Probabilistic Determination of Native State Ensembles of Proteins

How to learn from inconsistencies: Integrating molecular simulations with experimental data

Molecular simulations and biophysical experiments can be used to provide independent and complementary insights into the molecular origin of biological processes. A particularly useful strategy is to use molecular simulations as a modeling tool to interpret experimental measurements, and to use experimental data to refine our biophysical models. Thus, explicit integration and synergy between molecular simulations and experiments is fundamental for furthering our understanding of biological processes. This is especially true in the case where discrepancies between measured and simulated observables emerge. In this chapter, we provide an overview of some of the core ideas behind methods that were developed to improve the consistency between experimental information and numerical predictions. We distinguish between situations where experiments are used to refine our understanding and models of specific systems, and situations where experiments are used more generally to refine transferable models. We discuss different philosophies and attempt to unify them in a single framework. Until now, such integration between experiments and simulations have mostly been applied to equilibrium data, and we discuss more recent developments aimed to analyze time-dependent or time-resolved data.

Bayesian-Maximum-Entropy Reweighting of IDP Ensembles Based on NMR Chemical Shifts

Bayesian and Maximum Entropy approaches allow for a statistically sound and systematic fitting of experimental and computational data. Unfortunately, assessing the relative confidence in these two types of data remains difficult as several steps add unknown error. Here we propose the use of a validation-set method to determine the balance, and thus the amount of fitting. We apply the method to synthetic NMR chemical shift data of an intrinsically disordered protein. We show that the method gives consistent results even when other methods to assess the amount of fitting cannot be applied. Finally, we also describe how the errors in the chemical shift predictor can lead to an incorrect fitting and how using secondary chemical shifts could alleviate this problem.

Efficient Ensemble Refinement by Reweighting

Ensemble refinement produces structural ensembles of flexible and dynamic biomolecules by integrating experimental data and molecular simulations. Here we present two efficient numerical methods to solve the computationally challenging maximum-entropy problem arising from a Bayesian formulation of ensemble refinement. Recasting the resulting constrained weight optimization problem into an unconstrained form enables the use of gradient-based algorithms. In two complementary formulations that differ in their dimensionality, we optimize either the log-weights directly or the generalized forces appearing in the explicit analytical form of the solution. We first demonstrate the robustness, accuracy, and efficiency of the two methods using synthetic data. We then use NMR J-couplings to reweight an all-atom molecular dynamics simulation ensemble of the disordered peptide Ala-5 simulated with the AMBER99SB*-ildn-q force field. After reweighting, we find a consistent increase in the population of the polyproline-II conformations and a decrease of α-helical-like conformations. Ensemble refinement makes it possible to infer detailed structural models for biomolecules exhibiting significant dynamics, such as intrinsically disordered proteins, by combining input from experiment and simulation in a balanced manner.

Precision DEER Distances from Spin-Label Ensemble Refinement

Double electron-electron resonance (DEER) experiments probe nanometer-scale distances in spin-labeled proteins and nucleic acids. Rotamer libraries of the covalently attached spin-labels help reduce position uncertainties. Here we show that rotamer reweighting is essential for precision distance measurements, making it possible to resolve Ångstrom-scale domain motions. We analyze extensive DEER measurements on the three N-terminal polypeptide transport-associated (POTRA) domains of the outer membrane protein Omp85. Using the "Bayesian inference of ensembles" maximum-entropy method, we extract rotamer weights from the DEER measurements. Small weight changes suffice to eliminate otherwise significant discrepancies between experiments and model and unmask 1-3 Å domain motions relative to the crystal structure. Rotamer-weight refinement is a simple yet powerful tool for precision distance measurements that should be broadly applicable to label-based measurements including DEER, paramagnetic relaxation enhancement, and fluorescence resonance energy transfer (FRET).

