Skip to main content
SpringerLink
Log in

Simon-Ando decomposability and fitness landscapes

  • Published:
Theory in Biosciences Aims and scope Submit manuscript

Abstract

In this paper, we investigate fitness landscapes (under point mutation and recombination) from the standpoint of whether the induced evolutionary dynamics have a “fast-slow” time scale associated with the differences in relaxation time between local quasi-equilibria and the global equilibrium. This dynamical hevavior has been formally described in the econometrics literature in terms of the spectral properties of the appropriate operator matrices by Simon and Ando (Econometrica 29 (1961) 111), and we use the relations they derive to ask which fitness functions and mutation/recombination operators satisfy these properties. It turns out that quite a wide range of landscapes satisfy the condition (at least trivially) under point mutation given a sufficiently low mutation rate, while the property appears to be difficult to satisfy under genetic recombination. In spite of the fact that Simon-Ando decomposability can be realized over fairly wide range of parameters, it imposes a number of restriction on which landscape partitionings are possible. For these reasons, the Simon-Ando formalism does not appear to be applicable to other forms of decomposition and aggregation of variables that are important in evolutionary systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Altenberg, L., 2000. Comments on “adaptive characters: natural subsystem decomposition of Darwinian systems” by G.P. Wagner and J. Kim. Unpublished memo.

  • Ancel, L., Fontana, W., 2000. Plasticity, evolvability, and modularity in RNA. Journal of Experimental Zoology (Molecular and Developmental Evolution) 288, 242–283.

    Article  CAS  Google Scholar 

  • Ando, A., Fisher, F., 1963. Near-decomposability: partition and aggregation, and the relevance of stability discussions International Economic Review 4, 53–67.

    Article  Google Scholar 

  • Buerger, R., 1998. Mathematical properties of mutation-selection models. Genetica 103, 279–298.

    Article  Google Scholar 

  • Buerger, R., 2001. The Mathematical Theory of Selection, Recombination, and Mutation. Wiley, New York, NY.

    Google Scholar 

  • Bulmer, M.G., 1985. The mathematical theory of quantitative genetics. Clarendon Press. Oxford, UK.

    Google Scholar 

  • Bush, G.L., 1994. Sympatric speciation: old wine in new bottles. Trends in Ecology and Evolution 9, 285–288.

    Article  Google Scholar 

  • Carter, A., 1997. Unpublisdhed notes in appendix to Wagner and Kim (1996).

  • Coyne, J.A., Barton, N.H., Turelli, M., 2000. Critique of Wright's shifting balance theory of evolution. Evolution 51, 643–671.

    Article  Google Scholar 

  • Courtois, P.J., 1978. Decomposability: Queueing and Computer System Applications. Academic Press, New York.

    Google Scholar 

  • Courtois, P. J., Semal, P., 1984. Bounds for the positive eigenvectors of nonnegative matrices and for their approximations by decomposition. Journal of the Association for Computing Machinery 31, 804–825.

    Google Scholar 

  • Crow, J., Kimura, M., 1970. An Introduction to Population Genetics Theory. Harper and Row, New York, NY.

    Google Scholar 

  • Culberson, J., 1993. Crossover versus mutations: fueling the debate. In: Forrest, S., (Ed.), ICGA Proceedings of the Fifth International Conference on Genetic Algorithms, Urbana, IL, p. 632.

  • Deuflhard, P., Huisinga, W., Fischer, A., Schutte, C., 1998. Identification of almost invariant aggregates in nearly uncoupled Markov chains. Linear Algebra and its Applications 315, 39–59.

    Article  Google Scholar 

  • Eigen, M., McCaskill, J., Schuster, P., 1989. The molecular quasis-species. Advances in Chemistry and Physics 75, 149–263.

    Article  CAS  Google Scholar 

  • Ewens, W.J., 1979. Mathematical Population Genetics. Springer, New York.

    Google Scholar 

  • Fontana, W., Griesmacher, T., Schnabl, W., Stadler, P.F., Schuster, P., 1991. Statistics of landscapes based on free energey, replication and degradation rate constants of RNA secondary structures.Chemical Montly 122, 795–819.

