Sampling the Grand Canonical Ensemble with Multisite λ Dynamics

Abstract

Molecular simulation in the grand canonical ensemble is widely used to study a diverse range of systems and processes, such as water networks in biological macromolecules, drug binding, and the adsorption of molecules at an interface. Here, we develop grand canonical multisite lambda dynamics (GC-MSλD) to sample fluctuations in molecule number by coupling the molecules of interest to a dynamic λ variable. The chemical potential, set as a λ-dependent energetic bias, is used to control the number of molecules. We anticipate that GC-MSλD may equilibrate faster and with less computational overhead than some GCMC/MD algorithms. We demonstrate the use of the GC-MSλD framework to control the number of molecules in a box of TIP3P water. Next, we apply the methodology to sample crystallographic water occupancies in a protein cavity and to compute protein-ligand binding free energies involving water displacement.

Graphical Abstract

Computer simulation techniques such as Monte Carlo (MC) and molecular dynamics (MD) offer atomistic insight into studying the molecular compositions of condensed phase systems. However, sampling these molecular compositions may be kinetically bottlenecked by energetic barriers in conventional NPT and NV T simulations. To overcome these challenges, grand canonical Monte Carlo (GCMC) sampling is typically employed, allowing the number of molecules to fluctuate during the simulation via a series of molecule insertion and deletion moves at a given chemical potential μ. Applications of GCMC include—but are not limited to—investigating water occupancies in biomolecules,^1–3 drug binding,^4,5 and adsorption in surfaces and pores.^6–8 GCMC’s acceptance probabilities are typically low due to steric clashes, but can be increased with cavity biasing^9,10 or by generating nonequilibrium candidate configurations.¹¹ Additionally, configurational sampling in GCMC can be further improved by simulated annealing of the chemical potential,³ oscillating the chemical potential about a target value,¹² or doing Hamiltonian replica exchange across different chemical potentials.¹³

Grand canonical molecular dynamics (GCMD), first pioneered by Çağın and Pettitt,^14,15 and later developed in additional studies,^16–19 is an attractive alternative to GCMC that utilizes an extended Hamiltonian to control the number of molecules in a simulation. In this formulation, a molecule is coupled to a continuous dynamical variable λ that scales the molecule’s interactions with its environment, ranging from a non-interacting state (λ=0) to a fully interacting state (λ = 1). The λ variable—with its own potential energy and mass—is propagated as an additional degree of freedom to enable smooth molecule insertions and deletions, potentially leading to more efficient sampling than conventional Monte Carlo. The most common implementation of extended Hamiltonian Grand Canonical methods involves simulating until λ’s value becomes greater than 0.99 or less than 0.01. When λ > 0.99, the λ-coupled molecule becomes a ”real” molecule (no longer coupled to λ), and a new λ-coupled molecule is then inserted into the system. When λ < 0.01, the λ-coupled molecule is deleted from the system, and a different molecule is then selected to be coupled to λ. Although extended Hamiltonian Grand Canonical methods are promising, they still remain largely unexplored and are not widely used today—likely owing to higher implementation complexity relative to GCMC.

In the spirit of GCMD, we develop Grand Canonical Multisite λ Dynamics (GC-MSλD), which adapts multisite λ dynamics (MSλD) for grand canonical sampling. Although GC-MSλD is based on an extended Hamiltonian, it differs from existing GCMD methods in two aspects. First, GC-MSλD has a material reservoir containing a pre-defined number of molecules that are each coupled to a respective λ variable. This reservoir defines the region where molecules will be inserted or deleted. By propagating multiple λ variables during dynamics, GC-MSλD simultaneously evaluates multiple insertion and deletion ”moves” within the reservoir region. Second, unlike GCMD, the topology of the system does not change in GC-MSλD. A ”deleted” molecule is fully decoupled from the system (λ < 0.01) but is not actually removed from the topology, and likewise, an ”inserted” molecule is simply a molecule being fully coupled to the system (λ > 0.99) from an ideal gas state. Thus, the maximum number of molecules that can be ”inserted” or ”deleted” is bound by the pre-defined number of molecules in the material reservoir. GC-MSλD’s material reservoir is effectively infinite as long as the reservoir size is greater than the average number of molecules that can be inserted into the environment. In the event that the environment can hold more molecules than the reservoir provides, one can select an additional molecule to be λ-scaled or add additional molecules into the reservoir, as is done in GCMD. Like existing GCMD methods, the chemical potential is set as an energetic bias in λ-space.