Distance-independent Cross-correlated Relaxation and Isotropic Chemical Shift Modulation in Protein Dynamics Studies

Cross-correlated relaxation (CCR) in multiple-quantum coherences differs from other relaxation phenomena in its theoretical ability to be mediated across an infinite distance. The two interfering relaxation mechanisms may be dipolar interactions, chemical shift anisotropies, chemical shift modulations or quadrupolar interactions. These properties make multiple-quantum CCR an attractive probe for structure and dynamics of biomacromolecules not accessible from other measurements. Here, we review the use of multiple-quantum CCR measurements in dynamics studies of proteins. We compile a list of all experiments proposed for CCR rate measurements, provide an overview of the theory with a focus on protein dynamics, and present applications to various protein systems.

Molecular dynamics ensemble refinement of the heterogeneous native state of NCBD using chemical shifts and NOEs

Many proteins display complex dynamical properties that are often intimately linked to their biological functions. As the native state of a protein is best described as an ensemble of conformations, it is important to be able to generate models of native state ensembles with high accuracy. Due to limitations in sampling efficiency and force field accuracy it is, however, challenging to obtain accurate ensembles of protein conformations by the use of molecular simulations alone. Here we show that dynamic ensemble refinement, which combines an accurate atomistic force field with commonly available nuclear magnetic resonance (NMR) chemical shifts and NOEs, can provide a detailed and accurate description of the conformational ensemble of the native state of a highly dynamic protein. As both NOEs and chemical shifts are averaged on timescales up to milliseconds, the resulting ensembles reflect the structural heterogeneity that goes beyond that probed, e.g., by NMR relaxation order parameters. We selected the small protein domain NCBD as object of our study since this protein, which has been characterized experimentally in substantial detail, displays a rich and complex dynamical behaviour. In particular, the protein has been described as having a molten-globule like structure, but with a relatively rigid core. Our approach allowed us to describe the conformational dynamics of NCBD in solution, and to probe the structural heterogeneity resulting from both short- and long-timescale dynamics by the calculation of order parameters on different time scales. These results illustrate the usefulness of our approach since they show that NCBD is rather rigid on the nanosecond timescale, but interconverts within a broader ensemble on longer timescales, thus enabling the derivation of a coherent set of conclusions from various NMR experiments on this protein, which could otherwise appear in contradiction with each other.

Linking time-series of single-molecule experiments with molecular dynamics simulations by machine learning

Single-molecule experiments and molecular dynamics (MD) simulations are indispensable tools for investigating protein conformational dynamics. The former provide time-series data, such as donor-acceptor distances, whereas the latter give atomistic information, although this information is often biased by model parameters. Here, we devise a machine-learning method to combine the complementary information from the two approaches and construct a consistent model of conformational dynamics. It is applied to the folding dynamics of the formin-binding protein WW domain. MD simulations over 400 μs led to an initial Markov state model (MSM), which was then "refined" using single-molecule Förster resonance energy transfer (FRET) data through hidden Markov modeling. The refined or data-assimilated MSM reproduces the FRET data and features hairpin one in the transition-state ensemble, consistent with mutation experiments. The folding pathway in the data-assimilated MSM suggests interplay between hydrophobic contacts and turn formation. Our method provides a general framework for investigating conformational transitions in other proteins.

Utilizing dipole-dipole cross-correlated relaxation for the measurement of angles between pairs of opposing CαHα-CαHα bonds in anti-parallel β-sheets

