A systematic study of minima in alanine dipeptide

Multiple Parameter Replica Exchange Gaussian Accelerated Molecular Dynamics for Enhanced Sampling and Free Energy Calculation of Biomolecular Systems

This study introduces a novel method named multiple parameter replica exchange Gaussian accelerated molecular dynamics (MP-Rex-GaMD), building on the Gaussian accelerated molecular dynamics (GaMD) algorithm. GaMD enhances sampling and retrieves free energy information for biomolecular systems by adding a harmonic boost potential to smooth the potential energy surface without the need for predefined reaction coordinates. Our innovative approach advances the acceleration power and energetic reweighting accuracy of GaMD by incorporating a replica exchange algorithm that enables the exchange of multiple parameters, including the GaMD boost parameters of force constant and energy threshold, as well as temperature. Applying MP-Rex-GaMD to the three model systems of dialanine, chignolin, and HIV protease, we demonstrate its superior capability over conventional molecular dynamics and GaMD simulations in exploring protein conformations and effectively navigating various biomolecular states across energy barriers. MP-Rex-GaMD allows users to accurately map free energy landscapes through energetic reweighting, capturing the ensemble of biomolecular states from low-energy conformations to rare high-energy transitions within practical computational time scales.

Neighbor effect on conformational spaces of alanine residue in azapeptides

The conformational properties of Alanine (Ala) residue have been investigated to understand protein folding and develop force fields. In this work, we examined the neighbor effect on the conformational spaces of Ala residue using model azapeptides, Ac-Ala-azaGly-NHMe (3, AaG), and Ac-azaGly-Ala-NHMe (4, aGA1). Ramachandran energy maps were generated by scanning (φ, ψ) dihedral angles of the Ala residues in models with the fixed dihedral angles (φ = ±90°, ψ = ±0° or ±180°) of azaGly residue using LCgau-BOP and LCgau-BOP + LRD functionals in the gas and water phases. The integral-equation-formalism polarizable continuum model (IEF-PCM) and a solvation model density (SMD) were employed to mimic the solvation effect. The most favorable conformation of Ala residue in azapeptide models is found as the polyproline II (βP), inverse γ-turn (γ'), β-sheet (βS), right-handed helix (αR), or left-handed helix (αL) depending on the conformation of neighbor azaGly residue in isolated form. Solvation methods exhibit that the Ala residue favors the βP, δR, and αR conformations regardless of its position in azapeptides 3 and 4 in water. Azapeptide 5, Ac-azaGly-Ala-NH2 (aGA2), was synthesized to evaluate the theoretical results. The X-ray structure showed that azaGly residue adopts the polyproline II (βP) and Ala residue adopts the right-handed helical (αR) structure in aGA2. The conformational preferences of aGA2 and the dimer structure of aGA2 based on the X-ray structure were examined to assess the performance of DFT functionals. In addition, the local minima of azapeptide 6, Ac-Phe-azaGly-NH2 (FaG), were compared with the previous experimental results. SMD/LCgau-BOP + LRD methods agreed well with the reported experimental results. The results suggest the importance of weak dispersion interactions, neighbor effect, and solvent influence in the conformational preferences of Ala residue in model azapeptides.

emle-engine: A Flexible Electrostatic Machine Learning Embedding Package for Multiscale Molecular Dynamics Simulations

We present in this work the emle-engine package (https://github.com/chemle/emle-engine)─the implementation of a new machine learning embedding scheme for hybrid machine learning potential/molecular-mechanics (ML/MM) dynamics simulations. The package is based on an embedding scheme that uses a physics-based model of the electronic density and induction with a handful of tunable parameters derived from in vacuo properties of the subsystem to be embedded. This scheme is completely independent of the in vacuo potential and requires only the positions of the atoms of the machine learning subsystem and the positions and partial charges of the molecular mechanics environment. These characteristics allow emle-engine to be employed in existing QM/MM software. We demonstrate that the implemented electrostatic machine learning embedding scheme (named EMLE) is stable in enhanced sampling molecular dynamics simulations. Through the calculation of free energy surfaces of alanine dipeptide in water with two different ML options for the in vacuo potential and three embedding models, we test the performance of EMLE. When compared to the reference DFT/MM surface, the EMLE embedding is clearly superior to the MM one based on fixed partial charges. The configurational dependence of the electronic density and the inclusion of the induction energy introduced by the EMLE model leads to a systematic reduction in the average error of the free energy surface when compared to MM embedding. By enabling the usage of EMLE embedding in practical ML/MM simulations, emle-engine will make it possible to accurately model systems and processes that feature significant variations in the charge distribution of the ML subsystem and/or the interacting environment.