Throughout this work, we will use GC-MSλD to insert and delete water molecules. We first assess the GC-MSλD framework with a TIP3P²⁰ water box, where we demonstrate the ability to control the number of water molecules and show that it recaptitulates thermodynamic quantities consistent with NPT simulations. Next, GC-MSλD is applied to two biophysical test cases: 1). sampling crystallographic water occupancies in the Bovine pancreatic trypsin inhibitor (BPTI) and 2). computing relative binding free energies between heat shock protein 90 (HSP90) inhibitors that differ in bound state water occupancies.

Consistent with many previous GCMC studies, the Adams value B—rather than the chemical potential—will be used throughout most of the text:^2,4,21

$$B=\beta \mu +\text{ln}(\frac{V_{\mathit{\text{resv}}}}{V°})$$ (1)

where β = (k_BT)⁻¹, μ is the excess chemical potential of a TIP3P water molecule, V_resv is the volume of the GC-MSλD reservoir, and V ° is the standard state volume of water. The Adams value is convenient to use because it explicitly sets the density of the reservoir. In GC-MSλD, the $\text{ln}(\frac{V_{res\upsilon }}{V°})$ term is encoded by the density of λ-coupled molecules in the reservoir and should match bulk density. We add a correction factor of $\text{ln}(\frac{V_{res\upsilon }}{V°})-\text{ln}(N_{res\upsilon })$, with N_resv being the number of λ-scaled molecules, in cases where the density of the reservoir does not match bulk density. Throughout this work, −6.18 kcal/mol was used for μ and 29.78 ų was used for V ° (see the Methods section for details on computing μ and V °).

As a test of our implementation, we demonstrate that GC-MSλD recapitulates important NPT thermodynamic quantities for a box of TIP3P water molecules. The water-box simulated in GC-MSλD has a size of 38.87 Å with 1546 real water molecules and 864 λ-coupled water molecules, which allows the system to have a density ranging between 0.79 – 1.23 g/mL (see the Methods section for details on preparing the water boxes for the GC-MSλD and NPT simulations). The reservoir was defined to be the entire box. The corresponding Adams value used was −2.84 after correcting for the density of the reservoir region. As seen in Figure 1A, GC-MSλD initially starts at a density of 0.79 g/mL but recovers the NPT density and maintains it throughout the simulation. Owing to the ability of simultaneously evaluating multiple molecule insertions and deletions, GC-MSλD equilibrates to the NPT density within the first nanosecond of simulation, which is much faster than existing GCMC/MD implementations that can take upwards of five nanoseconds to equilibrate the density of a similarly-sized TIP3P water box.²² As an additional test of self-consistency, the water-water radial distribution function computed from the GC-MSλD simulation is consistent with the NPT radial distribution function (Figure 1B), demonstrating that the GC-MSλD water box is structurally equivalent to the NPT system.

Figure 1:

A). GC-MSλD water box density compared against the NPT density across multiple Adams value. B). Comparison of water-water radial distribution function from NPT and GC-MSλD.

Furthermore, we demonstrate GC-MSλD’s ability to precisely control the number of water molecules in the system via titration of the Adams value (Figure 2). At an Adams value of −2.84 (where μ = μ_bulk), the λ-coupled water molecules have a fractional occupancy of 0.492, which is in close agreement with the expected 0.50 occupancy in the case where the chemical potential is perfectly calibrated. The discrepancy in the occupancy, which accounts for ~0.03 kcal/mol, is well within the margin of error of our computed μ value.

Figure 2:

A). Fraction of λ-scaled water molecules as a function of the Adams value. B). Density of the GC-MSλD water box as a function of the Adams value.

