Unconstrained Enhanced Sampling for Free Energy Calculations of Biomolecules: A Review

Abstract

Free energy calculations are central to understanding the structure, dynamics and function of biomolecules. Yet insufficient sampling of biomolecular configurations is often regarded as one of the main sources of error. Many enhanced sampling techniques have been developed to address this issue. Notably, enhanced sampling methods based on biasing collective variables (CVs), including the widely used umbrella sampling, adaptive biasing force and metadynamics, have been discussed in a recent excellent review (Abrams and Bussi, Entropy, 2014). Here, we aim to review enhanced sampling methods that do not require predefined system-dependent CVs for biomolecular simulations and as such do not suffer from the hidden energy barrier problem as encountered in the CV-biasing methods. These methods include, but are not limited to, replica exchange/parallel tempering, self-guided molecular/Langevin dynamics, essential energy space random walk and accelerated molecular dynamics. While it is overwhelming to describe all details of each method, we provide a summary of the methods along with the applications and offer our perspectives. We conclude with challenges and prospects of the unconstrained enhanced sampling methods for accurate biomolecular free energy calculations.

1. Introduction

Ever since the first simulation of bovine pancreatic trypsin inhibitor (BPTI) in 1977[1], molecular dynamics (MD) has embraced a new era for its wide applications in computational modeling of biomolecules[2]. The simulated systems range from proteins to membrane lipids and nucleic acids. Based on a molecular force field model, MD solves the Newton’s equations of motion for all atoms in the system along time. From the atomic trajectory, one is able to extract structural and dynamical information on the biomolecules that can be compared and used to interpret experimental data obtained from nuclear magnetic resonance (NMR)[3–5] and neutron scattering[6, 7]. MD simulations have been also found use in refinement of structures determined by cryo-electron microscopy[8, 9], X-ray crystallography[10–12] and NMR[13, 14].

In addition, MD simulations have been used to calculate free energies that determine the behavior of biomolecules at or near equilibrium. In the simplest form, the free energy of biomolecules in a canonical ensemble is calculated as F = −k_BT ln Q, where k_B is the Boltzmann’s constant, T is the system temperature and Q is the system partition function. Free energies describe the spontaneity of biomolecular transitions from one configuration state to another. They are central to understanding the structure, dynamics and function of the biomolecules. Examples include ligand binding to proteins, conformational change of large proteins during enzyme catalysis or cell signaling, and permeation of small-molecule drugs through cell membrane as discussed in a long list of previous review articles[15–18].

Whereas MD simulations provide a detailed picture of the biomolecular dynamics in atomistic detail, current MD simulations are often limited to tens of microseconds for average-sized proteins or even shorter timescales for larger systems[19]. Different functional states of biomolecules are often separated by high energy barriers (e.g., 8–12 kcal/mol) and structural transitions across the energy barriers take place over milliseconds or even longer timescales[20, 21]. Therefore, there remains a large gap between the MD simulation timescales and those of biological processes. This often leads to insufficient sampling or unconverged simulations of the biomolecules. It has been described as one of the main sources of error in biomolecular free energy calculations[15, 17, 18].

2. Why unconstrained enhanced sampling simulations are needed

To overcome the insufficient sampling problem of free energy calculations, many enhanced sampling techniques have been developed during the last several decades as reviewed earlier[22–27]. The first class of these methods is based on biasing collective variables (CVs) of biomolecules. They include the widely used umbrella sampling[28, 29], blue moon sampling[30], adaptive biasing force[31, 32], metadynamics[33, 34], conformational flooding[35], string method[36, 37], λ-dynamics[38], adiabatic free energy dynamics[39], temperature accelerated MD[40] and targeted MD[41–43].

The above enhanced sampling methods greatly improve the convergence of free energy calculations, but they require predefined CVs and often suffer from hidden energy barriers. In Figure 1, a hidden energy barrier is illustrated for enhanced sampling simulations that are based on biasing CVs. Without the loss of generality, there are two local energy minimum states for a given system along the y direction. If only x is defined as the CV, apparently converged sampling of the free energy surface along the x direction will be obtained from the corresponding enhanced sampling simulations, but not for the hidden energy barrier along the y direction. Therefore, when important CVs are missed during the simulation setup, the CV-biasing enhanced sampling methods often suffer from insufficient sampling of the hidden energy barrier(s), which leads to poor convergence of the simulations. Abrams and Bussi noted in their recent review[22] that: “the drawback of all CV-biasing approaches is the risk that the chosen CV space does not provide the most faithful representation of the true spectrum of metastable subensembles and the barriers that separate them. Guaranteeing that sampling of CV space is not stymied by hidden barriers must be of paramount concern in the continued evolution of such methods”.