Conformational preference of dipeptide zwitterions in aqueous solvents

Proper description of solvent effects is challenging for theoretical methods, particularly if the solute is a zwitterion. Here, a series of theoretical procedures are used to determine the preferred solvated conformations of twelve hydrophobic dipeptides (Leu-Leu, Leu-Phe, Phe-Leu, Ile-Leu, Phe-Phe, Ala-Val, Val-Ala, Ala-Ile, Ile-Ala, Ile-Val, Val-Ile and Val-Val) in the zwitterionic state. First, the accuracy of density functional theory (DFT), combined with different implicit solvent models, for describing zwitterions in aqueous solvent is assessed by comparing the predicted against the experimental glycine tautomerization energy, i.e., the energetic difference between canonical and zwitterionic glycine in aqueous solvents. It is found that among the tested solvation schemes, the charge-asymmetric nonlocally determined local-electric solvation model (CANDLE) predicts an energetic difference in excellent agreement with the experimental value. Next, DFT-CANDLE is used to determine the most favorable solvated conformation for each of the investigated dipeptide zwitterions. The CANDLE-solvated structures are obtained by exploring the conformational space of each dipeptide zwitterion concatenating DFT calculations, in vacuum, with classical molecular dynamics simulations, in explicit solvents, and DFT calculations including explicit water molecules. It is found that the energetically most favorable conformations are similar to those of the dipeptide zwitterions in their respective crystal structures. Such structural agreement is indicative of the DFT-CANDLE accomplishment of the description of solvated zwitterions, and suggests that these biomolecules self-assemble as quasi-rigid objects.

Trajectory Statistical Learning of the Potential Mean of Force and Diffusion Coefficient from Molecular Dynamics Simulations

Central to studying the conformational changes of a complex protein is understanding the dynamics and energetics involved. Phenomenologically, structural dynamics can be formulated using an overdamped Langevin model along an observable, e.g., the distance between two residues in the protein. The Langevin model is specified by the deterministic force (the potential of mean force, PMF) and stochastic force (characterized by the diffusion coefficient, D). It is therefore of great interest to be able to extract both PMF and D from an observable time series but under the same computational framework. Here, we approach this challenge in molecular dynamics (MD) simulations by treating it as a missing-data Bayesian estimation problem. An important distinction in our methodology is that the entire MD trajectory, as opposed to the individual data elements, is used as the statistical variable in Bayesian imputation. This idea is implemented through an eigen-decomposition procedure for a time-symmetrized Fokker-Planck equation, followed by maximizing the likelihood for parameter estimation. The mathematical expressions for the functional derivatives used in learning PMF and D also provide new physical insights for the manner by which the information on both the deterministic and stochastic forces is encoded in the dynamics data. An all-atom MD simulation of a nontrivial biomolecule case is used to illustrate the application of this approach. We show that, interestingly, the results of trajectory statistical learning can motivate new order parameters for an improved description of the kinetic bottlenecks in conformational changes. Complementing purely data-driven or black-box methods, this work underscores the advantages of physics-based machine learning in gaining chemical insights from quantitative parameter estimation.

Toward a general neural network force field for protein simulations: Refining the intramolecular interaction in protein

