Acceleration of Biomolecular Kinetics in Gaussian Accelerated Molecular Dynamics

Abstract

Recent studies demonstrated that Gaussian accelerated molecular dynamics (GaMD) is a robust computational technique, which provides simultaneous unconstrained enhanced sampling and free energy calculations of biomolecules. However, the exact acceleration of biomolecular dynamics or speedup of kinetic rates in GaMD simulations and, more broadly, in enhanced sampling methods, remains a challenging task to be determined. Here, the GaMD acceleration is examined using alanine dipeptide in explicit solvent as a biomolecular model system. Relative to long conventional molecular dynamics (cMD) simulation, GaMD simulations exhibited ~36–67 times speedup for sampling of the backbone dihedral transitions. The acceleration depended on level of the GaMD boost potential. Furthermore, Kramers’ rate theory was applied to estimate GaMD acceleration using simulation-derived diffusion coefficients, curvatures and barriers of free energy profiles. In most cases, the calculations also showed significant speedup of dihedral transitions in GaMD, although the GaMD acceleration factors tended to be underestimated by ~3–96 fold. Because greater boost potential can be applied in GaMD simulations of systems with increased sizes, which potentially leads to higher acceleration, it is subject to future studies on accelerating the dynamics and recovering kinetic rates of larger biomolecules such as proteins and protein-protein/nucleic acid complexes.

Introduction

It is important to determine kinetic rates of biomolecular dynamics, such as protein folding, ligand binding and unbinding of target receptors and protein-protein/nucleic acid interactions, in order to understand life processes¹ and guide drug design². However, transitions of biomolecular conformations often take place over “mesoscales” of hundreds of nanoseconds to milliseconds in time and nanometers to micrometers in length³. They can be difficult to probe in both chemical and biophysical experiments (e.g., nuclear magnetic resonance⁴ and neutron scattering^{5, 6}) and computer simulations³.

With recent remarkable advances in supercomputers and GPU computing, molecular dynamics (MD) simulations provide a powerful approach to investigate biomolecular dynamics at an atomic level^7–9. MD simulations performed over tens to hundreds of microseconds have been successfully applied to capture folding of native proteins with ~10–100 amino acid residues¹⁰, spontaneous binding of drug molecules to target receptors^{11, 12}, and repeated binding and unbinding of millimolar drug fragments¹³. However, due to the limited simulation timescales, conventional MD (cMD) suffers from insufficient sampling of biomolecular processes that take place over longer timescales, e.g., unbinding of potent drug molecules from target receptors, protein-protein binding, gene editing, etc.

Many enhanced simulation techniques have been actively developed over the past several decades to address the sampling issue of biomolecules^14–19. These techniques include the umbrella sampling^{20, 21}, metadynamics^{22, 23}, adaptive biasing force^24–26, replica-exchange MD^{27, 28}, conformational flooding²⁹, accelerated MD (aMD)^{30, 31}, potential scaled MD^{32, 33}, and so on. Enhanced sampling has proven useful for improved simulation convergence and free energy calculations of various biomolecular systems^{14, 15, 34}. Despite remarkable developments and successful applications in a number of biological systems^{23, 26}, “constrained” enhanced sampling requiring the use of predefined collective variables can limit the applicability of these methods^{15, 16, 35}. On the other hand, “unconstrained” enhanced sampling can suffer from large statistical noise during reweighting to recover the original free energy landscapes³⁶, leading to limitations for certain biomolecular systems^{16, 34}.

To achieve both unconstrained enhanced sampling and proper energetic reweighting for biomolecular simulations, Gaussian accelerated MD (GaMD) has been recently developed by applying a harmonic boost potential to smooth the potential energy surface³⁷. Similar to aMD^{30, 37, 38}, GaMD greatly reduces the energy barriers and enhances biomolecular conformational sampling without the need of predefined collective variables^39–41. Moreover, because the boost potential follows a Gaussian distribution, the original free energy profiles of biomolecules can be recovered through a cumulant expansion to the second order³⁷. GaMD solves the energetic reweighting problem encountered in the previous aMD method³⁶. It allows us to characterize structural dynamics of complex biomolecules, including the fast-folding proteins^{37, 42}, GPCRs^{40, 43}, HIV protease⁴⁴ and CRISPR-Cas9 (clustered regularly interspaced short palindromic repeats-CRISPR associated protein 9)⁴¹.

