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

^{1}

^{2}

^{*}

## Abstract

**:**

^{18}O 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.

## 1. Introduction

^{18}O enrichment of cellulose in tree rings could reveal historical records of local climate and ecohydrological changes such as temperature, humidity, and precipitation [1,2,3,4]. Isotope abundances of light elements are reported as the relative ratio of the heavy to light isotope of the sample with respect to a standard of known composition [5]. For example, the

^{18}O isotope enrichment of a biomass is given by

^{18}O of acetone in water can be described by the isotope fractionation factor between the carbonyl compound and water in aqueous solution [6]. Remarkably, there is a global average ${\delta}^{18}\mathrm{O}$ of 27‰ enrichment in cellulose of trees relative to the source water during its biosynthesis [7]. However, correlation between isotope signature and climate parameters could be challenging since many dynamic, thermodynamic, and kinetic factors can influence the final composition [4], including the isotope signal of stem water reflecting precipitation, the

^{18}O enrichment of leaf water due to evaporation that is sensitive to environmental temperature and humidity [8], and biochemical fractionation during carbohydrate and cellulose biosynthesis [9]. A primary biochemical origin responsible for the

^{18}O fractionation of cellulose is the equilibrium isotope effects between carbonyl groups and water during cellulose biosynthesis [6,10]. Thus, it is of interest to develop and test a computational method for estimating isotope fractionation under different environmental conditions in solution and in biological systems.

^{16}O) in the present study (of course, there is no difference if one chooses to carry out the simulation using the heavy particle). Note also that “L” and “H” in Equation (2) are not limited to a single isotope exchange, but multiple substitutions of different atoms can be simultaneously treated.

## 2. Methods

#### 2.1. Potential Energy Surface

#### 2.2. Path Integral-Free Energy Perturbation

#### 2.3. Double Averaging

#### 2.4. Equilibrium Isotope Effects

## 3. Computational Details

^{18}O enrichment of carbohydrates in water. Thus, a total of three separate simulations are needed, the isotope exchange of acetone, acetaldehyde, and water in aqueous solution. A main goal is to test the convergence of solvent configurational sampling through double averaging. Thus, the semi-empirical parameterized model 3 (PM3) Hamiltonian [55] was used to represent the solute molecule. The accuracy of the computed absolute fractionation factor can certainly be improved if a more accurate QM model such as a DFT representation of the solute were used, but we have not further explored these choices. It would be of interest to perform a systematic investigation in future studies. The solvent molecules were described by the TIP3P three-point charge model for water [83].

^{3}, consisting of 1338 water molecules. Periodic boundary conditions along with particle mesh-Ewald was used to treat long-range electrostatic interactions using the isothermal-isobaric (NPT) ensemble at 25 °C and 1 atm to generate the classical trajectory for the outer averages of Equation (18). In FP path-integral sampling, a smoothing function was used to feather intermolecular interactions to zero between 13.0 and 13.5 Å since we do not expect long range effects are critical to NQE. Overall, at least 10 ns molecular dynamics simulations were discarded as equilibration for each system, followed by about 500 ps simulations for averaging. As in the past, atoms that are within two covalent bonds of the atom (oxygen) with isotope exchange are quantized and represented by 32 or 64 beads. We have previously carried out extensive tests of the convergence of FP sampling using different quasi-particles [22], and the present choice has consistently yielded converged results. In this work, we further evaluated the convergence of FPPI sampling with respect to the number of quasi-particles used to represent each quantized particle path. We have also tested the effect of different intervals for saving classical coordinates on the computed nuclear quantum effects.

## 4. Results and Discussion

^{18}O]/[

^{16}O] for water and for aldehyde, an alchemical equilibrium constant by mutating the naturally most abundant isotope

^{16}O into

^{18}O through free energy perturbation. The equilibrium constants are greater than unity because the compounds containing the lighter isotope have higher vibrational frequencies and zero-point energy than the heavy isotopologues. In both cases, the computed R quickly and monotonically approaches its average value. In comparison to the average values obtained using 128 beads, the deviations are 0.5 (0.5)%, 0.4 (0.3)%, and 0.05 (0.2)% for water (acetaldehyde) using 16, 32, and 64 beads, respectively. Clearly, in both systems, the use of four and eight beads is not sufficient for path integral convergence (having absolute errors of about 4% and 1.5%, respectively, in the two cases). We find that the use of 32 beads offers a good balancing of the computational costs and precision, consistent with previous studies of kinetic isotope effects in enzymatic reactions [77,88]. Interestingly, the estimated isotope fractionation factors of acetaldehyde relative to water, i.e., equilibrium enrichment of the aldehyde in aqueous solution, are relatively invariant with respect to different number of beads (Figure 3), although the smallest value, P = 4, tested has relatively larger deviations than the rest.

