Simulations of the large kinetic isotope effect and the temperature dependence of the hydrogen atom transfer in lipoxygenase

Elucidating the role of nuclear quantum mechanical (NQM) effects in enzyme catalysis is a topic of significant current interest. Despite the great experimental progress in this field it is important to have theoretical approaches capable of evaluating and analyzing nuclear quantum mechanical contributions to catalysis. In this study, we use the catalytic reaction of lipoxygenase, which is characterized by an extremely large kinetic isotope effect, as a challenging test case for our simulation approach. This is done by applying the quantum classical path (QCP) method with an empirical valence bond potential energy surface. Our computational strategy evaluates the relevant NQM corrections and reproduces the large observed kinetic isotope effect and the temperature dependence of the H atom transfer reaction while being less successful with the D atom transfer reaction. However, the main point of our study is not so much to explore the temperature dependence of the isotope effect but rather to develop and validate an approach for calculations of nuclear quantum mechanical contributions to activation free energies. Here, we find that the deviation between the calculated and observed activation free energies is small for both H and D at all investigated temperatures. The present study also explores the nature of the reorganization energy in the enzyme and solution reactions. It is found that the outer-sphere reorganization energy is extremely small. This reflects the fact that the considered reaction involves a very small charge transfer. The implication of this finding is discussed in the framework of the qualitative vibronic model. The main point of the present study is, however, that the rigorous QCP approach provides a reliable computational tool for evaluating NQM contributions to catalysis even when the given reaction includes large tunneling contributions. Interestingly, our results indicate that the NQM effects in the lipoxygenase reaction are similar in the enzyme and in the reference solution reactions, and thus do not contribute to catalysis. We also reached similar conclusions in studies of other enzymes.     

Simulation of tunneling in enzyme catalysis by combining a biased propagation approach and the quantum classical path method: application to lipoxygenase

The ability of using wave function propagation approaches to simulate isotope effects in enzymes is explored, focusing on the large H/D kinetic isotope effect of soybean lipoxygenase-1 (SLO-1). The H/D kinetic isotope effect (KIE) is calculated as the ratio of the rate constants for hydrogen and deuterium transfer. The rate constants are calculated from the time course of the H and D nuclear wave functions. The propagations are done using one-dimensional proton potentials generated as sections from the full multidimensional surface of the reacting system in the protein. The sections are obtained during a classical empirical valence bond (EVB) molecular dynamics simulation of SLO-1. Since the propagations require an extremely long time for treating realistic activation barriers, it is essential to use an effective biasing approach. Thus, we develop here an approach that uses the classical quantum path (QCP) method to evaluate the quantum free energy change associated with the biasing potential. This approach provides an interesting alternative to full QCP simulations and to other current approaches for simulating isotope effects in proteins. In particular, this approach can be used to evaluate the quantum mechanical transmission factor or other dynamical effects, while still obtaining reliable quantized activation free energies due to the QCP correction.     

A semiclassical study of the thermal conductivity of low temperature liquids

The conventional classical energy current auto-correlation function has been extended into a quantum mechanical version and then approximated by the linearized semiclassical initial value representation approach. Comparison of the thermal conductivity to simulation results shows that about 15% quantum correction to the classical molecular dynamics results for liquid neon are quantitatively predicted. For liquid para-hydrogen the quantum effects are sufficiently large that the linearized semiclassical approach is only 20% accurate, while for both liquid He(4) and He(3) the thermal conductivity disagrees by a factor of 2, although exchange effects appear to play a minor role.     

Quasi-harmonic approximation of thermodynamic properties of ice Ih, II, and III

Several thermodynamic properties of ice Ih, II, and III are studied by a quasi-harmonic approximation and compared to results of quantum path integral and classical simulations. This approximation allows to obtain thermodynamic information at a fraction of the computational cost of standard simulation methods, and at the same time permits studying quantum effects related to zero-point vibrations of the atoms. Specifically, we have studied the crystal volume, bulk modulus, kinetic energy, enthalpy, and heat capacity of the three ice phases as a function of temperature and pressure. The flexible q-TIP4P/F model of water was employed for this study, although the results concerning the capability of the quasi-harmonic approximation are expected to be valid independently of the employed water model. The quasi-harmonic approximation reproduces with reasonable accuracy the results of quantum and classical simulations showing an improved agreement at low temperatures (T< 100 K). This agreement does not deteriorate as a function of pressure as long as it is not too close to the limit of mechanical stability of the ice phases.     