Fig. 1.

Illustration of an example hidden energy barrier along the y direction between two local energy minima.

Improved enhanced sampling methodology has been developed to address the “hidden barrier” issue, notably the orthogonal space random walk (OSRW)[44, 45], the essence of which is to perform a simultaneous random walk in the orthogonal space along the directions of a CV and its generalized force. Moreover, there exists a second class of enhanced sampling methods that do not require predefined system-dependent CVs and as such do not suffer from the hidden barrier problem as encountered in the CV-biasing methods, such as replica exchange (RE)[46, 47] or parallel tempering (PT)[48], self-guided molecular/Langevin dynamics (SGMD and SGLD)[49–52], essential energy space random walk (EESRW)[53–55] and accelerated molecular dynamics (aMD)[56, 57]. Compared with the CV-biasing enhanced sampling methods, these methods are advantageous to explore biomolecular configurations without the need for a priori knowledge. They provide unconstrained enhanced sampling to explore biomolecular structural transition pathways (e.g., protein folding and ligand binding) and identify possible unknown intermediate configuration states of the biomolecules.

3. Unconstrained enhanced sampling methods

Here, we consider a list of unconstrained enhanced sampling methods, including the RE/PT, SGMD/SGLD, EESRW and aMD. It is not possible to describe all details of each method in this review. Instead, we provide a summary of the methods along with their applications and offer our perspectives.

3.1 Replica Exchange/Parallel Tempering

As illustrated in Figure 2, Replica Exchange/Parallel Tempering (RE/PT) methods are based on running multiple copies (replicas) of parallel MD (or Monte Carlo) simulations at different temperatures and exchanging system configurations (or equivalently exchanging temperatures) between pairs of replicas at a certain frequency to achieve enhanced sampling. The replicas are exchanged between each pair of neighboring temperatures at a probability that meets the Metropolis criterion. It ensures that the generalized ensemble of the system satisfies a detailed balance and converges towards an equilibrium distribution. Meanwhile, the replica exchanges avoid the trapping of biomolecules in local energy-minimum configurations as simulated at low, room temperature and facilitate barrier crossing events in the biomolecules.

Fig. 2.

Schematic illustration of Replica Exchange Molecular Dynamics (REMD): a number of M (e.g., 4 as shown here) non-interacting replicas of MD runs are performed in the canonical ensemble, each at a different temperature T_m (m = 1, ⋯, M). At a certain simulation time interval, we exchange each pair of replicas with neighboring temperatures at a probability that meets the Metropolis criterion. The generalized ensemble of the system then satisfies a detailed balance and converges towards an equilibrium distribution during the simulation.

Consider a system of N atoms with positions r ≡ {r₁, ⋯, r_N} and momenta p ≡ {p₁, ⋯, p_N}. In the canonical ensemble at temperature T, each state x ≡ (r,p) is weighted by the Boltzmann factor:

$$W_B(x;T)=\text{exp}\{-\beta H(r,p)\},$$ (1)

where β = 1/k_BT with k_B as the Boltzmann constant and H(r,p) is the system Hamiltonian that is given by the sum of the potential energy V(r) and kinetic energy K(p).

In RE/PT, the generalized ensemble consists of M non-interacting replicas of the original system in the canonical ensemble, each at a different temperature T_m (m = 1, ⋯, M). There is a one-to-one correspondence between the replicas and temperatures. The label i (i = 1, ⋯, M) for replicas is a permutation of the label m (m = 1, ⋯, M) for temperatures, i.e., i = i(m) ≡ f(m), and vice versa m = m(i) ≡ f⁻¹(i). f(m) is a permutation function of m and f⁻¹(i) is its inverse. Let $X=(x_{\mathit{1}}^{[i(l)]},\cdots ,x_M^{[i(M)]})$ stand for a “state” in the generalized ensemble. Each “substate” $x_m^{[i]}$ is specified by the coordinates r^[i] and momenta p^[i] of N atoms in replica i at temperature T_m: $x_m^{[i]}={(r^{[i]},p^{[i]})}_m$. Because the replicas are non-interacting, the weight factor for the state X in this generalized ensemble is given by the product of the Boltzmann factors of each replica:

