Combined Quantum Mechanical/Molecular Mechanical Methodologies Applied to Biomolecular Systems

Theoretical trends in the dynamics simulations of molecular machines across multiple scales

Over the past few decades, molecular machines have been extensively studied, since they are composed of single molecules for functional materials capable of responding to external stimuli, enabling motion at scales ranging from the microscopic to the macroscopic level within molecular aggregates. This advancement holds the potential to efficiently transform external resources into mechanical movement, achieved through precise control of conformational changes in stimuli-responsive materials. However, the underlying mechanism that links microscopic and macroscopic motions remains unclear, demanding computational development associated with simulating the construction of molecular machines from single molecules. This bottleneck has impeded the design of more efficient functional materials. Advancements in theoretical simulations have successfully been developed in various computational models to unveil the operational mechanisms of stimulus-responsive molecular machines, which could help us reduce the costs in experimental trial-and-error procedures. It opens doors to the computer-aided design of innovative functional materials. In this perspective, we have reviewed theoretical approaches employed in simulating dynamic processes involving conformational changes in molecular machines, spanning different scales and environmental conditions. In addition, we have highlighted current challenges and anticipated future trends in the collective control of aggregates within molecular machines. Our goal is to provide a comprehensive overview of recent theoretical advancements in the field of molecular machines, offering valuable insights for the design of novel smart materials.

Solvation Effects on the Thermal Helix Inversion of Molecular Motors from QM/MM Calculations

Molecular motors convert light and thermal energies into mechanical work, offering good opportunities to design novel molecular devices. Among them, molecular motors alternate a photoisomerization and a thermal helix inversion to achieve unidirectional rotation. The rotational speed is limited by the helix inversion step, which in turn is governed by a barrier in the electronic ground state. In this work, we systematically study the solvation effect on the thermal process of selected molecular motors, comparing reaction barriers obtained from both density functional theory (DFT) in the isolated system and umbrella sampling within a hybrid quantum mechanics/molecular mechanics (QM/MM) model in solution. We find more prominent solvation effects on those molecular motors with larger dipole moments. The results could provide insight into how to functionalize molecular motors to speed up their rotation.

Factors That Determine the Variation of Equilibrium and Kinetic Properties of QM/MM Enzyme Simulations: QM Region, Conformation, and Boundary Condition

To analyze the impact of various technical details on the results of quantum mechanical (QM)/molecular mechanical (MM) enzyme simulations, including the QM region size, catechol-O-methyltransferase (COMT) is studied as a model system using an approximate QM/MM method (DFTB3/CHARMM). The results show that key equilibrium and kinetic properties for methyl transfer in COMT exhibit limited variations with respect to the size of the QM region, which ranges from ∼100 to ∼500 atoms in this study. With extensive sampling, local and global structural characteristics of the enzyme are largely conserved across the studied QM regions, while the nature of the transition state (e.g., secondary kinetic isotope effect) and reaction exergonicity are largely maintained. Deviations in the free energy profile with different QM region sizes are similar in magnitude to those observed with changes in other simulation protocols, such as different initial enzyme conformations and boundary conditions. Electronic structural properties, such as the covariance matrix of residual charge fluctuations, appear to exhibit rather long-range correlations, especially when the peptide backbone is included in the QM region; this observation holds when a range-separated DFT approach is used as the QM region, suggesting that delocalization error is unlikely the origin. Overall, the analyses suggest that multiple simulation details determine the results of QM/MM enzyme simulations with comparable contributions.

Quantum-Mechanical/Molecular-Mechanical (QM/MM) Simulations for Understanding Enzyme Dynamics

Quantum mechanics/molecular mechanics (QM/MM) methods have become widely used for computational modeling of enzyme structure and mechanism. In these approaches, a portion of the enzyme of great interest (e.g., where a chemical reaction is occurring) is treated with QM, whereas the surrounding region is treated with MM. A critical challenge with these methods is the choice of the region to partition into QM and which to treat with MM along with numerous practical choices that must be made at each step of the modeling procedure. Here, we attempt to simplify this process by describing the steps involved in preparing protein structures, choosing the appropriate QM region size and electronic structure methods, preparing all necessary input files, and troubleshooting common errors for QM/MM simulations of enzymes.

Affordable Ab Initio Path Integral for Thermodynamic Properties via Molecular Dynamics Simulations Using Semiempirical Reference Potential

Path integral molecular dynamics (PIMD) is becoming a routinely applied method for incorporating the nuclear quantum effect in computer simulations. However, direct PIMD simulations at an ab initio level of theory are formidably expensive. Using the protonated 1,8-bis(dimethylamino)naphthalene molecule as an example, we show in this work that the computational expense for the intramolecular proton transfer between the two nitrogen atoms can be remarkably reduced by implementing the idea of reference-potential methods. The simulation time can be easily extended to a scale of nanoseconds while maintaining the accuracy on an ab initio level of theory for thermodynamic properties. In addition, postprocessing can be carried out in parallel on massive computer nodes. A 545-fold reduction in the total CPU time can be achieved in this way as compared to a direct PIMD simulation at the same ab initio level of theory.

