Simultaneous Construction of Free Energy Surfaces Via Multisite $\mathit{\lambda }$ Dynamics and Umbrella Sampling

Abstract

Umbrella sampling (US) is a cornerstone enhanced sampling technique that constructs potentials of mean force (PMF) by biasing a system of interest along a set of collective variables (CV). With a good choice of CVs, one can uncover the free energy landscapes governing a plethora of biological processes such as protein-ligand (un)binding, protein (un)folding, and enzyme catalysis. However, multiple sets of umbrella simulations have to be run when one is interested in exploring how the free energy landscape of a particular process varies with chemical space, such as the effects of different mutations on the (un)folding landscape or how different ligands (un)bind to the same protein. We propose an enhanced sampling framework called $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ that couples umbrella sampling with multisite $\lambda$ dynamics ($\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$) to leverage $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$’s hallmark feature of sampling multiple chemical species in one simulation. In doing so, we recast the characterization of free energy landscapes for multiple chemical species into a simultaneous search of conformational and chemical landscapes. We validate $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$’s ability to construct multiple PMFs from a single set of umbrella simulations using three test systems of increasing complexity, with the most complex system involving the (un)binding PMFs of two ligands from trypsin.

Graphical Abstract

2. Introduction

Important biological processes such as protein-ligand (un)binding and protein (un)folding can take from microseconds to upwards of hours to occur. Knowledge of the underlying free energy surfaces that govern these processes is important for understanding the mechanism of action, obtaining free energy differences, identifying plausible transition states, and inferring kinetics. However, even with the advent of fast computing power and sophisticated algorithms, many of these biological processes remain difficult to sample in conventional molecular dynamics (MD). Umbrella sampling¹ (US) is one of many enhanced sampling techniques that addresses the rare event problem by biasing a system along a set of collective variables (CV) and reconstructing the potential of mean force (PMF). Ideally, this is achieved by determining CVs that describe a biological process’s slowest relaxing degrees of freedom and performing biased sampling along the CV space to connect system states of interest through a series of umbrella windows. Each umbrella window consists of a molecular simulation with a bias centered on a value in CV space, and window spacings are chosen to allow good phase space overlap between umbrella windows. The unbiased probability distribution of the CVs, and therefore the free energy landscape of the process, is recovered by methods such as multistate Bennett acceptance ratio (MBAR) or the weighted histogram analysis method (WHAM).^2,3 US has been used to investigate the mechanisms of biological processes such as ligand binding and protein folding, and the PMFs constructed from these simulations can be used to compute free energies of binding, folding, and conformational change.^4–8

While umbrella sampling is popular for its robust and relative ease of use and implementation, it is typically only used to interrogate the PMF of a biological process for one kind of chemical species. If one is interested in how the free energy surface varies with chemical space, such as the effects of different mutations on a protein’s (un)folding free energy landscape, a set of umbrella sampling simulations has to be run for each chemical species of interest. To this end, we introduce an enhanced sampling framework called $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ that combines umbrella sampling with multisite $\lambda$ dynamics ($\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$),^9–11 which enables continuous sampling of chemical space within an umbrella window. Multisite $\lambda$ dynamics is recognized for its use in high-throughput computation of protein-ligand binding free energies. By integrating $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ with umbrella sampling, one can construct the PMF of different chemical species for a biological process within a single set of umbrella simulations. Continuous sampling of alchemical space is afforded by giving $\lambda$ a mass and allowing it to dynamically take on values between 0 to 1 in response to the fluctuations of the environment. There are many approaches that fall under the family of the $\lambda$-dynamics methods. For example, adiabatic free energy dynamics (AFED) achieves efficient sampling of $\lambda$-space by adiabatically separating $\lambda$ from the physical degrees of freedom and integrating $\lambda$ at a much higher temperature than the system temperature.¹² In contrast to continuous $\lambda$-dynamics methods, $\lambda$-dynamics with bias-updated Gibbs sampling (LaDyBUGS) utilizes Gibbs sampling and an aggressive biasing scheme to move $\lambda$ between discrete values.^13,14

The idea of exploring conformational and chemical space in conjunction is not a new one, and has been investigated in the context of 2-dimensional umbrella sampling¹⁵ and more recently in alchemical metadynamics.¹⁶ However, in both cases, chemical space is sampled discretely through individual windows or by Monte Carlo. On the other hand, efforts have been made to improve configurational sampling in $\lambda$-dynamics simulations. For example, replica exchange has been used to improve sampling of buried charges and for constant pH simulations.^17,18 Enhanced sampling along multiple configurational collective variables in AFED simulations can be enabled via the driven adiabatic free energy dynamics (d-AFED) approach. d-AFED has been applied in a recently published pH-AFED scheme to improve torsional sampling of titratable residues.¹⁹

In the present work, we outline the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ enhanced sampling framework and demonstrate its ability in constructing multiple PMFs from a single set of umbrella simulations on three test systems of increasing complexity. The simplest system involves recovering the separation PMFs of argon—$X$ in liquid argon, where $X$ equals neon, argon, krypton, or xenon. The second system involves recovering the separation PMFs between a TIP3 water molecule and a series of benzene derivatives, and the final system involves the (un)binding of benzamidine (BZMD) and trans-2-phenylcyclopropylamine (TPA) from trypsin.

