Why some fitness landscapes are fractal

Many biological and biochemical measurements, for example the "fitness" of a particular genome, or the binding affinity to a particular substrate, can be treated as a "fitness landscape", an assignment of numerical values to points in sequence space (or some other configuration space). As an alternative to the enormous amount of data required to completely describe such a landscape, we propose a statistical characterization, based on the properties of a random walk through the landscape and, more specifically, its autocorrelation function. Under assumptions roughly satisfied by two classes of simple model landscapes (the N-k model and the p-spin model) and by the landscape of estimated free energies of RNA secondary structures, this autocorrelation function, along with the mean and variance of individual points and the size of the landscape, completely characterize it. Having noted that these and other landscapes of estimated replication and degradation rates all have a well-defined correlation length, we propose a classification of landscapes depending on how the correlation length scales with the diameter of the landscape. The landscapes of some of the kinetic parameters of RNA molecules scale similarly to the model landscapes introduced into evolutionary studies from other fields, such as quadratic spin glasses and the traveling salesman problem, but the correlation length of RNA landscapes are considerably smaller. Nevertheless, both the model and some of the RNA landscapes satisfy a test of self-similarity proposed by Sorkin (1988).