Advancing Binding Affinity Calculations: A Non-Equilibrium Simulations Approach for Calculation of Relative Binding Free Energies in Systems with Trapped Waters

Abstract

The formation of protein-ligand complexes involves displacement of water molecules that were previously occupying the protein’s binding site. In some cases, however, some water molecules may not be displaced by the ligand’s binding and they can stabilize the complex by mediating the interactions between the ligand and the protein. A relative binding free energy (RBFE) calculation between two ligands, one of which binds to the protein with an intermediate water while the other ligand displaces it, can yield wrong results if the water fails to rearrange itself within the simulation timescale. Enhanced sampling methods have previously been used to address the sampling of such “trapped” waters, inserting or deleting waters in the protein’s binding site during ligand transformation. While sometimes effective, the enhanced sampling methods typically require long simulation times to converge and may lead to differences in RBFE estimates (i.e., hysteresis) based on initial water placement. In this study, we present a non-equilibrium switching (NES) method to calculate RBFEs in systems with trapped waters. Our approach requires the knowledge of the positions of the trapped waters prior to performing the free energy calculation for ligand transformation, and then uses this information to efficiently calculate the RBFE between the ligands. In our simulation protocol, we perform ligand transformation in the binding site of the target protein using three consecutive NES switches. The three NES switches implement restraints, transform the ligand, and then remove the restraints. We demonstrate that our NES simulations-based method results in RBFE estimates within 1.1 kcal mol⁻¹ of experimental RBFEs, with associated statistical errors under 0.4 kcal mol⁻¹, for eight systems involving trapped water displacement. Our method provides a computationally inexpensive alternative for estimating RBFEs for systems involving trapped waters by leveraging distributed computational resources.

Graphical Abstract

2. Introduction

In structure-based drug design, free energy calculations have a number of applications, such as calculating the binding free energies of protein-ligand complexes,^1,2 studying the effects of protein side chain mutations on complex formation,^3,4 calculating relative binding free energies between ligands with small⁵ and large⁶ modifications, etc. Modern computational capabilities have increased the scalability of free energy calculations, and their applications in reliable predictions of protein-ligand complex formation and molecular design are becoming mainstream.

One of the main applications of free energy calculations is estimating the binding affinities of small molecular ligands to their target proteins. The formation of a protein-ligand complex involves displacement of the water molecules that previously occupied the binding site of the protein, and formation of new interactions of the ligand with the amino acids at the protein’s binding site. In some cases, some water molecules may not be displaced by the ligand’s binding, and these waters can stabilize the protein-ligand complex by mediating the interactions between the ligand and the binding site.⁷ In ligand optimization, it is common to design ligands that can displace these localized waters,^8,9 as displacing energetically unfavorable waters can enhance binding affinity.¹⁰

A relative binding free energy (RBFE) calculation assesses the difference between binding affinities of two ligands to their target protein by alchemically mutating one ligand into the other. The calculation is performed in two parts– a complex leg of the calculation, i.e., the ligands are transformed in the binding site of the protein, and a solvent leg of the calculation, in which the ligand transformation is performed in solvent.

To correctly compute an RBFE between two ligands, the alchemical transformation in the complex must transform one ligand in its correct binding mode into another ligand in the latter’s correct binding mode. A ligand’s binding mode refers to its overall placement and conformation in the binding site, including key interactions such as hydrogen bonds, hydrophobic contacts, and the presence of structured water molecules. An RBFE calculation between two ligands, one that coordinates with a localized water molecule in its binding pose (referred to as ligand A) and another that displaces the water (referred to as ligand B), can present sampling challenges if the movement of the localized water is too slow on the timescale of the calculation. Such a localized water can be considered “trapped” in the binding site, and if the trapped water is not carefully treated, the free energy calculation may yield inaccurate results. The inaccuracy can result from one of these two scenarios: 1. By neglecting a water needed to model one of the ligands, the calculations end up considering a ligand binding mode that is missing an essential water molecule, which could lead to an unconverged calculation of the binding affinity; or 2. By retaining an extra water for one of the ligands, the retained trapped water disrupts the ligand’s interactions with the protein, potentially leading to an inaccurate depiction of the binding mode of the ligand in the calculation and an unconverged free energy estimate.

Enhanced sampling simulation methods, such as hybrid Monte Carlo/molecular dynamics (MC/MD)¹¹ and grand canonical Monte Carlo combined with MD (GCMC/MD),^12,13 attempt to address the challenge of trapped water sampling by inserting and/or deleting water molecules in the protein’s binding site while transforming the ligand. These methods divide the ligand transformation up into a series of small transformations between intermediate λ states connecting the physical end states of interest (with each end states representing the relevant ligand bound to the protein). Each λ state is simulated and the occupancies of the trapped waters are estimated at each λ state. However, these methods can be computationally expensive, requiring long simulation times to reach convergence. Additionally, they can sometimes result in different RBFE estimates (i.e., hysteresis) depending on whether the simulations are started with or without the trapped waters in the initial structures,¹⁴ suggesting incomplete water equilibration or insufficient convergence of the simulations. Recent studies on the accuracy and efficiency of these methods in estimating occupancies of trapped water sites indicate that, for several systems, the simulation times required to accurately estimate the location or population/fractional occupancies of the trapped waters can exceed the timescales of typical free energy calculations.^15,16

A recent study by Ge et al.¹⁷ suggested separating the sampling of trapped waters from the free energy calculation of ligand transformation. In this study, the authors proposed locating trapped waters in the system prior to performing the free energy calculation for the ligand transformation. Once the locations of the trapped waters are determined, they proposed to perform the RBFE calculation using a separation of states approach, in which the trapped waters are displaced from the binding site first and the ligand transformation takes place after that. The trapped waters in the binding site can be identified from previous studies of the system, protein-ligand crystal structures, solvation site analysis methods,¹⁸ enhanced sampling simulations (e.g. hybrid MC/MD or GCMC/MD), or from sufficiently long MD simulations. Prior to Ge et al.’s study, Jorgensen and coworkers had presented an MC simulations-based approach of displacing the trapped waters from the binding site prior to transforming the ligands,¹⁹ suggesting that separate handling of trapped waters might be a productive strategy to calculate free energy changes associated with coupled modifications of a ligand and associated water molecules.

