Distribution of Reciprocal of Interatomic Distances: A Fast Structural Metric

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

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

Early Aggregation Mechanism of SOD128-38 Based on Force Field Parameter of 5-Cyano-Tryptophan

The aggregation of superoxide dismutase 1 (SOD1) results in amyloid deposition and is involved in familial amyotrophic lateral sclerosis, a fatal motor neuron disease. There have been extensive studies of its aggregation mechanism. Noncanonical amino acid 5-cyano-tryptophan (5-CN-Trp), which has been incorporated into the amyloid segments of SOD1 as infrared probes to increase the structural sensitivity of IR spectroscopy, is found to accelerate the overall aggregation rate and potentially modulate the aggregation process. Despite these observations, the underlying mechanism remains elusive. Here, we optimized the force field parameters of 5-CN-Trp and then used molecular dynamics simulation along with the Markov state model on the SOD128-38 dimer to explore the kinetics of key intermediates in the presence and absence of 5-CN-Trp. Our findings indicate a significantly increased probability of protein aggregate formation in 5CN-Trp-modified ensembles compared to wildtype. Dimeric β-sheets of different natures were observed exclusively in the 5CN-Trp-modified peptides, contrasting with wildtype simulations. Free-energy calculations and detailed analyses of the dimer structure revealed augmented interstrand interactions attributed to 5-CN-Trp, which contributed more to peptide affinity than any other residues. These results explored the key events critical for the early nucleation of amyloid-prone proteins and also shed light on the practice of using noncanonical derivatives to study the aggregation mechanism.

Watching ion-driven kinetics of ribozyme folding and misfolding caused by energetic and topological frustration one molecule at a time

Folding of ribozymes into well-defined tertiary structures usually requires divalent cations. How Mg2+ ions direct the folding kinetics has been a long-standing unsolved problem because experiments cannot detect the positions and dynamics of ions. To address this problem, we used molecular simulations to dissect the folding kinetics of the Azoarcus ribozyme by monitoring the path each molecule takes to reach the folded state. We quantitatively establish that Mg2+ binding to specific sites, coupled with counter-ion release of monovalent cations, stimulate the formation of secondary and tertiary structures, leading to diverse pathways that include direct rapid folding and trapping in misfolded structures. In some molecules, key tertiary structural elements form when Mg2+ ions bind to specific RNA sites at the earliest stages of the folding, leading to specific collapse and rapid folding. In others, the formation of non-native base pairs, whose rearrangement is needed to reach the folded state, is the rate-limiting step. Escape from energetic traps, driven by thermal fluctuations, occurs readily. In contrast, the transition to the native state from long-lived topologically trapped native-like metastable states is extremely slow. Specific collapse and formation of energetically or topologically frustrated states occur early in the assembly process.

PROTHON: A Local Order Parameter-Based Method for Efficient Comparison of Protein Ensembles

The comparison of protein conformational ensembles is of central importance in structural biology. However, there are few computational methods for ensemble comparison, and those that are readily available, such as ENCORE, utilize methods that are sufficiently computationally expensive to be prohibitive for large ensembles. Here, a new method is presented for efficient representation and comparison of protein conformational ensembles. The method is based on the representation of a protein ensemble as a vector of probability distribution functions (pdfs), with each pdf representing the distribution of a local structural property such as the number of contacts between C_β atoms. Dissimilarity between two conformational ensembles is quantified by the Jensen-Shannon distance between the corresponding set of probability distribution functions. The method is validated for conformational ensembles generated by molecular dynamics simulations of ubiquitin, as well as experimentally derived conformational ensembles of a 130 amino acid truncated form of human tau protein. In the ubiquitin ensemble data set, the method was up to 88 times faster than the existing ENCORE software, while simultaneously utilizing 48 times fewer computing cores. We make the method available as a Python package, called PROTHON, and provide a GitHub page with the Python source code at https://github.com/PlotkinLab/Prothon.

Energy landscapes of Aβ monomers are sculpted in accordance with Ostwald's rule of stages

The transition from a disordered to an assembly-competent monomeric state (N*) in amyloidogenic sequences is a crucial event in the aggregation cascade. Using a well-calibrated model for intrinsically disordered proteins (IDPs), we show that the N* states, which bear considerable resemblance to the polymorphic fibril structures found in experiments, not only appear as excitations in the free energy landscapes of Aβ40 and Aβ42, but also initiate the aggregation cascade. For Aβ42, the transitions to the different N* states are in accord with Ostwald's rule of stages, with the least stable structures forming ahead of thermodynamically favored ones. The Aβ40 and Aβ42 monomer landscapes exhibit different extents of local frustration, which we show have profound implications in dictating subsequent self-assembly. Using kinetic transition networks, we illustrate that the most favored dimerization routes proceed via N* states. We argue that Ostwald's rule also holds for the aggregation of fused in sarcoma and polyglutamine proteins.