Finally, we compared the GC-MSλD-computed relative hydration free energy of benzene and phenol against the NPT-computed value. In this test, conventional λ-dynamics simulations were used to compute the ΔΔG_hydr in the NPT and in the GC-MSλD water boxes. The computed ΔΔG_hydr from GC-MSλD is 4.11 ± 0.01 kcal/mol, which is in good agreement with the NPT value of 4.22±0.01 kcal/mol. While the standard deviations between the two ΔΔG_hydr values do not overlap, an error of 0.11 ± 0.01 kcal/mol is consistent with similar hydration free energy analyses reported in past GCMC studies.^12,23

After verifying that the thermodynamic quantities computed from GC-MSλD are in good agreement with NPT simulations, we compare the water occupancies of the Bovine pancreatic trypsin inhibitor (BPTI) sampled by GC-MSλD to the occupancies seen in crystallography and NPT simulations of the system. BPTI contains a cavity stabilized by three well-resolved and tightly bound water molecules, making it a popular system for bench-marking grand canonical sampling methodologies.^13,24,25 As shown in Figure 3, GC-MSλD recaptitulates the three crystallographic water occupancies in the cavity. The three bound water molecules from GC-MSλD have an average RMSD of 0.36 ± 0.16 Å with respect to the crystal water positions, which is in excellent agreement with the NPT average RMSD of 0.33±0.16 Å and with the RMSDs reported in previous GCMC studies.^22,25 Additionally, since the reservoir volume used in the GC-MSλD simulation extends beyond the cavity of BPTI, two additional water molecules are readily sampled near the cavity entrance (see the two water molecules at the bottom in Figure 3). The first water molecule, stabilized by hydrogen bonds to a glutamic acid and backbone amines, is found across multiple crystal structures of BPTI (including the crystal structure used in this study).^26–28 The second water molecule—which acts as an interaction bridge between the first water molecule and one of the interior cavity water molecules—is reported in one other crystal structure.²⁸

Figure 3:

λ-scaled waters from GC-MSλD (blue sticks) with λ > 0.99 superimposed on top of 5PTI crystal structure waters (magenta sticks). The GC-MSλD water second from the bottom is not found in 5PTI, but is present in 6PTI.

When comparing against NPT, the average number of water molecules sampled in the GC-MSλD reservoir region is 4.17 ± 0.85 which is in agreement with the NPT average of 4.20 ± 0.83 water molecules observed in the same volume (Figures 4A and 4B).

Figure 4:

A). Number of physical λ-scaled water molecules in the reservoir region for the BPTI GC-MSλD production run. B). Number of water molecules in the BPTI NPT production run. The waters are counted within the same volume as the GC-MSλD reservoir region.

While GC-MSλD does not necessarily demonstrate an advantage over the NPT simulation in sampling water molecules for this system, this is expected since the cavity of BPTI is readily accessible by bulk water in conventional MD timescales. These results, however, further validate GC-MSλD’s ability to recaptitulate NPT and crystallographic water occupancies.

In the final biophysical test case, we apply GC-MSλD to compute binding free energies involving a change in water occupancy. Protein-ligand binding free energy calculations are well-developed and routinely done in molecular simulation, but can struggle to compute accurate binding free energies when the end states differ in water occupancies. There are multiple existing approaches to hydrate a binding site for free energy calculations, such as using GCMC/MD,²³ decoupling the ligand in a manner that is conducive for binding site rehydration,²⁹ or applying metadynamics-like biases.³⁰ For this application, we turn to two HSP90 inhibitors studied by Woodhead et al that differ in the number of bound water molecules.³¹ The first inhibitor—referred to as ”ligand 1” from the original study—contains a methanol functional group that interacts with a nearby aspartic acid via two bridging water molecules. The second inhibitor—referred to as ”ligand 2”—has a methoxy group in place of the methanol group that displaces both water molecules, effectively breaking the favorable interaction bridge (Figure 5). As a result, the ΔΔG_bind to go from ligand 1 to ligand 2 is 1.97 kcal/mol, as determined by cell assay measurements.³¹

Figure 5:

Frames sampled from the GC-MSλD HSP90 simulation superimposed on top of the 2xab crystal structure (ligand 1 bound). The crystal structure ligand is shown in a magenta-colored line representation, and the crystal water molecules are shown in a magenta-colored stick representation. The blue sticks are λ-scaled water molecules (λ > 0.99). A). Physical frame of ligand 1. B). Physical frame of ligand 2. One of the crystal water molecules is removed for clarity.