$$W_{RE}(X)=\prod _{i=\mathit{1}}^M\text{exp}\{-{\beta }_{m(i)}H(x_{m(i)}^{[i]}\left)\right\}=\text{exp}\{-\sum _{i=\mathit{1}}^M{\beta }_{m(i)}H(x_{m(i)}^{[i]}\left)\right\}.$$ (2)

To help the biomolecules escape from local energy minimum states, we exchange a pair of replicas with neighboring temperatures at probability w(X → X′). Such replica exchange needs to satisfy a detailed balance in order to converge towards an equilibrium distribution:

$$W_{RE}(X)w(X\to X\prime )=W_{RE}(X\prime )w(X\prime \to X).$$ (3)

This can be met when one uses the Metropolis criterion to calculate the exchange probability:

$$w(X\to X\prime )\equiv w(x_m^{[i]}|x_n^{[j]})=\text{min}(\mathit{1},\text{exp}(-\mathrm{\Delta })),$$ (4)

where Δ = (β_n − β_m)[V(r^[i]) − V(r^[j])], and i, j, m and n are related by the permutation functions before the exchange: i = f(m), j = f(n).

Compared with the simulated tempering (ST)[58, 59] and mulicanonical algorithms (MUCA)[60] that perform random walk in the temperature and potential energy spaces, respectively, the RE/PT methods are advantageous with a priori known weight factors. Canonical ensemble properties are readily obtained from the RE/PT simulations. Many variants of the RE/PT methods have been also developed for further enhanced sampling[47], such as the replica-exchange multicanonical algorithm[61, 62], replica-exchange simulated tempering[63], multicanonical replica-exchange[61], simulated tempering replica-exchange[63] and ab initio replica-exchange[64]. Moreover, multidimensional extension of REMD, multidimensional replica-exchange[65], has led to the development of the replica exchange umbrella sampling[66] and Hamiltonian REMD[67, 68] methods.

Instead of using spatial CVs, the RE/PT methods are based on exchanging temperature between parallel MD or Monte Carlo simulations. This allows for unconstrained conformational sampling of the biomolecules, which has enabled a wide range of applications, for example, in conformational transitions of Argon fluid[69] and peptides[70, 71], protein folding[46, 72–74], protein structural refinement[75], ligand binding free energy calculations[65], DNA base stacking[76], nucleic acid conformational change[67], etc.

However, it has been noted that the RE/PT methods do not work for processes with the first-order phase transitions. Moreover, the number of replicas required for converged RE/PT simulations is proportional with the square root of the system number of degrees of freedom. This suggests that for systems of increasing size, a significantly larger number of replicas are needed are needed to run the simulations. This often requires expensive computational resources. One solution for this is to first run a short RE/PT simulation to determine the mulitcanonical weight factor and then perform with this weight factor a regular multicanonical simulation with high statistics. This is reflected in the replica-exchange multicanonical algorithm[61, 62] and replica-exchange simulated tempering method[63, 77].

3.2 Self-guided molecular/Langevin Dynamics

Self-guided molecular dynamics (SGMD)[49, 50] was developed to improve the conformational searching efficiency of MD simulations. It is based on the rationale that the conformational searching efficiency is often limited by slow systematic motions. To accelerate these systematic motions, a self-guiding force was introduced into the Newton’s equation of motion:

$$m_i{\ddot{\mathit{r}}}_i={\mathit{f}}_i+{\mathit{g}}_i,$$ (5)

where m_i is the mass of atom i, r̈_i is the acceleration of atom i with altered systematic motion, f_i is the atom instantaneous force derived from force field, g_i is the guiding force calculated as:

$${\mathit{g}}_i=\lambda \frac{\mathit{1}}{t_{\mathit{1}}}{\int }_{t-t_{\mathit{1}}}^t({\mathit{f}}_i+{\mathit{g}}_i)d\tau ,$$ (6)

where λ is a guiding factor and t_L is the local averaging time. Moreover, bonded substructures were defined in SGMD to describe the local structural rigidity of a molecule[50]. It was suggested that most of the atomic motions due to bonded interactions within each substructure are fast. In contrast, systematic motions that result from non-bonded interactions between atoms of different substructures are generally slow. Thus the self-guiding force was applied to enhance these systematic motions only. With this, SGMD was shown to greatly improve the sampling of conformational transitions in the alanine dipeptide and a 16-residue synthetic polypeptide that undergoes conformational change between a fully extended conformation and a folded helix[50].

