Efficient Calculation of Free Energy Differences Associated with Isotopic Substitution Using Path-Integral Molecular Dynamics

Reducing the Cost of Neural Network Potential Generation for Reactive Molecular Systems

Although machine learning potentials have recently had a substantial impact on molecular simulations, the construction of a robust training set can still become a limiting factor, especially due to the requirement of a reference ab initio simulation that covers all the relevant geometries of the system. Recognizing that this can be prohibitive for certain systems, we develop the method of transition tube sampling that mitigates the computational cost of training set and model generation. In this approach, we generate classical or quantum thermal geometries around a transition path describing a conformational change or a chemical reaction using only a sparse set of local normal mode expansions along this path and select from these geometries by an active learning protocol. This yields a training set with geometries that characterize the whole transition without the need for a costly reference trajectory. The performance of the method is evaluated on different molecular systems with the complexity of the potential energy landscape increasing from a single minimum to a double proton-transfer reaction with high barriers. Our results show that the method leads to training sets that give rise to models applicable in classical and path integral simulations alike that are on par with those based directly on ab initio calculations while providing the computational speedup we have come to expect from machine learning potentials.

Computational approaches to delivery of anticancer drugs with multidimensional nanomaterials

Functionalized nanotubes (NTs), nanosheets, nanorods, and porous organometallic scaffolds are potential in vivo carriers for cancer therapeutics. Precise delivery through these agents depends on factors like hydrophobicity, payload capacity, bulk/surface adsorption, orientation of molecules inside the host matrix, bonding, and nonbonding interactions. Herein, we summarize advances in simulation techniques, which are extremely valuable in initial geometry optimization and evaluation of the loading and unloading behavior of encapsulated drug molecules. Computational methods broadly involve the use of quantum and classical mechanics for studying the behavior of molecular properties. Combining theoretical processes with experimental techniques, such as X-ray crystallography, NMR spectroscopy, and bioassays, can provide a more comprehensive understanding of the structure and function of biological molecules. This integrated approach has led to numerous breakthroughs in drug discovery, enzyme design, and the study of complex biological processes. This short review provides an overview of results and challenges described from erstwhile investigations on the molecular interaction of anticancer drugs with nanocarriers of different aspect ratios.

Comparison of Implicit and Explicit Solvent Approaches in Ab Initio Evaluation of Thermochemistry in Solution: Application in Studying Boron Isotope Fractionation in Water

Evaluation of thermochemistry in solution plays a key role in numerous fields. For this task, the solvent effects are commonly included in theoretical computations based on either implicit or explicit solvent approaches. In the present study, we evaluate and compare the performance of some of the most widely applied methods based on these two approaches. For studying the solvent effect on thermochemistry with an explicit solvent, we demonstrate that partial normal mode analysis with frozen geometry of solvent molecules for multiple solute-solvent configurations can yield quite accurate and reliable results for a drastically reduced computational cost. As a case study, we consider the evaluation of the equilibrium constant for the boron isotope exchange between boric acid and borate (k_3-4) in pure and saline water which is of high geochemical importance. Employing three different rigorous and high-precision theoretical approaches, we provide a reliable estimation of k_3-4 which is a value within 1.028 to 1.030 for both pure and saline water which is in excellent agreement with experimental data.

Dimension-free path-integral molecular dynamics without preconditioning

Convergence with respect to imaginary-time discretization (i.e., the number of ring-polymer beads) is an essential part of any path-integral-based molecular dynamics (MD) calculation. However, an unfortunate property of existing non-preconditioned numerical integration schemes for path-integral molecular dynamics-including essentially all existing ring-polymer molecular dynamics (RPMD) and thermostatted RPMD (T-RPMD) methods-is that for a given MD time step, the overlap between the exact ring-polymer Boltzmann-Gibbs distribution and that sampled using MD becomes zero in the infinite-bead limit. This has clear implications for hybrid Metropolis Monte Carlo/MD sampling schemes, and it also causes the divergence with bead number of the primitive path-integral kinetic-energy expectation value when using standard RPMD or T-RPMD. We show that these and other problems can be avoided through the introduction of "dimension-free" numerical integration schemes for which the sampled ring-polymer position distribution has non-zero overlap with the exact distribution in the infinite-bead limit for the case of a harmonic potential. Most notably, we introduce the BCOCB integration scheme, which achieves dimension freedom via a particular symmetric splitting of the integration time step and a novel implementation of the Cayley modification [R. Korol et al., J. Chem. Phys. 151, 124103 (2019)] for the free ring-polymer half-steps. More generally, we show that dimension freedom can be achieved via mollification of the forces from the external physical potential. The dimension-free path-integral numerical integration schemes introduced here yield finite error bounds for a given MD time step, even as the number of beads is taken to infinity; these conclusions are proven for the case of a harmonic potential and borne out numerically for anharmonic systems that include liquid water. The numerical results for BCOCB are particularly striking, allowing for nearly three-fold increases in the stable time step for liquid water with respect to the Bussi-Parrinello (OBABO) and Leimkuhler (BAOAB) integrators, while introducing negligible errors in the calculated statistical properties and absorption spectrum. Importantly, the dimension-free, non-preconditioned integration schemes introduced here preserve ergodicity and global second-order accuracy, and they remain simple, black-box methods that avoid additional computational costs, tunable parameters, or system-specific implementations.