Molecular dynamics (MD) is an extremely powerful, highly effective, and widely used approach to understanding the nature of chemical processes in atomic details for proteins. The accuracy of results from MD simulations is highly dependent on force fields. Currently, molecular mechanical (MM) force fields are mainly utilized in MD simulations because of their low computational cost. Quantum mechanical (QM) calculation has high accuracy, but it is exceedingly time consuming for protein simulations. Machine learning (ML) provides the capability for generating accurate potential at the QM level without increasing much computational effort for specific systems that can be studied at the QM level. However, the construction of general machine learned force fields, needed for broad applications and large and complex systems, is still challenging. Here, general and transferable neural network (NN) force fields based on CHARMM force fields, named CHARMM-NN, are constructed for proteins by training NN models on 27 fragments partitioned from the residue-based systematic molecular fragmentation (rSMF) method. The NN for each fragment is based on atom types and uses new input features that are similar to MM inputs, including bonds, angles, dihedrals, and non-bonded terms, which enhance the compatibility of CHARMM-NN to MM MD and enable the implementation of CHARMM-NN force fields in different MD programs. While the main part of the energy of the protein is based on rSMF and NN, the nonbonded interactions between the fragments and with water are taken from the CHARMM force field through mechanical embedding. The validations of the method for dipeptides on geometric data, relative potential energies, and structural reorganization energies demonstrate that the CHARMM-NN local minima on the potential energy surface are very accurate approximations to QM, showing the success of CHARMM-NN for bonded interactions. However, the MD simulations on peptides and proteins indicate that more accurate methods to represent protein-water interactions in fragments and non-bonded interactions between fragments should be considered in the future improvement of CHARMM-NN, which can increase the accuracy of approximation beyond the current mechanical embedding QM/MM level.

DeepWEST: Deep Learning of Kinetic Models with the Weighted Ensemble Simulation Toolkit for Enhanced Sampling

Recent advances in computational power and algorithms have enabled molecular dynamics (MD) simulations to reach greater time scales. However, for observing conformational transitions associated with biomolecular processes, MD simulations still have limitations. Several enhanced sampling techniques seek to address this challenge, including the weighted ensemble (WE) method, which samples transitions between metastable states using many weighted trajectories to estimate kinetic rate constants. However, initial sampling of the potential energy surface has a significant impact on the performance of WE, i.e., convergence and efficiency. We therefore introduce deep-learned kinetic modeling approaches that extract statistically relevant information from short MD trajectories to provide a well-sampled initial state distribution for WE simulations. This hybrid approach overcomes any statistical bias to the system, as it runs short unbiased MD trajectories and identifies meaningful metastable states of the system. It is shown to provide a more refined free energy landscape closer to the steady state that could efficiently sample kinetic properties such as rate constants.

Constructing Potential Energy Surface with Correlated Theory for Dipeptides Using Molecular Tailoring Approach

We demonstrate a cost-effective alternative employing the fragment-based molecular tailoring approach (MTA) for building the potential energy surface (PES) for two dipeptides viz. alanine-alanine and alanine-proline employing correlated theory, with augmented Dunning basis sets. About 1369 geometries are generated for each test dipeptide by systematically varying the dihedral angles Φ ${{\rm{\Phi }}}$ and Ψ ${{{\Psi }}}$ . These conformational geometries are partially optimized by relaxing all the other Z-matrix parameters, fixing the values of Φ ${{\rm{\Phi }}}$ and Ψ ${{{\Psi }}}$ . The MP2 level PES is constructed from the MTA-energies of chemically intact geometries using minimal hardware. The fidelity of MP2/aug-cc-pVDZ level PES is brought out by comparing it with its full calculation counterpart. Further, we bring out the power of the method by reporting the MTA-based CCSD/aug-cc-pVDZ level PES for these two dipeptides containing 498 and 562 basis functions respectively.

Sigmoid Accelerated Molecular Dynamics: An Efficient Enhanced Sampling Method for Biosystems

Gaussian accelerated molecular dynamics (GaMD) is recognized as a popular enhanced sampling method for tackling long-standing challenges in biomolecular simulations. Inspired by GaMD, Sigmoid accelerated molecular dynamics (SaMD) is proposed in this work by adding a Sigmoid boost potential to improve the balance between the highest acceleration and accurate reweighting. Compared with GaMD, SaMD extends the accessible time scale and improves the computational efficiency as tested in three tasks. In the alanine dipeptide task, SaMD can produce the free energy landscape with better accuracy and efficiency. In the chignolin folding task, the estimated Gibbs free energy difference can converge to the experimental value ∼30% faster. In the protein-ligand binding task, the bound conformations are closer to the crystal structure with a minimal ligand root-mean-square deviation of 1.7 Å. The binding of the ligand XK263 to the HIV protease is reproduced by SaMD in ∼60% less simulation time.