Cooperative multivalent receptor binding promotes exposure of the SARS-CoV-2 fusion machinery core

The molecular events that permit the spike glycoprotein of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) to bind and enter cells are important to understand for both fundamental and therapeutic reasons. Spike proteins consist of S1 and S2 domains, which recognize angiotensin-converting enzyme 2 (ACE2) receptors and contain the viral fusion machinery, respectively. Ostensibly, the binding of spike trimers to ACE2 receptors promotes dissociation of the S1 domains and exposure of the fusion machinery, although the molecular details of this process have yet to be observed. We report the development of bottom-up coarse-grained (CG) models consistent with cryo-electron tomography data, and the use of CG molecular dynamics simulations to investigate viral binding and S2 core exposure. We show that spike trimers cooperatively bind to multiple ACE2 dimers at virion-cell interfaces in a manner distinct from binding between soluble proteins, which processively induces S1 dissociation. We also simulate possible variant behavior using perturbed CG models, and find that ACE2-induced S1 dissociation is primarily sensitive to conformational state populations and the extent of S1/S2 cleavage, rather than ACE2 binding affinity. These simulations reveal an important concerted interaction between spike trimers and ACE2 dimers that primes the virus for membrane fusion and entry.

Early aggregation mechanism of Aβ16-22 revealed by Markov state models

Aβ_16-22 is believed to have critical role in early aggregation of full length amyloids that are associated with the Alzheimer's disease and can aggregate to form amyloid fibrils. However, the early aggregation mechanism is still unsolved. Here, multiple long-term molecular dynamics simulations combining with Markov state model were used to probe the early oligomerization mechanism of Aβ_16-22 peptides. The identified dimeric form adopted either globular random-coil or extended β-strand like conformations. The observed dimers of these variants shared many overall conformational characteristics but differed in several aspects at detailed level. In all cases, the most common type of secondary structure was intermolecular antiparallel β-sheets. The inter-state transitions were very frequent ranges from few to hundred nanoseconds. More strikingly, those states which contain fraction of β secondary structure and significant amount of extended coiled structures, therefore exposed to the solvent, were majorly participated in aggregation. The assembly of low-energy dimers, in which the peptides form antiparallel β sheets, occurred by multiple pathways with the formation of an obligatory intermediates. We proposed that these states might facilitate the Aβ_16-22 aggregation through a significant component of the conformational selection mechanism, because they might increase the aggregates population by promoting the inter-chain hydrophobic and the hydrogen bond contacts. The formation of early stage antiparallel β sheet structures is critical for oligomerization, and at the same time provided a flat geometry to seed the ordered β-strand packing of the fibrils. Our findings hint at reorganization of this part of the molecule as a potentially critical step in Aβ aggregation and will insight into early oligomerization for large β amyloids.

Cancer-Associated Mutations Perturb the Disordered Ensemble and Interactions of the Intrinsically Disordered p53 Transactivation Domain

Intrinsically disordered proteins (IDPs) are key components of regulatory networks that control crucial aspects of cell decision making. The intrinsically disordered transactivation domain (TAD) of tumor suppressor p53 mediates its interactions with multiple regulatory pathways to control the p53 homeostasis during the cellular response to genotoxic stress. Many cancer-associated mutations have been discovered in p53-TAD, but their structural and functional consequences are poorly understood. Here, by combining atomistic simulations, NMR spectroscopy, and binding assays, we demonstrate that cancer-associated mutations can significantly perturb the balance of p53 interactions with key activation and degradation regulators. Importantly, the four mutations studied in this work do not all directly disrupt the known interaction interfaces. Instead, at least three of these mutations likely modulate the disordered state of p53-TAD to perturb its interactions with regulators. Specifically, NMR and simulation analysis together suggest that these mutations can modulate the level of conformational expansion as well as rigidity of the disordered state. Our work suggests that the disordered conformational ensemble of p53-TAD can serve as a central conduit in regulating the response to various cellular stimuli at the protein-protein interaction level. Understanding how the disordered state of IDPs may be modulated by regulatory signals and/or disease associated perturbations will be essential in the studies on the role of IDPs in biology and diseases.

