Moments of the Boltzmann distribution for RNA secondary structures

We here present a dynamic programming algorithm which is capable of calculating arbitrary moments of the Boltzmann distribution for RNA secondary structures. We have implemented the algorithm in a program called RNA-VARIANCE and investigate the difference between the Boltzmann distribution of biological and random RNA sequences. We find that the minimum free energy structure of biological sequences has a higher probability in the Boltzmann distribution than random sequences. Moreover, we show that the free energies of biological sequences have a smaller variance than random sequences and that the minimum free energy of biological sequences is closer to the expected free energy of the rest of the structures than that of random sequences. These results suggest that biologically functional RNA sequences not only require a thermodynamically stable minimum free energy structure, but also an ensemble of structures whose free energies are close to the minimum free energy.