Comparison of Radii Sets, Entropy, QM Methods, and Sampling on MM-PBSA, MM-GBSA, and QM/MM-GBSA Ligand Binding Energies of F. tularensis Enoyl-ACP Reductase (FabI)

Abstract

To validate a method for predicting the binding affinities of FabI inhibitors, three implicit solvent methods, MM-PBSA, MM-GBSA and QM/MM GBSA were carefully compared using sixteen benzimidazole inhibitors in complex with F. tularensis FabI. The data suggests that the prediction results are sensitive to radii sets, GB methods, QM Hamiltonians, sampling protocols, and simulation length, if only one simulation trajectory is used for each ligand. In this case, QM/MM-GBSA using 6 ns MD simulation trajectories together with GB^neck2, PM3, and the mbondi2 radii set, generate the closest agreement with experimental values (r²= 0.88). However, if the three implicit solvent methods are averaged from six 1 ns MD simulations for each ligand (called “multiple independent sampling”), the prediction results are relatively insensitive to all the tested parameters. Moreover, MM/GBSA together with GB^HCT and mbondi, using 600 frames extracted evenly from six 0.25 ns MD simulations, can also provide accurate prediction to experimental values (r² = 0.84). Therefore, the multiple independent sampling method can be more efficient than a single, long simulation method. Since future scaffold expansions may significantly change the benzimidazole's physiochemical properties (charges, etc.) and possibly binding modes, which may affect the sensitivities of various parameters, the relatively insensitive “multiple independent sampling method” may avoid the need of an entirely new validation study. Moreover, due to large fluctuating entropy values, (QM/)MM-P(G)BSA were limited to inhibitors’ relative affinity prediction, but not the absolute affinity. The developed protocol will support an ongoing benzimidazole lead optimization program.

Introduction

Tularemia, a deadly zoonotic infection caused by the Gram-negative pathogen Francisella tularensis, is a viable bioweapon due to its ease of cultivation and aerosolization as well as its low infectious dose.¹ Although treatments for tularemia are available, including aminoglycoside antibiotics, streptomycin, ciprofloxacin and tetracycline, a widespread outbreak of this disease may still be unmanageable due to their requirement for intravenous use (aminoglycosides, streptomycin) or contraindication for use in pregnant women and children (ciprofloxacin, tetracyclines).² There is, therefore, a strong interest within the antibacterial research community in the identification and development of novel agents with improved physicochemical properties and activity against F. tularensis.

The bacterial fatty acid synthesis (FAS-II) metabolic pathway, which is responsible for the synthesis of fatty acid components of bacterial lipid membranes and energy stores, is an attractive antibacterial target. Distinct from its mammalian FAS-I counterpart, which consists of a single, large, multifunctional enzyme with low similarity, the bacterial FAS-II pathway is composed of separate enzyme steps with low similarity to FAS-I. The differences between FAS-I and FAS-II allow for the selective targeting of FAS-II enzymes, while minimizing disruption to the mammalian FAS-I pathway, a significant advantage in antibacterial drug design. The enoyl-[acyl-carrier-protein] reductase enzyme, FabI, catalyzes a key, rate-limiting reduction step in bacterial fatty acid synthesis and is considered to be one of the more attractive enzyme targets in the FAS-II pathway.

We have previously reported the identification of a series of benzimidazole compounds with FabI inhibitory activity as well as F. tularensis antibacterial activity using a novel shape/electrostatic virtual screening campaign.³ In addition to activity against F. tularensis, the benzimidazoles showed strong antibacterial activity against other Gram-positive and Gram-negative pathogens. Further, structural studies performed in-house revealed that the benzimidazole compounds bound to the FabI active site in a conformation that was unique from other known FabI inhibitors, including triclosan, a marketed antiseptic (Figure 1).^4-6 This suggested the possibility that the benzimidazole compounds might have utility against F. tularensis strains bearing resistance to other FabI targeting antibacterials, including triclosan. Additional metabolic and toxicity studies showed that the benzimidazole scaffold possessed moderate metabolic stability and low cell toxicity.⁷ Taken together, the biological, microbiological, and pharmacokinetic data collected to date justify the further biochemical optimization of the benzimidazole compounds as a lead series for treatment of F. tularensis and possibly other bacterial infection.

Figure 1.

Representative Benzimidazole FabI Inhibitors and Triclosan. ^4-6

The goal of the studies presented here was the development of a computational method that could predict the FabI binding affinity of benzimidazole compounds that were being proposed for synthesis and testing. The rationale was that a reliable computational affinity prediction protocol could allow for a more efficient and rapid lead optimization process by identifying compounds, prior to costly synthesis and testing, that were predicted to have high binding affinity to the FabI target. In previous work, we extensively studied various molecular docking and scoring algorithms for use in predicting relative FabI affinity, however these methods generally failed to accurately rank benzimidazole compounds by binding affinity in validation trials.³ This was likely due to insufficient conformational sampling of a flexible loop near the substrate/ligand binding site as well as inaccuracies in the scoring functions utilized. Herein, we report our studies of more advanced computational methods for predicting the binding affinity of the benzimidazole compounds to F. tularensis FabI, including MM-PBSA, MM-GBSA, and QM/MM-GBSA. Previous work has shown that the MM/P(G)BSA methods can accurately predict relative binding free energies of similar compounds using enhanced energy sampling from simulations combined with solvation energy estimations using implicit methods.⁸ We chose to explore these implicit solvent methods over more advanced explicit solvent methods, such as free energy perturbation and thermodynamic integration, as the higher computational expense of the latter methods would adversely impact the throughput of our planned lead optimization studies.⁹

Although MM-P(G)BSA methods have been used successfully in both virtual screening^10,11 and lead optimization programs^12-17, it has been shown that the results are sensitive to atomic charges, simulation length, entropy calculations, and sampling protocols which can lead to dramatic differences in affinity predictions using the same study system.^18-21 Studies have also suggested that prediction results of MM-GBSA methods might be influenced by radii settings.^22-29 Additionally, a recent study suggested that multiple independent simulations in MM-GBSA offered improved statistically converged results over one long MD simulation.³⁰ Thus, it was also of interest to see if multiple independent samplings offer a better agreement between experimental and calculated binding free energy than a single, long MD simulation for the studied system. Lastly, the recently developed hybrid QM/MM-GBSA method^31-34 has yet to be extensively compared with MM-GBSA methods with respect to the factors just mentioned.³⁵

Within this context and our ultimate goal of developing the most suitable method to support our lead optimization program, we have performed a series of comparative trials using the F. tularensis FabI (FtFabI) structure and sixteen benzimidazole compounds with known affinity from experimental studies and several experimentally confirmed binding conformations. These studies were specifically designed to answer the following questions: (1) What is the best combination of radii settings, QM Hamiltonians, implicit solvent methods, and simulation length for the studied system? (2) If entropy calculations are included, will they improve the prediction results? (3) Can multiple independent samplings improve the prediction of binding free energy over the use of one long simulation for the methods studied? And ultimately, (4) Which optimized method offers the best predictive power in terms of the absolute and relative binding affinity for our study system?

Methods

Complex Preparation

The binding conformation of FtFabI-benzimidazole complexes were taken from co-crystal structures solved in-house (PDB codes: 3UIC, 4J3F, 4J4T).^4,6 RESP atomic partial charges were assigned to benzimidazole ligands and the cofactor, NADH, with geometry optimization and the electrostatic potential calculations performed using HF/6-31G* and Gaussian 09³⁶ in the R.E.D. server.^37-39 The AMBER FF12SB force field and the general AMBER force field (GAFF)⁴⁰ parameters were assigned to the protein and the ligand using antechamber in AMBER v12.⁴¹ A 10Å TIP3P water molecule octahedron box was set to solvate the complex system along with Na⁺ and Cl⁻ counter-ions to neutralize the system.