However, several drawbacks were found during applications of SGMD. Among those noted by Wu and Brooks[51], the self-guiding force results in an unwanted alteration of the conformational distribution in SGMD simulations. Moreover, the guiding force derived from the local average of actual forces may not be sufficient to enhance conformational searching in stochastic dynamics simulations. To address these issues, the local average of friction forces that appear in a Langevin dynamics (LD) simulation was proposed to calculate the guiding force instead, which led to the development of the self-guided Langevin dynamics (SGLD) method. In SGLD, the self-guiding force was introduced into the Langevin equation:

$${\mathit{ṗ}}_i={\mathit{f}}_i+{\mathit{g}}_i-{\gamma }_i{\mathit{p}}_i+{\mathit{R}}_i,$$ (7)

where p_i is the momentum of atom i, f_i is the atom instantaneous force derived from the force field, γ_i is the collision frequency and R_i is the random force that is related to γ_i and the system temperature T by 〈R_i(0) R_i(t)〉 = 2m_ik_BTγ_iδ(t), where k_B is the Boltzmann constant and δ(t) is the Dirac delta function. In the case that the time derivative of momentum can be dropped from the Langevin equation (i.e., high friction regime), the guiding force g_i is calculated as:

$${\mathit{g}}_i=\lambda {\gamma }_i{\langle {\mathit{p}}_i\rangle }_L-\xi {\gamma }_i{\mathit{p}}_i,$$ (8)

where λ is a guiding factor that controls the strength of the guiding force. 〈p_i〉_L denotes local average of p_i over time t_L and can be calculated efficiently during simulation via:

$${\langle {\mathit{p}}_i\rangle }_L(t)\approx (\mathit{1}-\frac{\partial t}{t_{\mathbf{L}}}){\langle {\mathit{p}}_i\rangle }_L(t-\partial t)+\frac{\partial t}{t_{\mathbf{L}}}{\mathit{p}}_i(t).$$ (9)

The local averaging acts like a low frequency filter that reduces the high frequency components of the motion while keeping its low frequency contributions. The parameter, ξ, is an energy conservation factor used to cancel net energy input from the guiding force: $\xi =\frac{\lambda \sum _i{\langle {\mathit{p}}_i\rangle }_L·{\mathit{ṙ}}_i}{\sum _i{\mathit{p}}_i·{\mathit{ṙ}}_i}$.

According to Eqn. (7), when the collision frequency and random force are equal to zero, SGLD is then reduced to SGMD. In this context, MD and LD are special cases of SGMD and SGLD, respectively, when the guiding force is zero. Up to date a list of SGLD variants has been developed using different formula of the guiding force (see Ref. [78] for details). Particularly, SGMD calculates the guiding force with nonbonded interactions as described above [49, 50], SGLD uses momenta[51] and is also referred to as SGLDp, and SGLDfp uses both forces and momenta[79]. In the most recent development of SGLD, the self-guiding force is modified according to a generalized Langevin equation using ${\mathit{g}}_i=\lambda {\gamma }_i{\langle {\mathit{p}}_i\rangle }_L-(\mathit{1}-\sqrt{\mathit{1}-\lambda }){\langle {\mathit{R}}_i\rangle }_L$ to sample exactly the NVT and NPT ensembles, i.e., SGLD-GLE[52].

Along with the theoretical development of the SGMD and SGLD methods, they have been applied for enhanced sampling of a wide range of systems, such as peptide conformational change[50], surface adsorption of complementary peptides[80], flexible fitting of biomolecular structures into electron microscopic density maps[81], protein folding and conformational rearrangements[49, 51], molecular docking[82], ligand binding[83, 84], crystallization and phase transitions[85], and so on.

