Discrete path sampling

Mapping conformational landscape in protein folding: Benchmarking dimensionality reduction and clustering techniques on the Trp-Cage mini-protein

Quantitative characterization of protein conformational landscapes is a computationally challenging task due to their high dimensionality and inherent complexity. In this study, we systematically benchmark several widely used dimensionality reduction and clustering methods to analyze the conformational states of the Trp-Cage mini-protein, a model system with well-documented folding dynamics. Dimensionality reduction techniques, including Principal Component Analysis (PCA), Time-lagged Independent Component Analysis (TICA), and Variational Autoencoders (VAE), were employed to project the high-dimensional free energy landscape onto 2D spaces for visualization. Additionally, clustering methods such as K-means, hierarchical clustering, HDBSCAN, and Gaussian Mixture Models (GMM) were used to identify discrete conformational states directly in the high-dimensional space. Our findings reveal that density-based clustering approaches, particularly HDBSCAN, provide physically meaningful representations of free energy minima. While highlighting the strengths and limitations of each method, our study underscores that no single technique is universally optimal for capturing the complex folding pathways, emphasizing the necessity for careful selection and interpretation of computational methods in biomolecular simulations. These insights will contribute to refining the available tools for analyzing protein conformational landscapes, enabling a deeper understanding of folding mechanisms and intermediate states.

Decoding Solubility Signatures from Amyloid Monomer Energy Landscapes

This study investigates the energy landscapes of amyloid monomers, which are crucial for understanding protein misfolding mechanisms in Alzheimer's disease. While proteins possess inherent thermodynamic stability, environmental factors can induce deviations from native folding pathways, leading to misfolding and aggregation, phenomena closely linked to solubility. Using the UNOPTIM program, which integrates the UNRES potential into the Cambridge energy landscape framework, we conducted single-ended transition state searches and employed discrete path sampling to compute kinetic transition networks starting from PDB structures. These kinetic transition networks consist of local energy minima and the transition states that connect them, which quantify the energy landscapes of the amyloid monomers. We defined clusters within each landscape using energy thresholds and selected their lowest-energy structures for the structural analysis. Applying graph convolutional networks, we identified solubility trends and correlated them with structural features. Our findings identify specific minima with low solubility, characteristic of aggregation-prone states, highlighting the key residues that drive reduced solubility. Notably, the exposure of the hydrophobic residue Phe19 to the solvent triggers a structural collapse by disrupting the neighboring helix. Additionally, we investigated selected minima to determine the first passage times between states, thereby elucidating the kinetics of these energy landscapes. This comprehensive approach provides valuable insights into the thermodynamics and kinetics of Aβ monomers. By integration of multiple analytical techniques to explore the energy landscapes, our study investigates structural features associated with reduced solubility. These insights have the potential to inform future therapeutic strategies aimed at addressing protein misfolding and aggregation in neurodegenerative diseases.

Energy Landscapes for the Unitary Coupled Cluster Ansatz

The unitary coupled cluster (UCC) approach has been one of the most popular wavefunction parametrizations for the variational quantum eigensolver due to the relative ease of optimization compared to hardware-efficient ansätze. In this contribution, we explore the energy landscape of the unitary coupled cluster singles and doubles (UCCSD) wavefunction for two commonly employed benchmark systems, lithium hydride and the nitrogen dimer. We investigate the organization of the solution space in terms of local minima and show how it changes as the number and order of operators of the UCC ansatz are varied. Surprisingly, we find that in all cases, the UCCSD energy has numerous low-lying minima connected by high energy transition states. Additionally, the energy spread of the minima that lie in the same band as the global minimum may exceed chemical accuracy, making the location of the true global minimum especially challenging.

Investigation into the effect of phenylalanine gating on anaerobic haem breakdown using the energy landscape approach

We have recently demonstrated a novel anaerobic NADH-dependent haem breakdown reaction, which is carried out by a range of haemoproteins. The Yersinia enterocolitica protein, HemS, is the focus of further research presented in the current paper. Using conventional experimental methods, bioinformatics, and energy landscape theory (ELT), we provide new insight into the mechanism of the novel breakdown process. Of particular interest is the behavior of a double phenylalanine gate, which opens and closes according to the relative situations of haem and NADH within the protein pocket. This behavior suggests that the double phe-gate fulfills a regulatory role within the pocket, controlling the access of NADH to haem. Additionally, stopped-flow spectroscopy results provide kinetic comparisons between the wild-type and the selected mutants. We also present a fully resolved crystal structure for the F104AF199A HemS monomer, including its extensive loop, the first such structure to be completely resolved for HemS or any of its close homologues. The energy landscapes approach provided key information regarding the gating strategy employed by HemS, compensating for current limitations with conventional biophysical and molecular dynamics approaches. We propose that ELT become more widely used in the field, particularly in the investigation of the dynamics and interactions of weak-binding ligands, and for gating features, within protein cavities.

Vibrational Energy Landscapes and Energy Flow in GPCRs

We construct and analyze disconnectivity graphs to provide the first graphical representation of the vibrational energy landscape of a protein, in this study β2AR, a G-protein coupled receptor (GPCR), in active and inactive states. The graphs, which indicate the relative free energy of each residue and the minimum free energy barriers for energy transfer between them, reveal important composition, structural and dynamic properties that mediate the flow of energy. Prolines and glycines, which contribute to GPCR plasticity and function, are identified as bottlenecks to energy transport along the backbone from which alternative pathways for energy transport via nearby noncovalent contacts emerge, seen also in the analysis of first passage time (FPT) distributions presented here. Striking differences between the disconnectivity graphs and FPT distributions for the inactive and active states of β2AR are found where structural and dynamic changes occur upon activation, contributing to allosteric regulation.