Multiscale Quantum Refinement Approaches for Metalloproteins

Biomolecules with metal ion(s) (e.g., metalloproteins) play many important biological roles. However, accurate structural determination of metalloproteins, particularly those containing transition metal ion(s), is challenging due to their complicated electronic structure, complex bonding of metal ions, and high number of conformations in biomolecules. Quantum refinement, which was proposed to combine crystallographic data with computational chemistry methods by several groups, can improve the local structures of some proteins. In this study, a quantum refinement method combining several multiscale computational schemes with experimental (X-ray diffraction) information was developed for metalloproteins. Various quantum refinement approaches using different ONIOM (our own N-layered integrated molecular orbital and molecular mechanics) combinations of quantum mechanics (QM), semiempirical (SE), and molecular mechanics (MM) methods were conducted to assess the performance and reliability on the refined local structure in two metalloproteins. The structures for two (Cu- or Zn-containing) metalloproteins were refined by combining two-layer ONIOM2(QM1/QM2) and ONIOM2(QM/MM) and three-layer ONIOM3(QM1/QM2/MM) schemes with experimental data. The accuracy of the quantum-refined metal binding sites was also examined and compared in these multiscale quantum refinement calculations. ONIOM3(QM/SE/MM) schemes were found to give good results with lower computational costs and were proposed to be a good choice for the multiscale computational scheme for quantum refinement calculations of metal binding site(s) in metalloproteins with high efficiency. Additionally, a two-center ONIOM approach was employed to speed up the quantum refinement calculations for the Zn metalloprotein with two remote active sites/ligands. Moreover, a recent quantum-embedding wavefunction-in-density functional theory (WF-in-DFT) method was also adopted as the high-level method in unprecedented ONIOM2(CCSD-in-B3LYP/MM) and ONIOM3(CCSD-in-B3LYP/SE/MM) calculations, which can be regarded as novel pseudo-three- and pseudo-four-layer ONIOM methods, respectively, to refine the key Zn binding site at the coupled-cluster singles and doubles (CCSD) level. These refined results indicate that multiscale quantum refinement schemes can be used to improve the structural accuracy obtained for local metal binding site(s) in metalloproteins with high efficiency.

Accelerated Computation of Free Energy Profile at Ab Initio Quantum Mechanical/Molecular Mechanics Accuracy via a Semiempirical Reference Potential. 4. Adaptive QM/MM

Although quantum mechanical/molecular mechanics (QM/MM) methods are now routinely applied to the studies of chemical reactions in condensed phases and enzymatic reactions, they may experience technical difficulties when the reactive region is varying over time. For instance, when the solvent molecules are directly participating in the reaction, the exchange of water molecules between the QM and MM regions may occur on a time scale comparable to the reaction time. To cope with this situation, several adaptive QM/MM schemes have been proposed. However, these methods either add significantly to the computational cost or introduce artificial restraints to the system. In this work, we developed a novel adaptive QM/MM scheme and applied it to the study of a nucleophilic addition reaction. In this scheme, the configuration sampling was performed with a small QM region (without solvent molecules), and the thermodynamic properties under another potential energy function with a larger QM region (with a certain number of solvent molecules and/or different levels of QM theory) are computed via extrapolation using the reference-potential method. Our simulation results show that this adaptive QM/MM scheme is numerically stable, at least for the case studied in this work. Furthermore, this method also offers an inexpensive way to examine the convergence of the QM/MM calculation with respect to the size of the QM region.

Insight into the Catalytic Mechanism of GH11 Xylanase: Computational Analysis of Substrate Distortion Based on a Neutron Structure

The reaction mechanism of biomass decomposition by xylanases remains the subject of debate. To clarify the mechanism we investigated the glycosylation step of GH11 xylanase, an enzyme that catalyzes the hydrolysis of lignocellulosic hemicellulose (xylan). Making use of a recent neutron crystal structure, which revealed the protonation states of relevant residues, we used ab initio quantum mechanics/molecular mechanics (QM/MM) calculations to determine the detailed reaction mechanism of the glycosylation step. In particular, our focus is on the controversial question of whether or not an oxocarbenium ion intermediate is formed on the reaction pathway. The calculations support the validity of a basic retaining mechanism within a double-displacement scheme. The estimated free energy barrier of this reaction is ∼18 kcal/mol with QM/MM-CCSD(T)/6-31(+)G**//MP2/6-31+G**/AMBER calculations, and the rate-determining step of the glycosylation is scission of the glycosidic bond after proton transfer from the acidic Glu177. The estimated lifetime of the oxocarbenium ion intermediate (on the order of tens of ps) and the secondary kinetic isotope effect suggest that there is no accumulation of this intermediate on the reaction path, although the intermediate can be transiently formed. In the enzyme-substrate (ES) complex, the carbohydrate structure of the xylose residue at the -1 subsite has a rather distorted (skewed) geometry, and this xylose unit at the active site has an apparent half-chair conformation when the oxocarbenium ion intermediate is formed. The major catalytic role of the protein environment is to orient residues that take part in the initial proton transfer. Because of a fine alignment of catalytic residues, the enzyme can accelerate the glycosylation reaction without paying a reorganization energy penalty.

Multi-level free energy simulation with a staged transformation approach

Combining multiple levels of theory in free energy simulations to balance computational accuracy and efficiency is a promising approach for studying processes in the condensed phase. While the basic idea has been proposed and explored for quite some time, it remains challenging to achieve convergence for such multi-level free energy simulations as it requires a favorable distribution overlap between different levels of theory. Previous efforts focused on improving the distribution overlap by either altering the low-level of theory for the specific system of interest or ignoring certain degrees of freedom. Here, we propose an alternative strategy that first identifies the degrees of freedom that lead to gaps in the distributions of different levels of theory and then treats them separately with either constraints or restraints or by introducing an intermediate model that better connects the low and high levels of theory. As a result, the conversion from the low level to the high level model is done in a staged fashion that ensures a favorable distribution overlap along the way. Free energy components associated with different steps are mostly evaluated explicitly, and thus, the final result can be meaningfully compared to the rigorous free energy difference between the two levels of theory with limited and well-defined approximations. The additional free energy component calculations involve simulations at the low level of theory and therefore do not incur high computational costs. The approach is illustrated with two simple but non-trivial solution examples, and factors that dictate the reliability of the result are discussed.