Accurate Description of Nuclear Quantum Effects with High-Order Perturbed Path Integrals (HOPPI)

Imaginary time path-integral (PI) simulations that account for nuclear quantum effects (NQE) beyond the harmonic approximation are increasingly employed together with modern electronic-structure calculations. Existing PI methods are applicable to molecules, liquids, and solids; however, the computational cost of such simulations increases dramatically with decreasing temperature. To address this challenge, here, we propose to combine high-order PI factorization with perturbation theory (PT). Already for conventional second-order PI simulations, the PT ansatz increases the accuracy 2-fold compared to fourth-order schemes with the same settings. In turn, applying PT to high-order path integrals (HOPI) further improves the efficiency of simulations for molecular and condensed matter systems especially at low temperatures. We present results for bulk liquid water, the aspirin molecule, and the CH₅⁺ molecule. Perturbed HOPI simulations remain both efficient and accurate down to 20 K and provide a convenient method to estimate the convergence of quantum-mechanical observables.

Role of Microsolvation and Quantum Effects in the Accurate Prediction of Kinetic Isotope Effects: The Case of Hydrogen Atom Abstraction in Ethanol by Atomic Hydrogen in Aqueous Solution

Hydrogen abstraction from ethanol by atomic hydrogen in aqueous solution is studied using two theoretical approaches: the multipath variational transition state theory (MP-VTST) and a path-integral formalism in combination with free-energy perturbation and umbrella sampling (PI-FEP/UM). The performance of the models is compared to experimental values of H kinetic isotope effects (KIE). Solvation models used in this study ranged from purely implicit, via mixed–microsolvation treated quantum mechanically via the density functional theory (DFT) to fully explicit representation of the solvent, which was incorporated using a combined quantum mechanical-molecular mechanical (QM/MM) potential. The effects of the transition state conformation and the position of microsolvating water molecules interacting with the solute on the KIE are discussed. The KIEs are in good agreement with experiment when MP-VTST is used together with a model that includes microsolvation of the polar part of ethanol by five or six water molecules, emphasizing the importance of explicit solvation in KIE calculations. Both, MP-VTST and PI-FEP/UM enable detailed characterization of nuclear quantum effects accompanying the hydrogen atom transfer reaction in aqueous solution.

Accelerating equilibrium isotope effect calculations. II. Stochastic implementation of direct estimators

Path integral calculations of equilibrium isotope effects and isotopic fractionation are expensive due to the presence of path integral discretization errors, statistical errors, and thermodynamic integration errors. Whereas the discretization errors can be reduced by high-order factorization of the path integral and statistical errors by using centroid virial estimators, two recent papers proposed alternative ways to completely remove the thermodynamic integration errors: Cheng and Ceriotti [J. Chem. Phys. 141, 244112 (2015)] employed a variant of free-energy perturbation called "direct estimators," while Karandashev and Vaníček [J. Chem. Phys. 143, 194104 (2017)] combined the thermodynamic integration with a stochastic change of mass and piecewise-linear umbrella biasing potential. Here, we combine the former approach with the stochastic change in mass in order to decrease its statistical errors when applied to larger isotope effects and perform a thorough comparison of different methods by computing isotope effects first on a harmonic model and then on methane and methanium, where we evaluate all isotope effects of the form CH_4-xD_x/CH₄ and CH_5-xD_x ⁺/CH₅ ⁺, respectively. We discuss the reasons for a surprising behavior of the original method of direct estimators, which performed well for a much larger range of isotope effects than what had been expected previously, as well as some implications of our work for the more general problem of free energy difference calculations.