Cooperative multivalent receptor binding promotes exposure of the SARS-CoV-2 fusion machinery core

The molecular events that permit the spike glycoprotein of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) to bind, fuse, and enter cells are important to understand for both fundamental and therapeutic reasons. Spike proteins consist of S1 and S2 domains, which recognize angiotensin-converting enzyme 2 (ACE2) receptors and contain the viral fusion machinery, respectively. Ostensibly, the binding of spike trimers to ACE2 receptors promotes the preparation of the fusion machinery by dissociation of the S1 domains. We report the development of bottom-up coarse-grained (CG) models validated with cryo-electron tomography (cryo-ET) data, and the use of CG molecular dynamics simulations to investigate the dynamical mechanisms involved in viral binding and exposure of the S2 trimeric core. We show that spike trimers cooperatively bind to multiple ACE2 dimers at virion-cell interfaces. The multivalent interaction cyclically and processively induces S1 dissociation, thereby exposing the S2 core containing the fusion machinery. Our simulations thus reveal an important concerted interaction between spike trimers and ACE2 dimers that primes the virus for membrane fusion and entry.

EGCG binds intrinsically disordered N-terminal domain of p53 and disrupts p53-MDM2 interaction

Epigallocatechin gallate (EGCG) from green tea can induce apoptosis in cancerous cells, but the underlying molecular mechanisms remain poorly understood. Using SPR and NMR, here we report a direct, μM interaction between EGCG and the tumor suppressor p53 (KD = 1.6 ± 1.4 μM), with the disordered N-terminal domain (NTD) identified as the major binding site (KD = 4 ± 2 μM). Large scale atomistic simulations (>100 μs), SAXS and AUC demonstrate that EGCG-NTD interaction is dynamic and EGCG causes the emergence of a subpopulation of compact bound conformations. The EGCG-p53 interaction disrupts p53 interaction with its regulatory E3 ligase MDM2 and inhibits ubiquitination of p53 by MDM2 in an in vitro ubiquitination assay, likely stabilizing p53 for anti-tumor activity. Our work provides insights into the mechanisms for EGCG's anticancer activity and identifies p53 NTD as a target for cancer drug discovery through dynamic interactions with small molecules.

InfleCS: Clustering Free Energy Landscapes with Gaussian Mixtures

Free energy landscapes provide insights into conformational ensembles of biomolecules. In order to analyze these landscapes and elucidate mechanisms underlying conformational changes, there is a need to extract metastable states with limited noise. This has remained a formidable task, despite a plethora of existing clustering methods. We present InfleCS, a novel method for extracting well-defined core states from free energy landscapes. The method is based on a Gaussian mixture free energy estimator and exploits the shape of the estimated density landscape. The core states that naturally arise from the clustering allow for detailed characterization of the conformational ensemble. The clustering quality is evaluated on three toy models with different properties, where the method is shown to consistently outperform other conventional and state-of-the-art clustering methods. Finally, the method is applied to a temperature enhanced molecular dynamics simulation of Ca²⁺-bound Calmodulin. Through the free energy landscape, we discover a pathway between a canonical and a compact state, revealing conformational changes driven by electrostatic interactions.

Accommodation of a dimer in an Ar-like lattice: exploring the generic structural motifs

A global optimization strategy is applied to Lennard-Jones models describing the stable trapping sites of a dimer in the face-centered cubic Ar-like lattice. Effective volumes of the trapping sites, quantified as the number of host atoms dislodged from the lattice, are mapped onto the parameter space defined by the strength and range of the dimer interaction potentials. The two models considered differ in the host-guest interaction and give very different maps that reflect the effect of local lattice relaxation. A hierarchical complete-linkage clustering technique is applied to identify generic structural types of the dimer accommodations. The dominant types found and enlisted maintain the symmetry of the isolated dimer and possess high tetrahedral and octahedral symmetry of the host environment with respect to the dimer atoms or center and can be roughly classified as the "whole" or "per atom" dimer accommodations. The results are compared to the analysis of the analogous model for trapped atoms and realistic model for trapped alkaline-earth metal dimers. Implications for matrix isolation spectroscopy are discussed.