Experimental Enzymatic Activity

The FabI enzyme reduces butenyl-CoA to butyryl-CoA utilizing the cofactor NADH. Enzyme activity was monitored by following the rate of decrease in fluorescence of NADH at 450 nm (excitation wavelength 340 nm). Detailed methods for the determination of the IC₅₀ and K_i values of the benzimidazole compounds against FtFabI have been previously described.^3,4,6 The compounds used in this study are shown in Supplementary Table 1, along withexperimental inhibition data. The experimental free energies of binding (ΔG_bind) were calculated from K_i using Equation 1, where R is the ideal gas constant (1.9872×10⁻³ kcal K⁻¹ mol⁻¹) and T is the room temperature (300K).

$$\Delta Gbind=RTlnKi$$ (1)

Molecular Dynamics (MD) Simulations

The systems were first minimized using 5000 steps of steepest descent minimization using the Particle Mesh Ewald (PME) potential function. After minimization, the systems were heated from 0K to 300K over 50 picoseconds (ps) using the NVT ensemble with a 10 kcal/mol-Å weak restraint on the enzyme, cofactor and ligands. Following this, the systems were equilibrated over 50 ps at constant pressure (1 bar) and temperature (300K) using NPT equilibration. Next, a 6 nanosecond (ns) NPT production run was performed at 300 K and 1 bar. The following settings were activated in all of the equilibration and production run MD simulations: Langevin dynamics for temperature scaling, 2 ps as the pressure relaxation time, 8 Å electrostatic interactions cut off, the SHAKE bond length constraints of hydrogen atoms, and 1 fs time step. In the production runs, the MD simulation trajectories were saved every 2.5 ps for subsequent (QM/)MM-P(G)BSA analyses. The pmemd.MPI program in AMBER12 was used for all of the above minimizations and simulations.

MM-PBSA

The MM-PBSA calculations were performed using MMPBSA.py in AMBERTools13.⁴² When the PARSE, bondi, mbondi, and mbondi2 sets were applied, the MM-PBSA surface tension (α) and the non-polar free energy correction term (β) were set to 0.00542 kcal/mol-Ų and 0.92, respectively, following the recommendation of the original PARSE radii study⁴³ and the AMBER user manuals.⁴¹ An exterior dielectric constant of 80 and solute dielectric constant of 1 were used. 2,400 snapshots were taken evenly from the MD simulations trajectory from 0 to 6 ns in the MM-PBSA calculations.

MM-GBSA and QM/MM-GBSA

The (QM)/MM-GBSA calculations were performed using MMPBSA.py in AMBERTools13. We investigated several GB models in this study, including the pairwise model developed by Hawkins et al. (GB^HCT),²³ the model developed by Onufriev et al. (GB^OBC),²⁴ the optimized version of GB^OBC (GB^OBC2),²⁵ the model developed by Mogan et al. to solve the so-called “bottle-neck” issue (GB^Neck ),²⁶ and the optimized version of GB^Neck (GB^Neck2).²⁷ Additional details are provided in Supplementary Table 2. The bondi, mbondi, and mbondi2 radii sets were prepared using the antechamber program in AMBER12. In AMBERTools13, the default setting of MM-GBSA surface tension (α = 0.0072 kcal / mol Ų) and the non-polar free energy correction term (β = 0) were applied. In the QM/MM-GBSA, the benzimidazole ligand was treated as the QM region using the AM1, PM3 and PM6 semi-empirical Hamiltonian theories. The QM charge of the ligand was set to zero because none of the ligands in this study are expected to carry a formal charge at physiological pH. The remaining QM/MM-GBSA settings are identical to the above MM-GBSA section.

Entropy

The entropy calculations were performed using Normal Mode Analysis (NMA) in the MMPBSA.py program in AMBERTools13.⁴⁴ The following were the settings for the entropy calculations: The distance-dependent dielectric constant was set to 1.0, and the energy gradient of minimization was 0.001 with 10,000 minimization cycles per snapshot. Due to limited computational resources, we used only 48 frames, which were evenly extracted from 0 to 6 ns of the MD trajectories for entropy calculations.

Multiple Independent MD Simulations

The multiple independent MD simulations were prepared using the same settings and starting structures described above, with the exception that the random starting velocities were applied by turning on the pseudo-random starting velocity generator (the “ig” flag in AMBER12). In order to compare the differences between these and single 6 ns MD simulations, the MD simulations were separated into six, 1 ns components. For the MM-PBSA and (QM/)MM-GBSA, 400 snapshots were taken evenly from each of the six, 1-ns MD simulation trajectories from 0 to 1 ns. Other settings for the MM-PBSA, MM-GBSA, and QM/MM-GBSA calculations for multiple MD simulations were as described above.

Results and Discussion

The Effect of MD Simulation Length on MM-PBSA

The coefficient of determination between experimental and predicted binding free energies using different MD trajectory lengths and radii settings in MM-PBSA calculations are summarized in Table 1. The MM-PBSA calculations based on the 0.25 to 2.00 ns MD trajectories were satisfactory (r²>0.70) in all bondi, mbondi, mbondi2, and PARSE radii sets. However, using data from MD trajectories equal to or longer than 2 ns for the MM-PBSA calculations seemed to lower the accuracy of prediction in all four radii settings. To further analyze the nature of the decrease in predictive power with longer MD simulation times, we first checked the MM-PBSA energy terms plot of FtFabI- GRL-0056 (Figure 2) and the rest of fifteen ligands (Supplementary Figure 1). The yellow line was the instantaneous enthalpy of binding (ΔG_bind) and it can be readily seen that all the ΔG_bind in the sixteen ligands using all the radii settings were readily converged. Therefore, it is likely that the decrease in predictive power along the MD simulation time frames is due to an amplification of force field errors along the MD simulation time frames, instead of non-converged energy in the system, as reported in the previous study.²¹

Table 1.

The coefficient of determination of Experimental and Predicted Binding Free Energies of Different MD Trajectory Lengths and Radii Sets in MM/PBSA Calculations.^a

Radii Sets	R20.25ns	R20.5ns	R20.75ns	R21ns	R22ns	R23ns	R24ns	R25ns	R26ns
bondi	0.78	0.82	0.83	0.82	0.74	0.61	0.49	0.4	0.36
mbondi	0.71	0.75	0.76	0.77	0.72	0.62	0.54	0.48	0.45
mbondi2	0.79	0.82	0.83	0.82	0.74	0.61	0.49	0.43	0.37
PARSE	0.62	0.67	0.72	0.75	0.77	0.68	0.61	0.53	0.48

^aR²_n: “R²” stands for coefficient of determination in the MM/PBSA calculation. Here 100, 400, 1200, 2400 MD simulation frames were evenly extracted from one 0.25, 1, 3, 6 ns MD simulation trajectories, respectively.

Figure 2.

FtFabI: GRL-0056 energy components using MM-PBSA and the bondi radii setting. Black, red, blue, green, cyan and yellow lines indicate the ΔG_vdw, ΔG_GB, ΔG_solvation, ΔG_{electrostatic}, ΔG_nonpolar and ΔG_bind terms respectively. The x axis is the frame number. Each frame is 2.5 ps. The total frame number is 2,400 frames, equal to 6 ns MD simulations.

The Effect of Radii Sets on MM-PBSA

The impact of four different radii sets (bondi, mbondi, mbondi2, and PARSE) are also summarized in Table 1. It can be clearly seen that bondi and mbondi2 gave almost identical predictive power since bondi and mbondi2 were optimized via similar theoretical frame works.²⁷ Interestingly, the difference in predictive power between mbondi versus bondi and mbondi2 was also quite small. Considering the potential errors of the experimental binding free energy, these differences show little significance. Among the four radii sets, the PARSE radii offered the poorest correlation with experimental values using MD trajectories shorter than 2 ns (r²= 0.62 - 0.77). A possible explanation is that the GB radii sets (bondi, mbondi, and mbondi2) are more extensively parameterized in various atom types for each chemical element while the PARSE radii set has only one atom type for each chemical element, as discussed in previous studies. ^{21, 43} Finally, in Supplementary Table 3, one can clearly see that the three Born radii sets always generated smaller solvation energy terms (ΔG_solvation) and correspondingly more negative binding enthalpy values (ΔH_bind) than the PARSE radii sets. A similar effect was noted in a previous study.²¹ A possible reason is that the three Born radii sets contain larger radii and larger corresponding dielectric boundaries than the PARSE radii set and therefore result in higher solvation energy terms when applying the three bondi radii sets. ^{21,23-26,43,45}

The Effect of MD Simulation Length on MM-GBSA

The coefficient of determination of experimental and predicted binding free energies for different MD trajectory lengths and radii sets in MM-GBSA calculations are summarized in Table 2. The predictive power of the MM-GBSA method with longer simulations decreased in a manner similar to the MM-PBSA predictions. For all three radii sets, the MM-GBSA predictions based on MD trajectories less than or equal to 1 ns were in the satisfactory range (R² > 0.7), while the predictive power dropped significantly when using MD trajectories larger than 1 ns. As above, the decrease in predictive power may be due to the amplification of force field errors. ²¹

Table 2.