The Impact of Electron Correlation on Describing QM/MM Interactions in the Attendant Molecular Dynamics Simulations of CO in Myoglobin

The impact of the dispersion and electron correlation effects on describing quantum mechanics/molecular mechanics (QM/MM) interactions in QM/MM molecular dynamics (MD) simulations was explored by performing a series of up to 2 ns QM/MM MD simulations on the B states of the myoglobin-carbon monoxide (MbCO) system. The results indicate that both dispersion and electron correlations play significant roles in the simulation of the ratios of two B states (B₁/B₂), which suggests that the inclusion of the electron correlation effects is essential for accurately modeling the interactions between QM and MM subsystems. We found that the QM/MM interaction energies between the CO and the surroundings statistically present a linear correlation with the electric fields along the CO bond. This indicates that QM/MM interactions can be described by a simple physical model of a dipole with constant moment under the action of the electric fields. The treatment provides us with an accurate and effective approach to account for the electron correlation effects in QM/MM MD simulations.

Both Configuration and QM Region Size Matter: Zinc Stability in QM/MM Models of DNA Methyltransferase

Quantum-mechanical/molecular-mechanical (QM/MM) methods are essential to the study of metalloproteins, but the relative importance of sampling and degree of QM treatment in achieving quantitative predictions is poorly understood. We study the relative magnitude of configurational and QM-region sensitivity of energetic and electronic properties in a representative Zn²⁺ metal binding site of a DNA methyltransferase. To quantify property variations, we analyze snapshots extracted from 250 ns of molecular dynamics simulation. To understand the degree of QM-region sensitivity, we perform analysis using QM regions ranging from a minimal 49-atom region consisting only of the Zn²⁺ metal and its four coordinating Cys residues up to a 628-atom QM region that includes residues within 12 Å of the metal center. Over the configurations sampled, we observe that illustrative properties (e.g., rigid Zn²⁺ removal energy) exhibit large fluctuations that are well captured with even minimal QM regions. Nevertheless, for both energetic and electronic properties, we observe a slow approach to asymptotic limits with similarly large changes in absolute values that converge only with larger (ca. 300-atom) QM region sizes. For the smaller QM regions, the electronic description of Zn²⁺ binding is incomplete: the metal binds too tightly and is too stabilized by the strong electrostatic potential of MM point charges, and the Zn-S bond covalency is overestimated. Overall, this work suggests that efficient sampling with QM/MM in small QM regions is an effective method to explore the influence of enzyme structure on target properties. At the same time, accurate descriptions of electronic and energetic properties require a larger QM region than the minimal metal-coordinating residues in order to converge treatment of both metal-local bonding and the overall electrostatic environment.

Polarizable Embedding with a Transferable H2O Potential Function I: Formulation and Tests on Dimer

The incorporation of mutual polarization in multiscale simulations where different regions of the system are treated at different level of theory is important in studies of, for example, electronic excitations and charge transfer processes. We present here an energy functional for describing a quantum mechanics/molecular mechanics (QM/MM) scheme that includes reciprocal polarization between the two subsystems. The inclusion of polarization alleviates shortcomings inherent in electrostatic embedding QM/MM models based on point-charge force fields. A density functional theory (DFT) description of the QM subsystem is coupled to a single center multipole expansion (SCME) description of H₂O molecules in the MM subsystem that includes anisotropic dipole and quadrupole polarizability as well as static multipoles up to and including the hexadecapole. The energy functional and the coupling scheme is general and can be extended to arbitrary order in terms of both the static and induced moments. Tests of the energy surface for the H₂O dimer show that the QM/MM results lie in between the pure DFT and pure SCME values. The consistency of the many-body contributions to the energy and analytical forces is demonstrated for an H₂O pentamer.

Revealing quantum mechanical effects in enzyme catalysis with large-scale electronic structure simulation

Enzymes have evolved to facilitate challenging reactions at ambient conditions with specificity seldom matched by other catalysts. Computational modeling provides valuable insight into catalytic mechanism, and the large size of enzymes mandates multi-scale, quantum mechanical-molecular mechanical (QM/MM) simulations. Although QM/MM plays an essential role in balancing simulation cost to enable sampling with full QM treatment needed to understand electronic structure in enzyme active sites, the relative importance of these two strategies for understanding enzyme mechanism is not well known. We explore challenges in QM/MM for studying the reactivity and stability of three diverse enzymes: i) Mg2+-dependent catechol O-methyltransferase (COMT), ii) radical enzyme choline trimethylamine lyase (CutC), and iii) DNA methyltransferase (DNMT1), which has structural Zn2+ binding sites. In COMT, strong non-covalent interactions lead to long range coupling of electronic structure properties across the active site, but the more isolated nature of the metallocofactor in DNMT1 leads to faster convergence of some properties. We quantify these effects in COMT by computing covariance matrices of by-residue electronic structure properties during dynamics and along the reaction coordinate. In CutC, we observe spontaneous bond cleavage following initiation events, highlighting the importance of sampling and dynamics. We use electronic structure analysis to quantify the relative importance of CHO and OHO non-covalent interactions in imparting reactivity. These three diverse cases enable us to provide some general recommendations regarding QM/MM simulation of enzymes.

