Accelerating ab initio QM/MM Molecular Dynamics Simulations with Multiple Time Step Integration and a Recalibrated Semi-empirical QM/MM Hamiltonian

Abstract

Molecular dynamics (MD) simulations employing ab initio quantum mechanical and molecular mechanical (ai-QM/MM) potentials are considered to be the state of the art, but the high computational cost associated with the ai-QM calculations remains a theoretical challenge for their routine application. Here, we present a modified protocol of Multiple Time Step (MTS) method to accelerate ai-QM/MM MD simulations of condensed-phase reactions. Within a previous MTS protocol [Nam, J. Chem. Theory Comput., 2014, 10, 2175], reference forces are evaluated using a low-level (semi-empirical QM/MM) Hamiltonian and employed at inner time steps to propagate the nuclear motions. Correction forces, which arise from the force differences between high-level (ai-QM/MM) and low-level Hamiltonians, are applied at outer time steps, where the MTS algorithm allows the time-reversible integration of the correction forces. To increase the outer step size, which is bound by the highest-frequency component in the correction forces, the semi-empirical QM Hamiltonian is re-calibrated in this work to minimize the magnitude of the correction forces. The remaining high frequency modes, which are mainly bond stretches involving hydrogen atoms, are then removed from the correction forces. When combined with Langevin or SIN(R) thermostat, the modified MTS-QM/MM scheme remains robust with an up to 8 fs (with Langevin) or 10 fs (with SIN(R)) outer time step (with 1 fs inner time steps) for the chorismate mutase system. This leads to an over 5-fold speedup over standard ai-QM/MM simulations, without sacrificing the accuracy in the predicted free energy profile of the reaction.

Graphical TOC Entry

Introduction

The combined quantum mechanical and molecular mechanical (QM/MM) method has been established as an essential tool in the computational modeling of solution-phase and enzymatic reactions.^1–11 In recent years, ab initio QM/MM (ai-QM/MM) calculations,^12–15 especially those based on a density functional theory (DFT)^16–20 description of the reactive region, have shed light into the mechanism of many enzymatic reactions. However, due to their high computational cost, the routine application of the ai-QM/MM potentials is often limited to minimum energy pathway (MEP) calculations, where only $\mathcal{O}\left({10}^2\right)$ QM/MM energy/force calculations are required to optimize each point along a discretized reaction pathway. Less common is the direct employment of ai-QM/MM models in the minimum free energy pathway (MFEP) simulations,^21–30 which typically involve $\mathcal{O}\left({10}^4\right)$ QM/MM energy/force calculations to sample each point/window along the free energy pathway and thus require hundreds of thousands (if not millions) of computer core-hours. Consequently, direct QM/MM MFEP simulation often resort to less-accurate semi-empirical QM/MM (se-QM/MM) Hamiltonians based on AM1,³¹ PM3,³² PM6,³³ or tight-binding DFT^34–37 methods. However, the accuracy of each se-QM model is not always guaranteed for a specific enzyme system of interest.

Going beyond direct QM/MM MFEP calculations, Gao⁴ and Warshel³⁸ pioneered the dual-Hamiltonian strategy for simultaneously exploiting the accuracy of a high-level Hamiltonian and the efficiency of a low-level one. As illustrated in Figure 1, there are two general schemes to implement the dual-Hamiltonian strategy. In the reference potential method (i.e. indirect QM/MM simulations),^4,39–45 the configurational sampling is carried out using a reference low-level potential energy, and then the low-level free energy results are corrected in a post-processing process using thermodynamical perturbation.⁴⁶ This correction can also be achieved through non-equilibrium work simulations.^47–51

Figure 1:

A hierarchy of free energy simulations using a lower-level (LL) Hamiltonian and/or a high-level (HL) Hamiltonian.

In the alternative, multiple time step (MTS) approach,⁵² the configurational sampling using the low-level QM/MM Hamiltonian is corrected on-the-fly via regular computation of high-level forces.⁵³ Dating back to the reversible REference System Propagator Algorithm (rRESPA) developed by Tuckerman and coworkers,⁵⁴ the MTS method assumes a separation of the total force of each configuration of a system into fast and slow components (see Figure 2). Within the MTS integration, the fast forces are applied at each step of a molecular dynamics (MD) simulation, which becomes inner time steps with intervals δt (typically ranging between 0.5 and 2.0 fs). The slow forces, on the other hand, are applied only at outer time steps with intervals Δt = N * δt, i.e., N times longer. For the finite-temperature free energy simulations, the MTS algorithm can be combined with Nosé-Hoover,⁵⁴ Nosé-Hoover-Langevin thermostat with isokinetic constraints,^55–57 Langevin and Andersen thermostats.^57,58 Monte Carlo steps have also been introduced by Roux and coworkers to suppress the discretization error in MTS simulations.⁵⁸ For QM/MM simulations, MTS has been used by Nam to accelerate the modeling of solution-phase⁵³ and enzyme reaction pathways⁵⁹ and by Rothlisberger and coworkers to speed up the computation of solution-phase molecular vibrational frequencies.^60,61

Figure 2:

Illustration of MTS simulations with fast forces applied at each inner time step (with intervals δt) and slow forces at outer time steps (at intervals of Δt).

