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. 2021 Apr 29;125(18):4701–4713. doi: 10.1021/acs.jpcb.1c01862

Exploring the Minimum-Energy Pathways and Free-Energy Profiles of Enzymatic Reactions with QM/MM Calculations

Kiyoshi Yagi †, Shingo Ito †, Yuji Sugita †,‡,§,*
PMCID: PMC10986901  PMID: 33914537

Abstract

graphic file with name jp1c01862_0009.jpg

Understanding molecular mechanisms of enzymatic reactions is of vital importance in biochemistry and biophysics. Here, we introduce new functions of hybrid quantum mechanical/molecular mechanical (QM/MM) calculations in the GENESIS program to compute the minimum-energy pathways (MEPs) and free-energy profiles of enzymatic reactions. For this purpose, an interface in GENESIS is developed to utilize a highly parallel electronic structure program, QSimulate-QM (https://qsimulate.com), calling it as a shared library from GENESIS. Second, algorithms to search the MEP are implemented, combining the string method (E et al. J. Chem. Phys. 2007, 126, 164103. ) with the energy minimization of the buffer MM region. The method implemented in GENESIS is applied to an enzyme, triosephosphate isomerase, which converts dihyroxyacetone phosphate to glyceraldehyde 3-phosphate in four proton-transfer processes. QM/MM-molecular dynamics simulations show performances of greater than 1 ns/day with the density functional tight binding (DFTB), and 10–30 ps/day with the hybrid density functional theory, B3LYP-D3. These performances allow us to compute not only MEP but also the potential of mean force (PMF) of the enzymatic reactions using the QM/MM calculations. The barrier height obtained as 13 kcal mol–1 with B3LYP-D3 in the QM/MM calculation is in agreement with the experimental results. The impact of conformational sampling in PMF calculations and the level of electronic structure calculations (DFTB vs B3LYP-D3) suggests reliable computational protocols for enzymatic reactions without high computational costs.

1. Introduction

The role of molecular simulations in life science has been increased rapidly and significantly.1 The conformational dynamics of membrane proteins,2 ribosomes,3 and other biomolecular complexes4,5 has been investigated in molecular dynamics (MD) simulations based on atomistic6−9 or coarse-grained (CG)10−13 molecular models. Biomolecular interactions, such as protein–protein14,15 or protein–DNA/RNA interactions,16 are also examined in detail with large-scale biomolecular simulations in the cellular environments14,17 and viruses.18,19 MD-special supercomputers like Anton/Anton220 or MD-GRAPE21 allow us to simulate slow atomic motions of proteins on time scales up to milliseconds. However, most MD simulations are performed on the basis of classical mechanics, relying on the so-called “molecular force fields”,22−24 which can cause several limitations in the simulations in terms of their accuracy and applicability. The force-field accuracy has long been discussed in simulation communities.25,26 It is difficult to keep a good balance between the conformational stability of globular proteins and flexible structures of intrinsically disordered proteins/regions (IDP/IDR) even in the latest updates of molecular force fields.27,28 Divalent cations, namely, Mg2+ or Ca2+, often add another complexity to MD simulations.29,30 MD simulations using a “molecular force field” cannot describe chemical reactions or enzymatic reactions, since the motion of electrons including bond breaking/formation is totally neglected in the simulations.31−34

To overcome those difficulties in MD simulations with “force fields”, a hybrid quantum mechanical/molecular mechanical (QM/MM) calculation was proposed, at first, by Warshal, Levitt, and Karplus more than 40 years ago.35−37 The electronic structure calculation is applied to a small region where the most interesting chemical reaction happens, while the surrounding buffer regions containing a large number of atoms are treated with molecular force fields. Conventionally, QM/MM calculations were difficult to apply to many interesting biological systems, presumably, due to their high computational costs. The QM region had to be sufficiently small, just containing less than 50 atoms, and only very short-time dynamics (shorter than 100 ps in total) were available if ab initio electronic structure theory was used in the QM calculations. However, recent advances in simulation methodologies and growing computational resources in supercomputers and graphics processing unit (GPU) clusters are opening a new era in the QM/MM calculations. One of the promising future directions is the free-energy calculations using the hybrid QM/MM potential. Yang and co-workers developed methods to compute the free-energy and minimum free-energy path with QM/MM.38−40 Hayashi and his colleagues decoupled the QM/MM free-energy optimization from atomistic MD simulations to simulate large conformational changes between the reactants, transition states, and products in enzymes or motor proteins.41,42 Rosta and Hummer applied umbrella sampling (US) based on the QM/MM potentials to describe the free-energy profiles (or potential of mean forces (PMFs)) of adenosine triphosphate (ATP) hydrolysis in solution and in enzymes.43−45 Hammes-Shiffer and co-workers have also applied the method to various enzymatic reactions.46−50

In recent QM/MM simulations, ab initio electronic structure calculations with the QM program are combined with high-performance MD programs, such as AMBER51,52 and NAMD.53 We have also developed highly parallel MD software, GENESIS (https://www.r-ccs.riken.jp/labs/cbrt/), for simulating chemical and biological applications. One of the key features in GENESIS is the extensive parallelization for massively parallel supercomputers54 and GPU-parallel clusters.55 On the basis of our new computational techniques, such as the midpoint cell method,56 the volumetric decomposition of three-dimensional (3D) fast Fourier transform (FFT),57 huge biological systems, such as the cytoplasm of Mycoplasma genetalium,14 a small bacterial system, or an entire gene locus (GATA4)16 were simulated on supercomputers like the K computer, Fugaku (both in RIKEN) and Trinity in Los Alamos National Laboratory. The former contains more than 100 million atoms, and the latter includes over 1 billion atoms of biomolecules as well as solvent and ions. The second key feature is the availability of enhanced conformational sampling algorithms for extending the conformational space of biomolecules and accelerating the convergence of free-energy calculations. Replica-exchange MD (REMD),58,59 replica-exchange US (REUS),60 generalized replica exchange with solute tempering (gREST),61 Gaussian accelerated MD (GaMD),62,63 free-energy perturbation (FEP),64,65 the string method with mean forces,66,67 and combinations of these methods are now available. The CHARMM-GUI developed by Im and his colleagues prepares input files of MD simulations and free-energy calculations.68

These two key features in GENESIS are also useful in the QM/MM simulation, which has been implemented recently.69 The well-known QM software, namely, Gaussian,70 Q-Chem,71 TeraChem,72 and DFTB+,73 is available in GENESIS from the release of 1.5. We did not change the QM programs and just call each binary of the QM software from the QM/MM interface program in GENESIS. This simple approach is sufficient in many QM/MM calculations. However, overhead costs required in the system call and exchanges of information using input/output (I/O) files between GENESIS and QM software are non-negligible when we use DFTB+ in QM simulations. In our previous work, we performed a molecular vibrational analysis using the QM/MM program in GENESIS as well as SINDO,74 which can introduce the anharmonic effect of molecular vibrations. It was applied to an enzyme, P450 nitric oxide reductase, with the NO molecule bound to a ferric heme.69

In this study, we develop two new functions of the QM/MM calculations in GENESIS: (1) the interface with highly parallelized QM software, QSimulate-QM,75 and (2) a reaction path search algorithm using the string method.76,77 Note that the original string method in GENESIS67 can be used with QM/MM calculations as well to compute the minimum free-energy pathway (MFEP). However, the method requires QM/MM-MD simulations to obtain the mean force, which is costly in general. In the present study, we implement an alternative string method (the so-called zero-temperature version), which computes the minimum energy pathway (MEP) on the potential energy. In this method, the buffer MM region is energy minimized with the QM region replaced by atomic charges derived from QM calculation. The QM calculation is required only once in the iteration of string optimization, and thus the computational cost is greatly reduced. The developed method is demonstrated in a conversion reaction of dihyroxyacetone phosphate (DHAP) into glyceraldehyde 3-phosphate (GAP) catalyzed by an enzyme, triosephosphate isomerase (TIM). The mechanism for the reaction has been well-established in previous theoretical works.38,39,78−82 The whole reaction involves four proton-transfer (PT) processes. The MEP of each process is obtained by the string method, and the free-energy profiles along the PT reactions are calculated using the US method based on the QM/MM MD simulations. The high-performance computations using QSimulate-QM allow us to perform such free-energy calculations of enzymatic reactions in TIM with reasonable computational costs. Different levels of electronic structure calculations suggest promising schemes that are applicable to many other enzymatic reactions based on the QM/MM simulations.

2. Method

2.1. Interface with QSimulate-QM

We have previously implemented a general interface with electronic structure programs for QM/MM calculations in GENESIS.69 GENESIS sends the atomic coordinates and the MM charges to an electronic structure program and receives the energy and gradient upon return. The interface exchanges the information through external files. It (1) generates an input file of the electronic structure program, (2) invokes the electronic structure program using a system call function, and (3) reads output files of the electronic structure program. The advantage of such a scheme is that any electronic structure program can be readily adapted once the format of input/output files are known. The current version (ver. 1.6) supports DFTB+, Gaussian, Q-Chem, and TeraChem. However, the drawbacks of this scheme are that the file I/O is intensive and that the system call is not suitable for message passing interface (MPI) parallelized programs.