Dipole-dipole cross-correlated relaxation (CCR) between two spin pairs is rich with macromolecular structural and dynamic information on inter-nuclear bond vectors. Measurement of short range dipolar CCR rates has been demonstrated for a variety of inter-nuclear vector spin pairs in proteins and nucleic acids, where the multiple quantum coherence necessary for observing the CCR rate is created by through-bond scalar coupling. In principle, CCR rates can be measured for any pair of inter-nuclear vectors where coherence can be generated between one spin of each spin pair, regardless of both the distance between the two spin pairs and the distance of the two spins forming the multiple quantum coherence. In practice, however, long range CCR (lrCCR) rates are challenging to measure due to difficulties in linking spatially distant spin pairs. By utilizing through-space relaxation allowed coherence transfer (RACT), we have developed a new method for the measurement of lrCCR rates involving CαHα bonds on opposing anti-parallel β-strands. The resulting lrCCR rates are straightforward to interpret since only the angle between the two vectors modulates the strength of the interference effect. We applied our lrCCR measurement to the third immunoglobulin-binding domain of the streptococcal protein G (GB3) and utilize published NMR ensembles and static NMR/X-ray structures to highlight the relationship between the lrCCR rates and the CαHα-CαHα inter-bond angle and bond mobility. Furthermore, we employ the lrCCR rates to guide the selection of sub-ensembles from the published NMR ensembles for enhancing the structural and dynamic interpretation of the data. We foresee this methodology for measuring lrCCR rates as improving the generation of structural ensembles by providing highly accurate details concerning the orientation of CαHα bonds on opposing anti-parallel β-strands.

Detection of Correlated Protein Backbone and Side-Chain Angle Fluctuations

NMR methods for the characterization of local protein motions have attained a high level of sophistication. Measurement of the synchronization between those motions, however, poses a serious challenge. Such correlated motions are one of the underlying mechanisms for the propagation of local changes to remote sites and as such for information transfer. Here, we demonstrate the experimental detection of the synchronization of motion over an intermediate range. To that purpose, we designed pulse sequences for the measurement of cross-correlated relaxation between the backbone H^N -N and side-chain H^β -C^β dipoles in Ile, Thr, and Val in the protein GB3. These bonds are related through two and three intervening dihedral angles. We show that the correlated motions inherent in a structural ensemble obtained from a large and diverse array of NMR probes are in excellent agreement with our measurements.

Combining experimental and simulation data of molecular processes via augmented Markov models

Accurate mechanistic description of structural changes in biomolecules is an increasingly important topic in structural and chemical biology. Markov models have emerged as a powerful way to approximate the molecular kinetics of large biomolecules while keeping full structural resolution in a divide-and-conquer fashion. However, the accuracy of these models is limited by that of the force fields used to generate the underlying molecular dynamics (MD) simulation data. Whereas the quality of classical MD force fields has improved significantly in recent years, remaining errors in the Boltzmann weights are still on the order of a few [Formula: see text], which may lead to significant discrepancies when comparing to experimentally measured rates or state populations. Here we take the view that simulations using a sufficiently good force-field sample conformations that are valid but have inaccurate weights, yet these weights may be made accurate by incorporating experimental data a posteriori. To do so, we propose augmented Markov models (AMMs), an approach that combines concepts from probability theory and information theory to consistently treat systematic force-field error and statistical errors in simulation and experiment. Our results demonstrate that AMMs can reconcile conflicting results for protein mechanisms obtained by different force fields and correct for a wide range of stationary and dynamical observables even when only equilibrium measurements are incorporated into the estimation process. This approach constitutes a unique avenue to combine experiment and computation into integrative models of biomolecular structure and dynamics.

The Exact Nuclear Overhauser Enhancement: Recent Advances

Although often depicted as rigid structures, proteins are highly dynamic systems, whose motions are essential to their functions. Despite this, it is difficult to investigate protein dynamics due to the rapid timescale at which they sample their conformational space, leading most NMR-determined structures to represent only an averaged snapshot of the dynamic picture. While NMR relaxation measurements can help to determine local dynamics, it is difficult to detect translational or concerted motion, and only recently have significant advances been made to make it possible to acquire a more holistic representation of the dynamics and structural landscapes of proteins. Here, we briefly revisit our most recent progress in the theory and use of exact nuclear Overhauser enhancements (eNOEs) for the calculation of structural ensembles that describe their conformational space. New developments are primarily targeted at increasing the number and improving the quality of extracted eNOE distance restraints, such that the multi-state structure calculation can be applied to proteins of higher molecular weights. We then review the implications of the exact NOE to the protein dynamics and function of cyclophilin A and the WW domain of Pin1, and finally discuss our current research and future directions.

