Adaptive Restraints to Accelerate Geometry Optimizations of Large Biomolecular Systems

Abstract

Quantum mechanical/molecular mechanical geometry optimizations of large-scale biological systems, such as enzymes, proteins, membranes, and solutions, are typically computationally expensive to the point of being cost-prohibitive. By convention, an approximation is made to such calculations that atoms beyond a certain distance from the QM region provide only negligible improvements to the resulting optimization energy and geometry, and as such are restrained to reduce the number of degrees of freedom. These constraints are normally applied beyond a user-defined radius. Here we describe a new method of geometry optimization acceleration and automation which generates adaptive gradient-based restraints for QM/MM optimizations, leading to significantly faster optimizations and generally lower relative energies. The restraints are determined by an algorithm rather than a user, and can adapt to directional optimizations as well as differences in starting geometry. This flexibility is key to finding excited state minima and minimum energy conical intersections (MECIs) in complex protein environments. This algorithm was implemented as an external Python tool for use alongside TeraChem, with a modular interface that can be straightforwardly applied to other QM/MM packages. We tested on a green fluorescent protein (rsEGFP2) and two red fluorescent proteins (FusionRed, mScarlet) in water and a proton-swapping aspartic acid pair in explicit water. We are able to produce a nearly 50% reduction in computational time while maintaining appropriately optimized geometries and relative energies.

1 |. Introduction

The theoretical study of biological processes often involves simulations of large and explicitly solvated biomolecules, such as proteins, nucleic acid chains, and similarly complex systems. Large-scale molecular dynamics (MD) simulations are commonly used to investigate the structure-function relationships of these systems; however, most classical MD force fields cannot account for the formation or breaking of covalent bonds. To explore reaction mechanisms in these large systems, it is necessary to apply hybrid quantum mechanics/molecular mechanics (QM/MM) calculations and treat a relatively small portion of the system with quantum mechanics to account for the electronic structure of the reaction [1–9]. QM/MM molecular dynamics (QM/MM-MD) have been increasing in popularity for a number of theoretical applications, however, the transition from pure MM to reasonable QM/MM descriptions is often computationally expensive when obtaining starting structures from classical MD [10, 11]. Calculating the enthalpy of a reaction mechanism using QM/MM methods generally begins with optimized structures for both the reactant and product [12–14]. The optimization of large structures can become prohibitively expensive, as the number of degrees of freedom increases quickly with every added molecule. Even in a relatively small system, such as green fluorescent protein (GFP) solvated with water and neutralized with counterions, this can correspond to tens of thousands of atoms and hundreds of thousands of degrees of freedom (3N − 6) to be optimized.

To mitigate this computational expense, QM/MM optimizations commonly include restraints on atoms beyond a given distance [13, 15–19]. It is assumed in these cases that the long-range interactions are generally constant, that effects on relative energies will be driven almost entirely by local changes, and that the optimization will not have a large directional component. However, long-range effects can be nontrivial, especially in systems with charged regions or highly electronegative atoms [20–22]. Depending on the initial conformation or reaction, it is also possible that an optimization will progress along a vector, where much of the QM region moves in a single direction, such as proton transfer through an ion channel [23–28], or electron transfer through iron-sulfur clusters [29–31]. This can be of particular importance when QM/MM optimizations are used as a set of starting structures for QM/MM-MD simulations on ground and excited states, where it is desirable to start from a local minimum but not a prerequisite to start from a true, global minimum.

In addition to ground-and excited-state geometry optimizations, many photophysical systems are optimized to a minimum energy conical intersection (MECI), a geometry at which the ground and excited states are degenerate. Optimizations to the MECI for larger photoactive molecules, such as fluorescent proteins, are expensive and necessarily require the application of restraints to much of the system [32, 33]. MECI geometries can be chemically unintuitive and highly directional, making it challenging to know what restraints to include.