In this study, we developed an interface that directly combines GENESIS with QSimulate-QM through a library. QSimulate-QM has realized high-accuracy electronic structure calculations of large chemical systems via extensive parallelization. QSimulate-QM is first compiled to create a shared library (libqsimulate.so). Then, the QM/MM routine in GENESIS calls an entry function of QSimulate-QM included in the library to perform electronic structure calculations. Note that GENESIS and QSimulate-QM are written in Fortran and C++, respectively. The interface, therefore, implements a Fortran/C++ interlanguage call, which complies with standardized protocols in Fortran 2003 to be interoperable with C++. In this scheme, all the information is sent and received on memory without requiring the file I/O. GENESIS provides the atomic coordinates and MM charges, and QSimulate-QM returns the energy and gradient. The atomic charges of QM atoms derived from the intrinsic atomic orbitals (IAO)83 are optionally returned. Furthermore, one of the arguments of the entry function is set to an MPI communicator, which specifies MPI processes assigned to QSimulate-QM. This specification makes it feasible to run multiple electronic structure calculations in parallel, for example, over replicas in REMD or REUS, images in the string method, and so on.

The current QM/MM implementation, not only for QSimulate-QM but for any others, assumes that the system is set up as a cluster system, that is, a nonperiodic system. The electronic structure of a QM system is calculated in the presence of all MM charges. In other words, the electrostatic interaction between electrons, nuclei, and MM charges are included in the electronic Hamiltonian. QM programs solve the electronic Schrödinger equation to obtain the QM energy, the nuclear gradients, and the response of MM charges to the electron density. The information is returned to GENESIS and used as forces to propagate the atomic position.

2.2. Reaction-Path Search

We implemented an algorithm to find the MEP. The procedure is outlined in the following.

Step 0. Setup

The system is divided into three layers: active atoms in a path search, buffer atoms, and fixed atoms. One of the natural choices of these layers is to set the active atoms to QM atoms, the buffer atoms to MM atoms in the vicinity of a QM region (e.g., within ∼8.0 Å), and the fixed atoms to other MM atoms in the outer region. The reaction path is represented by atomic coordinates in a number of discretized images. The definition of three layers, the number of images (Nimg), and the coordinates of images along an initial path must be provided in the input by users.

Step 1. Relax the Buffer Atoms

The first step in the calculation is to relax the structure of buffer atoms with active atoms (QM atoms) held fixed. In this step, the QM/MM interaction is calculated by replacing the QM atoms with atomic charges derived from the electronic structure calculations. The gradient correction is applied by adding the difference between the QM/MM gradient and the approximate one, as is commonly done in the micro-iteration scheme.69,84

Step 2. QM/MM Calculations

A single-point QM/MM calculation is performed to update the energy, gradient, and atomic charges.

Step 3. Update the Coordinates of Active Atoms

The coordinates of active atoms are updated in two steps based on the simplified string method by E et al.77 At first, the image moves along the gradient

Step 3. 1

where n is an index of iterations, xni is the Cartesian coordinates of active atoms of the ith image in the nth iteration, V is the potential energy, and δt is a constant. Then, the images are reparameterized to be equidistant. Specifically, the path length is calculated as follows

Step 3. 2

and the coordinates, {x*i}, are interpolated as a function of path length, {si}, to construct an interpolation function, I(s). Then, the new coordinates are obtained as

Step 3. 3

Step 4. Check the Convergence

If the change in the energy profile and path length are both within threshold values, the calculation is terminated. Otherwise, the calculation goes back to the first step and relaxes the buffer atoms in the updated coordinates.

In the above procedure, the computational bottleneck usually comes from electronic structure calculations in step (2). Therefore, step (2) together with step (1) is parallelized over the image. One can calculate either all images or a subset of images in parallel by setting the number of MPI processes to Nimg × M (M is an integer) or a divisor of Nimg, respectively. The number of MPI processes per image is set to M in the former, while it is 1 in the latter.

2.3. Free-Energy Calculations

The free-energy profile of the enzymatic reaction is computed by an umbrella sampling (US) that uses QM/MM MD simulations after the MEP search. A set of interatomic distances, specified by the user, is used for collective variables (CVs). The window is generated at an equal distance along the MEP in the CV space by interpolation. Then, QM/MM-MD simulations are performed for each window with a bias potential, that is, harmonic restraints to CVs.

The analysis of MD trajectories gives the free-energy profile or PMF. The estimator f̂i is obtained by the multistate Bennett acceptance ratio (MBAR) method85 as

2.3. 4

where K is the number of windows, Ubiasi denotes the bias potential of the ith window, Nj is the number of snapshots in the jth window, and xjn is a sample from the nth snapshot and the jth window. Equation 4 is solved iteratively to obtain {f̂i}. Then, the PMF is obtained using MBAR

2.3. 5
2.3. 6

where Z is the partition function. In order to explore the free-energy profile along MEP, it is convenient to introduce pathCV,67,86,87 which characterizes the component of tangent vector along the path

2.3. 7

where rk is the CV at the kth window, and λ is a constant.

2.3. 8

3. Computational Details

3.1. Modeling and Equilibration of a System

The atomistic system of TIM was constructed from an X-ray crystal structure (PDBID: 7TIM, resolution 1.90 Å).88 The crystal structure is a dimer of TIM cocrystallized with an inhibitor, phosphoglycolohydroxamic acid (C2H6NO6P). The two TIM proteins were both included in the simulation system, because the reaction center was located at the interface of the dimer. The nitrogen atom of the inhibitor was replaced with a carbon atom to generate DHAP. All titratable residues were ionized at a neutral pH condition based on pKa values estimated by PROPKA 3.1.89 An exception is Glu165 with pKa of 7.75, which was kept ionized because it is an acceptor of a proton from DHAP. The proton of His95 was added at the ε position. CHARMM-GUI90,91 was used to add hydrogen atoms and to solvate the system in a box of water molecules (110 × 90 × 90 Å3). An initial equilibration was performed in a periodic boundary condition (PBC) for 300 ps by the conventional classical MD with positional restraints to heavy atoms of proteins and DHAP. Then, the system was trimmed to 15 Å around a TIM dimer to create a non-PBC system. The system was equilibrated for 2.6 ns by classical MD gradually reducing the force constants of restraint, followed by a further equilibration for 140 ps by QM/MM-MD using scc-DFTB.92 The protocol of the equilibration is summarized in Tables S1 and S2. Two spherical boundary potentials with a radius of 35 Å were applied around the center of mass of each TIM. The nonbonding interaction was reduced to zero between 16 and 18 Å employing a switching function. The neighbor list was updated every 20 fs. The force field parameters of TIM and water were CHARMM3693 and TIP3P,94 respectively, and those of DHAP were generated using the force field toolkit.95 The time integration was done using the velocity-Verlet integrator with a time step of 2.0 fs, and the temperature was controlled at 300 K using the Bussi thermostat.96 The bond length of covalent bonds involving hydrogen atoms was constrained using the RATTLE97 and SETTLE98 algorithms.

3.2. QM/MM Calculations

The QM region was set to DHAP and the side chain of Glu165 and His95, which consists of 35 QM atoms. The boundary between QM and MM regions is set between the Cα and Cβ atoms. The link hydrogen atoms were attached to the Cβ atoms excluding the charges of MM atoms that belong to the same group as the Cα atoms. The electronic structure calculations were mainly performed by density functional theory (DFT) at the level of B3LYP hybrid functionals99,100 with a D3 version of Grimm’s dispersion101 (B3LYP-D3) using Dunning’s aug-cc-pVDZ for the basis sets.102 The PBE functional103 with def2-SVP104 basis sets was also employed for benchmark calculations. The number of basis sets was 378 and 623 with def2-SVP and aug-cc-pVDZ, respectively. The scc-DFTB calculations were performed using the 3OB parameter sets.92 The QM/MM calculations were performed using a development version of GENESIS interfaced with QSimulate-QM.

3.3. MEP Search

The structure of a reactant (DHAP) was obtained with the QM/MM energy minimization from the initial one described in Section 3.1. The residues within 6.0 Å of DHAP were relaxed, whereas others were kept fixed. Then, the structure of a product was obtained in two steps. First, the structure was roughly optimized adding distant restraints to X–H bonds with the reference distance set to 1.7–2.0 Å for a dissociating bond and to 1.0 Å for a newly formed bond. Second, the structure was refined by restarting the energy minimization without restraint. In this step, the residues within 3.0 Å of DHAP were relaxed.

Once the reactant and product structures were obtained, the MEP between them was searched using the string method. The initial path was obtained by a linear interpolation of the reactant and product in Cartesian coordinates. The number of images was set to 16. The active atoms in a path search were the same as QM atoms (DHAP and side chains of His95 and Glu165), and the buffer atoms were set to residues within 6.0 Å of DHAP. The step size δt was set to 0.0005 Å. The relaxation of buffer atoms was performed using a limited memory version of Broyden-Fletcher-Goldfarb-Shanno (L-BFGS-B)105−107 and a macro/micro-iteration scheme.84 The IAO charges83 and the self-consistent charges were used for the micro-iteration in B3LYP-D3 and DFTB3, respectively. Convergence thresholds of the energy profile and the path length were set to 0.01 kcal mol–1 and 0.01 Å, respectively.

