Gaussian accelerated molecular dynamics in OpenMM

Abstract

Gaussian accelerated molecular dynamics (GaMD) is a computational technique that provides both unconstrained enhanced sampling and free energy calculations of biomolecules. Here, we present the implementation of GaMD in the OpenMM simulation package and validate it on model systems of alanine dipeptide and RNA folding. For alanine dipeptide, 30ns GaMD production simulations reproduced free energy profiles of 1000ns conventional molecular dynamics (cMD) simulations. In addition, GaMD simulations captured folding pathways of three hyperstable RNA tetraloops (UUCG, GCAA, and CUUG) and binding of the rbt203 ligand to the HIV-1 Tar RNA, both of which involved critical electrostatic interactions such as hydrogen bonding and base stacking. Together with previous implementations, GaMD in OpenMM will allow for wider applications in simulations of proteins, RNA, and other biomolecules.

Introduction

Molecular dynamics (MD) is a powerful computational technique for simulating biomolecular dynamics at an atomistic level¹. Due to advancements in computing hardware and software, timescales accessible to MD simulations have increased, while costs have decreased^2–3. However, conventional MD (cMD), which makes no use of any enhanced sampling schemes, is often limited to tens to hundreds of microseconds^3–10 for simulations of biomolecular systems, and cannot attain the timescales required to observe many biological processes of interest, which typically occur over milliseconds or longer, due to high energy barriers (e.g., 8–12 kcal/mol)^3–10.

Many enhanced sampling techniques have been developed during the last several decades to overcome the challenges mentioned above^11–15. One class of enhanced sampling techniques use predefined collective variables (CVs) or reaction coordinates (RCs), including umbrella sampling^16–17, metadynamics^18–19, adaptive biasing force^20–21 and steered MD²². However, it can be challenging to define proper CVs prior to simulation³, and predefined CVs might significantly limit the sampling of conformational space during simulations³. Another class of enhanced sampling techniques, including replica exchange MD (REMD)^23–24 or parallel tempering²⁵, self-guided Langevin MD^26–28 and accelerated MD (aMD)^29–30, do not require predefined CVs. The latter class of unconstrained enhanced sampling techniques remain attractive to improve the sampling of biomolecular dynamics and obtain sufficient accuracy in free energy calculations.

Gaussian accelerated molecular dynamics (GaMD) is an unconstrained enhanced sampling technique that works by applying a harmonic boost potential to smooth biomolecular potential energy surface³¹. Since this boost potential usually exhibits a near Gaussian distribution, cumulant expansion to the second order (“Gaussian approximation”) can be applied to achieve proper energy reweighting³². GaMD allows for simultaneous unconstrained enhanced sampling and free energy calculations of large biomolecules³¹. GaMD has been successfully demonstrated on enhanced sampling of ligand binding^{31, 33–36}, protein folding^{31, 35}, protein conformational changes^{34, 37–40}, protein-membrane⁴¹, protein-protein^42–44, and protein-nucleic acid^45–46 interactions. Furthermore, GaMD has been combined with REMD^47–48 to further improve conformational sampling and free energy calculations³. In addition, “selective GaMD” algorithms, including Ligand GaMD (LiGaMD)⁴⁹, Peptide GaMD (Pep-GaMD)⁵⁰, and Protein-Protein Interaction-GaMD (PPI-GaMD)⁴⁴ have been developed to enable repetitive binding and dissociation of small-molecule ligands, highly flexible peptides, and proteins within microsecond simulations, which allow for highly efficient and accurate calculations of ligand/peptide/protein binding free energy and kinetic rate constants³. Recently, GaMD has been combined with Deep Learning and free energy profiling in a workflow (GLOW) to predict molecular determinants and map free energy landscapes of biomolecules³⁷. GaMD has been implemented in widely used simulation packages including AMBER³¹, NAMD³⁵, GENESIS⁴⁸, and TINKER-HP⁵¹.

In this work, we have implemented GaMD in the OpenMM simulation package⁵². OpenMM is an open-source scientific software package for performing MD simulations on a range of high-performance computing architectures⁵². OpenMM was designed to be simple and easy to use, hardware independent, and extensible so that new hardware architectures can be accommodated and new functionality can be easily added⁵². In fact, accelerated MD (aMD) has been previously implemented in OpenMM⁵³. We validated the implementation on the model simulations of alanine dipeptide, three hyperstable RNA tetraloops of UUCG, GCAA, and CUUG, and rbt203 ligand binding to the HIV-1 Tar RNA.