Adaptive simulations, where various parameters are changed on the fly, provide promising directions to improve the computational efficiency of QM/MM simulations. Previous work by Waller et al. proposed an adaptive QM region defined by density of non-covalent interactions, e.g., which can lower bias in choosing QM regions and smooth the transition between partitioning methods [34]. Another common QM/MM scheme involves a macro/microiterations method, where the QM region is optimized with a frozen MM, then the MM region is optimized with a frozen QM [35]. When a quadratic coupling is included, this reduces the number of expensive QM optimization steps, but the scheme still requires computing two sets of gradients and sometimes results in oscillations in optimizations. In this work, we describe an algorithm that accelerates geometry optimizations by automatically choosing atomic restraints to reduce the degrees of freedom that are optimized during each iteration (see Figure 1).

FIGURE 1 |.

Adaptive restraint workflow for accelerated geometry optimizations of large QM/MM systems.

This algorithm is designed to be implemented with any quantum chemistry software package that can perform restrained geometry optimizations and gradient calculations. The process begins with the calculation of the gradient from the starting geometry. An array of booleans is constructed for all atoms with a gradient magnitude below a user-provided threshold (T₀). Atoms below the threshold are restrained or “frozen” while the rest remain unrestrained and allowed to optimize towards a local minimum (see Equation 1). Atoms in the QM region of the calculation are excluded from this list of restrained atoms, as they are often of greatest interest in QM/MM calculations.

$$F_i=\{\begin{array}{cc}0& \text{if}\left|\nabla E_i\right|\ge T_0\\ 1& \text{if}\left|\nabla E_i\right|<T_0\end{array}$$ (1)

As the optimization progresses, the gradient magnitudes are periodically checked against the threshold, and atoms are restrained or, in some cases, unrestrained. Generally speaking, the number of atoms that fall beneath the gradient threshold and are restrained increases, lowering the expense of the individual calculations proportionally to the number of atoms frozen. Additionally, the number of iterations required to reach the minimum usually decreases as well. The frequency of the gradient check is determined by the user, though in this work, we recommend particular parameters for generally optimal outcomes as discussed below. Once the optimization converges under restrained conditions, a final optimization is performed without restraints, allowing the system to reach a local minimum on the potential energy surface.

Two different selection algorithms were implemented and tested to determine their impact on optimization efficiency.

Static Single Threshold

The Static Single Threshold (SST) algorithm accepts a single threshold value, which is unchanged through the optimization. This value is set at the beginning of the calculation and remains unchanged for the duration.

Dynamic Single Threshold

The Dynamic Single Threshold (DST) algorithm takes a user-defined percentage value. During the restraint phase of each cycle, the numerical value of the threshold is recalculated with this percentage and the value of the largest gradient magnitude (MaxGrad) in the system. This ensures that atoms with unusually large gradients will be optimized quickly. As the maximum gradient value decreases, the numerical value of the threshold also decreases, until the threshold eventually falls below the convergence criteria and the system is able to converge.

2 |. Methods

The adaptive restraints algorithm was implemented as a Python tool for use with TeraChem. In its current form, it requires no non-standard Python libraries and may be used with standard TeraChem inputs. It is command-line-callable, allowing it to be used in batch jobs on High Performance Computing (HPC) systems. Because the algorithm was developed as an external tool and is open-source, it was designed to be easily modified to work with other QM/MM packages. Internal classes, functions, and variable names are written to ensure a clear understanding of purpose and scope, and users may choose to modify the internal contents of specific functions according to their specific QM/MM package of choice. At minimum, implementation of a new QM/MM package will require edits to the input file writer, gradient reader, and external system calls, as well as minor edits to known messages from successful output files from said package.

To test the different restraint selection algorithms, we performed QM/MM geometry optimizations on a green fluorescent protein variant system (rsEGFP2) with a chlorinated chromophore. The chromophore was placed in the QM region, and the remainder of the protein and solvent in the MM. We also used an aspartic acid pair system with both molecules in the QM region and the explicit solvent TIP3P water in the MM (Figure 2).

FIGURE 2 |.

(a) The chromophore of the GFP test system, shown in ball-and-stick representation, is in the QM region, while the remainder of the protein and all solvent waters are in the MM region, and (b) the diaspartate test system in the QM region amid MM solvent water molecules.

To prepare the rsEGFP2 variant, we took the crystal structure (PDB ID: 6PFR), protonated and checked it with MolProbity and propKa, and solvated with a 12Å pad of TIP3P water in an approximately spherical shape. We retained the crystal waters and prepared inputs for QM/MM with the Amber ff14SB force field and the tleap program in AmberTools18.

