# Exploring Routes to Enhance the Calculation of Free Energy Differences via Non-Equilibrium Work SQM/MM Switching Simulations Using Hybrid Charge Intermediates between MM and SQM Levels of Theory or Non-Linear Switching Schemes

^{1}

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^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Overview of Calculated Free Energy Differences and Paths

#### 2.2. Performances of 2 and 5 ps Linear Switching Protocols in Solution

**1-t1**(see further down), $\left|\delta \mathrm{\Delta}A\right|$ was $<0.1$ kcal/mol. While $\delta \mathrm{\Delta}A\left(\mathbf{1}\text{-}\mathbf{t}\mathbf{1}\right)\approx 0.35$ kcal/mol is slightly larger than the ideal maximum deviation of $\pm 0.25$ kcal/mol, this value is still acceptable for most practical applications. Thus, the protocol recommendations of our previous study [36] are applicable to the model compounds studied here. Based on the gas phase results, any noticeable deterioration of $\delta \mathrm{\Delta}A$ therefore has to be caused by differences in the description of solute–solvent interactions at the two levels of theory.

**1-t1**and

**8-t2**can be considered as acceptable, $\left|\delta \mathrm{\Delta}A\right|<0.4$ kcal/mole; furthermore, $\delta \mathrm{\Delta}A$ (

**1-t1**) is comparable to the value obtained in the gas phase (see Table S3). However, for

**5-t2**and

**6-t2**, the deviation from the reference result is larger than 1 kcal/mol and would lead to a sizable systematic error.

**6-t2**, $\delta \mathrm{\Delta}A\approx 0.8$ kcal/mol, remains. Thus, the relatively inexpensive protocol suggested in Ref. [36] cannot be recommended for calculations in solution. Even worse, the use of more costly switching simulations of 5 ps length does not help in all cases.

**5-t2**,

**6-t2**, and

**8-t2**, are lactams. Interestingly, the corresponding tautomeric lactim states do not cause any problems. The pair

**1-t1**/

**1-t2**belongs to the keto-enol class of tautomerism. For

**1-t1**, $\delta \mathrm{\Delta}A\approx 0.35$ kcal/mol in both the gas phase and in aqueous solution; therefore, different conformational preferences at the two levels of theory may be responsible for the poor convergence when using switching lengths of 2 ps.

#### 2.3. Performances of Hybrid Charge Intermediates

**5-t2**is still missed, $\delta \mathrm{\Delta}A=0.32$ kcal/mol, and the result for

**Ala**becomes worse ($\delta \mathrm{\Delta}A=-0.49$ kcal/mol). For MULL(solv), the blue squares, all results lie de facto within the $\pm 0.25$ kcal/mol threshold ($\delta \mathrm{\Delta}A\left(\mathbf{5}\text{-}\mathbf{t}\mathbf{2}\right)=-0.28$ kcal/mol, $\delta \mathrm{\Delta}A\left(\mathbf{Ser}\right)=-0.26$ kcal/mol). It should be noted that taking the statistical uncertainty of the results into account, none of these deviations from the $\pm 0.25$ kcal/mol threshold are statistically significant. Finally, all MULL(solv*) results (green diamonds) lie well within the $\pm 0.25$ kcal/mol threshold.

#### 2.4. Performances of Modified Switching Protocols

**5-t2**, all three stepwise linear protocols reduce $\delta \mathrm{\Delta}A$ considerably. However, in the case of

**6-t2**, it is difficult to speak of improvements. Two protocols,

**L2-1**and

**L3-1**, lower $\delta \mathrm{\Delta}A$, but the values are still far outside of the $\pm 0.25$ kcal/mol threshold. The third protocol,

**L3-2**, which worked extremely well for

**5-t2**, actually increases $\delta \mathrm{\Delta}A$ for

**6-t2**compared to

**L1**, the default linear protocol. The other three compounds included in the subset were chosen as negative controls because they already performed well with the JAR(2ps) protocol. The performance of the stepwise linear protocol is comparable, except for one poor result for

