Energy Landscapes for Proteins: From Single Funnels to Multifunctional Systems

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.

Multilevel Framework for Analysis of Protein Folding Involving Disulfide Bond Formation

In this study, a three-layered multicenter ONIOM approach is implemented to characterize the naive folding pathway of bovine pancreatic trypsin inhibitor (BPTI). Each layer represents a distinct level of theory, where the initial layer, encompassing the entire protein, is modeled by a general all-atom force-field GFN-FF. An intermediate electronic structure layer consisting of three multicenter fragments is introduced with the state-of-the-art semiempirical tight-binding method GFN2-xTB. Higher accuracy, specifically addressing the breaking and formation of the three disulfide bonds, is achieved at the innermost layer using the composite DFT method r2SCAN-3c. Our analysis sheds light on the structural stability of BPTI, particularly the significance of interlinking disulfide bonds. The accuracy and efficiency of the multicenter QM/SQM/MM approach are benchmarked using the oxidative formation of cystine. For the folding pathway of BPTI, relative stabilities are investigated through the calculation of free energy contributions for selected intermediates, focusing on the impact of the disulfide bond. Our results highlight the intricate trade-off between accuracy and computational cost, demonstrating that the multicenter ONIOM approach provides a well-balanced and comprehensive solution to describe electronic structure effects in biomolecular systems. We conclude that multiscale energy landscape exploration provides a robust methodology for the study of intriguing biological targets.

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.

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.

Protein conformational ensembles in function: roles and mechanisms

The sequence-structure-function paradigm has dominated twentieth century molecular biology. The paradigm tacitly stipulated that for each sequence there exists a single, well-organized protein structure. Yet, to sustain cell life, function requires (i) that there be more than a single structure, (ii) that there be switching between the structures, and (iii) that the structures be incompletely organized. These fundamental tenets called for an updated sequence-conformational ensemble-function paradigm. The powerful energy landscape idea, which is the foundation of modernized molecular biology, imported the conformational ensemble framework from physics and chemistry. This framework embraces the recognition that proteins are dynamic and are always interconverting between conformational states with varying energies. The more stable the conformation the more populated it is. The changes in the populations of the states are required for cell life. As an example, in vivo, under physiological conditions, wild type kinases commonly populate their more stable "closed", inactive, conformations. However, there are minor populations of the "open", ligand-free states. Upon their stabilization, e.g., by high affinity interactions or mutations, their ensembles shift to occupy the active states. Here we discuss the role of conformational propensities in function. We provide multiple examples of diverse systems, including protein kinases, lipid kinases, and Ras GTPases, discuss diverse conformational mechanisms, and provide a broad outlook on protein ensembles in the cell. We propose that the number of molecules in the active state (inactive for repressors), determine protein function, and that the dynamic, relative conformational propensities, rather than the rigid structures, are the hallmark of cell life.

Replica Exchange with Hybrid Tempering Efficiently Samples PGLa Peptide Binding to Anionic Bilayer

We evaluated the utility of a variant of the replica exchange method, a replica exchange with hybrid tempering (REHT), for all-atom explicit water biomolecular simulations and compared it with a more traditional replica exchange with the solute tempering (REST) algorithm. As a test system, we selected a 21-mer antimicrobial peptide PGLa binding to an anionic DMPC/DMPG lipid bilayer. Application of REHT revealed the following binding mechanism. Due to the strong hydrophobic moment, the bound PGLa adopts an extensive helical structure. The binding free energy landscape identifies two major bound states, a metastable surface bound state and a dominant inserted state. In both states, positively charged PGLa amino acids maintain electrostatic interactions with anionic phosphate groups by rotating the PGLa helix around its axis. PGLa binding causes an influx of anionic DMPG and an efflux of zwitterionic DMPC lipids from the peptide proximity. PGLa thins the bilayer and disorders the adjacent fatty acid tails. Deep invasion of water wires into the bilayer hydrophobic core is detected in the inserted peptide state. The analysis of charge density distributions indicated that peptide positive charges are nearly compensated for by lipid negative charges and water dipole ordering, whereas ions play no role in peptide binding. Thus, electrostatic interactions are the key energetic factor in binding cationic PGLa to an anionic DMPC/DMPG bilayer. Comparison of REHT and REST shows that due to exclusion of lipids from tempered partition, REST lags behind REHT in peptide equilibration, particularly, with respect to peptide insertion and helix acquisition. As a result, REST struggles to provide accurate details of PGLa binding, although it still qualitatively maps the bimodal binding mechanism. Importantly, REHT not only equilibrates PGLa in the bilayer faster than REST, but also with less computational effort. We conclude that REHT is a preferable choice for studying interfacial biomolecular systems.

