Models of single-molecule experiments with periodic perturbations reveal hidden dynamics in RNA folding

Traditionally, microscopic fluctuations of molecules have been probed by measuring responses of an ensemble to perturbations. Now, single-molecule experiments are capable of following fluctuations without introducing perturbations. However, dynamics not readily sampled at equilibrium should be accessible to nonequilibrium single-molecule measurements. In a recent study [Qu, X. et al. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 6602-6607], the efficiency of fluorescence resonance energy transfer (FRET) between probes on the L18 loop and 3' terminus of the 260 nucleotide RNase P RNA from Bacillus stearothermophilus was found to exhibit complex kinetics that depended on the (periodically alternating) concentration of magnesium ions ([Mg2+]) in solution. Specifically, this time series was found to exhibit a quasi-periodic response to a square-wave pattern of [Mg2+] changes. Because these experiments directly probe only one of the many degrees of freedom in the macromolecule, models are needed to interpret these data. We find that Hidden Markov Models are inadequate for describing the nonequilibrium dynamics, but they serve as starting points for the construction of models in which a discrete observable degree of freedom is coupled to a continuously evolving (hidden) variable. Consideration of several models of this general form indicates that the quasi-periodic response in the nonequilibrium experiments results from the switching (back and forth) in positions of the minima of the effective potential for the hidden variable. This switching drives oscillation of that variable and synchronizes the population to the changing [Mg2+]. We set the models in the context of earlier theoretical and experimental studies and conclude that single-molecule experiments with periodic peturbations can indeed yield qualitatively new information beyond that obtained at equilibrium.