes. J. Theo. Bio. 128, 11–45.MathSciNetGoogle Scholar
  22. Kimura, M. (1962). On the probability of fixation of mutant genes in a population. Genetics 47, 713–719.Google Scholar
  23. Kimura, M. (1964). Diffusion models in population genetics. J. Appl. Prob. 1, 177–232.zbMATHCrossRefGoogle Scholar
  24. Kimura, M. (1983). The Neutral Theory of Molecular Evolution, Cambridge University Press.Google Scholar
  25. Lande, R. (1985). Expected time for random genetic drift of a population between stable phenotype states. Proc. Natl. Acad. Sci. USA 82, 7641–7645.zbMATHMathSciNetCrossRefGoogle Scholar
  26. Macken, C. A. and A. S. Perelson (1989). Protein evolution in rugged fitness landscapes. Proc. Natl. Acad. Sci. USA 86, 6191–6195.MathSciNetCrossRefGoogle Scholar
  27. Maynard Smith, J. (1970). Natural selection and the concept of a protein space. Nature 225, 563–564.CrossRefGoogle Scholar
  28. Newman, C. M., J. E. Cohen and C. Kipnis (1985). Neo-darwinian evolution implies punctuated equilibrium. Nature 315, 400–401.CrossRefGoogle Scholar
  29. Newman, M and R. Engelhardt (1998). Effect of neutral selection on the evolution of molecular species. Proc. R. Soc. Lond. B. 256, 1333–1338.Google Scholar
  30. Nowak, M. and P. Schuster (1989). Error thresholds of replication in finite populations, mutation frequencies and the onset of Muller’s ratchet. J. Theor. Biol. 137, 375–395.Google Scholar
  31. Prügel-Bennett, A. and J. L. Shapiro (1994). Analysis of genetic algorithms using statistical mechanics. Phys. Rev. Lett. 72, 1305–1309.CrossRefGoogle Scholar
  32. Reidys, C. M., P. F. Stadler and P. Schuster (1997). Generic properties of combinatory maps—Neutral networks of RNA secondary structures. Bull. Math. Biol. 59, 339–397.CrossRefGoogle Scholar
  33. van Kampen, N. G. (1992). Stochastic Processes in Physics and Chemistry, Amsterdam: North-Holland.Google Scholar
  34. van Nimwegen, E. (1999). The statistical dynamics of epochal evolution, PhD thesis, University of Utrecht, Available electronically at:
  35. van Nimwegen, E. and J. P. Crutchfield (2000a). Optimizing epochal evolutionary search: Population-size dependent theory, in Machine Learning, Santa Fe Institute Working Paper 98-10-090. adap-org/9810004, in press.Google Scholar
  36. van Nimwegen, E. and J. P. Crutchfield (2000b). Optimizing epochal evolutionary search: Population-size independent theory, in the Special Issue on Evolutionary and Genetic Algorithms in Computational Mechanics and Engineering, D. Goldberg and K. Deb (Eds), Comput. Meth. Appl. Mech. Engng. 186, 171–194.Google Scholar
  37. van Nimwegen, E., J. P. Crutchfield and M. Mitchell (1997). Finite populations induce metastability in evolutionary search. Phys. Lett. A 229, 144–150.MathSciNetCrossRefGoogle Scholar
  38. van Nimwegen, E., J. P. Crutchfield and M. Mitchell (1999). Statistical dynamics of the Royal Road genetic algorithm, in the Special Issue on Evolutionary Computation, A. Eiben and G. Rudolph (Eds), Theoret. Comput. Sci. 229, 41–102Google Scholar
  39. Weisbuch, G. (1991). Complex Systems Dynamics: An Introduction to Automata Networks, volume 2 of Santa Fe Institute Studies in the Sciences of Complexity, Lecture Notes, Reading, MA: Addison-Wesley.Google Scholar
  40. Wright, S. (1932). The roles of mutation, inbreeding, crossbreeding and selection in evolution, in Proceedings of the Sixth International Congress of Genetics, Vol. 1, pp. 356–366.Google Scholar
  41. Wright, S. (1982). Character change, speciation, and the higher taxa. Evolution 36, 427–443.CrossRefGoogle Scholar

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© Society for Mathematical Biology 2000

Authors and Affiliations

  1. 1.Santa Fe InstituteSanta FeUSA
  2. 2.Bioinformatics GroupUniversity of UtrechtUtrechtThe Netherlands

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