Analysing ill-conditioned Markov chains

Discrete state Markov chains in discrete or continuous time are widely used to model phenomena in the social, physical and life sciences. In many cases, the model can feature a large state space, with extreme differences between the fastest and slowest transition timescales. Analysis of such ill-conditioned models is often intractable with finite precision linear algebra techniques. In this contribution, we propose a solution to this problem, namely partial graph transformation, to iteratively eliminate and renormalize states, producing a low-rank Markov chain from an ill-conditioned initial model. We show that the error induced by this procedure can be minimized by retaining both the renormalized nodes that represent metastable superbasins, and those through which reactive pathways concentrate, i.e. the dividing surface in the discrete state space. This procedure typically returns a much lower rank model, where trajectories can be efficiently generated with kinetic path sampling. We apply this approach to an ill-conditioned Markov chain for a model multi-community system, measuring the accuracy by direct comparison with trajectories and transition statistics. This article is part of a discussion meeting issue 'Supercomputing simulations of advanced materials'.

Molecular Energy Landscapes of Hardware-Efficient Ansätze in Quantum Computing

Rapid advances in quantum computing have opened up new opportunities for solving the central electronic structure problem in computational chemistry. In the noisy intermediate-scale quantum (NISQ) era, where qubit coherence times are limited, it is essential to exploit quantum algorithms with sufficiently short quantum circuits to maximize qubit efficiency. The procedural construction of hardware-efficient ansätze provides one approach to design such circuits. However, refining the accuracy of the global minimum by increasing circuit depth may lead to a proliferation of local minima that hinders global optimization. To investigate this phenomenon, we explore the energy landscapes of hardware-efficient circuits to identify ground-state energies of the hydrogen, lithium hydride, and beryllium hydride molecules. We also propose a simple dimensionality reduction procedure that reduces quantum gate depth while retaining high accuracy for the global minimum, simplifying the energy landscape, and hence speeding up optimization from both software and hardware perspectives.

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.

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.

Molecular insights into binding dynamics of tandem RNA recognition motifs (tRRMs) of human antigen R (HuR) with mRNA and the effect of point mutations in impaired HuR-mRNA recognition

Human antigen R (HuR) is a key regulatory protein with prominent roles in RNA metabolism and post-transcriptional gene regulation. Many studies have shown the involvement of HuR in plethora of human diseases, which are often manifestations of impaired HuR-RNA interactions. However, the inherent complexities of highly flexible protein-RNA interactions have limited our understanding of the structural basis of HuR-RNA recognition. In this study, we dissect the underlying molecular mechanism of interaction between N-terminal tandem RNA-recognition motifs (tRRMs) of HuR and mRNA using molecular dynamics simulation. We have also explored the effect of point mutations (T90A, R97A and R136A) of three reported critical residues in HuR-mRNA binding specificity. Our findings show that N-terminal tRRMs exhibit conformational stability upon RNA binding. We further show that R136A and R97A mutants significantly lose their binding affinity owing to the loss of critical interactions with mRNA. This may be attributed to the larger domain rearrangements in the mutant complexes, especially the β2β3 loops in both the tRRMs, leading to unfavourable conformations and loss of binding affinity. We have identified critical binding residues in tRRMs of HuR, contributing favourable binding energy in mRNA recognition. This study contributes significantly to understand the molecular mechanism of RNA recognition by tandem RRMs and provides a platform to modulate binding affinities through mutations. This may further guide in future structure-based drug-therapies targeting impaired HuR-RNA interactions.Communicated by Ramaswamy H. Sarma.