The procedure was repeated for all four PT reactions to obtain the structures of DHAP, three intermediates, and GAP and the MEPs connecting them.

3.4. Free-Energy Calculations

The umbrella sampling was performed along the MEP predicted from the string-method calculations. The bond distances r(X-H) involved in the reaction are taken as the CVs. The CVs used for each proton-transfer process are illustrated in Figures S1a–S3a. The force constant was set to 100 kcal/mol Å2, and the windows were set with an interval of δr = 0.1 Å following the convention in the literature.43−47,108 We checked that the probability distribution of each window has a sufficient overlap, as shown in Figures S1c–S3c. QM/MM-MD simulations were performed for 10 ps by DFTB3 and for 12 ps by B3LYP-D3. The hydrogen atoms (protons) involved in the reaction were unconstrained, and thus the time step of the integration was set to δt = 0.5 fs. The canonical ensemble (NVT) simulation was performed at T = 300 K with τT = 0.5 ps for the Bussi thermostat. The structure was sampled every 5 fs in the last 10 ps of the simulation. The trajectory data were analyzed using the MBAR and path CV analyses to obtain the PMF. The MBAR and PMF analyses were performed using the analysis tools of GENESIS.

4. Results

4.1. Performance

QM/MM calculations were performed for a water droplet (7014 atoms, 2338 water molecules) and TIM in solution (37 QM atoms and 37,182 MM atoms) to study the performance of GENESIS/QSimulate-QM. The timing data were obtained from MD simulations in an NVT condition with a time step of 2 fs for 100 steps (200 fs) and 5000 steps (10 ps) in DFT and DFTB, respectively.

In a water droplet system, water molecules within a radius r of a center were taken as a QM region, which was varied in size by changing r in a range of 3.0–6.0 Å (corresponding to 24–147 atoms). In Figure 1a, QM/MM-MD with DFTB3 shows a performance of more than 1 ns/day up to 114 atoms (or 380 electrons), which is achievable owing to the library interface developed in this work. The DFT calculations are an order of magnitude more expensive than DFTB3. Nonetheless, the performance is found to be more than 10 ps/day with PBE/def2-SVP and B3LYP-D3/aug-cc-pVDZ up to 96 and 45 atoms, respectively, using 2 nodes, and up to 147 and 66 atoms, respectively, using 8 nodes.

Figure 1.

Figure 1

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Performance of QM/MM-MD simulations combining GENESIS and QSimulate-QM. (a) Visualization of a water droplet and a QM region in the center (left). The timing data for a water droplet with increasing QM size using DFTB3, PBE/def2-SVP, and B3LYP/aug-cc-pVDZ. The total number of atoms is 7014. The computing node has two CPUs of Intel Xeon Gold 6148 (20 core 2.40 GHz), and 10 threads ×4MPI processes were assigned per node. (b) Visualization of TIM and an enlargement of the binding pocket (left). The timing data of TIM using 1 node (7 threads × 8 MPI processes) with various electronic structure methods (middle) and multiple nodes with B3LYP-D3/aug-cc-pVDZ (right). The number of QM and MM atoms are 35 and 37 182, respectively. The number of basis sets is 378 and 623 with def2-SVP and aug-cc-pVDZ, respectively. The computing node has two CPUs of Intel Xeon Platinum 8280 (28 core 2.70 GHz, AVX-512). The timing was measured in NVT condition with a time step of 2 fs.

The results of TIM are displayed in Figure 1b. QM/MM-MD with DFTB3 shows a performance of 1.42 ns/day. The breakdown has found that portions of 63 and 29% of the time are spent for the calculation of DFTB3 and a nonbonded MM interaction, respectively. Therefore, the cost of MM calculations is non-negligible in a system with more than 30 000 atoms.

Among the DFT results, the pure DFT with moderate basis sets (PBE/def2-SVP) exhibits the best performance of 36.9 ps/day. The performance of PBE/aug-cc-pVDZ is decreased to 20.8 ps/day, because the use of diffuse functions (aug-cc-pVDZ) not only increases the number of basis sets but also makes the self-consistent field (SCF) convergence slower. The hybrid DFT, B3LYP-D3/aug-cc-pVDZ, is the most time-consuming due to the need of the Hartree–Fock exchange. Nonetheless, as shown in Figure 1b, the performance of B3LYP-D3 scales well with respect to the number of nodes. The best performance is obtained as 31.0 ps/day using 16 nodes (28 threads ×32 MPI). Interestingly, the result of 7 threads × 8 MPI reaches the maximum at 4 nodes, and a further increase in nodes leads to a drastic decrease of the performance, indicating the importance of a good balance between the number of threads and MPI processes. These results show that QM/MM-MD simulations of picoseconds and nanoseconds order is feasible at the level of DFT and DFTB, respectively, using GENESIS/QSimulate-QM.

4.2. Proton-Transfer Reactions in TIM

The mechanism of the conversion of DHAP to GAP is illustrated in Figure 2. In the first step, the proton H31 is transferred from DHAP to Glu165 of TIM. It is notable that the proton transfer is coupled with an electron transfer, where the negative charge migrates from Glu165 to the carboxylic acid of DHAP, varying the C2–C3 bond from a single to double bond. The negative charge of O2 drives the subsequent PTs of HE2 from His95 and HO3 from the ligand. Finally, the H31 of Glu165 returns to the ligand at C2 to yield GAP.

Figure 2.

Figure 2

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Schematic illustration of the conversion of DHAP to GAP with Glu165 and His95 of TIM. Protons are indicated in blue, and the electrons as well as CC/CO double bonds are in red.

The five structures of the active site of TIM with the ligand, denoted I–V in the following, are shown in Figure 3. The X–H distances are listed in Table S3 of the Supporting Information. It is seen that the current procedure yields the intermediates and the product starting from the reactant. The energy minimization of each structure was converged in 20–50 iterations. These structures were obtained in 76.8 min using four computing nodes in total.

Figure 3.

Figure 3

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Structures of DHAP (I), intermediates (II–IV), and GAP (V) with His95 and Glu165 obtained by QM/MM calculations at the B3LYP-D3/aug-cc-pVDZ level of theory. The protons before and after the reaction are indicated in green and pink circles, respectively.

Figure 4 shows the convergence of the string-method calculations in the first PT step (I → II). Both the energy and the pathway rapidly change in the first 10 iterations. The change is less pronounced during 10–50 iterations, and the convergence is achieved at the 93rd iteration. The energy profile of I → II gives a barrier height of 15.5 kcal mol–1 and an endothermic reaction energy of 11.7 kcal mol–1.

Figure 4.

Figure 4

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Convergence of an MEP search in the first PT step (I → II) with the string-method calculations.

Although the pathway at the 50th iteration overlaps with the converged path around the transition state (TS) in Figure 4, notable changes are observed in the initial stage of the reaction. This stage corresponds to a rotation of a CH2 group, which makes the C3–H31 bond point toward the OE2 of Glu165 (see Movie S1). The path search in Cartesian coordinates enables to describe smooth changes in the structure space, thereby avoiding discontinuities caused by hidden coordinates.

Figures 5 and 6 compare the results of an MEP search in the whole PT processes obtained in the QM/MM calculations based on B3LYP-D3 and DFTB3. A stark difference is that the intermediate III, where His95 is deprotonated, is not a minimum but a TS in DFTB3. The error is ascribed to a known deficiency in DFTB3 to describe a deprotonated nitrogen due to the lack of d-type orbitals in the basis sets.92 Furthermore, the energetics obtained by DFTB3 is 2–3 times larger than those obtained by B3LYP-D3 as listed in Table 1. For example, the relative energy of II is 11.7 and 28.0 kcal mol–1 in B3LYP-D3 and DFTB3, respectively. The proton affinity (PA) of DHAP at the level of B3LYP-D3 and DFTB3 is calculated to be different by 15.3 kcal mol–1 (Tables S4–S6), indicating that the error in the PA is not the primary cause. As shown in Figure 6, the N–H group of His95 is hydrogen-bonded with an anionic oxygen and a hydroxyl group of the ligand during II to IV. The hydrogen-bond network presumably leads the nitrogen atom toward a deprotonated state and incurs the error in DFTB3. This is mirrored in the product state (V), where GAP is remote from His95. The relative energy is obtained with a reasonable agreement as 8.0 and 12.0 kcal mol–1 using B3LYP-D3 and DFTB3, respectively.

Figure 5.

Figure 5

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Interatomic distances (left) and energetics (right) along MEPs obtained by QM/MM calculations at the level of (a) B3LYP-D3 and (b) DFTB3. The definition of r1 – r8 is given in Figure 2.

Figure 6.

Figure 6

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Structure of the reactive site along the MEP and the relevant intermolecular distances (in Å) obtained by B3LYP-D3/aug-cc-pVDZ. Those of DFTB3 are also given in parentheses. Note that DFTB3 yields III as a transition state, so that TS2 and TS3 are not characterized.