Residual Structures and Transient Long-Range Interactions of p53 Transactivation Domain: Assessment of Explicit Solvent Protein Force Fields

Molecular dynamics simulations using physics-based atomistic force fields have been increasingly used to characterize the heterogeneous structural ensembles of intrinsically disordered proteins (IDPs). To evaluate the accuracy of the latest atomistic explicit-solvent force fields in modeling larger IDPs with nontrivial structural features, we focus on the 61-residue N-terminal transactivation domain (TAD) of tumor suppressor p53, an important protein in cancer biology that has been extensively studied, and abundant experimental data is available for evaluation of simulated ensembles. We performed extensive replica exchange with solute tempering simulations, in excess of 1.0 μs/replica, to generate disordered structural ensembles of p53-TAD using six latest explicit solvent protein force fields. Multiple local and long-range structural properties, including chain dimension, residual secondary structures, and transient long-range contacts, were analyzed and compared against available experimental data. The results show that IDPs such as p53-TAD remain highly challenging for atomistic simulations due to conformational complexity and difficulty in achieving adequate convergence. Structural ensembles of p53-TAD generated using various force fields differ significantly from each other. The a99SB-disp force field demonstrates the best agreement with experimental data at all levels and proves to be suitable for simulating unbound p53-TAD and how its conformational properties may be modulated by phosphorylation and other cellular signals or cancer-associated mutations. Feasibility of such detailed structural characterization is a key step toward establishing the sequence-disordered ensemble-function-disease relationship of p53 and other biologically important IDPs.

Molecular basis for the increased affinity of an RNA recognition motif with re-engineered specificity: A molecular dynamics and enhanced sampling simulations study

The RNA recognition motif (RRM) is the most common RNA binding domain across eukaryotic proteins. It is therefore of great value to engineer its specificity to target RNAs of arbitrary sequence. This was recently achieved for the RRM in Rbfox protein, where four mutations R118D, E147R, N151S, and E152T were designed to target the precursor to the oncogenic miRNA 21. Here, we used a variety of molecular dynamics-based approaches to predict specific interactions at the binding interface. Overall, we have run approximately 50 microseconds of enhanced sampling and plain molecular dynamics simulations on the engineered complex as well as on the wild-type Rbfox·pre-miRNA 20b from which the mutated systems were designed. Comparison with the available NMR data on the wild type molecules (protein, RNA, and their complex) served to establish the accuracy of the calculations.

Free energy calculations suggest that further improvements in affinity and selectivity are achieved by the S151T replacement.

RNA is an outstanding target for oncological intervention. Engineering the most common RNA binding motif in human proteins (called RRM) so as to bind to a specific RNA has an enormous pharmacological potential. Yet, it is highly non trivial to design RRM-bearing protein variants with RNA selectivity and affinity sufficiently high for clinical applications. Here we present an extensive molecular simulation study which shed light on the exquisite molecular recognition of the empirically-engineered complex between the RRM-bearing protein Rbfox and its RNA target pre-miR21. The simulations allow predicting a variant, the S151T, which may lead to further enhancement of selectivity and affinity for pre-miR21.

Conformational Ensembles Exhibit Extensive Molecular Recognition Features

Intrinsically disordered proteins (IDPs) are important for signaling and regulatory pathways. In contrast to folded proteins, they sample a diverse conformational space. IDPs have residue ranges within a sequence that have been referred to as molecular recognition features (MoRFs). A MoRF can be viewed as contiguous residues exhibiting a conformational disorder that become ordered upon binding to another protein or ligand. In this work, we introduce a structural characterization of MoRFs based on entropy and mutual information (MI). In this view, a MoRF is a set of contiguous residues that exhibit a large entropy (from rotameric residue sampling) and large MI, the latter indicating a dependence among the residues' rotameric sampling comprising the MoRF. The methodology is first applied to a number of ubiquitin ensembles that were obtained based on nuclear magnetic resonance experiments. One is a denatured Ub ensemble that has a large entropy for various unitSizes (number of contiguous residues) but essentially zero MI, indicting no dependence among the residue rotamer sampling. Another ensemble does exhibit extensive regions along the sequence where there are MoRFs centered on nonsecondary structure regions. The MoRFs are present for unitSizes 2-10. That a substantial number of MoRFs are present in Ub strongly suggests a conformational selection mechanism for this protein. Two additional ensembles for the cyclin-dependent kinase inhibitor Sic1 and for the amyloid protein α-synuclein, which have been shown to be IDPs, are also analyzed. Both exhibit MoRF-like character.