Following the MTS QM/MM enzyme simulations by Nam,⁵³ in this article we focus on the application of the rRESPA method to combine se-QM/MM and ai-QM/MM Hamiltonians in MFEP simulations of condensed-phase reactions. In particular, the se-QM/MM forces are employed as the “fast” forces to propagate the trajectory at inner time steps, whereas the difference between ai-QM/MM and se-QM/MM forces is used as the “slow” correction forces at outer time steps. As we show in this work, in comparison to the “total” (ai-QM/MM) force, the difference forces (as shown in Figures S1 and S4 for the reactant and transition-state regions of chorismate mutase catalysis pathway) indeed oscillate at substantially reduced magnitudes. Indeed, such small force differences allowed Nam to apply Hartree-Fock/3–21G/MM correction forces at outer time steps of up to 3 fs to accurately simulate the S_N2 reaction between CH₃Cl and Cl⁻ in water, in which the AM1/MM or PM3/MM energy function was used as the lower-level Hamiltonian.⁵³

In this work, we shall aim at a further increase in the outer time step of an MTS QM/MM free energy simulation of condensed-phase reactions and thus a more significant enhancement to the computational efficiency with these simulations. Clearly, the smaller the difference forces are (between se-QM/MM and ai-QM/MM models), the weaker the correction would be at the outer time steps, which can then make the MTS simulations more robust with larger outer time steps. Henceforth, we sought to reduce the difference forces via recalibrating the se-QM/MM Hamiltonian for the specific reaction of interest, following the idea of reaction-path force matching parameterization of Zhou and Pu.⁶² Such a recalibration of the se-QM/MM Hamiltonian and its application in MTS simulations are described in the Theory section, with our MTS simulation details provided in the Computational Details section. Computational results on the chorismate mutase system are presented and discussed in the Results and Discussion section. Finally, concluding remarks are made in the Conclusions section.

In comparison to the earlier MTS QM/MM simulations by Nam,⁵³ our implementation offers a couple of advantages. Firstly, the recalibration of the se-QM model should allow us to use longer outer timesteps, thus increasing the computational efficiency. Secondly, the use of our QM/MM-AC electrostatics model⁶³ enabled the use of the periodic boundary condition instead of a cluster model for force correction, thus ensuring a continuous potential energy surface. Finally, our MTS implementation utilizes the QMHub interface, which allows the use of QM methods in different QM software packages, such as those in the Q-Chem software package,⁶⁴ and a range of semi-empirical methods, such as those implemented in the SQM module in AmberTools.^65,66

Methods

In QM/MM calculations, the system is typically divided into a QM subsystem and an MM subsystem. The potential energy of the whole system can be expressed as

$$\begin{array}{l}E=E_{\text{QM}}+E_{\text{QM}-\text{MM}}+E_{\text{MM}}\hfill \\ =E_{\text{QM}}+E_{\text{QM}-\text{MM}}^{\text{elec}}+E_{\text{QM}-\text{MM}}^{\text{vdW}}+E_{\text{MM}},\hfill \end{array}$$ (1)

where the first two terms are evaluated together at a QM level of theory, while the last two terms are usually calculated at the MM level. In this work, we will focus on

$$F=-\frac{\partial E}{\partial R}$$ (2)

which is the corresponding force on both QM and MM atoms, with coordinates at R_QM and R_MM. When ai-QM method and se-QM are applied for the QM region, the force will be denoted as F^AI and F^SE, respectively.

Multiple Time Step (MTS) Integrator for AI-QM/MM MD Simulations

In our MTS QM/MM method, the ai-QM/MM forces for the entire system, F^AI, are decomposed into

$$F^{\text{AI}}=F^{\text{SE}}+\Delta F^{\text{SE}\to \text{AI}}$$ (3)

where F^SE refers to the se-QM/MM forces and the correction forces are defined as

$$\Delta F^{\text{SE}\to \text{AI}}=F^{\text{AI}}-F^{\text{SE}}$$ (4)

each with 3N_atom components. [In contrast to the original MTS QM/MM method by Nam,⁵³ where the correction forces are calculated only for atoms within the real-space cutoff, in this work, both F^AI and F^SE are calculated using the QM/MM with Augmentary Charges (QM/MM-AC) electrostatic embedding scheme we developed recently⁶³ to include the correction for the long-range QM/MM electrostatics.] Accordingly, the Kolmogorov operator is

$${\mathcal{L}}^{\text{AI}}={\mathcal{L}}_x+{\mathcal{L}}_p^{\text{SE}}+{\mathcal{L}}_p^{\text{SE}\to \text{AI}}$$ (5)