Although recent studies demonstrated the applications of GaMD for efficient enhanced sampling and free energy calculations of biomolecules^{37, 39–42}, the usage of GaMD simulations for extracting biomolecular kinetic rates has yet to be explored. In this context, there have been studies using several other enhanced sampling techniques to obtain biomolecular kinetics. In particular, Parrinello and co-workers have applied metadynamics to determine unbinding kinetics of inhibitors of trypsin⁴⁵ and the p38 MAP Kinase⁴⁶. De Oliveira et al.⁴⁷ and Hamelberg and co-workers^48–50 were able to extract kinetic rates of peptide conformational transitions from the aMD simulations using Kramers’ rate theory. Frank and Andricioaei applied Kramers’ rate theory to recover biomolecular kinetics from potential-scaled MD simulations³³. Here, GaMD is combined with Kramers’ rate theory with the aim of both enhancing biomolecular simulations and recovering the kinetic rates, as well as elucidating the exact acceleration of biomolecular kinetics in GaMD simulations.

As demonstrated on the alanine dipeptide biomolecular model system, GaMD simulations exhibited ~36–67 times speedup relative to long-timescale cMD for sampling transitions of the peptide backbone dihedrals. The acceleration depended on level of the GaMD boost potential. We also derived diffusion coefficients of the alanine dipeptide, curvatures and barriers of the dihedral free energy profiles from cMD and GaMD simulations. Application of Kramers’ rate equation showed mostly significant speedup of GaMD compared with cMD, although the acceleration factors tended to be underestimated. This study will provide important guidance for future applications of GaMD in studying kinetics of larger biomolecular systems.

Methods

Kramers’ Rate Theory

In 1940, Kramers pioneered the research on the rate of chemical reactions. For a particle climbing over potential energy barriers, he showed that the reaction rate depends on temperature and viscosity of the host medium⁵¹. He derived reaction rates for both limiting cases of small and large viscosity. In the context of biomolecular simulations in aqueous medium, it is relevant for us to focus on the large viscosity limiting case. Biomolecules move in the high friction (“overdamping”) regime and energy barriers are much greater than k_BT (k_B is the Boltzmann’s constant and T is temperature). In this case, the reaction rate is calculated as:

$$k_R\cong \frac{2\pi w_m{w}_b}{\xi }{e}^{-\mathrm{\Delta }F/k_{B}T},$$ (1)

where w_m and w_b are frequencies of the approximated harmonic oscillators (also referred to as curvatures of free energy surface^{33, 50}) near the energy minimum and barrier, respectively, ξ is the apparent friction coefficient and ΔF is the free energy barrier of transition.

Without the loss of generality, we consider a 1D free energy profile or potential of mean force (PMF) of a reaction coordinate F(A). Near minimum at A_m, the free energy can be approximated by a harmonic oscillator⁵¹ of frequency w_m, i.e., $F\left(A\right)=\frac{1}{2}{\left(2\pi w_m\right)}^2{\left(A-A_m\right)}^2$. Near barrier at A_b, the free energy is approximated as $\left(A\right)=F_b-\frac{1}{2}{\left(2\pi w_b\right)}^2{\left(A-A_b\right)}^2$, where F_b is the free energy at A_b and w_b is the frequency of the approximated harmonic oscillator. Then we can calculate w_m and w_b as:

$$w=\sqrt{\frac{\left|F"\left(A\right)\right|}{2\pi }},$$ (2)

Where $F"\left(A\right)$ is the second-order derivative of the PMF profile.

The apparent friction coefficient ξ or diffusion coefficient D with ξ = k_BT / D can be estimated as follows. First, we calculate a survival function S(t) as the probability that the system remains in an energy well longer than time t. In a direct approach⁴⁸, we count the events that the system visits the energy well throughout a simulation. We record and measure the time intervals of each visiting event until the system escapes over an energy barrier. Then we have a time series T_i, where i=1, 2, …, N, and N is the total number barrier transitions observed in the simulation. The time series is subsequently ordered such that ${\widehat{T}}_1\le {\widehat{T}}_2\le \cdots \le {\widehat{T}}_N$. With that, the survival function is estimated as $S\left({\widehat{T}}_i\right)\approx 1-i/N$, which is the probability that the system is trapped in the energy well for time longer than ${\widehat{T}}_i$. Alternatively, we can numerically calculate the time-dependent probability density of reaction coordinate A, ρ(A, t) by solving the Smoluchowski equation along 1D PMF profile of the reaction coordinate:

$$\frac{\partial \rho \left(A, t\right)}{\partial t}=D\frac{\partial }{\partial A}\left[e^{-F\left(A\right)/k_{B}T}\frac{\partial }{\partial A}\left(e^{F\left(A\right)/k_{B}T}\rho \right)\right].$$ (3)

Then the survival function is calculated as $S\left(t\right)= {\int }_t^{\infty }{\int }_{A_{b1}}^{A_{b2}}\rho \left(A, t\right)dAdt$, where A_b1 and A_b2 are two boundaries of the energy well. The initial condition is often set as the Boltzmann distribution of reaction coordinate A in the energy well, i.e., $\rho \left(A, 0\right)=e^{-F\left(A\right)/k_{B}T}$.