Ground-state properties of the retinal molecule: from quantum mechanical to classical mechanical computations of retinal proteins

Retinal proteins are excellent systems for understanding essential physiological processes such as signal transduction and ion pumping. Although the conjugated polyene system of the retinal chromophore is best described with quantum mechanics, simulations of the long-timescale dynamics of a retinal protein in its physiological, flexible, lipid-membrane environment can only be performed at the classical mechanical level. Torsional energy barriers are a critical ingredient of the classical force-field parameters. Here we review briefly current retinal force fields and discuss new quantum mechanical computations to assess how the retinal Schiff base model and the approach used to derive the force-field parameters may influence the torsional potentials.     

Calculation of quantum mechanical free energy at low temperature

The path integral method is used to calculate the quantum mechanical free energy at low temperature. Based on the variational harmonic reference system and implemented by the partial averaging technique, the path integral can be cast into the form of a classical configurational integral with the original potential replaced by an effective one. We compared this approach with other related methods and found that it gave better results than the others considered in this paper. Furthermore, the multidimensional implementation of this method is discussed. Received: 15 September 1997 / Accepted: 1 October 1997     

Origin of the temperature dependence of isotope effects in enzymatic reactions: the case of dihydrofolate reductase

The origin of the temperature dependence of kinetic isotope effects (KIEs) in enzyme reactions is a problem of general interest and a major challenge for computational chemistry. The present work simulates the nuclear quantum mechanical (NQM) effects and the corresponding KIE in dihydrofolate reductase (DHFR) and two of its mutants by using the empirical valence bond (EVB) and the quantum classical path (QCP) centroid path integral approach. Our simulations reproduce the overall observed trend while using a fully microscopic rather than a phenomenological picture and provide an interesting insight. It appears that the KIE increases when the distance between the donor and acceptor increases, in a somewhat counter intuitive way. The temperature dependence of the KIE appears to reflect mainly the temperature dependence of the distance between the donor and acceptor. This trend is also obtained from a simplified vibronic treatment, but as demonstrated here, the vibronic treatment is not valid at short and medium distances, where it is essential to use the path integral or other approaches capable of moving seamlessly from the adiabatic to the diabatic limits. It is pointed out that although the NQM effects do not contribute to catalysis in DHFR, the observed temperature dependence can be used to refine the potential of mean force for the donor and acceptor distance and its change due to distanced mutations.     

Future in biomolecular computation

Summary Large-scale computations for biomolecules are dominated by three levels of theory: rigorous quantum mechanical calculations for molecules with up to about 30 atoms, semi-empirical quantum mechanical calculations for systems with up to several hundred atoms, and force-field molecular dynamics studies of biomacromolecules with 10,000 atoms and more including surrounding solvent molecules. It can be anticipated that increased computational power will allow the treatment of larger systems of ever growing complexity. Due to the scaling of the computational requirements with increasing number of atoms, the force-field approaches will benefit the most from increased computational power. On the other hand, progress in methodologies such as density functional theory will enable us to treat larger systems on a fully quantum mechanical level and a combination of molecular dynamics and quantum mechanics can be envisioned. One of the greatest challenges in biomolecular computation is the protein folding problem. It is unclear at this point, if an approach with current methodologies will lead to a satisfactory answer or if unconventional, new approaches will be necessary. In any event, due to the complexity of biomolecular systems, a hierarchy of approaches will have to be established and used in order to capture the wide ranges of length-scales and time-scales involved in biological processes. In terms of hardware development, speed and power of computers will increase while the price/performance ratio will become more and more favorable. Parallelism can be anticipated to become an integral architectural feature in a range of computers. It is unclear at this point, how fast massively parallel systems will become easy enough to use so that new methodological developments can be pursued on such computers. Current trends show that distributed processing such as the combination of convenient graphics workstations and powerful general-purpose supercomputers will lead to a new style of computing in which the calculations are monitored and manipulated as they proceed. The combination of a numeric approach with artificial-intelligence approaches can be expected to open up entirely new possibilities. Ultimately, the most exciding aspect of the future in biomolecular computing will be the unexpected discoveries.     

