Abstract
The effect of recombination on genotypes can be represented in the form of P-structures, i.e., a map from the set of pairs of genotypes to the power set of genotypes. The interpretation is that the P-structure maps the pair of parental genotypes to the set of recombinant genotypes which result from the recombination of the parental genotypes. A recombination fitness landscape is then a function from the genotypes in a P-structure to the real numbers. In previous papers we have shown that the eigenfunctions of (a matrix associated with) the P-structure provide a basis for the Fourier decomposition of arbitrary recombination landscapes.
Here we generalize this framework to include the effect of genotype frequencies, assuming linkage equilibrium. We find that the autocorrelation of the eigenfunctions of the population-weighted P-structure is independent of the population composition. As a consequence we can directly compare the performance of mutation and recombination operators by comparing the autocorrelations on the finite set of elementary landscapes. This comparison suggests that point mutation is a superior search strategy on landscapes with a low order and a moderate order of interaction p < n/3 (n is the number of loci). For more complex landscapes 1-point recombination is superior to both mutation and uniform recombination, but only if the distance among the interacting loci (defining length) is minimal.
Furthermore we find that the autocorrelation on any landscape is increasing as the distribution of genotypes becomes more extreme, i.e., if the population occupies a location close to the boundary of the frequency simplex. Landscapes are smoother the more biased the distribution of genotype frequencies is. We suggest that this result explains the paradox that there is little epistatic interaction for quantitative traits detected in natural populations if one uses variance decomposition methods while there is evidence for strong interactions in molecular mapping studies for quantitative trait loci.
Similar content being viewed by others
References
Altenberg, L. (1995). The Schema theorem and the Price’s theorem, in Foundations of Genetic Algorithms, D. Whitley and M. D. Vose (Eds), Cambridge, MA: MIT Press, pp. 23–49.
Altenberg, L. (1997). Fitness distance correlation analysis: an instructive counterexample, in Proceedings of the 7th International Conference on Genetic Algorithms (ICGA 97), T. Baeck (Ed.), San Fransisco, CA: Morgan Kaufmann, pp. 57–64.
Altenberg, L. and M. W. Feldman (1987). Selection, generalized transmission, and the evolution of modifier genes. I. The reduction principle. Genetics 117, 559–572.
Baatz, M. and G. P. Wagner (1997). Adaptive inertia caused by hidden pleiotropic effects. Theor. Pop. Biol. 51, 49–66.
Babel, L., I. V. Chuvaeva, M. Klin and D. V. Pasechnik (1995). Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. II. Program Implementation of the Weisfeiler-Leman Algorithm, TUM-M9701, Garching, Germany: TU München.
Babel, L., S. Baumann, M. Lüdecke and G. Tinhofer (1997). STABCOL: Graph Isomorphism Testing Based on the Weisfeiler-Leman Algorithm, TUM-M9702, Garching, Germany: TU München.
Bennett, J. H. (1954). On the theory of random mating. Ann. Eugen. 18, 311–317.
Berge, C. (1989). Hypergraphs, Amsterdam NL: Elsevier.
Besag, J. (1974). Spatial Interactions and the Statistical Analysis of Lattice Systems. Amer. Math. Monthly 81, 192–236.
Bhattacharya, R. N. and E. C. Waymire (1990). Stochastic Processes with Applications, New York: Wiley.
Bryant, E. H., S. A. McCommas and L. M. Combs (1986). The effect of an experimental bottleneck upon genetic variation in the house fly. Genetics 114, 1191–1211.
Bürger, R. (1986). Constraints for the evolution of functionally coupled characters: a nonlinear analysis of a phenotypic model. Evolution 40, 182–193.
Bürger, R. (1998). Mathematical properties of mutation-selection models. Genetika 102/103, 279–298.
Cheverud, J. M. and E. J. Routman (1996). Epistasis as a source of increased additive genetic variance at population bottlenecks. Evolution 50, 1042–1051.
Crow, J. F. and M. Kimura (1965). Evolution in sexual and asexual populations. Am. Nat. 99, 439–450.
Crow, J. F. and M. Kimura (1970). An Introduction to Population Genetics Theory, New York: Harper and Row.
Culberson, J. C. (1995). Mutation-crossover isomorphism and the construction of discriminating functions. Evol. Comp. 2, 279–311.
