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Simulation methods like transition path sampling (TPS) generate an abundance of information buried in the collection of reactive trajectories that they generate. However, only limited use has been made of this information, mainly for the identification of the reaction coordinate. The standard TPS tools have been designed for monitoring the progress of the system from reactants to products. However, the reaction coordinate does not contain all the information regarding the mechanism. In our earlier work, we have used TPS on enzymatic systems and have identified important motions in the reactant well that prepares the system for the reaction. Since these events take place in the reactant well, they are beyond the reach of standard TPS postprocessing methods. We present a simple scheme for identifying the common trends in enzymatic trajectories. This scheme was designed for a specific class of enzymatic reactions: it can be used for identifying motions that guide the system to reaction-ready conformations. We have applied it to two enzymatic systems that we have studied in the past, formate dehydrogenase and purine nucleoside phosphorylase, and we were able to identify interactions, far from the transition state, that are important for preparing the system for the reaction but that had been overlooked in earlier work.

The influence of protein motions on enzyme catalysis remains a topic of active discussion. Protein motions occur across a variety of time scales, from vibrational fluctuations in femtoseconds, to collective motions in milliseconds. There have been numerous studies that show conformational motions may assist in catalysis, protein folding, and substrate specificity. It is also known through transition path sampling studies that rapid promoting vibrations contribute to enzyme catalysis. Human purine nucleoside phosphorylase (PNP) is one enzyme that contains both an important conformational motion and a rapid promoting vibration. The slower motion in this enzyme is associated with a loop motion, that when open allows substrate entry and product release but closes over the active site during catalysis. We examine the differences between an unconstrained PNP structure and a PNP structure with constraints on the loop motion. To investigate possible coupling between the slow and fast protein dynamics, we employed transition path sampling, reaction coordinate identification, electric field calculations, and free energy calculations reported here.

We have analyzed the reaction catalyzed by formate dehydrogenase using transition path sampling. This system has recently received experimental attention using infrared spectroscopy and heavy-enzyme studies. Some of the experimental results point to the possible importance of protein motions that are coupled to the chemical step. We found that the residue Val123 that lies behind the nicotinamide ring occasionally comes into van der Waals contact with the acceptor and that in all reactive trajectories, the barrier-crossing event is preceded by this contact, meaning that the motion of Val123 is part of the reaction coordinate. Experimental results have been interpreted with a two-dimensional formula for the chemical rate, which cannot capture effects such as the one we describe.

Heavy enzyme isotope effects occur in proteins substituted with (2)H-, (13)C-, and (15)N-enriched amino acids. Mass alterations perturb femtosecond protein motions and have been used to study the linkage between fast motions and transition-state barrier crossing. Heavy enzymes typically show slower rates for their chemical steps. Heavy bacterial methylthioadenosine nucleosidases (MTANs from Helicobactor pylori and Escherichia coli) gave normal isotope effects in steady-state kinetics, with slower rates for the heavy enzymes. However, both enzymes revealed rare inverse isotope effects on their chemical steps, with faster chemical steps in the heavy enzymes. Computational transition-path sampling studies of H. pylori and E. coli MTANs indicated closer enzyme–reactant interactions in the heavy MTANs at times near the transition state, resulting in an improved reaction coordinate geometry. Specific catalytic interactions more favorable for heavy MTANs include improved contacts to the catalytic water nucleophile and to the adenine leaving group. Heavy bacterial MTANs depart from other heavy enzymes as slowed vibrational modes from the heavy isotope substitution caused improved barrier-crossing efficiency. Improved sampling frequency and reactant coordinate distances are highlighted as key factors in MTAN transition-state stabilization.

In recent years, there has been much discussion regarding the origin of enzymatic catalysis and whether including protein dynamics is necessary for understanding catalytic enhancement. An important contribution in this debate was made with the application of the vibrational Stark effect spectroscopy to measure electric fields in the active site. This provided a window on electric fields at the transition state in enzymatic reactions. We performed computational studies on two enzymes where we have shown that fast dynamics is part of the reaction mechanism and calculated the electric field near the bond-breaking event. We found that the fast motions that we had identified lead to an increase of the electric field, thus preparing an enzymatic configuration that is electrostatically favorable for the catalytic chemical step. We also studied the enzyme that has been the subject of Stark spectroscopy, ketosteroid isomerase, and found electric fields of a similar magnitude to the two previous examples.

