On the scale-free nature of RNA secondary structure networks

A network is scale-free if its connectivity density function is proportional to a power-law distribution. Scale-free networks may provide an explanation for the robustness observed in certain physical and biological phenomena, since the presence of a few highly connected hub nodes and a large number of small- degree nodes may provide alternate paths between any two nodes on average -- such robustness has been suggested in studies of metabolic networks, gene interaction networks and protein folding. A theoretical justification for why biological networks are often found to be scale-free may lie in the well-known fact that expanding networks in which new nodes are preferentially attached to highly connected nodes tend to be scale-free. In this paper, we provide the first efficient algorithm to compute the connectivity density function for the ensemble of all secondary structures of a user-specified length, and show both by computational and theoretical arguments that preferential attachment holds when expanding the network from length n to length n + 1 structures. Since existent power-law fitting software, such as powerlaw, cannot be used to determine a power-law fit for our exponentially large RNA connectivity data, we also implement efficient code to compute the maximum likelihood estimate for the power-law scaling factor and associated Kolmogorov-Smirnov p-value. Statistical goodness-of-fit tests indicate that one must reject the hypothesis that RNA connectivity data follows a power-law distribution. Nevertheless, the power-law fit is visually a good approximation for the tail of connectivity data, and provides a rationale for investigation of preferential attachment in the context of macromolecular folding.