The coefficient of determination of Experimental and Predicted Binding Free Energies of Different MD Trajectory Lengths and Radii Sets in MM/GBSA Calculations.^a

Radii Sets	GB	R20.25nsa	R20.5ns	R20.75ns	R21ns	R22ns	R23ns	R24ns	R25ns	R26ns
bondi	GBHCT	0.85	0.88	0.85	0.81	0.65	0.53	0.45	0.41	0.41
mbondi	GBHCT	0.81	0.83	0.8	0.75	0.59	0.51	0.45	0.43	0.44
mbondi2	GBHCT	0.85	0.87	0.85	0.8	0.64	0.52	0.44	0.4	0.41
bondi	GBOBC	0.79	0.83	0.82	0.78	0.63	0.51	0.42	0.37	0.38
mbondi	GBOBC	0.68	0.71	0.7	0.67	0.52	0.43	0.37	0.35	0.36
mbondi2	GBOBC	0.77	0.81	0.8	0.77	0.61	0.48	0.4	0.36	0.36
bondi	GBOBC2	0.77	0.82	0.81	0.77	0.62	0.49	0.4	0.35	0.36
mbondi	GBOBC2	0.59	0.64	0.62	0.59	0.42	0.31	0.25	0.23	0.24
mbondi2	GBOBC2	0.73	0.78	0.77	0.74	0.57	0.44	0.35	0.31	0.32
bondi	GBNeck	0.57	0.58	0.57	0.56	0.4	0.26	0.18	0.15	0.16
mbondi2	GBNeck	0.52	0.53	0.53	0.52	0.36	0.23	0.16	0.13	0.15
bondi	GBNeck2	0.63	0.68	0.6	0.6	0.52	0.5	0.49	0.48	0.5
mbondi2	GBNeck2	0.63	0.67	0.6	0.6	0.51	0.5	0.5	0.49	0.51

^aR²_n: “R²” stands for coefficient of determination in the MM/GBSA calculation. Here 100, 400, 1200, 2400 MD simulation frames were evenly extracted from one 0.25, 1, 3, 6 ns MD simulation trajectories, respectively.

Interestingly, although the predictive power of GB^Neck2 also decreased with longer MD simulations, the extent of the decrease was less than for the other four GB methods (R² = 0.6 at 1 ns and R² = 0.51 at 6 ns). This might suggest that GB^Neck2 provided better agreement to the experimental values in terms of solvation energy and therefore offset the degree of predictive power diminishment. This will be discussed in more detail below in the QM/MM-GBSA section.

The Effect of Radii Sets on MM-GBSA

It is known that MM-GBSA is more sensitive to various radii sets than MM-PBSA, and earlier studies have suggested the optimal radii set (s) for each GB method (summarized in Supplementary Table 2).^23-26,45 The mbondi radii set was a non-recommended radii setting for GB^OBC. However, as shown in Table 2, in our studies the three different radii sets in GB^OBC resulted in only minor differences. Thus, for this system, considering the potential errors in experimental free energy of binding, the differences between the predictive powers of the three radii sets for the GB^OBC method can be safely ignored. In GB^OBC, GB^OBC2, GB^Neck and GB^Neck2, it is not surprising that the suggested radii sets, bondi and mbondi2, yielded nearly identical prediction results because mbondi2 is a modification of and very similar to bondi (Table 2).⁴⁵ We did not include the mbondi set in the GB^Neck and GB^Neck2 calculations, since earlier studies did not recommend the mbondi setting in the GB^Neck and GB^Neck2 methods, and MMPBSA.py in the AMBERTools13 suite did not allow the mbondi radii set in the default GB^Neck and GB^Neck2 methods. Additionally, in GB^OBC2, our data in Table 2 did suggest that the predicted binding free energy calculated by bondi and mbondi2 radii sets correlated better with the experimental binding free energy than the unfavorable mbondi settings.

The three radii sets and their corresponding effects on the solvation energy terms are summarized in Supplementary Figure 2 & Supplementary Table 4. When MM-GBSA calculations were performed under similar conditions (inhibitor, number of MD frames, GB method, and charges) using the different radii sets, the only energy terms showing significant changes are the polar energy term (ΔG_GB), the total solvation energy term (ΔG_solvation), and the total free energy of binding, ΔG_bind (the red, blue, yellow lines on top of each figure in Supplementary Figure 2 respectively and Equation 2). This is due to the fact that radii settings only affect ΔG_GB and ΔG_solvation. From the ΔG_GB energy curves in Supplementary Figure 2, bondi and mbondi2 result in nearly identical solvation energy values while mbondi consistently offered more positive ΔG_GB values than bondi and mbondi2. These clear differences allow us to visualize the reason why bondi and mbondi2 suggested highly similar prediction results in all five GB methods.

$$\Delta {\mathrm{G}}_{\text{solvation}}=\Delta {\mathrm{G}}_{\mathrm{PB}∕\mathrm{GB}}+\Delta {\mathrm{G}}_{\text{nonpolar solvation energy}}$$ (2)

Comparison of Generalized Born Methods

From Table 2, we can clearly see that GB^HCT offered the best agreement with the experimental binding free energies (higher coefficient of determination) amongst the five GB methods using MD simulation trajectory data from 0.25 ns to 2 ns (R² = 0.85 ^~ 0.65). However, if we used data from MD simulation trajectories longer than 2 ns, GB^HCT showed similar predictive performance compared to GB^OBC, GB^OBC2, and GB^Neck2. Moreover, because GB^OBC and GB^OBC2 have a similar computational foundations, with only subtle differences in parameter settings, it is not surprising that GB^OBC and GB^OBC2 resulted in similar predictive power for the FtFabI-benzimidazole system. With respect to the difference between GB^Neck and GB^Neck2, GB^Neck2 outperformed GB^Neck as would be expected since GB^Neck2 is an optimized version of GB^Neck, and includes a higher number of parameters and a broader range of scaling factors.²⁶

On the other hand, GB^Neck2 behaved differently compared to all of the other four GB methods, as summarized in Figure 3 and Supplementary Figure 2. Even though the ΔG_GB and ΔG_bind terms calculated by GB^HCT converged throughout the 6 ns trajectories (2,400 frames) and GB^Neck2 converged after 500 ps (200 frames), the ΔG_GB (the red line in Figure 3a & 3b) and ΔG_bind (the yellow line in Figure 3a & 3b) terms in GB^Neck2 had consistently greater fluctuation than with GB^HCT and the other GB methods (Supplementary Figure 2). This was seen with all sixteen benzimidazole ligands (data not shown). Figures 3c and 3d show the Q-Q plots for ΔG_GB using compound FabI 91, 6,000 ps (2,400 frames) using GB^HCT and GB^neck2 methods, respectively. Compared to the nearly straight line in Figure 3c (GB^HCT ), the two slightly rising tails in Figure 3d (GB^Neck2) suggest that the ΔG_GB population calculated by GB^Neck2 may be skewed, which might bias the average ΔG_GB in GB^Neck2. A similar effect can be observed in the Q-Q plot for ΔG_bind calculated by GB^HCT and GB^neck2: the straighter line in Figure 3e (GB^HCT ) suggests that ΔG_bind calculated by GB^HCT has a more normal distribution than that calculated using GB^Neck2 (Figure 3e). Again, this may indicate that in calculations using GB^Neck2, ΔG_GB, ΔG_solvation, and ΔG_bind might be slightly biased by the extreme values (outliers) and therefore provide less accurate prediction than the remaining four GB methods, at least using MD trajectories less than 3 ns.

Figure 3.

A. FtFabI: FabI 91 energy components using MM-GBSA, bondi radii and GB^HCT. Black, red, blue, green, cyan and yellow lines indicate the ΔG_vdw, ΔG_GB, ΔG_solvation, ΔG_{electrostatic}, ΔG_nonpolar and ΔG_bind terms respectively. The x axis is the frame number. Each frame is 2.5 ps. The total frame number is 2,400 frames, equal to 6 ns MD simulations.

B. FtFabI: FabI 91 energy components using MM-GBSA, bondi radii and GB^Neck2.

C. The quantile-quantile plot (Q-Q plot) of the ΔG_GB term in GB^HCT.

D. The Q-Q plot of the ΔG_GB term in GB^Neck2.

E. The Q-Q plot of the ΔG_bind term in GB^HCT.

F. The Q-Q plot of the ΔG_bind term in GB^Neck2.

Additionally, another reason why GB^HCT, GB^OBC, and GB^OBC2 offered better agreement with experimental binding free energies than GB^Neck1 and GB^Neck2, at least for this FtFabI-benzimidazole system, is that the recently released GB^Neck and GB^Neck2 methods have thus far only been optimized using peptide-protein systems, not protein-small molecule systems.²⁷ Future optimization of “neck region correction” GB methods covering non-peptide/non-protein systems will likely change this situation. As it stands, GB^Neck1 and GB^Neck2 are likely better suited than the GB^HCT GB^OBC, and GB^OBC2 methods in predicting protein-protein binding free energies.