Cross-correlated relaxation rates between protein backbone H-X dipolar interactions

The relaxation interference between dipole-dipole interactions of two separate spin pairs carries structural and dynamics information. In particular, when compared to individual dynamic behavior of those spin pairs, such cross-correlated relaxation (CCR) rates report on the correlation between the spin pairs. We have recently mapped out correlated motion along the backbone of the protein GB3, using CCR rates among and between consecutive H^N-N and H^α-C^α dipole-dipole interactions. Here, we provide a detailed account of the measurement of the four types of CCR rates. All rates were obtained from at least two different pulse sequences, of which the yet unpublished ones are presented. Detailed comparisons between the different methods and corrections for unwanted pathways demonstrate that the averaged CCR rates are highly accurate and precise with errors of 1.5-3% of the entire value ranges.

Bayesian inference of protein ensembles from SAXS data

The inherent flexibility of intrinsically disordered proteins (IDPs) and multi-domain proteins with intrinsically disordered regions (IDRs) presents challenges to structural analysis. These macromolecules need to be represented by an ensemble of conformations, rather than a single structure. Small-angle X-ray scattering (SAXS) experiments capture ensemble-averaged data for the set of conformations. We present a Bayesian approach to ensemble inference from SAXS data, called Bayesian ensemble SAXS (BE-SAXS). We address two issues with existing methods: the use of a finite ensemble of structures to represent the underlying distribution, and the selection of that ensemble as a subset of an initial pool of structures. This is achieved through the formulation of a Bayesian posterior of the conformational space. BE-SAXS modifies a structural prior distribution in accordance with the experimental data. It uses multi-step expectation maximization, with alternating rounds of Markov-chain Monte Carlo simulation and empirical Bayes optimization. We demonstrate the method by employing it to obtain a conformational ensemble of the antitoxin PaaA2 and comparing the results to a published ensemble.

The Exact NOE as an Alternative in Ensemble Structure Determination

The structure-function paradigm is increasingly replaced by the structure-dynamics-function paradigm. All protein activity is steered by the interplay between enthalpy and entropy. Conformational dynamics serves as a proxy of conformational entropy. Therefore, it is essential to study not only the average conformation but also the spatial sampling of a protein on all timescales. To this purpose, we have established a protocol for determining multiple-state ensembles of proteins based on exact nuclear Overhauser effects (eNOEs). We have recently extended our previously reported eNOE data set for the protein GB3 by a very large set of backbone and side-chain residual dipolar couplings and three-bond J couplings. Here, we demonstrate that at least four structural states are required to represent the complete data set by dissecting the contributions to the CYANA target function, which quantifies restraint violations in structure calculation. We present a four-state ensemble of GB3, which largely preserves the characteristics obtained from eNOEs only. Due to the abundance of the input data, the ensemble and χ(1) angles in particular are well suited for cross-validation of the input data and comparison to x-ray structures. Principal component analysis is used to automatically identify and validate relevant states of the ensembles. Overall, our findings suggest that eNOEs are a valuable alternative to traditional NMR probes in spatial elucidation of proteins.

Bayesian ensemble refinement by replica simulations and reweighting

We describe different Bayesian ensemble refinement methods, examine their interrelation, and discuss their practical application. With ensemble refinement, the properties of dynamic and partially disordered (bio)molecular structures can be characterized by integrating a wide range of experimental data, including measurements of ensemble-averaged observables. We start from a Bayesian formulation in which the posterior is a functional that ranks different configuration space distributions. By maximizing this posterior, we derive an optimal Bayesian ensemble distribution. For discrete configurations, this optimal distribution is identical to that obtained by the maximum entropy "ensemble refinement of SAXS" (EROS) formulation. Bayesian replica ensemble refinement enhances the sampling of relevant configurations by imposing restraints on averages of observables in coupled replica molecular dynamics simulations. We show that the strength of the restraints should scale linearly with the number of replicas to ensure convergence to the optimal Bayesian result in the limit of infinitely many replicas. In the "Bayesian inference of ensembles" method, we combine the replica and EROS approaches to accelerate the convergence. An adaptive algorithm can be used to sample directly from the optimal ensemble, without replicas. We discuss the incorporation of single-molecule measurements and dynamic observables such as relaxation parameters. The theoretical analysis of different Bayesian ensemble refinement approaches provides a basis for practical applications and a starting point for further investigations.