Semi-Empirical Born-Oppenheimer Molecular Dynamics (SEBOMD) within the Amber Biomolecular Package

Semi-empirical quantum methods from the neglect of differential diatomic overlap (NDDO) family such as MNDO, AM1, or PM3 are fast albeit approximate quantum methods. By combining them with linear scaling methods like the divide & conquer (D&C) method, it is possible to quickly evaluate the energy of systems containing hundreds to thousands of atoms. We here present our implementation in the Amber biomolecular package of a SEBOMD module that provides a way to run semi-empirical Born-Oppenheimer molecular dynamics. At each step of a SEBOMD, a fully converged self-consistent field (SCF) calculation is performed to obtain the semiempirical quantum potential energy of a molecular system encaged or not in periodic boundary conditions. We describe the implementation and the features of our SEBOMD implementation. We show the requirements to conserve the total energy in NVE simulations, and how to accelerate SCF convergence through density matrix extrapolation. Specific ways of handling periodic boundary conditions using mechanical embedding or electrostatic embedding through a tailored quantum Ewald summation is developed. The parallel performance of SEBOMD simulations using the D&C scheme are presented for liquid water systems of various sizes, and a comparison between the traditional full diagonalization scheme and the D&C approach for the reproduction of the structure of liquid water illustrates the potentiality of SEBOMD to simulate molecular systems containing several hundreds of atoms for hundreds of picoseconds with a quantum mechanical potential in a reasonable amount of CPU time.

Computational analysis of carbohydrate recognition based on hybrid QM/MM modeling: a case study of norovirus capsid protein in complex with Lewis antigen

Norovirus is a major pathogen of nonbacterial acute gastroenteritis in humans and animals. Carbohydrate recognition between norovirus capsid proteins and Lewis antigens is considered to play a critical role in initiating infection of eukaryotic cells. In this article, we first report a detailed atomistic simulation study of the norovirus capsid protein in complex with the Lewis antigen based on ab initio QM/MM combined with MD-FEP simulations. To understand the mechanistic details of ligand binding, we analyzed and compared the carbohydrate recognition mechanism of the wild-type P domain protein with a mutant protein. Small structural differences between two capsid proteins are observed on the weak interaction site of residue 389, which is located on the solvent exposed surface of the P domain. To further clarify affinity differences in ligand binding, we directly evaluated free energy changes of the ligand binding process. Although the mutant protein loses its interaction energy with the Lewis antigen, this small amount of energy penalty is compensated for by an increase in the solvation stability, which is induced by structural reorganization at the ligand binding site on the protein surface. As a sum of these opposite energy components, the mutant P domain obtains a slightly enhanced binding affinity for the Lewis antigen. The present computational study clearly demonstrated that a detailed free energy balance of the interaction energy between the capsid protein and the surrounding aqueous solvent is the mechanistic basis of carbohydrate recognition in the norovirus capsid protein.

Adaptive QM/MM for Molecular Dynamics Simulations: 5. On the Energy-Conserved Permuted Adaptive-Partitioning Schemes

In combined quantum-mechanical/molecular-mechanical (QM/MM) dynamics simulations, the adaptive-partitioning (AP) schemes reclassify atoms on-the-fly as QM or MM in a smooth manner. This yields a mobile QM subsystem with contents that are continuously updated as needed. Here, we tailor the Hamiltonian adaptive many-body correction (HAMBC) proposed by Boreboom et al. [J. Chem. Theory Comput.2016, 12, 3441] to the permuted AP (PAP) scheme. The treatments lead to the HAMBC-PAP method (HPAP), which both conserves energy and produces accurate solvation structures in the test of "water-in-water" model system.

Analysis of Phosphoryl-Transfer Enzymes with QM/MM Free Energy Simulations

We discuss the application of quantum mechanics/molecular mechanics (QM/MM) free energy simulations to the analysis of phosphoryl transfers catalyzed by two enzymes: alkaline phosphatase and myosin. We focus on the nature of the transition state and the issue of mechanochemical coupling, respectively, in the two enzymes. The results illustrate unique insights that emerged from the QM/MM simulations, especially concerning the interpretation of experimental data regarding the nature of enzymatic transition states and coupling between global structural transition and catalysis in the active site. We also highlight a number of technical issues worthy of attention when applying QM/MM free energy simulations, and comment on a number of technical and mechanistic issues that require further studies.

Large-scale QM/MM free energy simulations of enzyme catalysis reveal the influence of charge transfer

Hybrid quantum mechanical-molecular mechanical (QM/MM) simulations provide key insights into enzyme structure-function relationships. Numerous studies have demonstrated that large QM regions are needed to systematically converge ground state, zero temperature properties with electrostatic embedding QM/MM. However, it is not well known if ab initio QM/MM free energy simulations have this same dependence, in part due to the hundreds of thousands of energy evaluations required for free energy estimations that in turn limit QM region size. Here, we leverage recent advances in electronic structure efficiency and accuracy to carry out range-separated hybrid density functional theory free energy simulations in a representative methyltransferase. By studying 200 ps of ab initio QM/MM dynamics for each of five QM regions from minimal (64 atoms) to one-sixth of the protein (544 atoms), we identify critical differences between large and small QM region QM/MM in charge transfer between substrates and active site residues as well as in geometric structure and dynamics that coincide with differences in predicted free energy barriers. Distinct geometric and electronic structure features in the largest QM region indicate that important aspects of enzymatic rate enhancement in methyltransferases are identified with large-scale electronic structure.