In the study of Ge et al.,¹⁷ the authors also calculated the free energy to displace a trapped water from the binding site of the protein. However, their simulation protocol resulted in convergence issues because displacing the trapped water from the binding site resulted in a cavity which must be stabilized. In their protocol, they reported sampling challenges related to re-hydration of the cavity and conformational changes in binding site to occupy the space created by the cavity (i.e., “binding site collapse”).

Building on Ge et al.’s study,¹⁷ we previously developed an MD-based separation of states protocol for RBFE calculations that efficiently handles the displacement of a trapped water in the free energy calculation.²⁰ Our previous protocol uses a thermodynamic cycle that applies a harmonic restraint on the trapped water to keep it at its putative location in the binding site, position restraints on the binding site to prevent changes in its conformation once the cavity forms, and a solvent repulsion term to maintain a dry cavity. We applied our previous protocol to five different protein-ligand complexes and calculated accurate RBFEs for ligand transformations involving trapped water displacement. Our previous protocol served as a proof-of-concept study to test restraining schemes to stabilize the cavity.

In the present work, we build on our prior work to make it substantially more efficient, taking advantage of non-equilibrium switching (NES) techniques. In particular, we developed a protocol using NES simulations to calculate the RBFE between a ligand pair, where one ligand binds to the protein with a trapped water while the other ligand displaces the water. Identifying the location of the trapped water is a prerequisite for performing the RBFE calculation using our protocol.

In the current study, we have used the restraining schemes from our previous separation of states protocol,²⁰ i.e., the harmonic restraint on the trapped water and the solvent repulsion potential. In our previous separation of states protocol, we employed position restraints on the binding site to prevent binding site collapse during the decoupling of the trapped water. The decoupling of trapped water created a cavity in the binding site, into which a bigger ligand was grown via a separate leg of the thermodynamic cycle. In contrast, in our present study, we have eliminated the need of the position restraints on the binding site by simultaneously transforming the trapped water and the ligand. Because no cavity is introduced during this simultaneous transformation, the structural integrity of the binding site is naturally preserved, obviating the need for the binding site restraints. While our current study builds upon the general framework introduced in our previous separation of states protocol, this shift to simultaneous transformation of ligand and trapped water dramatically improves efficiency by changing how binding site stability is maintained.

Furthermore, in our previous separation of states protocol, we used a combination of equilibrium free energy simulations (to apply and remove the restraints) and NES simulations (to transform the trapped water and the ligand). In contrast, our present study relies exclusively on NES simulations for all steps—including the application and removal of any necessary restraints—streamlining the computational workflow. Additionally, each of the complex and solvent leg of the RBFE calculation is performed using a single NES simulation (Figure 2). This unified NES-based approach significantly reduces the total simulation time and implementation complexity, while maintaining high accuracy and convergence across the systems tested, and making our approach more suitable for large-scale RBFE campaigns.

Figure 2.

The non-equilibrium switching (NES) simulation protocol for relative binding free energy (RBFE) calculation between ligand A (green) and B (purple). Ligand A binds to the target protein (yellow) with a trapped water (red circle) in the binding site while ligand B displaces the trapped water. A decoupled water is shown in pale red. A black cross on the trapped water indicates a harmonic restraint, and a dashed circle represents a solvent repulsion potential applied at the position of the trapped water. The NES simulation for the complex leg of the RBFE calculation is divided in three parts: NES1, NES2 and NES3. Edge HR-Remove represents removing the harmonic restraint on the trapped water and transferring the water to bulk solvent at standard concentration, and edge W-Couple represents the coupling of a non-interacting water to bulk solvent at standard volume. The NES simulation for the solvent leg of the RBFE calculation is shown in a dashed rectangle in bottom left.

The main goal of this work is to compute accurate and precise RBFE estimates between ligand pairs, in which displacement of a trapped water plays a critical role in ligand binding, using this new, more efficient framework. To achieve this, we have developed a simulation protocol that efficiently leverages distributed computational resources.

3. Methods

3.1. Conceptual Framework.

Non-equilibrium free energy calculations can be used to estimate the free energy difference between two end states. In a non-equilibrium calculation, equilibrium simulations of the both the end states (referred to as end state simulations) are performed, from which structures are extracted along the simulation trajectories. These structures are then used to initiate rapid simulations (or switches) between the end states through alchemical space, driving the system from one end state to the other, both in forward and reverse directions, via non-equilibrium pathways. To perform the rapid transitions in forward and reverse directions, the Hamiltonians of the end states are linked to an external parameter λ that transitions between two discrete values (usually 0 and 1, representing the two end states) with every timestep of the switching simulations. Because the switches change the Hamiltonian of the system in a non-equilibrium manner, we need to account for work dissipation along the alchemical paths of the switches. Using this dissipated work, we can calculate the free energy difference between the two end states by solving the Crooks fluctuation theorem^21,22 with the Bennett’s Acceptance Ratio (BAR) estimator,²³ by numerically solving

$$\sum _{i=1}^{n_f}\frac{1}{1+\frac{n_f}{n_r}{\text{e}}^{\beta \left(W_f-\Delta G\right)}}=\sum _{i=1}^{n_r}\frac{1}{1+\frac{n_r}{n_f}{\text{e}}^{-\beta \left(W_r-\Delta G\right)}}$$ (1)

where n_f and n_r are the number of switches between the end states in the forward and reverse directions, respectively, β equals $\frac{1}{k_{B}T}$, where k_B is Boltzmann constant and T is the simulation temperature, and ΔG is the free energy difference between the end states. The work along the forward and reverse directions, W_f and W_r, is calculated by accumulating the changes in the Hamiltonian as the coupling parameter λ is changed during the switches $\left(W={\int }_0^1\frac{\delta H(\lambda )}{\delta \lambda }d\lambda \right)$, where H is the Hamiltonian of the system. In an accurate NES simulation, the work distributions for the forward and reverse switches should have sufficient overlap.

