Journal of Statistical Physics

, Volume 172, Issue 1, pp 226–278 | Cite as

Universality Classes of Interaction Structures for NK Fitness Landscapes

  • Sungmin Hwang
  • Benjamin Schmiegelt
  • Luca Ferretti
  • Joachim KrugEmail author


Kauffman’s NK-model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems. Genotypes are represented as sequences of L binary loci. The fitness assigned to a genotype is a sum of contributions, each of which is a random function defined on a subset of \(k \le L\) loci. These subsets or neighborhoods determine the genetic interactions of the model. Whereas earlier work on the NK model suggested that most of its properties are robust with regard to the choice of neighborhoods, recent work has revealed an important and sometimes counter-intuitive influence of the interaction structure on the properties of NK fitness landscapes. Here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible (that is, fitness-monotonic) mutational pathways. In particular, we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of L, and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large k. Moreover, we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in L for a large class of interaction structures that we characterize as locally bounded. Finally, we discuss the impact of the NK interaction structures on the dynamics of evolution using adaptive walk models.


Evolution Fitness landscapes Epistasis Adaptive walks 



We thank David Dean for useful discussions, and an anonymous reviewer for constructive remarks on the manuscript. JK acknowledges the kind hospitality of the MPI for Physics of Complex Systems (Dresden) and the Kavli Institute for Theoretical Physics (Santa Barbara) during the completion of the paper. This research was supported by DFG within SFB 680 Molecular basis of evolutionary innovations and SPP1590 Probabilistic structures in evolution, and in part by the National Science Foundation Grant No. NSF PHY-1125915, NIH Grant No. R25GM067110, and the Gordon and Betty Moore Foundation Grant No. 2919.01.


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Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUniversity of CologneCologneGermany
  2. 2.Integrative Biology GroupThe Pirbright InstituteWokingUK
  3. 3.LPTMS, Universite Paris-Sud 11OrsayFrance

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