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.

Evolution of the folding landscape of effector caspases

Caspases are a family of cysteinyl proteases that control programmed cell death and maintain homeostasis in multicellular organisms. The caspase family is an excellent model to study protein evolution because all caspases are produced as zymogens (procaspases [PCPs]) that must be activated to gain full activity; the protein structures are conserved through hundreds of millions of years of evolution; and some allosteric features arose with the early ancestor, whereas others are more recent evolutionary events. The apoptotic caspases evolved from a common ancestor (CA) into two distinct subfamilies: monomers (initiator caspases) or dimers (effector caspases). Differences in activation mechanisms of the two subfamilies, and their oligomeric forms, play a central role in the regulation of apoptosis. Here, we examine changes in the folding landscape by characterizing human effector caspases and their CA. The results show that the effector caspases unfold by a minimum three-state equilibrium model at pH 7.5, where the native dimer is in equilibrium with a partially folded monomeric (PCP-7, CA) or dimeric (PCP-6) intermediate. In comparison, the unfolding pathway of PCP-3 contains both oligomeric forms of the intermediate. Overall, the data show that the folding landscape was first established with the CA and was retained for >650 million years. Partially folded monomeric or dimeric intermediates in the ancestral ensemble provide mechanisms for evolutionary changes that affect stability of extant caspases. The conserved folding landscape allows for the fine-tuning of enzyme stability in a species-dependent manner while retaining the overall caspase–hemoglobinase fold.

Numerical analysis of first-passage processes in finite Markov chains exhibiting metastability

We describe state-reduction algorithms for the analysis of first-passage processes in discrete- and continuous-time finite Markov chains. We present a formulation of the graph transformation algorithm that allows for the evaluation of exact mean first-passage times, stationary probabilities, and committor probabilities for all nonabsorbing nodes of a Markov chain in a single computation. Calculation of the committor probabilities within the state-reduction formalism is readily generalizable to the first hitting problem for any number of alternative target states. We then show that a state-reduction algorithm can be formulated to compute the expected number of times that each node is visited along a first-passage path. Hence, all properties required to analyze the first-passage path ensemble (FPPE) at both a microscopic and macroscopic level of detail, including the mean and variance of the first-passage time distribution, can be computed using state-reduction methods. In particular, we derive expressions for the probability that a node is visited along a direct transition path, which proceeds without returning to the initial state, considering both the nonequilibrium and equilibrium (steady-state) FPPEs. The reactive visitation probability provides a rigorous metric to quantify the dynamical importance of a node for the productive transition between two endpoint states and thus allows the local states that facilitate the dominant transition mechanisms to be readily identified. The state-reduction procedures remain numerically stable even for Markov chains exhibiting metastability, which can be severely ill-conditioned. The rare event regime is frequently encountered in realistic models of dynamical processes, and our methodology therefore provides valuable tools for the analysis of Markov chains in practical applications. We illustrate our approach with numerical results for a kinetic network representing a structural transition in an atomic cluster.

Structural Transitions in Glassy Atactic Polystyrene Using Transition-State Theory

Transition pathways on the energy landscape of atactic polystyrene (aPS) glassy specimens are probed below its glass-transition temperature. Each of these transitions is considered an elementary structural relaxation event, whose corresponding rate constant is calculated by applying multidimensional transition-state theory. Initially, a wide spectrum of first-order saddle points surrounding local minima on the energy landscape is discovered by a stabilized hybrid eigenmode-following method. Then, (minimal-energy) “reaction paths” to the adjacent minima are constructed by a quadratic descent method. The heights of the free energy, the potential energy, and the entropy barriers are estimated for every connected triplet of transition state and minima. The resulting distribution of free energy barriers is asymmetric and extremely broad, extending to very high barrier heights (over 50 k_BT); the corresponding distribution of rate constants extends over 30 orders of magnitude, with well-defined peaks at the time scales corresponding to the subglass relaxations of polystyrene. Analysis of the curvature along the reaction paths reveals a multitude of different rearrangement mechanisms; some of them bearing multiple distinct phases. Finally, connections to theoretical models of the glass phenomenology allows for the prediction, based on first-principles, of the “ideal” glass-transition temperature entering the Vogel–Fulcher–Tammann (VFT) equation describing the super-Arrhenius temperature dependence of glassy dynamics. Our predictions of the time scales of the subglass relaxations and the VFT temperature are in favorable agreement with available experimental literature data for systems of similar molecular weight under the same conditions.

