Enhanced Sampling Approaches Research Articles

Predominant Binding Mode of Palmatine to DNA.

Palmatine is a protoberberine alkaloid, which may produce singlet oxygen under visible light irradiation and binds to DNA. The fact that singlet oxygen activation in palmatine may be triggered by environmental conditions, and in particular its interaction with nucleic acids, makes it a most suitable candidate for photodynamic therapy and DNA-targeted noninvasive anticancer strategies. Despite these remarkable properties, the actual binding mode between palmatine and DNA has not been resolved, yet. Its elucidation has indeed led to contrasting hypotheses. In this contribution, by using long-range molecular dynamic simulations and enhanced sampling approaches, we unequivocally identify that intercalation is the dominant binding mode of palmatine with DNA, from both a thermodynamic and kinetic point of view.

The Arabidopsis AtSWEET13 transporter discriminates sugars by selective facial and positional substrate recognition

Transporters are targeted by endogenous metabolites and exogenous molecules to reach cellular destinations, but it is generally not understood how different substrate classes exploit the same transporter’s mechanism. Any disclosure of plasticity in transporter mechanism when treated with different substrates becomes critical for developing general selectivity principles in membrane transport catalysis. Using extensive molecular dynamics simulations with an enhanced sampling approach, we select the Arabidopsis sugar transporter AtSWEET13 as a model system to identify the basis for glucose versus sucrose molecular recognition and transport. Here we find that AtSWEET13 chemical selectivity originates from a conserved substrate facial selectivity demonstrated when committing alternate access, despite mono-/di-saccharides experiencing differing degrees of conformational and positional freedom throughout other stages of transport. However, substrate interactions with structural hallmarks associated with known functional annotations can help reinforce selective preferences in molecular transport.

Modeling Chromatography Binding through Molecular Dynamics Simulations with Resin Fragments.

Accurate atomistic modeling of the interactions of a chromatography resin with a solute can inform the selection of purification conditions for a product, an important problem in the biotech and pharmaceutical industries. We present a molecular dynamics simulation-based approach for the qualitative prediction of interaction sites (specificity) and retention times (affinity) of a protein for a given chromatography resin. We mimicked the resin with an unrestrained ligand composed of the resin headgroup coupled with successively larger fragments of the agarose backbone. The interactions of the ligand with the protein are simulated in an explicit solvent using the Replica Exchange Molecular Dynamics enhanced sampling approach in conjunction with Hydrogen Mass Repartitioning (REMD-HMR). We computed the ligand interaction surface from the simulation trajectories and correlated the features of the interaction surface with experimentally determined retention times. The simulation and analysis protocol were first applied to a series of ubiquitin mutants for which retention times on Capto MMC resin are available. The ubiquitin simulations helped identify the optimal ligand that was used in subsequent simulations on six proteins for which Capto MMC elution times are available. For each of the six proteins, we computed the interaction surface and characterized it in terms of a range of simulation-averaged residue-level physicochemical descriptors. Modeling of the salt concentrations required for elution with respect to the descriptors resulted in a linear fit in terms of aromaphilicity and Kyte-Doolittle hydrophobicity that was robust to outliers, showed high correlation, and correctly ranked the protein elution order. The physics-based model building approach described here does not require a large experimental data set and can be readily applied to different resins and diverse biomolecules.

A semi-parametric claims reserving model with monotone splines

AbstractStochastic reserving models used in the insurance industry are usually based on an assumed distribution of claim amounts. Despite their popularity, such models may unavoidably be affected by the misspecification issue given that it is likely that the underlying distribution will be different from that assumed. In this paper, we incorporate monotone splines to ensure the expected monotonically increasing pattern of cumulative development factors (CDFs) to develop a new semi-parametric reserving model that does not require a density assumption. To allow the maximum utilization of available information, we also propose an enhanced sampling approach that greatly increases the size of unbiased CDFs, particularly in later development periods. Based on the enhanced samples, a bootstrap technique is employed in the estimation of monotone splines, from which incurred-but-not-reported (IBNR) reserves and prediction errors can be obtained. Associated technical features, such as the consistency of estimator, are discussed and demonstrated. Our simulation studies suggest that the new model improves the accuracy of IBNR reserving, compared with a range of classic competing models. A real data analysis produces many consistent findings, thus supporting the usefulness of the monotone spline model in actuarial and insurance practice.

