Constant pH Molecular Dynamics Simulations: Current Status and Recent Applications

Abstract

Many important protein functions are carried out through proton-coupled conformational dynamics. Thus, the ability to accurately model protonation states dynamically has wide-ranging implications. Over the past two decades, two main types of constant pH methods (discrete and continuous) have been developed to enable proton-coupled molecular dynamics (MD) simulations. In this short review, we discuss the current status of the development and highlight recent applications that have advanced our understanding of protein structure-function relationships. We conclude the review by outlining the remaining challenges in the method development and projecting important areas for future applications.

Graphical Abstract

Introduction

Many important biological functions are mediated by the so-called proton-coupled conformational dynamics, whereby titration of often one or two amino acid sidechains accompany a large conformational change in the protein. Such processes cannot be directly modeled by conventional molecular dynamics (cMD) simulations, as the protonation states are preassigned and fixed. Instead, it is desirable to perform MD at constant pH such that protonation states are allowed to switch in response to the change of conformational environment at a specified solution pH, i.e., simultaneous sampling of conformation and protonation states. Another use of constant pH MD is to determine protonation and tautomer states of protein sidechains, which are largely invisible to experimental techniques [1].

Current status of constant pH molecular dynamics methods

Since the report of arguably the first constant pH MD simulation by Baptista et al. in 1997 [2], two main types of constant pH methods have been developed. The discrete constant pH framework [2, 3, 4, 5, 6, 7, 8, 9], makes use of a hybrid MD/Monte Carlo (MC) approach, in which a MD trajectory is periodically interrupted by attempts to switch a protonation state based on the Metropolis criterion (Fig. 1a). A straightforward implementation of this approach is to perform MC sampling on protonation states based on the configurations taken directly from the MD trajectory; this has led to the implementation of the generalized Born (GB) [6, 10] and hybrid-solvent (explicit/GB) based discrete constant pH methods in Amber [8] as well as the hybrid-solvent explicit/Poisson-Boltzmann (PB) based constant pH method for GROMACS simulations [3, 11]. By combining the explicit-solvent conformational sampling with the implicit-solvent protonation state sampling, the hybrid-solvent scheme offers improved accuracy while avoiding potential convergence issue in explicit water [8]. However, it has significant drawbacks, for example, the effects of explicit water and ions cannot be fully modeled, which play important roles in biological phenomena such as protein ligand binding and ion transport [12]. In order to extend the MD/MC framework to fully explicit simulations, a hurdle needs to be overcome, namely, a trial move (protonation state switch) in explicit solvent is almost always rejected as it is associated with a large energy change. To overcome this hurdle, a nonequilibrium MD (neMD)/MC approach was developed for all-atom simulations [7, 9], in which a trial MD run via a time-dependent coupling parameter λ is performed to gradually shift the system towards the alternative protonation state before the Metropolis criterion is applied [7, 9, 13] (Fig. 1a). This all-atom neMD/MC approach has been recently implemented in the NAMD package [9, 13].

Figure 1: Schematics of the continuous and discrete constant pH MD frame-works.

a) In the λ-dynamics (λMD) based continuous constant pH MD (CpHMD) [14], an extended Hamiltonian is used to propagate a set of continuous λ coordinates, whose end points (λ ≈0 or 1) represent the protonated and deprotonated states, b) In the discrete constant pH framework, the conventional MD (cMD) trajectory is periodically interrupted (grey box) to attempt a protonation state change, e.g. λ from 0 to 1. This is accomplished by applying the Metropolis criterion directly to the trajectory configuration in the MD/MC approach [2] or following a short nonequilibrium MD (neMD) to gradually scale λ between 0 and 1 in nMD/MC approach [7, 9]. In both frameworks, a dummy proton remains covalently attached but its interactions are turned off in the deprotonated state (highlighted in yellow).