Because free energy is a state function, i.e., it depends solely on the state and not on the path taken to reach the state, alchemical free energy calculations can be divided into intermediate stages. In this work, we have divided the non-equilibrium transformation of the complex leg of the RBFE calculation in three parts (Figure 2). The NES simulation in the forward direction begins from the protein-bound state of the ligand A, with the trapped water present in the binding site (stage 1). In the first NES simulation (NES1), we restrain the trapped water to its approximate location in the binding site and apply a repulsion potential on the solvent to keep it from occupying the trapped water’s binding site as the trapped water is decoupled. In the second NES simulation (NES2), we transform the ligand and simultaneously decouple the trapped water. In the third NES simulation (NES3), we remove the repulsion potential on the solvent. In the reverse direction, we start from the protein-bound state of ligand B, with a decoupled trapped water restrained at the location of the trapped water in the binding mode of the ligand A (stage 1’). In the first NES simulation (NES1 in the reverse direction), we apply a repulsive potential on the solvent at the position of the decoupled trapped water. In the second NES simulation (NES2 in the reverse direction), we transform the ligand and couple the trapped water to the system at its restraining position in the binding site. In the third NES simulation (NES3 in the reverse direction), we remove the harmonic restraint on the trapped water and the solvent repulsion potential.

For each switch in both the forward and reverse directions, we continue the trajectory of NES1 into NES2, and the trajectory of NES2 into NES3, essentially making the NES1, NES2 and NES3 transitions a single switch (referred to as NES-Total, Figure 2) between the two end states (stage 1 and stage 1’). To continue the trajectory of an NES switch, we take the coordinates and velocities of all atoms in the system from the final state of the NES switch and use them as the starting point for the simulation of the subsequent NES switch. This continuity eliminates the need for equilibrium simulation of intermediate states (stages 2 and 2’), streamlining the protocol. Analysis requires accumulating the non-equilibrium work done in a given transformation; here, we accumulate work across the three NES switches, as we discuss further below in Section 3.4.

In our NES scheme, we apply a harmonic restraint on the trapped water to ensure that it stays at its putative location in the binding site while going through the alchemical decoupling during NES2. Without restraint, the trapped water can wander off in the solvent when its interactions with the system are sufficiently weak, which would make the convergence of NES2 (and therefore, of NES-Total) very challenging. The harmonic restraint on the trapped water is also necessary to ensure that a water is present in the binding site when the ligand is transformed in NES2 in the reverse direction. Without the restraint, a decoupled trapped water would have to find its way into the binding site during the reverse NES2 switches, which would be nearly impossible. Because all water molecules are indistinguishable in the simulation of stage 1, we apply the harmonic restraint to the water molecule that is closest to the putative location of the trapped water in the binding site, in case the trapped water exchanges with solvent water in the equilibrium simulation of stage 1.

In our NES scheme, we added a solvent repulsion potential to prevent solvent water from coming close to the restraining site of the trapped water. Without the repulsion potential, solvent water could replace the trapped water in NES2 when the trapped water’s interactions with the system are sufficiently weak and the binding site is somewhat accessible. In such an event, the end states of the NES2 switches would have additional water in the binding site, making the NES-Total protocol ill-defined.

After the ligand transformation, the harmonic restraint on the decoupled trapped water is removed (edge HR-Remove) and the trapped water is coupled to bulk solvent (edge W-Couple).

For simulating the solvent leg of the RBFE calculation, ligand transformation in solvent is performed using NES simulation.

3.2. Simulated systems.

In this study, we calculated the RBFEs for eight ligand pairs, each pair involving displacement of a trapped water in their protein-ligand complexes. We selected complexes where experimental binding affinities of the ligand pairs bound to their respective target proteins are known, and the experimental studies suggest that one ligand binds to the protein with a trapped water while the other ligand displaces the trapped water. We studied one pair of ligands for each of scytalone dehydratase, β-secretease 1 (BACE1), epidermal growth factor receptor kinase (EGFRK), heat shock protein 90 (HSP 90), factor Xa, and three ligand pairs of thrombin (Figure 3).

Figure 3.

Ligand transformations for the systems studied here. In each ligand pair, the ligand on the left binds with the trapped water in its binding pose, while the ligand on the right displaces the trapped water in its binding pose. The approximate location of the trapped water is also shown relative to the ligand on the left.

3.3. Simulation Details.

3.3.1. Preparation of Structures of Protein-Ligand Complexes.

The structures of the protein-ligand complexes were downloaded from the Protein Data Bank (PDB), according to the PDB IDs listed in Table 1. Binding modes of missing ligands were generated manually, using ligand coordinates taken from complexes containing closely related ligands. Missing residues and heavy atoms in the proteins were added using PDBFixer.^24,25 Protonation states of amino acids were determined according to the experimental pH reported on the PDB webpages of the complexes, using PROPKA algorithm available on the PDB2PQR server²⁶ to estimate the most likely protonation state of each residue at that pH. The protonation states of the ligands were determined using the Chemicalize utility of ChemAxon, and are depicted in the ligand structures in Figure 3. We have provided the structures of all the protein-ligand complexes used in our simulations in the GitHub repository associated with this paper waterNES.²⁷

Table 1.