Methods

Gaussian accelerated molecular dynamics (GaMD)

GaMD works by adding a harmonic boost potential to smooth the potential energy surface when the system potential drops below a reference energy E³¹:

$$\Delta V\left(r\right)=\{\begin{array}{rr}\frac{1}{2}k{\left(E-V\left(\stackrel{-}{r}\right)\right)}^2,& V\left(\stackrel{-}{r}\right)<E\\ 0,& V\left(\stackrel{-}{r}\right)\ge E,\end{array}$$ (1)

where k is the harmonic force constant. The two adjustable parameters E and k can be determined based on three enhanced sampling principles. First, for any two arbitrary potential values $V_1\left(\stackrel{⃑}{r}\right)$ and $V_2\left(\stackrel{⃑}{r}\right)$ found on the original energy surface, if $V_1\left(\stackrel{⃑}{r}\right)<V_2\left(\stackrel{⃑}{r}\right),\Delta V$ should be a monotonic function that does not change the relative order of the biased potential values; i.e., $V_1^{*}\left(\stackrel{⃑}{r}\right)<V_2^{*}\left(\stackrel{⃑}{r}\right)$. Second, if $V_1\left(\stackrel{⃑}{r}\right)<V_2\left(\stackrel{⃑}{r}\right)$, the potential difference observed on the smoothed energy surface should be smaller than that of the original, i.e., $V_2^{*}\left(\stackrel{⃑}{r}\right)-V_1^{*}\left(\stackrel{⃑}{r}\right)<V_2\left(\stackrel{⃑}{r}\right)-V_1\left(\stackrel{⃑}{r}\right)$. The reference energy needs to be set in the following range:

$$V_{max}\le E\le V_{min}+\frac{1}{k},$$ (2)

where V_max and V_min are the system minimum and maximum potential energies. To ensure that equation (2) is valid, k must satisfy: $k\le \frac{1}{V_{max}-V_{min}}$. Let us define $k\equiv k_0\frac{1}{V_{max}-V_{min}}$, then $0\le k_0\le 1$. Third, the standard deviation of $\Delta V$ needs to be small enough (i.e., narrow distribution) to ensure proper energetic reweighting³²: ${\sigma }_{\Delta V}=k\left(E-V_{avg}\right){\sigma }_V\le {\sigma }_0$, where V_avg and ${\sigma }_V$ are the average and standard deviation of the system potential energies, ${\sigma }_{\Delta V}$ is the standard deviation of $\Delta V$ with ${\sigma }_0$ as a user-specified upper limit (e.g., 10k_BT) for proper reweighting. When E is set to the lower bound E = V_max, k₀ can be calculated as:

$$k_0=\min \left(1.0,k_0^{\mathrm{\text{'}}}\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left(1.0,\frac{{\sigma }_0}{{\sigma }_V}\frac{V_{max}-V_{min}}{V_{max}-V_{avg}}\right),$$ (3)

Alternatively, when the threshold energy E is set to its upper bound $E\le V_{min}+\frac{1}{k}$, k₀ is set to:

$$k_0=k_0^{\mathrm{\text{'}}\mathrm{\text{'}}}\equiv \left(1.0-\frac{{\sigma }_0}{{\sigma }_V}\right)\frac{V_{max}-V_{min}}{V_{avg}-V_{min}},$$ (4)

if $k_0^{\text{'}\text{'}}$ is found to be between 0 and 1. Otherwise, k₀ is calculated using equation (3).

For energetic reweighting of GaMD simulations, the probability distribution along a selected reaction coordinate can be calculated from simulations as $p^{*}\left(A\right)$. Given the boost potential $\Delta V\left(r\right)$ of each frame in GaMD simulations, $p^{*}\left(A\right)$ can be reweighted to recover the canonical ensemble distribution, $p\left(A\right)$, as:

$$p\left(A_j\right)=p^{\mathrm{*}}\left(A_j\right)\frac{{〈e^{\beta \Delta V\left(\stackrel{-}{r}\right)}〉}_j}{\sum _{i=1}^M{〈{p^{\mathrm{*}}\left(A_i\right)e}^{\beta \Delta V\left(\stackrel{-}{r}\right)}〉}_i},j=1,\dots ,M$$ (5)

where M is the number of bins, $\beta =k_{B}T$ and ${〈e^{\beta \Delta V\left(\stackrel{-}{r}\right)}〉}_j$ is the ensemble-averaged Boltzmann factor of $\Delta V\left(\stackrel{-}{r}\right)$ for simulation frames found in the j^th bin. The ensemble-averaged reweighting factor can be approximated using cumulant expansion^31–32:

$$〈e^{\beta \Delta V\left(\stackrel{-}{r}\right)}〉=exp\left\{\sum _{k=1}^{\mathrm{\infty }}\frac{{\beta }^k}{k!}{C}_k\right\}$$ (6)

where the first two cumulants are given by:

$$\begin{array}{c}{C}_1=〈\Delta V〉,\\ C_2=〈\Delta V^2〉-{〈\Delta V〉}^2={\sigma }_v^2.\end{array}$$ (7)

The boost potential obtained from GaMD simulations usually shows near-Gaussian distribution⁵⁴. Cumulant expansion to the second order thus provides a good approximation for computing the reweighting factor^31–32. The reweighted free energy $F\left(A\right)={-k}_{B}T\ln p\left(A\right)$ is calculated as:

$$F\left(A\right)=F^{\mathrm{*}}\left(A\right)-\sum _{k=1}^2\frac{{\beta }^k}{k!}{C}_k+F_c$$ (8)

where $F^{*}\left(A\right)={-k}_{B}T\ln p^{*}\left(A\right)$ is the modified free energy obtained from GaMD simulation and $F_c$ is a constant.

Implementation of GaMD in OpenMM

In recent years, the OpenMM simulation engine⁵² has been developed to enable fast and extensible MD simulations. OpenMM features a convenient API layer, which allows users to access OpenMM’s functions from external programs, including code written in Python. OpenMM also possesses lower layers to make the most efficient use of CPU and GPU hardware capabilities.