A new class of enhanced kinetic sampling methods for building Markov state models

Markov state models (MSMs) and other related kinetic network models are frequently used to study the long-timescale dynamical behavior of biomolecular and materials systems. MSMs are often constructed bottom-up using brute-force molecular dynamics (MD) simulations when the model contains a large number of states and kinetic pathways that are not known a priori. However, the resulting network generally encompasses only parts of the configurational space, and regardless of any additional MD performed, several states and pathways will still remain missing. This implies that the duration for which the MSM can faithfully capture the true dynamics, which we term as the validity time for the MSM, is always finite and unfortunately much shorter than the MD time invested to construct the model. A general framework that relates the kinetic uncertainty in the model to the validity time, missing states and pathways, network topology, and statistical sampling is presented. Performing additional calculations for frequently-sampled states/pathways may not alter the MSM validity time. A new class of enhanced kinetic sampling techniques is introduced that aims at targeting rare states/pathways that contribute most to the uncertainty so that the validity time is boosted in an effective manner. Examples including straightforward 1D energy landscapes, lattice models, and biomolecular systems are provided to illustrate the application of the method. Developments presented here will be of interest to the kinetic Monte Carlo community as well.

Inference of Calmodulin's Ca2+-Dependent Free Energy Landscapes via Gaussian Mixture Model Validation

A free energy landscape estimation method based on the well-known Gaussian mixture model (GMM) is used to compare the efficiencies of thermally enhanced sampling methods with respect to regular molecular dynamics. The simulations are carried out on two binding states of calmodulin, and the free energy estimation method is compared with other estimators using a toy model. We show that GMM with cross-validation provides a robust estimate that is not subject to overfitting. The continuous nature of Gaussians provides better estimates on sparse data than canonical histogramming. We find that diffusion properties determine the sampling method effectiveness, such that diffusion-dominated apo calmodulin is most efficiently sampled by regular molecular dynamics, while holo calmodulin, with its rugged free energy landscape, is better sampled by enhanced sampling methods.

Enhanced unbiased sampling of protein dynamics using evolutionary coupling information

One of the major challenges in atomistic simulations of proteins is efficient sampling of pathways associated with rare conformational transitions. Recent developments in statistical methods for computation of direct evolutionary couplings between amino acids within and across polypeptide chains have allowed for inference of native residue contacts, informing accurate prediction of protein folds and multimeric structures. In this study, we assess the use of distances between evolutionarily coupled residues as natural choices for reaction coordinates which can be incorporated into Markov state model-based adaptive sampling schemes and potentially used to predict not only functional conformations but also pathways of conformational change, protein folding, and protein-protein association. We demonstrate the utility of evolutionary couplings in sampling and predicting activation pathways of the β 2-adrenergic receptor (β 2-AR), folding of the FiP35 WW domain, and dimerization of the E. coli molybdopterin synthase subunits. We find that the time required for β 2-AR activation and folding of the WW domain are greatly diminished using evolutionary couplings-guided adaptive sampling. Additionally, we were able to identify putative molybdopterin synthase association pathways and near-crystal structure complexes from protein-protein association simulations.

MSMBuilder: Statistical Models for Biomolecular Dynamics

MSMBuilder is a software package for building statistical models of high-dimensional time-series data. It is designed with a particular focus on the analysis of atomistic simulations of biomolecular dynamics such as protein folding and conformational change. MSMBuilder is named for its ability to construct Markov state models (MSMs), a class of models that has gained favor among computational biophysicists. In addition to both well-established and newer MSM methods, the package includes complementary algorithms for understanding time-series data such as hidden Markov models and time-structure based independent component analysis. MSMBuilder boasts an easy to use command-line interface, as well as clear and consistent abstractions through its Python application programming interface. MSMBuilder was developed with careful consideration for compatibility with the broader machine learning community by following the design of scikit-learn. The package is used primarily by practitioners of molecular dynamics, but is just as applicable to other computational or experimental time-series measurements.

Optimized parameter selection reveals trends in Markov state models for protein folding

As molecular dynamics simulations access increasingly longer time scales, complementary advances in the analysis of biomolecular time-series data are necessary. Markov state models offer a powerful framework for this analysis by describing a system's states and the transitions between them. A recently established variational theorem for Markov state models now enables modelers to systematically determine the best way to describe a system's dynamics. In the context of the variational theorem, we analyze ultra-long folding simulations for a canonical set of twelve proteins [K. Lindorff-Larsen et al., Science 334, 517 (2011)] by creating and evaluating many types of Markov state models. We present a set of guidelines for constructing Markov state models of protein folding; namely, we recommend the use of cross-validation and a kinetically motivated dimensionality reduction step for improved descriptions of folding dynamics. We also warn that precise kinetics predictions rely on the features chosen to describe the system and pose the description of kinetic uncertainty across ensembles of models as an open issue.