Second, using the above survival functions, we estimate the effective kinetic rates as the negative of the slopes in linear fitting of the ln[S(t)] versus t, i.e., $k=-dln\left[S\left(t\right)\right]/dt$. This is based on the assumption that the survival function exhibits exponential decay as observed in earlier studies^{48, 49}. Finally, the apparent diffusion coefficient D is obtained by dividing the kinetic rate calculated directly using the transition time series collected from the simulation by that using the probability density solution of the Smoluchowski equation⁴⁸.

Gaussian Accelerated Molecular Dynamics

Gaussian accelerated molecular dynamics (GaMD) is an enhanced sampling technique that works by adding a harmonic boost potential to reduce the system energy barriers. Details of the method have been described in previous studies^{37, 42}. A brief summary is provided here. Consider a system with N atoms at positions $\stackrel{\rightharpoonup }{r}=\left\{{\stackrel{\rightharpoonup }{r}}_1,\cdots ,{\stackrel{\rightharpoonup }{r}}_N\right\}$. When the system potential $V\left(\stackrel{\rightharpoonup }{r}\right)$ is lower than a reference energy E, the modified potential $V^{*}\left(\stackrel{\rightharpoonup }{r}\right)$ of the system is calculated as:

$$\begin{array}{l}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{V}^{*}\left(\stackrel{\rightharpoonup }{r}\right)=V\left(\stackrel{\rightharpoonup }{r}\right)+\mathrm{\Delta }V\left(\stackrel{\rightharpoonup }{r}\right),\\ \mathrm{\Delta }V\left(\stackrel{\rightharpoonup }{r}\right)=\{\begin{array}{c}\frac{1}{2}k{\left(E-V\left(\stackrel{\rightharpoonup }{r}\right)\right)}^2, V\left(\stackrel{\rightharpoonup }{r}\right)<E\hspace{0.17em}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}0, V\left(\stackrel{\rightharpoonup }{r}\right)\ge E\end{array}\end{array}$$ (4)

where k is the harmonic force constant. The two adjustable parameters E and k are automatically determined based on three enhanced sampling principles³⁷. The reference energy needs to be set in the following range:

$$V_{max}\le E\le V_{min}+\frac{1}{k},$$ (5)

where V_max and V_min are the system minimum and maximum potential energies. To ensure that Eqn. (5) is valid, k has to satisfy: $k\le \frac{1}{V_{max}-V_{min}}$ Let us define $\equiv k_0\frac{1}{V_{max}-V_{min}}$, then 0 < k₀ ≤ 1. The standard deviation of ∆V needs to be small enough (i.e., narrow distribution) to ensure proper energetic reweighting⁵²: ${\sigma }_{\mathrm{\Delta }V}=k\left(E-V_{avg}\right){\sigma }_V\le {\sigma }_0$ where V_avg and σ_V are the average and standard deviation of the system potential energies, σ_∆V is the standard deviation of ∆V with σ₀ 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:

$$k_0=\min \left(1.0, k_0^{\text{'}}\right)=\min \left(1.0, \frac{{\sigma }_0}{{\sigma }_V}\frac{V_{max}-V_{min}}{V_{max}-V_{avg}}\right).$$ (6)

Alternatively, when the threshold energy E is set to its upper bound $E=V_{min}+\frac{1}{k},k_0$ is set to:

$$k_0=k_0^{"}\equiv (1-\frac{{\sigma }_0}{{\sigma }_V})\frac{V_{max}-V_{min}}{V_{max}-V_{avg}},$$ (7)

if $k_0^{"}$ is found to be between 0 and 1. Otherwise, k₀ is calculated using Eqn. (6).

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

$$p\left(A_j\right)=p^{*}\left(A_j\right)\frac{e\beta \mathrm{\Delta }V\left(\stackrel{\rightharpoonup }{r}\right)j}{{\sum }_{i=1}^M{\langle p^{*}\left(A_i\right)e^{\beta \mathrm{\Delta }V\left(\stackrel{\rightharpoonup }{r}\right)}\rangle }_i}, j=1,\dots , M,$$ (8)

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

$$\langle e^{\beta \mathrm{\Delta }V\left(\stackrel{\rightharpoonup }{r}\right)}\rangle =exp\left\{{\sum }_{k=1}^{\infty }\frac{{\beta }^k}{k!}{C}_k\right\},$$ (9)

where the first two cumulants are given by:

$$\begin{array}{c}{C}_1=\langle \mathrm{\Delta }V\rangle ,\\ C_2=\langle \mathrm{\Delta }{V}^2\rangle -{\langle \mathrm{\Delta }V\rangle }^2={\sigma }_v^2.\end{array}$$ (10)