Machine learning based implicit solvent model for aqueous-solution alanine dipeptide molecular dynamics simulations

Inspired by the recent work from Noé and coworkers on the development of machine learning based implicit solvent model for the simulation of solvated peptides [Chen et al., J. Chem. Phys., 2021, 155, 084101], here we report another investigation of the possibility of using machine learning (ML) techniques to "derive" an implicit solvent model directly from explicit solvent molecular dynamics (MD) simulations. For alanine dipeptide, a machine learning potential (MLP) based on the DeepPot-SE representation of the molecule was trained to capture its interactions with its average solvent environment configuration (ASEC). The predicted forces on the solute deviated only by an RMSD of 0.4 kcal mol-1 Å-1 from the reference values, and the MLP-based free energy surface differed from that obtained from explicit solvent MD simulations by an RMSD of less than 0.9 kcal mol-1. Our MLP training protocol could also accurately reproduce combined quantum mechanical molecular mechanical (QM/MM) forces on the quantum mechanical (QM) solute in ASEC environment, thus enabling the development of accurate ML-based implicit solvent models for ab initio-QM MD simulations. Such ML-based implicit solvent models for QM calculations are cost-effective in both the training stage, where the use of ASEC reduces the number of data points to be labelled, and the inference stage, where the MLP can be evaluated at a relatively small additional cost on top of the QM calculation of the solute.

Capabilities and limits of autoencoders for extracting collective variables in atomistic materials science

Free energy calculations in materials science are routinely hindered by the need to provide reaction coordinates that can meaningfully partition atomic configuration space, a prerequisite for most enhanced sampling approaches. Recent studies on molecular systems have highlighted the possibility of constructing appropriate collective variables directly from atomic motions through deep learning techniques. Here we extend this class of approaches to condensed matter problems, for which we encode the finite temperature collective variable by an iterative procedure starting from 0 K features of the energy landscape i.e. activation events or migration mechanisms given by a minimum - saddle point - minimum sequence. We employ the autoencoder neural networks in order to build a scalar collective variable for use with the adaptive biasing force method. Particular attention is given to design choices required for application to crystalline systems with defects, including the filtering of thermal motions which otherwise dominate the autoencoder input. The machine-learning workflow is tested on body-centered cubic iron and its common defects, such as small vacancy or self-interstitial clusters and screw dislocations. For localized defects, excellent collective variables as well as derivatives, necessary for free energy sampling, are systematically obtained. However, the approach has a limited accuracy when dealing with reaction coordinates that include atomic displacements of a magnitude comparable to thermal motions, e.g. the ones produced by the long-range elastic field of dislocations. We then combine the extraction of collective variables by autoencoders with an adaptive biasing force free energy method based on Bayesian inference. Using a vacancy migration as an example, we demonstrate the performance of coupling these two approaches for simultaneous discovery of reaction coordinates and free energy sampling in systems with localized defects.

Iterative Landmark-Based Umbrella Sampling (ILBUS) Protocol for Sampling of Conformational Space of Biomolecules

Computer simulations of biomolecules such as molecular dynamics often suffer from insufficient sampling. Due to limited computational resources, insufficient sampling prevents obtaining proper equilibrium distributions of observed properties. To deal with this problem, we proposed a simulation protocol for efficient resampling of collected off-equilibrium trajectories. These trajectories are utilized for the initial mapping of the conformational space, which is later properly resampled by the introduced Iterative Landmark-Based Umbrella Sampling (ILBUS) method. Reconstruction of static equilibrium properties is achieved by the multistate Bennett acceptance ratio (MBAR) method, which enables efficient use of simulated data. The ILBUS protocol is geometry-based and does not demand any additional collective variable or a dimensional-reduction technique. The only requirement is a set of suitably spaced reference conformations, which serve as landmarks in the mapped conformational space. Additionally, the ILBUS protocol encompasses an iterative process that optimizes the force constant used in the umbrella sampling simulation. Such tuning is an inherent feature of the protocol and does not need to be performed by the user in advance. Furthermore, even the simulations with suboptimal force constants can be used in estimates by MBAR. We demonstrate the feasibility and the performance of this approach in the study of the conformational landscape of the alanine dipeptide, met-enkephalin, and adenylate kinase.