3.3 Essential Energy Space Random Walk

Essential Energy Space Random Walk (EESRW)[53, 54] is an enhanced sampling method based on scaling perturbations of the biomolecular essential potential energy. Here, the essential potential energy U_s is defined as the sum of the energy terms that determine the local conformations of a region of interest. In case that the solute in a solution system is selected as the target region, U_s can be decomposed to two parts: the self-energy component U_ss that is calculated as the sum of self-interaction terms within the solute except for the bond, angle and improper energy terms and the interaction energy component U_se that is calculated as the sum of all the interaction terms between the solute and solvent. The sum of the remaining energy terms gives the environmental energy component U_e with the system total energy U₀ = U_s + U_e. Provided the U_s essential energy, a biasing potential is added to the biomolecular system to flatten its energy surface for enhanced sampling[53, 54]:

$$U=U_{\mathit{0}}+f_m\left(U_s\right),$$ (10)

where U₀ is the original system potential energy, U_s is the essential potential energy and f_m(U_s) is a relatively small Gaussian-shaped repulsive potential calculated similar to the metadynamics[33, 34] approach:

$$f_m\left(U_s\right)=\sum _{\underset{t_i<t}{t_i=\tau ,2\tau ,\cdots }}h\text{ exp }(-\frac{{|U_s-U_s(t_i\left)\right|}^{\mathit{2}}}{2w^2}),$$ (11)

where τ is the frequency at which the Gaussian repulsive potential is added, h is the Gaussian height and w is the Gaussian width. EESRW has been implemented in CHARMM and successfully applied to accelerate conformational transitions in the alanine dipeptide, leucine sidechain and a “pentanelike” biomolecular model system[53, 54].

However, the U_ss and U_se essential energy components were found to fluctuate in various magnitudes with even opposite directions during many conformational transitions of biomolecules (e.g., alanine dipeptide). A one-dimensional potential biasing scheme in the original EESRW method is not able to capture such energy fluctuations effectively. To resolve this issue, EESRW was generalized to a two-dimension-EESRW (2D-EESRW) scheme[55], in which the modified system potential is given by:

$$U=U_{\mathit{0}}+f_m(U_{ss},U_{se}),$$ (12)

where U_ss and U_se are the essential self-energy and interaction energy components, respectively, and f_m(U_ss, U_se) is the negative of the free energy G(U_ss, U_se) that is obtained from the free energy profile along (U_ss, U_se). Similar to Eqn. (11), the f_m(U_ss, U_se) term is calculated by repetitively adding a relatively small Gaussian-shaped repulsive potential:

$$f_m(U_{ss},U_{se})=\sum _{\underset{t_i<t}{t_i=\tau ,2\tau ,\cdots }}h\text{exp}(-\frac{{|U_{ss}-U_{ss}(t_i)|}^{\mathit{2}}}{\mathit{2}{w}_{ss}^{\mathit{2}}})\text{exp}(-\frac{{|U_{se}-U_{se}(t_i)|}^{\mathit{2}}}{2w_{se}^2}),$$ (13)

where τ is the frequency at which the Gaussian repulsive potential is added, h is the Gaussian height, and w_ss and w_se are the widths of Gaussians corresponding to the essential potential energies U_ss and U_se, respectively.

The 2D-EESRW was shown to accelerate conformational sampling of the alanine dipeptide and aspartate-arginine peptide model systems[55]. Compared with the original EESRW, the 2D-EESRW simulations exhibit significantly higher rates of conformational transitions in the model systems. It was also suggested that the EESRW methods could be extended to higher dimension schemes when three or more essential energetic terms are needed to account for their distinct interplays during biomolecular conformational transitions.

3.4 Accelerated Molecular Dynamics (aMD)

Figure 3 depicts the central idea of the original Accelerated Molecular Dynamics (aMD) that was developed as a potential-biasing method for enhanced sampling[56]. It accelerates the conformational sampling of biomolecules often by adding a non-negative boost potential to the potential energy surface when the system potential is lower than a reference energy[56, 86, 87]:

$$V^{*}(r)=V(r),V(r)\ge E,$$

$$V^{*}(r)=V(r)+\mathrm{\Delta }V(r),V(r)<E,$$ (14)

where V(r) is the original potential, E is the reference energy, and V^* (r) is the modified potential. The boost potential, ΔV(r)is given by:

$$\mathrm{\Delta }V(r)=\frac{{(E-V(r))}^2}{\alpha +E-V(r)},$$ (15)

where α is the acceleration factor. As the acceleration factor α decreases, the potential energy surface is flattened and biomolecular transitions between the low-energy states are increased (Figure 3).