**4-t2**when using

**L3-1**.

**4-t2**, the stepwise linear protocols perform slightly poorer, and for

**6-t2**, $\delta \mathrm{\Delta}A$ obtained with the

**L3-1**is >1 kcal/mol.

#### 2.5. A Detailed Analysis of the Factors Affecting Convergence

#### 2.5.1. Effects of Charge Distribution on Solute Properties

**5-t2**vs.

**5-t1**, etc. One reason for this is that the differences between the MM and the SQM(/MM) descriptions are greater for the lactams than for the lactims. The analysis of partial charges points in this direction as well. Several differences can be discerned in the detailed charge distribution data for the individual molecules, e.g., Figure S4. For all lactim–lactam pairs (compounds

**2**,

**3**,

**4**,

**5**,

**6**,

**8**),the MM ${\mathrm{RMSD}}_{\mathrm{q}}$ of the lactam state is noticeably higher than that of the corresponding lactim state. Similarly, the MM ${\mathrm{RMSD}}_{\mathrm{q}}$ of each of the lactams (red circle) is at least 0.1e higher than for MULL(solv) (blue squares).

**5-t2**(top) and

**6-t2**(bottom). For each of the three charge representations, MM, MULL(gas), and MULL(solv), we display the differential atomic charges, both as labels, as well as a color gradient from blue to red, and the resulting differential dipole moment vectors (orange arrows). The magnitude of the differential dipole moment $\mathrm{\Delta}\mu =\left|\mathrm{\Delta}\overrightarrow{\mu}\right|$ and the angle ${\mathrm{\Theta}}_{\mathrm{SQM}}$ between the dipole moment of a charge distribution and that of the MULL(solv*) reference charge distribution are listed directly. The difference in partial charges and the MULL(solv*) charges can be large, e.g., the difference for the -C=O part of the lactam moiety is almost $\pm 0.5$e for

**5-t2**(top, left in Figure 5). One further sees that the charge differences become smaller between the MM and the two MULL charge sets, with atoms colored in clear blue or red for MM; e.g., the -C=O group (left side) has just a shade of blue or red for MULL(solv) (right side). Accordingly, the length of the orange arrows, i.e., $\mathrm{\Delta}\mu $ of the MULL charge states, is noticeably smaller than that for MM. Even more detailed plots, including the exact values of the charge difference for each atom, for

**3-t2**,

**4-t2**,

**5-t2**,

**6-t2**, and

**8-t2**, can be found in Figures S7–S11.

#### 2.5.2. Effects of Charge Distribution on the First Solvation Shell

**2-t2**,

**3-t2**,

**4-t2**,

**5-t2**) the difference in water molecules $\mathrm{\Delta}{N}_{Waters}\approx -1$. For the MULL(gas) charges (orange triangles), all $\mathrm{\Delta}{N}_{Waters}$ values are negative, i.e., on average, there are fewer water molecules in close contact with the solute, compared to SQM/MM. This is not too surprising, because the charges were derived from SCC-DFTB gas phase calculations; hence, the solute did not experience polarization from its interaction with the solvent. This may also explain why on average, the use of the MULL(gas) hybrid intermediate state performed worse than MULL(solv), even though it performed significantly better than MM (cf. Table 1).

**6-t2**, $\mathrm{\Delta}{N}_{Waters}$ is negative, whereas the lactims, i.e., have a positive $\mathrm{\Delta}{N}_{Waters}$. Thus, one sees again distinct differences in properties for the force field representations of lactims and lactams, respectively.

#### 2.5.3. Water Reorientation Dynamics

**5t-2**are shown in Figure 7; analogous plots for

**4t-2**,

**6t-2**, and

**8t-2**can be found in Figures S12–S14 in SI. All fit parameters are summarized in Table S13 of SI.

**6-t2**, very similar results were obtained; see Figure S13 and Table S13.

#### 2.5.4. Interplay between Charge Distribution and Conformational Preferences

**1-t1**and the blocked amino acids

**Ala**and

**Ser**. Modifying partial charges as for the MULL hybrid intermediate states may have an effect on conformational preferences. As can be seen in Figure 3, all three MULL hybrid intermediate states improve the convergence for