Energy landscapes for clusters of hexapeptides

We present the results for energy landscapes of hexapeptides obtained using interfaces to the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) program. We have used basin-hopping global optimization and discrete path sampling to explore the landscapes of hexapeptide monomers, dimers, and oligomers containing 10, 100, and 200 monomers modeled using a residue-level coarse-grained potential, Mpipi, implemented in LAMMPS. We find that the dimers of peptides containing amino acid residues that are better at promoting phase separation, such as tyrosine and arginine, have melting peaks at higher temperature in their heat capacity compared to phenylalanine and lysine, respectively. This observation correlates with previous work on the same uncapped hexapeptide monomers modeled using atomistic potential. For oligomers, we compare the variation in monomer conformations with radial distance and observe trends for selected angles calculated for each monomer. The LAMMPS interfaces to the GMIN and OPTIM programs for landscape exploration offer new opportunities to investigate larger systems and provide access to the coarse-grained potentials implemented within LAMMPS.

Energy Landscapes and Structural Ensembles of Glucagon-like Peptide-1 Monomers

While GLP-1 and its analogues are important pharmaceutical agents in the treatment of type 2 diabetes and obesity, their susceptibility to aggregate into amyloid fibrils poses a significant safety issue. Many factors may contribute to the aggregation propensity, including pH. While it is known that the monomeric structure of GLP-1 has a strong impact on primary nucleation, probing its diverse structural ensemble is challenging. Here, we investigated the monomer structural ensembles at pH 3, 4, and 7.5 using state-of-the-art computational methods in combination with experimental data. We found significant stabilization of β-strand structures and destabilization of helical structures at lower pH, correlating with observed aggregation lag times, which are lower under these conditions. We further identified helical defects at pH 4, which led to the fastest observed aggregation, in agreement with our far-UV circular dichroism data. The detailed atomistic structures that result from the computational studies help to rationalize the experimental results on the aggregation propensity of GLP-1. This work provides a new insight into the pH-dependence of monomeric structural ensembles of GLP-1 and connects them to experimental observations.

Assessing RNA atomistic force fields via energy landscape explorations in implicit solvent

Predicting the structure and dynamics of RNA molecules still proves challenging because of the relative scarcity of experimental RNA structures on which to train models and the very sensitive nature of RNA towards its environment. In the last decade, several atomistic force fields specifically designed for RNA have been proposed and are commonly used for simulations. However, it is not necessarily clear which force field is the most suitable for a given RNA molecule. In this contribution, we propose the use of the computational energy landscape framework to explore the energy landscape of RNA systems as it can bring complementary information to the more standard approaches of enhanced sampling simulations based on molecular dynamics. We apply the EL framework to the study of a small RNA pseudoknot, the Aquifex aeolicus tmRNA pseudoknot PK1, and we compare the results of five different RNA force fields currently available in the AMBER simulation software, in implicit solvent. With this computational approach, we can not only compare the predicted 'native' states for the different force fields, but the method enables us to study metastable states as well. As a result, our comparison not only looks at structural features of low energy folded structures, but provides insight into folding pathways and higher energy excited states, opening to the possibility of assessing the validity of force fields also based on kinetics and experiments providing information on metastable and unfolded states.

Supplementary Information

The online version contains supplementary material available at 10.1007/s12551-024-01202-9.

Prediction of 3D RNA Structures from Sequence Using Energy Landscapes of RNA Dimers: Application to RNA Tetraloops

Access to the three-dimensional structure of RNA enables an ability to gain a more profound understanding of its biological mechanisms, as well as the ability to design RNA-targeting drugs, which can take advantage of the unique chemical environment imposed by a folded RNA structure. Due to the dynamic and structurally complex properties of RNA, both experimental and traditional computational methods have difficulty in determining RNA's 3D structure. Herein, we introduce TAPERSS (Theoretical Analyses, Prediction, and Evaluation of RNA Structures from Sequence), a physics-based fragment assembly method for predicting 3D RNA structures from sequence. Using a fragment library created using discrete path sampling calculations of RNA dinucleoside monophosphates, TAPERSS can sample the physics-based energy landscapes of any RNA sequence with relatively low computational complexity. We have benchmarked TAPERSS on 21 RNA tetraloops, using a combinatorial algorithm as a proof-of-concept. We show that TAPERSS was successfully able to predict the apo-state structures of all 21 RNA hairpins, with 16 of those structures also having low predicted energies as well. We demonstrate that TAPERSS performs most accurately on GNRA-like tetraloops with mostly stacked loop-nucleotides, while having limited success with more dynamic UNCG and CUYG tetraloops, most likely due to the influence of the RNA force field used to create the fragment library. Moreover, we show that TAPERSS can successfully predict the majority of the experimental non-apo states, highlighting its potential in anticipating biologically significant yet unobserved states. This holds great promise for future applications in drug design and related studies. With discussed improvements and implementation of more efficient sampling algorithms, we believe TAPERSS may serve as a useful tool for a physics-based conformational sampling of large RNA structures.

100 Years of the Lennard-Jones Potential

It is now 100 years since Lennard-Jones published his first paper introducing the now famous potential that bears his name. It is therefore timely to reflect on the many achievements, as well as the limitations, of this potential in the theory of atomic and molecular interactions, where applications range from descriptions of intermolecular forces to molecules, clusters, and condensed matter.