Fig. 3.

Schematic illustration of Accelerated Molecular Dynamics (aMD): a non-negative boost potential is added to biomolecular potential surface when the system potential is lower than a threshold energy E. As the acceleration factor α decreases, the potential energy surface is flattened more and transitions between different low-energy states become increased.

In aMD, the boost potential was initially applied to only the dihedral energy term based on the finding that the dihedral torsions are determining factors of the protein conformations[56]. This “dihedral-boost” was found, however, to be still insufficient for conformational sampling of many biomolecules. Thus another boost was added to the system total potential energy, which accelerates diffusion of the solvent molecules and provides further enhanced sampling of the biomolecules. This led to the development of “dual-boost” aMD[86].

In addition, a number of different forms of aMD have been developed to improve the sampling statistics of bimolecular configurations, including the windowed aMD designed to protect the high-energy barriers[88], adaptive aMD[89], rotatable aMD or RaMD[90] and selective aMD[91]. With the enhanced sampling power, aMD has also been combined with many other techniques, including the thermodynamic integration[92], replica exchange[93–95], constant pH MD[96] and ab initio MD [97, 98], for various classical and quantum computational simulations of biomolecules.

A wide range of applications has shown that aMD works well for enhanced sampling of biomolecules[87]. For example, aMD simulations provide significant speed-up of the peptide conformational transitions[56, 99, 100], lipid diffusion and mixing[101], conformational transitions in proteins[89, 102–105], protein folding[106, 107] and protein-ligand binding[108]. Notably, hundreds-of-nanosecond aMD simulations are to capture millisecond-timescale events in both globular and membrane proteins[109–112].

While aMD is powerful for enhanced conformation sampling, its accuracy (particularly reweighting) for free energy calculations has attracted lots of attention[113]. In theory, frames of aMD simulations can be reweighted by the Boltzmann factors of the corresponding boost potential (i.e., e^ΔV/k_BT) and averaged over each bin of selected reaction coordinate(s) to obtain the canonical ensemble, an algorithm termed “exponential average”. However, exponential reweighting is known to suffer from large statistical noise in practical calculations[87, 111, 113, 114] because the Boltzmann reweighting factors are often dominated by a very few frames with high boost potential. To reduce the energetic noise, the RaMD[90] and selective aMD[91] methods were developed to apply boost potential to only rotatable dihedrals or certain regions of the simulated system. Alternatively, scaled MD flattens the biomolecular potential energy surface with a scaling factor[114], a similar notion used in EESRW[53, 54]. It enables population-based reweighting and significantly improves the free energy calculations. However, scaled MD does not allow selective acceleration of biomolecular conformational changes like aMD.

Overall, the boost potential in aMD simulations of proteins is on the order of tens to hundreds of kcal/mol, which is much greater in magnitude and wider in distribution than that of CV-biasing simulations (e.g., several kcal/mol in metadynamics). It has been a long-standing problem to accurately reweight aMD simulations and recover the original free energy landscapes, especially for large proteins[108, 111]. A recent study showed that when the boost potential follows near-Gaussian distribution, cumulant expansion to the second order provides improved reweighting of aMD simulations compared with the previously used exponential average and Maclaurin series expansion reweighting methods[115]. The reweighted free energy profiles are in good agreement with the long-timescale cMD simulations as demonstrated on alanine dipeptide and fast-folding proteins[107]. However, such improvement is limited to rather small systems (e.g., proteins with less than ~35 amino acid residues)[107]. In simulations of larger systems, the boost potential exhibits significantly wider distribution and does not allow for accurate reweighting.