These systems were chosen to demonstrate the usefulness of the algorithm with both larger biomolecules and small molecules in solvent. A set of twelve (12) static threshold values was selected, ranging from 0.5 to 1E-6, and gradient check frequencies from 10 to 100 steps in increments of 10 steps, producing a total of 120 different parameter sets for the SST tests. The DST algorithm was similarly tested, using a set of percentages from 1% to 50% rather than static threshold values, with the same array of gradient check frequencies, resulting in 70 different parameter sets.

All geometry optimizations were performed with TeraChem using the B3LYP functional and 6–31g** basis set.

Optimizations used the DL-FIND [36] library with LBFGS [37], with all inputs given as Cartesian coordinates, and all atomic positions (QM and MM) adjusted after each optimization step. TeraChem optimizes QM/MM geometries by computing the combined gradient of the system via an additive scheme and adjusting the geometry in one step, and includes polarization of the QM from the MM charges via electrostatic embedding, applying the Z1 scheme where the charge at the link atom is set to zero [38]. Cut bonds are treated with the link atom method, which caps the cut bond with a hydrogen. TeraChem provides the QM region calculations and uses OpenMM for the MM region. Convergence criteria was kept at the TeraChem defaults (energy change: 10⁻⁶ a.u., maximum step size: 1. 8 · 10⁻³ Bohr, RMS step size: 1. 2 · 10⁻³ Bohr, maximum gradient magnitude: 4. 5 · 10⁻⁴ Hartree/Bohr, and RMS gradient magnitude: 3. 0 · 10⁻⁴ Hartree/Bohr) with mixed precision and a maximum of 200 SCF iterations.

Testing for MECI calculations was done with published data on mScarlet by Pieri et al. [33] and FusionRed by Murphy et al. [11]. In these cases, the level of theory used was SA-α-CASSCF(4,3)/6–31g**(α = 0. 64). The mScarlet data was used to confirm that adaptive restraints do not affect the final geometry or computational cost, while FusionRed was used to demonstrate that an MECI may be calculated at various points from classical and QM/MM-MD sampling without additional preoptimization steps. We additionally computed S₀ and S₁ minima using the FusionRed system to test the BFGS algorithm with a Fischer-Almlof guess Hessian, and a separate S₀ minimum as a micro/macroiteration scheme test. The entire QM/MM gradient was used to optimize the MECI structures, and the algorithm used was the Lagrange-Newton method [39].

Full details of the inputs are found in the Supporting Information.

3 |. Results and Discussion

3.1 |. GFP in Water

The rsEGFP2 test system was optimized with no restraints to a local minimum. The computational time and final optimized energy were taken as the baseline values for rsEGFP2 in water against which our algorithms were compared. Most static threshold/frequency combinations yielded slightly lower optimized relative energy values, but some were more computationally expensive than the baseline (see Tables S1 and S2). The largest improvement to computational speed was obtained with a threshold of 0.5, recalculating restraints every 90 steps, which resulted in a time reduction of 48.75%. While the energy shows a decrease of 0.034% (0.53 a.u.), we consider this to indicate differences in the geometry of the MM region rather than a significant alteration of the final geometry vs. the unrestrained system. The QM region differs minimally in geometry and energy, as discussed further below. Nearly all combinations of threshold and gradient check frequency resulted in some degree of improvement in both computational time and similarly shifted optimized energies. Across all of the systems and types of calculations we explore here, a threshold of 0.01 in the gradient and 70 steps per cycle result in good improvements in speed, and we suggest this as a starting place for typical QM/MM systems.

Similar to the static threshold, we find that most combinations of threshold and gradient check frequency provide an improved computational speed with similar energetic changes (see Tables S3 and S4). Using a 50% dynamic threshold and gradient check frequency of 60 steps gave the best improvement to computational time (−44.55%).