Analysis and interpretation of first passage time distributions featuring rare events

In this contribution we consider theory and associated computational tools to treat the kinetics associated with competing pathways on multifunnel energy landscapes. Multifunnel landscapes are associated with molecular switches and multifunctional materials, and are expected to exhibit multiple relaxation time scales and associated thermodynamic signatures in the heat capacity. Our focus here is on the first passage time distribution, which is encoded in a kinetic transition network containing all the locally stable states and the pathways between them. This network can be renormalised to reduce the dimensionality, while exactly conserving the mean first passage time and approximately conserving the full distribution. The structure of the reduced network can be visualised using disconnectivity graphs. We show how features in the first passage time distribution can be associated with specific kinetic traps, and how the appearance of competing relaxation time scales depends on the starting conditions. The theory is tested for two model landscapes and applied to an atomic cluster and a disordered peptide. Our most important contribution is probably the reconstruction of the full distribution for long time scales, where numerical problems prevent direct calculations. Here we combine accurate treatment of the mean first passage time with the reliable part of the distribution corresponding to faster time scales. Hence we now have a fundamental understanding of both thermodynamic and kinetic signatures of multifunnel landscapes.

Combining experiment and energy landscapes to explore anaerobic heme breakdown in multifunctional hemoproteins

To survive, many pathogens extract heme from their host organism and break down the porphyrin scaffold to sequester the Fe2+ ion via a heme oxygenase. Recent studies have revealed that certain pathogens can anaerobically degrade heme. Our own research has shown that one such pathway proceeds via NADH-dependent heme degradation, which has been identified in a family of hemoproteins from a range of bacteria. HemS, from Yersinia enterocolitica, is the main focus of this work, along with HmuS (Yersinia pestis), ChuS (Escherichia coli) and ShuS (Shigella dysenteriae). We combine experiments, Energy Landscape Theory, and a bioinformatic investigation to place these homologues within a wider phylogenetic context. A subset of these hemoproteins are known to bind certain DNA promoter regions, suggesting not only that they can catalytically degrade heme, but that they are also involved in transcriptional modulation responding to heme flux. Many of the bacterial species responsible for these hemoproteins (including those that produce HemS, ChuS and ShuS) are known to specifically target oxygen-depleted regions of the gastrointestinal tract. A deeper understanding of anaerobic heme breakdown processes exploited by these pathogens could therefore prove useful in the development of future strategies for disease prevention.

The Effect of Pulling and Twisting Forces on Chameleon Sequence Peptides

Chameleon sequences are amino acid sequences found in several distinct configurations in experiment. They challenge our understanding of the link between sequence and structure, and provide insight into structural competition in proteins. Here, we study the energy landscapes for three such sequences, and interrogate how pulling and twisting forces impact the available structural ensembles. Chameleon sequences do not necessarily exhibit multiple structural ensembles on a multifunnel energy landscape when we consider them in isolation. The application of even small forces leads to drastic changes in the energy landscapes. For pulling forces, we observe transitions from helical to extended structures in a very small span of forces. For twisting forces, the picture is much more complex, and highly dependent on the magnitude and handedness of the applied force as well as the reference angle for the twist. Depending on these parameters, more complex and more simplistic energy landscapes are observed alongside more and less diverse structural ensembles. The impact of even small forces is significant, confirming their likely role in folding events. In addition, small forces exerted by the remaining scaffold of a protein may be sufficient to lead to the adoption of a specific structural ensemble by a chameleon sequence.

Energy landscapes for proteins described by the UNRES coarse-grained potential

The self-assembly of proteins is encoded in the underlying potential energy surface (PES), from which we can predict structure, dynamics, and thermodynamic properties. However, the corresponding analysis becomes increasingly challenging with larger protein sizes, due to the computational time required, which grows significantly with the number of atoms. Coarse-grained models offer an attractive approach to reduce the computational cost. In this Feature Article, we describe our implementation of the UNited RESidue (UNRES) coarse-grained potential in the Cambridge energy landscapes software. We have applied this framework to explore the energy landscapes of four proteins that exhibit native states involving different secondary structures. Here we have tested the ability of the UNRES potential to represent the global energy landscape of proteins containing up to 100 amino acid residues. The resulting potential energy landscapes exhibit good agreement with experiment, with low-lying minima close to the PDB geometries and to results obtained using the all-atom AMBER force field. The new program interfaces will allow us to investigate larger biomolecules in future work, using the UNRES potential in combination with all the methodology available in the computational energy landscapes framework.

Energy landscapes and heat capacity signatures for peptides correlate with phase separation propensity

Phase separation plays an important role in the formation of membraneless compartments within the cell and intrinsically disordered proteins with low-complexity sequences can drive this compartmentalisation. Various intermolecular forces, such as aromatic-aromatic and cation-aromatic interactions, promote phase separation. However, little is known about how the ability of proteins to phase separate under physiological conditions is encoded in their energy landscapes and this is the focus of the present investigation. Our results provide a first glimpse into how the energy landscapes of minimal peptides that contain - and cation- interactions differ from the peptides that lack amino acids with such interactions. The peaks in the heat capacity () as a function of temperature report on alternative low-lying conformations that differ significantly in terms of their enthalpic and entropic contributions. The analysis and subsequent quantification of frustration of the energy landscape suggest that the interactions that promote phase separation lead to features (peaks or inflection points) at low temperatures in . More features may occur for peptides containing residues with better phase separation propensity and the energy landscape is more frustrated for such peptides. Overall, this work links the features in the underlying single-molecule potential energy landscapes to their collective phase separation behaviour and identifies quantities ( and frustration metric) that can be utilised in soft material design.