    CAS  Google Scholar 

  • Frenken, K., Marengo, L., Valente, M., 1999. Interdependences, near-decomposability and adaptation. In: Brenner (Ed.), Computational Techniques to Model Learning in Economics. pp. 145–165.

  • Gavrilets, S., Gravner, J., 1997. Percolation on the fitness hypercube and the evolution of reproductive isolation. Journal of Theoretical Biology 184, 51–64.

    Article  PubMed  CAS  Google Scholar 

  • Gitchoff, P., Wagner, G.P., 1996. Recombination induced hypergraphs: a new approach to mutation-recombination isomorphism. Complexity 2, 43–47.

    Article  Google Scholar 

  • Goldberg, D.E., 1989. Genetic algorithms in search, optimization, and machine learning. Addison Wesley, New York, NY.

    Google Scholar 

  • Guckenheimer, J., Holmes, P., 1983. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York, NY.

    Google Scholar 

  • Hermisson, J., Redner, O., Wagner, H., Baake, E., 2002. Mutation-selection balance: ancestry, load, and maximum principle. Theoretical Population Biology 62, 9–46.

    Article  PubMed  Google Scholar 

  • Holland, J., 1975. Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor, MI.

    Google Scholar 

  • Jones, B.L., 1978. Some principles governing selection in self-reproducing macromolecular systems. Journal of Mathematical Biology 6, 169–175.

    Article  PubMed  CAS  Google Scholar 

  • Jones, T., 1995. One operator, one landscape. Santa Fe Technical Reports 95-02-021.

  • Kafeety, H.D., Meyer, C.D., Stewart, W.J., 1992. A general framework for iterative aggregation/disaggregation methods. Proceedings of the Fourth Copper Mountain Conference on Iterative Methods, Boulder, CO.

  • Kauffman, S.A., 1993. The Origins of Order. Oxford University Press, Oxford, UK.

    Google Scholar 

  • Kondrashov, A.S., Mina, M., 1986. Sympatric speciation: when is it possible?. Biological Journal of the Linnean Society 27, 201–223.

    Article  Google Scholar 

  • Lewontin, R.C., 1970. The units of selection. Annual Review of Ecology and Systematics 1, 1–14.

    Article  Google Scholar 

  • Meyer, C.D., 1989. Stochastic complementation, uncoupling markov chains, and the theory of nearly reducible systems. SIAM Review 31, 240–272.

    Article  Google Scholar 

  • Rabani, Y., Rabinovich, Y., Sinclair, A., 1995. A computational view of population genetics. Proceedings of the 27th ACM Symposium on the Theory of Computing 83–92. Las Vegas, Nevada.

  • Rabinovich, Y., Sinclair, A., Wigderson, A., 1992. Quadratic dynamical systems. Proceedings of 33rd IEEE Symposium on Foundations of Computer Science, pp. 304–313.

  • Reidys, C.M., Stadler, P.F., 2001. Neutrality in fitness landscapes. Applied Mathematics and Computation 117, 321–350.

    Article  Google Scholar 

  • Rosen, R., 1978. Fundamentals of measurement and representation of natural systems. North Holland Press, New York, NY.

    Google Scholar 

  • Rumschitzki, D., 1987. Spectral properties of Eigen's evolution matrices. Bulletin of Mathematical Biology 24, 667–680.

    CAS  Google Scholar 

  • Schuster, P., Swetina J., 1988. Stationary mutant distributions and evolutionary optimization. Bulletin of Mathematical Biology 50, 635–660.

    PubMed  CAS  Google Scholar 

  • Shpak, M., Stadler, P.F., Wagner, G.P., Hermisson, J., 2004. Aggregation of variables and system decomposition: applications to fitness landscapes. Theory in Biosciences, in review.

  • Simon, H.A., Ando, A., 1961. Aggregation of variables in dynamic systems. Econometrica 29, 111–138.

    Article  Google Scholar 

  • Simon, H.A., 1972. The Sciences of the Artificial. MIT Press, Cambridge, MA.