In the λ-dynamics [15] based continuous constant pH framework [14, 16, 17, 18, 19, 20], an extended Hamiltonian is used to propagate a set of λ coordinates with fictitious mass (Fig. 1b). Here the end points of λ (λ ≈1 or 0) represent the protonated and deprotonated states, which is different from a weak coupling scheme where λ represents the extent of protonation [21]. The implicit-solvent continuous constant pH MD (CpHMD) methods were implemented in CHARMM [14, 16] and Amber [22], while the hybrid-solvent CpHMD was implemented in CHARMM [23]. The all-atom CpHMD methods are available in CHARMM in the original CpHMD formulation [20] or following the multi-site λ dynamics (MSλD) method for relative free energy calculations, termed CpHMD^MSλD [18, 19]. An implementation similar to the latter is also available in GROMACS [17]. Most recently, the GPU-accelerated all-atom CpHMD has been implemented in Amber [24].

Comparison of constant pH methods

There are advantages and drawbacks to both constant pH frameworks. The major advantages of the discrete framework are: physical representation of protonation states; straightforward implementation and incorporation of nonbonded terms and tautomer states. A drawback is slower convergence [8, 25], since typically one protonation state can be switched in each MC cycle. An additional limitation of the all-atom neMD/MC approach is the need for the “inherent” (guess) pK_a’s which should be close to the actual pK_a’s [13]. A drawback of the continuous framework is that λ has to transiently assume values corresponding to partially protonated states which are unphysical but nonetheless necessary to allow transition between the end states (λ ≈ 1 or 0). As a result, in the post analysis, data associated with the unphysical states have to be discarded, which can become large at pH conditions where many sites titrate. The CpHMD methods, particularly with the particle-mesh Ewald (PME) scheme is nontrivial to implement [20, 24], and perturbation of the nonbonded terms has not been added yet [24]. However, compared to the discrete constant pH methods, CpHMD simulations converge protonation states much faster, as λ values of all titratable sites are updated at every step. With the pH replica-exchange enhanced sampling protocol [23], the pK_a’s typically converge ~1 ns with implicit solvent [22], ~10 ns with hybrid solvent [23], below 100 ns with the all-atom PME scheme [20, 24].

In addition to the discrete and continuous frameworks, alternative constant pH methods have been developed in recent years. For example, the EDS (enveloping distribution sampling) method makes use of a hybrid Hamiltonian to describe a system in multiple protonation states such that they can be sampled in a single MD simulation [26]. An approximate constant pH method based on scaling force field parameters has been recently developed [27]. The discrete constant pH framework has been used to develop constant pH methods for coarse-grained simulations. A MC/MC approach has been developed for structure-based Go model simulations, in which both conformational and protonation states are sampled with MC moves [28]. The MD/MC approach [2] with continuum electrostatics for MC sampling has been used to develop a constant pH method for coarse-grained simulations [29]. The neMD/MC approach [7, 9] has been recently adapted [30] for MARTINI coarse-grained simulations. A proof-of-concept approach that allows proton exchange between solvent and solute has also been proposed for MARTINI simulations [31].

The development of the aforementioned constant pH methods has enabled a variety of novel applications. Below we will discuss recent application studies of the discrete and continuous constant pH methods. Due to the concise nature of this review, we will mainly focus on simulation studies published within the past three to four years.

Determination of protonation and redox states

Protonation states of enzyme active sites.

A major application of constant pH MD is to predict the protonation states of enzyme active-site residues which often have highly shifted pK_a’s relative to model values in solution. While PB and empirical calculations are faster, the calculated pK_a’s are sensitive to input structures and parameters (e.g., protein dielectric constant in PB calculations) [32]. Using the hybrid-solvent MD/MC method [8], Hofer et al. [33] calculated the macroscopic pK_a’s of the active-site aspartates and histidines in several aspartic, cysteine, and serine proteases and found that the directions of the experimental pK_a shifts are mostly reproduced. Verma et al. [34] applied the implicit-solvent CpHMD [22] to calculate the pK_a’s of the SARS-CoV-2 main protease (a cysteine protease) and suggested that the protonation of an active-site histidine is responsible for the pH-induced decrease in enzyme activity.

