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To gain insight into the reaction mechanism of activated processes, we introduce an exact approach for quantifying the topology of high-dimensional probability surfaces of the underlying dynamic processes. Instead of Morse indexes, we study the homology groups of a sequence of superlevel sets of the probability surface over high-dimensional configuration spaces using persistent homology. For alanine-dipeptide isomerization, a prototype of activated processes, we identify locations of probability peaks and connecting ridges, along with measures of their global prominence. Instead of a saddle point, the transition state ensemble (TSE) of conformations is at the most prominent probability peak after reactants/products, when proper reaction coordinates are included. Intuition-based models, even those exhibiting a double-well, fail to capture the dynamics of the activated process. Peak occurrence, prominence, and locations can be distorted upon subspace projection. While principal component analysis accounts for conformational variance, it inflates the complexity of the surface topology and destroys the dynamic properties of the topological features. In contrast, TSE emerges naturally as the most prominent peak beyond the reactant/product basins, when projected to a subspace of minimum dimension containing the reaction coordinates. Our approach is general and can be applied to investigate the topology of high-dimensional probability surfaces of other activated processes.

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.

A theoretical study showed that the base excision repair enzyme uracil-DNA glycosylase (UDG) exploits electrostatic interactions with backbone phosphate groups in the substrate for catalysis. Although experiments performed to test the calculated results confirmed the predicted importance of the -2, -1, and +1 phosphate groups, there was an apparent disagreement with regard to the +2 phosphate group. The calculations indicated that it made an important contribution, while experimentally, the effect of its deletion or neutralization was small. The +2 phosphate group interacts directly with an active site histidine (H148 in humans) in the crystal structure of UDG in complex with double-stranded (ds) DNA. We previously calculated that H148 has a strong anticatalytic effect due to its protonation, and here we use alchemical free energy simulations to estimate its site-specific pKa. We find that it is positively charged over the entire experimental pH range (4-10), so its deprotonation cannot compensate for deletion or neutralization of the +2 phosphate group. The free energy simulations are facilitated by an efficient charge-scaling procedure that allows quantitative correction for the implicit treatment of solvent far from the active site; improvements are made to that method to account carefully for differences in the truncation of electrostatic interactions in the contributing molecular-mechanical and continuum-electrostatic approaches. Additional simulations are used to demonstrate that the +2 phosphate group is fully solvent exposed in complexes with single-stranded DNA substrates like those used in the experiments. In contrast, it is well-structured and buried in the dsDNA complex used in the original simulations. Differences in solvent shielding thus account for the apparent lack of an effect observed experimentally upon neutralization or deletion of this group.

To interpret simulations of a complex system to determine the physical mechanism of a dynamical process, it is necessary to identify the small number of coordinates that distinguish the stable states from the transition states. We develop an automatic method for identifying these degrees of freedom from a database of candidate physical variables. In the method neural networks are used to determine the functional dependence of the probability of committing to a stable state (committor) on a set of coordinates, and a genetic algorithm selects the combination of inputs that yields the best fit. The method enables us to obtain the first set of coordinates that is demonstrably sufficient to specify the transition state of the C(7eq)--> alpha(R) isomerization of the alanine dipeptide in the presence of explicit water molecules. It is revealed that the solute-solvent coupling can be described by a solvent-derived electrostatic torque around one of the main-chain bonds, and the collective, long-ranged nature of this interaction accounts for previous failures to characterize this reaction.