Table 1. Relative Energy of Minimum and TS Structures Relative to the Reactant (in kcal mol–1).

  I TS1 II TS2 III TS3 IV TS4 V
B3LYP-D3              
MEP 0.0 15.5 11.7 18.5 17.2 18.7 11.7 13.1 8.0
PMF 0.0 11.8 7.0 13.2 11.8 13.3 5.9 9.3 3.4
DFTB3              
MEP 0.0 31.1 28.0 51.6 49.1 51.7 32.0 33.3 12.0
PMF 0.0 25.6 20.2 50.7 50.5 51.6 33.5 35.7 15.5
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Despite the large errors in the energetics, the structural change along the MEP of DFTB3 is found to be similar to that of B3LYP-D3 as shown in the left panel of Figure 5a,b. The difference in the motion of the MEP obtained by B3LYP-D3 and DFTB3 (Movies S1 and S2, respectively) is hardly discernible. The result suggests the possibility of a multilevel approach where the MEP obtained by DFTB may be used to correct the energetics using DFT.

The umbrella sampling simulations were performed for the PT reactions using the distances r1 – r8, shown in Figure 2 as CVs. The details of the CVs, windows, and the probability distribution of path CV are given in Figures S1–S3. The PMFs obtained by B3LYP-D3 and DFTB3 are shown as a function of Path CV in Figure 7. DFTB3 gives the free-energy barrier of ∼50 kcal mol–1, nearly 4 times larger compared to that of B3LYP-D3. The comparison of MEP and PMF summarized in Table 1 indicates that the entropic effect on the energy is less than 10 kcal mol–1. The free-energy barrier is obtained as 13 kcal mol–1 by B3LYP-D3 in good agreement with the experimental result.109

Figure 7.

Figure 7

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PMF as a function of Path CV obtained by B3LYP-D3 (left) and DFTB3 (right).

Once the weight of each snapshot [eq 6] is obtained from MBAR, it can be used to derive the PMF in arbitrary coordinates. As an example, two-dimensional (2D) PMFs are plotted in Figure 8 as a function of various bond-forming/dissociating coordinates. The contour plots show that the configurations relevant for the reaction are sampled well. It is notable in Figure 8b that the minima around III is shallow and widely spread. The result indicates that two protons (HE2 and HO3) and a deprotonated His95 share an excess electron and stabilize the reactive motion to form a plateau potential. The PMF forms a channel in a diagonal direction, suggesting a concerted double proton transfer. Interestingly, the corresponding figure obtained by DFTB3 shown in Figure S4 shows a channel parallel to the reaction coordinates suggesting a stepwise mechanism.

Figure 8.

Figure 8

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2D PMF of PT reactions obtained by QM/MM calculations at the B3LYP-D3/aug-cc-pVDZ level.

5. Discussion and Perspectives

In this study, the minimum-energy pathways and the free-energy profiles of enzymatic reactions are predicted using the hybrid QM/MM calculations implemented in GENESIS software. The benchmark calculations using four different electronic structure theories, namely, DFTB3, PBE/def2-SVP, PBE/aug-cc-pVDZ, and B3LYP-D3/aug-cc-pVDZ, have revealed efficient computations using QSimulate-QM as the electronic structure calculations. Since the precompiled binary of QSimulate-QM is linked as a shared library from the QM/MM interface program in GENESIS, we can avoid the large overhead that is required to exchange the energy and gradient information through file I/O between GENESIS and externally coupled electronic structure programs. The direct combination between GENESIS and QSimulate-QM in this way is efficient, in particular, for the DFTB3-level calculations, since the computational cost for the electronic structure calculations is rather comparable to the molecular mechanics calculations. The benchmark performance in the combination of QSimulate-QM/GENESIS suggests that the QM/MM-MD simulations become a useful research tool in chemical and biological problems without spending high computational resources. The free-energy calculations based on the QM/MM umbrella sampling or other enhanced sampling methods are important to take into account the entropic effects in the enzymatic reactions.

Comparisons of free-energy profiles obtained from different levels of the electronic structure calculations (DFTB3 vs B3LYP-D3) give us valuable lessons. To obtain sufficiently accurate energetics, ab initio electronic structure theories, such as B3LYP-D3, are inevitable. If we investigate the other chemical reactions with more complicated changes in the electronic structures, for instance, the reactions involving metal irons in a protein or a heme group, higher-level QM theories are necessary. Interestingly, the structures of TIM obtained in the QM/MM US simulations with DFTB3 and B3LYP-D3 are very similar to each other, suggesting that both of them are able to give us reliable local geometry of the chemical reactions. This suggests a possibility to perform QM/MM MD simulations and free-energy calculations using low-cost electronic structure theories, such as DFTB3. After the simulations, we can calibrate the free-energy profiles, computing more reliable potential energies of simulation snapshots using higher-level calculations (B3LYP-D3). Lee et al.110−112 already examined the reliability and reproducibility of energetics using such schemes and pointed out the importance of overlapped distributions between high- and low-level QM or QM/MM MD simulations. Since the theoretical framework using MBAR as a reweighting method has been already established, it is important to increase our experiences in such calculations for examining the accuracy and reproducibility of the calibration using different levels of electronic calculations.

The QM/MM interface program in the GENESIS software has two unique features. In our previous paper,69 a high-precision anharmonic vibrational analysis is shown to be possible in combination with the SINDO program.74 For this application, a large number of the QM/MM single-point energy calculations on the grids near the energy minima are required to describe the potential energy surface. Accurate and high-performance calculations with QSimulate-QM are powerful for this purpose. Second, the QM/MM MD simulations with the speed of 1 ns/day are realized using DFTB3 as shown in this paper. This opens new possibilities to investigate the conformational dynamics of small- or medium-sized proteins on the time scales of 100 ns or longer using the QM/MM-MD simulations. The free-energy profiles of enzymatic reactions are now able to include the effect of conformational dynamics or an entropic effect more directly in the calculations. Enhanced conformational sampling algorithms and/or free-energy calculation methods in GENESIS, which have been originally implemented just for the classical MD simulations, are all available in the QM/MM calculations. Parallel MD simulation algorithms, such as REMD, gREST, and REUS, seem to be useful in the QM/MM-MD simulations with many replicas using massively parallel supercomputers like Fugaku.

The current version of GENESIS includes two different MD engines: ATDYN and SPDYN. ATDYN has been aimed to develop as a simple MD code with easy modifications for testing novel simulation algorithms. Therefore, CG and hybrid QM/MM models were easily implemented in ATDYN. However, if we aim to extend the simulations toward much larger systems, such as membrane proteins, ribosomes, DNA/RNA polymerases, and so on, the cost of molecular mechanics calculations are non-negligible. Also, the periodic boundary condition for QM/MM calculations113−116 is more important to simulate membrane proteins and/or long DNA chains. In the future, we plan to implement the QM/MM calculations into SPDYN, allowing efficient spatial decomposition schemes for the parallelization of a whole simulation system and making the QM/MM calculations of huge biological systems possible. These approaches should be more important in understanding molecular mechanisms underlying the enzymatic reactions, designing biomacromolecules with new functions, and developing novel drugs that can control the regulation of functions of proteins related to diseases.

Acknowledgments

This research is partially supported by RIKEN Pioneering Research Projects (Dynamic Structural Biology/Glycolipidologue Initiative) (to Y.S.), RIKEN Incentive Research Project (to K.Y.), Program for Promoting Research on the Supercomputer Fugaku (Biomolecular dynamics in a living cell/MD-driven Precision Medicine), MEXT/KAKENHI Grant No. JP19H05645 (to Y.S.) and JP20H02701 (to K.Y.). We used the computer system HOKUSAI, provided by the RIKEN Information System Division, and Oakbridge-CX and Octopus, provided by the University of Tokyo and Osaka University, respectively, through the HPCI System Research Project (hp200098). We thank Prof. Y. Matsunaga (Saitama Univ.) for his helpful comments on the free-energy calculations. Dr. T. Shiozaki (QSimulate Inc.) is acknowledged for helping us develop the GENESIS/QSimulate-QM interface program. Dr. Y. Akinaga (VINAS Co., Ltd.) is acknowledged for his help to implement the string-method routines into GENESIS.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcb.1c01862.

  • Protocols of equilibration, interatomic distances, calculated total energies and proton affinities, Cartesian coordinates, details of umbrella sampling, 2D PMF obtained by DFTB3 (PDF)

  • Animation for the visualization of minimum-energy pathways from DHAP to GAP using QM/MM calculations at the level of B3LYP-D3/aug-cc-pVDZ (MP4)

  • Animation for the visualization of minimum-energy pathways from DHAP to GAP using QM/MM calculations at the levels of DFTB3 (MP4)

The authors declare no competing financial interest.

Notes

Data Availability. The data that support the findings of this study are available from the corresponding author upon reasonable request.

Special Issue

Published as part of The Journal of Physical Chemistry virtual special issue “Computational Advances in Protein Engineering and Enzyme Design”.