In order to achieve both unconstrained enhanced sampling and accurate energetic reweighting for protein simulations, Gaussian Accelerated Molecular Dynamics (GaMD) was developed by applying a harmonic boost potential instead to smoothen the system potential energy surface[57]. As shown in Figure 4, when the system potential V(r⃑) drops below a threshold energy E, the energy surface is modified by adding a boost potential as:

$$\mathrm{\Delta }V(\stackrel{\rightharpoonup }{r})=\frac{\mathit{1}}{\mathit{2}}k{(E-V(\stackrel{\rightharpoonup }{r}))}^{\mathit{2}},V(\stackrel{\rightharpoonup }{r})<E,$$ (16)

where k is the harmonic force constant. The two adjustable parameters E and k are automatically determined by applying the following three criteria. First, for any two arbitrary potential values V₁(r⃑) and V₂(r⃑) found on the original energy surface, if V₁(r⃑)<V₂(r⃑), ΔV should be a monotonic function that does not change the relative order of the biased potential values, i.e., $V_{\mathit{1}}^{*}(\stackrel{\rightharpoonup }{r})<V_{\mathit{2}}^{*}(\stackrel{\rightharpoonup }{r})$. Secondly, if V₁(r⃑) < V₂(r⃑), the potential difference observed on the smoothened energy surface should be smaller than that of the original, i.e., $V_{\mathit{2}}^{*}(\stackrel{\rightharpoonup }{r})-V_{\mathit{1}}^{*}(\stackrel{\rightharpoonup }{r})<V_{\mathit{2}}(\stackrel{\rightharpoonup }{r})-V_{\mathit{1}}(\stackrel{\rightharpoonup }{r})$. By combining the first two criteria and plugging in the formula of V^*(r⃗) and ΔV, we obtain:

$$V_{\text{max}}\le E\le V_{\text{min}}+\frac{1}{k},$$ (17)

where V_min and V_max are the system minimum and maximum potential energies. To ensure that Eqn. (17) is valid, k has to satisfy: $k\le \frac{\mathit{1}}{V_{\text{max}}-V_{\text{min}}}$. Let us define $k\equiv k_{\mathit{0}}•\frac{\mathit{1}}{V_{\text{max}}-V_{\text{min}}}$, then 0 < k₀ ≤ 1. Thirdly, the standard deviation of ΔV needs to be small enough (i.e., narrow distribution) to ensure accurate reweighting using cumulant expansion to the second order[115]: σ_ΔV = k(E−V_avg)σ_V ≤ σ₀ where V_avg and σ_V are the average and standard deviation of the system potential energies, σ_ΔV is the standard deviation of ΔV with σ₀ as a user-specified upper limit (e.g., 10k_BT) for accurate reweighting. When E is set to the lower bound E = V_max according to Eqn. (17), k₀ can be calculated as:

$$k_{\mathit{0}}=\text{min}(\mathit{1.0},k_{\mathit{0}}^{\prime })=\text{min}(\mathit{1.0},\frac{{\sigma }_{\mathit{0}}}{{\sigma }_V}·\frac{V_{\text{max}}-V_{\text{min}}}{V_{\text{max}}-V_{\mathit{\text{avg}}}}).$$ (18)

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

$$k_{\mathit{0}}=k_{\mathit{0}}^{″}\equiv (\mathit{1}-\frac{{\sigma }_{\mathit{0}}}{{\sigma }_V})·\frac{V_{\text{max}}-V_{\text{min}}}{V_{\mathit{\text{avg}}}-V_{\text{min}}},$$ (19)

if $k_{\mathit{0}}^{″}$ is calculated between 0 and 1. Otherwise, k₀ is calculated using Eqn. (18), instead of being set to 1 directly as described in Ref. [57].

Fig. 4.

Schematic illustration of Gaussian Accelerated Molecular Dynamics (GaMD): when the threshold energy is set to the maximum potential (E = V_max), the system potential energy surface is smoothened by adding a harmonic boost potential that follows Gaussian distribution. The effective harmonic constant k₀ in the range of 0 to 1 determines the magnitude of the applied boost potential. With greater k₀, higher boost potential is added to the original energy surface in conventional molecular dynamics (cMD), which provides enhanced sampling of biomolecules across decreased energy barriers.

By adaptively adding a harmonic boost potential to smoothen the system 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. After the threshold energy E is calculated using Eqn. (17), the effective harmonic constant k₀ in the range of 0 to 1 determines the magnitude of the applied boost potential. With greater k₀, higher boost potential is added to the original energy surface, which provides enhanced sampling of biomolecules across the decreased energy barriers (Figure 4). The boost potential follows Gaussian distribution and accurate reweighting is obtained through cumulant expansion to the second order. Without the need to set predefined CVs, GaMD enables unconstrained enhanced sampling of the biomolecules. As demonstrated earlier, free energy profiles obtained from GaMD simulations allow us to identify distinct low energy states and characterize the protein folding and ligand binding pathways quantitatively[57].