Finding Our Way in the Dark Proteome

The traditional structure-function paradigm has provided significant insights for well-folded proteins in which structures can be easily and rapidly revealed by X-ray crystallography beamlines. However, approximately one-third of the human proteome is comprised of intrinsically disordered proteins and regions (IDPs/IDRs) that do not adopt a dominant well-folded structure, and therefore remain "unseen" by traditional structural biology methods. This Perspective considers the challenges raised by the "Dark Proteome", in which determining the diverse conformational substates of IDPs in their free states, in encounter complexes of bound states, and in complexes retaining significant disorder requires an unprecedented level of integration of multiple and complementary solution-based experiments that are analyzed with state-of-the art molecular simulation, Bayesian probabilistic models, and high-throughput computation. We envision how these diverse experimental and computational tools can work together through formation of a "computational beamline" that will allow key functional features to be identified in IDP structural ensembles.

The Dynamic Basis for Signal Propagation in Human Pin1-WW

Allostery is the structural manifestation of information transduction in biomolecules. Its hallmark is conformational change induced by perturbations at a distal site. An increasing body of evidence demonstrates the presence of allostery in very flexible and even disordered proteins, encouraging a thermodynamic description of this phenomenon. Still, resolving such processes at atomic resolution is difficult. Here we establish a protocol to determine atomistic thermodynamic models of such systems using high-resolution solution state nuclear magnetic resonance data and extensive molecular simulations. Using this methodology, we study information transduction in the WW domain of a key cell-cycle regulator Pin1. Pin1 binds promiscuously to phospho-Ser/Thr-Pro motifs, however, disparate structural and dynamic responses have been reported upon binding different ligands. Our model consists of two topologically distinct states whose relative population may be specifically skewed by an incoming ligand. This model provides a canonical basis for the understanding of multi-functionality in Pin1.

Direct Investigation of Slow Correlated Dynamics in Proteins via Dipolar Interactions

The synchronization of native state motions as they transition between microstates influences catalysis kinetics, mediates allosteric interactions, and reduces the conformational entropy of proteins. However, it has proven difficult to describe native microstates because they are usually minimally frustrated and may interconvert on the micro- to millisecond time scale. Direct observation of concerted equilibrium fluctuations would therefore be an important tool for describing protein native states. Here we propose a strategy that relates NMR cross-correlated relaxation (CCR) rates between dipolar interactions to residual dipolar couplings (RDCs) of individual consecutive H(N)-N and H(α)-C(α) bonds, which act as a proxy for the peptide planes and the side chains, respectively. Using Xplor-NIH ensemble structure calculations restrained with the RDC and CCR data, we observe collective motions on time scales slower than nanoseconds in the backbone for GB3. To directly access the correlations from CCR, we develop a structure-free data analysis. The resulting dynamic correlation map is consistent with the ensemble-restrained simulations and reveals a complex network. In general, we find that the bond motions are on average slightly correlated and that the local environment dominates many observations. Despite this, some patterns are typical over entire secondary structure elements. In the β-sheet, nearly all bonds are weakly correlated, and there is an approximately binary alternation in correlation intensity corresponding to the solvent exposure/shielding alternation of the side chains. For α-helices, there is also a weak correlation in the H(N)-N bonds. The degree of correlation involving H(α)-C(α) bonds is directly affected by side-chain fluctuations, whereas loops show complex and nonuniform behavior.