The Performance of QM/MM-GBSA

The Effect of Semi-Empirical QM Theory Levels on QM/MM-GBSA

The effects of three commonly used semi-empirical QM theory levels, Parameterized Model number 3 (PM3),^46,47 Austin Model 1 (AM1),⁴⁸ and Parameterized Model number 6 (PM6)⁴⁹ on QM/MM-GBSA are summarized in Tables 3 - 5. The PM3 Hamiltonian and QM-MM/GBSA together give the best agreement between experimental and predicted binding affinities, regardless of radii settings and GB methods (Table 3). The AM1 Hamiltonian and QM-MM/GBSA together resulted in a slightly worse accuracy than the PM3 Hamiltonian for all conditions (Table 4). Since the PM3 Hamiltonian is known to predict the intermolecular hydrogen bonds more accurately than the AM1 Hamiltonian,⁵⁰ the PM3 Hamiltonian would be a better choice than the AM1 Hamiltonian for the benzimidazole inhibitors, which forms key intermolecular hydrogen bonds with FtFabI Tyr156 and the ribose on the nicotinamide ring of the cofactor NADH (Figure 4).^4,6

Table 3.

The coefficient of determination of Experimental and Predicted Binding Free Energies of Different MD Trajectory Lengths and Radii Sets in QM-MM/GBSA Calculations using the PM3 Hamiltonian.^a

Radii Sets	GB	R20.25nsa	R20.5ns	R20.75ns	R21ns	R22ns	R23ns	R24ns	R25ns	R26ns
bondi	GBHCT	0.78	0.81	0.78	0.73	0.64	0.58	0.53	0.52	0.54
mbondi	GBHCT	0.76	0.79	0.75	0.69	0.59	0.56	0.54	0.55	0.57
mbondi2	GBHCT	0.78	0.81	0.78	0.73	0.64	0.58	0.54	0.52	0.54
bondi	GBOBC	0.76	0.79	0.76	0.72	0.64	0.6	0.56	0.54	0.56
mbondi	GBOBC	0.72	0.75	0.71	0.67	0.57	0.55	0.53	0.54	0.56
mbondi2	GBOBC	0.76	0.79	0.76	0.71	0.63	0.59	0.55	0.53	0.55
bondi	GBOBC2	0.75	0.79	0.75	0.71	0.62	0.56	0.51	0.49	0.51
mbondi	GBOBC2	0.72	0.75	0.71	0.64	0.51	0.47	0.44	0.44	0.46
mbondi2	GBOBC2	0.75	0.78	0.75	0.69	0.59	0.54	0.49	0.47	0.49
bondi	GBNeck	0.73	0.73	0.69	0.64	0.52	0.46	0.4	0.39	0.42
mbondi2	GBNeck	0.72	0.72	0.68	0.63	0.51	0.44	0.38	0.37	0.4
bondi	GBNeck2	0.62	0.66	0.62	0.65	0.69	0.76	0.79	0.82	0.86
mbondi2	GBNeck2	0.64	0.67	0.64	0.67	0.7	0.78	0.81	0.83	0.88

^aR²_n: “R²” stands for coefficient of determination in the QM-MM/GBSA calculation. Here 100, 400, 1200, 2400 MD simulation frames were evenly extracted from one 0.25, 1, 3, 6 ns MD simulation trajectories, respectively.

Table 5.

The coefficient of determination of Experimental and Predicted Binding Free Energies of Different MD Trajectory Lengths and Radii Sets in QM-MM/GBSA Calculations using the PM6 Hamiltonian.^a

Radii Sets	GB	R20.25nsa	R20.5ns	R20.75ns	R21ns	R22ns	R23ns	R24ns	R25ns	R26ns
bondi	GBHCT	0.54	0.61	0.57	0.52	0.41	0.28	0.18	0.14	0.14
mbondi	GBHCT	0.49	0.55	0.51	0.46	0.36	0.30	0.26	0.25	0.27
mbondi2	GBHCT	0.56	0.63	0.59	0.54	0.43	0.30	0.20	0.16	0.17
bondi	GBOBC	0.15	0.21	0.20	0.17	0.09	0.04	0.01	0.01	0.01
mbondi	GBOBC	0.13	0.18	0.16	0.13	0.07	0.05	0.04	0.03	0.04
mbondi2	GBOBC	0.17	0.24	0.22	0.19	0.11	0.06	0.02	0.01	0.02
bondi	GBOBC2	0.16	0.23	0.22	0.19	0.11	0.05	0.02	0.01	0.01
mbondi	GBOBC2	0.12	0.17	0.15	0.11	0.05	0.02	0.01	0.01	0.02
mbondi2	GBOBC2	0.17	0.24	0.23	0.19	0.11	0.05	0.02	0.01	0.02
bondi	GBNeck	0.24	0.26	0.23	0.20	0.13	0.07	0.03	0.02	0.03
mbondi2	GBNeck	0.26	0.29	0.26	0.23	0.15	0.09	0.05	0.04	0.05
bondi	GBNeck2	0.52	0.57	0.55	0.56	0.56	0.57	0.57	0.60	0.64
mbondi2	GBNeck2	0.53	0.57	0.54	0.56	0.56	0.57	0.58	0.60	0.64

^aR²_n: “R²” stands for coefficient of determination in the QM-MM/GBSA calculation. Here 100, 400, 1200, 2400 MD simulation frames were evenly extracted from one 0.25, 1, 3, 6 ns MD simulation trajectories, respectively.

Table 4.

The coefficient of determination of Experimental and Predicted Binding Free Energies of Different MD Trajectory Lengths and Radii Sets in QM-MM/GBSA Calculations using the AM1 Hamiltonian.^a

Radii Sets	GB	R20.25nsa	R20.5ns	R20.75ns	R21ns	R22ns	R23ns	R24ns	R25ns	R26ns
bondi	GBHCT	0.78	0.83	0.79	0.74	0.64	0.55	0.47	0.44	0.45
mbondi	GBHCT	0.74	0.78	0.73	0.69	0.58	0.53	0.49	0.48	0.50
mbondi2	GBHCT	0.79	0.83	0.79	0.74	0.64	0.56	0.48	0.44	0.46
bondi	GBOBC	0.67	0.75	0.72	0.69	0.60	0.54	0.45	0.41	0.42
mbondi	GBOBC	0.60	0.67	0.63	0.60	0.51	0.48	0.43	0.42	0.44
mbondi2	GBOBC	0.68	0.75	0.72	0.69	0.59	0.53	0.45	0.41	0.42
bondi	GBOBC2	0.65	0.74	0.72	0.68	0.57	0.50	0.41	0.37	0.37
mbondi	GBOBC2	0.61	0.68	0.64	0.58	0.46	0.40	0.34	0.33	0.35
mbondi2	GBOBC2	0.66	0.74	0.71	0.66	0.55	0.47	0.39	0.35	0.36
bondi	GBNeck	0.62	0.63	0.58	0.53	0.40	0.31	0.24	0.22	0.24
mbondi2	GBNeck	0.62	0.63	0.58	0.53	0.40	0.31	0.24	0.22	0.24
bondi	GBNeck2	0.55	0.61	0.60	0.62	0.64	0.69	0.70	0.74	0.79
mbondi2	GBNeck2	0.57	0.63	0.62	0.65	0.67	0.72	0.73	0.76	0.81

^aR²_n: “R²” stands for coefficient of determination in the QM-MM/GBSA calculation. Here 100, 400, 1200, 2400 MD simulation frames were evenly extracted from one 0.25, 1, 3, 6 ns MD simulation trajectories, respectively.

Figure 4.