Quantifying Unbiased Conformational Ensembles from Biased Simulations Using ShapeGMM.

Quantifying the conformational ensembles of biomolecules is fundamental to describing mechanisms of processes such as protein folding, interconversion between folded states, ligand binding, and allosteric regulation. Accurate quantification of these ensembles remains a challenge for conventional molecular simulations of all but the simplest molecules due to insufficient sampling. Enhanced sampling approaches, such as metadynamics, were designed to overcome this challenge; however, the nonuniform frame weights that result from many of these approaches present an additional challenge to ensemble quantification techniques such as Markov State Modeling or structural clustering. Here, we present rigorous inclusion of nonuniform frame weights into a structural clustering method entitled shapeGMM. The result of frame-weighted shapeGMM is a high dimensional probability density and generative model for the unbiased system from which we can compute important thermodynamic properties such as relative free energies and configurational entropy. The accuracy of this approach is demonstrated by the quantitative agreement between GMMs computed by Hamiltonian reweighting and direct simulation of a coarse-grained helix model system. Furthermore, the relative free energy computed from a shapeGMM probability density of alanine dipeptide reweighted from a metadynamics simulation quantitatively reproduces the underlying free energy in the basins. Finally, the method identifies hidden structures along the actin globular to filamentous-like structural transition from a metadynamics simulation on a linear discriminant analysis coordinate trained on GMM states, illustrating how structural clustering of biased data can lead to biophysical insight. Combined, these results demonstrate that frame-weighted shapeGMM is a powerful approach to quantifying biomolecular ensembles from biased simulations.

Efficient Sampling of Cavity Hydration in Proteins with Nonequilibrium Grand Canonical Monte Carlo and Polarizable Force Fields.

Prediction of the hydration levels of protein cavities and active sites is important to both mechanistic analysis and ligand design. Due to the unique microscopic environment of these buried water molecules, a polarizable model is expected to be crucial for an accurate treatment of protein internal hydration in simulations. Here we adapt a nonequilibrium candidate Monte Carlo approach for conducting grand canonical Monte Carlo simulations with the Drude polarizable force field. The GPU implementation enables the efficient sampling of internal cavity hydration levels in biomolecular systems. We also develop an enhanced sampling approach referred to as B-walking, which satisfies detailed balance and readily combines with grand canonical integration to efficiently calculate quantitative binding free energies of water to protein cavities. Applications of these developments are illustrated in a solvent box and the polar ligand binding site in trypsin. Our simulation results show that including electronic polarization leads to a modest but clear improvement in the description of water position and occupancy compared to the crystal structure. The B-walking approach enhances the range of water sampling in different chemical potential windows and thus improves the accuracy of water binding free energy calculations.

Adaptive Sampling Methods for Molecular Dynamics in the Era of Machine Learning.

Molecular dynamics (MD) simulations are fundamental computational tools for the study of proteins and their free energy landscapes. However, sampling protein conformational changes through MD simulations is challenging due to the relatively long time scales of these processes. Many enhanced sampling approaches have emerged to tackle this problem, including biased sampling and path-sampling methods. In this Perspective, we focus on adaptive sampling algorithms. These techniques differ from other approaches because the thermodynamic ensemble is preserved and the sampling is enhanced solely by restarting MD trajectories at particularly chosen seeds rather than introducing biasing forces. We begin our treatment with an overview of theoretically transparent methods, where we discuss principles and guidelines for adaptive sampling. Then, we present a brief summary of select methods that have been applied to realistic systems in the past. Finally, we discuss recent advances in adaptive sampling methodology powered by deep learning techniques, as well as their shortcomings.

