The Annals of Probability

Rigorous results for the N K model

Abstract

Motivated by the problem of the evolution of DNA sequences, Kauffman and Levin introduced a model in which fitnesses were assigned to strings of 0's and 1's of length N based on the values observed in a sliding window of length $K+1$. When $K\ge 1$, the landscape is quite complicated with many local maxima. Its properties have been extensively investigated by simulation but until our work and the independent investigations of Evans and Steinsaltz little was known rigorously about its properties except in the case $K=N-1$. Here, we prove results about the number of local maxima, their heights and the height of the global maximum. Our main tool is the theory of (substochastic) Harris chains.

Article information

Source
Ann. Probab., Volume 31, Number 4 (2003), 1713-1753.

Dates
First available in Project Euclid: 12 November 2003

https://projecteuclid.org/euclid.aop/1068646364

Digital Object Identifier
doi:10.1214/aop/1068646364

Mathematical Reviews number (MathSciNet)
MR2016598

Zentralblatt MATH identifier
1049.60037

Citation

Durrett, Richard; Limic, Vlada. Rigorous results for the N K model. Ann. Probab. 31 (2003), no. 4, 1713--1753. doi:10.1214/aop/1068646364. https://projecteuclid.org/euclid.aop/1068646364

References

• Athreya, K. and Ney, P. E. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493--501.
• Durrett, R. (1995). Probability: Theory and Examples. Wadsworth & Brooks/Cole, Belmont, CA.
• Evans, S. N. and Steinsaltz, D. (2002). Estimating some features of $NK$ fitness landscapes. Ann. Appl. Probab. To appear.
• Flyvbjerg, H. and Lautrup, B. (1992). Evolution in rugged fitness landscapes. Phys. Rev. A 46 6714--6723.
• Kauffman, S. A. (1993). The Origins of Order: Self-Organization and Selection in Evolution. Oxford Univ. Press.
• Kauffman, S. A. and Levin, S. A. (1987). Towards a general theory of adaptive walks on rugged landscapes. J. Theoret. Biol. 128 11--45.
• Kesten, H. (1981). Percolation Theory for Mathematicians. Birkhäuser, Boston.
• Macken, C. A., Hagan, P. S. and Perelson, A. S. (1991). Evolutionary walks on rugged landscapes. SIAM J. Appl. Math. 51 799--827.
• Macken, C. A. and Perelson, A. S. (1989). Protein evolution on rugged landscapes. Proc. Natl. Acad. Sci. USA 86 6191--6195.
• Riesz, F. and Sz.-Nagy, B. (1990). Functional Analysis. Dover, New York.
• Tweedie, R. (1974). $R$-theory for Markov chains on a general state space. I. Solidarity properties and $R$-recurrent chains. Ann. Probab. 2 840--864.
• Weinberger, E. D. (1989). A more rigorous derivation of some properties of uncorrelated fitness landscapes. J. Theoret. Biol. 134 125--129.
• Weinberger, E. D. (1991). Local properties of Kauffman's $NK$ model: A tunably rugged energy landscape. Phys. Rev. A 44 6399--6413.