Ensemble-based evaluation for protein structure models

Motivation: Comparing protein tertiary structures is a fundamental procedure in structural biology and protein bioinformatics. Structure comparison is important particularly for evaluating computational protein structure models. Most of the model structure evaluation methods perform rigid body superimposition of a structure model to its crystal structure and measure the difference of the corresponding residue or atom positions between them. However, these methods neglect intrinsic flexibility of proteins by treating the native structure as a rigid molecule. Because different parts of proteins have different levels of flexibility, for example, exposed loop regions are usually more flexible than the core region of a protein structure, disagreement of a model to the native needs to be evaluated differently depending on the flexibility of residues in a protein.

Results: We propose a score named FlexScore for comparing protein structures that consider flexibility of each residue in the native state of proteins. Flexibility information may be extracted from experiments such as NMR or molecular dynamics simulation. FlexScore considers an ensemble of conformations of a protein described as a multivariate Gaussian distribution of atomic displacements and compares a query computational model with the ensemble. We compare FlexScore with other commonly used structure similarity scores over various examples. FlexScore agrees with experts’ intuitive assessment of computational models and provides information of practical usefulness of models.

Availability and implementation:https://bitbucket.org/mjamroz/flexscore

Contact:dkihara@purdue.edu

Supplementary information:Supplementary data are available at Bioinformatics online.

A Monte Carlo Study of the Early Steps of Functional Amyloid Formation

In addition to their well-known roles in neurodegenerative diseases and amyloidoses, amyloid structures also assume important functional roles in the cell. Although functional amyloid shares many physiochemical properties with its pathogenic counterpart, it is evolutionarily optimized to avoid cytotoxicity. This makes it an interesting study case for aggregation phenomenon in general. One of the most well-known examples of a functional amyloid, E. coli curli, is an essential component in the formation of bacterial biofilm, and is primarily formed by aggregates of the protein CsgA. Previous studies have shown that the minor sequence variations observed in the five different subrepeats (R1-R5), which comprise the CsgA primary sequence, have a substantial influence on their individual aggregation propensities. Using a recently described diffusion-optimized enhanced sampling approach for Monte Carlo simulations, we here investigate the equilibrium properties of the monomeric and dimeric states of these subrepeats, to probe whether structural properties observed in these early stage oligomers are decisive for the characteristics of the resulting aggregate. We show that the dimerization propensities of these peptides have strong correlations with their propensity for amyloid formation, and provide structural insights into the inter- and intramolecular contacts that appear to be essential in this process.

Investigating protein conformational energy landscapes and atomic resolution dynamics from NMR dipolar couplings: a review

Nuclear magnetic resonance spectroscopy is exquisitely sensitive to protein dynamics. In particular inter-nuclear dipolar couplings, that become measurable in solution when the protein is dissolved in a dilute liquid crystalline solution, report on all conformations sampled up to millisecond timescales. As such they provide the opportunity to describe the Boltzmann distribution present in solution at atomic resolution, and thereby to map the conformational energy landscape in unprecedented detail. The development of analytical methods and approaches based on numerical simulation and their application to numerous biologically important systems is presented.

Side Chain Conformational Distributions of a Small Protein Derived from Model-Free Analysis of a Large Set of Residual Dipolar Couplings

Accurate quantitative measurement of structural dispersion in proteins remains a prime challenge to both X-ray crystallography and NMR spectroscopy. Here we use a model-free approach based on measurement of many residual dipolar couplings (RDCs) in differentially orienting aqueous liquid crystalline solutions to obtain the side chain χ₁ distribution sampled by each residue in solution. Applied to the small well-ordered model protein GB3, our approach reveals that the RDC data are compatible with a single narrow distribution of side chain χ₁ angles for only about 40% of the residues. For more than half of the residues, populations greater than 10% for a second rotamer are observed, and four residues require sampling of three rotameric states to fit the RDC data. In virtually all cases, sampled χ₁ values are found to center closely around ideal g^–, g⁺ and t rotameric angles, even though no rotamer restraint is used when deriving the sampled angles. The root-mean-square difference between experimental ³J_HαHβ couplings and those predicted by the Haasnoot-parametrized, motion-adjusted Karplus equation reduces from 2.05 to 0.75 Hz when using the new rotamer analysis instead of the 1.1-Å X-ray structure as input for the dihedral angles. A comparison between observed and predicted ³J_HαHβ values suggests that the root-mean-square amplitude of χ₁ angle fluctuations within a given rotamer well is ca. 20°. The quantitatively defined side chain rotamer equilibria obtained from our study set new benchmarks for evaluating improved molecular dynamics force fields, and also will enable further development of quantitative relations between side chain chemical shift and structure.