Part of OpenMM’s extensibility includes the built-in CustomIntegrator object, which allows developers to design integration algorithms from within the high-level API layer, not requiring them to delve into the complexities of the lower OpenMM code layers. The CustomIntegrator accepts a set of variables and instructions in the form of character strings. OpenMM passes these strings to a just-in-time compiler⁵⁵ to be converted to CPU or GPU platform code at runtime - enabling both highly efficient and highly customizable code. We used the CustomIntegrator to implement several variations of the GaMD algorithm within Python.

For GaMD in OpenMM, multiple modes are available for applying boost potential to biomolecules: (1) boosting the dihedral energetic term only, (2) boosting the total potential energy only, (3) boosting the non-bonded terms in the potential energy, and (4) boosting a combination of two of the aforementioned terms, called “dual-boost” (i.e., “total energy – dihedral energy dual boost”). The GaMD boost potential is computed based on statistics of the system as detailed in the previous section. In addition, both the “lower-bound” and “upper-bound” integration schemes are implemented in GaMD OpenMM.

GaMD simulations generally include three stages: (i) short cMD, (ii) GaMD equilibration and (iii) GaMD production. The program first collects potential statistics from a short cMD run. Subsequently, a boost potential is added to the system in the GaMD equilibration stage while updates of the potential statistics continue. After the equilibration stage, the statistics collected is assumed to be sufficient to represent the potential energy landscape of interest. Hence, the reference energy and harmonic force constant are fixed to calculate the boost potential for running the production simulation. Note that in both the cMD and equilibration stages, there are a small number of steps at the beginning of each stage during which we do not collect statistics. These steps, named preparation steps, are performed to allow the system to adapt to the simulation environment. The program starts collecting statistics of the potential energies after the preparation steps.

MD simulations frequently experience interruptions; therefore, it is helpful for simulation utilities to be able to easily restart incomplete simulations. This is accomplished in GaMD OpenMM by leveraging OpenMM’s checkpoint utility - the exact state of the simulation, including all variables related to the GaMD portion of the simulation, is saved with a frequent interval in time. Therefore, if an interruption occurs to the simulation, the GaMD OpenMM program can automatically recover the most recent checkpoint and continue the simulation from where it left off, regardless of which stage of the GaMD process it was in when the interruption occurred.

Our GaMD OpenMM package is open-source and available for download, along with documentation for installation and usage, as well as tutorials, at https://github.com/MiaoLab20/gamd-openmm.git.

Simulation Protocols and Benchmarks

For alanine dipeptide, the AMBER ff99SB force field parameters were used. The LEaP module in the AmberTools package^56–59 was used to build the simulation system for alanine dipeptide. The alanine dipeptide was solvated in a TIP3P⁶⁰ water box that extends ~8–10 Å from the solute surface. The final system contained 1912 atoms, with a total of 630 water molecules.

For GaMD simulations of RNA molecules, the AMBER RNA OL3⁶¹ and GAFF2⁶² force field parameters were used for the RNA and ligand molecules, respectively. The simulation systems of the UUCG, GCAA, and UUCG tetraloops were prepared starting from the 1F7Y⁶³, 1ZIH⁶⁴, and 1RNG⁶⁵ PDB structures, respectively. The PDB structures were solvated in octahedral TIP3P⁶⁰ water boxes that extended 12 Å from the RNA surfaces, with approximately 1M KCl added to the solutions by the LEaP module in the AmberTools package^{57–59, 66}. The final systems of UUCG, GCAA, and CUUG tetraloops contained 7,805, 6,218, and 7,538 atoms, respectively. Starting from the 1UUD⁶⁷ PDB structure, the bound rbt203 ligand was removed from the HIV-1 Tar RNA and placed at a ~15 Å distance away from the RNA surface to prepare the simulation system for ligand binding to the HIV-1 Tar RNA. The RNA-ligand complex was then solvated in a cubical TIP3P⁶⁰ water box that extended 15 Å from the solute surface by the CHARMM-GUI webserver^68–70. The system charge was neutralized with 0.15 M NaCl and 0.01 M Mg²⁺, which resulted in a final system size of 40,829 atoms. All RNA systems were simulated at a temperature of 300 K.

Periodic boundary conditions were applied for the simulation systems. Bonds containing hydrogen atoms were restrained with the SHAKE⁷¹ algorithm and a 2fs timestep was used. Weak coupling to an external temperature and pressure bath was used to control both temperature and pressure⁷². The electrostatic interactions were calculated using the particle mesh Ewald (PME) summation⁷³ with a cutoff of 8.0 Å for long-range interactions. After the initial energy minimization and thermalization, dual-boost GaMD was applied to simulate the systems. The system threshold energy for applying the boost potential was set V_max. The default parameter values were used for the GaMD simulations except stated otherwise. For alanine dipeptide, three independent simulations were performed with randomized initial atomic velocities, each of which consisted of 2ns short cMD, followed by 4ns GaMD equilibration and then 30 ns GaMD production simulation. After collecting the statistics, the threshold energy E and harmonic force constant k were computed according to equation (3).