PDB IDs of protein-ligand complexes used in the RBFE calculations for the ligand transformations

Protein	Ligand	PDB ID
scytalone	C3d	5STD*28
dehydratase	C5d	3STD29
HSP 90	D13b	4FCP*30
	D15b	4FCP*30
factor Xa	IID	2BQ731
	IIE	2BQW31
BACE1	C4j	4DJW32
	C4b	4DJV32
EGFRK	W63	1M17*33
	W8	1M17*33
thrombin	B5	2ZFF34
	B1a	2ZDV34
	B1b	2ZC934
	B3a	2ZF034

^*indicates the PDB ID of the complex containing a closely related ligand

3.3.2. Preparation of GROMACS Topology Files for Protein-Ligand Complexes.

We used the OpenMM software package³⁵ to generate the force field parameters for the proteins. Force field parameters for the ligands were generated using the Open Force Field Toolkit.³⁶ Protein structures were parameterized using the AMBER 14SB force field,³⁷ and ligands were parameterized using the OpenFF-2.0.0 force field (Sage)³⁸ and the AM1-BCC charge model implemented in the OpenEye toolkit.³⁹ The TIP3P water model⁴⁰ was used to solvate our systems, and 0.2 molar Na⁺ and Cl⁻ were added to the simulation box.

Ligand pairs were simulated using a hybrid topology approach. To generate the hybrid structures and topologies of the ligand pairs, we used the PMX software package.⁴¹

We modified the topology files of our complexes to define the distinguishable trapped water, the harmonic restraint on the trapped water, and the solvent repulsion potential (see Section 3.1 for details). Details of the topology file modifications are described in our prior study.²⁰ We have also provided the topology files for all the systems studied here in our GitHub repository waterNES.²⁷

For simulation of the solvent leg of the calculation, we solvated the structure of a hybrid ligand in a box of TIP3P water,⁴⁰ and added 0.2 molar Na⁺ and Cl⁻ to the simulation box.

3.3.3. End State Simulations.

All MD simulations were run using the GROMACS 2022.1 simulation package.⁴²

Water molecules in the system, except the trapped water, were constrained according to the SETTLE⁴³ implementation in GROMACS, using the settles directive. The trapped water molecule was defined using a separate moleculetype (see Section 3.3.2 for details), and hydrogen atoms of the trapped water were constrained according to the SETTLE implementation in GROMACS, using the constraints directive. In GROMACS, the settles directive can not be defined for two molecule types (in this case, the normal water and the trapped water).

Coulomb and vdW interactions were calculated for a 10 Å cut-off radius and long-range Coulomb interactions were calculated using Particle Mesh Ewald (PME)⁴⁴ with a Fourier grid spacing of 1 Å. A time step of 2 fs was used in MD simulations.

For each system, starting from the solvated structure of the complex, we performed an energy minimization for 5000 steps using the steepest descent algorithm. Then, we equilibrated the system in an NVT ensemble for 10 ps, stabilizing the temperature of the system at 298.15 K using the stochastic dynamics integrator and an inverse friction constant of 2 ps. In the NVT equilibration, we restrained each of the heavy atoms of the protein backbone and the ligand using a harmonic position restraint with a force constant of 1000 kJ mol⁻¹ nm⁻². Finally, we performed a 100 ps equilibration of the system in an NPT ensemble, by coupling the system to the Parrinello-Rahman barostat⁴⁵ with an isotropic pressure coupling of 1 bar using a time constant of 1 ps. During the NPT equilibration as well, each of the heavy atoms of the protein backbone and the ligand was restrained using a harmonic position restraint with a force constant of 1000 kJ mol⁻¹ nm⁻². After the NPT equilibration, both the end states of the complex were simulated for 6 ns production runs in the NPT ensemble using the stochastic dynamics integrator and the Parrinello-Rahman barostat.

In the end state simulation of the protein-bound ligand B (stage 1’, Figure 2), a decoupled water was restrained at the location of the trapped water in the binding mode of the ligand A (see Section 3.3.2 for details on the restraint).

The end states for the solvent leg were simulated using the hybrid topology of the ligands. For each system, the solvated structure of the hybrid ligand, obtained from the procedure described in Section 3.3.2, was equilibrated in the NVT and NPT ensembles using the same procedure as used for the protein-ligand complexes. For both end states of the solvent leg, 6 ns production runs were generated in the NPT ensemble using the stochastic dynamics integrator and the Parrinello-Rahman barostat.

For all the systems, we simulated three independent repeats of the end state simulations of the complex and solvent legs.

We have provided the input parameter files (GROMACS mdp files) for NVT and NPT equilibrations, and the production runs of the end states in the GitHub repository waterNES.²⁷

3.3.4. NES Simulations.

We used the free energy implementation in GROMACS to simulate the NES switches. For free energy calculation in GROMACS, states A and B can be defined for the system and the Hamiltonian of the system can be coupled to an external parameter λ that switches between 0 and 1 during the course of an NES switch, driving the system from state A to state B (or from state B to state A in an NES switch in reverse direction). We defined states A and B for the harmonic restraint on the trapped water, the solvent repulsion potential, and the vdW and Coulomb parameters of the trapped water in the topology files of the complexes, according to the procedure described in Section 3.3.2. States A and B of the ligand were defined using the hybrid topology files we prepared using PMX software package (see Section 3.3.2 for details).

NES switches were simulated for all three repeats of the complex and solvent legs of all the systems. All NES switches were simulated in the NPT ensemble with same parameters as those for the end state production runs (see Section 3.3.3 for details).

From the last 4 ns of the 6 ns production runs of the end state simulations (see Section 3.3.3 for details), 100 starting structures were extracted for the NES switches. In each starting structure obtained from the end state simulation of the protein-bound ligand A, the water molecule closest to the restraining site of the trapped water was designated as the trapped water (see Sections 3.1 and 3.3.2 for details).