3. Methods

3.1. Multisite $\mathit{\lambda }$-Dynamics

$\lambda$-dynamics, and its extension, multisite $\lambda$-dynamics, have been described in previous work.^9–11,20,21 In this study, the ligands were represented by a Ligand Overlay topology model whereby each ligand was represented explicitly and was tethered together by pinning their equivalent common core atoms with strong harmonic restraints.²² The potential energy for $N$ ligands is:

$$U_{total}=U_{env}+\sum _{i=1}^N{\lambda }_i\left(U_{env,lig}^i+U_{lig,lig}^i\right)+U_{{\lambda }_{bias}}(\{\lambda \})+U_{pin}$$ (1)

where $U_{\mathit{total}}$ is the total potential energy of the system, $U_{\mathit{env}}$ is the interaction energy between the environment atoms, $U_{\mathit{env},\mathit{lig}}^i$ is the interaction energy between the environment atoms and the atoms of ligand $i$, and $U_{\mathit{lig,lig}}^i$ is the internal energy of ligand $i$. $U_{\mathit{pin}}$ is the energy term for harmonically pinning the equivalent common core atoms. $U_{{\lambda }_{\mathit{bias}}}(\{\lambda \})$ is a set of $\lambda$-dependent biasing potentials to improve sampling in $\lambda$-space:

$$U_{{\lambda }_{\mathit{bias}}}(\{\lambda \})=U_{\mathit{fixed}}(\{\lambda \})+U_{\mathit{quadratic}}(\{\lambda \})+U_{\mathit{end}}(\{\lambda \})+U_{\mathit{skew}}(\{\lambda \})$$ (2)

where each $\lambda$ biasing term consists of a specific functional form:²¹

$$U_{\mathit{fixed}}(\{\lambda \})=\sum _s^M\sum _i^{N_s}{\varphi }_{si}{\lambda }_{si}$$ (3)

$$U_{\mathit{quadratic}}(\{\lambda \})=\sum _s^M\sum _i^{N_s}\sum _{j>i}^{N_s}{\psi }_{si,sj}{\lambda }_{si}{\lambda }_{sj}$$ (4)

$$U_{\mathit{end}}(\{\lambda \})=\sum _s^M\sum _i^{N_s}\sum _{j\ne i}^{N_s}\frac{{\omega }_{si,sj}{\lambda }_{si}{\lambda }_{sj}}{{\lambda }_{si}+0.017}$$ (5)

$$U_{\mathit{skew}}(\{\lambda \})=\sum _s^M\sum _i^{N_s}\sum _{j\ne i}^{N_s}{\chi }_{si,sj}{\lambda }_{sj}(1-e^{\frac{-{\lambda }_{si}}{0.18}})$$ (6)

While equations 3–6 are written for a more general $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ Hamiltonian, the Ligand Overlay topology model used in this study sets $M=1$ and $N_s=N$ ligands. Before production simulations, Adaptive Landscape Flattening (ALF)²⁰ is used to find the optimal fitting parameters $\left\{{\varphi }_{si}\right\},\left\{{\psi }_{si}\right\},\left\{{\omega }_{si,sj}\right\}$, and $\left\{{\chi }_{si,sj}\right\}$ that flattens the $\lambda$ free energy landscape. However, one should note that an optimal $U_{{\lambda }_{\mathit{bias}}}(\{\lambda \})$ is not a necessary condition to enable adequate sampling in $\lambda$ space.

3.2. $\mathbf{M}\mathbf{S}\mathbf{\lambda }\mathbf{D}+\mathbf{U}\mathbf{S}$ Framework

The $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ framework employs an $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ simulation (or, in the case of this study, a $\lambda$-dynamics simulation) in each umbrella window. Thus, for a set of $K$ umbrella windows, the potential energy at the $k^{\mathit{th}}$ umbrella window is:

$$U_{\mathit{total}}^k=U_{\mathit{env}}+\sum _{i=1}^N{\lambda }_i\left(U_{\mathit{env,lig}}^i+U_{\mathit{lig,lig}}^i\right)+U_{\mathit{pin}}+U_{{\lambda }_{\mathit{bias}}}^k(\{\lambda \})+U_{\xi \mathit{bias}}^k(x)$$ (7)

where $U_{{\xi }_{\mathit{bias}}}^k(x)$ is a harmonic umbrella bias between an atom in the environment and an atom in the reference ligand that focuses sampling on the collective variable $\xi$. Although the umbrella bias acts only on an atom from one of the $N$ ligands, this is a nonissue as long as $U_{\mathit{pin}}$ is strong enough to tether the ligands together and that the choice of the biased atom in the reference ligand enables adequate sampling of CV space for the other ligands. Finally, while ALF is run for each umbrella window in this study, this may not be necessary as neighboring umbrella windows may have similar $U_{{\lambda }_{\mathit{bias}}}(\{\lambda \})$. Thus, one can cut down on ALF’s computational expense by having umbrella windows that sample similar environments share the same $\lambda$-dependent biasing parameters.