For the simulations of RNA tetraloops, three independent dual-boost GaMD simulations were performed for each system, each of which consisted of 2 ns cMD, 8 ns GaMD equilibration after adding the boost potential and 3,000–5,000 ns GaMD production. The σ_0V values were lowered to 1.5 kcal/mol from the default 6.0 kcal/mol to observe semi-stable refolding of the RNA tetraloops. For the simulations of the rbt203 ligand binding to the HIV-1 Tar RNA, five independent dual-boost GaMD simulations were performed, each of which included 1.6 ns cMD, 6.4 ns GaMD equilibration after adding the boost potential and 400–500 ns GaMD production. GaMD simulation frames were saved every 0.1 ps. The VMD⁷⁴ and CPPTRAJ⁷⁵ tools were used for simulation trajectory analysis. Finally, the PyReweighting toolkit³² was applied to compute the potential of mean force (PMF) profiles of the backbone dihedrals Φ and Ψ in the alanine dipeptide. The heavy-atom RMSD of RNA tetraloops relative to respective PDB structures (1F7Y for UUCG, 1ZIH for GCAA, and 1RNG for CUUG) and the U3-G6, G3-A6, and C3-G6 center-of-mass (COM) distances were used as RCs to calculate the PMF profiles in the RNA tetraloop simulations. The COM distance between the rbt203 ligand and nucleotide A6 and the COM distance between RNA nucleotides A6 and U7 side chains were selected to compute the PMF profiles in the simulations of ligand binding to the HIV-1 Tar RNA.

Results

Free Energy Profiles of Alanine Dipeptide

For alanine dipeptide, outputs from the dual-boost GaMD simulations were used to compute free energy profiles of the Φ and Ψ dihedrals (Figure 1A). The boost potential from three independent 30 ns GaMD production simulations was 9.4 ± 2.7 kcal/mol. The 2^nd order cumulant expansion was applied to energetically reweight the GaMD simulations.

Figure 1.

(A) Schematic representation of backbone dihedrals Φ and Ψ in alanine dipeptide. (B-C) Potential of mean force (PMF) profiles of the (B) Φ and (C) Ψ dihedrals calculated from three 30 ns GaMD simulations combined using cumulant expansion to the 2^nd order. (D) The 2D PMF profile of backbone dihedrals (Φ, Ψ) from combined three 30ns GaMD simulations trajectories. The low energy wells are labeled corresponding to the right-handed α helix (α_R), left-handed α helix (α_L), β-sheet (β) and polyproline II (P_II) conformations. (E) The 2D PMF profile of backbone dihedrals (Φ, Ψ) from combined three 1000 ns cMD simulations trajectories. The low energy wells are labeled corresponding to the right-handed α helix (αR), left-handed α helix (αL), β-sheet (β) and polyproline II (PII) conformations.

The 1D free energy profiles obtained from three 30 ns GaMD simulations agreed quantitatively with the PMF profiles from the 1000 ns cMD simulations (Figures 1B-E and S1). For Φ, moderate fluctuations were observed near the energy barrier at 0°, and the free energy value increased slightly at ~50° (Figures 1B and S1A-B). The 2D free energy profiles of backbone dihedrals (Φ, Ψ) in 3 × 30ns GaMD simulations and 3 × 1000ns cMD simulations are shown in Figure 1D and 1E. Overall, GaMD in OpenMM was able to identify five low-energy conformational states, which centered around (−148°, 0°) and (−69°, −17°) for the right-handed α helix (α_R), (48°, −12°) for the left-handed α helix (α_L), (−150°, 159°) for the β-sheet and (−72°, 162°) for the polyproline II (P_II) conformation. The corresponding minimum free energies were approximately 0, 0.74, 3.15, 1.68, and 2.65 kcal/mol. The 2D free energy profile obtained from the GaMD and cMD simulations showed a high degree of similarity (Figure 1E).

Folding of the RNA tetraloop structures: UUCG, GCAA, and CUUG

Multiple independent dual-boost GaMD simulations were performed on three RNA tetraloops structures of UUCG (PDB: 1F7Y) ⁶³, GCAA (PDB: 1ZIH) ⁶⁴, and CUUG (PDB: 1RNG)⁶⁵. Similar averages and standard deviations of the added boost potentials were recorded for the systems, i.e., 9.3 ± 2.5 kcal/mol for UUCG, 10.1 ± 3.1 kcal/mol for GCAA, and 9.0 ± 2.9 kcal/mol for CUUG. Starting from the folded structures, GaMD simulations captured multiple unfolding and semi-stable folding events. A folding event was defined as attaining < 4Å heavy-atom RMSD relative to respective PDB structures⁷⁶ of the three RNA tetraloops for more than ~10 ns (Figures S2-S4). The 2D PMF free energy profiles were calculated from the respective heavy-atom RMSD to the PDB structures and COM distances between first and last residues of the RNA tetraloops to characterize their folding processes.