The FabI 138 inhibitor (pdb code: 3UIC) and other benzimidazole inhibitors (pdb codes: 4J1N, 4J3F, 4J4T) form key intermolecular hydrogen bonds with the FtFabI Tyr156 and the ribose on the nicotinamide ring of the cofactor NADH

The PM6 Hamiltonian is the recent advanced semi-empirical QM theory and it is believed to correct multiple defects in the AM1 and PM3 Hamiltonians.^49,51 However, use of the PM6 Hamiltonian in QM-MM/GBSA resulted in the worst binding affinity prediction of the three semi-empirical QM theory levels here (Table 5). The PM6 Hamiltonian in QM-MM/GBSA using GB^HCT, GB^OBC, GB^OBC2, and GB^Neck radii sets resulted in no observed correlation between the predicted and experimental binding affinities (Table 5). The PM6 Hamiltonian in QM-MM/GBSA using GB^Neck2 was the only case where a moderate accuracy was observed. The ΔG_bind, ΔG_GB, and the QM/MM-GBSA electrostatic term (Self Consistent Energy, ΔG_scf) using the PM3 Hamiltonian are all converged (Supplementary Figure 3), similar to the AM1 and PM6 Hamiltonians (data not shown). A possible reason that the PM6 Hamiltonian performed poorly here is because in AMBER, the parameters of the PM6 Hamiltonian describing electrostatic interactions between QM and MM regions haven't been optimized and are borrowed from the PM3 Hamiltonian.⁴¹ However, since the PM6 Hamiltonian is more advanced and covers broader cases than the PM3 Hamiltonian, the PM3 parameters might not be exclusive enough for the PM6 Hamiltonian. The missing parameters of the PM6 Hamiltonian are estimated, as detailed in the AMBER manual.⁴¹ Therefore, the PM6 Hamiltonian requires more extensive tests, as pointed out in a previous study³⁴ and the AMBER user manual.⁴¹ The future optimized PM6 parameters for electrostatic interactions between QM and MM regions are likely to significantly improve the PM6 Hamiltonian accuracy in the QM-MM/GBSA method within AMBER. Thus, since the PM3 Hamiltonian outperformed the AM1 and PM6 Hamiltonians in all conditions (MD simulation length, radii sets, etc.), only the PM3 Hamiltonian is considered in the following sections.

The Effect of MD Simulation Length on QM/MM-GBSA

The coefficient of determination between experimental and predicted binding free energies for different lengths of MD trajectories, various radii sets and semi-empirical QM methods in QM/MM-GBSA calculations, are summarized in Tables 3 - 5. It can be clearly seen that QM/MM-GBSA produced decreasing agreement with experimental binding free energies when longer MD trajectories were used with GB^HCT, GB^OBC1, GB^OBC2, and GB^Neck1 in all the tested semi-empirical methods. However, the extent of decreasing predictive power with simulation time was significantly less in QM/MM-GBSA than that observed in MM-GBSA when using GB^HCT, GB^OBC, GB^OBC2, and GB^Neck with PM3 and AM1 semi-empirical methods (Tables 2 - 4). This suggests that the extra QM and QM/MM terms in QM/MM-GBSA (Equation 3 & 4) may offset the amplification of “force field errors” and improve the predictive agreement with experimental binding free energies. With respect to GB^Neck2 in QM/MM-GBSA using all the tested semi-empirical methods, surprisingly, the predictive power actually increased as the MD simulation lengths grew. The reason for this reversed trend compared to the other four GB methods can be traced back to the GB^Neck2 results in MM-GBSA (Table 2). Although the predictive power of GB^Neck2 was seen to decrease with longer MD simulations, the extent of decrease was smaller than the other four GB methods. Moreover, the differences in behavior of GB^Neck2 between MM-GBSA and QM/MM-GBSA can be further visualized in Figures 3 & 5. The Q-Q plot for ΔG_GB using GB^neck2 suggests that ΔG_GB in MM-GBSA (Figure 3d) has more outliers than ΔG_GB in QM/MM-GBSA using the PM3 Hamiltonian (Figure 5d), which may bias the population mean and deteriorate the predictive results of the former. Similar results can be observed in the Q-Q plots of ΔG_bind using GB^neck2 (Figure 3f & 5f). It seems that the GB^Neck2 and extra QM energy terms in QM/MM-GBSA together provide a better description to ΔG_GB and ΔG_bind compared with the MM-GBSA results, and therefore offset the force field errors which could diminish the predictive power along the MD trajectory length.

$${\mathrm{E}}_{\text{total}}={\mathrm{E}}_{\mathrm{QM}}+{\mathrm{E}}_{\mathrm{MM}}+{\mathrm{E}}_{\mathrm{QM}∕\mathrm{MM}}+{\mathrm{G}}_{\mathrm{GB}}+{\mathrm{G}}_{\text{nonpolar solvation energy}}$$ (3)

$${\mathrm{G}}_{\mathrm{GB}}={\mathrm{G}}_{\text{polar}}={\mathrm{G}}_{\text{polor}<\mathrm{QM}>}+{\mathrm{G}}_{\text{polor}<\mathrm{MM}>}+{\mathrm{G}}_{\text{polor}<\mathrm{QM}-\mathrm{MM}>}$$ (4)

Figure 5.

A. FtFabI: FabI 91 energy components using QM/MM-GBSA, bondi radii and GB^HCT. Black, red, blue, green, cyan and yellow lines indicate the ΔG_vdw, ΔG_GB, ΔG_solvation, ΔG_SCF (Self Consistent Energy), ΔG_nonpolar and ΔG_bind terms respectively. The x axis is the frame number. Each frame is 2.5 ps. The total frame number is 2,400 frames, equal to 6 ns MD simulations.

B. FtFabI: FabI 91 energy components using QM/MM-GBSA, bondi radii and GB^Neck2. One can see from the ΔG_GB term (the red line) and the ΔG_bind term (the yellow line) in GB^Neck2 fluctuates wilder than the one in GB^HCT (Figure 5a).

C. The quantile-quantile plot (Q-Q plot) of the ΔG_GB term in GB^HCT.

D. The Q-Q plot of the ΔG_GB term in GB^Neck2.

E. The Q-Q plot of the ΔG_bind term in GB^HCT.

F. The Q-Q plot of the ΔG_bind term in GB^Neck2.

The Effect of Radii Sets on QM/MM-GBSA

It has been previously noted by both AMBER and CHARMM developers that optimized radii specifically for QM Hamiltonians will advance the predictive power of QM/MM-GBSA methods.^35,52 With an understanding of their potential limitations, we investigated the optimized MM-GBSA radii sets (bondi, mbondi, and mbondi2) with QM/MM-GBSA methods in these studies. Interestingly, all three radii sets were able to provide satisfactory agreements with experimental data (R²>0.70) using the five GB methods, various MD trajectory lengths, and the PM3 Hamiltonian, as discussed above and summarized in Table 3. Further, it can be clearly seen that the different radii settings yielded very similar results in both MM-GBSA (Table 2) and QM/MM-GBSA using the PM3 Hamiltonian (Table 3). The reason behind these similarities is that MM-GBSA and QM/MM-GBSA are analogous. Also, even though QM/MM-GBSA includes extra QM and QM/MM energy terms, some of these energy terms such as ΔG_polar<QM> and ΔG_polar<QM/MM> (Equation 3) require radii sets as part of their ΔG_polar mathematical functions. Therefore, QM/MM-GBSA showed a predictive sensitivity similar to MM-GBSA in several specific situations. First, the mbondi radii set yielded similar coefficient of determination to bondi and mbondi2 when using GB^HCT. Second, the predictive power of bondi was always similar to mbondi2 in GB^OBC, GB^OBC2, GB^Neck, and GB ^Neck2 since bondi and mbondi2 are alike in nature (discussed above). The only difference between MM-GBSA and QM/MM-GBSA in terms of radii sets can be seen in Table 3. In both GB^OBC1 and GB^OBC2 methods, the mbondi radii set showed similar predictive power to bondi and mbondi2 radii sets with QM/MM-GBSA, whereas mbondi results were always inferior to bondi and mbondi2 with MM-GBSA. The reason for these different behaviors of GB^OBC and GB^OBC2 in MM-GBSA and QM/MM-GBSA can be clearly visualized in Supplementary Figure 2 & Supplementary Table 4 (MM-GBSA) and Supplementary Figure 3 & Supplementary Table 5 (QM/MM-GBSA). While there is only one ΔG_polar term (MM ΔG_polar term) in MM-GBSA, QM/MM-GBSA includes extra QM and QM/MM ΔG_polar terms (Equation 3). Therefore, QM and QM/MM ΔG_polar terms together seemed to minimize the differences of the ΔG_polar term in the three tested radii sets, and the corresponding ΔG_solvation. On the other hand, while the MM-GBSA electrostatic term in AMBER is radii independent, the QM/MM-GBSA electrostatic term (Self Consistent Energy, ΔG_scf in Figure 5a,b) is radii sensitive, as noted by Walker and Case.³⁵ Therefore, the QM/MM-GBSA electrostatic terms varied slightly among the three tested radii sets (Figure 5 & Supplementary Table 5), while the MM-GBSA electrostatic terms were identical in the three tested radii sets. This also suggests that if one optimized radii sets for QM Hamiltonians, the QM/MM-GBSA should offer better descriptions, not only of ΔG_solvation, but also of ΔG_{electrostatic}.