Dual QM and MM Approach for Computing Equilibrium Isotope Fractionation Factor of Organic Species in Solution

A dual QM and MM approach for computing equilibrium isotope effects has been described. In the first partition, the potential energy surface is represented by a combined quantum mechanical and molecular mechanical (QM/MM) method, in which a solute molecule is treated quantum mechanically, and the remaining solvent molecules are approximated classically by molecular mechanics. In the second QM/MM partition, differential nuclear quantum effects responsible for the isotope effect are determined by a statistical mechanical double-averaging formalism, in which the nuclear centroid distribution is sampled classically by Newtonian molecular dynamics and the quantum mechanical spread of quantized particles about the centroid positions is treated using the path integral (PI) method. These partitions allow the potential energy surface to be properly represented such that the solute part is free of nuclear quantum effects for nuclear quantum mechanical simulations, and the double-averaging approach has the advantage of sampling efficiency for solvent configuration and for path integral convergence. Importantly, computational precision is achieved through free energy perturbation (FEP) theory to alchemically mutate one isotope into another. The PI-FEP approach is applied to model systems for the 18O enrichment found in cellulose of trees to determine the isotope enrichment factor of carbonyl compounds in water. The present method may be useful as a general tool for studying isotope fractionation in biological and geochemical systems.

Can Path Integral Molecular Dynamics Make a Good Approximation for Vapor Pressure Isotope Effects Prediction for Organic Solvents? A Comparison to ONIOM QM/MM and QM Cluster Calculation

Isotopic fractionation of volatile organic compounds (VOCs), which are under strict measures of control because of their potential harm to the environment and humans, has an important ecological aspect, as the isotopic composition of compounds may depend on the conditions in which such compounds are distributed in Nature. Therefore, detailed knowledge on isotopic fractionation, not only experimental but also based on theoretical models, is crucial to follow conditions and pathways within which these contaminants are spread throughout the ecosystems. In this work, we present carbon and, for the first time, bromine vapor pressure isotope effect (VPIE) on the evaporation process from pure-phase systems-dibromomethane and bromobenzene, the representatives of aliphatic and aromatic brominated VOCs. We combine isotope effects measurements with their theoretical prediction using three computational techniques, namely path integral molecular dynamics, QM cluster, and hybrid ONIOM models. While evaporation of both compounds resulted in normal bromine VPIEs, the difference in the direction of carbon isotopic fractionation is observed for the aliphatic and aromatic compounds, where VPIEs are inverse and normal, respectively. Even though theoretical models tested here turned out to be insufficient for quantitative agreement with the experimental values, cluster electronic structure calculations, as well as two-layer ONIOM computations, provided better reproduction of experimental trends.

Computational Methods for Ab Initio Molecular Dynamics

Ab initio molecular dynamics is an irreplaceable technique for the realistic simulation of complex molecular systems and processes from first principles. This paper proposes a comprehensive and self-contained review of ab initio molecular dynamics from a computational perspective and from first principles. Quantum mechanics is presented from a molecular dynamics perspective. Various approximations and formulations are proposed, including the Ehrenfest, Born–Oppenheimer, and Hartree–Fock molecular dynamics. Subsequently, the Kohn–Sham formulation of molecular dynamics is introduced as well as the afferent concept of density functional. As a result, Car–Parrinello molecular dynamics is discussed, together with its extension to isothermal and isobaric processes. Car–Parrinello molecular dynamics is then reformulated in terms of path integrals. Finally, some implementation issues are analysed, namely, the pseudopotential, the orbital functional basis, and hybrid molecular dynamics.

Kinetic isotope effects and how to describe them

We review several methods for computing kinetic isotope effects in chemical reactions including semiclassical and quantum instanton theory. These methods describe both the quantization of vibrational modes as well as tunneling and are applied to the ⋅H + H2 and ⋅H + CH4 reactions. The absolute rate constants computed with the semiclassical instanton method both using on-the-fly electronic structure calculations and fitted potential-energy surfaces are also compared directly with exact quantum dynamics results. The error inherent in the instanton approximation is found to be relatively small and similar in magnitude to that introduced by using fitted surfaces. The kinetic isotope effect computed by the quantum instanton is even more accurate, and although it is computationally more expensive, the efficiency can be improved by path-integral acceleration techniques. We also test a simple approach for designing potential-energy surfaces for the example of proton transfer in malonaldehyde. The tunneling splittings are computed, and although they are found to deviate from experimental results, the ratio of the splitting to that of an isotopically substituted form is in much better agreement. We discuss the strengths and limitations of the potential-energy surface and based on our findings suggest ways in which it can be improved.