Accurate prediction of acid and base components of a catalytic dyad is relevant for understanding enzyme mechanisms. Using five model enzymes, Huang et al. [35] demonstrated that the hybrid-solvent CpHMD [23] can recapitulate the NMR assignment of general acid (proton donor) and base (nucleophile). An interesting finding was that the deprotonated base always forms additional hydrogen bonds that are absent in the X-ray structure, which explains why the static structure-based PB or empirical calculations often fail to distinguish acid and base [35].

Using the hybrid-solvent MD/MC method [8], Gupta et al. [36] analyzed the pH-dependent titration and conformational dynamics of the active site of glycinamide ribonucleotide transformylase (GART) in complex with ligands. Their acid/base assignment [36], which is different from an early hypothesis, is supported by the agreement between the calculated pH profile of the combined population of the catalytically competent protonation states and the experimental pH-activity curve.

Protonation states of membrane-inserted peptides.

pK_a’s of membrane-inserted peptides are nontrivial to predict due to the lipid environment. The first such attempt was made by Panahi and Brooks III [37], who applied the all-atom CpHMD^MSλD to calculate the pK_a’s of the transmembrane GWALP23 peptide and study the pH-dependent helix tilting observed experimentally. Applying the hybrid-solvent MD/MC method [3, 11], Vila-Vicosa and Machuqueiro [38] obtained the position-dependent pK_a profiles of titratable sites in two pHLIP peptides. Despite the caveat of using PB calculations for lipid environment, the calculated pK_a’s are in good agreement with the measured membrane-insertion pH range of the peptide [38]. Using the same method [3, 11], Lousa et al. [39] studied the pH-dependent conformational dynamics of the influenza fusion peptide and found that the helix tilt angle is pH dependent.

Protonation states of membrane proteins.

Pieri et al. [25] combined the pK_a’s calculated from the hybrid-solvent MD/MC [8] and QM/MM simulations to understand the pH-dependent electronic absorption spectrum of Anabaena Sensory Rhodopsin. Vo et al. [40] used the pK_a’s from the membrane hybrid-solvent CpHMD [12] in the weighted-ensemble simulations to investigate the binding of fentanyl with the μ-opioid receptor. Using the same CpHMD method [12], Li et al. [41] calculated the pK_a’s followed by the multi-scale reactive MD simulations to investigate the mechanism of a proton-coupled peptide transporter.

The membrane hybrid-solvent CpHMD [12] has also been used to address the controversy regarding the identity of the second proton binding residue in the prototypical sodium-proton antiporter NhaA by Henderson et al. [42]. Most recently, Bignucolo et al. ([43]) applied the all-atom neMD/MC method [13] to calculate the pK_a of an acid residue in the human acid-sensing ion channel hASIC1a.

Reactive cysteines and lysines for covalent drug design.

In targeted covalent drug design, it is desirable to identify highly nucleophilic residues. Since the pK_a’s of cysteines and lysines are related to the apparent reaction rates, pK_a calculations can be used to assess their nucleophilicities [44]. Employing the implicit-solvent CpHMD [22], Liu et al. retrospectively and prospectively predicted reactive cysteines and lysines in many families of kinases [44]. Interestingly, in many examined cases deprotonation of cysteines is coupled to the formation of hydrogen bonds that are absent in the X-ray structures, consistent with the proton-coupled hydrogen bond formation of general bases observed in the simulations with the hybrid-solvent CpHMD and a different force field [35].

pK_a calculations for training machine learning models.

Another novel application of constant pH simulations is to provide training data for machine learning models, as experimental pK_a data set is rather small [45]. Cai et al. [46] computed the pK_a’s of 279 soluble proteins using the implicit-solvent CpHMD method [22] to train a convolutional neural network model for protein pK_a prediction.

pH-dependent redox potential.