The 2D PMF free energy profile of the UUCG folding was calculated from the heavy-atom RMSD of the RNA tetraloop relative to the 1F7Y PDB structure and the distance between nucleotides U3 and G6 (Figures 2A and S2). Four low-energy conformational states, including “Folded” (Figure 2B), “I1” (Figure 2C), “I2” (Figure 2D), and “Unfolded” (Figure 2E), were uncovered from the GaMD simulations of the UUCG tetraloop. All the low-energy conformational states were compared to the nuclear magnetic resonance (NMR) structure of folded UUCG (PDB: 1F7Y)⁶³ (Figure 2). In the “Folded” low-energy conformation, nucleotides U3 and G6 flipped in and formed hydrogen bonds with one another, and nucleotide U3 base stacked with nucleotide C5. The heavy-atom RMSD of UUCG relative to the 1F7Y PDB was ~1.1 Å, and the COM distance between nucleotides U3 and G6 was ~9.8 Å (Figure 2B). The COM distance between nucleotides U3 and G6 increased to ~14.2 Å, and the heavy-atom RMSD increased to ~4.1 Å in the “I1” low-energy conformational state. Nucleotide G6 flipped out, whereas nucleotide C5 flipped in to interact with nucleotide U3 (Figure 2C). The heavy-atom RMSD in the “I2” low-energy conformational state was similar to the “I1” low-energy conformational state, although the RNA backbone distorted heavily, which decreased the distance between nucleotides U3 and G6 to ~7.3 Å (Figure 2D). In the “Unfolded” low-energy conformational state, the heavy-atom RMSD relative to the 1F7Y PDB structure was ~5.8 Å, and the U3-G6 distance was ~8.0 Å (Figure 2E).

Figure 2. Folding of the UUCG RNA tetraloop captured by GaMD in OpenMM.

(A) 2D free energy profile of the heavy-atom RMSD of UUCG relative to the 1F7Y PDB structure and the center of mass (COM) distance between nucleotides U3 and G6. The low-energy RNA conformational states are labeled “Folded”, “I1”, “I2”, and “Unfolded”. (B) The “Folded” low-energy conformational state compared to the 1F7Y PDB structure, for which the RMSD is ~1.1 Å and the U3-G6 distance is ~9.8 Å. (C) The “I1” low-energy conformational state compared to the 1F7Y PDB structure, for which the RMSD is ~4.1 Å and the U3-G6 distance is ~14.2 Å. (D) The “I2” low-energy conformational state compared to the 1F7Y PDB structure, for which the RMSD is ~4.2 Å and the U3-G6 distance is ~7.3 Å. (E) The “Unfolded” low-energy conformational state compared to the 1F7Y PDB structure, for which the RMSD is ~5.8 Å and the U3-G6 distance is ~8.0 Å. The low-energy RNA conformations are colored orange, cyan, magenta, and yellow, and the 1F7Y PDB structure is colored gray.

For the GCAA RNA tetraloop, four low-energy conformational states were identified from 2D PMF calculated from the GaMD simulations (Figures 3A and S3), including “Folded” (Figure 3B), “I1” (Figure 3C), “I2” (Figure 3D), and “Unfolded” (Figure 3E). They were compared to the MMR structure of folded GCAA (PDB: 1ZIH) ⁶⁴ in Figure 3. In the “Folded” state, the side chains of nucleotides C4-A6 were base stacked and located on the opposite side of nucleotide G3. The heavy-atom RMSD relative to the 1ZIH PDB structure was ~1.0 Å, and the COM distance between nucleotides G3 and A6 was ~9.3 Å (Figure 3A). In the “I1” state, the side chain of nucleotide G3 flipped to the same side of nucleotides U4-A6, and the base stacking only existed between nucleotides A5 and A6. The heavy-atom RMSD was ~2.6 Å, and the distance between nucleotides G3 and A6 was ~8.8 Å (Figure 3B). In the “I2” state, the base stacking between nucleotides A5 and A6 remained stable. The heavy-atom RMSD relative to the 1ZIH PDB structure was ~3.9 Å, and the G3-A6 distance was ~10.5 Å (Figure 3C). In the “Unfolded” state, nucleotide C4 flipped out, while nucleotide G3 formed base stacking with nucleotides A5 and A6. The heavy-atom RMSD was ~4.6 Å, and the G3-A6 distance was ~7.6 Å (Figure 3D).

Figure 3. Folding of the GCAA RNA tetraloop captured by GaMD in OpenMM.

(A) 2D free energy profile of the heavy-atom RMSD of GCAA relative to the 1ZIH PDB structure and the COM distance between nucleotides G3 and A6. The low-energy RNA conformational states are labeled “Folded”, “I1”, “I2”, and “Unfolded”. (B) The “Folded” low-energy conformational state compared to the 1ZIH PDB structure, for which the RMSD is ~1.0 Å and the G3-A6 distance is ~9.0 Å. (C) The “I1” low-energy conformational state compared to the 1ZIH PDB structure, for which the RMSD is ~2.6 Å and the G3-A6 distance is ~8.8 Å. (D) The “I2” low-energy conformational state compared to the 1ZIH PDB structure, for which the RMSD is ~3.9 Å and the G3-A6 distance is ~11.0 Å. (E) The “Unfolded” low-energy conformational state compared to the 1ZIH PDB structure, for which the RMSD is ~4.5 Å and the G3-A6 distance is ~8.0 Å. The low-energy RNA conformations are colored orange, cyan, magenta, and yellow, and the 1ZIH PDB structure is colored gray.

