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Several formally exact expressions for quantum mechanical rate constants (i.e., bimolecular reactive cross sections suitably averaged and summed over initial and final states) are derived and their relation to one another analyzed. It is suggested that they may provide a useful means for calculating quantum mechanical rate constants accurately without having to solve the complete state-to-state quantum mechanical reactive scattering problem. Several ways are discussed for evaluating the quantum mechanical traces involved in these expressions, including a path integral evaluation of the Boltzmann operator/time propagator and a discrete basis set approximation. Both these methods are applied to a one-dimensional test problem (the Eckart barrier).

What are the atomic motions at enzymatic catalytic sites on the timescale of chemical change? Combined experimental and computational chemistry approaches take advantage of transitionstate analogs to reveal dynamic motions linked to transition-state formation. QM/MM transition path sampling from reactive complexes provides both temporal and dynamic information for barrier crossing. Fast (femtosecond to picosecond) dynamic motions provide essential links to enzymatic barrier crossing by local or promoting-mode dynamic searches through bondvibrational space. Transition-state lifetimes are within the femtosecond timescales of bond vibrations and show no manifestations of stabilized, equilibrated complexes. The slow binding and protein conformational changes (microsecond to millisecond) also required for catalysis are temporally decoupled from the fast dynamic motions forming the transition state. According to this view of enzymatic catalysis, transition states are formed by fast, coincident dynamic excursions of catalytic site elements, while the binding of transition-state analogs is the conversion of the dynamic excursions to equilibrated states.The catalytic power of enzymes to efficiently form transition states has been proposed, since the time of Pauling 1 , to involve an equilibrium between the Michaelis complex and a stabilized transition state, providing a rationale for tight binding of transition-state analogs 2 . Recent approaches explore enzymatic transition states from the viewpoints of molecular dynamics, geometry, electron distribution, the chemical lifetime of enzyme-bound transition-state species, and the motion of all atoms in the enzymatic ensemble as barrier crossing occurs [3][4][5][6][7][8][9][10] . Here, we summarize our perspective on the nature of barrier crossing with an emphasis on combined experimental and computational approaches to the nature of enzymatic transition states and dynamic motions. We conclude that the concept of a stabilized or thermodynamically equilibrated enzymatic transition state should be replaced by a new view in which fast protein dynamics dominates barrier crossing.Enzymatic catalysis presents daunting challenges of timescales. The catalytic cycle of most enzymes is on the scale of milliseconds, with typical turnovers of 1-100 ms 11 . Continuous quantum mechanical/molecular mechanics (QM/MM) calculations are typically limited to a few nanoseconds of protein-ligand motions, a small fraction of a single catalytic turnover 4 . Transition-state lifetimes have a lower limit of femtoseconds, corresponding to bond vern@aecom.yu.edu . Note: Supplementary information is available on the Nature Chemical Biology website.Reprints and permissions information is available online at http://npg.nature.com/reprintsandpermissions/.

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The transition path sampling method previously applied in our group to the reaction catalyzed by lactate dehydrogenase was used to generate a transition path ensemble for this reaction. Based on analysis of the reactive trajectories generated, important residues behind the active site were implicated in a compressional motion that brought the donor-acceptor atoms of the hydride closer together. In addition, residues behind the active site were implicated in a relaxational motion, locking the substrate in product formation. Although this suggested that the compressionrelaxation motions of these residues were important to catalysis, it remained unproven. In this work, we used committor distribution analysis to show that these motions are integral components of the reaction coordinate.protein dynamics ͉ catalysis T he relationship between enzymatic structure and function remains an area of intense research. Many studies have demonstrated that factors such as electrostatic and entropic effects are significant to enzyme function (1). The role of dynamics in catalysis, however, is less well understood. Protein motions such as conformational fluctuations on the millisecond timescale have been shown to influence substrate binding and the height of the barrier to reaction. The role of vibrational motions on the femtosecond to picosecond timescale and their connection to catalysis remain controversial. It has been suggested that the vibrational motions that exist within the protein are in equilibrium with the reaction and thermally averaged along the reaction coordinate (2, 3). But we and others have suggested that vibrational motions within the protein may in fact be coupled to the reaction coordinate (4). Previous work in our group has suggested the coupling of these vibrational motions to the reaction coordinate and has developed methods to study their effect on enzyme reactions (5).The search for these subpicosecond motions (''protein promoting vibrations'') was first applied to the enzyme horse liver alcohol dehydrogenase. This work suggested there existed motions, protein-promoting vibrations (PPVs), that coupled directly to the reaction coordinate and that were on the timescale of barrier crossing (6). The algorithm was then tested on another enzyme system lactate dehydrogenase (LDH), and similar PPVs were identified that, along with conformational fluctuations, helped explain the preference of the heart isoform to produce pyruvate and of muscle isoform to produce lactate (7).Further investigation into the mechanism of LDH was achieved with the transition path sampling (TPS) method (8). TPS is a computational method that can simulate rare events in complex systems. Using a Monte Carlo approach, a reactive path ensemble connecting reactants to products can be defined without prior knowledge of the reaction coordinate. This method allows mechanistic details to be identified from reactive paths generated with no bias (9). A reactive path ensemble was generated for LDH, and it was found that, in all reactive trajectori...

We have applied the Transition Path Sampling algorithm to the reaction catalyzed by the enzyme Lactate Dehydrogenase. This study demonstrates the ease of scaling Transition Path Sampling for applications on many degree of freedom systems, whose energy surface is a complex terrain of valleys and saddle points. As a Monte Carlo importance sampling method, transition path sampling is capable of surmounting barriers in path phase space and focuses simulation on the rare event of enzyme catalyzed atom transfers. Generation of the transition path ensemble, for this reaction, resolves a paradox in the literature in which some studies exposed the catalytic mechanism of hydride and proton transfer by lactate dehydrogenase to be concerted and others stepwise. Transition path sampling has confirmed both mechanisms as possible paths from reactants to products. With the objective to identify a generalized, reduced reaction coordinate, time series of both donor-acceptor distances and residue distances from the active site have been examined. During the transition from pyruvate to lactate, residues located behind the transferring hydride collectively compress toward the active site causing residues located behind the hydride acceptor to relax away. It is demonstrated that an incomplete compression/relaxation transition across the donor-acceptor axis compromises the reaction.

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