Energy Landscapes and Heat Capacity Signatures for Monomers and Dimers of Amyloid-Forming Hexapeptides

Amyloid formation is a hallmark of various neurodegenerative disorders. In this contribution, energy landscapes are explored for various hexapeptides that are known to form amyloids. Heat capacity (CV) analysis at low temperature for these hexapeptides reveals that the low energy structures contributing to the first heat capacity feature above a threshold temperature exhibit a variety of backbone conformations for amyloid-forming monomers. The corresponding control sequences do not exhibit such structural polymorphism, as diagnosed via end-to-end distance and a dihedral angle defined for the monomer. A similar heat capacity analysis for dimer conformations obtained using basin-hopping global optimisation shows clear features in end-to-end distance versus dihedral correlation plots, where amyloid-forming sequences exhibit a preference for larger end-to-end distances and larger positive dihedrals. These results hold true for sequences taken from tau, amylin, insulin A chain, a de novo designed peptide, and various control sequences. While there is a little overall correlation between the aggregation propensity and the temperature at which the low-temperature CV feature occurs, further analysis suggests that the amyloid-forming sequences exhibit the key CV feature at a lower temperature compared to control sequences derived from the same protein.

Theoretical and computational methodologies for understanding coordination self-assembly complexes

This perspective highlights three theoretical and computational methods to capture the coordination self-assembly processes at the molecular level: quantum chemical modeling, molecular dynamics, and reaction network analysis. These methods cover the different scales from the metal-ligand bond to a more global aspect, and approaches that are best suited to understand the coordination self-assembly from different perspectives are introduced. Theoretical and numerical researches based on these methods are not merely ways of interpreting the experimental studies but complementary to them.

Revealing Free Energy Landscape From MD Data via Conditional Angle Partition Tree

Deciphering the free energy landscape of biomolecular structure space is crucial for understanding many complex molecular processes, such as protein-protein interaction, RNA folding, and protein folding. A major source of current dynamic structure data is Molecular Dynamics (MD) simulations. Several methods have been proposed to investigate the free energy landscape from MD data, but all of them rely on the assumption that kinetic similarity is associated with global geometric similarity, which may lead to unsatisfactory results. In this paper, we proposed a new method called Conditional Angle Partition Tree to reveal the hierarchical free energy landscape by correlating local geometric similarity with kinetic similarity. Its application on the benchmark alanine dipeptide MD data showed a much better performance than existing methods in exploring and understanding the free energy landscape. We also applied it to the MD data of Villin HP35. Our results are more reasonable on various aspects than those from other methods and very informative on the hierarchical structure of its energy landscape.

Diffusive exit rates through pores in membrane-enclosed structures

The function of many membrane-enclosed intracellular structures relies on release of diffusing particles that exit through narrow pores or channels in the membrane. The rate of release varies with pore size, density, and length of the channel. We propose a simple approximate model, validated with stochastic simulations, for estimating the effective release rate from cylinders, and other simple-shaped domains, as a function of channel parameters. The results demonstrate that, for very small pores, a low density of channels scattered over the boundary is sufficient to achieve substantial rates of particle release. Furthermore, we show that increasing the length of passive channels will both reduce release rates and lead to a less steep dependence on channel density. Our results are compared to previously-measured local calcium release rates from tubules of the endoplasmic reticulum, providing an estimate of the relevant channel density responsible for the observed calcium efflux.

Chemical bonds in collagen rupture selectively under tensile stress

Collagen fibres are the main constituent of the extracellular matrix, and fulfil an important role in the structural stability of living multicellular organisms. An open question is how collagen absorbs pulling forces, and if the applied forces are strong enough to break bonds, what mechanisms underlie this process. As experimental studies on this topic are challenging, simulations are an important tool to further our understanding of these mechanisms. Here, we present pulling simulations of collagen triple helices, revealing the molecular mechanisms induced by tensile stress. At lower forces, pulling alters the configuration of proline residues leading to an effective absorption of applied stress. When forces are strong enough to introduce bond ruptures, these are located preferentially in X-position residues. Reduced backbone flexibility, for example through mutations or cross linking, weakens tensile resistance, leading to localised ruptures around these perturbations. In fibre-like segments, a significant overrepresentation of ruptures in proline residues compared to amino acid contents is observed. This study confirms the important role of proline in the structural stability of collagen, and adds detailed insight into the molecular mechanisms underlying this observation.

Enhancing Biomolecular Simulations with Hybrid Potentials Incorporating NMR Data

Some recent advances in biomolecular simulation and global optimization have used hybrid restraint potentials, where harmonic restraints that penalize conformations inconsistent with experimental data are combined with molecular mechanics force fields. These hybrid potentials can be used to improve the performance of molecular dynamics, structure prediction, energy landscape sampling, and other computational methods that rely on the accuracy of the underlying force field. Here, we develop a hybrid restraint potential based on NapShift, an artificial neural network trained to predict protein nuclear magnetic resonance (NMR) chemical shifts from sequence and structure. In addition to providing accurate predictions of experimental chemical shifts, NapShift is fully differentiable with respect to atomic coordinates, which allows us to use it for structural refinement. By employing NapShift to predict chemical shifts from the protein conformation at each simulation step, we can compute an energy penalty and the corresponding hybrid restraint forces based on the difference between the predicted values and the experimental chemical shifts. The performance of the hybrid restraint potential was benchmarked using both basin-hopping global optimization and molecular dynamics simulations. In each case, the NapShift hybrid potential improved the accuracy, leading to better structure prediction via basin-hopping and increased local stability in molecular dynamics simulations. Our results suggest that neural network hybrid potentials based on NMR observables can enhance a broad range of molecular simulation methods, and the prediction accuracy will improve as more experimental training data become available.