3.3. Computing Potentials of Mean Force

The free energy offset $f$ for the $k^{\mathit{th}}$ umbrella window was solved with the multistate Bennett acceptance ratio² (MBAR) using the FastMBAR python library:²³

$$e^{f_k}=\sum _{n=1}^N\frac{e^{-w_k\left(x_n,\{\lambda {\}}_n\right)}}{\sum _{j=1}^J{N}_j{e}^{f_j-w_j\left(x_n,\{\lambda {\}}_n\right)}}$$ (8)

where $N$ is the total number of snapshots across all umbrella windows, $N_j$ is the number of snapshots in the $j^{\mathit{th}}$ window, and $w_k\left(x_n,\{\lambda {\}}_n\right)$ is the sum of the reduced umbrella and $\lambda$-dependent biases at the $n^{\mathit{th}}$ snapshot:

$$w_k(x,\{\lambda \})=\beta [U_{{\xi }_{\mathit{bias}}}^k(x)+U_{\lambda ,\mathit{bias}}^k(\{\lambda \})]$$ (9)

The weight $W(x,\{\lambda \})$ to reweight each snapshot into the unbiased ensemble is computed by:²⁴

$$W(x,\{\lambda \})=\frac{1}{\sum _{k=1}^K{N}_k{e}^{f_k-w_k(x,\{\lambda \})}}$$ (10)

$\lambda >0.99$ is used as the cut-off for a physical ligand state. The unbiased probability distribution of $\xi$ projected on the $l^{\mathit{th}}$ ligand is:

$$P_{{\lambda }_l}(\xi )=\sum _{n=1}^N{\delta }_{\xi }\left(x_n\right){\delta }_{{\lambda }_l}\left(\{\lambda {\}}_n\right)W\left(x_n,\{\lambda {\}}_n\right)$$ (11)

where the delta functions ${\delta }_{\xi }(\dots )$ and ${\delta }_{{\lambda }_l}(\dots )$ are defined to equal to 1 for snapshots that satisfy $\xi$ and $\left\{{\lambda }_l>0.99\right\}$ respectively. Accordingly, the unbiased free energy landscape is then:

$$G_{{\lambda }_l}(\xi )=-{\beta }^{-1}\mathrm{l}\mathrm{n}{P}_{{\lambda }_l}(\xi )$$ (12)

All potentials of mean force shown in this work were computed with a bin width of 0.10 Å. To enable comparison of PMFs computed from the radial distribution function, the Jacobian term $+2k_{B}T\mathrm{l}\mathrm{n}(\xi )$ was added to equation 12 for the noble gas and benzene derivative systems.²⁵

3.4. $\mathit{\lambda }$-Reweighting of Snapshots

The projection of the $\xi$ free energy surface onto the $l^{\mathit{th}}$ chemical species, $G_{{\lambda }_l}(\xi )$, does not include snapshots where $\left\{{\lambda }_l\le 0.99\right\}$ (equation 11). In an effort to incorporate these snapshots into the estimation of $G_{{\lambda }_l}(\xi )$, we considered a $\lambda$-reweighting procedure in which snapshots contribute to $P_{{\lambda }_l}(\xi )$ by a factor of:

$${\omega }_{\xi }(x,\lambda )=e^{+\beta \mathrm{\Delta }{G}_{\xi ,\{\lambda \}\to \left\{{\lambda }_l\right\}}(x,\{\lambda \})}$$ (13)

In equation 13, $\mathrm{\Delta }{G}_{\xi ,\{\lambda \}\to \left\{{\lambda }_l\right\}}(x,\{\lambda \})$ is the free energy difference between $\{\mathrm{\lambda }\}$ and the desired physical ligand $\left\{{\lambda }_l>0.99\right\}$ at $\xi$. This free energy difference is computed from the unbiased (i.e., MBAR-reweighted) probability distribution with $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$’s implicit constraints removed (Figure S1A). Thus, equation 11 is reformulated to:

$$P_{{\lambda }_l}(\xi )=\sum _{n=1}^N{\delta }_{\xi }\left(x_n\right)W\left(x_n,\{\lambda {\}}_n\right){\omega }_{\xi }\left(x_n,\{\lambda {\}}_n\right)$$ (14)

where all snapshots, regardless of their $\{\lambda \}$ values, are now used to estimate $P_{{\lambda }_l}(\xi )$.