^{18}/Q

^{16}ratios for water, acetaldehyde, and acetone in water, obtained over 1000-configuration blocks saved at different intervals of MD steps, ranging from every 1 fs to 100 fs (1 fs integration steps). For each configuration, 200 random closed paths after discarding the first 10 configurations, represented by 32 quasi-particles for acetaldehyde and acetone and 64 for water, were used in the free-particle path-integral averaging. Thus, each point in Figure 4 represents an average over 200,000 path integral configurations (12.8 to 25.6 M QM/MM energy calculations to obtain the heavy to light isotope ratio). These configurations were divided into 10 separate blocks, from which standard deviations were determined based on these block-averages. The standard deviations for the carbonyl compounds were small in Figure 4. The variations of these eight separate averages, which were used together to determine the final isotope fractionation factor discussed next, reflect the dynamic fluctuations of the solvent configurations. For these small molecules in a rather homogeneous solvent, the fluctuations are relatively small, although they are certainly noticeable and cannot be neglected.

^{18}O isotope is enriched at the carbonyl position relative to that of water. As is well-known, carbonyl compounds can rapidly undergo isotope exchange with water, especially for unhindered carbonyls, through formation of hydrates. This reaction has been attributed to be a primary biochemical factor responsible for the

^{18}O enrichment found in carbohydrates such as celluloses in trees. As a result, the much higher

^{18}O enrichment in CO

_{2}introduced in carbon fixation by RuBisCO is quickly lost through the carbonyl-hydrate equilibrium from small sugar compounds. An experimental measurement of the

^{18}O enrichment of acetone in water yielded a fractionation factor of 1.027 [6], coincidentally the same as the average enrichment in trees. We were not able to find an experimental isotope fractionation value for acetaldehyde. The computed value of δ

^{18}O about 39‰ for acetone is in reasonable agreement with experimental data, in view of the semi-empirical potential energy function used. The present semi-empirical PM3 method is perhaps among the fastest electronic structure models, allowing for adequate condensed-phase sampling. However, the harmonic vibrational frequencies for carbonyl groups are about 14% higher than experimental data (1979 cm

^{−1}from PM3 vs. 1731 cm

^{−1}from experiments for acetone), contributing to the absolute errors in the computed isotope effects. The computational accuracy may be improved by using a higher level of theory that can provide a good description of molecular vibrations.

