We have developed a hidden Markov model and optimization procedure for photon-based single-molecule FRET data, which takes into account the trace-dependent background intensities. This analysis technique reveals an unprecedented amount of detail in the folding kinetics of the Diels–Alderase ribozyme. We find a multitude of extended (low-FRET) and compact (high-FRET) states. Five states were consistently and independently identified in two FRET constructs and at three Mg2+ concentrations. Structures generally tend to become more compact upon addition of Mg2+. Some compact structures are observed to significantly depend on Mg2+ concentration, suggesting a tertiary fold stabilized by Mg2+ ions. One compact structure was observed to be Mg2+-independent, consistent with stabilization by tertiary Watson–Crick base pairing found in the folded Diels–Alderase structure. A hierarchy of time scales was discovered, including dynamics of 10 ms or faster, likely due to tertiary structure fluctuations, and slow dynamics on the seconds time scale, presumably associated with significant changes in secondary structure. The folding pathways proceed through a series of intermediate secondary structures. There exist both compact pathways and more complex ones, which display tertiary unfolding, then secondary refolding, and, subsequently, again tertiary refolding.

Although Markov state models have proven to be powerful tools in resolving the complex features of biomolecular kinetics, the discretization of the conformational space has been a bottleneck since the advent of the method. A recently introduced variational approach, which uses basis functions instead of crisp conformational states, opened up a route to construct kinetic models in which the discretization error can be controlled systematically. Here, we develop and test a basis set for peptides to be used in the variational approach. The basis set is constructed by combining local residue-centered kinetic modes that are obtained from kinetic models of terminally blocked amino acids. Using this basis set, we model the conformational kinetics of two hexapeptides with sequences VGLAPG and VGVAPG. Six basis functions are sufficient to represent the slow kinetic modes of these peptides. The basis set also allows for a direct interpretation of the slow kinetic modes without an additional clustering in the space of the dominant eigenvectors. Moreover, changes in the conformational kinetics due to the exchange of leucine in VGLAPG to valine in VGVAPG can be directly quantified by comparing histograms of the basis set expansion coefficients.

We investigate barrier-crossing processes corresponding to collective hydrogen-bond rearrangements in liquid water using Markov state modeling techniques. The analysis is based on trajectories from classical molecular dynamics simulations and accounts for the full dynamics of relative angular and separation coordinates of water clusters and requires no predefined hydrogen bond criterium. We account for the complete 12-dimensional conformational subspace of three water molecules and distinguish five well-separated slow dynamic processes with relaxation times in the picosecond range, followed by a quasi-continuum spectrum of faster modes. By analysis of the Markov eigenstates, these processes are shown to correspond to different collective interchanges of hydrogen-bond donors and acceptors. Using a projection onto hydrogen-bond states, we also analyze the switching of one hydrogen bond between two acceptor water molecules and derive the complete transition network. The most probable pathway corresponds to a direct switch without an intermediate, in agreement with previous studies. However, a considerable fraction of paths proceeds along alternative routes that involve different intermediate states with short-lived alternative hydrogen bonds or weakly bound states.