The average time improvement for static and dynamic threshold tests was 16.44% and 13.18%, respectively. Comparing the two methods, we find that both provide similar optimization energies and geometries, however, the static threshold demonstrates better computational time improvements than the dynamic thresholds. We also performed frequency calculations on the selected fastest/best optimized structures and compared them to the unrestrained optimized structure. As shown in Figure 3, the atomic positions of atoms in the QM region of each of the selected systems differ by less than 0.01 Å, indicating that the optimized geometry of the region of primary interest in QM/MM calculations is not significantly affected by the adaptive restraint algorithm.

FIGURE 3 |.

(a) Overlaid QM region for all GFP test systems, aligned to fluorescent chromophore and surrounded by MM environment within 3Å. Pairwise RMSD of all QM regions against each other yields a maximum variation of 0.122 Å, and (b) 2D representation of chromophore represented in QM region. Low-valence carbonyl and amine groups occur at the border between QM and MM regions of the system, where the chromophore connects to the rest of the protein.

For each test system, a single-point energy calculation was performed on the optimized QM region atoms to determine the energetic results of the optimization algorithms. The results of these single-point energy calculations indicate that the energies of all GFP test systems are consistent to the limit of precision, regardless of the optimization parameters selected. Interestingly, the QM regions deviated by less than 0.122 Å in a pairwise RMSD calculation, while the surrounding MM environments exhibit slightly more variation in their final optimized positions (0.934 Å maximum deviation).

To explore the utility of the adaptive restraint algorithm on frequency calculations, we optimized ten additional frames from the original classical MD simulation using SST (threshold value of 0.01 and 70 steps per cycle) with a QM region composed of the chromophore, 6 amino acid side chains hydrogen bonded to the chromophore, and a water molecule, totalling 103 atoms. After adaptive restraint optimizations were completed, we performed a frequency calculation on the chromophore only, totalling 32 atoms. All ten snapshots resulted in similar frequency profiles, each with one imaginary frequency value between −30 and −65 cm⁻¹. Visualization of these frequencies reveals they correspond to small conformational twisting of the chromophore, similar to the motion expected during photoisomerization of the chromophore (see Supporting Information Tables S5–S7 and Figure S1). The presence of small imaginary frequencies in QM/MM simulations of proteins is known to relate to minor conformational changes [40], truncated Hessians, numerical noise in DFT calculations, or the use of coordinate-locking methods [41–43].

3.2 |. mNeongreen and FusionRed

Another fluorescent protein system based on mNeonGreen was tested as well (see SI for system details), resulting in no imaginary frequencies. We take from this that the choice of system and snapshot is the predominant factor in obtaining highly optimized minima, but that the adaptive gradient model achieves comparable or better results in this arena compared to conventional QM/MM optimization methods.

As a point of comparison, we computed optimizations for ground and excited state minima for FusionRed with BFGS and separately with a macro/microiteration scheme, as described in the Methods. With BFGS and including our adaptive restraint algorithm, we obtained similar final optimized geometries and energies, but the calculation time increased by about 4x. The micro/macroiteration method, using a scheme with alternating frozen MM with frozen QM, did not fully converge, and the calculation time increased by about 10x. We did not include the quadratic coupling between the QM and MM regions, which would no doubt improve the convergence. Nonetheless, we believe these results support the use of our algorithm with a variety of optimization routines and that the adaptive gradient model is an improvement.

3.3 |. Aspartic Acid Proton Transfer in Water

To explore the limitations and effects of the accelerator on small molecules and the results of reaction enthalpy calculations, an aspartic acid test system was generated (DiASP). This initial conformation was considered the reactant system (Figure 4a). We performed an unrestrained optimization of the DiASP system to establish a baseline for comparisons (Figure 4b). Using the same test parameter sets as with rsEGFP2 and static thresholds, we found that the best improvement in calculation time was with a threshold of 0.1 and a gradient check frequency of 90 steps, which required 48.48% less computational time. (see Tables S8 and S9).

FIGURE 4 |.

Optimization progression of diaspartate test system showing (a) initial geometry, (b) unrestrained optimized geometry, (c) initial product guess, and (d) final unrestrained optimized product.

By comparison, the dynamic thresholds produced the best improvement in computational time of 49.04% with a 50% threshold and 100 steps gradient check frequency (see Tables S10 and S11).