_{2}DCHO and CH

_{3}CDO, and we have computed both pairs in the gas phase and in aqueous solution. The computed results are listed in Table 2. For the D/H ratio, the values are much greater than those of oxygen isotopes because of the large mass difference, leading to greater difference in vibrational frequency and zero-point energy. Both in the gas phase and aqueous solution, we found that the methyl position is enriched by more than 200‰ per hydrogen of the methyl group. If the symmetry factor of three equivalent positions is considered, the present PM3/TIP3P model yields a predicted enhancement of about 700‰ at the C2 position over that at the C1 position. Interestingly, there is little solvent effect on this isotope fractionation factor.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Barbour, M.M. Stable oxygen isotope composition of plant tissue: A review. Funct. Plant Biol.
**2007**, 34, 83–94. [Google Scholar] [CrossRef] - Lehmann, M.M.; Gamarra, B.; Kahmen, A.; Siegwolf, R.T.W.; Saurer, M. Oxygen isotope fractionations across individual leaf carbohydrates in grass and tree species. Plant Cell Environ.
**2017**, 40, 1658–1670. [Google Scholar] [CrossRef] [PubMed] - Sternberg, L.; Ellsworth, P.F.V. Divergent Biochemical Fractionation, Not Convergent Temperature, Explains Cellulose Oxygen Isotope Enrichment across Latitudes. PLoS ONE
**2011**, 6, e28040. [Google Scholar] [CrossRef] [PubMed] - Saurer, M.; Kirdyanov, A.V.; Prokushkin, A.S.; Rinne, K.T.; Siegwolf, R.T.W. The impact of an inverse climate-isotope relationship in soil water on the oxygen-isotope composition of Larix gmelinii in Siberia. New Phytol.
**2016**, 209, 955–964. [Google Scholar] [CrossRef] [PubMed] - Gonfiantini, R.; Stichler, W.; Rozanski, K. Standards and Intercomparison Materials Distributed by the International Atomic Energy Agency for Stable Isotope Measurements; Reference and Intercomparison Materials for Stable Isotopes of Light Elements; IAEA: Vienna, Austria, 1995. [Google Scholar]
- Sternberg, L.D.L.O.; Deniro, M.J.D. Biogeochemical Implications of the Isotopic Equilibrium Fractionation Factor between the Oxygen-Atoms of Acetone and Water. Geochim. Cosmochim. Acta
**1983**, 47, 2271–2274. [Google Scholar] [CrossRef] - Sternberg, L.D.L.O. Oxygen stable isotope ratios of tree-ring cellulose: The next phase of understanding. New Phytol.
**2009**, 181, 553–562. [Google Scholar] [CrossRef] [PubMed] - Cernusak, L.A.; Barbour, M.M.; Arndt, S.K.; Cheesman, A.W.; English, N.B.; Feild, T.S.; Helliker, B.R.; Holloway-Phillips, M.M.; Holtum, J.A.M.; Kahmen, A.; et al. Stable isotopes in leaf water of terrestrial plants. Plant Cell Environ.
**2016**, 39, 1087–1102. [Google Scholar] [CrossRef] [PubMed] - Buchanan, B.B.; Gruissem, W.; Jones, R.I. Biochemistry and Molecular Biology of Plants; Wiley: New York, NY, USA, 2015. [Google Scholar]
- Schmidt, H.L.; Werner, R.A.; Rossmann, A. O-18 pattern and biosynthesis of natural plant products. Phytochemistry
**2001**, 58, 9–32. [Google Scholar] [CrossRef] - Pu, J.; Gao, J.; Truhlar, D.G. Multidimensional tunneling, recrossing, and the transmission coefficient for enzymatic reactions. Chem. Rev.
**2006**, 106, 3140–3169. [Google Scholar] [CrossRef] [PubMed] - MacKerell, A.D., Jr.; Bashford, D.; Bellott, M.; Dunbrack, R.L.; Evanseck, J.D.; Field, M.J.; Fischer, S.; Gao, J.; Guo, H.; Ha, S.; et al. All-Atom Empirical Potential for Molecular Modeling and Dynamics Studies of Proteins. J. Phys. Chem. B
**1998**, 102, 3586–3616. [Google Scholar] [CrossRef] [PubMed] - Bowman, J.M.; Wang, D.Y.; Huang, X.C.; Huarte-Larranaga, F.; Manthe, U. The importance of an accurate CH4 vibrational partition function in full dimensionality calculations of the H + CH4 -> H2 + CH3 reaction. J. Chem. Phys.
**2001**, 114, 9683–9684. [Google Scholar] [CrossRef] - Gao, J. Methods and applications of combined quantum mechanical and molecular mechanical potentials. In Reviews in Computational Chemistry; Lipkowitz, K.B., Boyd, D.B., Eds.; VCH: New York, NY, USA, 1995; pp. 119–185. [Google Scholar]
- Liu, M.Y.; Wang, Y.J.; Chen, Y.K.; Field, M.J.; Gao, J.L. QM/MM through the 1990s: The First Twenty Years of Method Development and Applications. Israel J. Chem.