Supplementary Material

jp1c01862_si_001.pdf (949.9KB, pdf)
jp1c01862_si_002.mp4 (374KB, mp4)
jp1c01862_si_003.mp4 (326.8KB, mp4)

References

  1. Brini E.; Simmerling C.; Dill K. Protein storytelling through physics. Science 2020, 370, eaaz3041. 10.1126/science.aaz3041. [DOI] [PMC free article] [PubMed] [Google Scholar]
  2. Muller M. P.; Jiang T.; Sun C.; Lihan M.; Pant S.; Mahinthichaichan P.; Trifan A.; Tajkhorshid E. Characterization of Lipid–Protein Interactions and Lipid-Mediated Modulation of Membrane Protein Function through Molecular Simulation. Chem. Rev. 2019, 119, 6086–6161. 10.1021/acs.chemrev.8b00608. [DOI] [PMC free article] [PubMed] [Google Scholar]
  3. Whitford P. C.; Geggier P.; Altman R. B.; Blanchard S. C.; Onuchic J. N.; Sanbonmatsu K. Y. Accommodation of aminoacyl-tRNA into the ribosome involves reversible excursions along multiple pathways. RNA 2010, 16, 1196–1204. 10.1261/rna.2035410. [DOI] [PMC free article] [PubMed] [Google Scholar]
  4. Wang B.; Opron K.; Burton Z. F.; Cukier R. I.; Feig M. Five checkpoints maintaining the fidelity of transcription by RNA polymerases in structural and energetic details. Nucleic Acids Res. 2015, 43, 1133–1146. 10.1093/nar/gku1370. [DOI] [PMC free article] [PubMed] [Google Scholar]
  5. Da L.-T.; Pardo-Avila F.; Xu L.; Silva D.-A.; Zhang L.; Gao X.; Wang D.; Huang X. Bridge helix bending promotes RNA polymerase II backtracking through a critical and conserved threonine residue. Nat. Commun. 2016, 7, 11244. 10.1038/ncomms11244. [DOI] [PMC free article] [PubMed] [Google Scholar]
  6. Brooks B. R.; et al. CHARMM: The biomolecular simulation program. J. Comput. Chem. 2009, 30, 1545–1614. 10.1002/jcc.21287. [DOI] [PMC free article] [PubMed] [Google Scholar]
  7. Lee T.-S.; Cerutti D. S.; Mermelstein D.; Lin C.; LeGrand S.; Giese T. J.; Roitberg A.; Case D. A.; Walker R. C.; York D. M. GPU-Accelerated Molecular Dynamics and Free Energy Methods in Amber18: Performance Enhancements and New Features. J. Chem. Inf. Model. 2018, 58, 2043–2050. 10.1021/acs.jcim.8b00462. [DOI] [PMC free article] [PubMed] [Google Scholar]
  8. Abraham M. J.; Murtola T.; Schulz R.; Páll S.; Smith J. C.; Hess B.; Lindahl E. GROMACS: High performance molecular simulations through multi-level parallelism from laptops to supercomputers. SoftwareX 2015, 1–2, 19–25. 10.1016/j.softx.2015.06.001. [DOI] [Google Scholar]
  9. Phillips J. C.; Braun R.; Wang W.; Gumbart J.; Tajkhorshid E.; Villa E.; Chipot C.; Skeel R. D.; Kalé L.; Schulten K. Scalable Molecular Dynamics with NAMD. J. Comput. Chem. 2005, 26, 1781–1802. 10.1002/jcc.20289. [DOI] [PMC free article] [PubMed] [Google Scholar]
  10. Marrink S. J.; Risselada H. J.; Yefimov S.; Tieleman D. P.; de Vries A. H. The MARTINI Force Field: Coarse Grained Model for Biomolecular Simulations. J. Phys. Chem. B 2007, 111, 7812–7824. 10.1021/jp071097f. [DOI] [PubMed] [Google Scholar]
  11. Kenzaki H.; Koga N.; Hori N.; Kanada R.; Li W.; Okazaki K.-i.; Yao X.-Q.; Takada S. CafeMol: A Coarse-Grained Biomolecular Simulator for Simulating Proteins at Work. J. Chem. Theory Comput. 2011, 7, 1979–1989. 10.1021/ct2001045. [DOI] [PubMed] [Google Scholar]
  12. Han W.; Schulten K. Further Optimization of a Hybrid United-Atom and Coarse-Grained Force Field for Folding Simulations: Improved Backbone Hydration and Interactions between Charged Side Chains. J. Chem. Theory Comput. 2012, 8, 4413–4424. 10.1021/ct300696c. [DOI] [PMC free article] [PubMed] [Google Scholar]
  13. Seo S.; Shinoda W. SPICA Force Field for Lipid Membranes: Domain Formation Induced by Cholesterol. J. Chem. Theory Comput. 2019, 15, 762–774. 10.1021/acs.jctc.8b00987. [DOI] [PubMed] [Google Scholar]
  14. Yu I.; Mori T.; Ando T.; Harada R.; Jung J.; Sugita Y.; Feig M. Biomolecular Interactions Modulate Macromolecular Structure and Dynamics in Atomistic Model of a Bacterial Cytoplasm. eLife 2016, 5, 18457. 10.7554/eLife.19274. [DOI] [PMC free article] [PubMed] [Google Scholar]
  15. von Bülow S.; Siggel M.; Linke M.; Hummer G. Dynamic cluster formation determines viscosity and diffusion in dense protein solutions. Proc. Natl. Acad. Sci. U. S. A. 2019, 116, 9843–9852. 10.1073/pnas.1817564116. [DOI] [PMC free article] [PubMed] [Google Scholar]
  16. Jung J.; Nishima W.; Daniels M.; Bascom G.; Kobayashi C.; Adedoyin A.; Wall M.; Lappala A.; Phillips D.; Fischer W.; Tung C. S.; Schlick T.; Sugita Y.; Sanbonmatsu K. Y. Scaling molecular dynamics beyond 100,000 processor cores for large-scale biophysical simulations. J. Comput. Chem. 2019, 40, 1919–1930. 10.1002/jcc.25840. [DOI] [PMC free article] [PubMed] [Google Scholar]
  17. Singharoy A.; et al. Atoms to Phenotypes: Molecular Design Principles of Cellular Energy Metabolism. Cell 2019, 179, 1098–1111. 10.1016/j.cell.2019.10.021. [DOI] [PMC free article] [PubMed] [Google Scholar]
  18. Zhao G.; Perilla J. R.; Yufenyuy E. L.; Meng X.; Chen B.; Ning J.; Ahn J.; Gronenborn A. M.; Schulten K.; Aiken C.; Zhang P. Mature HIV-1 capsid structure by cryo-electron microscopy and all-atom molecular dynamics. Nature 2013, 497, 643–646. 10.1038/nature12162. [DOI] [PMC free article] [PubMed] [Google Scholar]
  19. Casalino L.; et al. AI-Driven Multiscale Simulations Illuminate Mechanisms of SARS-CoV-2 Spike Dynamics. bioRxiv 2020, 10.1101/2020.11.19.390187. [DOI] [PMC free article] [PubMed]
  20. Shaw D. E.; et al. In Anton 2: Raising the Bar for Performance and Programmability in a Special-Purpose Molecular Dynamics Supercomputer, SC ’14: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, Nov 16–21, 2014; IEEE, 2014; pp 41–53. 10.1109/SC.2014.9 [DOI]
  21. Ohmura I.; Morimoto G.; Ohno Y.; Hasegawa A.; Taiji M. MDGRAPE-4: a special-purpose computer system for molecular dynamics simulations. Philos. Trans. R. Soc., A 2014, 372, 20130387. 10.1098/rsta.2013.0387. [DOI] [PMC free article] [PubMed] [Google Scholar]
  22. Harder E.; Damm W.; Maple J.; Wu C.; Reboul M.; Xiang J. Y.; Wang L.; Lupyan D.; Dahlgren M. K.; Knight J. L.; Kaus J. W.; Cerutti D. S.; Krilov G.; Jorgensen W. L.; Abel R.; Friesner R. A. OPLS3: A Force Field Providing Broad Coverage of Drug-like Small Molecules and Proteins. J. Chem. Theory Comput. 2016, 12, 281–296. 10.1021/acs.jctc.5b00864. [DOI] [PubMed] [Google Scholar]
  23. Huang J.; Rauscher S.; Nawrocki G.; Ran T.; Feig M.; de Groot B. L.; Grubmüller H.; MacKerell A. D. CHARMM36m: an improved force field for folded and intrinsically disordered proteins. Nat. Methods 2017, 14, 71–73. 10.1038/nmeth.4067. [DOI] [PMC free article] [PubMed] [Google Scholar]
  24. Tian C.; Kasavajhala K.; Belfon K. A. A.; Raguette L.; Huang H.; Migues A. N.; Bickel J.; Wang Y.; Pincay J.; Wu Q.; Simmerling C. ff19SB: Amino-Acid-Specific Protein Backbone Parameters Trained against Quantum Mechanics Energy Surfaces in Solution. J. Chem. Theory Comput. 2020, 16, 528–552. 10.1021/acs.jctc.9b00591. [DOI] [PubMed] [Google Scholar]
  25. Best R. B.; Zheng W.; Mittal J. Balanced Protein–Water Interactions Improve Properties of Disordered Proteins and Non-Specific Protein Association. J. Chem. Theory Comput. 2014, 10, 5113–5124. 10.1021/ct500569b. [DOI] [PMC free article] [PubMed] [Google Scholar]
