Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities
Janko Gravner, Damien Pitman, and Sergey Gavrilets
Department of Mathematics, University of California, Davis, CA 95616,
Departments of Ecology and Evolutionary Biology
University of Tennessee, Knoxville, TN 37996, USA.
We study how correlations in the random fitness assignment may affect the structure of fitness landscapes, in three classes of fitness models.
The first is a phenotype space in which individuals are characterized
by a large number n of continuously varying traits. In a simple model of random fitness assignment, viable phenotypes are likely to form
a giant connected cluster percolating throughout the phenotype space
provided the viability probability is larger than
1/2^n. The second model explicitly describes genotype-to-phenotype
and phenotype-to-fitness maps, allows for neutrality at both phenotype and fitness levels, and results in a fitness landscape with tunable
correlation length. Here, phenotypic neutrality and correlation between fitnesses can reduce the percolation threshold, and correlations at
the point of phase transition between local and global are
most conducive to the formation of the giant cluster.
In the third class of models, particular combinations of alleles or
values of phenotypic characters are ``incompatible'' in the sense that the
resulting genotypes or phenotypes have zero fitness.
This setting can be viewed as a generalization of the canonical
Bateson-Dobzhansky-Muller model of speciation and is related to
K-SAT problems, prominent in computer science.
We analyze the conditions for the
existence of viable genotypes, their number, as well as the structure and the number of
connected clusters of viable genotypes.
We show that analysis based on expected values can easily lead to wrong conclusions,
especially when fitness correlations are strong.
We focus on pairwise incompatibilities between
diallelic loci, but we also address multiple alleles, complex incompatibilities, and
continuous phenotype spaces. In the case of diallelic loci, the number
of clusters is stochastically bounded and each cluster contains a very large sub-cube.
Finally, we demonstrate that the discrete NK model shares some signature properties
of models with high correlations.