**2014**, 54, 1250–1263. [Google Scholar] [CrossRef] [PubMed][Green Version] - Garcia-Viloca, M.; Nam, K.; Alhambra, C.; Gao, J. Solvent and Protein Effects on the Vibrational Frequency Shift and Energy Relaxation of the Azide Ligand in Carbonic Anhydrase. J. Phys. Chem. B
**2004**, 108, 13501–13512. [Google Scholar] [CrossRef][Green Version] - Major, D.T.; Gao, J. An Integrated Path Integral and Free-Energy Perturbation-Umbrella Sampling Method for Computing Kinetic Isotope Effects of Chemical Reactions in Solution and in Enzymes. J. Chem. Theory Comput.
**2007**, 3, 949–960. [Google Scholar] [CrossRef] [PubMed] - Zwanzig, R. High-temperature equation of state by a perturbation method. I. Nonpolar gases. J. Chem. Phys.
**1954**, 22, 1420–1426. [Google Scholar] [CrossRef] - Feynman, R.P.; Hibbs, A.R. Quantum Mechanics and Path Integrals; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Wong, K.-Y.; Gao, J. Systematic Approach for Computing Zero-Point Energy, Quantum Partition Function, and Tunneling Effect Based on Kleinert’s Variational Perturbation Theory. J. Chem. Theory Comput.
**2008**, 4, 1409–1422. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gao, J. Enzymatic Kinetic Isotope Effects from Path-Integral Free Energy Perturbation Theory. Methods Enzymol.
**2016**, 577, 359–388. [Google Scholar] [PubMed][Green Version] - Major, D.T.; Gao, J. Implementation of the bisection sampling method in path integral simulations. J. Mol. Graph. Model.
**2005**, 24, 121–127. [Google Scholar] [CrossRef] [PubMed] - Wong, K.-Y.; Gao, J. An automated integration-free path-integral method based on Kleinert’s variational perturbation theory. J. Chem. Phys.
**2007**, 127, 211103. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cao, J.; Voth, G.A. A unified framework for quantum activated rate processes. I. General theory. J. Chem. Phys.
**1996**, 105, 6856–6870. [Google Scholar] - Hernandez, R.; Cao, J.; Voth, G.A. On the Feynman path centroid density as a phase space distribution in quantum statistical mechanics. J. Chem. Phys.
**1995**, 103, 5018–5026. [Google Scholar] [CrossRef] - Hwang, J.K.; Warshel, A. A quantized classical path approach for calculations of quantum mechanical rate constants. J. Phys. Chem.
**1993**, 97, 10053–10058. [Google Scholar] [CrossRef] - Sprik, M.; Klein, M.L.; Chandler, D. Staging: A sampling technique for the Monte Carlo evaluation of path integrals. Phys. Rev. B
**1985**, 31, 4234–4244. [Google Scholar] [CrossRef] - Gao, J.; Ma, S.; Major, D.T.; Nam, K.; Pu, J.; Truhlar, D.G. Mechanisms and free energies of enzymatic reactions. Chem. Rev.
**2006**, 106, 3188–3209. [Google Scholar] [CrossRef] [PubMed] - Warshel, A.; Levitt, M. Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. J. Mol. Biol.
**1976**, 103, 227–249. [Google Scholar] [CrossRef] - Field, M.J.; Bash, P.; Karplus, M. A combined quantum mechanical and molecular mechanical potential for molecular dynamics simulations. J. Comput. Chem.
**1990**, 11, 700–733. [Google Scholar] [CrossRef] - Gao, J. Absolute free energy of solvation from Monte Carlo simulations using combined quantum and molecular mechanical potentials. J. Phys. Chem.
**1992**, 96, 537–540. [Google Scholar] [CrossRef] - Gao, J.; Xia, X. A prior evaluation of aqueous polarization effects through Monte Carlo QM-MM simulations. Science
**1992**, 258, 631–635. [Google Scholar] [CrossRef] [PubMed] - Car, R.; Parrinello, M. Unified approach for molecular dynamics and density-functional theory. Phys. Rev. Lett.