  26. van der Spoel D. Systematic design of biomolecular force fields. Curr. Opin. Struct. Biol. 2021, 67, 18–24. 10.1016/j.sbi.2020.08.006. [DOI] [PubMed] [Google Scholar]
  27. Robustelli P.; Piana S.; Shaw D. E. Developing a molecular dynamics force field for both folded and disordered protein states. Proc. Natl. Acad. Sci. U. S. A. 2018, 115, E4758–E4766. 10.1073/pnas.1800690115. [DOI] [PMC free article] [PubMed] [Google Scholar]
  28. Piana S.; Robustelli P.; Tan D.; Chen S.; Shaw D. E. Development of a Force Field for the Simulation of Single-Chain Proteins and Protein–Protein Complexes. J. Chem. Theory Comput. 2020, 16, 2494–2507. 10.1021/acs.jctc.9b00251. [DOI] [PubMed] [Google Scholar]
  29. Li P.; Merz K. M. Metal Ion Modeling Using Classical Mechanics. Chem. Rev. 2017, 117, 1564–1686. 10.1021/acs.chemrev.6b00440. [DOI] [PMC free article] [PubMed] [Google Scholar]
  30. Zhang A.; Yu H.; Liu C.; Song C. The Ca2+ permeation mechanism of the ryanodine receptor revealed by a multi-site ion model. Nat. Commun. 2020, 11, 922. 10.1038/s41467-020-14573-w. [DOI] [PMC free article] [PubMed] [Google Scholar]
  31. Senn H. M.; Thiel W. QM/MM Methods for Biomolecular Systems. Angew. Chem., Int. Ed. 2009, 48, 1198–1229. 10.1002/anie.200802019. [DOI] [PubMed] [Google Scholar]
  32. van der Kamp M. W.; Mulholland A. J. Combined Quantum Mechanics/Molecular Mechanics (QM/MM) Methods in Computational Enzymology. Biochemistry 2013, 52, 2708–2728. 10.1021/bi400215w. [DOI] [PubMed] [Google Scholar]
  33. Cui Q. Perspective: Quantum Mechanical Methods in Biochemistry and Biophysics. J. Chem. Phys. 2016, 145, 140901. 10.1063/1.4964410. [DOI] [PMC free article] [PubMed] [Google Scholar]
  34. Cui Q.; Pal T.; Xie L. Biomolecular QM/MM Simulations: What Are Some of the ″burning Issues″?. J. Phys. Chem. B 2021, 125, 689–702. 10.1021/acs.jpcb.0c09898. [DOI] [PMC free article] [PubMed] [Google Scholar]
  35. Warshel A.; Karplus M. Calculation of Ground and Excited State Potential Surfaces of Conjugated Molecules. I. Formulation and Parametrization. J. Am. Chem. Soc. 1972, 94, 5612–5625. 10.1021/ja00771a014. [DOI] [Google Scholar]
  36. 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. 10.1016/0022-2836(76)90311-9. [DOI] [PubMed] [Google Scholar]
  37. Field M. J.; Bash P. A.; Karplus M. A Combined Quantum Mechanical and Molecular Mechanical Potential for Molecular Dynamics Simulations. J. Comput. Chem. 1990, 11, 700–733. 10.1002/jcc.540110605. [DOI] [Google Scholar]
  38. Zhang Y.; Liu H.; Yang W. Free energy calculation on enzyme reactions with an efficient iterative procedure to determine minimum energy paths on a combined ab initio QM/MM potential energy surface. J. Chem. Phys. 2000, 112, 3483. 10.1063/1.480503. [DOI] [Google Scholar]
  39. Hu H.; Lu Z.; Yang W. QM/MM Minimum Free-Energy Path: Methodology and Application to Triosephosphate Isomerase. J. Chem. Theory Comput. 2007, 3, 390–406. 10.1021/ct600240y. [DOI] [PMC free article] [PubMed] [Google Scholar]
  40. Hu H.; Yang W. Free Energies of Chemical Reactions in Solution and in Enzymes with Ab Initio Quantum Mechanics/Molecular Mechanics Methods. Annu. Rev. Phys. Chem. 2008, 59, 573–601. 10.1146/annurev.physchem.59.032607.093618. [DOI] [PMC free article] [PubMed] [Google Scholar]
  41. Kosugi T.; Hayashi S. QM/MM Reweighting Free Energy SCF for Geometry Optimization on Extensive Free Energy Surface of Enzymatic Reaction. J. Chem. Theory Comput. 2012, 8, 322–334. 10.1021/ct2005837. [DOI] [PubMed] [Google Scholar]
  42. Hayashi S.; Uchida Y.; Hasegawa T.; Higashi M.; Kosugi T.; Kamiya M. QM/MM Geometry Optimization on Extensive Free-Energy Surfaces for Examination of Enzymatic Reactions and Design of Novel Functional Properties of Proteins. Annu. Rev. Phys. Chem. 2017, 68, 135–154. 10.1146/annurev-physchem-052516-050827. [DOI] [PubMed] [Google Scholar]
  43. Rosta E.; Woodcock H. L.; Brooks B. R.; Hummer G. Artificial reaction coordinate “tunneling” in free-energy calculations: The catalytic reaction of RNase H. J. Comput. Chem. 2009, 30, 1634–1641. 10.1002/jcc.21312. [DOI] [PMC free article] [PubMed] [Google Scholar]
  44. Rosta E.; Nowotny M.; Yang W.; Hummer G. Catalytic Mechanism of RNA Backbone Cleavage by Ribonuclease H from Quantum Mechanics/Molecular Mechanics Simulations. J. Am. Chem. Soc. 2011, 133, 8934–8941. 10.1021/ja200173a. [DOI] [PMC free article] [PubMed] [Google Scholar]
  45. Rosta E.; Yang W.; Hummer G. Calcium Inhibition of Ribonuclease H1 Two-Metal Ion Catalysis. J. Am. Chem. Soc. 2014, 136, 3137–3144. 10.1021/ja411408x. [DOI] [PMC free article] [PubMed] [Google Scholar]
  46. Ganguly A.; Thaplyal P.; Rosta E.; Bevilacqua P. C.; Hammes-Schiffer S. Quantum mechanical/molecular mechanical free energy simulations of the self-cleavage reaction in the hepatitis delta virus ribozyme. J. Am. Chem. Soc. 2014, 136, 1483–1496. 10.1021/ja4104217. [DOI] [PMC free article] [PubMed] [Google Scholar]
  47. Zhang S.; Ganguly A.; Goyal P.; Bingaman J. L.; Bevilacqua P. C.; Hammes-Schiffer S. Role of the active site guanine in the glmS ribozyme self-cleavage mechanism: Quantum mechanical/molecular mechanical free energy simulations. J. Am. Chem. Soc. 2015, 137, 784–798. 10.1021/ja510387y. [DOI] [PMC free article] [PubMed] [Google Scholar]
  48. Li P.; Soudackov A. V.; Hammes-Schiffer S. Fundamental Insights into Proton-Coupled Electron Transfer in Soybean Lipoxygenase from Quantum Mechanical/Molecular Mechanical Free Energy Simulations. J. Am. Chem. Soc. 2018, 140, 3068–3076. 10.1021/jacs.7b13642. [DOI] [PMC free article] [PubMed] [Google Scholar]
  49. Stevens D. R.; Hammes-Schiffer S. Exploring the Role of the Third Active Site Metal Ion in DNA Polymerase η with QM/MM Free Energy Simulations. J. Am. Chem. Soc. 2018, 140, 8965–8969. 10.1021/jacs.8b05177. [DOI] [PMC free article] [PubMed] [Google Scholar]
  50. Li P.; Rangadurai A.; Al-Hashimi H. M.; Hammes-Schiffer S. Environmental Effects on Guanine-Thymine Mispair Tautomerization Explored with Quantum Mechanical/Molecular Mechanical Free Energy Simulations. J. Am. Chem. Soc. 2020, 142, 11183–11191. 10.1021/jacs.0c03774. [DOI] [PMC free article] [PubMed] [Google Scholar]
  51. Walker R. C.; Crowley M. F.; Case D. A. The Implementation of a Fast and Accurate QM/MM Potential Method in Amber. J. Comput. Chem. 2008, 29, 1019–1031. 10.1002/jcc.20857. [DOI] [PubMed] [Google Scholar]
  52. Götz A. W.; Clark M. A.; Walker R. C. An Extensible Interface for QM/MM Molecular Dynamics Simulations with AMBER. J. Comput. Chem. 2014, 35, 95–108. 10.1002/jcc.23444. [DOI] [PMC free article] [PubMed] [Google Scholar]
  53. Melo M. C. R.; Bernardi R. C.; Rudack T.; Scheurer M.; Riplinger C.; Phillips J. C.; Maia J. D. C.; Rocha G. B.; Ribeiro J. V.; Stone J. E.; Neese F.; Schulten K.; Luthey-Schulten Z. NAMD goes quantum: an integrative suite for hybrid simulations. Nat. Methods 2018, 15, 351–354. 10.1038/nmeth.4638. [DOI] [PMC free article] [PubMed] [Google Scholar]