This system is a particularly good demonstration for applying GC-MSλD to protein-ligand free energy calculations because, as was shown in a previous GCMC study by Ross et al,²³ the ΔΔG has a sign flip (i.e., becomes favorable) when the transformation from ligand 1 to ligand 2 is computed without a hydrated binding site. We ran two calculations: the first being a conventional NPT MSλD simulation with the initial binding site waters deleted (referred to as the ”dry” MSλD run), and the second being an MSλD at constant volume with the binding site hydrated via a GC-MSλD reservoir (referred to as the ”wet” MSλD run). The transformation from ligand 1 to ligand 2 in the dry MSλD run is ΔΔG_bind = −1.18±0.04 kcal/mol, whereas the wet MSλD simulation computed ΔΔG_bind = 5.20 ± 0.05 kcal/mol. Although the unsigned experimental error is ~ 3 kcal/mol for both the dry and wet runs, the presence of the GC-MSλD reservoir recaptitulates the correct direction of the binding free energy change. As expected, the number of water molecules in the reservoir correlates with the physical ligand state, with an average of 1.75 water molecules appearing when ligand 1’s physical end state is sampled and an average of 0.87 water molecules appearing when ligand 2 is sampled (Figure S2). GC-MSλD correctly predicts the positions of the two water molecules in the ligand 1 ensemble, but ligand 2’s bulkier methoxy group is predicted to only displace one of the two water molecules (Figure 5). Although inconsistent with ligand 2’s crystal structure, the GC-MSλD result is physically plausible and is consistent with Ross et al’s GCMC results that also predicted the displacement of only one water molecule.²³

As insertion and deletion moves are sampled along a continuous λ coordinate, one concern of the GC-MSλD framework is that the presence of non-physical molecules (i.e., λ ∈ (0.01,0.99)) may perturb the grand canonical distribution. However, the effects of sampling non-physical molecules are limited as evidenced by the close agreement between the GC-MSλD and NPT experiments. The λ-scaled water molecules sample a physical end state (λ < 0.01 or λ > 0.99) more than 85% of the time in the water box GC-MSλD simulations and more than 95% in the BPTI and HSP90 simulations (Figure S1). This targeted sampling of the physical state is attributed to the current implementation of multisite λ-dynamics that utilizes a coordinate transform on λ to suppress the sampling of non-physical states.³² Further suppression of sampling non-physical states can be achieved by using a recently developed bias.³³

GC-MSλD simulations can be slower than conventional NPT simulations because the additional λ degree of freedom has to be integrated at every time step. However, at least in the case of binding free energy calculations, complementing MSλD simulations with a GC-MSλD reservoir is not expected to introduce an unreasonable slowdown. For example, a spherical reservoir with a 5.0 Å radius—a reasonable volume to partially hydrate a binding site—needs approximately ~ 17 water molecules (or 34 λ variables) to match bulk density and is within the normal range of λ variables used in conventional MSλD runs.^33–35 Relative to NPT simulations, the current implementation of GC-MSλD is about 25% slower for the water box system and 7% slower for BPTI. The computational overhead introduced by GC-MSλD is significantly less than that of GCMC/MD algorithms reported in previous literature.²²