Design of self-assembling mesoscopic Goldberg polyhedra

Palladium ions complexed with nonlinear bidentate ligands have been shown to form hollow, spherical shells with high symmetries. We show that such structures can be reproduced using model anisotropic mesoscale building blocks featuring excluded volume and long-range ionic interactions. A linear building block with a central charged particle, in combination with a bent 'ligand' particle with opposite charges at the ends is sufficient to drive the system towards planar coordination, and the charge ratio determines the coordination number. Similar to the molecular systems, the bend in the 'ligand' particle determines the curvature of the shells that these building blocks prefer. Besides reproducing exotic structures such as M₃₀L₆₀ and M₄₈L₉₆ tetravalent Goldberg polyhedra, we identify highly cooperative single transition state rearrangements between low-energy competing structures as well, corresponding to rotatory motions of a planar subunit within the spherical shell.

A robust and memory-efficient transition state search method for complex energy landscapes

Locating transition states is crucial for investigating transition mechanisms in wide-ranging phenomena, from atomistic to macroscale systems. Existing methods, however, can struggle in problems with a large number of degrees of freedom, on-the-fly adaptive remeshing and coarse-graining, and energy landscapes that are locally flat or discontinuous. To resolve these challenges, we introduce a new double-ended method, the Binary-Image Transition State Search (BITSS). It uses just two states that converge to the transition state, resulting in a fast, flexible, and memory-efficient method. We also show that it is more robust compared to existing bracketing methods that use only two states. We demonstrate its versatility by applying BITSS to three very different classes of problems: Lennard-Jones clusters, shell buckling, and multiphase phase-field models.

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.

Methanol Synthesis from CO2/CO Mixture on Cu-Zn Catalysts from Microkinetics-Guided Machine Learning Pathway Search

Methanol synthesis on industrial Cu/ZnO/Al₂O₃ catalysts via the hydrogenation of CO and CO₂ mixture, despite several decades of research, is still puzzling due to the nature of the active site and the role of CO₂ in the feed gas. Herein, with the large-scale machine learning atomic simulation, we develop a microkinetics-guided machine learning pathway search to explore thousands of reaction pathways for CO₂ and CO hydrogenations on thermodynamically favorable Cu-Zn surface structures, including Cu(111), Cu(211), and Zn-alloyed Cu(211) surfaces, from which the lowest energy pathways are identified. We find that Zn decorates at the step-edge at Cu(211) up to 0.22 ML under reaction conditions with the Zn-Zn dimeric sites being avoided. CO₂ and CO hydrogenations occur exclusively at the step-edge of the (211) surface with up to 0.11 ML Zn coverage, where the low coverage of Zn (0.11 ML) does not much affect the reaction kinetics, but the higher coverages of Zn (0.22 ML) poison the catalyst. It is CO₂ hydrogenation instead of CO hydrogenation that dominates methanol synthesis, agreeing with previous isotope experiments. While metallic steps are identified as the major active site, we show that the [-Zn-OH-Zn-] chains (cationic Zn) can grow on Cu(111) surfaces under reaction conditions, which suggests the critical role of CO in the mixed gas for reducing the cationic Zn and exposing metal sites for methanol synthesis. Our results provide a comprehensive picture on the dynamic coupling of the feed gas composition, the catalyst active site, and the reaction activity in this complex heterogeneous catalytic system.

Dynamical Signatures of Multifunnel Energy Landscapes

Multifunctional systems, such as molecular switches, exhibit multifunnel energy landscapes associated with the alternative functional states. In this contribution the multifunnel organization is decoded from dynamical signatures in the first passage time distribution between reactants and products. Characteristic relaxation rates are revealed by analyzing the kinetics as a function of the observation time scale, which scans the underlying distribution. Extracting the corresponding dynamical signatures provides direct insight into the organization of the molecular energy landscape, which will facilitate a rational design of target functionality. Examples are illustrated for multifunnel landscapes in biomolecular systems and an atomic cluster.

Evaluating Geometric Definitions of Stacking for RNA Dinucleoside Monophosphates Using Molecular Mechanics Calculations

RNA modulation via small molecules is a novel approach in pharmacotherapies, where the determination of the structural properties of RNA motifs is considered a promising way to develop drugs capable of targeting RNA structures to control diseases. However, due to the complexity and dynamic nature of RNA molecules, the determination of RNA structures using experimental approaches is not always feasible, and computational models employing force fields can provide important insight. The quality of the force field will determine how well the predictions are compared to experimental observables. Stacking in nucleic acids is one such structural property, originating mainly from London dispersion forces, which are quantum mechanical and are included in molecular mechanics force fields through nonbonded interactions. Geometric descriptions are utilized to decide if two residues are stacked and hence to calculate the stacking free energies for RNA dinucleoside monophosphates (DNMPs) through statistical mechanics for comparison with experimental thermodynamics data. Here, we benchmark four different stacking definitions using molecular dynamics (MD) trajectories for 16 RNA DNMPs produced by two different force fields (RNA-IL and ff99OL3) and show that our stacking definition better correlates with the experimental thermodynamics data. While predictions within an accuracy of 0.2 kcal/mol at 300 K were observed in RNA CC, CU, UC, AG, GA, and GG, stacked states of purine-pyrimidine and pyrimidine-purine DNMPs, respectively, were typically underpredicted and overpredicted. Additionally, population distributions of RNA UU DNMPs were poorly predicted by both force fields, implying a requirement for further force field revisions. We further discuss the differences predicted by each RNA force field. Finally, we show that discrete path sampling (DPS) calculations can provide valuable information and complement the MD simulations. We propose the use of experimental thermodynamics data for RNA DNMPs as benchmarks for testing RNA force fields.