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.

High resolution ensemble description of metamorphic and intrinsically disordered proteins using an efficient hybrid parallel tempering scheme

Mapping free energy landscapes of complex multi-funneled metamorphic proteins and weakly-funneled intrinsically disordered proteins (IDPs) remains challenging. While rare-event sampling molecular dynamics simulations can be useful, they often need to either impose restraints or reweigh the generated data to match experiments. Here, we present a parallel-tempering method that takes advantage of accelerated water dynamics and allows efficient and accurate conformational sampling across a wide variety of proteins. We demonstrate the improved sampling efficiency by benchmarking against standard model systems such as alanine di-peptide, TRP-cage and β-hairpin. The method successfully scales to large metamorphic proteins such as RFA-H and to highly disordered IDPs such as Histatin-5. Across the diverse proteins, the calculated ensemble averages match well with the NMR, SAXS and other biophysical experiments without the need to reweigh. By allowing accurate sampling across different landscapes, the method opens doors for sampling free energy landscape of complex uncharted proteins.

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.

Degeneracy in molecular scale organization of biological membranes

The scale-rich spatiotemporal organization in biological membranes has its origin in the differential inter- and intra-molecular interactions among their constituents. In this work, we explore the molecular-origin behind that variety and possible degeneracy in lateral organization in membranes. For our study, we post-process microsecond long all-atom molecular dynamics trajectories for three systems that exhibit fluid phase coexistence: (i) PSM/POPC/Chol (0.47/0.32/0.21), (ii) PSM/DOPC/Chol (0.43/0.38/0.19) and (iii) DPPC/DOPC/Chol (0.37/0.36/0.27). To distinguish the liquid ordered and disordered regions at molecular scales, we calculate the degree of non-affineness of individual lipids in their neighbourhood and track their topological rearrangements. Disconnectivity graph analysis with respect to membrane organization shows that the DPPC/DOPC/Chol and PSM/DOPC/Chol systems exhibit funnel-like energy landscapes as opposed to a highly frustrated energy landscape for the more biomimetic PSM/POPC/Chol system. We use these measurements to develop a continuous lattice Hamiltonian and evolve that using Monte Carlo simulated annealing to explore the possibility of structural degeneracy in membrane organization. Our data show that model membranes with lipid constituents that are biomimetic (PSM/POPC/Chol) have the ability to access a large range of membrane sub-structure space (higher degeneracy) as compared to the other two systems, which form only one kind of substructure even with changing composition. Since the spatiotemporal organization in biological membranes dictates the "molecular encounters" and in turn larger scale biological processes such as molecular transport, trafficking and cellular signalling, we posit that this structural degeneracy could enable access to a larger repository to functionally important molecular organization in systems with physiologically relevant compositions.

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.

Defining, Calculating, and Converging Observables of a Kinetic Transition Network

Kinetic transition networks (KTNs) of local minima and transition states are able to capture the dynamics of numerous systems in chemistry, biology, and materials science. However, extracting observables is numerically challenging for large networks and generally will be sensitive to additional computational discovery. To have any measure of convergence for observables, these sensitivities must be regularly calculated. We present a matrix formulation of the discrete path sampling framework for KTNs, deriving expressions for branching probabilities, transition rates, and waiting times. Using the concept of the quasi-stationary distribution, a clear hierarchy of expressions for network observables is established, from exact results to steady-state approximations. We use these results in combination with the graph transformation method to derive the sensitivity, with respect to perturbations of the known KTN, giving explicit terms for the pairwise sensitivity and discussing the pathwise sensitivity. These results provide guidelines for converging the network, with respect to additional sampling, focusing on the estimates obtained for the overall rate coefficients between product and reactant states. We demonstrate this procedure for transitions in the double-funnel landscape of the 38-atom Lennard-Jones cluster.