In recent years, there has been encouraging progress in the engineering of enzymes that are designed to catalyze reactions not accelerated by natural enzymes. We tested the possibility of reengineering an existing enzyme by introducing a fast protein motion that couples to the reaction. Aromatic amine dehydrogenase is a system that has been shown to use a fast substrate motion as part of the reaction mechanism. We identified a mutation that preserves this fast motion but also introduces a favorable fast motion near the active site that did not exist in the native enzyme. Transition path sampling was used for the analysis of the atomic details of the mechanism.

The design of artificial enzymes is an emerging field of research. Although progress has been made, the catalytic proficiency of many designed enzymes is low compared to natural enzymes. Nevertheless, recently Hilvert et al. ( Nat. Chem. 2017, 9, 50-56) created a series of five artificial retro-aldolase enzymes via directed evolution, with the final variant exhibiting a rate comparable to the naturally occurring enzyme fructose 1,6 bisphosphate aldolase. We present a study of this system in atomistic detail that elucidates the effects of mutational changes on the chemical step. Transition path sampling is used to create ensembles of reactive trajectories, and committor analysis is used to identify the stochastic separatrix of each ensemble. The application of committor distribution analysis to constrained trajectories allows the identification of changes in important protein motions coupled to reaction across the generated series of the artificial retro-aldolases. We observed two different reaction mechanisms and analyzed the role of the residues participating in the reaction coordinate of each enzyme. However, only in the most evolved variant we identified a fast motion that promotes catalysis, suggesting that this rate promoting vibration was introduced during directed evolution. This study provides further evidence that protein dynamics must be taken into account in designing efficient artificial enzymes.

The relevance of sub-picosecond protein motions to the catalytic event remains a topic of debate. Heavy enzymes (isotopically substituted) provide an experimental tool for bond-vibrational links to enzyme catalysis. A recent transition path sampling study with heavy purine nucleoside phosphorylase (PNP) characterized the experimentally observed mass-dependent slowing of barrier crossing (Antoniou, D.; Ge, X.; Schramm, V. L.; Schwartz, S. D. J. Phys. Chem. Lett. 2012, 3, 3538). Here we computationally identify second-sphere amino acid residues predicted to influence the freedom of the catalytic site vibrational modes linked to heavy enzyme effects in PNP. We mutated heavy and light PNPs to increase the catalytic site vibrational freedom. Enzymatic barrier-crossing rates were converted from mass-dependent to mass-independent as a result of the mutations. The mutagenic uncoupling of femtosecond motions between catalytic site groups and reactants decreased transition state barrier crossing by 2 orders of magnitude, an indication of the femtosecond dynamic contributions to catalysis.

The definition of a transition state on an individual reactive trajectory is made via a committor analysis. In the past, the bottleneck definition has often been applied in configuration space. This is an approximation, and in order to expand this definition, we are revisiting an enzyme in which we had identified a fast subpicosecond motion that makes the reaction possible. First we used a time-series analysis method to identify the exact time when this motion initiates donor-acceptor compression. Then we modified the standard committor analysis of transition path sampling to identify events in phase space and found that there is a dividing surface in phase space significantly earlier than the configurationally defined transition-state crossing.

Transition path-sampling calculations with several enzymes have indicated that local catalytic site femtosecond motions are linked to transition state barrier crossing. Experimentally, femtosecond motions can be perturbed by labeling the protein with amino acids containing 13C, 15N, and nonexchangeable 2H. A slowed chemical step at the catalytic site with variable effects on steady-state kinetics is usually observed for heavy enzymes. Heavy human purine nucleoside phosphorylase (PNP) is slowed significantly (kchemlight/kchemheavy = 1.36). An asparagine (Asn243) at the catalytic site is involved in purine leaving-group activation in the PNP catalytic mechanism. In a PNP produced with isotopically heavy asparagines, the chemical step is faster (kchemlight/kchemheavy = 0.78). When all amino acids in PNP are heavy except for the asparagines, the chemical step is also faster (kchemlight/kchemheavy = 0.71). Substrate-trapping experiments provided independent confirmation of improved catalysis in these constructs. Transition path-sampling analysis of these partially labeled PNPs indicate altered femtosecond catalytic site motions with improved Asn243 interactions to the purine leaving group. Altered transition state barrier recrossing has been proposed as an explanation for heavy-PNP isotope effects but is incompatible with these isotope effects. Rate-limiting product release governs steady-state kinetics in this enzyme, and kinetic constants were unaffected in the labeled PNPs. The study suggests that mass-constrained femtosecond motions at the catalytic site of PNP can improve transition state barrier crossing by more frequent sampling of essential catalytic site contacts.