**1985**, 55, 2471–2474. [Google Scholar] [CrossRef] [PubMed] - Titmuss, S.J.; Cummins, P.L.; Rendell, A.P.; Bliznyuk, A.A.; Gready, J.E. Comparison of linear-scaling semiempirical methods and combined quantum mechanical/molecular mechanical methods for enzymic reactions. II. An energy decomposition analysis. J. Comput. Chem.
**2002**, 23, 1314–1322. [Google Scholar] [PubMed] - Rohrig, U.F.; Guidoni, L.; Rothlisberger, U. Solvent and protein effects on the structure and dynamics of the rhodopsin chromophore. ChemPhysChem
**2005**, 6, 1836–1847. [Google Scholar] [CrossRef] [PubMed] - Stewart, J.J.P. Application of localized molecular orbitals to the solution of semiempirical self-consistent field equations. Int. J. Quantum Chem.
**1996**, 58, 133–146. [Google Scholar] [CrossRef] - Gao, J.; Zhang, J.Z.H.; Houk, K.N. Beyond QM/MM: Fragment Quantum Mechanical Methods. Acc. Chem. Res.
**2014**, 47, 2711. [Google Scholar] [CrossRef] [PubMed] - Byun, K.; Mo, Y.; Gao, J. New Insight on the Origin of the Unusual Acidity of Meldrum’s Acid from ab Initio and Combined QM/MM Simulation Study. J. Am. Chem. Soc.
**2001**, 123, 3974–3979. [Google Scholar] [CrossRef] [PubMed] - Hudson, P.S.; Woodcock, H.L.; Boresch, S. Use of Nonequilibrium Work Methods to Compute Free Energy Differences Between Molecular Mechanical and Quantum Mechanical Representations of Molecular Systems. J. Phys. Chem. Lett.
**2015**, 6, 4850–4856. [Google Scholar] [CrossRef] [PubMed] - Kearns, F.L.; Hudson, P.S.; Woodcock, H.L.; Boresch, S. Computing converged free energy differences between levels of theory via nonequilibrium work methods: Challenges and opportunities. J. Comput. Chem.
**2017**, 38, 1376–1388. [Google Scholar] [CrossRef] [PubMed] - Jia, X.Y.; Wang, M.T.; Shao, Y.H.; Konig, G.; Brooks, B.R.; Zhang, J.Z.H.; Mei, Y. Calculations of Solvation Free Energy through Energy Reweighting from Molecular Mechanics to Quantum Mechanics. J. Chem. Theory Comput.
**2016**, 12, 499–511. [Google Scholar] [CrossRef] [PubMed] - Konig, G.; Mei, Y.; Pickard, F.C.; Simmonett, A.C.; Miller, B.T.; Herbert, J.M.; Woodcock, H.L.; Brooks, B.R.; Shao, Y.H. Computation of Hydration Free Energies Using the Multiple Environment Single System Quantum Mechanical/Molecular Mechanical Method. J. Chem. Theory Comput.
**2016**, 12, 332–344. [Google Scholar] [CrossRef] [PubMed] - Olsson, M.A.; Ryde, U. Comparison of QM/MM Methods To Obtain Ligand-Binding Free Energies. J. Chem. Theory Comput.
**2017**, 13, 2245–2253. [Google Scholar] [CrossRef] [PubMed] - Steinmann, C.; Olsson, M.A.; Ryde, U. Relative Ligand-Binding Free Energies Calculated from Multiple Short QM/MM MD Simulations. J. Chem. Theory Comput.
**2018**, 14, 3228–3237. [Google Scholar] [CrossRef] [PubMed] - Gao, J. An Automated Procedure for Simulating Chemical Reactions in Solution. Application to the Decarboxylation of 3-Carboxybenzisoxazole in Water. J. Am. Chem. Soc.
**1995**, 117, 8600–8607. [Google Scholar] - Gao, J. Computation of intermolecular interactions with a combined quantum mechanical and classical approach. ACS Symp. Ser.
**1994**, 569, 8–21. [Google Scholar] - Freindorf, M.; Gao, J. Optimization of the Lennard-Jones parameters for a combined ab initio quantum mechanical and molecular mechanical potential using the 3–21 G basis set. J. Comput. Chem.
**1996**, 17, 386–395. [Google Scholar] [CrossRef] - Orozco, M.; Luque, F.J.; Habibollahzadeh, D.; Gao, J. The polarization contribution to the free energy of hydration. J. Chem. Phys.
**1995**, 103, 9112. [Google Scholar] [CrossRef] - Gao, J. Hybrid Quantum Mechanical/Molecular Mechanical Simulations: An Alternative Avenue to Solvent Effects in Organic Chemistry. Acc. Chem. Res.
**1996**, 29, 298–305. [Google Scholar] [CrossRef] - Major, D.T.; York, D.M.; Gao, J. Solvent Polarization and Kinetic Isotope Effects in Nitroethane Deprotonation and Implications to the Nitroalkane Oxidase Reaction. J. Am. Chem. Soc.
**2005**, 127, 16374–16375. [Google Scholar] [CrossRef] [PubMed] - Gao, J. Monte Carlo Quantum Mechanical-Configuration Interaction and Molecular Mechanics Simulation of Solvent Effects on the n. fwdarw.. pi.