The boost potential obtained from GaMD simulations usually follows near-Gaussian distribution³⁹. Cumulant expansion to the second order thus provides a good approximation for computing the reweighting factor^{37, 52}. The reweighted free energy $F\left(A\right)=-k_{B}T \text{ln }p\left(A\right)$ is calculated as:

$$F\left(A\right)=F^{*}\left(A\right)-{\sum }_{k=1}^2\frac{{\beta }^k}{k!}{C}_k+F_c,$$ (11)

Where $F^{*}\left(A\right)=-k_{B}T \text{ln }{p}^{*}\left(A\right)$ is the modified free energy obtained from GaMD simulation and F_c is a constant. Finally, curvatures and energy barriers of the reweighted and modified free energy profiles will be used in Kramers’ rate equation to determine the GaMD acceleration of biomolecular dynamics.

Simulations of Alanine Dipeptide

For simulations of alanine dipeptide (Fig. 1A), the AMBER ff99SB force field was used and the system was built using the Xleap module in the AMBER package^{9, 53–55} as described previously^{32, 37}. The serial GPU version of GaMD implemented in AMBER 16³⁷ was applied for running the simulations. By solvating alanine dipeptide in a TIP3P⁵⁶ water box that extends 8 to 10 Å from the surface, the system contained 630 water molecules and a total number of 1,912 atoms. Periodic boundary conditions were applied. Bonds containing hydrogen atoms were restrained with the SHAKE algorithm⁵⁷ and a 2fs timestep was used. Weak coupling to an external temperature and pressure bath was used to control both temperature and pressure (NPT ensemble) using Langevin dynamics⁵⁸. The electrostatic interactions were calculated using the PME (particle mesh Ewald summation)⁵⁹ with a cutoff of 8.0 Å for long-range interactions.

Fig. 1.

(A) Schematic representation of backbone dihedrals Φ and Ψ in alanine dipeptide. (B) 2D potential of mean force (PMF) of backbone dihedrals (Φ, Ψ) calculated from 1000 ns cMD simulation. 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. (C) Time course and (D) 1D PMF of dihedral Φ obtained from the 1000 ns cMD simulation.

The system was initially minimized for 2,000 steps using the conjugate gradient minimization algorithm and then the solvent was equilibrated for 50 ps with the solute atoms fixed. Another minimization was performed with all atoms free and the system was slowly heated to 300 K over 500 ps. System equilibration was further achieved by a 400 ps NPT run to assure that the water box of simulated system had reached appropriate density. Then the fully equilibrated system was taken as the restarting structure for cMD and GaMD simulations.

Using the NPT ensemble with Langevin dynamics, 1000 ns cMD production simulation was performed as reference. For GaMD simulations, the threshold energy for applying boost potential is set to E=V_max and the default 6 kcal/mol was used for σ₀. The maximum, minimum, average and standard deviation values of the system potential (V_max, V_min, V_avg and σ_V) were obtained from an initial 2 ns cMD simulation with no boost potential. Then two versions of GaMD simulations were performed, one with boost potential applied to the dihedral energy term only (i.e., dihedral GaMD) and the other with boost potential applied to both the dihedral and total potential energy terms (i.e., dual-boost GaMD). Each GaMD simulation proceeded with a 2 ns equilibration run after adding the boost potential, followed by three independent 30 ns production runs with randomized initial atomic velocities. A list of the cMD and GaMD simulations are summarized in Table 1.

Table 1.

The number of forward (“Fwd”) and backward (“Bwd”) transitions of dihedral Φ in alanine dipeptide across the 0º energy barrier observed in 1000 ns cMD, three 30 ns dihedral GaMD and three 30 ns dual-boost GaMD simulations, along with the acceleration obtained in GaMD simulations relative to the cMD.

Sim	Length (ns)	Ntransitions		kR (x106 s−1)		Acceleration
		Fwd	Bwd	Fwd	Bwd	Fwd	Bwd
cMD	1000	4	3	4	3	-	-
Dihedral GaMD	30 × 3	6, 4, 3	5, 4, 4	144±50	144±19	36.1±12.7	48.1±6.4
Dual-boost GaMD	30 × 3	6, 5, 7	5, 5, 8	200±33	200±58	50.0±8.3	66.7±19.2

For simulation analysis, the CPPTRAJ⁶⁰ in AMBER was applied to calculate backbone dihedrals (Φ, Ψ) in alanine dipeptide. The “PyReweighting” toolkit⁵² was used to reweight the GaMD simulations for calculating 1D and 2D PMF profiles of dihedrals in alanine dipeptide. A bin size of 6° was selected to balance between reducing the anharmonicity and increasing the bin resolution as discussed earlier⁵². For the kinetics, the number of forward and backward transitions of dihedral Φ in alanine dipeptide across the 0º energy barrier were counted in the 1000 ns cMD, three 30 ns dihedral GaMD and three 30 ns dual-boost GaMD simulations. The transition rates were calculated as the number of dihedral transitions divided by the length of each simulation. In case of the multiple GaMD simulations, the average and standard deviation of dihedral transition rates were obtained. Finally, the rates were used to estimate acceleration of dihedral transitions obtained in GaMD simulations compared with cMD (Table 1).