To compare the efficiency of the GC-MSλD to GCMC, we use the transition rate in λ-space as an analogous metric to GCMC’s acceptance probabilities. The λ transition rate is defined as the number of times a λ-scaled water molecule makes a full transition from one end state to the opposite end state per unit time and can be interpreted as the number of accepted ”insertions” and ”deletions” moves. Based on this interpretation, we report the transition rate as the sum across all of the λ-scaled water molecules, rather than the average. By converting GCMC’s acceptance probability to the number of acceptances per unit time, the efficiency between GC-MSλD and GCMC algorithms can be directly compared. The λ-scaled water transition rates are 695.93, 0.41, and 2.28 per nanosecond for the water box, BPTI, and HSP90, respectively. For reference, the GCMC/MD Python module implemented by Samways et al²² had an average of 50 acceptances per ns for a similarly-sized water box system and 5 acceptances per ns for the BPTI hydration cavity. It is worth noting that the results reported from Samways et al are from a ”baseline” GCMC/MD algorithm (i.e., no enhanced sampling, cavity biasing, etc.), allowing adequate grounds for comparison against GC-MSλD. While all of the test cases shown in this study had converged water sampling, GC-MSλD’s efficiency can be improved by adding biases to lower the barriers in λ-space, such as adaptive landscape flattening or metadynamics.^36,37 However, considering that the perturbation of a water molecule is relatively small, we show that a quadratic bias with a moderate barrier height of 4.0 kcal/mol is enough to significantly improve the λ transition rate by nearly two orders of magnitude in BPTI (Figure S3). We do not anticipate the need to add a λ bias in a majority of the applications; however, we have observed that the presence of the GC-MSλD water reservoir reduces the λ transition rates of the HSP90 ligands from 35.80±1.12 to 2.61±0.39 per nanosecond. This implies that the GC-MSλD reservoir, when run without λ biases on the reservoir molecules, may introduce a slow degree of freedom that can hinder ligand transition rates in binding free energy calculations. Although the reduced ligand transition rates did not affect the convergence of the GC-MSλD HSP90 simulations (Figure S4), we advise users to monitor ligand transition rates—and to add appropriate biases to the λ-scaled reservoir molecules if necessary—when supplementing MSλD free energy binding calculations with GC-MSλD. As conventional MSλD is unlikely to sample the correct bound water occupancies within normal simulation timescales, we argue that MSλD augmented with a GC-MSλD reservoir will always be more efficient for binding free energy calculations involving water displacement despite the reduced ligand transition rates.

In this study, we introduced grand canonical multisite λ dynamics, a method based on multisite λ dynamics to sample molecule number fluctuations at a given chemical potential. Like other existing enhanced sampling methodologies,^38,39 this method can also be used to improve configurational sampling in conventional λ-dynamics. GC-MSλD enables simultaneous insertions and deletion moves that may equilibrate the system faster than some conventional GCMC algorithms. From a technical standpoint, this method does not require swapping between MD and MC sampling (as done with GCMC/MD) and can be easily implemented in existing λ-dynamics engines. Thus, significantly less computational overhead (relative to NPT) is introduced when compared to GCMC/MD. When tested in a water box system, we showed that GC-MSλD recapitulates important thermodynamic quantities from NPT, including the density, the water-water radial distribution function, and hydration free energies. Additionally—like any other grand canonical sampling method—GC-MSλD can precisely control the average number of molecules via the Adams value (i.e., chemical potential). Next, we demonstrated that GC-MSλD sampling mirrors crystallographic and NPT water occupancies in the Bovine pancreatic trypsin inhibitor. Lastly, we showed that pairing GC-MSλD with a conventional MSλD binding free energy calculation recovers the correct ΔΔG_bind directionality between two HSP90 inhibitors that differ in bound state water occupancies. A future direction is to benchmark GC-MSλD’s on a large-scale protein-ligand binding free energy dataset involving water displacement, and to test its performance with popular enhanced sampling methods such as replica exchange and chemical potential oscillation. GC-MSλD can be useful for fragment-based drug discovery, where the smooth insertion and deletions may be beneficial for sampling the binding of multiple chemical fragments.

Methods

Grand Canonical Multisite λ-Dynamics.

Grand canonical multisite λ-dynamics is formally the same as multisite λ-dynamics^32,34,40 but is adapted to sample molecule number fluctuations. In a system with M λ-scaled molecules, each molecule is harmonically pinned to a dummy atom. Using the standard MSλD notation, this means there are M sites with 2 substituents (the molecule and the dummy atom) per site. The volume that these λ-scaled molecules can occupy is referred to as the reservoir region (analogous to GCMC’s insertion/deletion region). The reservoir region spans the entire simulation volume by default; however, restraints can be used to localize the sampling to a specific area, such as a protein binding site. The potential energy for the M λ-scaled molecules is then:

$$V(x,\{\lambda \})=\text{V(}{x}_0,x_0)+\sum _{i=1}^M{\lambda }_i(V(x_0,x_i)+V(x_i,x_i)-\mu )+\sum _{i=1}^M\sum _{j=i+1}^M{\lambda }_i{\lambda }_{j}V(x_i,x_j)+V_{pin}$$ (2)

where x₀ are the environment atom cartesian coordinates, λ_i is the scaling factor for the i^th molecule, μ is the excess chemical potential, and V_pin is the energy term to harmonically pin each molecule to its respective dummy atom. The excess chemical potential, instead of the Adams value, is used in Equation 2 because the density of λ-scaled molecules in the reservoir should already account for the second term in the Adams value equation (Equation 1). Any missing density can be corrected for, as mentioned in the Main Text. In this study, the dummy atoms were pinned to the oxygen atom of the λ-scaled water molecules. As the dummy atoms are non-interacting, the interaction terms involving the dummy atoms are not included in Equation 2 for clarity.

For systems involving a ligand perturbation, such as the relative binding free energy calculation of the HSP90 inhibitors, it is useful to provide a distinction between the λ-scaled molecules that are part of the GC-MSλD framework (i.e., involved in sampling number fluctuations) and the rest of the λ-scaled molecules (e.g., the HSP90 small molecule inhibitors). As we only investigated water occupancies in this study, the λ-scaled molecules involved in GC-MSλD sampling are referred to as λ-scaled water molecules, whereas the λ variables involving the ligand perturbations are referred to as the ligand λ’s. The ligands were represented via the ligand overlay topology model⁴¹ and Adaptive landscape flattening (ALF)³⁶ was used to enhance sampling of the ligand λ space. It should be noted that ALF was not used to flatten the λ variables of the GC-MSλD water molecules.

Protein Structure Preparation and Simulation Details.

Missing residues in protein structures were added via crimm⁴² and protonation states at a pH of 7.0 were assigned by PROPKA.⁴³ MMTSB⁴⁴ was used to solvate the proteins in a cubic box of water with an 8.0 Å cutoff and a NaCl concentration of 100 mM. The protein systems were minimized with 500 steps of steepest descent in pyCHARMM^45–47 using the CHARMM36m protein force field and CHARMM General Force Field (CGENFF) to parametrize small molecules.^48,49 Appropriate disulfide bridges were added via CHARMM’s patching facility.

All simulations were run using pyCHARMM with the BLaDE engine for GPU acceleration.⁵⁰ The simulation temperature was 298.15 Kelvin, and Langevin dynamics was integrated at a 2 fs timestep with a friction coefficient of 5 ps⁻¹. Periodic boundary conditions were imposed, and hydrogens were constrained with SHAKE.^51,52 Electrostatics were treated with particle mesh Ewald.⁵³ Van der Waals interactions were modeled with a switching function that starts at 9 Å and zeros at 12 Å (vswitch). NPT simulations were kept at 1.0 atm using a Monte Carlo barostat with a sampling frequency of 100 steps.⁵⁴

Computing the Excess Chemical Potential and Volume of a TIP3P Molecule.

A cubic box of 1978 TIP3P molecules was first equilibrated for 10 ns with NPT, resulting in a final box size of 38.87 Å and a density of 1.01 g/mL. The volume occupied by a TIP3P molecule (V₀) was computed by averaging over the 10 ns NPT run. To compute the excess chemical potential of a TIP3P molecule (μ), the final frame of the NPT run was used as the starting configuration. The Van der Waals and electrostatics of a singular water molecule were concurrently decoupled over 11 evenly-spaced λ windows of 15 ns. Softcore potentials were used to facilitate numerical stability at the endpoints.³⁶ FastMBAR was used to compute μ, which was determined to be 6.18 ± 0.07 kcal/mol.^55,56

Water Box System.