The final optimized structure from each parameter set was used as the starting guess for the product by moving the protons along the O· · ·O vector such that each acidic proton was moved to the opposite aspartic acid with the same O–H distance, resulting in a “swap”. These values were chosen to ensure consistency in the product guess geometries (see schematic in Figure 4). Similar performance was observed in the product optimizations (see Tables S12–S15). By comparing the optimized products to their reactants from the same parameter sets, we are able to calculate the resulting reaction enthalpies across all test systems (see Tables S16–S19).

Several parameter sets with high thresholds (>0.005) resulted in unexpectedly large enthalpy values for the proton exchange reaction, as high as −122.44 kcal· mol⁻¹. Most parameter sets, however, gave results much closer to chemical accuracy, below 1 kcal· mol⁻¹ (see Figure S2). The distribution of values is also entirely negative due to the continued optimization of the entire system following the initial product-guess modifications—the final unrestrained optimization step of the algorithm allows everything in the system to adjust slightly, compounding the energetic changes in the large MM region, especially. For these types of systems, we recommend a threshold of 0.0001 and checking every 70 steps, though if speed is a factor, one can benchmark to determine if a higher level threshold is suitable without compromising accuracy. Much of the additional energy change can be attributed to solvent reorganization as shown in Figure 5. The large changes persist even when preoptimizing with, e.g., 1,000 steps of steepest descent to try to avoid local minima. We conclude from this that the optimization algorithm cannot overcome the general issue of sampling in large systems, where small changes in the QM environment can result in local minima that have comparatively large energetic changes that arise from the MM optimization. That said, the changes in the QM region remain very small. Given the observed differences between parameter sets, it is recommended that users of this method benchmark a few combinations to identify combinations of threshold and recheck frequency that best serve their purposes before moving on to large-scale calculations.

FIGURE 5 |.

Diaspartic acid product optimization representative structure. Solvent waters within 3.0Å are shown to illustrate close-range solvent rearrangements during optimization of the product guess. The initial geometry is shown in transparency, with the final optimized structure shown in opaque color.

This implies that statistically converged sets of optimizations and/or thermodynamic sampling around the optimized structures will be required to obtain more accurate enthalpies for small molecules in explicit solvent environments. However, we suggest that this is a promising new direction for rapidly obtaining local minima of small molecules in complex condensed-phase systems.

We extracted the QM atoms from each system for secondary single-point-energy calculations using the same parameters as above to compare the energies, pairwise RMSD, and the final optimized geometries. We find that across all reactant test systems using the SST (DST), the standard deviation of all QM energies is 2. 1636 · 10⁻⁵ kcal·mol^–1 (2. 1997 · 10⁻⁵ kcal · mol⁻¹) with an overall maximum pairwise RMSD of 1.218Å. For all product test systems using SST (DST), however, the standard deviation is 9.2080 kcal · mol^–1 (9.4590 kcal · mol^–1) with an overall maximum pairwise RMSD of 1.224 Å. A perfect arrangement of reactant and product geometries would produce essentially the same set of standard deviations in energy, as in this case, the “reactant” and “product” are technically the same in the absence of solvent. Thus, the change in optimization of product guess is the primary source of error.

3.4 |. MECI of mScarlet and FusionRed

The calculation of conical intersections necessarily requires constraints simply to ensure that the limits of available computing resources are not exceeded. We compared the spherical selection method to our adaptive method.

A total of 33,141 atoms were constrained within the spherical region, leaving only 310 atoms free to move during the entire optimization. By comparison, the adaptive restraint algorithm was applied with a threshold of 0.001 and 10 steps per cycle and had as many as 894 unrestrained atoms in the system. Calculation times were comparable, with the spherical region running for 11 h 13 min, and the adaptive restraints for 13 h 36 min, but with significantly less of the system constrained. A completed MECI optimization should have a state energy gap as close to 0 as possible.

The original calculation led to a ground-to-excited state energy difference of 3. 5604 · 10⁻⁴ kcal · mol^–1, while the adaptive restraints gave a difference of 1. 3686 · 10⁻¹ kcal·mol⁻¹. The energy differences between states are considerably larger when using adaptive restraints. This difference may be attributed in large part to the increased number of atoms which are free to move, as the final geometries of both systems are comparable (see Figure 6).

FIGURE 6 |.