We tested this $\lambda$-reweighting procedure on the data obtained from a conventional $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ simulation (i.e., without an umbrella bias) and $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations of benzene/phenol in water. In the conventional $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ simulation, we computed the ${\mathrm{O}\mathrm{H}}_{\mathit{phen}}-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$ potential of mean force using equation 14 with the MBAR weights $W\left(x_n,\left\{{\lambda }_n\right\}\right)$ set to 1 (i.e., we $\lambda$-reweighted the radial distribution function). To quantify the effect of equation 14 in estimating $G_{{\lambda }_l}(\xi )$, we computed PMF with and without reweighting and calculated their RMSD against the ”ground truth” PMF obtained from MD simulation. This procedure was bootstrapped 1000 times across a range of simulation times.

3.5. Simulation Preparation and Simulation

Test Systems.

The $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ framework was validated on three systems: liquid argon, benzene derivatives in water, and trypsin. In all of these systems, PMFs computed from $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ were compared against PMFs computed by conventional umbrella sampling. The liquid argon and benzene derivative systems were chosen because they are simple enough for the radial distribution functions to be computed, allowing the PMFs from $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ to be compared against the ”ground truth.” The BZMD/TPA-Trypsin system was chosen as a realistic test case of ligand unbinding, as it has been extensively studied in past computational studies.^26–30

All system preparation and simulations were run on PyCHARMM, which is a Python wrapper for the CHARMM molecular simulation program.^31–33 BLaDE was used to run GPU-accelerated simulations for all molecular dynamics, umbrella sampling, and $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations.³⁴ The CHARMM36m force field and CGenFF were used to parametrize the proteins and small molecules.^35,36 Van der Waals interactions were force-switched to smoothly turn off the interaction energy from 9.0 to 12.0 Å, and long-range electrostatics were computed with particle mesh Ewald.³⁷ The SHAKE algorithm, hydrogen mass repartitioning, and periodic boundary conditions were applied.^38–40 Simulations were run with Langevin dynamics with a friction coefficient of 0.1 ps⁻¹. The pressure in NPT simulations was maintained using a Monte Carlo barostat with a sampling frequency of 100 steps.⁴¹

Liquid Argon.

The system consists of one atom of either argon, neon, krypton, or xenon solvated in 511 argon atoms. The atoms were encased in a 29 Å box, which results in a number density of 0.021 ų. For $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations, the neon, krypton, and xenon atoms were all present in the simulation and harmonically restrained to a reference argon atom. All simulations were run in an NV T ensemble at 86 Kelvin with a 10 fs time step. Snapshots were saved every 0.1 ps. The umbrella simulations had 20 windows with umbrella centers of [3.0,13.0] in 0.5 Å increments and a force constant of 2.5 kcal/molŲ. In the conventional US simulations, the umbrella bias was placed on the distance between an $X$ atom and an arbitrary argon atom, where $X$ equals argon, neon, krypton, or xenon. The US simulations were equilibrated for 100 ps, followed by a 3 ns production run. In the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations, the umbrella bias was placed on the distance between the $\lambda$-scaled reference argon atom and another arbitrary argon atom. The $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations had production runs of 20 ns per window. Radial distribution function for each $\mathrm{A}\mathrm{r}—\mathit{X}$ pair was also computed from 5 ns conventional MD simulations.

Benzene derivatives in water.

The system consists of a benzene, phenol, toluene, or anisole molecule solvated in a 32 Å box of TIP3P water.⁴² All simulations were run in an NPT ensemble at 298.15 Kelvins with a pressure of 1.0 atm at a time step of 4 fs. Snapshots were saved every 1 ps. The umbrella simulations had 43 windows with umbrella centers of [2.50,13.00] with intervals of 0.25 Å increments and a force constant of 10.0 kcal/molŲ. The conventional umbrella simulations were biased along the distance between the first atom of the benzene derivative R-group and the OH2 oxygen atom of a TIP3P molecule. Specifically, the collective variable for benzene is the distance between the CH hydrogen atom and the OH2 TIP3P atom (${\mathrm{C}\mathrm{H}}_{\mathit{benz}}-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$), the collective variable for phenol is the distance between the hydroxyl oxygen and the TIP3P molecule (${\mathrm{O}\mathrm{H}}_{\mathit{phen}}-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$), the collective variable for toluene is the distance between the methyl carbon and the TIP3P molecule (${\mathrm{C}}_{\mathit{tolu}}-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$), and the collective variable for anisole is the distance between the methoxy group oxygen and the TIP3P molecule (${\mathrm{O}}_{\mathit{anso}}-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$). In the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations, the benzene derivatives were tethered together by pinning equivalent carbon atoms in the six-membered ring with harmonic restraints. The MS$\lambda$D+US simulations were biased along the ${\mathrm{O}\mathrm{H}}_{\mathit{phen}}—\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$ distance. To compute the PMF projected on a collective variable ”$X-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$” (e.g., ${\mathrm{C}\mathrm{H}}_{\mathit{benz}}—\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$), physical snapshots of the molecule of interest ($\left\{{\lambda }_{\mathit{benz}}>\right.0.99\}$) were binned by $X—\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$ distances instead of the biased collective variable $\xi$ in equation 11.⁴³ The windows were equilibrated for 1 ns. The conventional US simulations were run for 50 ns per window, while $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations were run for 100 ns per window. A conventional MD simulation of 50 ns was run for each molecule to compute the radial distribution functions of the ${\mathrm{C}\mathrm{H}}_{\mathit{benz}}—\mathrm{O}\mathrm{H}{2}_{\mathit{water}},{\mathrm{O}\mathrm{H}}_{\mathit{phen}}—\mathrm{O}\mathrm{H}{2}_{\mathit{water}},{\mathrm{C}}_{\mathit{tolu}}—\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$, and ${\mathrm{O}}_{\mathit{anso}}—\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$ distances.