Results and Discussion

Free energy landscape and kinetics of alanine dipeptide

Free energy profiles were computed from 1000 ns cMD simulation to quantitatively characterize different conformations of alanine dipeptide (Fig. 1A). Five free energy wells were identified in 2D PMF of backbone dihedrals (Φ, Ψ), which are centered around (−144°, 0°) and (−72°, −18°) for the right-handed α helix (α_R), (48°, −6°) for the left-handed α helix (α_L), (−150°, 156°) for the β-sheet and (−72°, 162°) for the polyproline II (P_II) conformation (Fig. 1B).

Time course of dihedral Φ obtained from the 1000 ns cMD simulation was plotted in Fig. 1C. It visited the α_R helix conformation (48° energy well) for a total of five times. Two close but separate visits took place during ~ 652.5–654.8 ns (see Fig. S1). Next, we examined conformational transitions of dihedral Φ across the 0° energy barrier. A transition going from −72° to 48° (α_R→ α_L) was denoted a “forward transition”, while one from 48° to −72° (α_L→ α_R) was a “backward transition”. During the 1000 ns cMD simulation, dihedral Φ underwent 4 times of forward transitions and another 3 times of backward transitions across the 0° energy barrier (Fig. 1C and Table 1). Note that three transitions of dihedral Φ also occurred across a 120° energy barrier, one into the 48° energy well near 652.7 ns (Fig. S1A) and another two out of it near 653.7 ns and 815.8 ns (Fig. S1A and Fig. 1C). Therefore, the rates of forward and backward transitions of dihedral Φ across the 0° energy barrier are 4 × 10⁶ s⁻¹ and 3 × 10⁶ s⁻¹, respectively (Table 1).

Fig. 1D shows a 1D PMF profile of dihedral Φ. It exhibits an energy barrier at 0° that separates two energy wells centered at −72° (α_R helix) and 48° (α_L helix). The energy barrier for forward transition of dihedral Φ from −72° to 48° is 6.07 kcal/mol and 3.45 kcal/mol for the backward transition from 48° to −72°. Curvature of the PMF profile was 2.40 near the −72° energy minimum, 2.95 near the 0° energy maximum and 2.88 near the 48° energy minimum (Table 2).

Table 2.

Energy barriers of forward (“Fwd”) and backward (“Bwd”) transitions of dihedral Φ at 0º, curvatures of free energy profiles calculated near −72º, 0º and 48º, and the apparent friction coefficient (ξ) calculated from cMD, dihedral GaMD and dual-boost GaMD simulations of alanine dipeptide, along with the acceleration of dihedral transitions in GaMD simulations derived using the Kramers’ theory. For GaMD simulations, ∆F and w are the free energy barriers and curvatures calculated from the reweighted PMF profiles, while ∆F* and w* are those calculated from the modified PMF profiles without energetic reweighting.

Sim	∆F (kcal/mol)		∆F* (kcal/mol)		w (x10−2)			w* (x10−2)			D (x1014 deg2·s−1)		Acceleration
	Fwd	Bwd	Fwd	Bwd	−72º	0º	48º	−72º	0º	48º	Fwd	Bwd	Fwd	Bwd
cMD	6.07	3.45	-	-	2.40	2.95	2.88	-	-	-	2.21±0.08	2.11±0.21	-	-
Dihedral GaMD	6.07±0.22	2.39±0.27	4.57±0.34	2.20±0.36	2.39	3.76	3.51	1.99	2.88	2.28	3.06±0.19	1.52±0.10	10.95±0.69	0.50±0.76
Dual-Boost GaMD	6.09±1.05	3.43±1.11	4.40±0.08	2.19±0.22	2.32	4.39	2.19	1.94	3.08	2.15	1.98±0.12	1.30±0.05	8.96±1.78	3.36±1.91

Acceleration obtained through dihedral GaMD simulations

With the three 30 ns dihedral GaMD simulations of alanine dipeptide, energetic reweighting was applied to calculate free energy profiles of backbone dihedrals Φ and Ψ. In comparison, reweighted PMF profiles obtained from the 30 ns dihedral GaMD trajectories agreed quantitatively with the original profiles from the 1000 ns cMD simulation. The reweighted 2D PMF profile showed that same energy wells corresponding to the α_R helix, α_L helix, β-sheet and P_II conformations as depicted in Fig. 2A. The GaMD derived PMF profile of Φ essentially overlapped with the original profile of the 1000 ns cMD simulation (Fig. 2D). This was similar for the dihedral Ψ as shown in Fig. S2A.