Comparison of Generalized Born Methods on QM/MM-GBSA

With QM/MM-GBSA using the PM3 Hamiltonian (Table 3), it can be seen that the GB^OBC method offered similar predictive power to GB^OBC2. This is likely because both GB methods share similar underlying principles with respect to their scaling functions to treat the radii sets.^24,25 Likewise, while GB^HCT offered coefficient of determination similar to those of GB^OBC and GB^OBC2 in all the MD simulation time frames, GB^HCT behaved differently to GB^OBC and GB^OBC2 in MM-GBSA. This was quite surprising at first glance since the method to treat radii sets in GB^HCT is different from that of GB^OBC and GB^OBC2. One possible explanation for this behavior is that the extra QM and QM/MM ΔG_polar terms in QM/MM-GBSA are able to offset and minimize the differences between GB^HCT, GB^OBC and GB^OBC2 in QM/MM-GBSA.³⁵ On the other hand, in QM/MM-GBSA using the PM3 Hamiltonian (Table 3), the predictive power differences between the “neck correction” GB methods (GB^Neck1 and GB^Neck2) and “non-neck correction” methods (GB^HCT, GB^OBC1 and GB^OBC2) were significant. It is likely that the underlying theories behind the “neck correction” GB methods and “non-neck correction” methods are dissimilar to the extent that the extra QM and QM/MM ΔG_polar terms in QM/MM-GBSA were unable to offset the differences between “neck correction” GB methods and “non-neck correction” methods. Finally, in both MM-GBSA and QM/MM-GBSA using the PM3 Hamiltonian, GB^Neck2 still outperformed GB^Neck, as GB^Neck2 is the optimized version of the “neck correction” GB method.²⁷

The Effect of Conformational Entropy

The effect of conformational entropy (48 frames) on MM-PBSA (2,400 frames) is summarized in Table 6 and on MM-GBSA and QM/MM-GBSA (2,400 frames) in Table 7. The numerical values for conformational entropy and the corresponding absolute binding free energy of MM-PBSA, MM-GBSA and QM/MM-GBSA are summarized in Supplementary Table 3, 4, & 5, respectively. It can be seen that after including entropy, the predictive power (coefficient of determination) of MM/PBS and MM-GBSA decreased significantly, regardless of radii sets or GB methods (Table 6 & 7). Interestingly, while the inclusion of entropy considerably reduced the predictive power of QM/MM-GBSA with GB^Neck and GB^Neck2, the negative effect of entropy inclusion on the predictive power of QM/MM-GBSA with GB^HCT, GB^OBC and GB^OBC2 was less substantial (Table 7). It appears that the conformational entropy compromised the predictive power in the FtFabI-benzimidazole system due to large fluctuations.

Table 6.

Summary of the Impact of Entropy on Predicted Binding Free Energies of MD Trajectories of Different Lengths and Radii Sets in MM/PBSA Calculations.

1	2	3	4	5
Radii Sets	R26nsa,b	R26nsc	R26nsd	R26nse
bondi	0.36	0.06	0.21	0
mbondi	0.45	0.12	0.29	0
mbondi2	0.37	0.06	0.22	0
PARSE	0.48	0.15	0.27	0.01

^aR²_n: “R²” stands for coefficient of determination in the MM/PBSA calculation. Here 2,400 MD simulation frames were evenly extracted from 6 ns MD simulation trajectories.

^b2,400 frames calculated enthalpy & no calculated entropy

^c2,400 frames calculated enthalpy & 48 frames calculated entropy

^d48 frames calculated enthalpy & no calculated entropy

^e48 frames calculated enthalpy & 48 frames calculated entropy

Table 7.

Summary of Impact of Entropy on Predicted Binding Free Energies of MD Trajectories of Different Lengths and Radii Sets in MM/GBSA and QM-MM/GBSA Calculations

MM/GBSA						QM-MM/GBSA
1	2	3	4	5	6	7	8	9	10	11	12
Radii Sets	GB	R26nsa,b	R26nsc	R26nsd	R26nse	Radii Sets	GB	R26nsa,b	R26nsc	R26nsd	R26nse
bondi	GBHCT	0.41	0.24	0.37	0.14	bondi	GBHCT	0.54	0.52	0.54	0.48
mbondi	GBHCT	0.44	0.3	0.42	0.22	mbondi	GBHCT	0.57	0.59	0.59	0.58
mbondi2	GBHCT	0.41	0.24	0.37	0.14	mbondi2	GBHCT	0.54	0.54	0.54	0.50
bondi	GBOBC	0.38	0.12	0.31	0.04	bondi	GBOBC	0.56	0.51	0.56	0.41
mbondi	GBOBC	0.36	0.12	0.30	0.04	mbondi	GBOBC	0.56	0.54	0.57	0.49
mbondi2	GBOBC	0.36	0.12	0.29	0.03	mbondi2	GBOBC	0.55	0.52	0.55	0.43
bondi	GBOBC2	0.36	0.13	0.27	0.04	bondi	GBOBC2	0.51	0.45	0.49	0.36
mbondi	GBOBC2	0.24	0.05	0.16	0.00	mbondi	GBOBC2	0.46	0.42	0.43	0.35
mbondi2	GBOBC2	0.32	0.1	0.23	0.02	mbondi2	GBOBC2	0.49	0.45	0.46	0.35
bondi	GBNeck	0.16	0.00	0.07	0.02	bondi	GBNeck	0.42	0.28	0.35	0.14
mbondi2	GBNeck	0.15	0.00	0.06	0.02	mbondi2	GBNeck	0.4	0.28	0.34	0.15
bondi	GBNeck2	0.5	0.07	0.42	0.06	bondi	GBNeck2	0.86	0.47	0.75	0.42
mbondi2	GBNeck2	0.51	0.07	0.43	0.07	mbondi2	GBNeck2	0.88	0.52	0.76	0.46

^aR²_n: “R²” stands for coefficient of determination in the MM/PBSA calculation. Here 2,400 MD simulation frames were evenly extracted from 6 ns MD simulation trajectories.

^b2,400 frames calculated enthalpy & no calculated entropy

^c2,400 frames calculated enthalpy & 48 frames calculated entropy

^d48 frames calculated enthalpy & no calculated entropy

^e48 frames calculated enthalpy & 48 frames calculated entropy

Further analysis of the effect of conformational entropy on absolute binding free energy is summarized in Figure 6. Figure 6a shows that, although the cumulative average of conformational entropy (upper dashed red line) converged in the 48 frames entropy calculation, the instantaneous conformational entropy (upper solid black line) fluctuated greatly (± 10 kcal/mol) compared to the cumulative average. Such huge fluctuations suggest that a higher number of frames in entropy calculation are needed to improve the quality of entropy predictions and therefore yield more accurate absolute binding free energy, as suggested in a previous study.²¹ Unfortunately, the high computational cost of entropy calculations prevented our further pursuit. For example, each FtFabI-benzimizadole entropy calculation in 48 frames took 120-140 hours on TACC Lonestar's 48 large memory nodes, which is equivalent to 23,000-26,000 XSEDE Service Units per FtFabI- benzimizadole set.

Figure 6.

A. The enthalpy, entropy and binding free energy components (48 frames) in the 6 ns MD simulation of FtFabI: FabI-135 using MM/GBSA, the bondi radii setting and GB^HCT. The upper solid black and dash red line represents instanenous and cumulative average entropy respectively and the lower solid black and dash red line represents instanenous and cumulative average enthalpy. The blue line represents binding free energy.

B. The histogram view of the enthalpy, entropy and binding free energy (48 frames) in the 6 ns MD simulation of FtFabI: FabI-135 using MM/GBSA, the bondi radii setting and GB^HCT.

C. The histogram view of the enthalpy (2,400 frames) in the 6 ns MD simulation of FtFabI: FabI-135 using MM/GBSA, the bondi radii setting and GB^HCT.