Trypsin.

The PDBIDs 3PTB and 1TNL were used for the BZMD/trypsin and TPA/trypsin complexes, respectively.^44,45 The structures were prepared using crimm, a Python library for protein structure preparation that has integration with pyCHARMM.⁴⁶ crimm was used to fill missing gaps in the structures, and the crimm-integrated PROPKA module was used to assign proper side chain protonation states at physiological pH.⁴⁷ Additionally, disulfide bonds were constructed using CHARMM’s PATCH facility. The protein-ligand complexes were minimized in vacuum using 500 steps of stochastic gradient descent with the $\alpha$-carbon backbone harmonically restrained to its initial position. Next, the complexes were solvated in a box of TIP3P water molecules at a 14.0 Å cutoff and with a sodium chloride concentration of 10 mM using the MMTSB toolset.⁴⁸ Finally, the water molecules were minimized with 500 steps of stochastic gradient descent while the protein was constrained. For $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations, the 3PTB structure was used as the starting structure. The six-membered ring of TPA was superimposed onto the ring of BZMD with the R-group facing ASP189. The equivalent carbon atoms between both ligands in the ring were pinned together with harmonic restraints, and the structure preparation was done as described above with $\lambda =0.50$ to obtain a reasonable starting point for $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations.

All simulations were run in the NPT ensemble at 298.15 Kelvin and 1.0 atm with a time step of 4 fs. Snapshots were saved every 20 ps. The umbrella simulations were biased on the distance between the ASP189 $\gamma$-carbon and the first carbon in the R-group of BZMD or TPA ($C_{ASP189}-C_{\mathit{lig}}$) (Figure 1). For $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations, the R-group carbon of BZMD was chosen as the biasing point. There were 32 windows with umbrella centers of [2.00,12.50] with increments of 0.50 Å for the first 22 windows, followed by umbrella centers of [13.00,22.00] with increments of 1.00 Å for the last 10 windows. The initial conditions for each window were created by translating the ligand outside the binding site such that the $C_{ASP189}-C_{\mathit{l}\mathit{i}\mathit{g}}$ distance matches the umbrella center distance. The translation vector was pointed in a direction that allowed the ligand to be seeded along a natural opening of the binding site as seen in the crystal structure (Figure 2). Each window was equilibrated for 1 ns. Conventional US simulations ran 200 ns per window for each ligand, and the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations ran for 150 ns × 5 replicas per window.

Figure 1:

A). Ligand structure of BZMD and TPA. B). Binding modes of BZMD and TPA superimposed. An arrow is pointing to the ASP189 $\gamma$-carbon. An umbrella bias is placed between the ASP189 $\gamma$-carbon and the R-group carbon of the ligands.

Figure 2:

A). Surface representation of the 3PTB crystal structure reveals an entry in the binding site. The ligand is placed along the direction of the translation vector. B). Initial ligand placement for some of the umbrella windows.

4. Results and Discussion

Potentials of mean force of multiple chemical spaces can be constructed from conventional $\mathbf{M}\mathbf{S}\mathrm{\lambda }\mathbf{D}$ simulations.

Before formulating the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ framework, a preliminary—yet rather obvious—first step is to demonstrate the ability to construct potentials of mean force of multiple chemical spaces along some collective variable from conventional $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ simulations. As a proof of concept, we start with a liquid argon system in which one argon atom is $\lambda$-coupled to xenon. The pairwise $\mathrm{A}\mathrm{r}—\mathrm{X}$ radial distribution function (where $\mathrm{X}=\mathrm{A}\mathrm{r}$, or $\mathrm{X}\mathrm{e}$) and the resulting potential of mean force can be computed by only considering snapshots where the chemical species of interest $X$ is physical (i.e., $\left\{{\lambda }_X>0.99\right\}$). Figure 3 shows that potentials of mean force computed from conventional $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ simulation agree with conventional MD simulation. While this is not a surprising result, it serves as a sanity check for the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ framework.

Figure 3:

Comparison between the PMFs of $\mathrm{A}\mathrm{r}—\mathrm{A}\mathrm{r}$ and $\mathrm{A}\mathrm{r}—\mathrm{X}\mathrm{e}$ separation distances obtained from conventional MD simulation and conventional $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ simulations.

Approaches to computing potentials of mean force from $\mathbf{M}\mathbf{S}\mathrm{\lambda }\mathbf{D}+\mathbf{U}\mathbf{S}$ simulations.