Fig. 2.

Accelerated conformational sampling of alanine dipeptide in dihedral GaMD simulations: (A) Recovered 2D PMF of backbone dihedrals (Φ, Ψ) calculated from reweighting three 30 ns dihedral GaMD simulations combined using cumulant expansion to the 2^nd order. Low energy wells are labeled as in Fig. 1B. (B) 2D PMF of backbone dihedrals (Φ, Ψ) calculated directly from the three 30 ns dihedral GaMD simulations combined without energetic reweighting. (C) Time course of dihedral Φ obtained from the three 30 ns dihedral GaMD simulations. (D) 1D PMF of dihedral Φ obtained from the three 30 ns dihedral GaMD simulations without and with energetic reweighting (denoted “Modified” and “Reweighted”, respectively) compared with that of the 1000 ns cMD simulation. For GaMD, 1D PMF profiles were calculated separately for each simulation, and their average and standard deviation (error bars) were plotted here.

Time course of dihedral Φ obtained from the three 30 ns dihedral GaMD simulations was plotted in Fig. 2C. In the three GaMD simulations, dihedral Φ underwent 6, 4 and 3 (13 in total) forward transitions of α_R→ α_L and 5, 4 and 4 (13 in total) backward transitions of α_L→ α_R across the 0° energy barrier, respectively. It also climbed over the 120° energy barrier twice, one into the 48° energy well near 5.0 ns in “Sim3” and the other out of it near 2.75 ns in “Sim1” (Fig. S1B). Thus, the rates of forward and backward transitions of dihedral Φ across the 0° energy barrier were 144±50 × 10⁶ s⁻¹ and 144±19 × 10⁶ s⁻¹, respectively. Relative to cMD, the speedup obtained from dihedral GaMD simulations was 36.1±12.7 for the forward transitions and 48.1±6.4 for the backward transitions of dihedral Φ across the 0° energy barrier (Table 1).

Additionally, PMF profiles were computed without energetic reweighting from the dihedral GaMD simulations. Compared with cMD or GaMD-reweighted free energy profiles, broader energy wells and lower energy barriers were found in the GaMD-modified free energy profiles, such as the 2D PMF shown in Fig. 2B and 1D modified PMF of Φ (Fig. 2D) and Ψ (Fig. S2A).

In the reweighted PMF profile of dihedral Φ, energy barriers of α_R→ α_L forward transitions and α_L→ α_R backward transitions were 6.07±0.22 kcal/mol and 2.39±0.27 kcal/mol, respectively. Curvature of the reweighted PMF profile was 2.39 near the −72° energy minimum, 3.76 near the 0° energy maximum and 3.51 near the 48° energy minimum (Table 2). In the modified PMF profile without energetic reweighting, the energy barrier of the forward transitions was decreased to 4.57±0.34 kcal/mol and 2.20±0.36 kcal/mol for the backward transitions. Moreover, modified PMF curvatures were decreased 1.99, 2.88 and 2.28 near the −72° energy minimum, 0° energy maximum and 48° energy minimum, respectively (Table 2).

Higher acceleration in dual-boost GaMD simulations

Similar to the above analysis of dihedral GaMD simulations, we calculated the free energy profiles and kinetic rates of backbone dihedrals Φ and Ψ in alanine dipeptide from the three 30 ns dual-boost GaMD simulations. In comparison, the reweighted PMF profiles obtained from dual-boost GaMD simulations also agreed quantitatively with the original profiles from 1000 ns cMD simulation. The reweighted 2D PMF of dihedrals Φ and Ψ showed that same energy wells corresponding to the α_R helix, α_L helix, β-sheet and P_II conformations (Fig. 3A). The reweighted PMF of Φ overlapped with the original profile from 1000 ns cMD simulation (Fig. 1D), similarly for Ψ (Fig. S2B).

Fig. 3.