4. Future directions

It remains a grand challenge to bridge the gap between the timescales of MD simulations and those of biological processes in the near future. Enhanced sampling, especially the unconstrained methods as discussed in this review, will continue to help computational modeling of biomolecules. Along with the theoretical development, there is a need to extend their applications to systems of increasing size and complexity that are more relevant to biological studies and computer-aided drug design. Most currently simulated systems are often limited to short peptides (e.g., alanine dipeptide) and small proteins. We need to reach for larger systems of biological and pharmaceutical interest, e.g., functional enzymes, membrane proteins, cellular protein complexes, virus particles, etc.

While the enhanced sampling methods discussed above have been found successful in various applications, they often suffer from one or more limitations. In most cases, it will be beneficial to combine two or more of the above techniques for further enhanced sampling. For example, it has been shown that the combination of RE with ST and MUCA in the REXST and REMUCA algorithms is able to reduce the computational cost without requiring too many MD replicas in the pure RE simulations. Based on a potential-biasing approach, GaMD mainly accelerates transitions across enthalpic energy barriers. Improvement for its application to systems with high entropic barriers is still needed. On this regard, GaMD can be potentially combined with the RE/PT algorithms like in REXAMD[93, 94] for further enhanced sampling. Notably, the combination of parallel tempering and metadynamics (PT-MetaD)[116] has been shown to facilitate enhanced sampling of biomolecules over entropic barriers. Moreover, the REXAMD that combines RE and aMD methods have been found helpful in free energy calculations. Similarly, one can combine GaMD that provides improved reweighting and RE for more accurate free energy calculations. In addition, we can use the essential potential energy as described in the EESRW method to improve the potential-biasing sampling methods such as GaMD.

Finally, in order to obtain accurate free energy calculations, rigorous error analysis (particularly the reweighting) is still needed for the enhanced sampling methods. Take the reweighting of aMD/GaMD simulations as an example. Because the boost potential ΔV is physically equivalent to nonequilibrium work W, reweighting of aMD simulations for calculating free energies can be expressed by the Jarzynski equality[117]: e^−βΔF = 〈e^−βΔV〉. When the three ΔF estimators and different aMD reweighting techniques examined in Refs. [118] and [115] are compared, the mean work estimator is found equivalent to the approximation using Maclaurin series expansion, the fluctuation-dissipation (FD) theorem estimator corresponds to the cumulant expansion to the 2^nd order (also referred to as the “Gaussian approximation”[18]) and the Jarzynski estimator corresponds to direct “exponential average” calculation. Detailed analysis showed that in the near-equilibrium regime, the mean work estimator gives comparatively larger errors and the Jarzynski estimator is more accurate than the FD estimator when the number of W (or ΔV) data points N≤16. However, the FD estimator gives the smallest error with increasing N (see Figure 4 in Ref. [118]). This is consistent with our previous aMD reweighting study that shows the cumulant expansion to the 2^nd order is the most accurate compared with the Maclaurin series expansion and exponential average, for which the number of ΔV values N used for aMD reweighting is on the order of 10²−10⁶[115]. In summary, the bias and error of free energy calculations were examined rigorously in Ref. [118]. Similar error estimates need to be performed for GaMD simulations, as well as the other enhanced sampling methods, when they are applied for free energy calculations.

5. Conclusions

Enhanced sampling has been found highly useful in the last several decades for exploring the configuration space of biomolecules and improving the convergence of free energy calculations. In this review, we focused on the enhanced sampling methods that do not require predefined system-dependent CVs for biomolecular simulations and as such do not suffer from the hidden energy barrier problem as encountered in the CV-biasing methods. These methods are based on generalized ensemble simulations of temperature (for RE/PT), local average force (SGMD and SGLD) and potential energy (EESRW, aMD and GaMD). This allows for generally unconstrained enhanced sampling of biomolecules, particularly the pathways during their biological function such as protein folding and ligand binding. While theoretical and computational work is still needed to improve the sampling statistics and accuracy, wider applications are expected for the unconstrained enhanced sampling methods in free energy calculations of biomolecules and computer-aided drug design.