Escherichia coli dihydrofolate reductase (ecDHFR) is used to study fundamental principles of enzyme catalysis. It remains controversial whether fast protein motions are coupled to the hydride transfer catalyzed by ecDHFR. Previous studies with heavy ecDHFR proteins labeled with (13)C, (15)N, and nonexchangeable (2)H reported enzyme mass-dependent hydride transfer kinetics for ecDHFR. Here, we report refined experimental and computational studies to establish that hydride transfer is independent of protein mass. Instead, we found the rate constant for substrate dissociation to be faster for heavy DHFR. Previously reported kinetic differences between light and heavy DHFRs likely arise from kinetic steps other than the chemical step. This study confirms that fast (femtosecond to picosecond) protein motions in ecDHFR are not coupled to hydride transfer and provides an integrative computational and experimental approach to resolve fast dynamics coupled to chemical steps in enzyme catalysis.

The mechanisms involved in enzymatic hydride transfer have been studied for years, but questions remain due, in part, to the difficulty of probing the effects of protein motion and hydrogen tunneling. In this study, we use transition path sampling (TPS) with normal mode centroid molecular dynamics (CMD) to calculate the barrier to hydride transfer in yeast alcohol dehydrogenase (YADH) and human heart lactate dehydrogenase (LDH). Calculation of the work applied to the hydride allowed for observation of the change in barrier height upon inclusion of quantum dynamics. Similar calculations were performed using deuterium as the transferring particle in order to approximate kinetic isotope effects (KIEs). The change in barrier height in YADH is indicative of a zero-point energy (ZPE) contribution and is evidence that catalysis occurs via a protein compression that mediates a near-barrierless hydride transfer. Calculation of the KIE using the difference in barrier height between the hydride and deuteride agreed well with experimental results.

Recent studies of the bacterial enzymes EcMTAN and VcMTAN showed that they have different binding affinities for the same transition state analogue. This was surprising given the similarity of their active sites. We performed transition path sampling simulations of both enzymes to reveal the atomic details of the catalytic chemical step, which may be the key for explaining the inhibitor affinity differences. Even though all experimental data would suggest the two enzymes are almost identical, subtle dynamic differences manifest in differences of reaction coordinate, transition state structure, and eventually significant differences in inhibitor binding. Unlike EcMTAN, VcMTAN has multiple distinct transition states, which is an indication that multiple sets of coordinated protein motions can reach a transition state. Reaction coordinate information is only accessible from transition path sampling approaches, since all experimental approaches report averages. Detailed knowledge could have a significant impact on pharmaceutical design.

Significance Protein design from first principles is developing rapidly for structural elements, binding domains, and protein–protein interactions. Design of structural elements to generate predictable changes in the fundamental properties of enzymatic catalysis remains challenging, requiring input from protein dynamics and the quantum chemical effects of transition state formation and barrier crossing. Human purine nucleoside phosphorylase (PNP) has a well-understood mechanism of catalysis, which includes rapid protein dynamics. PNP was used in a design program to alter the catalytic-site response to heavy-atom substitution in the enzyme protein. Native PNP exhibits slowed chemistry when made heavy with 2 H, 13 C, and 15 N. We succeeded in designing a second-sphere mutation with improved promoting vibrations to catalyze faster chemistry in response to heavy PNP.

It has been accepted for many years that functionally important motions are crucial to binding properties of ligands in such molecules as hemoglobin and myoglobin. In enzymatic reactions, theory and now experiment are beginning to confirm the importance of motions on a fast (ps) time scale in the chemical step of the catalytic process. What is missing is a clear physical picture of how slow conformational fluctuations are related to the fast motions that have been identified as crucial. This paper presents a theoretical analysis of this issue for human heart lactate dehydrogenase. We will examine how slow conformational motions bring the system to conformations that are distinguished as catalytically competent because they favor specific fast motions.