For the CUUG RNA tetraloop, three distinct low-energy conformational states were identified from the 2D PMF (Figures 4A and S4), namely “Folded” (Figure 4B), “I1” (Figure 4C), and “Unfolded” (Figure 4D). They were also compared to the MMR structure of folded CUUG (PDB: 1RNG) ⁶⁵ (Figure 4). In the “Folded” state, nucleotides C3 and G6 flipped in and formed a hydrogen bond with one another. The heavy-atom RMSD relative to the 1RNG PDB structure was ~1.1 Å, and the COM distance between nucleotides C3 and G6 was ~10.8 Å (Figure 4B). The RNA backbone distorted in the “I1” state. The heavy-atom RMSD was ~3.9 Å, and the C3-G6 distance was ~6.9 Å (Figure 4C). In the “Unfolded” state, the heavy-atom RMSD relative to the 1RNG PDB structure was ~4.3 Å, and the C3-G6 distance was ~13.2 Å (Figure 4D).

Figure 4. Folding of the CUUG RNA tetraloop captured by GaMD in OpenMM.

(A) 2D free energy profile of the heavy-atom RMSD of CUUG relative to the 1RNG PDB structure and the COM distance between nucleotides C3 and G6. The low-energy RNA conformational states are labeled “Folded”, “I1”, and “Unfolded”. (B) The “Folded” low-energy conformational state compared to the 1RNG PDB structure, for which the RMSD is ~1.1 Å and the C3-G6 distance is ~10.9 Å. (C) The “I1” low-energy conformational state compared to the 1RNG PDB structure, for which the RMSD is ~3.9 Å and the C3-G6 distance is ~6.9 Å. (D) The “Unfolded” low-energy conformational state compared to the 1RNG PDB structure, for which the RMSD is ~4.1 Å and the C3-G6 distance is ~13.1 Å. The low-energy RNA conformations are colored orange, cyan, magenta, and yellow, and the 1RNG PDB structure is colored gray.

Binding of the rbt203 ligand to the HIV-1 Tar RNA

Starting from the 1UUD⁶⁷ PDB structure, the bound rbt203 ligand was removed and placed at a ~15 Å distance from the HIV-1 Tar RNA. Five independent 400–500 ns GaMD simulations captured multiple stable binding events of the rbt203 ligand to the HIV-1 Tar RNA (Figure S5). The average added boost potentials were recorded to be 9.1 ± 3.0 kcal/mol. The 2D PMF free energy profiles was calculated from the COM distance between the rbt203 ligand and nucleotide A6 side chain and the COM distance between nucleotides A6 and U7 side chains to characterize ligand binding to the HIV-1 Tar RNA (Figures 5 and S6).

Figure 5. Binding of the rbt203 ligand to the HIV-1 Tar RNA captured by GaMD in OpenMM.

(A) 2D free energy profile of the COM distance between the rbt203 ligand (Lig) and RNA nucleotide A6 and the COM distance between RNA nucleotides A6 and U7 side chains. The low-energy conformational states are labeled “B1”, “B2”, “I1”, “I2”, “I3”, and “U”. (B) The “B1” low-energy conformational state, for which the A6-P14 ligand distance is ~8.0 Å and the A6-U7 distance is ~3.5 Å. (C) The “B2” low-energy conformational state, for which the A6-P14 ligand distance is ~10.1 Å and the A6-U7 distance is ~10.0 Å. (D) The “I1” low-energy conformational state, for which the A6-P14 ligand distance is ~20.1 Å and the A6-U7 distance is ~4.1 Å. (E) The “I2” low-energy conformational state, for which the A6-P14 ligand distance is ~20.0 Å and the A6-U7 distance is ~13.5 Å. (F) The “I3” low-energy conformational state, for which the A6-P14 ligand distance is ~41.8 Å and the A6-U7 distance is ~12.7 Å. (G) The “U” low-energy conformational state, for which the A6-P14 ligand distance is ~34.1 Å and the A6-U7 distance is ~12.2 Å. The low-energy RNA-ligand conformational states are colored orange, green, cyan, magenta, yellow, pink, and marine, and the 1UUD PDB is colored gray.