Accelerated conformational sampling of alanine dipeptide in dual-boost GaMD simulations: (A) Recovered 2D PMF of backbone dihedrals (Φ, Ψ) calculated from reweighting three 30 ns dual-boost GaMD simulations combined using cumulant expansion to the 2^nd order. Low energy wells are labeled as in Fig. 1B. (B) 2D PMF of backbone dihedrals (Φ, Ψ) calculated directly from the three 30 ns dual-boost GaMD simulations combined without energetic reweighting. (C) Time course of dihedral Φ obtained from the three 30 ns dual-boost GaMD simulations. (D) 1D PMF of dihedral Φ obtained from the three 30 ns dual-boost GaMD simulations without and with energetic reweighting (denoted “Modified” and “Reweighted”, respectively) compared with that of the 1000 ns cMD simulation. For GaMD, 1D PMF profiles were calculated separately for each simulation, and their average and standard deviation (error bars) were plotted here.

Furthermore, PMF profiles were computed without energetic reweighting from the dual-boost GaMD simulations (Figs. 3B, 3D and S2B). Similarly, we observed decreases of free energy barriers and curvatures in the GaMD-modified PMF profiles than in the original cMD or GaMD-reweighted PMF profiles (Table 2). Particularly for dihedral Φ at 0°, reweighted free energy barriers for the α_R→ α_L forward transitions and α_L→ α_R backward transitions were 6.09±1.05 kcal/mol and 3.43±1.11 kcal/mol, respectively. Curvature of the reweighted PMF profile was 2.32 near the −72° energy minimum, 4.39 near the 0° energy maximum and 2.19 near the 48° energy minimum. In the GaMD-modified PMF profile, the energy barriers were decreased to 4.40±0.08 kcal/mol for the forward transitions and 2.19±0.22 kcal/mol for the backward transitions. Modified PMF curvatures were decreased to 1.94, 3.08 and 2.15 near −72°, 0° and 48°, respectively (Table 2).

In the three 30 ns dihedral GaMD simulations, dihedral Φ underwent 6, 4 and 7 (18 in total) forward transitions of α_R→ α_L and 5, 5 and 8 (18 in total) backward transitions of α_L→ α_R across the 0° energy barrier (Fig. 3C). It climbed over the 120° energy barrier twice, one into the 48° energy well near 23.6 ns in “Sim3” and the other out of it near 24.3 ns in “Sim1” (Fig. S1C). The rates of forward and backward transitions of Φ across the 0° energy barrier were 200±33 × 10⁶ s⁻¹ and 200±58 × 10⁶ s⁻¹, respectively. Higher acceleration was thus achieved in dual-boost GaMD simulations than in dihedral GaMD simulations. Relative to cMD, the speedup in dual-boost GaMD simulations is 50.0±8.3 for the forward transitions and 66.7±19.2 for the backward transitions of dihedral Φ across the 0° energy barrier (Table 1).

Application of Kramers’ rate theory to estimate GaMD acceleration

In addition to the free energy barriers and curvatures, we calculated apparent diffusion coefficients of dihedral Φ in alanine dipeptide from the cMD and GaMD simulations. Survival functions S(t) were computed both directly using the transition time series collected from the simulations and numerically by solving the Smoluchowski equation along the 1D PMF profile of Φ (Methods). Fig. 4A plots the survival functions calculated directly for the forward transition of dihedral Φ across the 0º energy barrier in the cMD and GaMD simulations. The dihedral and dual-boost GaMD simulations exhibited significantly faster decay of the survival functions compared the cMD simulation. The faster decay of survival functions and thus higher kinetic rates resulted from decreased energy barriers in the GaMD enhanced simulations. Similar results were observed in the survival functions calculated by solving the Smoluchowski equation (Fig. 4B). For the Smoluchowski equation, a reflective boundary was set near the left-side barrier of the energy well at −216 º with the dihedral periodicity taken into account and an absorbing boundary was set on top of the right-side barrier at 0º.

Fig. 4.

(A-B) Exponential decay of survival functions S(t) for the forward transition of dihedral Φ across the 0º energy barrier obtained from: (A) direct calculation using the transition time series collected from the simulations and (B) numerically solving the Smoluchowski equation along the 1D PMF profile of Φ. (C-D) Exponential decay of the survival functions S(t) for backward transition of dihedral Φ across the 0º energy barrier obtained from: (C) direct calculation using the transition time series collected from the simulations and (D) numerically solving the Smoluchowski equation along the 1D PMF profile of Φ. The survival functions were calculated using the cMD, dihedral GaMD and dual-boost GaMD simulations.

With the survival functions S(t), we estimated the effective dihedral kinetic rates as the negative of the slopes in linear fitting of ln[S(t)] versus t. They were then used to calculate the apparent diffusion coefficients. Results showed that the diffusion coefficients for the forward transition of dihedral Φ across the 0º energy barrier were 2.21±0.08 × 10¹⁴ deg²·s⁻¹ in cMD simulation, 3.06±0.19 × 10¹⁴ deg²·s⁻¹ in dihedral GaMD simulations and 1.98±0.12 × 10¹⁴deg²·s⁻¹ in the dual-boost GaMD simulations.