Structural transitions in the RNA 7SK 5' hairpin and their effect on HEXIM binding

7SK RNA, as part of the 7SK ribonucleoprotein complex, is crucial to the regulation of transcription by RNA-polymerase II, via its interaction with the positive transcription elongation factor P-TEFb. The interaction is induced by binding of the protein HEXIM to the 5′ hairpin (HP1) of 7SK RNA. Four distinct structural models have been obtained experimentally for HP1. Here, we employ computational methods to investigate the relative stability of these structures, transitions between them, and the effects of mutations on the observed structural ensembles. We further analyse the results with respect to mutational binding assays, and hypothesize a mechanism for HEXIM binding. Our results indicate that the dominant structure in the wild type exhibits a triplet involving the unpaired nucleotide U40 and the base pair A43-U66 in the GAUC/GAUC repeat. This conformation leads to an open major groove with enough potential binding sites for peptide recognition. Sequence mutations of the RNA change the relative stability of the different structural ensembles. Binding affinity is consequently lost if these changes alter the dominant structure.

Energy Landscapes of a Pair of Adsorbed Peptides

The wide relevance of peptide adsorption in natural and synthetic contexts means it has attracted much attention. Molecular dynamics (MD) simulation has been widely used in these endeavors. Much of this has focused on single peptides due to the computational effort required to capture the rare events that characterize their adsorption. This focus is, however, of limited practical relevance as in reality, most systems of interest operate in the nondilute regime where peptides will interact with other adsorbed peptides. As an alternative to MD simulation, we have used energy landscape mapping (ELM) to investigate two met-enkephalin molecules adsorbed at a gas/graphite interface. Major conformations of the adsorbed peptides and the connecting transition states are elucidated along with the associated energy barriers and rates of exchange. The last of these makes clear that MD simulations are currently of limited use in probing the co-adsorption of two peptides, let alone more. The constant volume heat capacity as a function of temperature is also presented. Overall, this study represents a significant step toward characterizing peptide adsorption beyond the dilute limit.

Energy Landscapes and Hybridization Pathways for DNA Hexamer Duplexes

Strand hybridization is not only a fundamental molecular mechanism underlying the biological functions of nucleic acids but is also a key step in the design of efficient nanodevices. Despite recent efforts, the microscopic rules governing the hybridization mechanisms remain largely unknown. In this study, we exploit the energy landscape framework to assess how sequence-specificity modulates the hybridization mechanisms in DNA. We find that GG-tracts hybridize much more rapidly compared to GC-tracts, via either zippering or slithering pathways. For the hybridization of GG-tracts, both zippering and slithering mechanisms appear to be kinetically relevant. In contrast, for the GC-tracts, the zippering mechanism is dominant. Our work reveals that even for the relatively small systems considered, the energy landscapes feature multiple metastable states and kinetic traps, which is at odds with the conventional "all-or-nothing" model of DNA hybridization formulated on the basis of thermodynamic arguments alone. Interestingly, entropic effects are found to play an important role in determining the thermal stability of competing conformational ensembles and in determining the preferred hybridization pathways.

Large-Scale Conformational Changes and Protein Function: Breaking the in silico Barrier

Large-scale conformational changes are essential to link protein structures with their function at the cell and organism scale, but have been elusive both experimentally and computationally. Over the past few years developments in cryo-electron microscopy and crystallography techniques have started to reveal multiple snapshots of increasingly large and flexible systems, deemed impossible only short time ago. As structural information accumulates, theoretical methods become central to understand how different conformers interconvert to mediate biological function. Here we briefly survey current in silico methods to tackle large conformational changes, reviewing recent examples of cross-validation of experiments and computational predictions, which show how the integration of different scale simulations with biological information is already starting to break the barriers between the in silico, in vitro, and in vivo worlds, shedding new light onto complex biological problems inaccessible so far.