**1-t1**. For the two blocked amino acids, MULL(gas) results in a poorer convergence for

**Ala**, and MULL(solv) results in a slightly poorer convergence for

**Ser**. Given that

**Ala**and

**Ser**are the smallest possible peptide-like molecules with protein backbone-like $\varphi $ and $\psi $ dihedral angles, we performed some analyses on the dihedral angle conformational preferences. In earlier work [31], we showed that purely classical Hamiltonians and SQM/MM Hamiltonians resulted in different preferred conformations of $\varphi $ and $\psi $, as well as ${\chi}_{1}$ in the

**Ser**case. In Figure 8 and Figure 9, $\varphi /\psi $ maps for

**Ala**and

**Ser**are shown for MM, MULL(solv), and SQM. The differences between MM (left) and SQM (right) are clearly visible. For both blocked amino acids, MM has a single narrow minimum at $\varphi \approx {150}^{\circ}$, $\psi \approx -{50}^{\circ}$, whereas SQM has a broader distribution at $\varphi \approx {150}^{\circ}$, and $-{150}^{\circ}<\psi <-{50}^{\circ}$. While the MULL $\varphi /\psi $ maps (middle plots) are more similar to MM than to SQM, a second minimum at $\varphi \approx {150}^{\circ}/\psi \approx -{150}^{\circ}$ has appeared. Thus, although the effect is small, the use of hybrid charge intermediates also makes this state slightly more similar to a high-level state, in terms of conformational preferences.

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Computing Free Energy Differences between the Levels of Theory Using NEW Methods

#### 4.2. Choice of Model Systems

**Ala**) and N-acetyl-serine-methylamide (

**Ser**) as model solutes, since these were the two compounds for which we noticed slow convergences in Ref. [31]. All model compounds used in this work are shown in Figure 10.

#### 4.3. Strategies to Improve the Convergences of NEW Simulations

#### 4.3.1. Hybrid Charge Intermediates

#### 4.3.2. Stepwise Linear Switching Protocols

**1-t2**,

**4-t2**,

**5-t2**,

**6-t2**, and

**8-t1**). This selection was motivated by the results obtained with the linear protocol, and includes both “easy” and “difficult” solutes with respect to convergence (cf. Results, Section 2.2).

#### 4.3.3. Analyses Carried Out

#### Characterizing Charge Distributions

`COOR DIPOle`command of CHARMM (see https://academiccharmm.org/documentation/version/c47b1/corman (accessed on 29 April 2023)); if the difference between two charge sets is assigned as partial charges, one obtains the corresponding differential dipole moment.

#### Characterization of the First Solvation Shell

#### Dynamics of Solvent Reorientation

**4-t2**,

**5-t2**,

**6-t2**and

**8-t2**), we computed 800 simulations of 10 ps length; based on the experimental findings, it is reasonable to expect that solvent reorientation is completed after this time [38,39]. At each timestep, we saved $\mathrm{\Delta}U\left(t\right)={U}^{SQM/MM}\left(t\right)-{U}^{method}\left(t\right)$, where method was either MM or MULL(solv). This task was facilitated by a locally modified version of CHARMM, but could equally well be accomplished through the post-processing of trajectories saved during the simulations. The 800 $\mathrm{\Delta}U\left(t\right)$ time series were then averaged, resulting in the averaged time series $\overline{\mathrm{\Delta}U\left(t\right)}$. Next, we estimated $\overline{\mathrm{\Delta}U(\infty )}$, i.e., the limit of $t\to \infty $, by averaging (again) over the last 2000 entries of $\overline{\mathrm{\Delta}U\left(t\right)}$ ($8\phantom{\rule{0.166667em}{0ex}}\mathrm{ps}\le t\le 10\phantom{\rule{0.166667em}{0ex}}\mathrm{ps}$).

#### 4.4. Overview of Simulations Carried Out

#### 4.4.1. Simulation Details

#### Preparation and Initial Equilibration

^{−1}, Langevin piston bath temperature: 300 K). The final 20 ps of these equilibration runs were used to determine the average box size. In Table S1 of the SI, we list the size of the simulation box determined in this manner, as well as the number of water molecules present, for each of the compounds studied.