While developing the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ framework, we recognized that there were two possible approaches to computing potentials of mean force using the data obtained from $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations. The first approach involves only using the physical snapshots of the $l^{\mathit{th}}$ ligand ($\left\{{\lambda }_l>0.99\right\}$) to compute the free energy offset for the $k^{\mathit{th}}$ window ($f_k$) (equation 8) followed by reweighting of the physical snapshots to obtain the potential of mean force (equation 10 and 11). In contrast, the second approach (which was described in the methods section above) utilizes all physical and nonphysical snapshots to estimate the free energy offsets in the MBAR equation, followed by reweighting of the snapshots and computing the potential of mean force for the $l^{\mathit{th}}$ ligand by projecting onto physical ${\lambda }_l$ snapshots. To provide a quantitative comparison between these two approaches, we bootstrapped the data from an $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulation of benzene and phenol in water across a range of simulation times. That is—for a given simulation time—we randomly sampled snapshots with replacement, computed a PMF using both approaches, and computed the RMSDs against the ”ground truth” PMF from MD simulation. This process was repeated 1000 times for each simulation time point, and the RMSDs were averaged. Figure S2 demonstrates that while both approaches converge to the ”ground truth” with enough sampling, the second approach (orange curve) produces PMFs with better agreement and much higher precision.

$\mathbf{M}\mathbf{S}\mathrm{\lambda }\mathbf{D}+\mathbf{U}\mathbf{S}$ recapitulates Potentials of Mean Force from Umbrella Sampling and the Radial Distribution Function.

After reconciling aspects of the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ framework, we validated it against potentials of mean force from umbrella sampling and conventional MD for liquid argon and the benzene derivative system. As seen in Figures 4 and 5, the PMFs produced by $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ match those produced by umbrella sampling and conventional MD in the liquid argon and benzene derivative systems, respectively.

Figure 4:

Comparison of PMFs obtained from $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$, conventional US, and MD for the liquid argon system. A). PMFs of argon — argon separation distances. B). PMFs of argon — neon separation distances. C). PMFs of argon — krypton separation distances. D). PMFs of argon — xenon separation distances.

Figure 5:

Comparison of PMFs obtained from $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$, conventional US, and MD for the benzene derivative system. A). PMFs of ${\mathrm{C}\mathrm{H}}_{\mathit{benz}}-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$ separation distances. B). PMFs of ${\mathrm{O}\mathrm{H}}_{\mathit{phen}}-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$ separation distances. C). PMFs of ${\mathrm{C}}_{\mathit{tolu}}-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$ separation distances. D). PMFs of ${\mathrm{O}}_{\mathit{anso}}-\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$ separation distances.

Next, we chose to use Trypsin as an additional validation for the framework since ligand (un)binding processes of Trypsin have been extensively studied with a wide range of enhanced sampling and conventional molecular dynamics techniques.^{26,28–30,49} Furthermore, this system conveniently serves as a preliminary step towards applying $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ to study protein-ligand unbinding processes for kinetics-oriented drug design in the future. The BZMD/TPA ligand’s initial seeding for the umbrella simulations most closely resembles ”Pathway 1” observed in the REVO weighted ensemble simulations done by Donyapour et al.²⁷ However, as the collective variable used for the umbrella biasing is pathway agnostic, we observed BZMD sampling an alternative pathway in the conventional US simulation windows, which led to a disagreement between the BZMD unbinding PMFs computed by conventional US and $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$. This alternate pathway—similar to ”Pathway 4” reported by Donyapour et al.—is characterized by ASP189 reorienting towards the solvent, which makes room for BZMD to traverse deeper into the binding site and escape from the back (Figure S3). The sampling of this alternative pathway leads to an underestimation of the bound and the transition state free energies in the US PMF compared to the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ PMF. To construct a PMF along the intended unbinding pathway, we manually inspected trajectories to determine collective variables that separate configurations involving the intended pathway from the alternative pathway. From this inspection, we concluded that a shorter distance between the BZMD center-of-mass and the LYS188 $C\alpha (C{V}_{LYS188})$ indicates BZMD is en route to escaping through the alternative pathway, whereas a shorter distance between the BZMD center-of-mass and CYS191 $C\alpha (C{V}_{CYS191})$ indicates a traversal through the intended pathway (Figure S4). To classify configurations as either the intended or alternate pathway, we fitted two-state hidden Markov models (HMM) to the $C{V}_{CYS191}$ and $C{V}_{LYS188}$ traces that have been concatenated across the first 13 umbrella windows (Figure S5). From there, only snapshots that are classified as the intended pathway by both the $C{V}_{CYS191}$ and $C{V}_{LYS188}$ HMMs are used to compute the PMF. Applying the pathway filtering protocol to the conventional US simulations led to a better estimation of the bound and transition state regions of the PMF, and improved the agreement between the US and $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ PMFs for BZMD (compare the dashed red curve in Figure S6A to the solid black curve in Figure S6B). As a control test, we also applied the pathway filtering protocol to the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ data and demonstrated that the resulting unbinding PMF for BZMD was largely unaffected, which indicates that the alternate pathway had limited sampling in the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations (Figure S6B). Visual inspections of the trajectories revealed that TPA only sampled the intended pathway in both conventional US and $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations; thus, the pathway filtering protocol was not applied for the TPA (un)binding PMFs. Figure 6 shows a side-by-side comparison of the (un)binding PMFs of BZMD and TPA computed by conventional US and $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$. $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ was able to recapitulate the bound state minimum for both ligands and produced PMF curves that have an RMSD of 0.60 kcal/mol or less when compared to the conventional US PMFs. To further quantify the agreement between the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ and conventional US PMFs, we also compared the relative $\mathrm{\Delta }\mathrm{\Delta }{G}_{\mathit{bind}}$ and $\mathrm{\Delta }\mathrm{\Delta }{G}^{‡}$ between TPA and BZMD that one would get from the $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ and conventional US PMFs. To do so, we partitioned each PMF into bound, unbound, and transition states and computed the $\mathrm{\Delta }\mathrm{\Delta }{G}_{\mathit{bind}}=-k_{B}T\mathrm{l}\mathrm{n}\left(\frac{P_{\mathit{bound}}}{P_{\mathit{trans}+\mathit{unbound}}}\right)$ and $\mathrm{\Delta }\mathrm{\Delta }{G}^{‡}=-k_{B}T\mathrm{l}\mathrm{n}\left(\frac{P_{\mathit{trans}}}{P_{\mathit{bound}}}\right)$. The $\mathrm{\Delta }\mathrm{\Delta }{G}_{\mathit{bind}}$ and $\mathrm{\Delta }\mathrm{\Delta }{G}^{‡}$ from $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ are −0.98 kcal/mol, and 2.00 kcal/mol, respectively, which are within 1 kcal/mol of the conventional US values of −0.18 kcal/mol and 1.17 kcal/mol.