Conserve Water: A Method for the Analysis of Solvent in Molecular Dynamics

Molecular dynamics with explicit solvent is favored for its ability to more correctly simulate aqueous biological processes and has become routine thanks to increasingly powerful computational resources. However, analysis techniques including Markov state models (MSMs) ignore solvent atoms and focus solely on solute coordinates despite solvent being implicated in myriad biological phenomena. We present a unified framework called "solvent-shells featurization" for including solvent degrees of freedom in analysis and show that this method produces better models. We apply this method to simulations of dewetting in the two-domain protein BphC to generate a predictive MSM and identify functional water molecules. Furthermore, the proposed methodology could be easily extended for building MSMs of any systems with indistinguishable components.

Kinetic distance and kinetic maps from molecular dynamics simulation

Characterizing macromolecular kinetics from molecular dynamics (MD) simulations requires a distance metric that can distinguish slowly interconverting states. Here, we build upon diffusion map theory and define a kinetic distance metric for irreducible Markov processes that quantifies how slowly molecular conformations interconvert. The kinetic distance can be computed given a model that approximates the eigenvalues and eigenvectors (reaction coordinates) of the MD Markov operator. Here, we employ the time-lagged independent component analysis (TICA). The TICA components can be scaled to provide a kinetic map in which the Euclidean distance corresponds to the kinetic distance. As a result, the question of how many TICA dimensions should be kept in a dimensionality reduction approach becomes obsolete, and one parameter less needs to be specified in the kinetic model construction. We demonstrate the approach using TICA and Markov state model (MSM) analyses for illustrative models, protein conformation dynamics in bovine pancreatic trypsin inhibitor and protein-inhibitor association in trypsin and benzamidine. We find that the total kinetic variance (TKV) is an excellent indicator of model quality and can be used to rank different input feature sets.

Equilibration of Protein States: A Time Dependent Free-Energy Disconnectivity Graph

The process of equilibration of protein states in a three-stranded antiparallel β-sheet miniprotein is studied using a time-dependent free energy disconnectivity graph. To determine the rates of transitions, the molecular dynamics simulation results of a recent work (Kalgin, I. V.; J. Phys. Chem. B 2013, 117, 6092) are employed. The vertices of the graph are the free energies of characteristic states of the protein, and the edges are the transition state free energies. To determine the latter, the "complete" partition function (Eyring, 1935) is used, which includes the translational partition function corresponding to the ballistic motion of the system along the reaction coordinate. The distance along the reaction coordinate that enters the translational partition function is taken to be proportional to the observation time and thus measures the number of representative points that cross the transition state surface during given time. As the time increases, the free energy barriers between the clusters of characteristic conformations (native-like, helical, and β-sheet conformations of different degree of organization) decrease and (local) equilibrium between the clusters is established. With time, these clusters are grouped into larger clusters, extending the equilibrium to a larger portion of protein states.

Free energy surface of the Michaelis complex of lactate dehydrogenase: a network analysis of microsecond simulations

It has long been recognized that the structure of a protein creates a hierarchy of conformations interconverting on multiple time scales. The conformational heterogeneity of the Michaelis complex is of particular interest in the context of enzymatic catalysis in which the reactant is usually represented by a single conformation of the enzyme/substrate complex. Lactate dehydrogenase (LDH) catalyzes the interconversion of pyruvate and lactate with concomitant interconversion of two forms of the cofactor nicotinamide adenine dinucleotide (NADH and NAD(+)). Recent experimental results suggest that multiple substates exist within the Michaelis complex of LDH, and they show a strong variance in their propensity toward the on-enzyme chemical step. In this study, microsecond-scale all-atom molecular dynamics simulations were performed on LDH to explore the free energy landscape of the Michaelis complex, and network analysis was used to characterize the distribution of the conformations. Our results provide a detailed view of the kinetic network of the Michaelis complex and the structures of the substates at atomistic scales. They also shed light on the complete picture of the catalytic mechanism of LDH.

Variational cross-validation of slow dynamical modes in molecular kinetics

Markov state models are a widely used method for approximating the eigenspectrum of the molecular dynamics propagator, yielding insight into the long-timescale statistical kinetics and slow dynamical modes of biomolecular systems. However, the lack of a unified theoretical framework for choosing between alternative models has hampered progress, especially for non-experts applying these methods to novel biological systems. Here, we consider cross-validation with a new objective function for estimators of these slow dynamical modes, a generalized matrix Rayleigh quotient (GMRQ), which measures the ability of a rank-m projection operator to capture the slow subspace of the system. It is shown that a variational theorem bounds the GMRQ from above by the sum of the first m eigenvalues of the system's propagator, but that this bound can be violated when the requisite matrix elements are estimated subject to statistical uncertainty. This overfitting can be detected and avoided through cross-validation. These result make it possible to construct Markov state models for protein dynamics in a way that appropriately captures the tradeoff between systematic and statistical errors.