#### Force Field-Based Equilibrium Simulations

^{−1}) were carried out in the gas phase and in solution, respectively. Each of these simulations was initialized with different random velocities drawn from a Maxwell-Boltzmann distribution at 300 K. In each gas phase run, 5 ns of simulation time were discarded as equilibration, followed by 10 ns of production, during which the restart files were saved every 1000th step. In the analogous solution simulations at constant volume, 0.5 ns were discarded as equilibration. During the subsequent 1 ns production phase, restart files were saved every 1000th step. Thus, during the cumulative simulation length of 8 ns (solution)/80 ns (gas phase), 8000 (solution)/80,000 (gas phase) restart files were saved. These served as the pool of configurations sampled in the canonical ensemble, from which non-equilibrium switching simulations to the high (SQM(/MM)) level of theory were started.

^{−1}, spline interpolation to order 6, FFT grid size of 32).

#### SQM(/MM) Equilibrium Simulations

#### 4.4.2. Non-Equilibrium Work Simulations

#### 4.4.3. Calculation of $\mathrm{\Delta}{A}_{Xsolv}^{\mathrm{MM}\phantom{\rule{0.277778em}{0ex}}\leftrightarrow \phantom{\rule{0.277778em}{0ex}}\mathrm{MULL}}$:

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MD | Molecular Dynamics |

MM | Molecular Mechanics |

(S)QM | (Semi-empirical-)Quantum Mechanics |

(S)QM/MM | (Semi-empirical-)Quantum Mechanical/ Molecular Mechanical hybrid methods |

FES | Free energy simulation |

FEP | Free energy perturbation |

TI | Thermodynamic integration |

BAR | Bennett’s acceptance ratio |

NEW | Non-equilibrium work methods |

JAR | Jarzynski’s Equation |

CRO | Crook’s Equation |

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**Figure 1.**(

**a**) Indirect alchemical thermodynamic cycle to compute $\mathrm{\Delta}{A}_{X\phantom{\rule{0.277778em}{0ex}}gas\to solv}^{SQM/MM}$ (dashed arrow) in three steps according to Equation (1). (

**b**) Indirect alchemical thermodynamic cycle to compute $\mathrm{\Delta}{A}_{X\phantom{\rule{0.277778em}{0ex}}solv}^{MM\to SQM/MM}$ (red arrow) via hybrid charge intermediates; cf. Equation (8).

**Figure 2.**Comparison of $\delta \mathrm{\Delta}{A}_{Xsolv}^{MM\to SQM}$ using three protocols without intermediate state: JAR(2ps): 2 ps forward switches from MM to SQM, JAR(5ps): 5 ps forward switches from MM to SQM, and CRO(2ps): 2 ps forward and backward switches. In all cases, the CRO results obtained from 5 ps switches were considered as the reference values CRO${}_{\mathrm{Ref}}$. For

**5t-2**and

**6t-2**, the JAR(2ps) results are off-scale, which is indicated by the red arrows.

**Figure 3.**Comparison of $\delta \mathrm{\Delta}A$ obtained with three hybrid intermediate charge states: MULL(gas), orange triangles; MULL(solv), blue squares; MULL(solv*), green diamonds. All results were obtained from 200 NEW switches of 2 ps length to compute $\mathrm{\Delta}{A}^{\mathrm{MULL}\to \mathrm{SQM}}$ and include the correction $\mathrm{\Delta}{A}^{\mathrm{MM}\leftrightarrow \mathrm{MULL}}$; cf. Equation (2b). The JAR(2ps) results already shown in Figure 2 (red circles) are included for comparison purposes.

**Figure 4.**Performances of the three modified switching protocols for a subset of the compounds. Results for the direct NEW switching protocols MM → SQM are shown as red circles; the results for calculations using the MULL(solv) hybrid charge intermediate state are blue squares. The fill color of the circles/squares indicates the stepwise linear switching protocol used (green =

**L2-1**, orange =

**L3-1**and salmon =

**L3-2**). For comparison purposes, the MM/JAR(2ps) (red circles) and MULL(solv) results (blue squares), both using the default switching protocol

**L1**, are included as well.

**Figure 5.**Graphical representation of differences in partial atomic charges compared to MULL(solv*) for

**5-t2**(

**top**) and

**6-t2**(

**bottom**). Charge differences are indicated as a color gradient from blue ($\delta q=-0.2e$) to red ($\delta q=+0.4e$). The differential dipole moment $\mathrm{\Delta}\overrightarrow{\mu}$ is displayed as an orange arrow. The magnitude $\mathrm{\Delta}\mu =\left|\mathrm{\Delta}\overrightarrow{\mu}\right|$ and the angle ${\mathrm{\Theta}}_{\mathrm{SQM}}$ between the dipole moment of the respective charge distribution with that of the MULL(solv*) reference distribution are given below each structure.