Starting from the final frame of the 10 ns NPT water box simulation, 432 water molecules were removed, and 864 λ-scaled water molecules were randomly added, which allows the GC-MSλD water box to have a density ranging between 0.79 – 1.23 g/mL. The reservoir region spanned the entire simulation box. Since the reservoir density does not match the bulk, the Adams value was corrected by a value of ln(1972.16) − ln(864) = 0.83, resulting in B = −2.84. To ensure that the results from the GC-MSλD simulation were not dependent on how the water box was prepared, a 2 ns NV T simulation was done with the λ-scaled water molecules fully decoupled from the environment (λ = 0.00) for each trial. Three 5 ns GC-MSλD production runs were run, in which all the λ values were initialized to 0.01.

Relative Hydration Free Energy of Benzene and Phenol.

To run the relative hydration free energy calculation with GC-MSλD, benzene was first placed in the GC-MSλD equilibrated water box and minimized at constant volume. Phenol was superimposed onto benzene and pinned together with restraints. The Adams value was set to −2.84. ALF was used to flatten the benzene and phenol λ potential energy surfaces with 60 cycles of 100 ps runs, 10 cycles of 1 ns runs, and 5 × 10 ns production runs at constant volume. For the NPT free energy calculation, the same system setup was prepared, except the NPT-equilibrated water box was used instead of the GC-MSλD equilibrated water box. Conventional NPT MSλD was run on this system using the same number of ALF cycles and production length. The vacuum simulation of benzene and phenol was run with the same ALF protocol.

Bovine Pancreatic Trypsin Inhibitor.

The 5PTI PDB was used as the starting structure for BPTI.²⁶ The GC-MSλD reservoir region was defined as a sphere of 5 Å radius placed at the center of mass of the TYR10 and ARG42 alpha carbons, and all crystallized water molecules in this volume were deleted. The λ-scaled water molecules were confined to the reservoir region via a strong quadratic potential that activates after a distance of 5 Å from the center of the sphere. Although not necessary, a repulsive quadratic potential was placed on the bulk water molecules to prevent entry into the reservoir region. To match bulk density, ~ 17.58 λ-scaled water molecules were placed in the reservoir region. 17 λ-scaled water molecules were placed in this region, and the missing density (0.58 water molecules) was corrected for, resulting in an Adams value of B = −7.57. Absolute restraints were placed on the alpha carbons of CYS10, CYS30, CYS51, and CYS55 to prevent the protein from drifting away from the GC-MSλD reservoir. These atoms were selected to minimize perturbing the overall dynamics of the protein, since the CYS alpha carbons are relatively far away from the hydration cavity and are already rigid due to disulfide bonds. The GC-MSλD simulation was run under NPT conditions for 2 ns to equilibrate the box volume, followed by a 30 ns production run with the volume set as constant. The λ-scaled water molecules were deleted in the NPT run. The NPT equilibration and production runs were the same length as the GC-MSλD runs.

Heat Shock Protein 90 Inhibitors.

The 2XAB PDB structure was used as the starting structure, which contains the bound state of ligand 1.³¹ Ligand 2 was superimposed onto ligand 1, and harmonic restraints were used to pin the common atoms together. A spherical reservoir with a 5 Å radius was placed at the center of mass of the LYS170 and LEU33 alpha carbons, and crystallized water molecules within this sphere were removed. The Adams value and restraints involving the reservoir were the same as BPTI. To prevent the protein from drifting away, absolute restraints were placed on the backbone atoms of residues 9–20 and 185–195. The system was first equilibrated under NPT conditions for 2 ns to relax the box size. The last frame was used as a starting point for GC-MSλD, which consisted of 200 phase 1 ALF cycles (100 ps each), 40 phase 2 ALF cycles (2 ns each), and finally a production of 5 × 30 ns at constant volume. Solvent seeding was used to improve the convergence speed of ALF (unpublished results).⁵⁷ In the conventional NPT MSλD simulations (in which the GC-MSλD reservoir is not present), the λ-scaled water molecules were deleted, and the simulations were run for the same amount of time as the GC-MSλD ones.

Supplementary Material

Figure S1 shows a density histogram of water reservoir λ values for the GC-MSλD simulations. Figure S2 compares the time series of the HSP90 ligand 1 λ value and the number of water molecules in the reservoir region. Figure S3 is the transition rate of the λ-scaled water molecules in BPTI as a function of a quadratic barrier height. Figure S4 compares the convergence of the HSP90 dry and wet runs.