Importance of a Fully Anharmonic Treatment of Equilibrium Isotope Fractionation Properties of Dissolved Ionic Species As Evidenced by Li+(aq)

Equilibrium fractionation of stable isotopes is critically important in fields ranging from chemistry, including medicinal chemistry, electrochemistry, geochemistry, and nuclear chemistry, to environmental science. The dearth of reliable estimates of equilibrium fractionation factors, from experiment or from natural observations, has created a need for accurate computational approaches. Because isotope fractionation is a purely quantum mechanical phenomenon, exact calculation of fractionation factors is nontrivial. Consequently, a severe approximation is often made, in which it is assumed that the system can be decomposed into a set of independent harmonic oscillators. Reliance on this often crude approximation is one of the primary reasons that theoretical prediction of isotope fractionation has lagged behind experiment. A class of problems for which one might expect the harmonic approximation to perform most poorly is the isotopic fractionation between solid and solution phases. In order to illustrate the errors associated with the harmonic approximation, we have considered the fractionation of Li isotopes between aqueous solution and phyllosilicate minerals, where we find that the harmonic approximation overestimates isotope fractionation factors by as much as 30% at 25 °C. Lithium is a particularly interesting species to examine, as natural lithium isotope signatures provide information about hydrothermal processes, carbon cycle, and regulation of the Earth's climate by continental alteration. Further, separation of lithium isotopes is of growing interest in the nuclear industry due to a need for pure ⁶Li and ⁷Li isotopes. Moving beyond the harmonic approximation entails performing exact quantum calculations, which can be achieved using the Feynman path integral formulation of quantum statistical mechanics. In the path integral approach, a system of quantum particles is represented as a set of classical-like ring-polymer chains, whose interparticle interactions are determined by the rules of quantum mechanics. Because a classical isomorphism exists between the true quantum system and the system of ring-polymers, classical-like methods can be applied. Recent developments of efficient path integral approaches for the exact calculation of isotope fractionation now allow the case of the aforementioned dissolved Li fractionation properties to be studied in detail. Applying this technique, we find that the calculations yield results that are in good agreement with both experimental data and natural observations. Importantly, path integral methods, being fully atomistic, allow us to identify the origins of anharmonic effects and to make reliable predictions at temperatures that are experimentally inaccessible yet are, nevertheless, relevant for natural phenomena.

Quantum Nuclear Dynamics of Protons within Layered Hydroxides at High Pressure

Studies of the structure of hydroxides under pressure using neutron diffraction reveal that the high concentration of hydrogen is distributed in a disordered network. The disorder in the hydrogen-bond network and possible phase transitions are reported to occur at pressures within the range accessible to experiments for layered calcium hydroxides, which are considered to be exemplary prototype materials. In this study, the static and dynamical properties of these layered hydroxides are investigated using a quantum approach describing nuclear motion, shown herein to be required particularly when studying diffusion processes involving light hydrogen atoms. The effect of high-pressure on the disordered hydrogen-bond network shows that the protons tunnel back and forth across the barriers between three potential minima around the oxygen atoms. At higher pressures the structure has quasi two-dimensional layers of hydrogen atoms, such that at low temperatures this causes the barrier crossing of the hydrogen to be significantly rarefied. Furthermore, for moderate values of both temperature and pressure this process occurs less often than the usual mechanism of proton transport via vacancies, limiting global proton diffusion within layers at high pressure.

Enzymatic Kinetic Isotope Effects from Path-Integral Free Energy Perturbation Theory

Path-integral free energy perturbation (PI-FEP) theory is presented to directly determine the ratio of quantum mechanical partition functions of different isotopologs in a single simulation. Furthermore, a double averaging strategy is used to carry out the practical simulation, separating the quantum mechanical path integral exactly into two separate calculations, one corresponding to a classical molecular dynamics simulation of the centroid coordinates, and another involving free-particle path-integral sampling over the classical, centroid positions. An integrated centroid path-integral free energy perturbation and umbrella sampling (PI-FEP/UM, or simply, PI-FEP) method along with bisection sampling was summarized, which provides an accurate and fast convergent method for computing kinetic isotope effects for chemical reactions in solution and in enzymes. The PI-FEP method is illustrated by a number of applications, to highlight the computational precision and accuracy, the rule of geometrical mean in kinetic isotope effects, enhanced nuclear quantum effects in enzyme catalysis, and protein dynamics on temperature dependence of kinetic isotope effects.