MECI geometries of mScarlet using spherical and adaptive restraints.

The MECI for the FusionRed test systems was calculated using the same restraint threshold as mScarlet. We tested the starting geometry for the excited state (S₁) Born-Oppenheimer molecular dynamics trajectory from Murphy et al. as well as several frames later in the trajectory which were reported as having low S₀ → S₁ energies. For the first frame, it is interesting to note that the initial optimization cycle had only 30 atoms restrained, likely due to the starting geometry being taken from S₀ dynamics and excited. Throughout the subsequent cycles, the total number of restrained atoms ranged from 20,816 to 21,354 out of a total of 21,401 atoms, and the total wall time was over 7 h. By comparison, lower energy frames required considerably less computational expense to reach the MECI using the same optimization settings. The initial and final geometries are shown in Figure 7. The final energy difference between states was 3. 3156 · 10⁻¹ kcal · mol⁻¹.

FIGURE 7 |.

MECI geometries of FusionRed using spherical and adaptive restraints.

4 |. Conclusions

The tests run on the rsEGFP2 system indicate that for large protein systems, a QM/MM optimization using the gradient-adapted restraint algorithm described herein provides a marked improvement in computational speed while maintaining consistent final QM/MM geometry and the final geometric results and subsequent energy of the QM region, with small changes in the potential energy and the MM region. We therefore suggest that this technique could be generally applied to QM/MM optimizations at scale for initiating QM/MM-MD sampling as well as identification of MECI geometries for photoactive biomolecules, particularly since the algorithm does not require manual selection of frozen atoms.

With the smaller diaspartate system, stable relative energies can come at a cost of increased calculation time when the frozen atoms are recomputed every 60 steps or less and when the threshold is set to below 0.01, likely due to proportionally higher file I/O cost compared to larger systems. However, we obtain consistently accurate relative energy results with a threshold of 0.0001 and recomputing every 90 steps or less. Additionally, comparisons of the QM regions indicate that the optimized geometries are generally conserved across test systems, with changes in the MM region contributing most of the change in energy. Indeed, we observed that across all test systems, the QM energy calculated does not meaningfully differ between different optimization acceleration parameter sets. As a result, we find that the adaptive restraints are most effective with larger systems, such as proteins in water, and may require stronger restraints or more careful tuning with small molecule models, where the initial computational cost is already lower.

The exception is in cases where the starting geometry is based on a modification of a previously optimized structure (such as with the product form of the diaspartic acid system—see Table 1). In this case, the speedup is considerably larger since the environment has been pre-optimized. However, given the larger energetic variations in the product test systems, we suggest that the use of this algorithm for reaction paths may not provide sufficient accuracy given low-quality starting geometries. This algorithm, while demonstrated using TeraChem, may be applied to other QM/MM optimization packages for improved computational efficiency. It can also be applied straightforwardly to other methods that accelerate QM/MM optimizations, such as fragmentation schemes. It is also not restricted to any specific functionals or basis sets, ensuring that future improvements in those areas may be easily incorporated here.

TABLE 1 |.

Improvements demonstrated by acceleration SST (DST) algorithm with 0.01 (20%) threshold and 70 (80) steps per cycle in multiple test systems compared to fully unrestrained control optimizations.

	Speedup
SST-rsEGFP2	−13.97%
SST-Diasp. React.	−34.75%
SST-Diasp. Prod.	−91.71%
SST-Diasp. Enth.	—
DST-rsEGFP2	−36.20%
DST-Diasp. React.	−15.77%
DST-Diasp. Prod.	−73.52%
DST-Diasp. Enth.	—

Overall, the benefits of the adaptive restraint algorithm lie in the decreased computational costs with little effect on final optimized geometries and resultant relative energies. The computational improvements shown here suggest that future work may be better able to take advantage of larger QM regions and higher-accuracy functionals and basis sets for overall improvements in calculated values.

Supplementary Material

Additional supporting information can be found online in the Supporting Information section.

Data Availability Statement

The source code for the Adaptive Restraints for TeraChem is available at GitHub (DOI: doi.org/10.5281/zenodo.8057628). All tested structures are available on Zenodo via the following DOI: 10.5281/zenodo.8057240.