Investigating the structural changes due to adenosine methylation of the Kaposi’s sarcoma-associated herpes virus ORF50 transcript

Kaposi’s sarcoma-associated herpes virus (KSHV) is a human oncovirus. KSHV relies on manipulating the host cell N6-methyl adenosine (m6A) RNA modification pathway to enhance virus replication. Methylation within a RNA stem loop of the open reading frame 50 (ORF50) increases transcript stability via the recruitment of the m6A reader, SND1. In this contribution we explore the energy landscapes of the unmethylated and methylated RNA stem loops of ORF50 to investigate the effect of methylation on the structure of the stem loop. We observe a significant shift upon methylation between an open and closed configuration of the top of the stem loop. In the unmethylated stem loop the closed configuration is much lower in energy, and, as a result, exhibits higher occupancy.

In this article we present the investigation of the change in structure of an RNA regulatory molecule upon a change in the chemistry of one of its bases. Eukaryotic RNAs contain more than 100 different types of chemical modifications, which can fine-tune the structure and function of RNA. Since RNA systems need to adopt a specific 3D shape to be functional, it is important to understand how a chemical modification impacts the structure adopted. Using the computational technique of energy landscape explorations, that is exploring what structures are available to the system at a given energy, we are able to characterise the RNA before and after the modification, and understand what the main differences between the ensembles of structures, which can be adopted by the system, are.

In this work, we present our results of this investigation on an oncogenic virus-encoded RNA. We show how a chemical modification at a precise location of the native structure affects the system globally, inducing a rearrangement of parts of the structure, which are far away from the modification site.

Energy Landscapes for Base-Flipping in a Model DNA Duplex

We explore the process of base-flipping for four central bases, adenine, guanine, cytosine, and thymine, in a deoxyribonucleic acid (DNA) duplex using the energy landscape perspective. NMR imino-proton exchange and fluorescence correlation spectroscopy studies have been used in previous experiments to obtain lifetimes for bases in paired and extrahelical states. However, the difference of almost 4 orders of magnitude in the base-flipping rates obtained by the two methods implies that they are exploring different pathways and possibly different open states. Our results support the previous suggestion that minor groove opening may be favored by distortions in the DNA backbone and reveal links between sequence effects and the direction of opening, i.e., whether the base flips toward the major or the minor groove side. In particular, base flipping along the minor groove pathway was found to align toward the 5' side of the backbone. We find that bases align toward the 3' side of the backbone when flipping along the major groove pathway. However, in some cases for cytosine and thymine, the base flipping along the major groove pathway also aligns toward the 5' side. The sequence effect may be caused by the polar interactions between the flipping-base and its neighboring bases on either of the strands. For guanine flipping toward the minor groove side, we find that the equilibrium constant for opening is large compared to flipping via the major groove. We find that the estimated rates of base opening, and hence the lifetimes of the closed state, obtained for thymine flipping through small and large angles along the major groove differ by 6 orders of magnitude, whereas for thymine flipping through small angles along the minor groove and large angles along the major groove, the rates differ by 3 orders of magnitude.

The Energy Landscape Perspective: Encoding Structure and Function for Biomolecules

The energy landscape perspective is outlined with particular reference to biomolecules that perform multiple functions. We associate these multifunctional molecules with multifunnel energy landscapes, illustrated by some selected examples, where understanding the organisation of the landscape has provided new insight into function. Conformational selection and induced fit may provide alternative routes to realisation of multifunctionality, exploiting the possibility of environmental control and distinct binding modes.

Computer-aided comprehensive explorations of RNA structural polymorphism through complementary simulation methods

While RNA folding was originally seen as a simple problem to solve, it has been shown that the promiscuous interactions of the nucleobases result in structural polymorphism, with several competing structures generally observed for non-coding RNA. This inherent complexity limits our understanding of these molecules from experiments alone, and computational methods are commonly used to study RNA. Here, we discuss three advanced sampling schemes, namely Hamiltonian-replica exchange molecular dynamics (MD), ratchet-and-pawl MD and discrete path sampling, as well as the HiRE-RNA coarse-graining scheme, and highlight how these approaches are complementary with reference to recent case studies. While all computational methods have their shortcomings, the plurality of simulation methods leads to a better understanding of experimental findings and can inform and guide experimental work on RNA polymorphism.

The effects of glycine to alanine mutations on the structure of GPO collagen model peptides

Collagen proteins are the main constituents of the extracellular matrix (ECM), and fulfil a number of wide-ranging functions, including contributions to the mechanical and biological behaviour of the ECM. Due to the heterogeneous nature of collagen in tissue samples it is difficult to fully explain the experimental observation, and hence the study of shorter model peptides is common place. Here, the computational energy landscape framework is employed to study Gly to Ala mutations in a GPO model peptide. The results show good agreement with the experimental observations for the GPO reference and a triply mutated peptide, demonstrating the validity of the approach. The modelling predicts that changes in structure are moderate and localised, with an increased dynamic in the backbone and alterations to the hydrogen bonding pattern. Two mechanisms for adjusting to the mutations are observed, with potential consequences regarding protein binding. Finally, in line with a hypothesis that proline puckering allows controlled flexibility (Chow et al., Sci. Rep., 2018, 8, 13809), alterations in the puckering preferences are observed in the strained residues surrounding the mutational sites.