Dihedral angle entropy measures for intrinsically disordered proteins

Protein stability is based on a delicate balance between energetic and entropic factors. Intrinsically disordered proteins (IDPs) interacting with a folded partner protein in the act of binding can order the IDP to form the correct functional interface by decrease in the overall free energy. In this work, we evaluate the part of the entropic cost of ordering an IDP arising from their dihedral states. The IDP studied is a leucine zipper dimer that we simulate with molecular dynamics and find that it does show disorder in six phi and psi dihedral angles of the N terminal sequence of one monomer. Essential to ascertain is the degree of disorder in the IDP, and we do so by considering the entire, discretized probability distribution function of N dihedrals with M conformers per dihedral. A compositional clustering method is introduced, whereby the NS = N(M) states are formed from the Cartesian product of each dihedral's conformational space. Clustering is carried out with a version of a k-means algorithm that accounts for the circular nature of dihedral angles. For the 12 dihedrals each found to have three conformers, among the resulting 531441 states, their populations show that the first 100 (500) most populated states account for ∼65% (∼90%) of the entire population, indicating that there are strong dependencies among the dihedrals' conformations. These state populations are used to evaluate a Kullback-Leibler divergence entropy measure and obtain the dihedral configurational entropy S. At 300 K, TS ∼ 3 kcal/mol, showing that IDP entropy, while roughly half that would be expected from independently distributed dihedrals, can be a decisive contributor to the free energy of this IDP binding and ordering.

Modeling molecular kinetics with tICA and the kernel trick

The allure of a molecular dynamics simulation is that, given a sufficiently accurate force field, it can provide an atomic-level view of many interesting phenomena in biology. However, the result of a simulation is a large, high-dimensional time series that is difficult to interpret. Recent work has introduced the time-structure based Independent Components Analysis (tICA) method for analyzing MD, which attempts to find the slowest decorrelating linear functions of the molecular coordinates. This method has been used in conjunction with Markov State Models (MSMs) to provide estimates of the characteristic eigenprocesses contained in a simulation (e.g., protein folding, ligand binding). Here, we extend the tICA method using the kernel trick to arrive at nonlinear solutions. This is a substantial improvement as it allows for kernel-tICA (ktICA) to provide estimates of the characteristic eigenprocesses directly without building an MSM.

The role of nucleobase interactions in RNA structure and dynamics

The intricate network of interactions observed in RNA three-dimensional structures is often described in terms of a multitude of geometrical properties, including helical parameters, base pairing/stacking, hydrogen bonding and backbone conformation. We show that a simple molecular representation consisting in one oriented bead per nucleotide can account for the fundamental structural properties of RNA. In this framework, canonical Watson-Crick, non-Watson-Crick base-pairing and base-stacking interactions can be unambiguously identified within a well-defined interaction shell. We validate this representation by performing two independent, complementary tests. First, we use it to construct a sequence-independent, knowledge-based scoring function for RNA structural prediction, which compares favorably to fully atomistic, state-of-the-art techniques. Second, we define a metric to measure deviation between RNA structures that directly reports on the differences in the base–base interaction network. The effectiveness of this metric is tested with respect to the ability to discriminate between structurally and kinetically distant RNA conformations, performing better compared to standard techniques. Taken together, our results suggest that this minimalist, nucleobase-centric representation captures the main interactions that are relevant for describing RNA structure and dynamics.

Perspective: Markov models for long-timescale biomolecular dynamics

Molecular dynamics simulations have the potential to provide atomic-level detail and insight to important questions in chemical physics that cannot be observed in typical experiments. However, simply generating a long trajectory is insufficient, as researchers must be able to transform the data in a simulation trajectory into specific scientific insights. Although this analysis step has often been taken for granted, it deserves further attention as large-scale simulations become increasingly routine. In this perspective, we discuss the application of Markov models to the analysis of large-scale biomolecular simulations. We draw attention to recent improvements in the construction of these models as well as several important open issues. In addition, we highlight recent theoretical advances that pave the way for a new generation of models of molecular kinetics.