Six low-energy conformational states were uncovered from the GaMD simulations of ligand binding to HIV-1 Tar RNA, including “B1”, “B2”, “I1”, “I2”, “I3”, and “U”. The “B1” and “B2” low-energy conformational states represented the bound conformation of rbt203 in the HIV-1 Tar RNA, while the “U” low-energy conformation state represented the unbound conformation. In the “B1” low-energy conformational state, the distance between rbt203 ligand and nucleotide A6 was ~7.1 Å, and the distance between nucleotides A6 and U7 was ~3.8 Å (Figure 5A). Nucleotide U7 flipped in and pointed towards the core of the HIV-1 Tar RNA. The rbt203 ligand interacted with nucleotides A6, U7, C8, U9, G10, C23, and U24 in this low-energy conformational state (Figure 5A). In the “B2” low-energy conformational state, the distance between rbt203 ligand and nucleotide A6 was ~9.1 Å, and the distance between nucleotides A6 and U7 was ~8.1 Å (Figure 5B). Similar to the “B1” low-energy conformational state, nucleotide U7 also pointed towards the ligand. The rbt203 ligand interacted with nucleotides G5, A6, U7, G10, A11, G12, C13, A19, G20, C21, U22, C23, and U24 (Figure 5C). In the “I1” low-energy conformational state, the A6-rbt203 ligand distance was ~20.1 Å, and the A6-U7 distance was ~4.1 Å (Figure 5D). The rbt203 ligand was located at the terminal nucleotides of HIV-1 Tar RNA. The interacting nucleotides with rbt203 ligand were G1, G2, C3, C23, U24, G27, C28, and C29 (Figure 5D). In the “I2” low-energy conformational state, the A6-rbt203 ligand distance was ~20.0 Å, and the A6-U7 distance was ~12.4 Å (Figure 5E). The rbt203 ligand was at a similar location as in the “I1” low-energy conformational state. The interacting nucleotides in the “I2” low-energy conformational state were G1, G2, C3, A4, U26, G27, C28, and C29 (Figure 5E). In the “I3” low-energy conformational state, the A6-rbt203 ligand distance was ~34.1 Å, and the A6-U7 distance was ~12.2 Å (Figure 5F). The rbt203 ligand was located at the U15-G18 RNA tetraloop. The rbt203 ligand interacted with nucleotides U15, G16, and G17 of the HIV-1 Tar RNA (Figure 5F). In the “U” low-energy conformational state, the rbt203 ligand was found in the bulk solvent, and nucleotide U7 flipped outwards. The distance between nucleotide A6 and rbt203 ligand was ~41.8 Å, and the distance between nucleotides A6 and U7 was ~12.7 Å in this low-energy conformational state (Figure 5G).

Discussion

By adding a harmonic boost potential to smoothen the potential energy surface, GaMD provides both unconstrained enhanced sampling and free energy calculation of biomolecules. Important statistical properties of the system potential, such as the average, maximum, minimum and standard deviation values, are used to calculate the simulation acceleration parameters, particularly the threshold energy and force constant for applying the boost potential. In this study, we have implemented GaMD in the OpenMM package. “Selective GaMD” algorithms, including Ligand GaMD, Peptide GaMD and Protein-Protein Interaction GaMD, have not been implemented in OpenMM, although they are planned to be implemented in the future.

Three independent 30 ns GaMD simulations were able to capture five different low-energy conformational states of the backbone dihedrals (Φ, Ψ) in alanine dipeptide, which were in good agreement with the cMD simulations (Figure 1D,E). In addition, the 1D free energy profiles of GaMD and cMD mostly overlapped, except the elevated free energy value at ~50° for Φ and minor fluctuations in the energy barriers at 0° for Φ and −120° for Ψ (Figures 1 and S1). Notably, both the 1D and 2D free energy profiles of GaMD in OpenMM were highly similar to those from previous implementations of GaMD in AMBER³¹ and NAMD³⁵ in terms of the low-energy states and free energy profiles. The alanine dipeptide system provides a sort of benchmark or validation of the correctness of the GaMD approach and any of its implementations. Our present results show that GaMD in OpenMM can reproduce the correct free energy profiles for alanine dipeptide, as we have shown for previous implementations of GaMD^{31, 35, 48, 51}, providing evidence that we have completed a correct implementation of GaMD in OpenMM, providing users confidence in applying GaMD OpenMM for their own systems of interest.

GaMD in OpenMM successfully captured the unfolding and semi-stable refolding of three hyperstable RNA tetraloops of UUCG, GCAA, and CUUG⁷⁶. The low-energy conformational states obtained illustrated the unfolding pathways of the three hyperstable tetraloops, which were mostly the reverses of the folding pathways uncovered by Chen et. al⁷⁶. For UUCG, starting from the “Folded” low-energy conformational state where the two nucleotides U3 and G6 pointed inwards and interacted with each other (Figure 2B), the backbone of the UUCG tetraloop skewed to the right as the nucleotide C5 flipped in, pushed G6 outwards, and formed interactions with U3 in the “I1” low-energy conformation (Figure 2C). As UUCG transited from the “I1” to “I2” conformation, the U3-C5 interaction was broken, and both nucleotides flipped outwards. The RNA core was solely occupied by nucleotide G6, heavily distorting the tetraloop (Figure 2D). Finally, the RNA stretched out and became unfolded in the “Unfolded” low-energy conformational state (Figure 2E). For GCAA, the unfolding pathway started with the “Folded” low-energy conformation where base stacking was observed between the three nucleotides C4-A6, and only G3 pointed inwards (Figure 3B). As GCAA transited from the “Folded” to “I1” conformation, the stacking between nucleotides C4 and A5 was broken, while the tetraloop shrunk in size (Figure 3C). The base stacking between nucleotides A5 and A6 remained stable in the “I2” low-energy conformational state, as the RNA began stretching out and nucleotide C4 flipped to the opposite side (Figure 3D). Finally, GCAA stretched out and coiled into the “Unfolded” low-energy conformation, where nucleotide G3 flipped outwards and formed base stacking with nucleotide A5, which in turn remained base-stacked with nucleotide A6 (Figure 3E). For CUUG, nucleotides C3 and G6 pointed inwards and formed hydrogen bonds with each other in the “Folded” low-energy conformational state (Figure 4B). As CUUG transited from the “Folded” to “I1” low-energy conformation, all four nucleotides in the tetraloop pointed outwards as the RNA skewed left and shrunk in its size significantly (Figure 4C). To completely unfold, CUUG straightened out its terminal nucleotides and became stretched out in the “Unfolded” low-energy conformation (Figure 4D). Overall, the low-energy conformational states and unfolding pathways uncovered from GaMD simulations in OpenMM agreed well with a previous study carried out by Chen et al.⁷⁶, particularly the “Folded” conformations, “I1” of UUCG, “I2” of GCAA, and “I1” of CUUG. Nevertheless, it is also worth noting that GaMD in OpenMM was only able to capture semi-stable refolding events of all three RNA tetraloops, where the heavy-atom RMSD relative to respective PDB structures were in the range of ~1.8–2.5 Å (Figures S2-S4). This was primarily because the RNA force field parameters were biased to favor rigid, highly stacked conformations, as described in the previous study⁷⁶. The independent GaMD simulations of each RNA tetraloop have not achieved proper convergences within the 4–5μs simulation time windows as indicated by the different free energy profiles across the simulations of each tetraloop (Figures S2-S4). Longer GaMD simulations combined with more accurate RNA force field parameter sets are required to achieve consistent simulations of RNA.