Nearly reducible finite Markov chains: Theory and algorithms

Finite Markov chains, memoryless random walks on complex networks, appear commonly as models for stochastic dynamics in condensed matter physics, biophysics, ecology, epidemiology, economics, and elsewhere. Here, we review exact numerical methods for the analysis of arbitrary discrete- and continuous-time Markovian networks. We focus on numerically stable methods that are required to treat nearly reducible Markov chains, which exhibit a separation of characteristic timescales and are therefore ill-conditioned. In this metastable regime, dense linear algebra methods are afflicted by propagation of error in the finite precision arithmetic, and the kinetic Monte Carlo algorithm to simulate paths is unfeasibly inefficient. Furthermore, iterative eigendecomposition methods fail to converge without the use of nontrivial and system-specific preconditioning techniques. An alternative approach is provided by state reduction procedures, which do not require additional a priori knowledge of the Markov chain. Macroscopic dynamical quantities, such as moments of the first passage time distribution for a transition to an absorbing state, and microscopic properties, such as the stationary, committor, and visitation probabilities for nodes, can be computed robustly using state reduction algorithms. The related kinetic path sampling algorithm allows for efficient sampling of trajectories on a nearly reducible Markov chain. Thus, all of the information required to determine the kinetically relevant transition mechanisms, and to identify the states that have a dominant effect on the global dynamics, can be computed reliably even for computationally challenging models. Rare events are a ubiquitous feature of realistic dynamical systems, and so the methods described herein are valuable in many practical applications.

RNA Modeling with the Computational Energy Landscape Framework

The recent advances in computational abilities, such as the enormous speed-ups provided by GPU computing, allow for large scale computational studies of RNA molecules at an atomic level of detail. As RNA molecules are known to adopt multiple conformations with comparable energies, but different two-dimensional structures, all-atom models are necessary to better describe the structural ensembles for RNA molecules. This point is important because different conformations can exhibit different functions, and their regulation or mis-regulation is linked to a number of diseases. Problematically, the energy barriers between different conformational ensembles are high, resulting in long time scales for interensemble transitions. The computational potential energy landscape framework was designed to overcome this problem of broken ergodicity by use of geometry optimization. Here, we describe the algorithms used in the energy landscape explorations with the OPTIM and PATHSAMPLE programs, and how they are used in biomolecular simulations. We present a recent case study of the 5'-hairpin of RNA 7SK to illustrate how the method can be applied to interpret experimental results, and to obtain a detailed description of molecular properties.

Graph transformation and shortest paths algorithms for finite Markov chains

The graph transformation (GT) algorithm robustly computes the mean first-passage time to an absorbing state in a finite Markov chain. Here we present a concise overview of the iterative and block formulations of the GT procedure and generalize the GT formalism to the case of any path property that is a sum of contributions from individual transitions. In particular, we examine the path action, which directly relates to the path probability, and analyze the first-passage path ensemble for a model Markov chain that is metastable and therefore numerically challenging. We compare the mean first-passage path action, obtained using GT, with the full path action probability distribution simulated efficiently using kinetic path sampling, and with values for the highest-probability paths determined by the recursive enumeration algorithm (REA). In Markov chains representing realistic dynamical processes, the probability distributions of first-passage path properties are typically fat-tailed and therefore difficult to converge by sampling, which motivates the use of exact and numerically stable approaches to compute the expectation. We find that the kinetic relevance of the set of highest-probability paths depends strongly on the metastability of the Markov chain, and so the properties of the dominant first-passage paths may be unrepresentative of the global dynamics. Use of a global measure for edge costs in the REA, based on net productive fluxes, allows the total reactive flux to be decomposed into a finite set of contributions from simple flux paths. By considering transition flux paths, a detailed quantitative analysis of the relative importance of competing dynamical processes is possible even in the metastable regime.

Is the H4 histone tail intrinsically disordered or intrinsically multifunctional?

The structural versatility of histone tails is one of the key elements in the organisation of chromatin, which allows for the compact storage of genomic information. However, this structural diversity also complicates experimental and computational studies. Here, the potential and free energy landscape for the isolated and bound H4 histone tail are explored. The landscapes exhibit a set of distinct structural ensembles separated by high energy barriers, with little difference between isolated and bound tails. This consistency is a desirable feature that facilitates the formation of transient interactions, which are required for the liquid-like chromatin organisation. The existence of multiple, distinct structures on a multifunnel energy landscape is likely to be associated with multifunctionality, i.e. a set of evolved, distinct functions. Contrasting it with previously reported results for other disordered peptides, this type of landscape may be associated with a conformational selection based binding mechanism. Given the similarity to other systems exhibiting similar multifunnel energy landscapes, the disorder in histone tails might be better described in context of multifunctionality.

Diffusive search and trajectories on tubular networks: a propagator approach

Several organelles in eukaryotic cells, including mitochondria and the endoplasmic reticulum, form interconnected tubule networks extending throughout the cell. These tubular networks host many biochemical pathways that rely on proteins diffusively searching through the network to encounter binding partners or localized target regions. Predicting the behavior of such pathways requires a quantitative understanding of how confinement to a reticulated structure modulates reaction kinetics. In this work, we develop both exact analytical methods to compute mean first passage times and efficient kinetic Monte Carlo algorithms to simulate trajectories of particles diffusing in a tubular network. Our approach leverages exact propagator functions for the distribution of transition times between network nodes and allows large simulation time steps determined by the network structure. The methodology is applied to both synthetic planar networks and organelle network structures, demonstrating key general features such as the heterogeneity of search times in different network regions and the functional advantage of broadly distributing target sites throughout the network. The proposed algorithms pave the way for future exploration of the interrelationship between tubular network structure and biomolecular reaction kinetics.

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.

Reaction prediction via atomistic simulation: from quantum mechanics to machine learning

It is an ultimate goal in chemistry to predict reaction without recourse to experiment. Reaction prediction is not just the reaction rate determination of known reactions but, more broadly, the reaction exploration to identify new reaction routes. This review briefly overviews the theory on chemical reaction and the current methods for computing/estimating reaction rate and exploring reaction space. We particularly focus on the atomistic simulation methods for reaction exploration, which are benefited significantly by recently emerged machine learning potentials. We elaborate the stochastic surface walking global pathway sampling based on the global neural network (SSW-NN) potential, developed in our group since 2013, which can explore complex reactions systems unbiasedly and automatedly. Two examples, molecular reaction and heterogeneous catalytic reactions, are presented to illustrate the current status for reaction prediction using SSW-NN.