Phanto-IDP: compact model for precise intrinsically disordered protein backbone generation and enhanced sampling.

The biological function of proteins is determined not only by their static structures but also by the dynamic properties of their conformational ensembles. Numerous high-accuracy static structure prediction tools have been recently developed based on deep learning; however, there remains a lack of efficient and accurate methods for exploring protein dynamic conformations. Traditionally, studies concerning protein dynamics have relied on molecular dynamics (MD) simulations, which incur significant computational costs for all-atom precision and struggle to adequately sample conformational spaces with high energy barriers. To overcome these limitations, various enhanced sampling techniques have been developed to accelerate sampling in MD. Traditional enhanced sampling approaches like replica exchange molecular dynamics (REMD) and frontier expansion sampling (FEXS) often follow the MD simulation approach and still cost a lot of computational resources and time. Variational autoencoders (VAEs), as a classic deep generative model, are not restricted by potential energy landscapes and can explore conformational spaces more efficiently than traditional methods. However, VAEs often face challenges in generating reasonable conformations for complex proteins, especially intrinsically disordered proteins (IDPs), which limits their application as an enhanced sampling method. In this study, we presented a novel deep learning model (named Phanto-IDP) that utilizes a graph-based encoder to extract protein features and a transformer-based decoder combined with variational sampling to generate highly accurate protein backbones. Ten IDPs and four structured proteins were used to evaluate the sampling ability of Phanto-IDP. The results demonstrate that Phanto-IDP has high fidelity and diversity in the generated conformation ensembles, making it a suitable tool for enhancing the efficiency of MD simulation, generating broader protein conformational space and a continuous protein transition path.

An Enhanced Sampling Approach for Computing the Free Energy of Solid Surface and Solid–Liquid Interface

AbstractFree energies of a solid surface and a solid–liquid interface play significant roles in thermodynamics. Due to the limited availability of experimental data, computational methods offer effective alternatives for calculating these properties. This study adopts advanced frameworks of the logarithmic mean force dynamics method to present an enhanced sampling approach for the calculation of the free energy at different temperatures. To achieve this, the free energy profile is constructed along with a pre‐established collective variable within the melting transition and cleavage processes. The values of the solid surface and solid–liquid interface free energies are then extrapolated from the excess free energy related to the formation and persistence of the solid surface or the solid–liquid interface. Furthermore, this methodology is employed to calculate the temperature dependence of the free energy measurements for the (100) and (110) surfaces and interfaces of Cu. It is shown that this methodology is robust and readily applicable in contemporary models of atomic interactions and various systems.

Hybrid enhanced sampling approaches to gain structural insights into leucine-rich repeat kinase 2 (LRRK2) Parkinson's disease mutation.

Parkinson's disease (PD) is a neurodegenerative disease affecting millions worldwide. A commonly mutated gene in PD is leucine-rich repeat kinase 2 (LRRK2), which contains multiple domains, including but not limited to kinase, ATPase, and GTPase. LRRK2 has been found to increase kinase activity and colocalize with microtubules. LRRK2 protein dynamics are affected by its conformation, regulating microtubule interactions and potentially enhancing intracellular trafficking. Despite recent advances in the structural insights of LRRK2, detailed studies on the effects of single-point mutations on the dynamics and conformations of LRRK2 are limited. Consequently, an all-atomistic molecular dynamics simulation is required to understand the allosteric and dynamical effects of single-point disease mutations in LRRK2, but it would be computationally expensive. We thereby employ enhanced molecular dynamics simulations to study single-point mutations in LRRK2, particularly the G2019S, R1441C, R1441G, R1441H, and Y1699C mutations. Our two-phase approach incorporates implementing Gaussian-accelerated molecular dynamics (GaMD) for enhanced sampling, followed by the weighted ensemble (WE) method. GaMD is an enhanced MD approach where a boost potential is added to the total potential energy of the system to accelerate the conformational search, while the WE method readjusts the weight of the simulation trajectories by frequent resampling to obtain the kinetic properties of interest accurately. Investigating the dynamics of such single-point mutations in LRRK2 could explain why specific mutations result in increased kinase activation and dynamicity of LRRK2.