It is now widely accepted that there is a class of enzymatic proton transfer reactions, which proceed through quantum tunneling. In a series of papers we have argued that some experimental features of these reactions can be explained by assuming the presence of a "rate-promoting" vibration which brings donor and acceptor closer together, thus leading to rate enhancement. There has never been a study of this effect for classical systems. We used transition path sampling to study the equivalent classical problem and found a complicated dynamical behavior that cannot be captured by transition state theory. Slow promoting vibrations lead to reactive trajectories that overshoot the saddle point, but on the other hand the short period of fast oscillations allows the reactants to stay only briefly in a low-barrier regime. There is a competition between these effects, which results to an intermediate value for the frequency of the rate-promoting vibration that is optimal for enhancing the rate.

The generalized Langevin equation encapsulates in the friction kernel the influence of the environment on the reaction coordinate. We have decomposed the Langevin equation in a Fourier transform representation and analyzed how different Fourier components contribute to the friction kernel. We identified that a long-wavelength contribution to the friction kernel originates from collective solvent motions, and we compared it with contributions coming from short-range scattering of the complex with the surrounding solvent atoms.

This review describes studies of particular enzymatically catalyzed reactions to investigate the possibility that catalysis is mediated by protein dynamics. That is, evolution has crafted the protein backbone of the enzyme to direct vibrations in such a fashion to speed reaction. The review presents the theoretical approach we have used to investigate this problem, but it is designed for the nonspecialist. The results show that in alcohol dehydrogenase, dynamic protein motion is in fact strongly coupled to chemical reaction in such a way as to promote catalysis. This result is in concert with both experimental data and interpretations for this and other enzyme systems studied in the laboratories of the two other investigators who have published reviews in this issue.

The role of protein dynamics on different time scales in enzyme catalysis remains an area of active debate. The connection between enzyme dynamics on the femtosecond time scale and transition state formation has been demonstrated in human purine nucleoside phosphorylase (PNP) through the study of a mass-altered enzyme. Isotopic substitution in human PNP (heavy PNP) decreased the rate of on-enzyme chemistry but did not alter either the transition state structure or steady-state kinetic parameters. Here we investigate the underlying atomic motions associated with altered barrier crossing probability for heavy PNP. Transition path sampling was employed to illuminate the molecular differences between barrier crossing in light and heavy enzymes. The mass effect is apparent in promoting vibrations that polarize the N-ribosidic bond, and that promote the stability of the purine leaving group. These motions facilitate barrier crossing.

In an earlier paper we developed a collective motion description of atomic liquids, by expressing the quantum Hamiltonian in terms of rotated components of the density Fourier transform, which allowed the treatment of potential interaction terms without approximations. In the present paper, we further explore the physical content of our formalism and we show how it can be used for calculations in a real system.

The standard view of the origin of the catalytic properties of enzymes focuses on the binding energy differences between the ground state and the transition state arising from the arrangement of residues in the active site (i.e., statics). There is an alternative view that suggests that protein motions (i.e., dynamics) might play a role in catalysis. Klinman and co-workers recently published (Kohen, A.; Cannio, R.; Bartolucci, S.; Klinman, J. Nature 1999, 399, 496−499) findings on rate measurements in thermophilic alcohol dehydrogonase (ADH). At lower temperatures (below 30 °C), this enzyme undergoes a transition to a more rigid structure, and it was found that the corresponding apparent activation energy increases and the primary kinetic isotope effect (KIE) increases and becomes temperature-dependent. Explaining these results presents a challenge to theory. We show that a model of the reaction coordinate for the rate-determining step, coupled to an enzymatic environment and a specific strongly coupled a...

We study the intramolecular proton transfer in the amino acid glycine in aqueous solution. We show that this system is an example of nonequilibrium solvation, where the proton-transfer step is fast and the solvent relaxes afterward. We show how the physical picture of equilibrium vs nonequilibrium solvation arises naturally within the framework of the quantized Zwanzig Hamiltonian.

In a recent paper [D. Antoniou and S. D. Schwartz, J. Chem. Phys. 110, 465 (1999)] we calculated the reaction rate for a proton transfer reaction in liquid methyl chloride. In that work, we used a spectral density obtained from a molecular dynamics simulation as input to a quantum Zwanzig Hamiltonian which we solved using our exponential resummation method. In the present paper we perform a similar calculation, allowing for a position dependent friction using the method of G. Haynes, G. Voth, and E. Pollak [J. Chem. Phys. 101, 7811 (1994)]. Compared with the results of our previous work, we found that including spatial dependence to the friction led to enhancement of the reaction rate and to reduction of the H/D kinetic isotope effect.