Eigen, M., J. McCaskill and P. Schuster (1989). The molecular quasispecies. Adv. Chem. Phys. 75, 149–263.
Eigen, M. and P. Schuster (1979). The Hypercycle, New York, Berlin: Springer.
Fisher, R. A. (1930). The Genetical Theory of Natural Selection, Oxford: Clarendon Press.
Flamm, C. (1998). RNA Folding Kinetics, PhD Thesis, University of Vienna.
Fogel, D. B. (1995). Evolutionary Computation, New York: IEEE Press.
Fontana, W., P. F. Stadler, E. G. Bornberg-Bauer, T. Griesmacher, I. L. Hofacker, M. Tacker, P. Tarazona, E. D. Weinberger and P. Schuster (1993). RNA folding and combinatory landscapes. Phys. Rev. E 47, 2083–2099.
Frazzetta, T. H. (1975). Complex Adaptations in Evolving Populations, Sunderland, MA: Sinauer.
García-Pelayo, R. and P. F. Stadler (1997). Correlation length, isotropy, and meta-stable states. Physica D 107, 240–254.
Gillespie, J. H. (1991). The Causes of Molecular Evolution, New York, Oxford: Oxford University Press.
Gitchoff, P. and G. P. Wagner (1996). Recombination induced hypergraphs: a new approach to mutation-recombination isomorphism. Complexity 2, 47–43.
Goodnight, C. J. (1987). On the effect of founder events on epistatic genetic variance. Evolution 41, 80–91.
Goodnight, C. J. (1988). Epistasis and the effect of founder events on the additive genetic variance. Evolution 42, 441–454.
Goodnight, C. J. (1995). Epistasis and the increase in additive genetic variance: implications for phase 1 of Wright’s shifting-balance process. Evolution 49, 502–511.
Happel, R. and P. F. Stadler (1996). Canonical approximation of fitness landscapes. Complexity 2, 53–58.
Hofbauer, J. and K. Sigmund (1988). Dynamical Systems and the Theory of Evolution, Cambridge: Cambridge University Press.
Holland, J. H. (1993). Adaptation in Natural and Artificial Systems, Cambridge, MA: MIT Press.
Hordijk, W. (1996). A measure of landscapes. Evol. Comput. 4, 335–360.
Hordijk, W. (1997). Correlation analysis of the synchronizing-CA landscape. Physica D 107, 255–264.
Hordijk, W. and P. F. Stadler (1998). Amplitude spectra of fitness landscapes. J. Complex Syst. 1, 39–66.
Huynen, M. A., P. F. Stadler and W. Fontana (1996). Smoothness within ruggedness: the role of neutrality in adaptation. Proc. Natl. Acad. Sci. (USA) 93, 397–401.
Jones, T. (1995a). One operator, one landscape, Technical Report #95-02-025, Santa Fe Institute.
Jones, T. (1995b). Evolutionary algorithms, fitness landscapes, and search, PhD thesis, University of New Mexico, Albuquerque, NM.
Kauffman, S. A. (1993). The Origin of Order, New York, Oxford: Oxford University Press.
Kauffman, S. A. and S. Levin (1987). Towards a general theory of adaptive walks on rugged landscapes. J. Theor. Biol. 128, 11–45.
Laubichler, M. (1997). Identifying units of selection: conceptual and methodological issues, PhD thesis, Yale University, New Haven, CT.
Lovász, L. (1975). Spectra of graphs with transitive groups. Periodica Math. Hung. 6, 191–195.
Lyubich, Y. I. (1992). Mathematical Structures in Population Genetics, Berlin: Springer.
Mézard, M., G. Parisi and M. Virasoro (1987). Spin Glass Theory and Beyond, Singapore: World Scientific.
Morgan, T. H. (1913). Heredity and Sex, New York: Columbia University Press.
Nagylaki, T. (1992). Introduction to Theoretical Population Genetics, Berlin: Springer.
Otto, S. P. and N. H. Barton (1997). The evolution of recombination: removing the limits to natural selection. Genetics 147, 879–906.
Papadimitrou, C. H. and K. Steiglitz (1982). Combinatorial Optimization: Algorithms and Complexity, Englewood Cliffs, NJ: Prentice-Hall.
Pearl, J. (1984). Heuristics: Intelligent Search Strategies for Computer Problem Solving, Reading, MA: Addison-Wesley.