Rare Event Kinetics from Adaptive Bias Enhanced Sampling.

We introduce a novel enhanced sampling approach named on-the-fly probability enhanced sampling (OPES) flooding for calculating the kinetics of rare events from atomistic molecular dynamics simulation. This method is derived from the OPES approach [Invernizzi and Parrinello, J. Phys. Chem. Lett. 2020, 11, 7, 2731-2736], which has been recently developed for calculating converged free energy surfaces for complex systems. In this paper, we describe the theoretical details of the OPES flooding technique and demonstrate the application on three systems of increasing complexity: barrier crossing in a two-dimensional double-well potential, conformational transition in the alanine dipeptide in the gas phase, and the folding and unfolding of the chignolin polypeptide in an aqueous environment. From extensive tests, we show that the calculation of accurate kinetics not only requires the transition state to be bias-free, but the amount of bias deposited should also not exceed the effective barrier height measured along the chosen collective variables. In this vein, the possibility of computing rates from biasing suboptimal order parameters has also been explored. Furthermore, we describe the choice of optimum parameter combinations for obtaining accurate results from limited computational effort.

Multiple Poses and Thermodynamics of Ligands Targeting Protein Surfaces: The Case of Furosemide Binding to mitoNEET in Aqueous Solution.

Human NEET proteins, such as NAF-1 and mitoNEET, are homodimeric, redox iron-sulfur proteins characterized by triple cysteine and one histidine-coordinated [2Fe-2S] cluster. They exist in an oxidized and reduced state. Abnormal release of the cluster is implicated in a variety of diseases, including cancer and neurodegeneration. The computer-aided and structure-based design of ligands affecting cluster release is of paramount importance from a pharmaceutical perspective. Unfortunately, experimental structural information so far is limited to only one ligand/protein complex. This is the X-ray structure of furosemide bound to oxidized mitoNEET. Here we employ an enhanced sampling approach, Localized Volume-based Metadynamics, developed by some of us, to identify binding poses of furosemide to human mitoNEET protein in solution. The binding modes show a high variability within the same shallow binding pocket on the protein surface identified in the X-ray structure. Among the different binding conformations, one of them is in agreement with the crystal structure’s one. This conformation might have been overstabilized in the latter because of the presence of crystal packing interactions, absent in solution. The calculated binding affinity is compatible with experimental data. Our protocol can be used in a straightforward manner in drug design campaigns targeting this pharmaceutically important family of proteins.

Fast estimation of ion-pairing for screening electrolytes: A cluster can approximate a bulk liquid.

The propensity for ion-pairing can often dictate the thermodynamic and kinetic properties of electrolyte solutions. Fast and accurate estimates of ion-pairing can thus be extremely valuable for supplementing design and screening efforts for novel electrolytes. We introduce an efficient cluster model to estimate the local ion-pair potential-of-mean-force between ionic solutes in electrolytes. The model incorporates an ion-pair and a few layers of explicit solvent in a gas-phase cluster and leverages an enhanced sampling approach to achieve high efficiency and accuracy. We employ harmonic restraints to prevent solvent escape from the cluster and restrict sampling of large inter-ion distances. We develop a cluster ion-pair sampling tool that implements our cluster model and demonstrate its potential utility for screening simple and poly-electrolyte systems.

An Efficient Gaussian-Accelerated Molecular Dynamics (GaMD) Multilevel Enhanced Sampling Strategy: Application to Polarizable Force Fields Simulations of Large Biological Systems.