* Blue Shift of Acetone. J. Am. Chem. Soc.
**1994**, 116, 9324–9328. [Google Scholar] [CrossRef] - Gao, J.; Byun, K. Solvent effects on the n -> pi* transition of pyrimidine in aqueous solution. Theor. Chem. Acc.
**1997**, 96, 151–156. [Google Scholar] [CrossRef] - Lin, Y.-L.; Gao, J. Solvatochromic Shifts of the n -> pi* Transition of Acetone from Steam Vapor to Ambient Aqueous Solution: A Combined Configuration Interaction QM/MM Simulation Study Incorporating Solvent Polarization. J. Chem. Theory Comput.
**2007**, 3, 1484–1493. [Google Scholar] [CrossRef] [PubMed] - Dewar, M.J.S.; Zoebisch, E.G.; Healy, E.F.; Stewart, J.J.P. Development and use of quantum mechanical molecular models. 76. AM1: A new general purpose quantum mechanical molecular model. J. Am. Chem. Soc.
**1985**, 107, 3902–3909. [Google Scholar] - Stewart, J.J.P. Optimization of Parameters for Semiempirical Methods I. Method. J. Comp. Chem.
**1989**, 10, 209–220. [Google Scholar] [CrossRef] - Cui, Q.; Elstner, M.; Kaxiras, E.; Frauenheim, T.; Karplus, M. A QM/MM Implementation of the Self-Consistent Charge Density Functional Tight Binding (SCC-DFTB) Method. J. Phys. Chem. B
**2001**, 105, 569–585. [Google Scholar] [CrossRef] - Gaus, M.; Cui, Q.A.; Elstner, M. DFTB3: Extension of the Self-Consistent-Charge Density-Functional Tight-Binding Method (SCC-DFTB). J. Chem. Theory Comput.
**2011**, 7, 931–948. [Google Scholar] [CrossRef] [PubMed][Green Version] - Chandler, D.; Wolynes, P.G. Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids. J. Chem. Phys.
**1981**, 74, 4078–4095. [Google Scholar] [CrossRef] - Ceperley, D.M. Path integrals in the theory of condensed helium. Rev. Mod. Phys.
**1995**, 67, 279–355. [Google Scholar] [CrossRef] - Jorgensen, W.L.; Ravimohan, C. Monte Carlo simulation of differences in free energies of hydration. J. Chem. Phys.
**1985**, 83, 3050–3054. [Google Scholar] [CrossRef] - McCammon, J.A.; Roux, B.; Voth, G.; Yang, W. Special Issue on Free Energy. J. Chem. Theory Comput.
**2014**, 10, 2631. [Google Scholar] [CrossRef] [PubMed] - Gao, J.; Wong, K.-Y.; Major, D.T. Combined QM/MM and path integral simulations of kinetic isotope effects in the proton transfer reaction between nitroethane and acetate ion in water. J. Comput. Chem.
**2008**, 29, 514–522. [Google Scholar] [CrossRef] [PubMed] - Feynman, R.P.; Kleinert, H. Effective classical partition functions. Phys. Rev. A
**1986**, 34, 5080. [Google Scholar] [CrossRef] - Gillan, M.J. The quantum simulation of hydrogen in metals. Philos. Mag. A
**1988**, 58, 257–283. [Google Scholar] [CrossRef] - Voth, G.A.; Chandler, D.; Miller, W.H. Rigorous formulation of quantum transition state theory and its dynamical corrections. J. Chem. Phys.
**1989**, 91, 7749–7760. [Google Scholar] [CrossRef] - Messina, M.; Schenter, G.K.; Garrett, B.C. Centroid-density, quantum rate theory: Variational optimization of the dividing surface. J. Chem. Phys.
**1993**, 98, 8525–8536. [Google Scholar] [CrossRef] - Cao, J.; Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys.
**1994**, 101, 6184–6192. [Google Scholar] - Hwang, J.-K.; Warshel, A. How Important Are Quantum Mechanical Nuclear Motions in Enzyme Catalysis? J. Am. Chem. Soc.
**1996**, 118, 11745–11751. [Google Scholar] [CrossRef] - Major, D.T.; Garcia-Viloca, M.; Gao, J. Path Integral Simulations of Proton Transfer Reactions in Aqueous Solution Using Combined QM/MM Potentials. J. Chem. Theory Comput.
**2006**, 2, 236–245. [Google Scholar] [CrossRef] [PubMed][Green Version] - Valleau, J.P.; Torrie, G.M. A guide to Monte Carlo for stastistical mechanics: 2. Byways. In Modern Theoretical Chemistry; Berne, B.J., Ed.; Plenum: New York, NY, USA, 1977; pp. 169–194. [Google Scholar]
- Makarov, D.E.; Topaler, M. Quantum transition-state theory below the crossover temperature. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top.