Perelson, A. S. and S A. Kauffman (Eds) (1991). Molecular Evolution on Rugged Landscapes: Proteins, RNA, and the Immune System, Vol. 9, Santa Fe Institute Studies, Reading, MA: Addison-Wesley.
Provine, W. B. (1986). Sewall Wright and Evolutionary Biology, Chicago, London: University of Chicago Press.
Riedl, R. (1977). A systems-analytical approach to macroevolutionary phenomena. Q. Rev. Biol. 52, 351–370.
Robbins, R. B. (1918). Some applications of mathematics to breeding problems. III. Genetics 3, 375–389.
Schuster, P. and P. F. Stadler (1994). Landscapes: complex optimization problems and biopolymer structures. Computers & Chem. 18, 295–314.
Schuster, P., P. F. Stadler and A. Renner (1997). RNA structures and folding: from conventional to new issues in structure predictions. Curr. Opinions Struct. Biol. 7, 229–235.
Serre, J.-P. (1977). Linear Representations of Finite Groups, New York, Heidelberg, Berlin: Springer.
Stadler, P. F. (1995). Random walks and orthogonal functions associated with highly symmetric graphs. Discrete Math. 145, 229–238.
Stadler, P. F. (1995). Towards a theory of landscapes, in Complex Systems and Binary Networks, R. Lopéz-Peña, R. Capovilla, R. García-Pelayo, H. Waelbroeck and F. Zertuche (Eds), Berlin, New York: Springer, pp. 77–163.
Stadler, P. F. (1996). Landscapes and their correlation functions. J. Math. Chem. 20, 1–45.
Stadler, P. F. and R. Happel (1999). Random field models for fitness landscapes. J. Math. Biol. 38, 435–478.
Stadler, P. F. and G. P. Wagner (1998). The algebraic theory of recombination spaces. Evol. Comput. 5, 241–275.
Stadler, P. F. (1999). Spectral landscape theory, in Evolutionary Dynamics—Exploring the Interplay of Selection, Neutrality, Accident, and Function, J. P. Crutchfield and P. Schuster (Eds), Oxford University Press, in press.
Stephens, C. R. and H. Waelbroeck (1998). Effective degrees of freedom in genetic algorithms. Phys. Rev. E 57, 3251–3264.
Vose, M. D. and A. H. Wright (1998a). The simple genetic algorithm and the Walsh transform. Part I: theory. Evol. Comput. 6, 253–274.
Vose, M. D. and A. H. Wright (1998b). The simple genetic algorithm and the Walsh transform. Part II: The inverse. Evol. Comput. 6, 275–289.
Wagner, G. P. (1988a). The influence of variation and of developmental constraints on the rate of multivariate phenotypic evolution. J. Evol. Biol. 1, 45–66.
Wagner, G. P. (1988b). The significance of developmental constraints for phenotypic evolution by natural selection, in Population Genetics and Evolution, G. deJong (Ed.), Berlin, Heidelberg: Springer, pp. 222–229.
Wagner, G. P. and L. Altenberg (1995). Complex adaptations and the evolution of evolvability. Evolution 50, 967–976.
Wagner, G. P. and P. F. Stadler (1998). Complex adaptations and the structure of recombination spaces, in Algebraic Engineering, C Nehaniv and M Ito (Eds), Singapore: World Scientific, pp. 151–170. Proceedings of the Conference on Semi-Groups and Algebraic Engineering, University of Aizu, Japan.
Weinberger, E. D. (1990). Correlated and uncorrelated fitness landscapes and how to tell the difference. Biol. Cybern. 63, 325–336.
Weinberger, E. D. (1991). Fourier and Taylor series on fitness landscapes. Biol. Cybern. 65, 321–330.
Whitlock, M. C., P. C. Phillips, F. B. Moore and S. J. Tonsor (1995). Multiple fitness peaks and epistasis. Ann. Rev. Ecol. Syst. 26, 601–629.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stadler, P.F., Seitz, R. & Wagner, G.P. Population dependent Fourier decomposition of fitness landscapes over recombination spaces: Evolvability of complex characters. Bull. Math. Biol. 62, 399–428 (2000). https://doi.org/10.1006/bulm.1999.0167
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1006/bulm.1999.0167