Determination of accurate backbone chemical shift tensors in microcrystalline proteins by integrating MAS NMR and QM/MM

Chemical shifts are highly sensitive probes of local conformation and overall structure. Both isotropic shifts and chemical shift tensors are readily accessible from NMR experiments but their quantum mechanical calculations remain challenging. In this work, we report and compare accurately measured and calculated 15NH and 13Cα chemical shift tensors in proteins, using the microcrystalline agglutinin from Oscillatoria agardhii (OAA). Experimental 13Cα and 15NH chemical tensors were obtained by solid-state NMR spectroscopy, employing tailored recoupling sequences, and for their quantum mechanics/molecular mechanics (QM/MM) calculations different sets of functionals were evaluated. We show that 13Cα chemical shift tensors are primarily determined by backbone dihedral angles and dynamics, while 15NH tensors mainly depend on local electrostatic contributions from solvation and hydrogen bonding. In addition, the influence of including crystallographic waters, the molecular mechanics geometry optimization protocol, and the level of theory on the accuracy of the calculated chemical shift tensors is discussed. Specifically, the power of QM/MM calculations in accurately predicting the unusually upfield shifted 1HN G26 and G93 resonances is highlighted. Our integrated approach is expected to benefit structure refinement of proteins and protein assemblies.

A Strategically Located Arg/Lys Residue Promotes Correct Base Paring During Nucleic Acid Biosynthesis in Polymerases

Polymerases (Pols) synthesize the double-stranded nucleic acids in the Watson-Crick (W-C) conformation, which is critical for DNA and RNA functioning. Yet, the molecular basis to catalyze the W-C base pairing during Pol-mediated nucleic acids biosynthesis remains unclear. Here, through bioinformatics analyses on a large data set of Pol/DNA structures, we first describe the conserved presence of one positively charged residue (Lys or Arg), which is similarly located near the enzymatic two-metal active site, always interacting directly with the incoming substrate (d)NTP. Incidentally, we noted that some Pol/DNA structures showing the alternative Hoogsteen base pairing were often solved with this specific residue either mutated, displaced, or missing. We then used quantum and classical simulations coupled to free-energy calculations to illustrate how, in human DNA Pol-η, the conserved Arg61 favors W-C base pairing through defined interactions with the incoming nucleotide. Taken together, these structural observations and computational results suggest a structural framework in which this specific residue is critical for stabilizing the incoming (d)NTP nucleotide and base pairing during Pol-mediated nucleic acid biosynthesis. These results may benefit enzyme engineering for nucleic acid processing and encourage new drug discovery strategies to modulate Pols function.

Grid-Based Projector Augmented Wave (GPAW) Implementation of Quantum Mechanics/Molecular Mechanics (QM/MM) Electrostatic Embedding and Application to a Solvated Diplatinum Complex

A multiscale density functional theory-quantum mechanics/molecular mechanics (DFT-QM/MM) scheme is presented, based on an efficient electrostatic coupling between the electronic density obtained from a grid-based projector augmented wave (GPAW) implementation of density functional theory and a classical potential energy function. The scheme is implemented in a general fashion and can be used with various choices for the descriptions of the QM or MM regions. Tests on H₂O clusters, ranging from dimer to decamer show that no systematic energy errors are introduced by the coupling that exceeds the differences in the QM and MM descriptions. Over 1 ns of liquid water, Born-Oppenheimer QM/MM molecular dynamics (MD) are sampled combining 10 parallel simulations, showing consistent liquid water structure over the QM/MM border. The method is applied in extensive parallel MD simulations of an aqueous solution of the diplatinum [Pt₂(P₂O₅H₂)₄]^4- complex (PtPOP), spanning a total time period of roughly half a nanosecond. An average Pt-Pt distance deviating only 0.01 Å from experimental results, and a ground-state Pt-Pt oscillation frequency deviating by <2% from experimental results were obtained. The simulations highlight a remarkable harmonicity of the Pt-Pt oscillation, while also showing clear signs of Pt-H hydrogen bonding and directional coordination of water molecules along the Pt-Pt axis of the complex.

Theoretical approaches for dynamical ordering of biomolecular systems

BACKGROUND

Living systems are characterized by the dynamic assembly and disassembly of biomolecules. The dynamical ordering mechanism of these biomolecules has been investigated both experimentally and theoretically. The main theoretical approaches include quantum mechanical (QM) calculation, all-atom (AA) modeling, and coarse-grained (CG) modeling. The selected approach depends on the size of the target system (which differs among electrons, atoms, molecules, and molecular assemblies). These hierarchal approaches can be combined with molecular dynamics (MD) simulation and/or integral equation theories for liquids, which cover all size hierarchies.

SCOPE OF REVIEW

We review the framework of quantum mechanical/molecular mechanical (QM/MM) calculations, AA MD simulations, CG modeling, and integral equation theories. Applications of these methods to the dynamical ordering of biomolecular systems are also exemplified.

MAJOR CONCLUSIONS

The QM/MM calculation enables the study of chemical reactions. The AA MD simulation, which omits the QM calculation, can follow longer time-scale phenomena. By reducing the number of degrees of freedom and the computational cost, CG modeling can follow much longer time-scale phenomena than AA modeling. Integral equation theories for liquids elucidate the liquid structure, for example, whether the liquid follows a radial distribution function.

GENERAL SIGNIFICANCE

These theoretical approaches can analyze the dynamic behaviors of biomolecular systems. They also provide useful tools for exploring the dynamic ordering systems of biomolecules, such as self-assembly. This article is part of a Special Issue entitled "Biophysical Exploration of Dynamical Ordering of Biomolecular Systems" edited by Dr. Koichi Kato.

Perspective: Quantum mechanical methods in biochemistry and biophysics

In this perspective article, I discuss several research topics relevant to quantum mechanical (QM) methods in biophysical and biochemical applications. Due to the immense complexity of biological problems, the key is to develop methods that are able to strike the proper balance of computational efficiency and accuracy for the problem of interest. Therefore, in addition to the development of novel ab initio and density functional theory based QM methods for the study of reactive events that involve complex motifs such as transition metal clusters in metalloenzymes, it is equally important to develop inexpensive QM methods and advanced classical or quantal force fields to describe different physicochemical properties of biomolecules and their behaviors in complex environments. Maintaining a solid connection of these more approximate methods with rigorous QM methods is essential to their transferability and robustness. Comparison to diverse experimental observables helps validate computational models and mechanistic hypotheses as well as driving further development of computational methodologies.