Because accurate absolute binding free energy calculations rely on simultaneous error cancellation between the entropy and enthalpy terms, enthalpy calculated from 2,400 frames together with entropy calculated from 48 frames may not offer good quality absolute binding free energy predictions (Table 6, column 3 & Table 7, column 4 & 10). To investigate this, we calculated the absolute binding free energy using 48 frames MM-PBSA and (QM/)MM-GBSA enthalpy and 48 frames entropy. The corresponding calculated absolute binding free energy was then correlated with the experimental binding free energy, as summarized in Table 6, column 5 & Table 7, column 6 & 12. The results showed that the enthalpy from 48 frames (Table 6, column 4 & Table 7, column 5 & 11) did not correlate well with the experimental data. This is most likely because the sample size of 48 frames was insufficient. The 48-frame enthalpy from MM-GBSA, the bondi radii setting and GB^HCT shown in the black histogram in Figure 6b, deviates significantly from the normal distribution seen for the 2,400-frame enthalpy histogram in Figure 6c. This suggests that the small sample size of 48 frames might offer a biased population mean. When the 48-frame enthalpy value was used with the 48-frame entropy value to calculate absolute binding free energy, the resulting value gave even worse agreement with the experimental binding free energy values (Table 6, column 5 & Table 7, column 6 & 12). This suggests that errors in entropy calculation using normal mode analysis here cannot be adequately cancelled using the enthalpy values from MM-PBSA and (QM/)MM-GBSA, even using the same MD frame samples for both calculations. Again, Figure 6a visualized the scenario here: Even though the instantaneous enthalpy from MM-GBSA (lower solid black line) was relatively stable, the “wild” instantaneous entropy (upper solid black line) still resulted in largely swinging and biased absolute free energy (blue dash line in Figure 6a) The histogram in Figure 6b visualized the distribution of 48 frames enthalpy from MM-GBSA (black), 48 frames entropy from normal mode analysis (red), and absolute free energy calculated from enthalpy minus entropy (blue). One can clearly see that entropy values in 48 frames distributed in a highly skewed shape with long double tails, which suggests the existence of extreme values. The extreme entropy values in some frames thus resulted in a skewed distribution with long double tails in absolute free binding energy (blue in Figure 6b). Thus, we will not include entropy calculation in future large scale benzimidazole prediction due to its high computational costs and relative inaccuracy.

Single MD Simulation Efficiency and Accuracy

MM-PBSA is about 60 times more computationally expensive (about 1 real world hour per 100 frames on TACC Lonestar using 48 CPU processors) than MM-GBSA and QM/MM-GBSA (about 1 real world minute per 100 frames on TACC Lonestar using 48 CPU processors). Therefore, MM-PBSA, together with bondi or mbondi2 settings using one 0.75-1 ns MD simulation trajectory (300-400 frames) can offer satisfactory predictive power (R²= 0.82-0.83) with 3-4 hours of computation. Alternatively, MM-GBSA using bondi/mbondi2 settings with a 0.5-0.75 ns MD simulation trajectory (200-300 frames) is able to achieve R²=0.85-0.88 with only 3-4 minutes of computation. Similarly, QM/MM-GBSA using GB^HCT and bondi/mbondi2 settings with 0.5 ns MD simulation trajectory (200 frames) can achieve R²=0.81 with only 2 minutes of computation. This suggests that using one long MD simulation for large scale FtFabI-benzimidazole predictions, MM-GBSA and QM/MM-GBSA will be more economical choices than MM-PBSA.

Multiple MD Simulation Efficiency and Accuracy

The MM-PBSA and (QM/)MM-GBSA results based on multiple independent trajectories together with various radii settings are summarized in Table 8-10 accordingly. Table 8-10 show that the differences between the length of MD simulations and different radii settings were not significant in all MM-PBSA and (QM/)MM-GBSA runs. Using multiple independent samplings, the differences between various GB methods in (QM-)MM-GBSA were also trivial. However, compared to one long MD simulation trajectory (Tables 1-5), the predictive power using several short MD trajectories didn't decrease as it did when using longer MD simulation time frames (Tables 8-10), suggesting that multiple independent samplings can offset force field errors.

Table 8.

The coefficient of determination of Experimental and Predicted Binding Free Energies of Different Multiple MD Trajectory Lengths and Radii Sets in MM/PBSA Calculations.

Radii Sets	R20.25n	R20.5ns	R20.75	R21ns
bondi	0.77	0.73	0.73	0.75
mbondi	0.76	0.74	0.74	0.76
mbondi2	0.77	0.73	0.73	0.76
PARSE	0.6	0.59	0.61	0.65

^a R²_n: “R²” stands for coefficient of determination in the MM/PBSA calculation. Here 600, 1200, 1800, 2400 MD simulation frames were evenly extracted from six 0.25, 0.5, 0.75, 1 ns MD simulation trajectories, respectively.

Table 10.

The coefficient of determination of Experimental and Predicted Binding Free Energies of Multiple MD of Different Trajectory Lengths and Radii Sets in QM-MM/GBSA Calculations.^a

Radii	GB	R20.25nsa	R20.5ns	R20.75ns	R21ns
bondi	GBHCT	0.76	0.75	0.75	0.75
mbondi	GBHCT	0.76	0.75	0.76	0.75
mbondi2	GBHCT	0.76	0.75	0.75	0.75
bondi	GBOBC	0.73	0.72	0.72	0.72
mbondi	GBOBC	0.73	0.73	0.73	0.72
mbondi2	GBOBC	0.73	0.72	0.72	0.72
bondi	GBOBC2	0.73	0.72	0.72	0.72
mbondi	GBOBC2	0.73	0.73	0.73	0.72
mbondi2	GBOBC2	0.73	0.72	0.72	0.71
bondi	GBNeck	0.68	0.67	0.67	0.67
mbondi2	GBNeck	0.68	0.67	0.67	0.66
bondi	GBNeck2	0.71	0.69	0.7	0.69
mbondi2	GBNeck2	0.71	0.7	0.71	0.7

^aR²_n: “R²” stands for coefficient of determination in the QM-MM/GBSA calculation. Here 600, 1200, 1800, 2400 MD simulation frames were evenly extracted from six 0.25, 0.5, 0.75, 1 ns MD simulation trajectories, respectively.

Unlike using one long MD simulation, the predictive performance from multiple independent samplings was insensitive to various radii settings in all MM-PBSA and (QM/)MM-GBSA methods (Table 8-10). In MM-PBSA (Table 8), the three Born radii sets offered satisfactory predictive power (R² > 0.7) while the PARSE radii setting was slightly worse (R² in the 0.6 range). Using multiple independent samplings, the differences between (QM/)MM-GBSA were minimal. In (QM/)MM-GBSA together with multiple independent samplings (Table 9 & 10), the three “non-neck” correction GB methods (GB^HCT, GB^OBC, GB^OBC2) behaved similarly but showed slight differences compared to the two neck correction GB methods (GB^Neck and GB^Neck2), which were similar to each other.

Table 9.

The coefficient of determination of Experimental and Predicted Binding Free Energies of Different Multiple MD Trajectory Lengths and Radii Sets in MM/GBSA Calculations.^a

Radii Sets	GB	R20.25nsa	R20.5ns	R20.75ns	R21ns
bondi	GBHCT	0.84	0.83	0.82	0.82
mbondi	GBHCT	0.84	0.84	0.83	0.82
mbondi2	GBHCT	0.84	0.83	0.82	0.82
bondi	GBOBC	0.81	0.8	0.8	0.8
mbondi	GBOBC	0.81	0.8	0.8	0.8
mbondi2	GBOBC	0.81	0.8	0.8	0.8
bondi	GBOBC2	0.81	0.79	0.79	0.79
mbondi	GBOBC2	0.78	0.77	0.76	0.76
mbondi2	GBOBC2	0.8	0.78	0.79	0.78
bondi	GBNeck	0.73	0.7	0.71	0.71
mbondi2	GBNeck	0.72	0.69	0.7	0.7
bondi	GBNeck2	0.72	0.68	0.68	0.66
mbondi2	GBNeck2	0.72	0.69	0.68	0.66

^aR²_n: “R²” stands for coefficient of determination in the MM/GBSA calculation. Here 600, 1200, 1800, 2400 MD simulation frames were evenly extracted from six 0.25, 0.5, 0.75, 1 ns MD simulation trajectories, respectively.

In terms of predictive power between the single, long MD simulation (Table 1-5) and multiple independent MD simulation (Table 8-10) methods, both provided similar predictive power using lower numbers of frames (600 and 1,200 frames). However, when larger numbers of frames were used, the predictive power decrease was negligible with the multiple independent simulation method but not with the single simulation. Taken together, these results suggest that methodology differences and errors may be minimized and averaged out using multiple independent samplings. With respect to computational efficiency, MM-PBSA is about 60-fold more computationally expensive than (QM/)MM-GBSA. Therefore, MM-GBSA together with GB^HCT using six 0.25 ns multiple independent MD simulations (600 frames) can offer very good agreement, r²= 0.84 (Figure 7b), with experimental data after just 6 real world minutes of computation. Therefore, MM-GBSA and QM/MM-GBSA offer a better balance in terms of predictive power and computational efficiency than MM-PBSA with multiple independent samplings.