  54. Jung J.; Kobayashi C.; Kasahara K.; Tan C.; Kuroda A.; Minami K.; Ishiduki S.; Nishiki T.; Inoue H.; Ishikawa Y.; Feig M.; Sugita Y. New parallel computing algorithm of molecular dynamics for extremely huge scale biological systems. J. Comput. Chem. 2021, 42, 231–241. 10.1002/jcc.26450. [DOI] [PMC free article] [PubMed] [Google Scholar]
  55. Jung J.; Naurse A.; Kobayashi C.; Sugita Y. Graphics Processing Unit Acceleration and Parallelization of GENESIS for Large-Scale Molecular Dynamics Simulations. J. Chem. Theory Comput. 2016, 12, 4947–4958. 10.1021/acs.jctc.6b00241. [DOI] [PubMed] [Google Scholar]
  56. Jung J.; Mori T.; Sugita Y. Midpoint cell method for hybrid (MPI+OpenMP) parallelization of molecular dynamics simulations. J. Comput. Chem. 2014, 35, 1064–1072. 10.1002/jcc.23591. [DOI] [PubMed] [Google Scholar]
  57. Jung J.; Kobayashi C.; Imamura T.; Sugita Y. Parallel implementation of 3D FFT with volumetric decomposition schemes for efficient molecular dynamics simulations. Comput. Phys. Commun. 2016, 200, 57–65. 10.1016/j.cpc.2015.10.024. [DOI] [Google Scholar]
  58. Sugita Y.; Okamoto Y. Replica-exchange molecular dynamics method for protein folding. Chem. Phys. Lett. 1999, 314, 141–151. 10.1016/S0009-2614(99)01123-9. [DOI] [Google Scholar]
  59. Sugita Y.; Okamoto Y. Replica-exchange multicanonical algorithm and multicanonical replica-exchange method for simulating systems with rough energy landscape. Chem. Phys. Lett. 2000, 329, 261–270. 10.1016/S0009-2614(00)00999-4. [DOI] [Google Scholar]
  60. Sugita Y.; Kitao A.; Okamoto Y. Multidimensional replica-exchange method for free-energy calculations. J. Chem. Phys. 2000, 113, 6042–6051. 10.1063/1.1308516. [DOI] [Google Scholar]
  61. Kamiya M.; Sugita Y. Flexible selection of the solute region in replica exchange with solute tempering: Application to protein-folding simulations. J. Chem. Phys. 2018, 149, 72304. 10.1063/1.5016222. [DOI] [PubMed] [Google Scholar]
  62. Miao Y.; Feher V. A.; McCammon J. A. Gaussian Accelerated Molecular Dynamics: Unconstrained Enhanced Sampling and Free Energy Calculation. J. Chem. Theory Comput. 2015, 11, 3584–3595. 10.1021/acs.jctc.5b00436. [DOI] [PMC free article] [PubMed] [Google Scholar]
  63. Oshima H.; Re S.; Sugita Y. Replica-Exchange Umbrella Sampling Combined with Gaussian Accelerated Molecular Dynamics for Free-Energy Calculation of Biomolecules. J. Chem. Theory Comput. 2019, 15, 5199–5208. 10.1021/acs.jctc.9b00761. [DOI] [PubMed] [Google Scholar]
  64. Tembre B. L.; Mc Cammon J. A. Ligand-receptor interactions. Comput. Chem. 1984, 8, 281–283. 10.1016/0097-8485(84)85020-2. [DOI] [Google Scholar]
  65. Oshima H.; Re S.; Sugita Y. Prediction of Protein-Ligand Binding Pose and Affinity Using the gREST+FEP Method. J. Chem. Inf. Model. 2020, 60, 5382–5394. 10.1021/acs.jcim.0c00338. [DOI] [PubMed] [Google Scholar]
  66. Maragliano L.; Fischer A.; Vanden-Eijnden E.; Ciccotti G. String method in collective variables: Minimum free energy paths and isocommittor surfaces. J. Chem. Phys. 2006, 125, 24106. 10.1063/1.2212942. [DOI] [PubMed] [Google Scholar]
  67. Matsunaga Y.; Komuro Y.; Kobayashi C.; Jung J.; Mori T.; Sugita Y. Dimensionality of Collective Variables for Describing Conformational Changes of a Multi-Domain Protein. J. Phys. Chem. Lett. 2016, 7, 1446–1451. 10.1021/acs.jpclett.6b00317. [DOI] [PubMed] [Google Scholar]
  68. Kim S.; Oshima H.; Zhang H.; Kern N. R.; Re S.; Lee J.; Roux B.; Sugita Y.; Jiang W.; Im W. CHARMM-GUI Free Energy Calculator for Absolute and Relative Ligand Solvation and Binding Free Energy Simulations. J. Chem. Theory Comput. 2020, 16, 7207–7218. 10.1021/acs.jctc.0c00884. [DOI] [PMC free article] [PubMed] [Google Scholar]
  69. Yagi K.; Yamada K.; Kobayashi C.; Sugita Y. Anharmonic Vibrational Analysis of Biomolecules and Solvated Molecules Using Hybrid QM/MM Computations. J. Chem. Theory Comput. 2019, 15, 1924–1938. 10.1021/acs.jctc.8b01193. [DOI] [PMC free article] [PubMed] [Google Scholar]
  70. Frisch M. J.; et al. Gaussian 16 , rev. C.01; Gaussian, Inc.: Wallingford, CT, 2016.
  71. Shao Y.; et al. Advances in molecular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys. 2015, 113, 184–215. 10.1080/00268976.2014.952696. [DOI] [Google Scholar]
  72. Seritan S.; Bannwarth C.; Fales B. S.; Hohenstein E. G.; Isborn C. M.; Kokkila-Schumacher S. I. L.; Li X.; Liu F.; Luehr N.; Snyder J. W. Jr; Song C.; Titov A. V.; Ufimtsev I. S.; Wang L.-P.; Martínez T. J. TeraChem: A graphical processing unit-accelerated electronic structure package for large-scale ab initio molecular dynamics. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2021, 11, e1494. 10.1002/wcms.1494. [DOI] [Google Scholar]
  73. Hourahine B.; et al. DFTB+, a software package for efficient approximate density functional theory based atomistic simulations. J. Chem. Phys. 2020, 152, 124101. 10.1063/1.5143190. [DOI] [PubMed] [Google Scholar]
  74. Yagi K. SINDO 4.0 beta. 2020, https://tms.riken.jp/en/research/software/sindo/.
  75. QSimulate-QM . Quantum Simulation Technologies, Inc.2020, https://qsimulate.com/.
  76. E W.; Ren W.; Vanden-Eijnden E. String method for the study of rare events. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 66, 52301. 10.1103/PhysRevB.66.052301. [DOI] [Google Scholar]
  77. E W.; Ren W.; Vanden-Eijnden E. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys. 2007, 126, 164103. 10.1063/1.2720838. [DOI] [PubMed] [Google Scholar]
  78. Cui Q.; Karplus M. Quantum mechanics/molecular mechanics studies of triosephosphate isomerase-catalyzed reactions: Effect of geometry and tunneling on proton-transfer rate constants. J. Am. Chem. Soc. 2002, 124, 3093–3124. 10.1021/ja0118439. [DOI] [PubMed] [Google Scholar]
  79. Zhang X.; Harrison D. H. T.; Cui Q. Functional specificities of methylglyoxal synthase and triosephosphate isomerase: A combined QM/MM analysis. J. Am. Chem. Soc. 2002, 124, 14871–14878. 10.1021/ja027063x. [DOI] [PubMed] [Google Scholar]
  80. Cui Q.; Karplus M. Quantum mechanical/molecular mechanical studies of the triosephosphate isomerase-catalyzed reaction: Verification of methodology and analysis of reaction mechanisms. J. Phys. Chem. B 2002, 106, 1768–1798. 10.1021/jp012659c. [DOI] [Google Scholar]
  81. Lennartz C.; Schäfer A.; Terstegen F.; Thiel W. Enzymatic reactions of triosephosphate isomerase: A theoretical calibration study. J. Phys. Chem. B 2002, 106, 1758–1767. 10.1021/jp012658k. [DOI] [Google Scholar]
  82. Mendieta-Moreno J. I.; Walker R. C.; Lewis J. P.; Gómez-Puertas P.; Mendieta J.; Ortega J. FIREBALL/AMBER: An Efficient Local-Orbital DFT QM/MM Method for Biomolecular Systems. J. Chem. Theory Comput. 2014, 10, 2185–2193. 10.1021/ct500033w. [DOI] [PubMed] [Google Scholar]
  83. Knizia G. Intrinsic atomic orbitals: An unbiased bridge between quantum theory and chemical concepts. J. Chem. Theory Comput. 2013, 9, 4834–4843. 10.1021/ct400687b. [DOI] [PubMed] [Google Scholar]
  84. Kästner J.; Thiel S.; Senn H. M.; Sherwood P.; Thiel W. Exploiting QM/MM Capabilities in Geometry Optimization: A Microiterative Approach Using Electrostatic Embedding. J. Chem. Theory Comput. 2007, 3, 1064–1072. 10.1021/ct600346p. [DOI] [PubMed] [Google Scholar]