How Large Should the QM Region Be in QM/MM Calculations? The Case of Catechol O-Methyltransferase

Hybrid quantum mechanical-molecular mechanical (QM/MM) simulations are widely used in studies of enzymatic catalysis. Until recently, it has been cost prohibitive to determine the asymptotic limit of key energetic and structural properties with respect to increasingly large QM regions. Leveraging recent advances in electronic structure efficiency and accuracy, we investigate catalytic properties in catechol O-methyltransferase, a prototypical methyltransferase critical to human health. Using QM regions ranging in size from reactants-only (64 atoms) to nearly one-third of the entire protein (940 atoms), we show that properties such as the activation energy approach within chemical accuracy of the large-QM asymptotic limits rather slowly, requiring approximately 500-600 atoms if the QM residues are chosen simply by distance from the substrate. This slow approach to asymptotic limit is due to charge transfer from protein residues to the reacting substrates. Our large QM/MM calculations enable identification of charge separation for fragments in the transition state as a key component of enzymatic methyl transfer rate enhancement. We introduce charge shift analysis that reveals the minimum number of protein residues (approximately 11-16 residues or 200-300 atoms for COMT) needed for quantitative agreement with large-QM simulations. The identified residues are not those that would be typically selected using criteria such as chemical intuition or proximity. These results provide a recipe for a more careful determination of QM region sizes in future QM/MM studies of enzymes.

A Self-Activated Mechanism for Nucleic Acid Polymerization Catalyzed by DNA/RNA Polymerases

The enzymatic polymerization of DNA and RNA is the basis for genetic inheritance for all living organisms. It is catalyzed by the DNA/RNA polymerase (Pol) superfamily. Here, bioinformatics analysis reveals that the incoming nucleotide substrate always forms an H-bond between its 3'-OH and β-phosphate moieties upon formation of the Michaelis complex. This previously unrecognized H-bond implies a novel self-activated mechanism (SAM), which synergistically connects the in situ nucleophile formation with subsequent nucleotide addition and, importantly, nucleic acid translocation. Thus, SAM allows an elegant and efficient closed-loop sequence of chemical and physical steps for Pol catalysis. This is markedly different from previous mechanistic hypotheses. Our proposed mechanism is corroborated via ab initio QM/MM simulations on a specific Pol, the human DNA polymerase-η, an enzyme involved in repairing damaged DNA. The structural conservation of DNA and RNA Pols supports the possible extension of SAM to Pol enzymes from the three domains of life.

A quantum-mechanics molecular-mechanics scheme for extended systems

We introduce and discuss a hybrid quantum-mechanics molecular-mechanics (QM-MM) approach for Car-Parrinello DFT simulations with pseudopotentials and planewaves basis, designed for the treatment of periodic systems. In this implementation the MM atoms are considered as additional QM ions having fractional charges of either sign, which provides conceptual and computational simplicity by exploiting the machinery already existing in planewave codes to deal with electrostatics in periodic boundary conditions. With this strategy, both the QM and MM regions are contained in the same supercell, which determines the periodicity for the whole system. Thus, while this method is not meant to compete with non-periodic QM-MM schemes able to handle extremely large but finite MM regions, it is shown that for periodic systems of a few hundred atoms, our approach provides substantial savings in computational times by treating classically a fraction of the particles. The performance and accuracy of the method is assessed through the study of energetic, structural, and dynamical aspects of the water dimer and of the aqueous bulk phase. Finally, the QM-MM scheme is applied to the computation of the vibrational spectra of water layers adsorbed at the TiO2 anatase (1 0 1) solid-liquid interface. This investigation suggests that the inclusion of a second monolayer of H2O molecules is sufficient to induce on the first adsorbed layer, a vibrational dynamics similar to that taking place in the presence of an aqueous environment. The present QM-MM scheme appears as a very interesting tool to efficiently perform molecular dynamics simulations of complex condensed matter systems, from solutions to nanoconfined fluids to different kind of interfaces.

Extension of the Effective Fragment Potential Method to Macromolecules

The effective fragment potential (EFP) approach, which can be described as a nonempirical polarizable force field, affords an accurate first-principles treatment of noncovalent interactions in extended systems. EFP can also describe the effect of the environment on the electronic properties (e.g., electronic excitation energies and ionization and electron-attachment energies) of a subsystem via the QM/EFP (quantum mechanics/EFP) polarizable embedding scheme. The original formulation of the method assumes that the system can be separated, without breaking covalent bonds, into closed-shell fragments, such as solvent and solute molecules. Here, we present an extension of the EFP method to macromolecules (mEFP). Several schemes for breaking a large molecule into small fragments described by EFP are presented and benchmarked. We focus on the electronic properties of molecules embedded into a protein environment and consider ionization, electron-attachment, and excitation energies (single-point calculations only). The model systems include chromophores of green and red fluorescent proteins surrounded by several nearby amino acid residues and phenolate bound to the T4 lysozyme. All mEFP schemes show robust performance and accurately reproduce the reference full QM calculations. For further applications of mEFP, we recommend either the scheme in which the peptide is cut along the Cα-C bond, giving rise to one fragment per amino acid, or the scheme with two cuts per amino acid, along the Cα-C and Cα-N bonds. While using these fragmentation schemes, the errors in solvatochromic shifts in electronic energy differences (excitation, ionization, electron detachment, or electron-attachment) do not exceed 0.1 eV. The largest error of QM/mEFP against QM/EFP (no fragmentation of the EFP part) is 0.06 eV (in most cases, the errors are 0.01-0.02 eV). The errors in the QM/molecular mechanics calculations with standard point charges can be as large as 0.3 eV.