with the individual operators being

$${\mathcal{L}}_x=\sum _{i=1}^{3N_{\text{atom}}}{\dot{r}}_i\frac{\partial }{\partial r_i}$$ (6a)

$${\mathcal{L}}_p^{\text{SE}}=\sum _{i=1}^{3N_{\text{atom}}}{F}_i^{\text{SE}}\frac{\partial }{\partial p_i}$$ (6b)

$${\mathcal{L}}_p^{\text{SE}\to \text{AI}}=\sum _{i=1}^{3N_{\text{atom}}}\Delta F_i^{\text{SE}\to \text{AI}}\frac{\partial }{\partial p_i}$$ (6c)

where r_i and p_i refer to the coordinates and momentum of each atom in the system.

A leapfrog step for molecular dynamics simulations,

$$e^{\mathcal{L}\delta t}\approx e^{{\mathcal{L}}_x\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_p\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_p\delta t}=e^{{\mathcal{L}}_x\delta t}{e}^{{\mathcal{L}}_p\delta t}$$ (7)

involves one force evaluation, two half-time step momentum updates, i.e., $p\left(t-\frac{1}{2}\delta t\right)\to p(t)$ and $p(t)\to p\left(t+\frac{1}{2}\delta t\right)$, and one full-time step coordinate update, r(t) → r(t + δt). Note that the two momentum updates utilize the same force and can thus be combined. With the decomposition of the Liouville operator in Eq. 5, the propagation of an MTS trajectory (as illustrated in Figure 2) within the leapfrog algorithm follows

$$e^{{\mathcal{L}}^{\text{AI}}\Delta t}\approx e^{{\mathcal{L}}_x\delta t}{e}^{({\mathcal{L}}_p^{\text{SE}}+N{\mathcal{L}}_p^{\text{SE}}\to \text{AI}})\delta t{\left[e^{{\mathcal{L}}_x\delta t}{e}^{{\mathcal{L}}_p^{\text{SE}}\delta t}\right]}^{N-1}$$ (8)

This will generate a microcanonical (NVE) ensemble of our enzyme system.

MTS ai-QM/MM Canonical Ensemble Simulations with Langevin Dynamics

When Langevin thermostat is adopted, one applies a Kolmogorov operator, ${\mathcal{L}}_T$, which corresponds to an additional (Ornstein-Uhlenbeck) update of the atomic momenta

$$p(t)\leftarrow e^{-\gamma \delta t}p(t)+\sqrt{\frac{1-e^{-2\gamma \delta t}}{\beta }}{M}^{\frac{1}{2}}\tilde{\eta }$$ (9)

and depends on γ, the friction constant, $\tilde{\eta }$, a random vector within a unit-variance Gaussian distribution, β, the inverse temperature 1/(k_BT), and M, a diagonal matrix with atomic masses.⁶⁷ With this thermostat added to our MTS leapfrog integration in Eq. 8, the evolution within each outer time step becomes

$$e^{{\mathcal{L}}^{\text{AI}}\Delta t}\approx {\left[e^{{\mathcal{L}}_T\delta t}{e}^{{\mathcal{L}}_x\delta t}{e}^{{\mathcal{L}}_p^{\text{SE}}\delta t}\right]}^{N-1}\times e^{{\mathcal{L}}_T\delta t}{e}^{{\mathcal{L}}_x\delta t}{e}^{\left({\mathcal{L}}_p^{\text{SE}}+N{\mathcal{L}}_p^{\text{SE}\to \text{AI}}\right)\delta t}$$ (10)

within the “end” scheme for the thermostat,⁶⁷ or

$$e^{{\mathcal{L}}^{\text{AI}}\Delta t}\approx {\left[e^{\frac{1}{2}{\mathcal{L}}_x\delta t}{e}^{{\mathcal{L}}_T\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_x\delta t}{e}^{{\mathcal{L}}_p^{\text{SE}}\delta t}\right]}^{N-1}\times e^{\frac{1}{2}{\mathcal{L}}_x\delta t}{e}^{{\mathcal{L}}_T\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_x\delta t}{e}^{\left({\mathcal{L}}_p^{\text{SE}}+N{\mathcal{L}}_p^{\text{SE}\to \text{AI}}\right)\delta t}$$ (11)

within the “middle” scheme.⁶⁷

MTS ai-QM/MM Simulations with SIN(R)