We present a quantum study of a proton transfer reaction AH–B⇌A−–H+B in liquid methyl chloride, where the AH–B complex corresponds to phenol-amine. We use the same intramolecular potentials that were used in two earlier studies of this system [H. Azzouzz and D. Borgis, J. Chem. Phys. 98, 7361 (1993); S. Hammes-Schiffer and J. C. Tully, J. Chem. Phys. 101, 4657 (1994).] The former study employed a Landau-Zener approach and a molecular dynamics centroid method, while the latter a surface-hopping method. These studies obtained results that differ by an order of magnitude. In the present work, we first performed a molecular dynamics simulation to obtain the spectral density, which was then used as an input to the method we have developed for the study of the quantum Kramers problem [S. D. Schwartz, J. Chem. Phys. 105, 6871 (1996)]. Thus, in this work both the reaction coordinate and the bath are treated quantum mechanically.

The variability with temperature of spectral densities and rates calculated with quantum activated rate theory is investigated. Classical spectral densities at two temperatures are computed via molecular dynamics for a model of proton transfer in methyl chloride. In addition, quantum dynamics is computed for spectral densities which artificially boost variability at low frequency. We find significant variation in computed spectral densities at moderate frequency. These variations, however, have little effect on overall computed quantum dynamics. In contradistinction, artificial variation in spectral densities at the lowest frequencies can generate fairly significant effects on quantum dynamics. Detailed flux correlation function calculations are presented which illustrate this phenomenon.

We present a calculation of the rate of synchronous double proton transfer in benzoic acid crystals. Experiments on these systems have been performed over a wide range of temperatures (roughly 10–400 °K). Even though the energetic barrier for proton transfer is rather high, the observed activation energy is low, while kinetic isotope experiments seem to indicate classical transfer. The system exhibits significant quantum character even at high temperatures and we show that the observed low activation energies can be reproduced assuming that the reaction is “assisted” by a low-frequency intramolecular mode, as has been suggested in different contexts by Benderskii [V. A. Benderskii, S. Yu. Grebenshchikov, and G. V. Mil’nikov, Chem. Phys. 194, 1 (1995)], Hynes [D. Borgis and J. Hynes, J. Chem. Phys. 94, 3619 (1991)] and Silbey [A. Suarez and R. Silbey, J. Chem. Phys. 94, 4809 (1991)]. We use our previous work on the quantum Kramers problem to perform a fully quantum calculation that incorporates symmetric coupling to the intramolecular mode and coupling to the condensed environment to all orders. We calculate the activation energies for hydrogen and deuterium transfer and we show that our results are in quantitative agreement with the experiment.

In the gas phase, tunneling reaction rates can be significantly enhanced if the reaction coordinate is symmetrically coupled to a harmonic oscillation, as has been emphasized by Benderskii and co-workers [Adv. Chem. Phys. 88, 1 (1994)]. This is due to the fact that the symmetric coupling leads to modulation of the barrier height. Similar effects have been observed in reactions in model condensed phase studies, as in the Hamiltonians that have been studied by Borgis and Hynes [J. Chem. Phys. 94, 3619 (1991)] and Suarez and Silbey [J. Chem. Phys. 94, 4809 (1991)]. All of these works assume that tunneling proceeds from the ground state. In this paper, we use the exponential resummation technique that we used in our recent work on the quantum Kramers problem, to study the case when there can be excitations to higher states and activated transmission over a barrier. We present a general methodology to exactly include direct coupling between the reaction coordinate and the symmetrically coupled promoting vibrat...

We propose an interpretation of the experimental findings of Klinman and coworkers [Cha, Y., Murray, C. J. & Klinman, J. P. (1989) Science 243, 1325–1330; Grant, K. L. & Klinman, J. P. (1989) Biochemistry 28, 6597–6605; and Bahnson, B. J. & Klinman, J. P. (1995) Methods Enzymol. 249, 373–397], who showed that proton transfer reactions that are catalyzed by bovine serum amine oxidase proceed through tunneling. We show that two different tunneling models are consistent with the experiments. In the first model, the proton tunnels from the ground state. The temperature dependence of the kinetic isotope effect is caused by a thermally excited substrate mode that modulates the barrier, as has been suggested by Borgis and Hynes [Borgis, D. & Hynes, J. T. (1991) J. Chem. Phys. 94, 3619–3628]. In the second model, there is both over-the-barrier transfer and tunneling from excited states. Finally, we propose two experiments that can distinguish between the possible mechanisms.