Due to the mathematical similarity between Henderson-Hasselbalch and Nernst equations, constant pH methods can be extended to calculate constant redox potentials [47, 48]. Using the hybrid-solvent MD/MC constant pH and constant redox potential implementation in Amber with two-dimensional Hamiltonian replica exchange [48], Cruzeiro et al. [49] demonstrated the calculation of the pH-dependent standard redox potentials for four proteins, which are in good agreement with experiment.

Protein-ligand and protein-protein binding

Protonation state change of ligand.

The aforementioned hybrid-solvent MD/MC simulations [8] of GART [36] showed that binding induced protonation state change of the ligand which in turn perturbed the local hydrogen bonding.

pH-dependent protein-ligand and protein-protein binding.

According to the Wyman linkage relation [50], the slope of the pH-dependent binding free energy change is proportional to the difference in the net charge (or number of bound protons) between the bound and unbound states. Analytic integration of this relation [51, 52, 53] allows calculation of the pH-dependent relative binding free energy from the pK_a values of the bound and unbound states. Using all-atom CpHMD^MSλD simulations, Paul et al. [54] found that different analogs of benzimidazole (guest molecule) have different pH-dependent binding free energy profiles with the host molecule CB7.

Using the hybrid-solvent MD/MC simulations [11], da Rocha et al. [55] examined the pH-dependent dimer dissociation energies and conformational dynamics of β-lactoglobulin (BLG). Interestingly, instead of using the pK_a’s to calculate the pH-dependent relative binding free energies as in the previous studies [51, 52, 53], they integrated the spline-fitted titration curves [55].

Proton-coupled conformational dynamics

pH- or protonation-dependent enzyme activities.

Enzyme function often involves a protonation-state-dependent conformational change of an important loop, for example, the “flap” in aspartyl proteases. The hybrid-solvent CpHMD [23] simulations revealed that pepstatin binding introduces a pH-dependent dynamics for the flap in plasmepsin II [56].

The aforementioned hybrid-solvent MD/MC simulations of GART [36]. suggested that the pH-dependent activity change is due to the order-disorder transition of the activation loop which is promoted by the titration of a histidine. In the aforementioned hybrid-solvent MD/MC study of BLG [55], a pH-dependent loop opening and closing movement was also demonstrated. Using the implicit-solvent CpHMD [22], Henderson et al. [57] compared the pK_a’s of SARS and MERS papain like proteases (PLpros) and they also found that the block loop 2 in the binding site in the SARS-CoV-2/SARS-CoV PLpros can open and close in response to pH, which is consistent with the X-ray structures.

The DFG motif located at the beginning of the activation loop is an important feature of all kinases, as it can undergo a dramatic crankshaft backbone change associated with the active DFG-in and inactive DFG-out states. Applying the hybrid-solvent CpHMD [23], Tsai et al. [58] were able to sample the unbiased conformational transition between the DFG-in and -out states as well as the movement of the αC helix and activation loop in the SRC kinase. This study demonstrated that shifts in the distribution of protonation states modulate the conformational transitions.

Estrogen receptor α (ERα) is a nuclear receptor protein that regulates cell growth, and its overactivation is related to breast cancer proliferation. Using a dual-basin Go-model simulations with the MD/MC constant pH approach, de Oliveira et al. [59] found that the cancer mutation D538G enhances the pH effect on the stability of the active conformation of the ERα ligand binding domain. Their analysis suggested that the deprotonation of His372 is responsible for the stabilization of the active state.

Proton-coupled ion and peptide transport through membrane transporters..

By combining the membrane hybrid-solvent CpHMD [12] and umbrella sampling, Yue et al. [60] were able to test a pH-partition hypothesis and more accurately calculate the potential of mean force (PMF) of membrane permeation of a cationic drug propranolol.In another study, Yue et al. combined CpHMD and umbrella sampling to understand the mechanism of fluoride permeation in the E. coli. fluoride channel [61]. Using a similar protocol as Yue et al. [60] but with the hybrid-solvent MD/MC constant pH method with PB calculations for protonation states, Oliveira and Machuqeiro calculated the PMF of ATP/ADP when pulled through a membrane transporter the ADP/ATP carrier protein [62].

pH-dependent dynamics of proton channels..