The Stochastic-Iso-NH-RESPA [SIN(R)] algorithm employs a set of isokinetic constraints to control the resonance problem in MTS, leading to an increase of the outer timestep.⁵⁵ When the “middle” thermostat scheme is applied to SIN(R), it yields more stable and accurate results, especially as the inner timestep increases.⁵⁷ When the “middle”-scheme SIN(R) with a leap frog integrator is used, the propagator reads

$$e^{{\mathcal{L}}^{\text{AI}}\Delta t}\approx {\left[e^{\frac{1}{2}{\mathcal{L}}_x\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_N\delta t}{e}^{{\mathcal{L}}_O\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_N\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_x\delta t}{e}^{{\mathcal{L}}_p^{\text{SE}}\delta t}\right]}^{N-1}\times e^{\frac{1}{2}{\mathcal{L}}_x\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_N\delta t}{e}^{{\mathcal{L}}_O\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_N\delta t}{e}^{\frac{1}{2}{\mathcal{L}}_x\delta t}{e}^{\left({\mathcal{L}}_p^{\text{SE}}+N{\mathcal{L}}_p^{\text{SE}\to \text{AI}}\right)\delta t}$$ (12)

where the relevant Kolmogorov operators (${\mathcal{L}}_x$, ${\mathcal{L}}_p$, ${\mathcal{L}}_N$, and ${\mathcal{L}}_O$) are defined in Eqs. 89–93 of Reference 57. Specifically, operator ${\mathcal{L}}_p$ updates the momentum, p_i, and the first set of thermodynamic variables, $v_{1,i}^{(j)}(j=1,2,\cdots ,L)$, for each nuclear degree of freedom (i = 1, 2, ⋯ , 3N_atom), i.e., its propagation is

$$p_i\leftarrow \frac{p_i+F_i{s}_i\left(\frac{\delta t}{2}\right)}{{\dot{s}}_i\left(\frac{\delta t}{2}\right)}$$ (13a)

$$v_{1,i}^{(j)}\leftarrow \frac{v_{1,i}^{(j)}}{{\dot{s}}_i\left(\frac{\delta t}{2}\right)}$$ (13b)

with

$$F_i{s}_i\left(\frac{\delta t}{2}\right)=\sqrt{m_{i}L{k}_{B}T}\text{sinh}\left(\frac{F_i}{\sqrt{m_{i}L{k}_{B}T}}\frac{\delta t}{2}\right)+p_i\left[\text{cosh}\left(\frac{F_i}{\sqrt{m_{i}L{k}_{B}T}}\frac{\delta t}{2}\right)-1\right]$$ (14a)

$${\dot{s}}_i\left(\frac{\delta t}{2}\right)=\frac{p_i}{\sqrt{m_{i}L{k}_{B}T}}\text{sinh}\left(\frac{F_i}{\sqrt{m_{i}L{k}_{B}T}}\frac{\delta t}{2}\right)+\text{cosh}\left(\frac{F_i}{\sqrt{m_{i}L{k}_{B}T}}\frac{\delta t}{2}\right)$$ (14b)

that is slightly different from Eq. S77 in Reference 57, namely the argument of hyperbolic functions was changed from $\sqrt{\frac{F_i^2}{m_{i}L{k}_{B}T}}\frac{\delta t}{2}$ to $\frac{F_i}{\sqrt{m_{i}L{k}_{B}T}}\frac{\delta t}{2}$ in order for F_is_i to retain the sign of F_i. Propagators of ${\mathcal{L}}_N$ and ${\mathcal{L}}_O$ follow Eqs. S78 and S79 of Reference 57.

Recalibrating se-QM/MM Models with Force Matching

In order to improve the efficiency of MTS simulations, it is preferable to set the outer time step to be as large as possible, while maintaining the stability and accuracy of the simulations. In MTS ai-QM/MM MD simulations, the length of the outer time step is bound by the magnitude and nature of the correction forces ΔF^SE→AI in Eq. 4. To achieve longer outer time steps, the correction forces ΔF^SE→AI need to be sufficiently small for the relevant configurations. In the limit of a “perfect” se-QM model which can fully reproduce the high-level forces for any configuration, i.e., ΔF^SE→AI → 0, the outer step would never be needed, i.e., Δt → ∞. In the other limit, where a null se-QM model with substantial and random errors in the atomic forces, the outer step size would have to be equal to the inner step size to correct the forces at each time step, i.e., Δt = δt, achieving no speedup at all. In practice, the accuracy of an se-QM/MM model for a specific system will fall between the two limits. By minimizing ΔF^SE→AI, more accurate and efficient samplings can thus be attained for MTS ai-QM/MM MD.