**1995**, 52, 178–188. [Google Scholar] [CrossRef] - Messina, M.; Schenter, G.K.; Garrett, B.C. A variational centroid density procedure for the calculation of transmission coefficients for asymmetric barriers at low temperature. J. Chem. Phys.
**1995**, 103, 3430–3435. [Google Scholar] [CrossRef] - Mills, G.; Schenter, G.K.; Makarov, D.E.; Jonsson, H. Generalized path integral based quantum transition state theory. Chem. Phys. Lett.
**1997**, 278, 91–96. [Google Scholar] [CrossRef][Green Version] - Jang, S.; Voth, G.A. A relationship between centroid dynamics and path integral quantum transition state theory. J. Chem. Phys.
**2000**, 112, 8747–8757. [Google Scholar] [CrossRef] - Habershon, S.; Manolopoulos, D.E.; Markland, T.E.; Miller, T.F. Ring-Polymer Molecular Dynamics: Quantum Effects in Chemical Dynamics from Classical Trajectories in an Extended Phase Space. Ann. Rev. Phys. Chem.
**2013**, 64, 387–413. [Google Scholar] [CrossRef] [PubMed] - Villa, J.; Warshel, A. Energetics and Dynamics of Enzymatic Reactions. J. Phys. Chem. B
**2001**, 105, 7887–7907. [Google Scholar] [CrossRef] - Major, D.T.; Heroux, A.; Orville, A.M.; Valley, M.P.; Fitzpatrick, P.F.; Gao, J. Differential quantum tunneling contributions in nitroalkane oxidase catalyzed and the uncatalyzed proton transfer reaction. Proc. Natl. Acad. Sci. USA
**2009**, 106, 20736–20739. [Google Scholar] [CrossRef] [PubMed] - Marsalek, O.; Chen, P.Y.; Dupuis, R.; Benoit, M.; Meheut, M.; Bacic, Z.; Tuckerman, M.E. Efficient Calculation of Free Energy Differences Associated with Isotopic Substitution Using Path-Integral Molecular Dynamics. J. Chem. Theory Comput.
**2014**, 10, 1440–1453. [Google Scholar] [CrossRef] [PubMed] - Cheng, B.Q.; Ceriotti, M. Direct path integral estimators for isotope fractionation ratios. J. Chem. Phys.
**2014**, 141, 244112. [Google Scholar] [CrossRef] [PubMed][Green Version] - Poltavsky, I.; DiStasio, R.A.; Tkatchenko, A. Perturbed path integrals in imaginary time: Efficiently modeling nuclear quantum effects in molecules and materials. J. Chem. Phys.
**2018**, 148, 102325. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wang, L.; Fried, S.D.; Boxer, S.G.; Markland, T.E. Quantum delocalization of protons in the hydrogen-bond network of an enzyme active site. Proc. Natl. Acad. Sci. USA
**2014**, 111, 18454–18459. [Google Scholar] [CrossRef] [PubMed] - Ceriotti, M.; Markland, T.E. Efficient methods and practical guidelines for simulating isotope effects. J. Chem. Phys.
**2013**, 138, 014112. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jorgensen, W.L.; Chandrasekhar, J.; Madura, J.D.; Impey, R.W.; Klein, M.L. Comparison of simple potential functions for simulating liquid water. J. Chem. Phys.
**1983**, 79, 926–935. [Google Scholar] [CrossRef] - Pollock, E.L.; Ceperley, D.M. Simulation of quantum many-body systems by path-integral methods. Phys. Rev. B
**1984**, 30, 2555–2568. [Google Scholar] [CrossRef] - Lin, Y.-L.; Gao, J.; Rubinstein, A.; Major, D.T. Molecular dynamics simulations of the intramolecular proton transfer and carbanion stabilization in the pyridoxal 5′-phosphate dependent enzymesl-dopa decarboxylase and alanine racemase. Biochim. Biophys. Acta Prot. Proteom.
**2011**, 1814, 1438–1446. [Google Scholar] [CrossRef] [PubMed] - Fan, Y.; Cembran, A.; Ma, S.; Gao, J. Connecting Protein Conformational Dynamics with Catalytic Function As Illustrated in Dihydrofolate Reductase. Biochemistry
**2013**, 52, 2036–2049. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brooks, B.R.; Brooks, C.L.; Mackerell, A.D.; Nilsson, L.; Petrella, R.J.; Roux, B.; Won, Y.; Archontis, G.; Bartels, C.; Boresch, S.; et al. CHARMM: The Biomolecular Simulation Program. J. Comput. Chem.
**2009**, 30, 1545–1614. [Google Scholar] [CrossRef] [PubMed] - Lin, Y.-L.; Gao, J. Kinetic Isotope Effects of L-Dopa Decarboxylase. J. Am. Chem. Soc.
**2011**, 133, 4398–4403. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Computed ratio of