**Figure 6.**Difference in the average number of water molecules closer than 3 Å, compared to SQM/MM. The average number of water molecules in the SQM/MM simulations was used as the reference value.

**Figure 7.**Water reorientation dynamics of TIP3P solvent for

**5-t2**when switching from MM and MULL(solv) to SQM/MM. Raw data are shown in gray; fit for MM in red and fit for MULL(solv) in blue. The fit parameters ($\mathrm{\Delta}{U}_{0}$ in kcal/mol, $\tau $ in fs) are listed in the inset; cf. Equation (14).

**Figure 11.**The standard linear (L1) and three modified switching protocols investigated. The total switching length $\tau $ was 2 ps in all four cases. The color code used for L2-1, L3-1, and L3-2 is also used in Results.

**Table 1.**MAD (mean absolute deviation) for the results shown in Figure 3, as well as the spread of $\delta \mathrm{\Delta}A$ (respectively, the smallest and largest absolute deviations).

Pathway | MAD [kcal/mol] | Spread MAD [kcal/mol] | |
---|---|---|---|

Min | Max | ||

MM | 0.29 | 0.01 | 1.80 |

MULL(gas) | 0.12 | 0.01 | 0.49 |

MULL(solv) | 0.09 | 0.01 | 0.28 |

MULL(solv*) | 0.07 | 0.01 | 0.20 |

**Table 2.**MAD (Mean Absolute Deviation) over all compounds for ${\mathrm{RMSD}}_{\mathrm{q}}$, the magnitude of the differential dipole moment $\mathrm{\Delta}\mu $, and the angle ${\mathrm{\Theta}}_{\mathrm{SQM}}$ between the dipole moment vector of the respective charge distribution and that of the MULL(solv*) charge distribution.

Pathway | MAD RMSD${}_{\mathit{q}}\left[\mathit{e}\right]$ | MAD $\mathbf{\Delta}\mathit{\mu}$ [D] | MAD ${\mathit{\theta}}_{\mathit{S}\mathit{Q}\mathit{M}}$[°] |
---|---|---|---|

MM | 0.14 | 3.38 | 29.8 |

MULL(gas) | 0.06 | 2.44 | 7.6 |

MULL(solv) | 0.04 | 1.81 | 16.0 |

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## Share and Cite

**MDPI and ACS Style**

Schöller, A.; Woodcock, H.L.; Boresch, S.
Exploring Routes to Enhance the Calculation of Free Energy Differences via Non-Equilibrium Work SQM/MM Switching Simulations Using Hybrid Charge Intermediates between MM and SQM Levels of Theory or Non-Linear Switching Schemes. *Molecules* **2023**, *28*, 4006.
https://doi.org/10.3390/molecules28104006

**AMA Style**

Schöller A, Woodcock HL, Boresch S.
Exploring Routes to Enhance the Calculation of Free Energy Differences via Non-Equilibrium Work SQM/MM Switching Simulations Using Hybrid Charge Intermediates between MM and SQM Levels of Theory or Non-Linear Switching Schemes. *Molecules*. 2023; 28(10):4006.
https://doi.org/10.3390/molecules28104006

**Chicago/Turabian Style**

Schöller, Andreas, H. Lee Woodcock, and Stefan Boresch.
2023. "Exploring Routes to Enhance the Calculation of Free Energy Differences via Non-Equilibrium Work SQM/MM Switching Simulations Using Hybrid Charge Intermediates between MM and SQM Levels of Theory or Non-Linear Switching Schemes" *Molecules* 28, no. 10: 4006.
https://doi.org/10.3390/molecules28104006