From the starting structures, NES1 switches were simulated for 10 ps, in which the harmonic restraint on the trapped water and the solvent repulsion potential were turned on by updating the λ value at every time step (Δλ = 0.0002). In the NES1 switches in the reverse direction, only the solvent repulsion potential was turned on, while the harmonic restraint on the trapped water was kept on during the switches.

Starting from the final structures of the NES1 switches, NES2 switches were simulated for 50 ps. In the NES2 switches, the hybrid ligand’s vdW and Coulomb parameters were transformed, and the vdW and Coulomb parameters of the trapped water were turned off, with λ updating at every time step (Δλ = 0.00004). In the NES2 switches in the reverse direction, the hybrid ligand’s vdW and Coulomb parameters were transformed, and the vdW and Coulomb parameters of the trapped water were turned on. The Gapsys soft-core potential⁴⁶ was applied to the vdW interactions to avoid singularities in the alchemical transformations. In NES2 simulations, the harmonic restraint and the solvent repulsion potential were both kept on.

Starting from the final structures of the NES2 switches, NES3 switches were simulated for 10 ps with the λ updating at every time step (Δλ = 0.0002). In the NES3 switches in the forward direction the solvent repulsion potential was turned off, while in the reverse NES3 switches, the harmonic restraint on the trapped water and the solvent repulsion potential were both turned off.

For the NES-Solvent simulations (Figure 2), 100 starting structures were extracted from the last 4 ns of the 6 ns production runs of the end states of the hybrid ligand in solvent (see Section 3.3.3 for details). From these structures, NES switches were simulated for 50 ns with the λ updating at every time step (Δλ = 0.00004), in which the vdW and Coulomb parameters of the hybrid ligand were transformed. The Gapsys soft-core potential⁴⁶ was applied to the vdW interactions to avoid singularities.

The input parameter files (GROMACS mdp files) for all the NES switches are provided in the GitHub repository waterNES.²⁷

3.4. Free Energy Calculation and Estimation of Statistical Error.

After simulating the complex and solvent legs of the RBFE calculation (Figure 2), the RBFE between a ligand pair was calculated using equation

$$\begin{gathered} \mathit{\text{RBFE}}=\Delta G_{\text{NES-Total}}+\Delta G_{\text{HR-Remove}} \\ +\Delta G_{\text{W-Couple}}-\Delta G_{\text{NES-Solvent}} \end{gathered}$$ (2)

For the free energy calculation of the NES-Total (ΔG_NES-Total) and the NES-Solvent (ΔG_NES-Solvent) simulations, we used the BAR implementation in the PMX software package.⁴¹ BAR requires the calculation of the non-equilibrium work in the forward and reverse switches (Equation 1).

We calculated the non-equilibrium work for NES-Total by adding the work values of NES1, NES2 and NES3, i.e., we added the work values of i^th NES1, i^th NES2 and i^th NES3 switches for calculating the work of i^th NES-Total switch.

Edge HR-Remove computes the free energy to remove the harmonic restraint on the decoupled trapped water and transfer the water to bulk solvent at the standard concentration (Figure 2). Since the water is non-interacting, changing its environment does not change the free energy and thus the overall free energy of this edge (ΔG_HR-Remove) can be calculated analytically^47–49 using equation

$$\Delta G_{\text{HR-Remove}}=RT\text{log}\left(C^o{\left(\frac{2\pi RT}{k}\right)}^{3/2}\right)$$ (3)

where R is the molar gas constant, T is the simulation temperature, C^o is the standard concentration of bulk solvent (55.5 molar), and k is the force constant of the harmonic restraint. A detailed derivation of the Equation 3 is described in our previous study.²⁰ In this study, we used a harmonic force constant (k) of 0.5 kcal mol⁻¹ Å⁻² to restrain the trapped water, for which ΔG_HR-Remove is −0.229 kcal mol⁻¹.

The free energy to couple a decoupled water to bulk solvent (edge W-Couple) is independent of the protein-ligand complex, and depends only on the water model used. This free energy is also independent of simulation box size for sufficiently large boxes.⁵⁰ In the thermodynamic cycle employed here (Figure 2), the decoupled water molecule is taken to the correct standard concentration by edge HR-Remove. Thus, in principle, edge W-Couple should be computed at a simulation box volume corresponding to the standard volume to complete the thermodynamic cycle. However, given that this free energy is independent of simulation box volume,⁵⁰ we simply used our free energy value precomputed for a box size of approximately 125 nm³. For the TIP3P water model, used in this study, we calculated the free energy for edge W-Couple to be −6.12 ± 0.03 kcal mol⁻¹, which agrees well with previous computational⁵¹ as well as experimental⁵² studies.

For each system studied here, we simulated three independent repeats of the complex and solvent legs (i.e., NES-Total and NES-Solvent in Figure 2) and calculated RBFE for each repeat using Equation 2. The final RBFE and its statistical error were taken as the mean and the standard deviation, respectively, of the RBFEs across the three repeats.

4. Results

In this work, we calculated the RBFEs for eight ligand pairs (Figure 3) using our NES-Total simulation protocol (Figure 2). In each of the ligand pairs studied here, one ligand binds to the target protein with a trapped water while the other ligand displaces the water. In the following sections, we discuss the accuracy of our calculations and compare our RBFE values with experimental and literature values.

4.1. Our NES Protocol Resulted in Accurate Estimates of RBFEs for Ligand Pairs.

We ran three independent repeats of the NES-Total and NES-Solvent simulations (Figure 2) for each of the eight ligand transformations studied here (Figure 3). In every repeat of the NES-Total and NES-Solvent simulations, we found sufficient overlap between the forward and reverse work distributions across all the eight systems.