In the 1UUD⁶⁷ PDB structure of HIV-1 Tar RNA, the distance between nucleotide A6 and rbt203 ligand is ~8.9 Å, and the distance between nucleotides A6 and U7 is ~7.4 Å. Nucleotide U7 points towards the core of the HIV-1 Tar RNA, and the rbt203 ligand interacts with nucleotides A6, U7, U9, G10, A11, G12, C13, A19, G20, C21, U22, C23, and U24. The distance between nucleotide A6 and rbt203 ligand in the 1UUD PDB structure is comparable to those in the “B1” and “B2” low-energy conformational states, while the distance between nucleotides A6 and U7 is the middle between those in the “B1” and “B2” low-energy conformational states (Figure 5B,C). The interacting nucleotides of the rbt203 ligand are highly similar between the GaMD-bound conformations and the 1UUD PDB structure, demonstrating the agreements between GaMD simulation results and experimental data⁶⁷.

One recent study by Tang et al.³⁶ demonstrated that base stacking between ligands and nucleotides is the key interaction that drives ligand binding in single-stranded nucleic acids. Furthermore, Chen et al.⁷⁶ found that preformed G1-A4 and C1-G4 base pairs played a significant role in the accurate folding of the GCAA and CUUG RNA tetraloops.

In addition, we observed that nucleotide U7 flips inwards and points towards the core of the HIV-1 Tar RNA in both the bound “B1” and “B2” low-energy conformations (Figure 5B,C) and the 1UUD⁶⁷ PDB structure, while flips outwards in the unbound “U” low-energy conformation (Figure 5G). The observation of nucleotide U7 “base-flipping”⁷⁷ phenomenon during ligand binding illustrated the importance of this nucleotide in the ligand binding to the HIV-1 Tar RNA. Furthermore, two slightly different binding pathways the rbt203 ligand to the HIV-1 Tar RNA could be observed from the free energy profile in Figure 5. While both pathways started from the “U”, “I3”, and “I2” low-energy conformational states, the second pathway arrived abruptly at the bound “B2” low-energy conformational state, whereas the dominant pathway involved a stabilization of the intermediate state as indicated by the transition from “I2” to “I1”, before ending at the bound “B1” low-energy conformation (Figures 5 and S5). The dominant pathway is described in detail as follow. Starting from the bulk solvent (Figure 5G), the rbt203 ligand approached the HIV-1 Tar RNA first through interactions with the U15-G18 tetraloop (Figure 5F). The rbt203 ligand then dissociated back to the bulk solvent and relocated to the terminal nucleotides of the HIV-1 Tar RNA (Figure 5E). At this stage, nucleotide U7 flipped inwards and became ready to interact with the rbt203 ligand (Figure 5D). Finally, the rbt203 ligand moved from the terminal of HIV-1 Tar RNA to its binding pocket, located at the core nucleotides of the RNA (Figures 5B and 5C). The ligand drew closer to nucleotide A6 in the “B2” low-energy conformational state (Figure 5C) and nucleotide U7 in the “B1” low-energy conformational state (Figure 5B). Given the fact that the RNA conformational changes took place after ligand bound, the binding of rbt203 ligand to the HIV-1 Tar RNA is an induced-fit process. On the other hand, it is also worth noting that similar to the RNA folding simulations, GaMD simulations of RNA-ligand binding has not converged within the 500ns simulation time windows as shown by the different free energy profiles calculated from the individual simulations (Figure S6). Furthermore, as mentioned above, more accurate RNA force field parameter sets are required to achieve consistent simulations of RNA molecules.

In summary, we have implemented GaMD in OpenMM. It is complementary to previous implementations of GaMD in AMBER³¹, NAMD³⁵, GENESIS⁴⁸, and TINKER-HP⁵¹. As demonstrated on model systems, results of the current work will facilitate the applications of GaMD in enhanced sampling and free energy calculations of a wide range of large biomolecules, especially RNA structures that involve critical electrostatic interactions.