Quantum and classical study of surface characterization by three-dimensional helium atom scattering

Exact time-dependent wavepacket calculations of helium atom scattering from model symmetric, chiral, and hexagonal surfaces are presented and compared with their classical counterparts. Analysis of the momentum distribution of the scattered wavepacket provides a convenient method to obtain the resulting energy and angle resolved scattering distributions. The classical distributions are characterized by standard rainbow scattering from corrugated surfaces. It is shown that the classical results are closely related to their quantum counterparts and capture the qualitative features appearing therein. Both the quantum and classical distributions are capable of distinguishing between the structures of the three surfaces.     

The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors

This work demonstrates that the quantum mechanical moments of a state described by the density matrix correspond to discrete spherical harmonic moments of the classical multipole expansion of the spatial distribution of the angular momentum vectors. For the diagonal density matrix elements, this work exploits the fact that the quantum mechanical vector coupling (Clebsch-Gordan) coefficients become increasingly accurate discrete representations of spherical harmonics as j increases. A Schwinger-type basis accounts for nonaxially symmetric angular distributions, which result in nonzero off-diagonal elements of the density matrix. The resulting discrete minimum uncertainty picture of the classical moments has a stringent equivalence with the quantum mechanical one for all j and provides an unambiguous connection for the classical and quantum moments in the large j limit. The equivalence is numerically tested for simple models, and there is a satisfying equivalence even for small j. Applications, implications, and extensions are indicated, and the relevance of this work for the interpretation of classical mechanical simulations of inelastic and reactive molecular collisions will be documented elsewhere.     

Evaluation of quantum correlation functions from classical data: Anharmonic models

The previously introduced method of evaluating quantum mechanical time correlation functions using as input only classical simulation data is generalized and applied to two anharmonic model systems, as a further test. The quantum correction approach utilizes the relation between a general quantum correlation function and its classical analog. For the tested models, we obtain numerical results of nonlinear correlation functions with comparable accuracy to that of the centroid molecular dynamics method, although the present method is much simpler to implement and not limited to real valued quantum correlation functions.     

Ar+ ArH+ Reactive Collisions of Astrophysical Interest: The Case of 36Ar

The reactive collision between ³⁶Ar and the ³⁶ArH⁺ species has been investigated by means of quantum mechanical (QM), quasiclassical trajectories (QCT) and statistical quantum mechanical (SQM) approaches. Reaction probabilities, cross sections as a function of the energy and rate constants in terms of the temperature have been obtained. Cumulative distributions as a function of the collision time and the inspection of selected QCT corresponding to specific dynamical mechanisms have been analysed. Predictions by means of the SQM method are in good agreement with the QM results, thus supporting the complex-forming nature of the process.     

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. I. Theories

We have reformulated and generalized our recent work [J. Liu and W. H. Miller, J. Chem. Phys. 126, 234110 (2007)] into an approach for generating a family of trajectory-based dynamics methods in the phase space formulation of quantum mechanics. The approach (equilibrium Liouville dynamics) is in the spirit of Liouville's theorem in classical mechanics. The trajectory-based dynamics is able to conserve the quantum canonical distribution for the thermal equilibrium system and approaches classical dynamics in the classical (? → 0), high temperature (β → 0), and harmonic limits. Equilibrium Liouville dynamics provides the framework for the development of novel theoretical∕computational tools for studying quantum dynamical effects in large∕complex molecular systems.     