We have included nonadiabatic effects in the calculation of the dynamical evolution of a system where a quantum particle in a double well is coupled to a classical oscillator. By performing an exponential resummation of the evolution operator we have included ‘‘polarization’’ effects (similar to the self‐energy corrections for an electron that moves in a polarizable medium) which lead to a renormalization of the energy of the quantum particle.

In this paper we report quantum calculations of the survival probability in linear hydrocarbon chains. We have performed both adiabatic gauge transform calculations and calculations that include corrections beyond the adiabatic approximation. We have managed to perform intermediate steps of the calculations analytically. We require the initial basis set expansion and final summations to be performed numerically. The corrections beyond the adiabatic approximation are shown to be small for this system for multiple time step calculations and large for single time step calculations. We have proved an identity that allows the extension of the calculations for HC2 to longer chains at little computational cost. In particular, we have proved that the quantum solution for any linear hydrocarbon chain can be obtained from the solution of a problem with 3 degrees of freedom. We have performed multi‐step adiabatic calculations for HC2 and HC6 that converge at up to 35–40 fs. We have devised a simple diagrammatic scheme that summarizes our method in a very compact form. Finally, we propose an alternative strategy of calculation that might lead to very fast solutions of the quantum dynamics of this system.

We have studied atomic motions during the chemical reaction catalysed by the enzyme dihydrofolate reductase of Escherichia coli (EcDHFR), an important enzyme for nucleic acid synthesis. In our earlier work on the enzymes human lactate dehydrogenase and purine nucleoside phosphorylase, we had identified fast sub-ps motions that are part of the reaction coordinate. We employed Transition Path Sampling (TPS) and our recently developed reaction coordinate identification methodology to investigate if such fast motions couple to the reaction in DHFR on the barrier-crossing timescale. While we identified some protein motions near the barrier crossing event, these motions do not constitute a compressive promoting vibration, and do not appear as a clearly identifiable protein component in reaction.

After many decades of investigation, the manner in which enzymes increase the rate of chemical reactions, at times by a factor of 10(17) compared to the rate of the corresponding solution phase reaction, is still opaque. A topic of significant discussion in the literature of the past 5-10 years has been the importance of protein dynamics in this process. This Feature Article will discuss the authors' work on this still controversial topic with focus on both methodology and application to real systems. The end conclusion of this work has been that for specific enzymes under study protein dynamics on both rapid time scales of barrier crossing (termed promoting vibrations by the authors) and of conformational fluctuations are central to the function of biological catalysts. In another enzyme we will discuss, the results are far less clear. The manner of the coupling of chemistry to protein dynamics has deep implications for protein architecture, both natural and created, and recent results reinforce the complexity of the protein form that has evolved to support these functions.

We propose a new method for analyzing an ensemble of transition states in order to extract components of the reaction coordinate. We use the kernel Principal Component Analysis (kPCA), which is a generalization of the ordinary PCA that does not make a linearization approximation We applied this method to a TPS study of human LDH we had previously published [S. Quaytman and S. D. Schwartz, Proc. Natl. Acad. Sci. USA 104, 12253 (2007)] and extracted a reasonable representation for the reaction coordinate.

We propose a method for incorporating nuclear quantum effects in transition path sampling studies of systems that consist of a few degrees of freedom that must be treated quantum mechanically, while the rest are classical-like. We used the normal mode centroid method to describe the quantum subsystem, which is a method that is not CPU intensive but still reasonably accurate. We applied this mixed centroid/classical transition path sampling method to a model system that has nontrivial quantum behavior, and showed that it can capture the correct quantum dynamical features.

We present a new approach to the identification of degrees of freedom which comprise a reaction coordinate in a complex system. The method begins with the generation of an ensemble of reactive trajectories. Each trajectory is analyzed for its equicommittor position or transition state; then the transition state ensemble is identified as the stochastic separatrix. Numerical analysis of the points along the separatrix for variability of coordinate location correctly identifies the components of the reaction coordinate in a test system of a double well coupled to a promoting vibration and a bath of linearly coupled oscillators.