Design of Peptides that Fold and Self-Assemble on Graphite

The graphite-water interface provides a unique environment for polypeptides that generally favors ordered structures more than in solution. Therefore, systems consisting of designed peptides and graphitic carbon might serve as a convenient medium for controlled self-assembly of functional materials. Here, we computationally designed cyclic peptides that spontaneously fold into a β-sheet-like conformation at the graphite-water interface and self-assemble, and we subsequently observed evidence of such assembly by atomic force microscopy. Using a novel protocol, we screened nearly 2000 sequences, optimizing for formation of a unique folded conformation while discouraging unfolded or misfolded conformations. A head-to-tail cyclic peptide with the sequence GTGSGTGGPGGGCGTGTGSGPG showed the greatest apparent propensity to fold spontaneously, and this optimized sequence was selected for larger scale molecular dynamics simulations, rigorous free-energy calculations, and experimental validation. In simulations ranging from hundreds of nanoseconds to a few microseconds, we observed spontaneous folding of this peptide at the graphite-water interface under many different conditions, including multiple temperatures (295 and 370 K), with different initial orientations relative to the graphite surface, and using different molecular dynamics force fields (CHARMM and Amber). The thermodynamic stability of the folded conformation on graphite over a range of temperatures was verified by replica-exchange simulations and free-energy calculations. On the other hand, in free solution, the folded conformation was found to be unstable, unfolding in tens of picoseconds. Intermolecular hydrogen bonds promoted self-assembly of the folded peptides into linear arrangements where the peptide backbone exhibited a tendency to align along one of the six zigzag directions of the graphite basal plane. For the optimized peptide, atomic force microscopy revealed growth of single-molecule-thick linear patterns of 6-fold symmetry, consistent with the simulations, while no such patterns were observed for a control peptide with the same amino acid composition but a scrambled sequence.

Nonparametric Bayesian inference for meta-stable conformational dynamics

Analyses of structural dynamics of biomolecules hold great promise to deepen the understanding of and ability to construct complex molecular systems. To this end, both experimental and computational means are available, such as fluorescence quenching experiments or molecular dynamics simulations, respectively. We argue that while seemingly disparate, both fields of study have to deal with the same type of data about the same underlying phenomenon of conformational switching. Two central challenges typically arise in both contexts: (i) the amount of obtained data is large, and (ii) it is often unknown how many distinct molecular states underlie these data. In this study, we build on the established idea of Markov state modeling and propose a generative, Bayesian nonparametric hidden Markov state model that addresses these challenges. Utilizing hierarchical Dirichlet processes, we treat different meta-stable molecule conformations as distinct Markov states, the number of which we then do not have to set a priori. In contrast to existing approaches to both experimental as well as simulation data that are based on the same idea, we leverage a mean-field variational inference approach, enabling scalable inference on large amounts of data. Furthermore, we specify the model also for the important case of angular data, which however proves to be computationally intractable. Addressing this issue, we propose a computationally tractable approximation to the angular model. We demonstrate the method on synthetic ground truth data and apply it to known benchmark problems as well as electrophysiological experimental data from a conformation-switching ion channel to highlight its practical utility.

Chasing Collective Variables Using Autoencoders and Biased Trajectories

Free energy biasing methods have proven to be powerful tools to accelerate the simulation of important conformational changes of molecules by modifying the sampling measure. However, most of these methods rely on the prior knowledge of low-dimensional slow degrees of freedom, i.e., collective variables (CVs). Alternatively, such CVs can be identified using machine learning (ML) and dimensionality reduction algorithms. In this context, approaches where the CVs are learned in an iterative way using adaptive biasing have been proposed: at each iteration, the learned CV is used to perform free energy adaptive biasing to generate new data and learn a new CV. In this paper, we introduce a new iterative method involving CV learning with autoencoders: Free Energy Biasing and Iterative Learning with AutoEncoders (FEBILAE). Our method includes a reweighting scheme to ensure that the learning model optimizes the same loss at each iteration and achieves CV convergence. Using the alanine dipeptide system and the solvated chignolin mini-protein system as examples, we present results of our algorithm using the extended adaptive biasing force as the free energy adaptive biasing method.