One way to improve the accuracy of an se-QM model is to recalibrate the se parameters through force-matching. In this work, the force-matching protocol was improved upon the one from our previous paper,⁴³ which was in turn based on the Reaction Path Force Matching(RP-FM) method.⁶² Inspired by the ForceBalance method,⁶⁸ where the objective function is defined as the weighted sum of a set of scaled properties of the system, such as energies, forces, and dipole moments, we defined the objective function in this work to include errors in energies, bond forces, Cartesian forces with bond forces removed, and a quadratic penalty term that restrains the parameters to their initial values. The final form of the objective function is

$$\begin{gathered} {\mathcal{X}}^2=W_E\frac{{\sum }_{i\in N_{\text{s}}}{\left(\left(E_i^{\text{SE}}-\langle E^{\text{SE}}\rangle \right)-\left(E_i^{\text{AI}}-\langle E^{\text{AI}}\rangle \right)\right)}^2}{{\sum }_{i\in N_{\text{s}}}{\left(E_i^{\text{AI}}-\langle E^{\text{AI}}\rangle \right)}^2} \\ +W_F\frac{{\sum }_{i\in N_{\text{s}}}{\sum }_{a\in N_{\text{bond}}}{w}_a{\left|F_{i,a}^{\text{SE}}-F_{i,a}^{\text{AI}}\right|}^2}{{\sum }_{i\in N_{\text{s}}}{\sum }_{a\in N_{\text{bond}}}{w}_a{\left|F_{i,a}^{\text{AI}}\right|}^2} \\ +W_F^{\prime }\frac{{\sum }_{i\in N_s}{\sum }_{b\in N_{\text{atom }}}{\left|F{\prime }_{i,b}^{\text{SE}}-F{\prime }_{i,b}^{\text{AI}}\right|}^2}{{\sum }_{i\in N_{\text{s}}}{\sum }_{b\in N_{\text{atom}}}{\left|F{\prime }_{i,b}^{\text{AI}}\right|}^2} \\ +\lambda \sum _c{\left|\frac{{\xi }_c-{\xi }_{c,0}}{{\xi }_{c,0}}\right|}^2 \end{gathered}$$ (15)

where N_s is the number of sampled configurations, 〈E^AI〉 and 〈E^SE〉 are the average ai-QM/MM and se-QM/MM energies for the samples, F_i,a refer to the forces on the a-th bond of the i-th configuration, F′_i,b correspond to the Cartesian forces on the b-th atom (with associated bond forces projected out), and ξ_c are the empirical parameters for the se-QM model. In this work, we set W_E = 1, W_F = 1, $W_F^{\prime }=1$, λ = 10⁻⁵ and w_a = 1/N_a, and reparameterized the PM3 se-QM model following the procedure described below. Henceforth recalibrated PM3 model is hereafter denoted as PM3* and the optimized parameters can be found in Table S1 of Reference 69.

The Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm as implemented in the SciPy package was used to perform the minimization, without any bounds on the parameters. The first-order derivatives with respect to the parameters were calculated using the finite difference method. It should be noted that, like in our previous paper,⁴³ the procedure will only find a local minimum that is close to the standard parameter set, which is not necessarily the global minimum in the parameter space. However, it is not an issue in practice for a couple of reasons. First, the standard parameter set has been calibrated for a wide range of systems, so moving too far from it (to reach the global minimum) could increase the risk of going into nonphysical regions of the parameter space. Second, the se-QM model in MTS ai-QM/MM MD simulations is only used as the low-level method for the configurational sampling, so the inaccuracy of the model (at a local minimum) will be corrected by the high-level corrections at outer time steps. Nevertheless, if necessary, for example when the accuracy of the recalibrated se-QM model is not sufficient, more elaborate non-linear parameter optimization algorithms, such as genetic algorithm, can be employed.

Computational Details

In this study, chorismate mutase, which catalyzes the conversion from chorismate to prephenate,^70–72 was chosen as the model system to test the proposed MTS method for computing the free energy profile of enzyme reactions. Due to a relatively small substrate with 24 atoms, the enzyme has been one of the most widely-used testing systems for QM/MM and method developments.^38,73–85

Starting from the X-ray crystal structure (PDB ID: 2CHT⁷⁰) of Bacillus subtilis chorismate mutase complexed with a transition state analog, the setup and equilibration of the system followed the same protocol in our previous study⁴³ using the AMBER software.^65,66 The last frame of a 2 ns classical trajectory was used as the starting structure for the QM/MM MD simulations in this study.

We first conducted short ai-QM/MM MD simulations under periodic boundary conditions, which were used as the reference calculations for se-QM/MM parameter recalibration. The QM subsystem consisted of the substrate chorismate (i.e. 24 QM atoms) and were described by the B3LYP/6–31G* level of theory^86–89 as implemented in the Q-Chem software package.⁶⁴ The MM subsystem included the enzyme, water solvent molecules, and sodium counter ions in the center cell, as well as all atoms in the image cells (including the QM images). The MM subsystem was described by ff14SB/GAFF/TIP3P forces fields^90–92 as implemented in the Amber software package.^65,66