For the backward transition of dihedral Φ across the 0º energy barrier, survival functions S(t) were calculated as the probability that Φ was trapped in the 48º energy well longer than time t (Fig. 4C). For the Smoluchowski equation, the top of the left-side barrier of the energy well (0º) was set as an absorbing boundary and a reflective boundary was set near the top of the right-side barrier at 96º. Fig. 4D shows exponential decay of the survival functions that are calculated using solutions of the Smoluchowski equation. Next, we also estimated the effective dihedral kinetic rates and then the apparent diffusion coefficients. The final diffusion coefficients for the backward transition of dihedral Φ across the 0º energy barrier were 2.11±0.21 × 10¹⁴ deg²·s⁻¹ in cMD simulation, 1.52±0.10 × 10¹⁴ deg²·s⁻¹ in dihedral GaMD simulations and 1.30±0.05 × 10¹⁴deg²·s⁻¹ in the dual-boost GaMD simulations. Overall, the apparent diffusion coefficients of dihedral Φ obtained in the present study of the alanine dipeptide appeared to be smaller than those of the dihedral ω in cis-trans isomerization of N-methylacetamide (D=1.38±0.02 × 10¹⁵ deg²·s⁻¹), although the N-methylacetamide was simulated in implicit solvent instead⁴⁹.

Finally, we applied Kramers’ rate theory to estimate acceleration of dihedral transitions in GaMD simulations using changes in the diffusion coefficients and free energy barriers and curvatures of alanine dipeptide. Relative to cMD, the calculated speedup in dihedral GaMD simulations was 10.95±0.69 for sampling forward transitions of dihedral Φ across the 0° energy barrier, while only 0.50±0.76 for backward transitions of dihedral Φ. The latter with large standard deviation likely resulted from insufficient sampling of the backward dihedral transitions in cMD as well as dihedral GaMD simulations. A significantly lower free energy barrier (2.39±0.27 kcal/mol) was obtained from reweighting the dihedral GaMD simulations compared with that of long-timescale cMD simulation (3.45 kcal/mol) or dual-boost GaMD simulation (3.43±1.11 kcal/mol) as shown in Table 2. In the dual-boost GaMD simulations, speedup over cMD estimated by applying Kramers theory was 8.96±1.78 for the forward transitions and 3.36±1.91 for the backward transitions of Φ (Table 2). Therefore, except backward transition of Φ in the dihedral GaMD simulations, application of Kramers’ rate theory also showed significant acceleration of dihedral transitions in the GaMD simulations. Depending on the studied dihedral transition, the acceleration factors tended to be underestimated by ~3–96 fold by applying Kramers’ rate theory. Nevertheless, significant acceleration of the biomolecular kinetics was successfully achieved through GaMD as shown for the dihedral transitions rates in Table 1.

In a previous study, De Oliveira et al. estimated the kinetic transition rates of dihedral Ψ in alanine dipeptide using shorter (20 ns) cMD and aMD simulations⁴⁷. In the aMD simulations, boost potential was added to only the total potential energetic term (“total-boost” aMD). The transition of Ψ across the 90° energy barrier from the α helix to β strand (Fig. 1B and S2) was examined. Relative to cMD, the transition rate of Ψ was increased by ~2–11 in the aMD simulations⁴⁷. Although dihedral³⁰ and dual-boost aMD³¹ may give better performance, the present dihedral and dual-boost GaMD simulations appear to provide higher acceleration (~36–67) of conformational transitions in the same system.

Conclusions

In this study, we have examined acceleration of biomolecular kinetics in the GaMD simulations, using alanine dipeptide in explicit solvent as a model system. The small system allowed us to perform converged cMD simulation lasting 1000 ns for comparison. For transitions of backbone dihedral Φ in alanine dipeptide, dihedral GaMD simulations exhibited ~36–48 times speedup and ~50–67 times speedup in dual-boost GaMD simulations relative to cMD. Furthermore, we applied Kramer’s rate theory to estimate GaMD acceleration using the simulation-derived friction coefficients, curvatures and barriers of free energy profiles. In most cases, we also obtained significant speedup of dihedral transitions in the GaMD simulations, although the acceleration factors were underestimated by ~3–96 fold compared with direct calculations using the available dihedral transition rates. Previous studies showed that greater boost potential can be applied in GaMD simulations of systems with increased sizes and more rugged energy landscapes^{37, 39–41}. Higher acceleration is thus expected in GaMD simulations larger biomolecules. It is subject to future studies on accelerating the dynamics and recovering kinetic rates of the larger biomolecules such as proteins and protein-protein/nucleic acid complexes.