# 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

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**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/).

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**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