QM/MM free energy simulations: recent progress and challenges

Due to the higher computational cost relative to pure molecular mechanical (MM) simulations, hybrid quantum mechanical/molecular mechanical (QM/MM) free energy simulations particularly require a careful consideration of balancing computational cost and accuracy. Here we review several recent developments in free energy methods most relevant to QM/MM simulations and discuss several topics motivated by these developments using simple but informative examples that involve processes in water. For chemical reactions, we highlight the value of invoking enhanced sampling technique (e.g., replica-exchange) in umbrella sampling calculations and the value of including collective environmental variables (e.g., hydration level) in metadynamics simulations; we also illustrate the sensitivity of string calculations, especially free energy along the path, to various parameters in the computation. Alchemical free energy simulations with a specific thermodynamic cycle are used to probe the effect of including the first solvation shell into the QM region when computing solvation free energies. For cases where high-level QM/MM potential functions are needed, we analyze two different approaches: the QM/MM-MFEP method of Yang and co-workers and perturbative correction to low-level QM/MM free energy results. For the examples analyzed here, both approaches seem productive although care needs to be exercised when analyzing the perturbative corrections.

A comparison of X-ray and calculated structures of the enzyme MTH1

Modern computational chemistry methods provide a powerful tool for use in refining the geometry of proteins determined by X-ray crystallography. Specifically, computational methods can be used to correctly place hydrogen atoms unresolved by this experimental method and improve bond geometry accuracy. Using the semiempirical method PM7, the structure of the nucleotide-sanitizing enzyme MTH1, complete with hydrolyzed substrate 8-oxo-dGMP, was optimized and the resulting geometry compared with the original X-ray structure of MTH1. After determining hydrogen atom placement and the identification of ionized sites, the charge distribution in the binding site was explored. Where comparison was possible, all the theoretical predictions were in good agreement with experimental observations. However, when these were combined with additional predictions for which experimental observations were not available, the result was a new and alternative description of the substrate-binding site interaction. An estimate was made of the strengths and weaknesses of the PM7 method for modeling proteins on varying scales, ranging from overall structure to individual interatomic distances. An attempt to correct a known fault in PM7, the under-estimation of steric repulsion, is also described. This work sheds light on the specificity of the enzyme MTH1 toward the substrate 8-oxo-dGTP; information that would facilitate drug development involving MTH1. Graphical Abstract Overlay of the backbone traces of the two MTH1 protein chains (green and orange respectively) in PDB 3ZR0 and the equivalent PM7 structures (magenta and cyan respectively) each optimized separately.

QM/MM Analysis of Transition States and Transition State Analogues in Metalloenzymes

Enzymology is approaching an era where many problems can benefit from computational studies. While ample challenges remain in quantitatively predicting behavior for many enzyme systems, the insights that often come from computations are an important asset for the enzymology community. Here we provide a primer for enzymologists on the types of calculations that are most useful for mechanistic problems in enzymology. In particular, we emphasize the integration of models that range from small active-site motifs to fully solvated enzyme systems for cross-validation and dissection of specific contributions from the enzyme environment. We then use a case study of the enzyme alkaline phosphatase to illustrate specific application of the methods. The case study involves examination of the binding modes of putative transition state analogues (tungstate and vanadate) to the enzyme. The computations predict covalent binding of these ions to the enzymatic nucleophile and that they adopt the trigonal bipyramidal geometry of the expected transition state. By comparing these structures with transition states found through free energy simulations, we assess the degree to which the transition state analogues mimic the true transition states. Technical issues worth treating with care as well as several remaining challenges to quantitative analysis of metalloenzymes are also highlighted during the discussion.

CPMD/GULP QM/MM interface for modeling periodic solids: Implementation and its application in the study of Y-zeolite supported Rhn clusters

We report here the development of hybrid quantum mechanics/molecular mechanics (QM/MM) interface between the plane-wave density functional theory based CPMD code and the empirical force-field based GULP code for modeling periodic solids and surfaces. The hybrid QM/MM interface is based on the electrostatic coupling between QM and MM regions. The interface is designed for carrying out full relaxation of all the QM and MM atoms during geometry optimizations and molecular dynamics simulations, including the boundary atoms. Both Born-Oppenheimer and Car-Parrinello molecular dynamics schemes are enabled for the QM part during the QM/MM calculations. This interface has the advantage of parallelization of both the programs such that the QM and MM force evaluations can be carried out in parallel to model large systems. The interface program is first validated for total energy conservation and parallel scaling performance is benchmarked. Oxygen vacancy in α-cristobalite is then studied in detail and the results are compared with a fully QM calculation and experimental data. Subsequently, we use our implementation to investigate the structure of rhodium cluster (Rhn ; n = 2 to 6) formed from Rh(C2 H4 )2 complex adsorbed within a cavity of Y-zeolite in a reducible atmosphere of H2 gas. © 2016 Wiley Periodicals, Inc.

Quantum Mechanical Studies on the Photophysics and the Photochemistry of Nucleic Acids and Nucleobases

The photophysics and photochemistry of DNA is of great importance due to the potential damage of the genetic code by UV light. Quantum mechanical studies have played a key role in interpretating the results of modern time-resolved pump-probe spectroscopy, and in elucidating the main photoactivated reactive paths. This review provides a concise, complete picture of the computational studies carried out, approximately, in the past decade. We start with an overview of the photophysics of the nucleobases in the gas phase and in solution. We discuss the proposed mechanisms for ultrafast decay to the ground state, that involve conical intersections, consider the role of triplet states, and analyze how the solvent modulates the photophysics. Then we move to larger systems, from dinucleotides to single- and double-stranded oligonucleotides. We focus on the possible role of charge transfer and delocalized or excitonic states in the photophysics of these systems and discuss the main photochemical paths. We finish with an outlook on the current challenges in the field and future directions of research.