C(α) torsion angles as a flexible criterion to extract secrets from a molecular dynamics simulation

Given the increasing complexity of simulated molecular systems, and the fact that simulation times have now reached milliseconds to seconds, immense amounts of data (in the gigabyte to terabyte range) are produced in current molecular dynamics simulations. Manual analysis of these data is a very time-consuming task, and important events that lead from one intermediate structure to another can become occluded in the noise resulting from random thermal fluctuations. To overcome these problems and facilitate a semi-automated data analysis, we introduce in this work a measure based on C(α) torsion angles: torsion angles formed by four consecutive C(α) atoms. This measure describes changes in the backbones of large systems on a residual length scale (i.e., a small number of residues at a time). Cluster analysis of individual C(α) torsion angles and its fuzzification led to continuous time patches representing (meta)stable conformations and to the identification of events acting as transitions between these conformations. The importance of a change in torsion angle to structural integrity is assessed by comparing this change to the average fluctuations in the same torsion angle over the complete simulation. Using this novel measure in combination with other measures such as the root mean square deviation (RMSD) and time series of distance measures, we performed an in-depth analysis of a simulation of the open form of DNA polymerase I. The times at which major conformational changes occur and the most important parts of the molecule and their interrelations were pinpointed in this analysis. The simultaneous determination of the time points and localizations of major events is a significant advantage of the new bottom-up approach presented here, as compared to many other (top-down) approaches in which only the similarity of the complete structure is analyzed.

Learning Kinetic Distance Metrics for Markov State Models of Protein Conformational Dynamics

Statistical modeling of long timescale dynamics with Markov state models (MSMs) has been shown to be an effective strategy for building quantitative and qualitative insight into protein folding processes. Existing methodologies, however, rely on geometric clustering using distance metrics such as root mean square deviation (RMSD), assuming that geometric similarity provides an adequate basis for the kinetic partitioning of phase space. Here, inspired by advances in the machine learning community, we introduce a new approach for learning a distance metric explicitly constructed to model kinetic similarity. This approach enables the construction of models, especially in the regime of high anisotropy in the diffusion constant, with fewer states than was previously possible. Application of this technique to the analysis of two ultralong molecular dynamics simulations of the FiP35 WW domain identifies discrete near-native relaxation dynamics in the millisecond regime that were not resolved in previous analyses.

New insights into the folding of a β-sheet miniprotein in a reduced space of collective hydrogen bond variables: application to a hydrodynamic analysis of the folding flow

A new analysis of the 20 μs equilibrium folding/unfolding molecular dynamics simulations of the three-stranded antiparallel β-sheet miniprotein (beta3s) in implicit solvent is presented. The conformation space is reduced in dimensionality by introduction of linear combinations of hydrogen bond distances as the collective variables making use of a specially adapted principal component analysis (PCA); i.e., to make structured conformations more pronounced, only the formed bonds are included in determining the principal components. It is shown that a three-dimensional (3D) subspace gives a meaningful representation of the folding behavior. The first component, to which eight native hydrogen bonds make the major contribution (four in each beta hairpin), is found to play the role of the reaction coordinate for the overall folding process, while the second and third components distinguish the structured conformations. The representative points of the trajectory in the 3D space are grouped into conformational clusters that correspond to locally stable conformations of beta3s identified in earlier work. A simplified kinetic network based on the three components is constructed, and it is complemented by a hydrodynamic analysis. The latter, making use of "passive tracers" in 3D space, indicates that the folding flow is much more complex than suggested by the kinetic network. A 2D representation of streamlines shows there are vortices which correspond to repeated local rearrangement, not only around minima of the free energy surface but also in flat regions between minima. The vortices revealed by the hydrodynamic analysis are apparently not evident in folding pathways generated by transition-path sampling. Making use of the fact that the values of the collective hydrogen bond variables are linearly related to the Cartesian coordinate space, the RMSD between clusters is determined. Interestingly, the transition rates show an approximate exponential correlation with distance in the hydrogen bond subspace. Comparison with the many published studies shows good agreement with the present analysis for the parts that can be compared, supporting the robust character of our understanding of this "hydrogen atom" of protein folding.

Improvements in Markov State Model Construction Reveal Many Non-Native Interactions in the Folding of NTL9

Markov State Models (MSMs) provide an automated framework to investigate the dynamical properties of high-dimensional molecular simulations. These models can provide a human-comprehensible picture of the underlying process, and have been successfully used to study protein folding, protein aggregation, protein ligand binding, and other biophysical systems. The MSM requires the construction of a discrete state-space such that two points are in the same state if they can interconvert rapidly. In the following, we suggest an improved method, which utilizes second order Independent Components Analysis (also known as time-structure based Independent Components Analysis, or tICA), to construct the state-space. We apply this method to simulations of NTL9 (provided by Lindorff-Larsen et al. Science2011), and show that the MSM is an improvement over previously built models using conventional distance metrics. Additionally, the resulting model provides insight into the role of non-native contacts by revealing many slow timescales associated with compact, non-native states.