Figure 6:

Comparison of (un)binding PMFs obtained from $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ (black curve) and conventional US (red curve) for trypsin. A). (Un)binding PMF of BZMD. B). (Un)binding PMF of TPA.

$\lambda$-Reweighting does not improve the estimation of $G_{{\lambda }_l}(\xi )$.

Earlier, we showed that including all $\lambda$ snapshots in the MBAR calculation (equation 8 and 10) produces higher quality PMFs when compared to only using physical snapshots. However, the projection of the $\xi$ free energy surface onto the $l^{\mathit{th}}$ chemical species (equation 11) only considers snapshots where $\left\{{\lambda }_l>0.99\right\}$. In theory, the proposed $\lambda$-reweighting procedure (equation 14) can improve the estimation of the $G_{{\lambda }_l}(\xi )$ surface by incorporating $\left\{{\lambda }_l\le 0.99\right\}$ snapshots, thus necessitating less simulation time to obtain a converged PMF. However, our bootstrapping suggests that the reweighting procedure tends to worsen or have no effect on the estimation of $G_{{\lambda }_l}(\xi )$ in both the conventional $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ and $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ test cases (Figure 7). Based on this result, we identified several reasons as to why the reweighting procedure may be impractical. For one, the contribution of a $\{\lambda \}$ snapshot to the $\left\{{\lambda }_l>0.99\right\}$ target ensemble drops off exponentially which severely limits the use of the reweighting procedure for realistic systems where the cost of an unfavorable ligand perturbation can be upwards of 15 kcal/mol. Secondly, the reweighting procedure requires adequate sampling of $P(\xi ,\{\lambda \})$. There could be cases where sampling of $P(\xi ,\{\lambda >0.99\})$ converges prior to $P(\xi ,\{\lambda \le 0.99\})$, thus $\left\{{\lambda }_l\le 0.99\right\}$ snapshots would no longer be needed to improve the $G_{{\lambda }_l}(\xi )$ estimation. Alternatively, inadequate sampling of $P(\xi ,\{\lambda \le 0.99\})$ may lead to erroneous weights that worsen the $G_{{\lambda }_l}(\xi )$ estimation, as seen in conventional $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ test case (compare dashed blue line with solid blue line in Figure 7). In this case, the initial 5 ns of simulation produced discontinuous/unconverged $\lambda$-dependent free energy curves for some of the $\xi$ bins (Figure S1B), resulting in erroneous $\lambda$-weights. Cases like this can occur since $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ has built-in implicit constraints that bias the sampling to physical $\lambda$ states, thus limiting the sampling of $P(\xi ,\{\lambda \le 0.99\})$ (refer to Knight and Brooks 2011 for details on $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$’s implicit constraints¹⁰).

Figure 7:

Bootstrapping results of the proposed $\lambda$-reweighting procedure (dashed) compared with no reweighting (solid). This bootstrapping was applied to computing the ${\mathrm{O}\mathrm{H}}_{\mathit{phen}}—\mathrm{O}\mathrm{H}{2}_{\mathit{water}}$ separation PMF from conventional $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ (blue) and $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ (orange) simulations.

