Residue network in protein native structure belongs to the universality class of three dimensional critical percolation cluster

A single protein molecule is regarded as a contact network of amino-acid residues. Some studies have indicated that this network is a small world network (SWN), while other results have implied that this is a fractal network (FN). However, SWN and FN are essentially different in the dependence of the shortest path length on the number of nodes. In this paper, we investigate this dependence in the residue contact networks of proteins in native structures, and show that the networks are not SWN but FN. FN is generally characterized by several dimensions. Among them, we focus on three dimensions; the network topological dimension $D_c$, the fractal dimension $D_f$, and the spectral dimension $D_s$. We find that proteins universally yield $D_c \approx 1.9$, $D_f \approx 2.5$ and $Ds \approx 1.3$. These values are in surprisingly good coincidence with those in three dimensional critical percolation cluster. Hence the residue contact networks in the protein native structures belong to the universality class of three dimensional percolation cluster. The criticality is relevant to the ambivalent nature of the protein native structures, i.e., the coexistence of stability and instability, both of which are necessary for a protein to function as a molecular machine or an allosteric enzyme.