The QM/MM-AC electrostatic embedding scheme⁶³ was used to handle both short- and long-range QM-MM electrostatic interactions. The particle mesh Ewald (PME) method^93,94 was used to treat the MM-MM electrostatic interactions with a real-space cutoff of 10Å. Both the QM-MM and MM-MM van der Waals interactions were treated classically using Lennard-Jones potential as defined in the Amber force fields,⁹⁰ and truncated at a cutoff of 10Å. The SHAKE algorithm⁹⁵ was used to constrain all the bonds involving hydrogen atoms in the MM subsystem. Langevin dynamics with a friction coefficient of 5 ps⁻¹ was performed at 300 K and a time step of 1 fs was used for the MD integration. As noted in the Theory section, both “end” and “middle” schemes were used for the thermostat in the leapfrog integration within the AMBER software package. For the SIN(R) simulations, L = 4 sets of thermostat variables were coupled to each degree of freedom with Q₁ = Q₂ = k_BT(0.05 ps)² for the thermostat coupling parameters and γ = 1 ps⁻¹ for the friction coefficient, respectively. For the MTS ai-QM/MM molecular dynamics simulations, the same setup was used as the regular ai-QM/MM MD, except that 1 fs time step was used for the inner steps, whereas the time step for the outer steps varied from 2 to 10 fs.

The umbrella sampling technique⁹⁶ was employed to construct the direct ai-QM/MM free energy profile along the reaction coordinate. In the present study, the reaction coordinate was defined as the difference between the bond lengths of the breaking C–O and forming C–C bonds. 40 windows were evenly distributed along the reaction coordinate ranged from −1.95 to 1.95Å, and the force constant of the harmonic biasing potential was set to be 300 kcal mol⁻¹ Å⁻² for all the windows. For each window, 60 ps ai-QM/MM MD simulation was performed. The snapshots were collected every 40 steps during the last 40 ps simulation, which resulted in 1000 snapshots for post-processing analysis. Multistate Bennett acceptance ratio (MBAR)⁹⁷ as implemented in the pymbar package was utilized to estimate the free energy profile.

Results and Discussion

Time Scale Separation of Reference and Correction Forces

The fundamental assumption of an MTS QM/MM simulation is a time scale separation of the reference and correction forces. Especially, the correction forces oscillate with a small amplitude and/or slowly than that of the reference forces. To test this assumption, we post-processed a 200 fs NVT trajectory in both reactant and transition-state regions that was generated using direct B3LYP/MM simulations (i.e. without MTS), and calculated the atomic forces on the QM atoms with both PM3/MM and PM3*/MM models.

Figures S1 and S4 showed the x-component of the B3LYP/MM forces on all QM atoms, as well as the correction forces to PM3/MM, and PM3*/MM models (i.e. difference between B3LYP/MM forces and those from these models). A clear separation of time scales between the reference and correction forces is observed for both PM3/MM and PM3*/MM models. This is even more evident after a Fourier transformation is performed, where Figures S2 and S5 showed that the high-frequency (>2,000 cm⁻¹) oscillations become quite small for the correction forces. More important are the correction forces of hydrogen atoms, as their oscillation determines the time step for MD simulation. Between PM3/MM and PM3*/MM, the recalibrated potential (green line) shows smaller and more smoothly varying correction forces than the uncalibrated one (orange line). Together, this allows for longer outer integration steps and the generation of more accurate trajectories.

Removing Fast Components from the Correction Force

In order to determine the maximum outer time steps allowed, we have investigated the stability of the MTS QM/MM trajectory with different outer time steps. As expected, the MTS QM/MM trajectory became unstable with the increase of the outer time step. When the outer time step increased to a certain point (4 fs for PM3 and 6 fs for PM3*), we observed hydrogen atoms from C–H and O–H bond detached from the C or O atoms.

In order to mitigate this, we first tested using the hydrogen mass repartitioning method⁹⁸ with a scaling factor of 3 or 4 to reduce the stretching frequencies of C–H and O–H bonds, which has been used to increase the outer time step for MTS simulations with polarizable force fields.⁹⁹ However, unlike in classical force fields where bonds are not able to break because of the use of harmonic potentials, we observed the bonds between heavy atoms were broken during the MTS QM/MM simulations with larger outer time steps, because of the reduced masses of the heavy atoms that were bonded to hydrogen atoms. This suggested that the hydrogen mass repartitioning method is not a viable strategy to increase the time step for QM/MM simulations.

Subsequently, we adopted an alternative strategy, where the force corrections associated with C–H and O–H bonds in the system were projected out. As shown in the left column of Figures S3 and S6, the PM3→B3LYP force corrections (blue line) for most of these bonds were reduced after the PM3 model recalibration (orange line). However, the remaining forces still caused the breaking of C–H and O–H bonds with larger outer time steps. When these bond force components were removed from the force corrections in the outer time steps, it improved the stability of the simulation (i.e. keeping the C–H and O–H bonds intact) even with larger outer time step. Hereafter, we denote it as the “Remove Force” (RF) version of our MTS QM/MM simulations.

Alternatively, we applied the SHAKE algorithm⁹⁵ to freeze the lengths of O–H and C–H bonds in the QM region (at values from the GAFF force field). While some encouragingly results will be shown later, this strategy is not necessarily universally applicable in part because many reactions involve proton transfer and in part because C–H and O–H bond lengths might change substantially in some reactions (such as Menshutkin reactions).