Imaging of femtosecond bond breaking and charge dynamics in ultracharged peptides

X-ray free-electrons lasers have revolutionized the method of imaging biological macromolecules such as proteins, viruses and cells by opening the door to structural determination of both single particles and crystals at room temperature. By utilizing high intensity X-ray pulses on femtosecond timescales, the effects of radiation damage can be reduced. Achieving high resolution structures will likely require knowledge of how radiation damage affects the structure on an atomic scale, since the experimentally obtained electron densities will be reconstructed in the presence of radiation damage. Detailed understanding of the expected damage scenarios provides further information, in addition to guiding possible corrections that may need to be made to obtain a damage free reconstruction. In this work, we have quantified the effects of ionizing photon-matter interactions using first principles molecular dynamics. We utilize density functional theory to calculate bond breaking and charge dynamics in three ultracharged molecules and two different structural conformations that are important to the structural integrity of biological macromolecules, comparing to our previous studies on amino acids. The effects of the ultracharged states and subsequent bond breaking in real space are studied in reciprocal space using coherent diffractive imaging of an ensemble of aligned biomolecules in the gas phase.

Side-Chain Polarity Modulates the Intrinsic Conformational Landscape of Model Dipeptides

The intrinsic conformational preferences of small peptides may provide additional insight into the thermodynamics and kinetics of protein folding. In this study, we explore the underlying energy landscapes of two model peptides, namely, Ac-Ala-NH₂ and Ac-Ser-NH₂, using geometry-optimization-based tools developed within the context of energy landscape theory. We analyze not only how side-chain polarity influences the structural preferences of the dipeptides, but also other emergent properties of the landscape, including heat capacity profiles, and kinetics of conformational rearrangements. The contrasting topographies of the free energy landscape agree with recent results from Fourier transform microwave spectroscopy experiments, where Ac-Ala-NH₂ was found to exist as a mixture of two conformers, while Ac-Ser-NH₂ remained structurally locked, despite exhibiting an apparently rich conformational landscape.

PSIQUE: Protein Secondary Structure Identification on the Basis of Quaternions and Electronic Structure Calculations

The secondary structure is important in protein structure analysis, classification, and modeling. We have developed a novel method for secondary structure assignment, termed PSIQUE, based on the potential energy surface (PES) of polyalanine obtained using an infinitely long chain model and density functional theory calculations. First, uniform protein segments are determined in terms of a difference of quaternions between neighboring amino acids along the protein backbone. Then, the identification of the secondary structure motifs is carried out based on the minima found in the PES. PSIQUE shows good agreement with other secondary structure assignment methods. However, it provides better discrimination of subtle secondary structures (e.g., helix types) and termini and produces more uniform segments while also accounting for local distortions. Overall, PSIQUE provides a precise and reliable assignment of secondary structures, so it should be helpful for the detailed characterization of the protein structure.

The emerging role of computational design in peptide macrocycle drug discovery

Drug discovery is a laborious process with rising cost per new drug. Peptide macrocycles are promising therapeutics, though conformational flexibility can reduce target affinity and specificity. Recent computational advancements address this problem by enabling rational design of rigidly folded peptide macrocycles.

AREAS COVERED

This review summarizes currently approved peptide macrocycle therapeutics and discusses advantages of mesoscale drugs over small molecules or protein therapeutics. It describes the history, rationale, and state of the art of computational tools, such as Rosetta, that allow the design of rigidly structured peptide macrocycles. The emerging pipeline for designing peptide macrocycle drugs is described, including current challenges in designing permeable molecules that can emulate the chameleonic behavior of natural macrocycles. Prospects for reducing computational cost and improving accuracy with emerging computational technologies are also discussed.

EXPERT OPINION

To embrace computational design of peptide macrocycle drugs, we must shift current attitudes regarding the role of computation in drug discovery, and move beyond Lipinski's rules. This technology has the potential to shift failures to earlier in silico stages of the drug discovery process, improving success rates in costly clinical trials. Given the available tools, now is the time for drug developers to incorporate peptide macrocycle design into drug discovery pipelines.

Geometry Optimization, Transition State Search, and Reaction Path Mapping Accomplished with the Fragment Molecular Orbital Method

Recent development of the fragment molecular orbital (FMO) method related to energy gradients, geometry optimization, transition state search, and chemical reaction mapping is summarized. The frozen domain formulation of FMO is introduced in detail, and the structure of related GAMESS input files for FMO is described.