Our calculations resulted in RBFEs for all the ligand pairs within 0.6 kcal mol⁻¹ of the experimental RBFEs (Table 2), with the exception of the BACE 1 and factor Xa systems, where our values differed by approximately 1.1 kcal mol⁻¹ from the experimental values. In the BACE1 system, we observed that the large error in our RBFE estimate resulted from sampling challenges related to flexibility of a BACE1 ligand; we have discussed this system in detail in Section 4.3. In our simulation of the factor Xa system, we did not find any evident source for large error. A possible reason for the large error, however, can be the large - R group change between the factor Xa ligands, which is the largest among all the ligand pairs studied here (Figure 3). The root mean square error (RMSE) in our RBFE estimates across all the eight ligand transformations, relative to the experimental data, was 0.6 kcal mol⁻¹.

Table 2.

Calculated and experimental relative binding free energies (RBFEs) (in kcal mol⁻¹) for ligand pairs studied here

System	Ligand pair	${\text{RBFE}}_{\mathit{\text{calc}}.}^{*}$	RBFEexp.	Error
SDa	C3d-C5d	−2.27 ± 0.3	−1.9829	0.3 ± 0.3
BACE1	C4j-C4b	0.46 ± 0.7	−0.6132	1.07 ± 0.7
EGFRK	W63-W8	0.76 ± 0.2	0.6057	0.2 ± 0.2
HSP 90	D13b-D15b	−1.97 ± 0.1	−1.4130	0.6 ± 0.1
FXab	IID-IIE	−3.07 ± 0.4	−1.9831	1.1 ± 0.4
Thrombin	B5-B1a	−0.58 ± 0.1	−0.6134	0.03 ± 0.1
	B5-B1b	−2.20 ± 0.1	−2.4534	0.2 ± 0.1
	B5-B3a	−1.51 ± 0.2	−1.3234	0.3 ± 0.2

^*statistical errors are calculated from three simulation repeats. See Section 3.4 for details.

^aSD = Scytalone Dehydratase;

^bFXa = Factor Xa

With the exception of the BACE1 system (see Section 4.3 for details), the statistical errors in our RBFE estimates remained below 0.4 kcal mol⁻¹.

We studied the effect of longer switching times on the statistical errors in our RBFE estimates. In our NES-Total calculations, we simulated NES1, NES2 and NES3 switches for 10 ps, 50 ps and 10 ps (see Section 3.3.4 for details), respectively, resulting in a total simulation time of 70 ps for one switch between the end states of the complex leg of the calculation (stages 1 and 1’ in Figure 2). We simulated the NES-Solvent switches for 50 ps. To study the effect of longer switching times on the statistical errors in our RBFE estimates, we also simulated a longer version of the protocol, in which we simulated the NES1, NES2 and NES3 switches for 50 ps, 500 ps and 50 ps, respectively, and the NES-Solvent switches for 500 ps. The longer switching times did not lead to any significant improvement in the statistical errors in our RBFE estimates (Table S1). The RBFE estimates with the longer switching times were within 0.2 kcal mol⁻¹ of the estimates obtained using our original shorter simulation protocol.

4.2. We Compared Our RBFE Estimates with Literature Values.

Previously, we have studied four of the eight ligand transformations studied here with a separation of states protocol.²⁰ For those four ligand transformations, i.e., C3d-C5d, C4j-C4b, IID-IIE, and B5-B1a (Figure 3), our separation of states protocol resulted in RBFE estimates of −1.59 ± 0.5 kcal mol⁻¹, 0.32 ± 0.6 kcal mol⁻¹, −2.46 ± 0.5 kcal mol⁻¹, and −0.97 ± 0.4 kcal mol⁻¹, respectively. For the four ligand transformations, our RBFE values calculated here (Table 2) are within statistical error of the RBFEs calculated in our previous separation of states study.

In this study, we aim to develop a method that separates the sampling of the trapped waters from the free energy calculation of ligand transformation. This naturally led us to compare our RBFE values with approaches that combine enhanced sampling of the trapped water with the free energy calculation. By comparing our RBFE estimates with those from the previous studies using the enhanced sampling approaches, we aim to validate our approach and identify potential biases in it. The goal of these comparisons is to gain a comprehensive understanding of possible limitations of our approach. Many of the systems studied here were also previously studied using enhanced sampling approaches such as hybrid MC/MD¹¹ and GCMC/FEP+.^14,53 While comparing our RBFEs with those reported in these previous studies, we acknowledge that differences in simulation methodologies such as the software used, differences in force field parameters and water model, and the exact simulation details across studies can result in discrepancies. A previous benchmarking study by Hahn et al.,⁵⁴ investigating the accuracies of the force field on RBFE estimates on a set of 599 ligands across 22 target proteins, reported that force fields Openff-2.0 (used here), GAFF (used in the hybrid MC/MD study¹¹) and OPLS3e/FEP+ (used in the GCMC/FEP+ study¹⁴) resulted in RMSEs of approximately 1.7 kcal mol⁻¹ relative to experiments. Our RBFE estimates for most of the systems studied here are within 0.6 kcal mol⁻¹ from experiments, i.e., well below the RMSE reported by Hahn et al.. Therefore, even in the worst-case scenario, i.e., in which our errors relative to experiments are in the exact opposite direction than the errors in the previous estimates (relative to experiments), we expect our estimates to be within 2.3 kcal mol⁻¹ of the RBFEs from the enhanced sampling studies.