Instead of removing the fast components of the correction force, one can in principle apply the MTS correction only on key slow degrees of freedom associated with the reaction coordinate, in the spirit of the reaction-path force-matching in the collective variables (RF-FM-CV) method.¹⁰⁰ The effectiveness of this strategy will be examined in future studies.

Microcanonical Ensemble MD Simulations

We tested the stability of the MTS ai-QM/MM MD simulations with different combinations of inner step models and outer step sizes using microcanonical (NVE) ensemble MD simulations. The setup is the same as the one described in the Computational Details section, and no thermostat was applied. All NVE simulations started from the same initial condition and ran for 10 ps. For the Δt = 3 fs case, the energy fluctuation is shown in Figure 3.

Figure 3:

Fluctuation in the total energy (in kcal/mol) of the chorismate mutase system during a 10 ps MTS simulation with PM3*/MM at 1 fs inner time steps and B3LYP/6–31G*/MM force corrections at 3 fs outer time steps.

The energy fluctuation, which is measured by the standard deviation of the total energy along the trajectory, is listed in Table 1. Separately, the energy drifts, which correspond to the slope of the linear least-squares regression of the same data, are also listed. Only the trajectories that were stable during the 10 ps simulations were presented.

Table 1:

Energy fluctuation and drift in 10 ps NVE simulations with B3LYP/MM as well as several MTS schemes.

	Fluctuation (kcal/mol)	Drift(kcal/mol/ps)
B3LYP, 1 fs	0.74	0.042
B3LYP/PM3, 2 fs/1 fs	0.78	0.006
B3LYP/PM3*, 2 fs/1 fs	0.74	−0.043
B3LYP/PM3*, 3 fs/1 fs	0.82	0.028
B3LYP/PM3*, 4 fs/1 fs, SHAKE	0.63	−0.058
B3LYP/PM3*, 5 fs/1 fs, SHAKE	0.76	0.017
B3LYP/PM3*, 6 fs/1 fs, SHAKE	0.88	−0.067

For the PM3 model, the MTS ai-QM/MM NVE MD was numerically stable up to Δt = 2 fs. Beyond that, the reactive region gets seriously distorted due to extra broken bonds. When the recalibrated PM3 model (PM3*) was used, the outer time step Δt could reach as far as 3 fs without artificial breakage of chemical bonds involving hydrogen atoms. When the SHAKE algorithm is applied to constrain the C–H and O–H bonds in the QM region, the outer time steps could be increased up to 6 fs.

Overall, the energy fluctuation in all MTS simulations ranged from 0.63 to 0.88 kcal/mol, which was comparable to the 0.74 kcal/mol fluctuation for the single-Hamiltonian simulation using the B3LYP/MM energy function. The energy drifts were also comparable. This confirmed that our implementation of the MTS algorithm maintained the energy conservation, largely due to a rigorous treatment of QM/MM electrostatics in a periodic system using our QM/MM-AC model.

Free Energy Profile

Next, we calculated the free energy profile for each of the umbrella sampling simulations with our MTS QM/MM method, where the outer time steps were stretched to 8 fs with the use of Langevin dynamics without the issue of C–H and H–O bond breakages. With the “end” scheme for the thermostat, the free energy profiles (Fig. 4A) are all in good agreement with the direct QM/MM simulations. In terms of the sampled pathways, overall the MTS ones were in agreement with the direct QM/MM one (Fig. 4B). However, noticeable deviations were observed in the transition state regions. Especially, the position of the highest-energy point on the free energy profiles was off by ~0.1Å in terms of the C–O bond length for the MTS simulations.

Figure 4:

(A) Free energy profiles and (B) sampled pathways of chorismate mutase reaction from direct and MTS simulations with “end” scheme Langevin dynamics, while (C) and (D) are the results from “middle” scheme Langevin dynamics, respectively. (E) and (F) are the results from MTS simulations with QM SHAKE, respectively. (G) and (H) are the results from MTS simulations with “middle” scheme SIN(R) thermostat, respectively. In MTS simulations, PM3*/MM forces are evaluated at each inner step (1 fs), while B3LYP/6–31G*/MM force evaluations are carried out at 2–10 fs intervals. The stars show the locations of the transition states on the pathways.

