The Annals of Applied Probability

Estimating some features of $NK$ fitness landscapes

Steven N. Evans and David Steinsaltz

Full-text: Open access

Abstract

Kauffman and Levin introduced a class of models for the evolution of hereditary systems which they called $NK$ fitness landscapes. Inspired by spinglasses, these models have the attractive feature of being tunable, with regard to both overall size (through the parameter $N$) and connectivity (through $K$). There are $N$ genes, each of which exists in two possible alleles [leading to a system indexed by $\{0,1\}^{N}$]; the fitness score of an allele at a given site is determined by the alleles of $K$ neighboring sites. Otherwise the fitnesses are as simple as possible, namely i.i.d., and the fitnesses of different sites are simply averaged.

Much attention has been focused on these fitness landscapes as paradigms for investigating the interaction between size and complexity in making evolution possible. In particular, the effect of the interaction parameter $K$ on the height of the global maximum and the heights of local maxima has attracted considerable interest, as well as the behavior of a "hill-climbing" walk from a random starting point. Nearly all of this work has relied on simulations, not on rigorous mathematics.

In this paper, some asymptotic features of $NK$ fitness landscapes are reduced to questions about eigenvalues and Lyapunov exponents. When $K$ is fixed, the expected number of local maxima grows exponentially with $N$ at a rate depending on the top eigenvalue of a kernel derived from the distribution of the fitnesses, and the average height of a local maximum converges to a value determined by the corresponding eigenfunction.

The global maximum converges in probability as $N \to \infty$ to a constant given by the top Lyapunov exponent for a system of i.i.d. max-plus random matrices, and this constant is nondecreasing with $K$. Various such quantities are computed for certain special cases when $K$ is small, and these calculations can, in principle, be extended to larger $K$.

Article information

Source
Ann. Appl. Probab., Volume 12, Number 4 (2002), 1299-1321.

Dates
First available in Project Euclid: 12 November 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1037125864

Digital Object Identifier
doi:10.1214/aoap/1037125864

Mathematical Reviews number (MathSciNet)
MR1936594

Zentralblatt MATH identifier
1040.60043

Subjects
Primary: 92D15: Problems related to evolution 60G60: Random fields 60G70: Extreme value theory; extremal processes 47A75: Eigenvalue problems [See also 47J10, 49R05]

Keywords
Genetics evolution eigenvalue Perron--Frobenius Lyapunov exponent max-plus algebra random field spinglass extreme value

Citation

Evans, Steven N.; Steinsaltz, David. Estimating some features of $NK$ fitness landscapes. Ann. Appl. Probab. 12 (2002), no. 4, 1299--1321. doi:10.1214/aoap/1037125864. https://projecteuclid.org/euclid.aoap/1037125864


Export citation

References

  • [1] BACCELLI, F. (1992). Ergodic theory of stochastic Petri networks. Ann. Probab. 20 375-396.
  • [2] DURRETT, R. and LIMIC, V. (2001). Rigorous results for the NK model. Preprint, Dept. Mathematics, Cornell Univ.
  • [3] HORN,R. A. and JOHNSON, C. R. (1985). Matrix Analy sis. Cambridge Univ. Press.
  • [4] HÖGNÄS, G. and MUKHERJEA, A. (1995). Probability Measures on Semigroups. Plenum Press, New York.
  • [5] JEAN-MARIE, A. (1999). Analy tical computation of Ly apunov exponents in stochastic event graphs. In Performance Evaluation of Parallel and Distributed Sy stems: Solution Methods II 309-341. Math. Centrum Wisk. Inform., Amsterdam.
  • [6] KAHANE, J.-P. (1985). Some Random Series of Functions 2nd ed. Cambridge Univ. Press.
  • [7] KAUFFMAN, S. (1993). The Origins of Order. Oxford Univ. Press.
  • [8] KAUFFMAN, S. A. and LEVIN, S. A. (1987). Towards a general theory of adaptive walks on rugged landscapes. J. Theoret. Biol. 128 11-45.
  • [9] SOLOW, D., BURNETAS, A., TSAI, M.-C. and GREENSPAN, N. S. (1999). Understanding and attenuating the complexity catastrophe in Kauffman's NK model of genome evolution. Complexity 5 53-66.
  • [10] STAUFFER, D. and JAN, N. (1994). Size effects in Kauffman ty pe evolution for rugged fitness landscapes. J. Theoret. Biol. 168 211-218.
  • [11] WEINBERGER, E. D. (1991). Local properties of Kauffman's NK model: a tunably rugged energy landscape. Phy s. Rev. A 44 6399-6413.
  • [12] WILKE, C. O. (1998). Evolution in time-dependent fitness landscapes. Technical Report 98-09, Institut für Neuroinformatik, Ruhr-Univ., Bochum.
  • [13] WRIGHT, S. (1932). The roles of mutation, inbreeding, crossbreeding, and selection in evolution. In Proceedings of the VI International Congress of Genetics 1 (D. Jones, ed.) 356-366. [Reprinted in Evolution (M. Ridley, ed.) (1997) Oxford Univ. Press.]
  • BERKELEY, CALIFORNIA 94720-3860 E-MAIL: evans@stat.berkeley.edu DEPARTMENT OF DEMOGRAPHY #2120 UNIVERSITY OF CALIFORNIA 2232 PIEDMONT AVENUE
  • BERKELEY, CALIFORNIA 94720-2120 E-MAIL: dstein@demog.berkeley.edu