Inverse Temperature Dependence of Nuclear Quantum Effects in DNA Base Pairs

Despite the inherently quantum mechanical nature of hydrogen bonding, it is unclear how nuclear quantum effects (NQEs) alter the strengths of hydrogen bonds. With this in mind, we use ab initio path integral molecular dynamics to determine the absolute contribution of NQEs to the binding in DNA base pair complexes, arguably the most important hydrogen-bonded systems of all. We find that depending on the temperature, NQEs can either strengthen or weaken the binding within the hydrogen-bonded complexes. As a somewhat counterintuitive consequence, NQEs can have a smaller impact on hydrogen bond strengths at cryogenic temperatures than at room temperature. We rationalize this in terms of a competition of NQEs between low-frequency and high-frequency vibrational modes. Extending this idea, we also propose a simple model to predict the temperature dependence of NQEs on hydrogen bond strengths in general.

Stability of Complex Biomolecular Structures: van der Waals, Hydrogen Bond Cooperativity, and Nuclear Quantum Effects

Biomolecules are complex systems stabilized by a delicate balance of weak interactions, making it important to assess all energetic contributions in an accurate manner. However, it is a priori unclear which contributions make more of an impact. Here, we examine stacked polyglutamine (polyQ) strands, a peptide repeat often found in amyloid aggregates. We investigate the role of hydrogen bond (HB) cooperativity, van der Waals (vdW) dispersion interactions, and quantum contributions to free energies, including anharmonicities through density functional theory and ab initio path integral simulations. Of these various factors, we find that the largest impact on structural stabilization comes from vdW interactions. HB cooperativity is the second largest contribution as the size of the stacked chain grows. Competing nuclear quantum effects make the net quantum contribution small but very sensitive to anharmonicities, vdW, and the number of HBs. Our results suggest that a reliable treatment of these systems can only be attained by considering all of these components.

Modeling quantum nuclei with perturbed path integral molecular dynamics

Here we combine perturbation theory with the Feynman–Kac imaginary-time path integral approach to quantum mechanics for modeling quantum nuclear effects.

The quantum nature of nuclear motions plays a vital role in the structure, stability, and thermodynamics of molecules and materials. The standard approach to model nuclear quantum fluctuations in chemical and biological systems is to use path-integral molecular dynamics. Unfortunately, conventional path-integral simulations can have an exceedingly large computational cost due to the need to employ an excessive number of coupled classical subsystems (beads) for quantitative accuracy. Here, we combine perturbation theory with the Feynman–Kac imaginary-time path integral approach to quantum mechanics and derive an improved non-empirical partition function and estimators to calculate converged quantum observables. Our perturbed path-integral (PPI) method requires the same ingredients as the conventional approach, but increases the accuracy and efficiency of path integral simulations by an order of magnitude. Results are presented for the thermodynamics of fundamental model systems, an empirical water model containing 256 water molecules within periodic boundary conditions, and ab initio simulations of nitrogen and benzene molecules. For all of these examples, PPI simulations with 4 to 8 classical beads recover the nuclear quantum contribution to the total energy and heat capacity at room temperature within a 3% accuracy, paving the way toward seamless modeling of nuclear quantum effects in realistic molecules and materials.

Quantum delocalization of protons in the hydrogen-bond network of an enzyme active site

Enzymes use protein architectures to create highly specialized structural motifs that can greatly enhance the rates of complex chemical transformations. Here, we use experiments, combined with ab initio simulations that exactly include nuclear quantum effects, to show that a triad of strongly hydrogen-bonded tyrosine residues within the active site of the enzyme ketosteroid isomerase (KSI) facilitates quantum proton delocalization. This delocalization dramatically stabilizes the deprotonation of an active-site tyrosine residue, resulting in a very large isotope effect on its acidity. When an intermediate analog is docked, it is incorporated into the hydrogen-bond network, giving rise to extended quantum proton delocalization in the active site. These results shed light on the role of nuclear quantum effects in the hydrogen-bond network that stabilizes the reactive intermediate of KSI, and the behavior of protons in biological systems containing strong hydrogen bonds.