The previous study of Ben-Shalom et al.¹¹ investigated ligand transformations in systems with trapped waters using a hybrid MC/MD approach. In the previous study, the authors inserted/deleted water molecules in the protein binding site using MC, while performing ligand transformation using MD. Five of the eight systems studied in our work—HSP 90, scytalone dehydratase, and the three thrombin systems—were also investigated in their study. For all systems except HSP 90, our RBFE values lie within 1 kcal mol⁻¹ of the values reported in the previous study. In their study, the authors also evaluated a variant of their method, in which the MC water sampling was only performed in the initial equilibration of the systems. When comparing our RBFE values with those calculated using the variant, we observe even closer agreement, with differences within 0.5 kcal mol⁻¹. For the HSP 90 system, the authors reported challenges in correctly estimating the occupancies of water molecules in the binding site. For this particular system, our RBFE estimate differs by more than 1.5 kcal mol⁻¹ from their value, with our value being more accurate (error of 0.6 kcal mol⁻¹ relative to experiment, Table 2).

Wahl et al.¹⁴ conducted ligand transformation free energy perturbation (FEP) calculations with replica exchange solute tempering (REST) in FEP+, following equilibration of the trapped waters in the protein-ligand complexes using GCMC. Their data suggest that pre-equilibrating trapped waters with GCMC did not result in better accuracy in RBFE estimates, when compared to calculations where the trapped waters were not equilibrated with GCMC prior to the FEP. Furthermore, they observed discrepancies in their RBFE values (i.e., hysteresis) for the same system depending on whether or not the trapped waters were included in the initial structure of the simulation.

We compared our RBFEs for the scytalone dehydratase, HSP 90 and the BACE1 ligand pairs with those reported previously by Wahl et al.¹⁴ For the scytalone dehydratase and HSP 90 systems, our RBFE estimates lie within approximately 1 kcal mol⁻¹ of the values reported by Wahl et al.. In contrast, for the BACE1 system, both our and the previous studies encountered sampling challenges related to protein motion and ligand flexibility (see Section 4.3 for details). Consequently, our RBFE estimate for BACE1 differs by more than 2 kcal mol⁻¹ from that of Wahl et al..

A follow-up study by Ross et al.⁵³ revisited the systems investigated by Wahl et al.¹⁴ and recommended integrating GCMC directly with FEP calculations, rather than using GCMC solely for initial water equilibration. Our RBFE value for the HSP 90 system agrees closely with the RBFEs reported by Ross et al., differing by 0.5 kcal mol⁻¹. However, for the scytalone dehydratase system, our RBFE estimate differs by more than 2.5 kcal mol⁻¹ from their reported values (with and without the trapped water in the initial structure). Notably, our RBFE estimate for the scytalone dehydratase system (−2.27±0.3 kcal mol⁻¹) is closer to the experimental RBFE of −1.98 kcal mol⁻¹, compared to the values reported by Ross et al. for this system (0.37±0.17 kcal mol⁻¹ with trapped water and 1.04 ± 0.34 kcal mol⁻¹ without the trapped water in the initial structure). Notably, the hysteresis in their RBFE estimates for the scytalone dehydratase system (0.67 ± 0.51 kcal mol⁻¹) was larger than that reported by Wahl et al.¹⁴ (0.40 ± 0.27 kcal mol⁻¹).

Our comparisons with these two previous GCMC-based studies indicate that our simulation protocol yields improved convergence and agreement with the experimental data, particularly when compared to instances that incorporated GCMC for sampling trapped waters in conjunction with the FEP calculations. While enhanced sampling methods have the advantage of not requiring prior knowledge of the locations of the trapped waters, our protocol offers better convergence and accuracy when these locations are already known or easily identifiable.

Some of the systems investigated here have also been studied previously by various other studies.^19,55,56 However, methodological differences—such as restraining schemes, sampling techniques, and differences in the underlying conceptual frameworks—make direct comparison with these studies challenging.

In summary, we found that our method yields RBFE estimates within 1 kcal mol⁻¹ of the values reported by previous studies combining enhanced sampling methods for trapped water sampling with free energy calculations for ligand transformations, except for one system where the difference in our RBFE estimate was more than 2.5 kcal mol⁻¹ from a previous GCMC-based study. Notably, our protocol resulted in well-converged RBFE estimates and better agreement with experimental data compared to the previous GCMC-based studies, suggesting advantages in treating water sampling separately from ligand transformation in free energy calculations.

4.3. Conformational Flexibility of Ligand Affects the RBFE Estimate.

In the RBFE estimation between the BACE1 ligand pair, we observed a significant statistical error of 0.7 kcal mol⁻¹, which is the largest among all the ligand pairs studied here. We calculated the statistical error in the RBFE estimate by taking standard deviation of the RBFE values calculated from three independent simulation repeats (see Section 3.4 for details).

Previous studies have suggested that insufficient sampling of the opening and closing of the 10s loop, as well as the motion of the flap region in BACE1, can have a detrimental effect on RBFE estimates for BACE1 systems.^58,59 Therefore, we extended the end state simulations for the BACE1 system to 20 ns, instead of the usual 6 ns (see Section 3.3.3 for details on end state simulations). In the end state simulations, we only observed the motion of the flap region, with no opening and closing of the 10s loop.

However, the RBFE estimates in our three simulation repeats were highly dependent on the conformation of the C4j ligand in the end state simulations of the complex. In the binding mode of the BACE1 ligand C4j, the nitrogen of the pyridine ring of the ligand forms a hydrogen bond with the trapped water. While the binding site of BACE1 can accommodate a rotation of the pyridine ring of the ligand C4j, such a rotation of the ring breaks its hydrogen bond with the trapped water. In one simulation repeat, we observed rotation of the pyridine ring early in the end state simulation, and most of the subsequent NES switches for this simulation were initiated from the rotated pyridine structures. This simulation repeat resulted in an RBFE estimate of 1.32 ± 0.2 kcal mol⁻¹. In another simulation repeat, where the rotation of the pyridine occurred relatively later and some NES switches were started from the correct binding pose of the ligand, the RBFE estimate was −0.38 ± 0.1 kcal mol⁻¹. The large difference in the RBFE estimates between the simulation repeats explains the significant statistical error in the RBFE estimate between the BACE1 ligand pair. The rotation of the pyridine ring of the ligand C4j has been previously reported by us²⁰ and another study.¹⁴ While the simulation of the correct binding pose is not the primary focus of this work, it remains critical for achieving accurate results. In our simulation repeat where the ligand adopted the correct binding pose, our protocol successfully calculated the correct RBFE, which suggests that, when the binding pose is correctly simulated, the protocol achieves high accuracy. This provides strong evidence for the overall accuracy of our approach.