Figure 7.

A. The correlation plot between experimental and predicited binding free energy using QM-MM/GBSA, GB^Neck2, mbondi2 & 2,400 frames evenly extracted from 6 ns long MD simulation trajectories (R² = 0.88).

B. The correlation plot between experimental and predicited binding free energy using MM/GBSA, GB^HCT, mbondi & 600 frames evenly extracted from six 0.25 ns long MD simulation trajectories (R² = 0.84).

Conclusions

In this study, we have extensively compared three implicit solvent methods: MM-PBSA, MM-GBSA, and QM/MM-GBSA and the effects of radii sets, entropy, GB methods, QM Hamiltonians, sampling and simulation lengths using sixteen carefully chosen benzimidazole inhibitors of F. tularensis FabI. In the first part of the this study using one long MD simulation for each ligand (the traditional method), the various GB methods, QM Hamiltonians, and simulation lengths resulted in significantly different predictive power in MM-PBSA and (QM/)MM-GBSA results, while the effect of radii sets on prediction power is relatively milder. The QM/MM-GBSA method using GB^Neck2, mbondi2, and the PM3 Hamiltonian setting with 2,400 frames extracted evenly from single 6 ns MD simulations offered the best agreement between predicted and experimental binding free energy (R² = 0.88, Figure 7a).

The inclusion of conformational entropy compromised the agreement between predicted absolute binding free energy and experimental binding free energy due to large fluctuations in the calculated entropy values. Theoretically, using a larger number of frames in the entropy calculations might increase its accuracy, but the associated high computational costs might not be easily affordable. Therefore, the implicit solvent free energy methods in this study (MM-PBSA and (QM/)MM-GBSA) were limited to comparing relative binding free energies of similar ligands due to the absence of accurate entropies. However, for the purposes of developing a rapid tool for energy predictions for a lead optimization program focusing on highly similar compounds, conformational entropy may be safely ignored as the implicit solvent methods used in this study can still be powerful tools in compound prioritization.

In the second part of the study, the prediction is based on the average of six 1 ns MD simulations for each FtFabI-benzimidazole system. We called this type of sampling, “multiple independent sampling” or “Monte Carlo like MD sampling”. In general, implicit solvent methods based on multiple independent samplings are less sensitive on radii sets, MD simulation length, and GB methods. This feature might minimize the future need of an entirely new validation study, when (1) a new scaffold of the same therapeutic target is pursued and (2) the same scaffold with additional moieties, which result in distinct physiochemical properties (charges, etc.) and binding modes from the previous validation sets, bring in different sensitivities on various parameters.

Because the predictive power of the multiple sampling method showed little decrease with MD simulation time, it can offer the same or better predictive power compared with the use of a single, long simulation with lower number of MD frames. For example, MM/GBSA together with GB^HCT and either bondi, mbondi or mbondi2, using 600 frames extracted evenly from six 0.25 ns MD simulations, can achieve a comparable agreement between predicted and experimental binding free energy (r² = 0.84, Figure 7b) to QM/MM-GBSA using GB^Neck2, mbondi2, and the PM3 Hamiltonian setting with 2,400 frames extracted evenly from 6 ns MD simulation (r² = 0.88, Figure 7a). Therefore, the multiple independent sampling method can be more computationally efficient than a single, long simulation method.

Different implicit solvents methods bring varied accuracy. In this study, MM-GBSA and QM/MM-GBSA brings better accuracy and higher efficiency than MM-PBSA, in either the multiple independent sampling method, or the tradition single, long simulation method (Table 11a).

Table 11.

Summary of the Effect of Parameters on Predictive Power

		Traditional method	Multiple Independent Sampling
a. Accuracy of Implicit Solvent Methods		QM/MM-GBSA > MM-GBSA > MM-PBSA	QM/MM-GBSA ≈ MM-GBSA > MM-PBSA
		MM-GBSA & QM/MM-GBSA
		Traditional method	Multiple Independent Sampling
b. Accuracy of GB Methods		GBNeck2 > GBHTC >= GBOBC = GBOBC2 > GBNeck	Differences are minimized
		MM-PBSA
		Traditional method	Multiple Independent Sampling
c. Accuracy of Radii Settings		bondi = mbondi2 ≈ mbondi > PARSE	Differences are minimized
		MM-GBSA & QM/MM-GBSA
GB Method Names	AMBER igb flag	Traditional method	Multiple Independent Sampling
GBHCT	1	bondi = mbondi2 ≈ mbondia	Differences are minimized
GBOBC	2	bondia = mbondi2a ≈ mbondi	Differences are minimized
GBOBC2	5	bondia = mbondi2a >≈b mbondi	Differences are minimized
GBNeck	7	mbondi2a,c ≈ bondi	Differences are minimized
GBNeck2	8	mbondi2a,d ≈ bondi	Differences are minimized
		QM/MM-GBSA
d. Accuracy of PM Hamiltonians		PM3 > AM1 > PM6

^aAMBER user manual recommended radii sets for the GB methods.

^bIn MM-GBSA and GB^OBC2: bondi^a = mbondi2^a > mbondi. In QM/MM-GBSA and GB^OBC2: bondi^a = mbondi2^a ≈ mbondi

^cbondi might also work, but it might cause instability in the native peptide REMD type of simulation.

^dmbondi3 is recommended in the MD/REMD type of simulation.

The effect of GB methods on accuracy (Table 11b) is more evident in the traditional one long simulation method. Unlike four other GB methods, GB^Neck2 can offer the best accuracy in the 6 ns time frame, without a decrease in accuracy along the MD simulation length in both MM-GBSA and QM/MM-GBSA. GB^HTC could offer slightly better performance than GB^OBC and GB^OBC2. GB^OBC and GB^OBC2 offer the comparable results, since they share highly similar methodologies. Therefore, if the GB methods are applied on one MD trajectory for each ligand, the accuracy is GB^Neck2 > GB^HTC >= GB^OBC = GB^OBC2 > GB^Neck. However, the differences between GB methods are largely minimized in the multiple independent sampling.

The effect of radii sets on small molecule binding free energy prediction is relatively mild in the traditional long MD method, comparing to other parameters (Table 11c). In MM-PBSA, the performance is bondi = mbondi2 ≈ mbondi > PARSE. In MM/GBSA together with GB^HCT,GB^OBC, GB^Neck, and GB^Neck2, the differences between radii sets are negligible. In MM/GBSA and GB^OBC2, the non-recommended mbondi setting shows inferior performance than recommended bondi and mbondi2 settings. Moreover, the effect of radii sets on QM/MM-GBSA is highly similar to MM-GBSA, besides in GB^OBC2, mbondi settings showed similar predictive power to bondi and mbondi2 with QM/MM-GBSA. On the other hand, the differences between radii sets are neutralized in the multiple sampling method.

QM Hamiltonians have a strong impact on QM/MM-GBSA results (Table 11d). In general, the performance is PM3 > AM1 > PM6. However, future optimized PM6 parameters may enhance the performance of the PM6 Hamiltonian in QM/MM-GBSA in AMBER.

To summarize, it is important, when optimizing computational protocols to predict binding free energy, to carefully investigate the optimal radii sets, entropy, sampling methods, GB methods, and QM Hamiltonians in (QM/)MM-P(G)BSA. Our results suggest that small deviations from optimized settings can have a significant impact on energy prediction results, not only for our FtFabI-benzimidazole systems, but potentially other similar systems amenable to such lead optimization methods. Lastly, because MM/GBSA together with GB^HCT, mbondi, and 600 frames extracted evenly from six 0.25 ns MD simulations, provided accurate predictions efficiently, with less sensitivity on various parameters, this method will be applied to future large-scale lead optimizations of benzimidazole FabI inhibitors.

Abbreviations

AM1

Austin Model 1

MD

Molecular Dynamics

MM-PBSA

Molecular Mechanics - Poisson-Boltzmann Surface Area

MM-GBSA

Molecular Mechanics - Generalized Born Surface Area

QM/MM-GBSA

Quantum Mechanics / Molecular Mechanics - Generalized Born Surface Area

FAS

Fatty Acid Synthesis

FtFabI

Francisella tularensis FabI

GAFF

general AMBER force field

PME

Particle Mesh Ewald

PM3

Parameterized Model number 3

PM6

Parameterized Model number 6

NMA

Normal Mode Analysis

RMSD

root-mean-square deviation

ΔG_bind

binding free energy

ΔH_bind

binding enthalpy

ΔG_solvation

solvation free energy

ΔG_GB

Generalized Born polar free energy

ΔG_PB

Poisson-Boltzmann polar free energy