J Stat Phys (2018) 172:226–278
https://doi.org/10.1007/s10955-018-1979-z
Universality Classes of Interaction Structures for NK
Fitness Landscapes
Sungmin Hwang
1,3
· Benjamin Schmiegelt
1
·
Luca Ferretti
2
· Joachim Krug
1
Received: 19 August 2017 / Accepted: 30 January 2018 / Published online: 13 February 2018
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract 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 ≤ 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 con-
cerning 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.
B
Joachim Krug
krug@thp.uni-koeln.de
Sungmin Hwang
hwang@thp.uni-koeln.de; sungmin.hwang@lptms.u-psud.fr
Benjamin Schmiegelt
schmiegb@thp.uni-koeln.de
Luca Ferretti
luca.ferretti@pirbright.ac.uk
1
Institute for Theoretical Physics, University of Cologne, Cologne, Germany
2
Integrative Biology Group, The Pirbright Institute, Woking, UK
3
Present Address: LPTMS, Universite Paris-Sud 11, Orsay, France
123
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