Stereoelectronic effects of porphyrin ligand on the oxygen transfer efficiency of high valent manganese-oxo porphyrin species: A DFT study

The structure and properties of the N[Formula: see text]–Mn–O bonds of high valent manganese-oxo of the second and third generation porphyrins in the presence of imidazole have been studied by means of density functional (DFT) method with 6-31G* basis set in the gas phase as well as water solution. The geometric structures, frontier molecular orbitals, thermodynamic parameters, aromaticity indices and physical properties such as chemical potential and chemical hardness of [(TPP)(ImH)MnO][Formula: see text] and its derivatives were calculated. The obtained results showed that [(TPP)(ImH)MnO][Formula: see text] bearing halogen atoms at the [Formula: see text]-pyrrole positions had a saddle conformation with low Mn–O strength. The electron density ([Formula: see text] and Laplacian ([Formula: see text] properties at critical points of the N[Formula: see text]–Mn–O bonds, estimated by AIM calculations, indicate that Mn–O bonds in third generation porphyrins have lower [Formula: see text] and Mn–N[Formula: see text] distances have higher [Formula: see text] than the second generation ones. The calculations of aromaticity indices for chelated rings at the porphyrin center show that HOMA and NICS in third generation porphyrins are generally lower than that of second ones which is in agreement with their saddle conformation. These results are supported by natural bond orbital (NBO) analysis.

Practically Efficient QM/MM Alchemical Free Energy Simulations: The Orthogonal Space Random Walk Strategy

The difference between free energy changes occurring at two chemical states can be rigorously estimated via alchemical free energy (AFE) simulations. Traditionally, most AFE simulations are carried out under the classical energy potential treatment; then, accuracy and applicability of AFE simulations are limited. In the present work, we integrate a recent second-order generalized ensemble strategy, the orthogonal space random walk (OSRW) method, into the combined quantum mechanical/molecular mechanical (QM/MM) potential based AFE simulation scheme. Thereby, within a commonly affordable simulation length, accurate QM/MM alchemical free energy simulations can be achieved. As revealed by the model study on the equilibrium of a tautomerization process of hydrated 3-hydroxypyrazole and by the model calculations of the redox potentials of two flavin derivatives, lumichrome (LC) and riboflavin (RF) in aqueous solution, the present OSRW-based scheme could be a viable path toward the realization of practically efficient QM/MM AFE simulations.

Excited States and Photochemistry of Chromophores in the Photoactive Proteins Explored by the Combined Quantum Mechanical and Molecular Mechanical Calculations

A photoactive protein usually contains a unique chromophore that is responsible for the initial photoresponse and functions of the photoactive protein are determined by the interaction between the chromophore and its protein surroundings. The combined quantum mechanical and molecular mechanical (QM/MM) approach is demonstrated to be a very useful tool for exploring structures and functions of a photoactive protein with the chromophore and its protein surroundings treated by the QM and MM methods, respectively. In this review, we summarize the basic formulas of the QM/MM approach and emphasize its applications to excited states and photoreactions of chromophores in rhodopsin protein, photoactive yellow protein, and green fluorescent protein.

Mediation of donor-acceptor distance in an enzymatic methyl transfer reaction

Enzymatic methyl transfer, catalyzed by catechol-O-methyltransferase (COMT), is investigated using binding isotope effects (BIEs), time-resolved fluorescence lifetimes, Stokes shifts, and extended graphics processing unit (GPU)-based quantum mechanics/molecular mechanics (QM/MM) approaches. The WT enzyme is compared with mutants at Tyr68, a conserved residue that is located behind the reactive sulfur of cofactor. Small (>1) BIEs are observed for an S-adenosylmethionine (AdoMet)-binary and abortive ternary complex containing 8-hydroxyquinoline, and contrast with previously reported inverse (<1) kinetic isotope effects (KIEs). Extended GPU-based computational studies of a ternary complex containing catecholate show a clear trend in ground state structures, from noncanonical bond lengths for WT toward solution values with mutants. Structural and dynamical differences that are sensitive to Tyr68 have also been detected using time-resolved Stokes shift measurements and molecular dynamics. These experimental and computational results are discussed in the context of active site compaction that requires an ionization of substrate within the enzyme ternary complex.

A DFT investigation of axial N-donor ligands effects on the high valent manganese-oxo meso-tetraphenyl porphyrin

The effect of different electronic and structural nitrogen donors (L) on the N ( ax )- Mn - O properties in the high valent manganese-oxo meso-tetraphenylporphyrin intermediate is investigated by the density functional B3LYP method with 6-31g* basis set. The geometric structures, frontier molecular orbitals, thermodynamic parameters and physical properties such as chemical potential and chemical hardness of [( TPP )( L ) MnO ]+ complexes in the gas phase as well as water solution are calculated. Our theoretical results confirm that the Mn - O distances in [( TPP )( L ) MnO ]+ species decrease by replacing imidazoles with pyridines, amins as well as electron-poor and hindered nitrogen donors resulting from extending the Mn – N ( ax ) bonds. These results are supported by vibrational frequencies, atoms in molecules (AIM) and natural bond orbital (NBO) analysis. It is worth mentioning that the co-catalytic activity trend of different nitrogen donors observed experimentally in the presence of title Mn porphyrin is consistent with the calculated Mn – N ( ax ) and Mn – O bond lengths in this work.