Quantum delocalization of benzene in the ring puckering coordinates

The effect of quantum mechanical delocalization of atomic nuclei on the conformation of the six‐membered ring structure in two hydrocarbons, cyclohexane and benzene, is investigated using ab initio path integral approach. A striking feature of benzene species is revealed using ring puckering coordinate representation, which demonstrates that the zero point motion of the heavy atom skeleton dominates over the out‐of‐plane thermal motions of the ring. Even more unexpected is the fact, that this is true not only at low temperature of 150 K, at which such behavior would not be surprising, but also at room temperature, where the nuclear quantum effects are usually of lesser importance, especially in the case of such heavy nuclei as carbon. In view of this finding the planar conformation of benzene, whose equilibrium (T = 0 K) geometry results from the well‐known properties of the electronic structure, can be elucidated also at nonzero temperature. According to our simulations, it appears as a consequence of quantum delocalization of the carbon nuclei rather than a trivial time average over the classical configurations of the puckered ring. This interesting behavior is contrasted with the clearly nonplanar structure of cyclohexane, whose ring puckering states can be unequivocally assigned even if the nuclear delocalization is taken into account. © 2014 Wiley Periodicals, Inc.     

Dynamical quantum-electrodynamics embedding: Combining time-dependent density functional theory and the near-field method

We develop an approach for dynamical (ω > 0) embedding of mixed quantum mechanical (QM)∕classical (or more precisely QM∕electrodynamics) systems with a quantum sub-region, described by time-dependent density functional theory (TDDFT), within a classical sub-region, modeled here by the recently proposed near-field (NF) method. Both sub-systems are propagated simultaneously and are coupled through a common Coulomb potential. As a first step we implement the method to study the plasmonic response of a metal film which is half jellium-like QM and half classical. The resulting response is in good agreement with both full-scale TDDFT and the purely classical NF method. The embedding method is able to describe the optical response of the whole system while capturing quantum mechanical effects, so it is a promising approach for studying electrodynamics in hybrid molecules-metals nanostructures.     

Few-body quantum and many-body classical hyperspherical approaches to reactions and to cluster dynamics

The hyperspherical method is a widely used and successful approach for the quantum treatment of elementary chemical processes. It has been mostly applied to three-atomic systems, and current progress is here outlined concerning the basic theoretical framework for the extension to four-body bound state and reactive scattering problems. Although most applications only exploit the advantages of the hyperspherical coordinate systems for the formulation of the few-body problem, the full power of the technique implies representations explicitly involving quantum hyperangular momentum operators as dynamical quantities and hyperspherical harmonics as basis functions. In terms of discrete analogues of these harmonics one has a universal representation for the kinetic energy and a diagonal representation for the potential (hyperquantization algorithm). Very recently, advances have been made on the use of the approach in classical dynamics, provided that a hyperspherical formulation is given based on “classical” definitions of the hyperangular momenta and related quantities. The aim of the present paper is to offer a retrospective and prospective view of the hyperspherical methods both in quantum and classical dynamics. Specifically, regarding the general quantum hyperspherical approaches for three- and four-body systems, we first focus on the basis set issue, and then we present developments on the classical formulation that has led to applications involving the implementations of hyperspherical techniques for classical molecular dynamics simulations of simple nanoaggregates.     

Ab initio centroid path integral molecular dynamics: application to vibrational dynamics of diatomic molecular systems

An ab initio centroid molecular dynamics (CMD) method is developed by combining the CMD method with the ab initio molecular orbital method. The ab initio CMD method is applied to vibrational dynamics of diatomic molecules, H2 and HF. For the H2 molecule, the temperature dependence of the peak frequency of the vibrational spectral density is investigated. The results are compared with those obtained by the ab initio classical molecular dynamics method and exact quantum mechanical treatment. It is shown that the vibrational frequency obtained from the ab initio CMD approaches the exact first excitation frequency as the temperature lowers. For the HF molecule, the position autocorrelation function is also analyzed in detail. The present CMD method is shown to well reproduce the exact quantum result for the information on the vibrational properties of the system.