Rare events and first passage time statistics from the energy landscape

We analyze the probability distribution of rare first passage times corresponding to transitions between product and reactant states in a kinetic transition network. The mean first passage times and the corresponding rate constants are analyzed in detail for two model landscapes and the double funnel landscape corresponding to an atomic cluster. Evaluation schemes based on eigendecomposition and kinetic path sampling, which both allow access to the first passage time distribution, are benchmarked against mean first passage times calculated using graph transformation. Numerical precision issues severely limit the useful temperature range for eigendecomposition, but kinetic path sampling is capable of extending the first passage time analysis to lower temperatures, where the kinetics of interest constitute rare events. We then investigate the influence of free energy based state regrouping schemes for the underlying network. Alternative formulations of the effective transition rates for a given regrouping are compared in detail to determine their numerical stability and capability to reproduce the true kinetics, including recent coarse-graining approaches that preserve occupancy cross correlation functions. We find that appropriate regrouping of states under the simplest local equilibrium approximation can provide reduced transition networks with useful accuracy at somewhat lower temperatures. Finally, a method is provided to systematically interpolate between the local equilibrium approximation and exact intergroup dynamics. Spectral analysis is applied to each grouping of states, employing a moment-based mode selection criterion to produce a reduced state space, which does not require any spectral gap to exist, but reduces to gap-based coarse graining as a special case. Implementations of the developed methods are freely available online.

Energy Landscape for the Membrane Fusion Pathway in Influenza A Hemagglutinin From Discrete Path Sampling

The conformational change associated with membrane fusion for Influenza A Hemagglutinin is investigated with a model based upon pre- and post-fusion structures of the HA2 component. We employ computational methods based on the potential energy landscape framework to obtain an initial path connecting these two end points, which provides the starting point for refinement of a kinetic transition network. Here we employ discrete path sampling, which provides access to the experimental time and length scales via geometry optimization techniques to identify local minima and the transition states that connect them. We then analyse the distinct phases of the predicted pathway in terms of structure and energetics, and compare with available experimental data and previous simulations. Our results provide the foundations for future work, which will address the effect of mutations, changes in pH, and incorporation of additional components, especially the HA1 chain and the fusion peptide.

Improving double-ended transition state searches for soft-matter systems

Transitions between different stable configurations of biomolecules are important in understanding disease mechanisms, structure-function relations, and novel molecular-scale engineering. The corresponding pathways can be characterized efficiently using geometry optimization schemes based on double-ended transition state searches. An interpolation is first constructed between the known states and then refined, yielding a band that contains transition state candidates. Here, we analyze an example where various interpolation schemes lead to bands with a single step transition, but the correct pathway actually proceeds via an intervening, low-energy minimum. We compare a number of different interpolation schemes for this problem. We systematically alter the number of discrete images in the interpolations and the spring constants used in the optimization and test two schemes for adjusting the spring constants and image distribution, resulting in a total of 2760 different connection attempts. Our results confirm that optimized bands are not necessarily a good description of the transition pathways in themselves, and further refinement to actually converge transition states and establish their connectivity is required. We see an improvement in the optimized bands if we employ the adjustment of spring constants with doubly-nudged elastic band and a smaller improvement from the image redistribution. The example we consider is representative of numerous cases we have encountered in a wide variety of molecular and condensed matter systems.

Affinity-Selected Bicyclic Peptide G-Quadruplex Ligands Mimic a Protein-like Binding Mechanism

The study of G-quadruplexes (G4s) in a cellular context has demonstrated links between these nucleic acid secondary structures, gene expression, and DNA replication. Ligands that bind to the G4 structure therefore present an excellent opportunity for influencing gene expression through the targeting of a nucleic acid structure rather than sequence. Here, we explore cyclic peptides as an alternative class of G4 ligands. Specifically, we describe the development of de novo G4-binding bicyclic peptides selected by phage display. Selected bicyclic peptides display submicromolar affinity to G4 structures and high selectivity over double helix DNA. Molecular simulations of the bicyclic peptide-G4 complexes corroborate the experimental binding strengths and reveal molecular insights into G4 recognition by bicyclic peptides via the precise positioning of amino acid side chains, a binding mechanism reminiscent of endogenous G4-binding proteins. Overall, our results demonstrate that selection of (bi)cyclic peptides unlocks a valuable chemical space for targeting nucleic acid structures.

Energy Landscapes of Deoxyxylo- and Xylo-Nucleic Acid Octamers

Artificial analogues of the natural nucleic acids have attracted interest as a diverse class of information storage molecules capable of self-replication. In this study, we use the computational potential energy landscape framework to investigate the structural and dynamical properties of xylo- and deoxyxylo-nucleic acids (XyNA and dXyNA), which are derived from their respective RNA and DNA analogues by inversion of a single chiral center in the sugar moiety of the nucleotides. For an octameric XyNA sequence and the analogue dXyNA, we observe facile conformational transitions between a left-handed helix, which is the free energy global minimum, and a ladder-type structure with approximately zero helicity. The competing ensembles are better separated in the dXyNA, making it a more suitable candidate for a molecular switch, whereas the XyNA exhibits additional flexibility. Both energy landscapes exhibit greater frustration than we observe in RNA or DNA, in agreement with the higher degree of optimization expected from the principle of minimal frustration in evolved biomolecules.