  85. Shirts M. R.; Chodera J. D. Statistically optimal analysis of samples from multiple equilibrium states. J. Chem. Phys. 2008, 129, 124105. 10.1063/1.2978177. [DOI] [PMC free article] [PubMed] [Google Scholar]
  86. Branduardi D.; Gervasio F. L.; Parrinello M. From A to B in free energy space. J. Chem. Phys. 2007, 126, 054103. 10.1063/1.2432340. [DOI] [PubMed] [Google Scholar]
  87. Bonomi M.; Branduardi D.; Bussi G.; Camilloni C.; Provasi D.; Raiteri P.; Donadio D.; Marinelli F.; Pietrucci F.; Broglia R. A.; Parrinello M. PLUMED: A portable plugin for free-energy calculations with molecular dynamics. Comput. Phys. Commun. 2009, 180, 1961–1972. 10.1016/j.cpc.2009.05.011. [DOI] [Google Scholar]
  88. Davenport R. C.; Bash P. A.; Seaton B. A.; Karplus M.; Petsko G. A.; Ringe D. Structure of the Triosephosphate Isomerase-Phosphoglycolohydroxamate Complex: An Analogue of the Intermediate on the Reaction Pathway. Biochemistry 1991, 30, 5821–5826. 10.1021/bi00238a002. [DOI] [PubMed] [Google Scholar]
  89. Olsson M. H. M.; Søndergaard C. R.; Rostkowski M.; Jensen J. H. PROPKA3: Consistent Treatment of Internal and Surface Residues in Empirical pKa predictions. J. Chem. Theory Comput. 2011, 7, 525–537. 10.1021/ct100578z. [DOI] [PubMed] [Google Scholar]
  90. Jo S.; Kim T.; Iyer V. G.; Im W. CHARMM-GUI: A web-based graphical user interface for CHARMM. J. Comput. Chem. 2008, 29, 1859–1865. 10.1002/jcc.20945. [DOI] [PubMed] [Google Scholar]
  91. Lee J.; Cheng X.; Swails J. M.; Yeom M. S.; Eastman P. K.; Lemkul J. A.; Wei S.; Buckner J.; Jeong J. C.; Qi Y.; Jo S.; Pande V. S.; Case D. A.; Brooks C. L.; MacKerell A. D.; Klauda J. B.; Im W. CHARMM-GUI Input Generator for NAMD, GROMACS, AMBER, OpenMM, and CHARMM/OpenMM Simulations Using the CHARMM36 Additive Force Field. J. Chem. Theory Comput. 2016, 12, 405–413. 10.1021/acs.jctc.5b00935. [DOI] [PMC free article] [PubMed] [Google Scholar]
  92. Gaus M.; Cui Q.; Elstner M. DFTB3: Extension of the Self-Consistent-Charge Density-Functional Tight-Binding Method (SCC-DFTB). J. Chem. Theory Comput. 2011, 7, 931–948. 10.1021/ct100684s. [DOI] [PMC free article] [PubMed] [Google Scholar]
  93. Best R. B.; Zhu X.; Shim J.; Lopes P. E. M.; Mittal J.; Feig M.; Mackerell A. D. Optimization of the Additive CHARMM All-Atom Protein Force Field Targeting Improved Sampling of the Backbone φ, ψ and Side-Chain χ1 and χ2 Dihedral Angles. J. Chem. Theory Comput. 2012, 8, 3257–3273. 10.1021/ct300400x. [DOI] [PMC free article] [PubMed] [Google Scholar]
  94. 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. 10.1063/1.445869. [DOI] [Google Scholar]
  95. Mayne C. G.; Saam J.; Schulten K.; Tajkhorshid E.; Gumbart J. C. Rapid parameterization of small molecules using the force field toolkit. J. Comput. Chem. 2013, 34, 2757–2770. 10.1002/jcc.23422. [DOI] [PMC free article] [PubMed] [Google Scholar]
  96. Bussi G.; Donadio D.; Parrinello M. Canonical sampling through velocity rescaling. J. Chem. Phys. 2007, 126, 014101. 10.1063/1.2408420. [DOI] [PubMed] [Google Scholar]
  97. Andersen H. C. Rattle: A “velocity” version of the shake algorithm for molecular dynamics calculations. J. Comput. Phys. 1983, 52, 24–34. 10.1016/0021-9991(83)90014-1. [DOI] [Google Scholar]
  98. Miyamoto S.; Kollman P. A. Settle: An analytical version of the SHAKE and RATTLE algorithm for rigid water models. J. Comput. Chem. 1992, 13, 952–962. 10.1002/jcc.540130805. [DOI] [Google Scholar]
  99. Lee C.; Yang W.; Parr R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 785–789. 10.1103/PhysRevB.37.785. [DOI] [PubMed] [Google Scholar]
  100. Becke A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648–5652. 10.1063/1.464913. [DOI] [Google Scholar]
  101. Grimme S.; Antony J.; Ehrlich S.; Krieg H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 2010, 132, 154104. 10.1063/1.3382344. [DOI] [PubMed] [Google Scholar]
  102. Dunning T. H. Jr. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007–1023. 10.1063/1.456153. [DOI] [Google Scholar]
  103. Perdew J. P.; Burke K.; Ernzerhof M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865–3868. 10.1103/PhysRevLett.77.3865. [DOI] [PubMed] [Google Scholar]
  104. Weigend F.; Ahlrichs R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297–3305. 10.1039/b508541a. [DOI] [PubMed] [Google Scholar]
  105. Byrd R. H.; Lu P.; Nocedal J.; Zhu C. A Limited Memory Algorithm for Bound Constrained Optimization. SIAM J. Sci. Stat. Comp. 1995, 16, 1190–1208. 10.1137/0916069. [DOI] [Google Scholar]
  106. Zhu C.; Byrd R.; Nocedal J.; Morales J. L. L-BFGS-B (ver. 3.0), http://users.iems.northwestern.edu/~nocedal/lbfgsb.html.
  107. Zhu C.; Byrd R. H.; Lu P.; Nocedal J. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization. ACM Transactions on Mathematical Software 1997, 23, 550–560. 10.1145/279232.279236. [DOI] [Google Scholar]
  108. Lence E.; van der Kamp M. W.; González-Bello C.; Mulholland A. J. QM/MM simulations identify the determinants of catalytic activity differences between type II dehydroquinase enzymes. Org. Biomol. Chem. 2018, 16, 4443–4455. 10.1039/C8OB00066B. [DOI] [PMC free article] [PubMed] [Google Scholar]
  109. Albery W. J.; Knowles J. R. Free-Energy Profile for the Reaction Catalyzed by Triosephosphate Isomerase. Biochemistry 1976, 15, 5627–5631. 10.1021/bi00670a031. [DOI] [PubMed] [Google Scholar]
  110. 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. 10.1021/acs.jpclett.5b02164. [DOI] [PubMed] [Google Scholar]
  111. 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. 10.1002/jcc.24706. [DOI] [PubMed] [Google Scholar]
  112. Hudson P. S.; Woodcock H. L.; Boresch S. Use of Interaction Energies in QM/MM Free Energy Simulations. J. Chem. Theory Comput. 2019, 15, 4632–4645. 10.1021/acs.jctc.9b00084. [DOI] [PMC free article] [PubMed] [Google Scholar]
  113. Holden Z. C.; Richard R. M.; Herbert J. M. Periodic boundary conditions for QM/MM calculations: Ewald summation for extended Gaussian basis sets. J. Chem. Phys. 2013, 139, 244108. 10.1063/1.4850655. [DOI] [PubMed] [Google Scholar]
  114. Giese T. J.; York D. M. Ambient-Potential Composite Ewald Method for ab Initio Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simulation. J. Chem. Theory Comput. 2016, 12, 2611–2632. 10.1021/acs.jctc.6b00198. [DOI] [PMC free article] [PubMed] [Google Scholar]
  115. Vasilevskaya T.; Thiel W. Periodic Boundary Conditions in QM/MM Calculations: Implementation and Tests. J. Chem. Theory Comput. 2016, 12, 3561–3570. 10.1021/acs.jctc.6b00269. [DOI] [PubMed] [Google Scholar]
  116. Kawashima Y.; Ishimura K.; Shiga M. Ab initio quantum mechanics/molecular mechanics method with periodic boundaries employing Ewald summation technique to electron-charge interaction: Treatment of the surface-dipole term. J. Chem. Phys. 2019, 150, 124103. 10.1063/1.5048451. [DOI] [PubMed] [Google Scholar]

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Data Citations

  1. Casalino L.; et al. AI-Driven Multiscale Simulations Illuminate Mechanisms of SARS-CoV-2 Spike Dynamics. bioRxiv 2020, 10.1101/2020.11.19.390187. [DOI] [PMC free article] [PubMed]

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