  The adaptive evolution of a population under the influence of mutation and
selection is strongly influenced by the structure of the underlying fitness
landscape, which encodes the interactions between mutations at different
genetic loci. Theoretical studies of such landscapes have been carried out for
several decades, but only recently experimental fitness measurements
encompassing all possible combinations of small sets of mutations have become
available. The empirical studies have spawned new questions about the
accessibility of optimal genotypes under natural selection. Depending on
population dynamic parameters such as mutation rate and population size,
evolutionary accessibility can be quantified through the statistics of
accessible mutational pathways (along which fitness increases monotoncially),
or through the study of the basin of attraction of the optimal genotype under
greedy (steepest ascent) dynamics. Here we investigate these two measures of
accessibility in the framework of Kauffman's $LK$-model, a paradigmatic family
of random fitness landscapes with tunable ruggedness. The key parameter
governing the strength of genetic interactions is the number $K$ of interaction
partners of each of the $L$ sites in the genotype sequence. In general,
accessibility increases with increasing genotype dimensionality $L$ and
decreases with increasing number of interactions $K$. Remarkably, however, we
find that some measures of accessibility behave non-monotonically as a function
of $K$, indicating a special role of the most sparsely connected, non-trivial
cases K=1 and 2. The relation between models for fitness landscapes and spin
glasses is also addressed.