Advantages of Coupling $\lambda$-Dynamics to Umbrella Sampling.

Sampling of configurational space via CV-based sampling methods, as in the case of umbrella sampling, may be hindered by orthogonal degrees of freedom such as sampling of water molecules in the binding site, ligand torsionals, or global and local protein conformational dynamics. Sampling of these dynamics can be improved by coupling CV-based sampling methodologies with additional ones, such as replica exchange or grand canonical Monte Carlo. Similarly, sampling of alchemical intermediates can also aid in relaxing orthogonal degrees of freedom by allowing the system to bypass barriers in configurational space. This idea is analogous to the established four-dimensional molecular dynamics and energy embedding methods,^50–52 whereby an extra spatial dimension is added to the potential energy function to facilitate barrier crossing. Hahn et al. have demonstrated that orthogonal barriers in alchemical free energy calculations can be overcome by use of ”$\lambda$-variation” strategies—that is, when $\lambda$’s value is allowed to change, like in the case of $\lambda$-dynamics, Hamiltonian replica exchange, Monte Carlo, and the conveyor belt thermodynamic integration scheme.^53,54 Diffusive sampling of $\lambda$ in $\lambda -$dynamics, enabled by recent strategies such as depositing history-dependent biases in $\lambda$-space or via an adaptive biasing force, has been observed to promote orthogonal relaxation.^55–57 Recently, there have been efforts to incorporate alchemical intermediates in existing enhanced sampling frameworks to improve orthogonal sampling, such as alchemical metadynamics, alchemical steered molecular dynamics, and multiple topology replica exchange of expanded.^16,58–60 In the same vein, we anticipate that adding $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}$ to umbrella sampling can improve configurational sampling, though this remains to be investigated as the ligand perturbations done in this study were not likely large enough to promote relaxation of orthogonal degrees of freedom. It should also be noted that $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ also enjoys enhanced sampling of the ligand torsionals due to the ligand topology model used.^22,61,62 One can envision a case where perturbing between ligands of varying molecular volumes may improve configurational sampling in narrow bottleneck regions along an unbinding pathway. The PMFs computed by $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$, on average, converge at a similar rate but with higher precision compared to conventional US (Figure S7 and Figure S8, see SI for the protocol to assess convergence). Given that PMF quality is dependent on the sampling of physical $\lambda$ states, we expect the convergence rate of $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ to increase as more techniques to improve the sampling physical $\lambda$ states are developed.⁶³

5. Conclusion

Protein-ligand binding, protein folding, and a multitude of other critical biological processes typically evolve on a timescale that is inaccessible to conventional molecular dynamics simulations. Uncovering the potentials of mean force that govern these biological processes is paramount for drug design, protein design, and a host of other applications. In this paper, we presented an enhanced sampling framework named $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ that extends conventional umbrella sampling with continuous sampling of chemical space to simultaneously sample both configurational and chemical space. Integrating an alchemical degree of freedom can improve configurational sampling that may be orthogonal to the chosen collective variables. We validated $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ using three test systems by demonstrating that the framework can recover the potentials of mean force of multiple chemical species within a set of umbrella simulations. Since the potentials of mean force from $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ are computed by projecting onto the physical snapshots of the $l^{\mathit{th}}$ chemical species of interest, we also investigated whether the incorporation of snapshots at other $\{\lambda \}$ values can improve the convergence of computing a PMF. To do this, we used a reweighting procedure to compute the contribution of snapshots at other $\{\lambda \}$ values to the desired physical $\left\{{\lambda }_l>0.99\right\}$ distribution. However, our testing indicates that the reweighting procedure tends to worsen or have no effect on computing PMFs.

In the current iteration of $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$, adaptive landscape flattening was run for each umbrella window to improve sampling of alchemical space, which can be time-consuming. ALF convergence can be sped up by utilizing $\lambda$ populations sampled across all umbrella windows to construct the ALF biases. Kinetics-oriented drug design is a future application of $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$, whereby chemical space is scanned for ligands with (un)binding PMFs that produce favorable binding kinetics.

Supplementary Material

Discussion of computing convergence rates; Figure S1 showing an example of free energy curves for $\lambda$-reweighting; Figure S2 showing bootstrap comparison between the two approaches to computing potentials of mean force in $\mathrm{M}\mathrm{S}\mathrm{\lambda }\mathrm{D}+\mathrm{U}\mathrm{S}$ simulations; Figure S3 showing comparison between the two BZMD unbinding pathways; Figure S4 showing the collective variables used to distinguish the two pathways; Figure S5 showing the hidden markov model fit to identify BZMD snapshots that take the intended pathway and the alternative pathway; Figure S6 showing comparison of BZMD PMFs before and after pathway filtering using a hidden markov model; Figures S7 and S8 bootstrapping convergence plots for benzene derivative and trypsin systems.