The authors developed a new method for calculating the quantum evolution of multidimensional systems, for cases in which the system can be assumed to consist of a quantum subsystem and a bath subsystem of heavier atoms. The method combines two ideas: starting from a simple frozen Gaussian description of the bath subsystem, then calculate quantum corrections to the propagation of the quantum subsystem. This follows from recent work by one of them, showing how one can calculate corrections to approximate evolution schemes, even when the Hamiltonian that corresponds to these approximate schemes is unknown. Then, they take the limit in which the width of the frozen Gaussians approaches zero, which makes the corrections to the evolution of the quantum subsystem depend only on classical bath coordinates. The test calculations they present use low-dimensional systems, in which comparison to exact quantum dynamics is feasible.

The catalytic site of the homotrimeric enzyme human purine nucleoside phosphorylase enzyme (hPNP) features residue F200 and the 241-265 loop directly skirting the purine base and a residue belonging to the adjacent monomer, F159, immediately conterminous to the ribosyl moiety. Crystallographic B-factors of apo human purine nucleoside phosphorylase, and hPNP complexed with substrate/transition state (TS) analogues, show that residue E250 is the centroid of a highly mobile loop region. Furthermore, superimposition of apo hPNP and hPNP complexed with TS analogue Immucillin-H shows a tightening of the active site, caused by the ligand-dependent 241-265 loop rearrangement taking place upon substrate/inhibitor binding, suggesting a putative dynamic role of the loop in binding/catalysis. However, crystallographic structures reveal only average atomic positions, and more detailed information is needed to discern the dynamic behavior of hPNP. The Essential Dynamics (ED) method is used here to investigate the existence of correlated motions in hPNP and consequently proposes mutagenesis assays to estimate the relative importance of these motions in the phosphorolytic efficiency of the reaction catalyzed by hPNP. We compare the concerted motions obtained from multiple molecular dynamics simulations of apo and Michaelis complex of hPNP both in vacuo and in solution. The results of the principal component analysis for the apo hPNP indicate the existence of strong correlations predominantly in the vicinity of residue F159. However, for the Michaelis complex, concerted motions are seen mostly around both active site residue F200 and loop residue E250. Additionally, for a simulation depicting the relaxation of tight complexed hPNP with a TS analogue, toward its relaxed apo form (after removal of the TS analog), a combination of the apo hPNP and Michaelis complex motions is found, with prominent concerted modes centered around neighboring residues F159, F200, and E250. Finally, we probed the extent to which these concerted motions bear an intrinsic catalytic role by performing experimental site-directed mutagenesis on some residues, followed by kinetic analysis. The F159G and F200G mutants displayed a strong increase in K M and modest decrease in k cat , suggesting that these concerted motions may provide dynamical roles in substrate binding and/or catalysis. However, further structural data for the hPNP mutants are needed to confirm our hypothesis.

This chapter will describe recent advances in the study of quantum particle transfer in condensed phase. In the Introduction we will discuss some concepts and results from the classical theory of reaction rates. The starting point for our quantum theory is the generalized Langevin equation and the equivalent formulation due to Zwanzig that allows for a natural extension to the quantum case. We also show how one can perform calculations for realistic systems using a MD simulation as input. This forms the basis of our quantum Kramers calculations. In the second section we discuss a method that we have developed for the solution of quantum many-particle Hamiltonians. We then discuss whether the Hamiltonians that are based on the quantum Kramers problem are appropriate models for realistic proton transfer problems. In the final three sections we describe some cases when the GLE-quantum Kramers framework is not sufficient: symmetric coupling to a solvent oscillation. position dependent friction and strong dependence on low-frequency modes of the solvent. In each case we describe physical/chemical examples when such complexities are present, and approaches one may use to overcome the challenges these problems present.