    Google Scholar 

  • Simon, H.A., 2000. Near decomposability and the speed of evolution. unpublished manuscript.

  • Stadler, P.F., 1996. Landscapes and their correlation functions. Journal of Mathematical Chemistry 20, 1–45.

    Article  CAS  Google Scholar 

  • Stadler, P.F., Seitz, R., Wagner, G.P., 2000. Population dependent Fourier decomposition of fitness landscapes over recombination spaces. Bulletin of Mathematicl Biology 62, 399–428.

    Article  CAS  Google Scholar 

  • Stadler, P.F., Wagner, G.P., 1998. The algebraic theory of recombination spaces. Evolutionary Computation 5, 241–275.

    Google Scholar 

  • Strogatz, S., 1994. Nonlinear dynamics and Chaos. Addison Wesley, New York, NY.

    Google Scholar 

  • Thompson, C.J., McBride, J.L., 1974. On Eigen's theory of self-organization of matter and evolution of biological macromolecules. Mathematical Biosciences 27, 127–142.

    Google Scholar 

  • Turelli, M., Barton, N., 1994. Genetic and statistical analysis on selection for quantitative traits: what, me normal?. Genetics 138, 913–941.

    PubMed  CAS  Google Scholar 

  • van Nimwegen, E., Crutchfield, J.P., Mitchell, M., 1997. Statistical dynamics of the royal road algorithm. Theoretical Computer Science 229, 41–102.

    Article  Google Scholar 

  • Wagner, G.P., Booth, G., Bagheri-Chaichian, H., 1997. A population genetic theory of canalization. Evolution 51, 329–347.

    Article  Google Scholar 

  • Wagner, G.P., Laubichler, M., 2000. Character identification in evolutionary biology: the role of the organism. Theory of Biosceines 119, 20–40.

    Google Scholar 

  • Watson, R.A., 2002. Compositional evolution: interdisciplinary investigations in evolvability, modularity, and symbiosis. Ph.D. Dissertation, Brandeis University.

  • Weinberger, E.D., 1991. Fourier and Taylor series on fitness landscapes. Biological Cybernetics 65, 321–330.

    Article  Google Scholar 

  • Wilke, C., 2001a. Adaptive evolution on neutral networks Bulletin of Mathematical Biology 63, 715–730.

    Article  PubMed  CAS  Google Scholar 

  • Wilke, C., 2001b. Selection for fitness versus selection for robustness in RNA secondary structure folding. Evolution 55, 2412–2420.

    PubMed  CAS  Google Scholar 

  • Wimsatt, W., 1981. Units of selectio and the structure of the multi-level genome. In: Asquith, P.D., Grere, R.N. (Eds.), PSA-180, vol. 2. Philosophy of Science Association, pp. 122–183.

  • Wolfram, S., 1997.The Mathematica Book. Wolfram Media, Cambridge, UK.

    Google Scholar 

  • Wright, S., 1932. The roles of mutation, inbreeding, cross-breeding, and selection in evolution. Proceedings of the VI International Congress of Genetics vol. 1, Menasha, WI, pp. 356–366.

Download references

Author information

Authors and Affiliations

  1. Department of Ecology and Evolutionary Biology, University of Tennessee, 37996, Knoxville, TN, USA

    Max Shpak

  2. Institute of Bioinformatics, University of Leipzig, Kreuzstrasse 7b, D-04103, Leipzig, Germany

    Peter Stadler

  3. Department of Ecology and Evolutionary Biology, Yale University, 06520-8106, New Haven, CT, USA

    Gunter P. Wagner

  4. Department of Information and Computer Sciences, University of Hawaii, 96753, Manoa, HI, USA

    Lee Altenberg

Authors
  1. Max Shpak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Stadler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gunter P. Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lee Altenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Max Shpak.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shpak, M., Stadler, P., Wagner, G.P. et al. Simon-Ando decomposability and fitness landscapes. Theory Biosci. 123, 139–180 (2004). https://doi.org/10.1016/j.thbio.2004.04.001

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1016/j.thbio.2004.04.001

Keywords

Search

Navigation