Spectral reconstruction of protein contact networks

In this work, we present a method for generating an adjacency matrix encoding a typical protein contact network. This work constitutes a follow-up to our recent work (Livi et al., 2015), whose aim was to estimate the relative contribution of different topological features in discovering of the unique properties of protein structures. We perform a genetic algorithm based optimization in order to modify the matrices generated with the procedures explained in (Livi et al., 2015). Our objective here is to minimize the distance with respect to a target spectral density, which is elaborated using the normalized graph Laplacian representation of graphs. Such a target density is obtained by averaging the kernel-estimated densities of a class of experimental protein maps having different dimensions. This is possible given the bounded-domain property of the normalized Laplacian spectrum. By exploiting genetic operators designed for this specific problem and an exponentially-weighted objective function, we are able to reconstruct adjacency matrices representing networks of varying size whose spectral density is indistinguishable from the target. The topological features of the optimized networks are then compared to the real protein contact networks and they show an increased similarity with respect to the starting networks. Subsequently, the statistical properties of the spectra of the newly generated matrices are analyzed by employing tools borrowed from random matrix theory. The nearest neighbors spacing distribution of the spectra of the generated networks indicates that also the (short-range) correlations of the Laplacian eigenvalues are compatible with those of real proteins.