5. Discussion

In this study, we focused on computing relative binding free energies in systems where ligand transformation displaces a single trapped water, introducing a more efficient protocol to handle slow water rearrangement in such cases. Our protocol can be extended to systems involving multiple displaced waters. Our efforts are directed towards separating the challenges of sampling of trapped waters from the challenges of free energy calculation of ligand transformation. Our study specifically addresses the free energy calculation and requires prior knowledge of the locations of the trapped waters in the binding site, which must be displaced when the ligand is transformed. We are conducting a separate study to predict the number and locations of the trapped waters for each ligand involved in the free energy calculation. Alternatively, other methods, such as enhanced sampling simulations (e.g., GCMC/MD and hybrid MC/MD), solvation site analysis techniques like 3D-RISM,¹⁸ or literature searches through previous computational and experimental studies on the systems under investigation, can provide useful information to determine the number and locations of the trapped waters. Data from earlier studies^11,53 suggests that enhanced sampling methods, such as GCMC/MD and hybrid MC/MD, are most advantageous when used for equilibrating the systems to determine the locations of trapped waters. Equilibrating the systems with enhanced sampling methods, prior to performing free energy calculations, is computationally far more efficient than directly combining these enhanced sampling methods with the ligand transformation.

The only true limitation of our protocol may be related to the efficiency of the NES simulations. NES simulations are highly sensitive to the local environment of the alchemical transformation being performed. For an efficient non-equilibrium free energy calculation, the conformations sampled by the local environment across the end states should be very similar, i.e., the conformations should have very high overlap in phase space, because the short NES switches cannot effectively bridge end states that sample distant conformations of the local environment. Large conformational changes in the protein and/or the ligand immediately adjacent to the region being alchemically transformed can be nearly impossible to sample in short NES switches, which can adversely affect the overlap of the non-equilibrium work distributions in the forward and reverse directions, ultimately leading to poor convergence of the free energy estimate. In one of the systems studied here, we observed that the conformation of a ligand in its protein-bound state changes because of a ring rotation in the ligand, ultimately pushing the conformations sampled by the end states too far apart in phase space. In two independent simulation repeats of this system, only one of which simulated the correct orientation of the ring in the protein-bound conformation of the ligand, we observed a difference of 1.7 kcal mol⁻¹ in the RBFE estimates from the two simulation repeats. A possible solution for such cases, where the sampled ligand conformations across the end states are too distant in phase space, can be to apply restraints on the rotatable dihedrals of the ligands in the end state simulations. The free energy to apply/remove these restraints would then ultimately be included in the RBFE estimate.

6. Conclusions

In this study, we have developed a non-equilibrium simulation-based protocol to calculate RBFEs between ligand pairs, where one ligand binds to the protein with a trapped water while the other ligand displaces the water. We calculated the RBFEs for eight such ligand pairs and our calculations resulted in RBFEs within 1.1 kcal mol⁻¹ of the experimental RBFEs. The statistical errors in our RBFE estimates were 0.4 kcal mol⁻¹ or better for most systems. Overall, our simulation protocol achieved an RMSE of 0.6 kcal mol⁻¹.

Our simulation protocol relies on prior knowledge of the locations of the trapped waters for each ligand in the ligand pair; this knowledge is then leveraged to efficiently compute the RBFE between the ligands– eliminating the need for long simulations to achieve convergence in water sampling during ligand transformation.

We also compared our RBFE values with those calculated by previous studies combining enhanced sampling of trapped water directly with free energy calculation of ligand transformation (hybrid MC/MD and GCMC/MD). We found that our RBFE estimates lie within 1 kcal mol⁻¹ of the previously calculated values for most cases. Notably, our approach consistently produced well-converged RBFE estimates across all systems, in contrast to the lack of convergence observed in previous GCMC-based methodologies at least for some cases.

The development of this protocol progresses our idea that, in systems with trapped waters, separating the sampling of the trapped waters from the free energy calculation of ligand transformation might be advantageous compared to performing both simultaneously. Performing both simultaneously effectively mixes water sampling challenges into the binding free energy calculation, whereas separating them allows the two problems to be solved individually/separately. In this study, we have presented an effective and computationally inexpensive method that can calculate RBFEs for such systems with appropriate treatment of the trapped waters involved by efficiently utilizing distributed computational resources.

Supplementary Material

Supporting Information Available

We provide the input files (mdp files, GROMACS topology files, and PDB structures of the protein-ligand complexes) for running the NES-Total and NES-Solvent simulations in the GitHub repository waterNES (https://github.com/MobleyLab/waterNES).

RBFE estimates for our systems using a 700 ps simulation time for the NES-Total simulation protocol is provided in Table S1 in the Supporting Document.

Figure 1.

Relative binding free energy (RBFE) calculation between two ligands, where one ligand (ligand A, green) binds to the protein (yellow) with a trapped water (red circle) while the other ligand (ligand B, purple) displaces the water. The vertical processes represent absolute binding free energy (ABFE) calculations of the bindings of the ligands to the protein. The horizontal processes represent the transformations of the ligands in solvent (solvent leg of the RBFE calculation, $\Delta G_{A->B}^{\mathit{\text{Solvent}}}$) and in the protein’s binding site (complex leg of the RBFE calculation, $\Delta G_{A\to B}^{\mathit{\text{Complex}},\mathit{\text{Water}}}$), the two components of the RBFE calculation.