We introduce a novel multilevel enhanced sampling strategy grounded on Gaussian-accelerated Molecular Dynamics (GaMD). First, we propose a GaMD multi-GPUs-accelerated implementation within the Tinker-HP molecular dynamics package. We introduce the new "dual-water" mode and its use with the flexible AMOEBA polarizable force field. By adding harmonic boosts to the water stretching and bonding terms, it accelerates the solvent-solute interactions while enabling speedups, thanks to the use of fast multiple-time step integrators. To further reduce the time-to-solution, we couple GaMD to Umbrella Sampling (US). The GaMD─US/dual-water approach is tested on the 1D Potential of Mean Force (PMF) of the solvated CD2-CD58 system (168 000 atoms), allowing the AMOEBA PMF to converge within 1 kcal/mol of the experimental value. Finally, Adaptive Sampling (AS) is added, enabling AS-GaMD capabilities but also the introduction of the new Adaptive Sampling-US-GaMD (ASUS-GaMD) scheme. The highly parallel ASUS-GaMD setup decreases time to convergence by, respectively, 10 and 20 times, compared to GaMD-US and US. Overall, beside the acceleration of PMF computations, Tinker-HP now allows for the simultaneous use of Adaptive Sampling and GaMD-"dual water" enhanced sampling approaches increasing the applicability of polarizable force fields to large-scale simulations of biological systems.

Enhanced Sampling Approach to the Induced-Fit Docking Problem in Protein-Ligand Binding: The Case of Mono-ADP-Ribosylation Hydrolase Inhibitors.

Enhanced sampling methods can predict free-energy landscapes associated with protein/ligand binding, characterizing the involved intermolecular interactions in a precise way. However, these in silico approaches can be challenged by induced-fit effects. Here, we present a variant of volume-based metadynamics tailored to tackle this problem in a general and efficient way. The validity of the approach is established by applying it to substrate/enzyme complexes of pharmacological relevance: mono-ADP-ribose (ADPr) in complex with mono-ADP-ribosylation hydrolases (MacroD1 and MacroD2), where induced-fit phenomena are known to be significant. The calculated binding free energies are consistent with experiments, with an absolute error smaller than 0.5 kcal/mol. Our simulations reveal that in all circumstances, the active loops, delimiting the boundaries of the binding site, undergo significant conformation rearrangements upon ligand binding. The calculations further provide, for the first time, the molecular basis of ADPr specificity and the relative changes in its experimental binding affinity on passing from MacroD1 to MacroD2 and all its mutants. Our study paves the way to the quantitative description of induced-fit events in molecular recognition.

Reweighted Jarzynski Sampling: Acceleration of Rare Events and Free Energy Calculation with a Bias Potential Learned from Nonequilibrium Work.

We introduce a simple enhanced sampling approach for the calculation of free energy differences and barriers along a one-dimensional reaction coordinate. First, a small number of short nonequilibrium simulations are carried out along the reaction coordinate, and the Jarzynski equality is used to learn an approximate free energy surface from the nonequilibrium work distribution. This free energy estimate is represented in a compact form as an artificial neural network and used as an external bias potential to accelerate rare events in a subsequent molecular dynamics simulation. The final free energy estimate is then obtained by reweighting the equilibrium probability distribution of the reaction coordinate sampled under the influence of the external bias. We apply our reweighted Jarzynski sampling recipe to four processes of varying scales and complexities─spanning chemical reaction in the gas phase, pair association in solution, and droplet nucleation in supersaturated vapor. In all cases, we find reweighted Jarzynski sampling to be a very efficient strategy, resulting in rapid convergence of the free energy to high precision.

Collective variable-based enhanced sampling and machine learning.

Collective variable-based enhanced sampling methods have been widely used to study thermodynamic properties of complex systems. Efficiency and accuracy of these enhanced sampling methods are affected by two factors: constructing appropriate collective variables for enhanced sampling and generating accurate free energy surfaces. Recently, many machine learning techniques have been developed to improve the quality of collective variables and the accuracy of free energy surfaces. Although machine learning has achieved great successes in improving enhanced sampling methods, there are still many challenges and open questions. In this perspective, we shall review recent developments on integrating machine learning techniques and collective variable-based enhanced sampling approaches. We also discuss challenges and future research directions including generating kinetic information, exploring high-dimensional free energy surfaces, and efficiently sampling all-atom configurations.Graphic abstract

The seven deadly sins: When computing crystal nucleation rates, the devil is in the details.