Crystallographic studies of human purine nucleoside phosphorylase (hPNP) with several transition-state (TS) analogues in the immucillin family showed an unusual geometric arrangement of the atoms O-5', O-4', and O(P), the nucleophilic phosphate oxygen, lying in a close three-oxygen stack. These observations were corroborated by extensive experimental kinetic isotope effect analysis. We propose that protein-facilitated dynamic modes in hPNP cause this stack, centered on the ribosyl O-4' oxygen, to squeeze together and push electrons toward the purine ring, stabilizing the oxacarbenium character of the TS. As the N-ribosidic bond is cleaved during the reaction, the pK(a) values of N-7 and O-6 increase by the electron density expelled by the oxygen-stack compression toward the purine ring. Increased electron density in the purine ring improves electrostatic interactions with nearby residues and facilitates the abstraction of a proton from a solvent proton or an unidentified general acid, making the purine a better leaving group, and accelerating catalysis. Classical and mixed quantum/classical molecular dynamics (MD) simulations of the Michaelis complex of hPNP with the substrates guanosine and phosphate were performed to assess the existence of protein-promoting vibrations (PPVs). Analogous simulations were performed for the substrates in aqueous solution. In the catalytic site, the O-5', O-4', and O(P) oxygens vibrate at frequencies of ca. 125 and 465 cm(-1), as opposed to 285 cm(-1) in the absence of hPNP. The hybrid quantum mechanical/molecular mechanical method was used to assess whether this enzymatic vibration pushing the oxygens together is coupled to the reaction coordinate, and thus has a direct positive impact on catalysis. The potential energy surface for the phosphorolysis reaction for several snapshots taken from the classical MD simulation showed substantial differences in oxygen compression. Our calculations showed the existence of PPVs coupled to the reaction coordinate, which effect electronic alterations in the active site by pushing the three oxygen centers together in proximity, and accelerate substrate turnover in the phosphorolysis reaction catalyzed by hPNP.

This review describes studies of particular enzymatically catalyzed reactions to investigate the possibility that catalysis is mediated by protein dynamics. That is, evolution has crafted the protein backbone of the enzyme to direct vibrations in such a fashion to speed reaction. The review presents the theoretical approach we have used to investigate this problem, but it is designed for the nonspecialist. The results show that in alcohol dehydrogenase, dynamic protein motion is in fact strongly coupled to chemical reaction in such a way as to promote catalysis. This result is in concert with both experimental data and interpretations for this and other enzyme systems studied in the laboratories of the two other investigators who have published reviews in this issue.

The comment by Peters on our recent paper consists of misunderstandings regarding the algorithm used and of simple factual mistakes. We will address them below. The first statement that Peters makes is that “competing methods” are more general. This is simply not true - the methods he references (either his own or related to his own) involve fitting of the committor. For his method to work, one needs a range of committor values. As we have stated in our manuscript this is almost impossible when the process being studied is a direct chemical reaction with a rapidly rising committor. The only values of the committor reasonably obtained are zero or unity. In LDH the rise occurs in less than 10 femtosecondsi, and it would be impossible to obtain a broad range of coordinate values to which one could fit the committor. The fitting methods, such as the work of Dinner and Maii and the Trout groupiii, are far better suited to diffusive transition regions. Both his and our methods have their merits, but they are not “competing” since they are designed for different types of barrier crossing dynamics.

We discuss the interplay of disorder and interactions in determining the spectrum of inter-Landau-level excitations of a two-dimensional electron gas in a strong magnetic field. Our work is based in part on some new exact sum rules for the inter-Landau-level response functions of noninteracting electrons in a strong magnetic field and in part on perturbative approximations for interaction and disorder. We emphasize the importance of including vertex corrections in the response functions which are consistent with the approximation that has been used for the self-energy. In the absence of disorder or in the absence of interactions our perturbative approximations reduce to those used by earlier workers for those limits.

The spin-polarization of nuclei near a two-dimensional electron gas (2D EG) may be relaxed by spin-flip excitations of the electron system. The spectrum of low-energy electronic spin-flip excitations depends on the disorder broadening of Landau levels and on the interaction enhancement of the Landau-level spin splitting. Disorder vertex corrections to the spin-flip response function capture the fact that the nuclear relaxation rate depends on local rather than thermodynamic Landau-level broadening, while interaction vertex corrections can strongly enhance the relaxation rate. We illustrate these effects by summing the disorder and interaction ladder diagrams for the spin-flip response function in the strong-magnetic-field limit. Our approach is also able to describe the effect of disorder on the spin-wave collective modes of a spin-polarized 2D EG.

Presentation d'un nouveau type de mode collectif qui resulte du couplage des plasmons aux excitations transverses par la force de Lorentz. Il emerge au-dessus du continuum des excitations particule-trou dans le niveau de Landau quand ce dernier est pratiquement a demi-rempli