^{18}O and

^{16}O partition functions (Q

^{18}/Q

^{16}) versus the number of quasi-particles P (= 4, 8, 16, 32, 64, and 128) used in path integral simulations of water in aqueous solution. Error bars were obtained from 10-block averages over the entire simulation.

**Figure 2.**Computed ratio of

^{18}O and

^{16}O partition functions (Q

^{18}/Q

^{16}) versus the number of quasi-particles P used in path integral simulations of acetaldehyde in aqueous solution. For most points, the error bars are small and hidden behind the data symbols.

**Figure 3.**Computed

^{18}O/

^{16}O equilibrium isotope fractionation for acetaldehyde in water versus the number of quasi-particles P used in path integral simulations.

**Figure 4.**Computed

^{18}O/

^{16}O equilibrium isotope fractionation for water, acetaldehyde, and acetone in water from eight sequential separate averages, in which the solvent configurations (outer average of Equation (18)) are saved ranging from every 1 to 100 time steps. For each configuration generated from Newtonian molecular dynamics at 25 °C and 1 atm, free-particle path-integral simulations are carried out to sample 210 random closed paths, of which the first 10 are discarded. For each block, 1000 configurations are included. Thus, each point in the figure corresponds to a FPPI averaging over 200,000 paths. 32 quasi-particles for acetaldehyde and acetone and 64 beads are used to represent each discretized path. The error bars for each point are estimated from 10 separate block-averages; those for acetaldehyde are smaller than the size of the symbols displayed, whereas those for water from 32-bead simulation are not shown for similarity to 64-bead. The fluctuations of the block averages mainly feature solvent configuration fluctuations.

**Table 1.**Computed ratio of partition functions and

^{18}O enrichments of CH

_{3}CHO and CH

_{3}COCH

_{3}in aqueous solution at 25 °C and 1 atm using the PM3/TIP3P QM/MM potential.

H_{2}O | CH_{3}CHO | CH_{3}COCH_{3} | |
---|---|---|---|

Q^{18}/Q^{16} | 1.0828 ± 0.0024 | 1.1204 ± 0.0014 | 1.1251 ± 0.0007 |

${\delta}^{18}O$ | 1 | 0.035 | 0.039 |

**Table 2.**Computed ratio of

^{2}H/

^{1}H partition functions and equilibrium isotope fractionation factor of CH

_{3}CHO in the gas phase and in aqueous solution at 25 °C and 1 atm using the PM3/TIP3P QM/MM potential.

CH_{3}C[D/H]O | [D/H]CH_{2}CHO | |||
---|---|---|---|---|

Gas | Aqueous | Gas | Aqueous | |

Q^{2}/Q^{1} | 19.20 ± 0.36 | 19.63 ± 0.98 | 23.72 ± 0.86 | 24.01 ± 1.12 |

${\alpha}_{\mathrm{C}2/\mathrm{C}1}$ | 1 | 1 | 1.24 | 1.22 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, M.; Youmans, K.N.; Gao, J. Dual QM and MM Approach for Computing Equilibrium Isotope Fractionation Factor of Organic Species in Solution. *Molecules* **2018**, *23*, 2644.
https://doi.org/10.3390/molecules23102644

**AMA Style**

Liu M, Youmans KN, Gao J. Dual QM and MM Approach for Computing Equilibrium Isotope Fractionation Factor of Organic Species in Solution. *Molecules*. 2018; 23(10):2644.
https://doi.org/10.3390/molecules23102644

**Chicago/Turabian Style**

Liu, Meiyi, Katelyn N. Youmans, and Jiali Gao. 2018. "Dual QM and MM Approach for Computing Equilibrium Isotope Fractionation Factor of Organic Species in Solution" *Molecules* 23, no. 10: 2644.
https://doi.org/10.3390/molecules23102644