The formation of crystals has proven to be one of the most challenging phase transformations to quantitatively model-let alone to actually understand-be it by means of the latest experimental technique or the full arsenal of enhanced sampling approaches at our disposal. One of the most crucial quantities involved with the crystallization process is the nucleation rate, a single elusive number that is supposed to quantify the average probability for a nucleus of critical size to occur within a certain volume and time span. A substantial amount of effort has been devoted to attempt a connection between the crystal nucleation rates computed by means of atomistic simulations and their experimentally measured counterparts. Sadly, this endeavor almost invariably fails to some extent, with the venerable classical nucleation theory typically blamed as the main culprit. Here, we review some of the recent advances in the field, focusing on a number of perhaps more subtle details that are sometimes overlooked when computing nucleation rates. We believe it is important for the community to be aware of the full impact of aspects, such as finite size effects and slow dynamics, that often introduce inconspicuous and yet non-negligible sources of uncertainty into our simulations. In fact, it is key to obtain robust and reproducible trends to be leveraged so as to shed new light on the kinetics of a process, that of crystal nucleation, which is involved into countless practical applications, from the formulation of pharmaceutical drugs to the manufacturing of nano-electronic devices.

A Deep Graph Network-Enhanced Sampling Approach to Efficiently Explore the Space of Reduced Representations of Proteins.

The limits of molecular dynamics (MD) simulations of macromolecules are steadily pushed forward by the relentless development of computer architectures and algorithms. The consequent explosion in the number and extent of MD trajectories induces the need for automated methods to rationalize the raw data and make quantitative sense of them. Recently, an algorithmic approach was introduced by some of us to identify the subset of a protein’s atoms, or mapping, that enables the most informative description of the system. This method relies on the computation, for a given reduced representation, of the associated mapping entropy, that is, a measure of the information loss due to such simplification; albeit relatively straightforward, this calculation can be time-consuming. Here, we describe the implementation of a deep learning approach aimed at accelerating the calculation of the mapping entropy. We rely on Deep Graph Networks, which provide extreme flexibility in handling structured input data and whose predictions prove to be accurate and-remarkably efficient. The trained network produces a speedup factor as large as 105 with respect to the algorithmic computation of the mapping entropy, enabling the reconstruction of its landscape by means of the Wang–Landau sampling scheme. Applications of this method reach much further than this, as the proposed pipeline is easily transferable to the computation of arbitrary properties of a molecular structure.

Energetics of Substrate Binding and Conformational Change of the Bile Acid Transporter ASBTNm

The apical sodium-dependant bile acid transporter (ASBT) utilises the sodium gradient to drive the re-absorption of bile acids from the intestine, operating through an alternating access mechanism. It is of interest as a potential target for the treatment of hypercholesterolaemia as well as for drug delivery. Several structures of bacterial homologues are available, including that from Neisseria meningitidis (ASBTNM) in an inward-facing conformation with sodium and the bile acid taurocholate bound. However, less is known about the dynamic details of the ASBT transport cycle, including the nature of the conformational transition and the interplay of the substrates. We studied the behaviour of ASBTNM on an atomistic scale with molecular dynamics (MD) simulations. Multi-microsecond simulations were used to investigate the stable inward- and outward-facing states and substrate binding sites. The energetics of the conformational transition and of sodium and taurocholate binding to ASBTNM, and how this is altered in different substrate-bound configurations, were calculated using a two-stage enhanced sampling approach, in which semi-quantitative energetic landscapes were first obtained from short metadynamics simulations and then used to guide parameter selection for more accurate Bias Exchange Umbrella Sampling (BEUS) simulations. Together, these results represent an important step towards understanding the complete transport cycle of ASBT.