Using the all-atom CpHMD^MSλD method, Torabifarda et al. studied the influenza M2 channel and found that the low pH induced channel opening can be recapitulated for the construct that includes the amphipathic helices (M2CD) and not the transmembrane domains alone (M2TM) [63]. Jardin et al. applied the hybrid-solvent MD/MC method [8] to address the pH-dependent gating mechanism of the human Hv1 channel [64]. Despite the limitation of the GB model for sampling protonation states of transmembrane protein, the simulations revealed a pH-dependent conformational change which supports the experimental hypothesis that gating involves a small outward movement of the S4 helix [64].

pH-dependent protein diffusion near a charged surface.

Using Brownian dynamics with the MD/MC constant pH method, Antosiewicz and Długosz studied the pH-dependent diffusion of lysozyme on a charged surface and found that the pK_a’s and diffusion behavior of the protein are perturbed relative to the surface-free values [65].

Challenges and Outlook

The application studies discussed in this review demonstrated the utilities and enormous potential of constant pH MD to provide novel insights and advance the understanding of important biological phenomena that are difficult to elucidate with wet-lab experiments alone. Despite the progress, further development of constant pH methods is needed. While the implicit-and hybrid-solvent constant pH methods are useful for certain applications, all-atom constant pH methods are in principle most accurate and can be applied to any system with a force field representation. For example, for heterogeneous dielectric systems such as transmembrane proteins, all-atom constant pH simulations are most appropriate. Nonetheless, several challenges remain. The first challenge is related to the fluctuating system net charge, which can be compensated with co-titrating water/ions; however, it slows down convergence [20, 66, 13, 43]. An alternative is to rely on the background plasma to neutralize the system and correct titration free energies due to the plasma-related offset potential [20, 43, 24]. This solution is feasible for systems with a small net charge, but details, e.g., how to deal with coupled titrating residues, remain unclear. A second challenge is related to the force field. For implementations with the Amber force fields [6, 10, 8], the dependence of backbone charges on the sidechain protonation state presents a problem and requires the modification of the partial charges, which is not ideal [6, 22]. Unpublished data from the Shen group suggests that protonation states of deeply buried residues overly favor neutral state, which is a limitation of the additive force fields. Further improvement may require either an empirical correction or use of polarizable force fields. Another force field related issue has to do with partial charges. Recent studies [54, 67] suggested that binding induced pK_a shifts may be sensitive to the partial charge set; however, further investigation is needed.

A third challenge is in sampling and computational speed. Although the pH replica-exchange protocol [23, 8] offers boosting in many cases, proton-coupled large conformational changes remain extremely challenging to sample. On the other hand, adoption of biased sampling methods is challenging, as they often distort the distributions of protonation states (Shen and coworkers, unpublished data). In order to fully understand the sampling and force field related issues, efficient GPU implementation is needed to minimize the computational cost of sampling protonation states [68, 24]. Finally, the protonation state dependent bonded terms are absent in most current constant pH implementations (e.g., MD/MC [11, 8] and CpHMD methods [23, 19, 24]), and they need to be added to further improve accuracy. We envision that future development will tackle these challenges and application studies will address more complex problems, e.g., modeling proton-coupled dynamic process of protein-ligand and protein-protein binding; elucidating the mechanisms of proton-coupled membrane channels and transporters where many open questions await computational investigations to address.

Highlights.

• The discrete and continuous constant pH methods allow protonation states to respond to conformational dynamics and pH during molecular dynamics (MD) simulations.

• Many application studies have focused on the determination of protonation states of various enzymes and explanation of their pH-dependent activities.

• Applications to proton-coupled protein-ligand binding and conformational changes of transmembrane proteins are emerging.

• The all-atom constant pH methods are most promising but several challenges remain.