We also tested the “middle” scheme Langevin dynamics for the QM/MM MTS simulations using 6 and 8 fs outer time steps, and obtained similar free energy profiles (Fig. 4C). Very encouragingly, the sampled pathways for the “middle” scheme in Fig. 4D were in better agreement with the direct QM/MM one than the ones from the “end” scheme simulations(Fig. 4A). This supports that the “middle” scheme Langevin dynamics can give more accurate ensembles in the configurational space than its “end” scheme counterpart.^57,67

So far, these free energy profiles were obtained from MTS simulations within our RF scheme, where the correction forces associated with O–H and C–H bonds in the QM region were removed at outer timesteps. Also shown in Fig. 4 were MTS simulation results with SHAKE constraints on O–H and C–H bonds in the QM region. The free energy profile in Fig. 4E with an 6 fs outer timestep also well matched direct ai-QM/MM results. When the outer timestep was further increased to 8 fs, the free energy profile yielded a barrier that was ~1.0 kcal/mol higher. This suggested that an adoption of the SHAKE algorithm, when applicable, could produce less accurate results for MTS simulations with longer time steps than our RF scheme. We note, however, this is not a completely consistent comparison, because we have yet to perform the direct simulation (i.e., no-MTS ai-QM/MM simulation) with SHAKE.

Finally, as shown in Fig. 4G and H, MTS QM/MM simulations with the SIN(R) thermostat remained stable with an outer time step up to 10 fs. When a 6 fs outer timestep was used, both predicted free energy barrier and reaction free energy were well within 1 kcal/mol from the direct ai-QM/MM values. For 8 fs outer timestep simulations, the predicted free energy barrier was ~1.2 kcal/mol higher than the direct ai-QM/MM value. When the outer timestep was increased to 10 fs, the differences in free energy barrier and reaction free energy increased to c.a. 1.8 kcal/mol and −2.5 kcal/mol, respectively.

To further test the accuracy of the MTS simulations, we calculated the average potential energy per atom^57,67,101 for direct and MTS simulations in the transition state window of the umbrella sampling simulations (Fig. S7). For all the models, the distributions of the average potential energy per atom were found to be consistent across the various outer time steps in use, which shows the robustness of our MTS model.

Speedup

We can use $t_{\text{QM}}^{\text{high}}$ and $t_{\text{QM}}^{\text{low}}$ to represent the time required for a single energy-and-force calculation for the QM region and the QM/MM interactions at high- and low-level methods, respectively. Meanwhile, t_MM corresponds to all other time required for a single MD step (including MM force calculation, MD integration, and communication overhead of calling external QM programs). The speedup of an MTS ai-QM/MM MD simulation relative to a standard ai-QM/MM one can then be estimated using

$$s=\frac{t_{\text{QM}}^{\text{high}}+t_{\text{MM}}}{\frac{1}{N}{t}_{\text{QM}}^{\text{high}}+t_{\text{QM}}^{\text{low}}+t_{\text{MM}}},$$ (16)

where the denominator — the average cost of each MTS time step — contains the cost of computing high-level QM/MM force at the outer time steps (divided evenly over all the time steps).

In the limit that $t_{\text{QM}}^{\text{low}}$ and t_MM are negligible compared to $t_{\text{QM}}^{\text{high}}$, the maximum speedup will tend to N. In practice, the speedup will be smaller than N, and the actual speedup depends on the ratio between $t_{\text{QM}}^{\text{high}}$ and $t_{\text{QM}}^{\text{low}}$ if t_MM is negligible. Figure 5 shows the maximum speedup of different $t_{\text{QM}}^{\text{high}}/t_{\text{QM}}^{\text{low}}$ ratios and actual speedup of the system under study. Our current implementation of the MTS ai-QM/MM method achieved a 5-fold speedup for the chorismate mutase system, when N = 8. For larger systems, more significant speedup is expected due to higher $t_{\text{QM}}^{\text{high}}/t_{\text{QM}}^{\text{low}}$ ratios.

Figure 5:

Theoretical and actual speedups for NVT calculations with MTS integrator. The theoretical speedups are shown for various $t_{\text{QM}}^{\text{high}}/t_{\text{QM}}^{\text{low}}$ ratios. The actual speedup comes from MTS simulations of chorismate mutase reaction with PM*/MM calculations at inner time steps and B3LYP/6–31G*/MM calculations at outer time steps.

Conclusions

In this work, we explored the use of recalibrated PM3 model for the inner time steps of MTS QM/MM molecular dynamics simulations of the chorismate mutase catalysis. The main findings were:

• The recalibrated PM3 model led to smaller correct forces at the outer steps, which allowed the use of longer outer time steps.

• In order to prevent nonphysically breakage of the C–H and O–H bonds with the use of longer time steps (such as 8 fs), one can either project out the force corrections associated with these bonds, or when applicable, apply SHAKE constraints to these bonds in the QM region.

• There are a couple of alternative ways to increase the outer time steps in NVT MTS QM/MM simulations. For the simulation of the chorimate mutase reaction, the outer time step of Langevin dynamics simulations could be increased to 6 fs with SHAKE and 8 fs with the removal of C–H/O–H force corrections. The outer timestep can reach 10 fs with SIN(R), if C–H/O–H force corrections are removed.

• With a 8 fs outer timestep, it enabled a 5-fold speedup in the computational time for acquiring the chorismate mutase reaction free energy profile.

We are applying the QM/MM MTS simulation in the modeling of several other enzymes with larger QM regions, which will be reported in the future.