Protein flexibility in the light of structural alphabets

Protein structures are valuable tools to understand protein function. Nonetheless, proteins are often considered as rigid macromolecules while their structures exhibit specific flexibility, which is essential to complete their functions. Analyses of protein structures and dynamics are often performed with a simplified three-state description, i.e., the classical secondary structures. More precise and complete description of protein backbone conformation can be obtained using libraries of small protein fragments that are able to approximate every part of protein structures. These libraries, called structural alphabets (SAs), have been widely used in structure analysis field, from definition of ligand binding sites to superimposition of protein structures. SAs are also well suited to analyze the dynamics of protein structures. Here, we review innovative approaches that investigate protein flexibility based on SAs description. Coupled to various sources of experimental data (e.g., B-factor) and computational methodology (e.g., Molecular Dynamic simulation), SAs turn out to be powerful tools to analyze protein dynamics, e.g., to examine allosteric mechanisms in large set of structures in complexes, to identify order/disorder transition. SAs were also shown to be quite efficient to predict protein flexibility from amino-acid sequence. Finally, in this review, we exemplify the interest of SAs for studying flexibility with different cases of proteins implicated in pathologies and diseases.

Molecular Dynamics of Biomolecules through Direct Analysis of Dipolar Couplings

Residual dipolar couplings (RDCs) are important probes in structural biology, but their analysis is often complicated by the determination of an alignment tensor or its associated assumptions. We here apply the maximum entropy principle to derive a tensor-free formalism which allows for direct, dynamic analysis of RDCs and holds the classic tensor formalism as a special case. Specifically, the framework enables us to robustly analyze data regardless of whether a clear separation of internal and overall dynamics is possible. Such a separation is often difficult in the core subjects of current structural biology, which include multidomain and intrinsically disordered proteins as well as nucleic acids. We demonstrate the method is tractable and self-consistent and generalizes to data sets comprised of observations from multiple different alignment conditions.

Extending the eNOE data set of large proteins by evaluation of NOEs with unresolved diagonals

The representation of a protein's spatial sampling at atomic resolution is fundamental for understanding its function. NMR has been established as the best-suited technique toward this goal for small proteins. However, the accessible information content rapidly deteriorates with increasing protein size. We have recently demonstrated that for small proteins distance restraints with an accuracy smaller than 0.1 Å can be obtained by replacing traditional semi-quantitative Nuclear Overhauser Effects (NOEs) with exact NOEs (eNOE). The high quality of the data allowed us to calculate structural ensembles of the small model protein GB3 consisting of multiple rather than a single state. The analysis has been limited to small proteins because NOEs of spins with unresolved diagonal peaks cannot be used. Here we propose a simple approach to translate such NOEs into correct upper distance restraints, which opens access to larger biomolecules. We demonstrate that for 16 kDa cyclophilin A the collection of such restraints extends the original 1254 eNOEs to 3471.

Time-averaged order parameter restraints in molecular dynamics simulations

A method is described that allows experimental S(2) order parameters to be enforced as a time-averaged quantity in molecular dynamics simulations. The two parameters that characterize time-averaged restraining, the memory relaxation time and the weight of the restraining potential energy term in the potential energy function used in the simulation, are systematically investigated based on two model systems, a vector with one end restrained in space and a pentapeptide. For the latter it is shown that the backbone N-H order parameter of individual residues can be enforced such that the spatial fluctuations of quantities depending on atomic coordinates are not significantly perturbed. The applicability to realistic systems is illustrated for the B3 domain of protein G in aqueous solution.

Application of the maximum entropy principle to determine ensembles of intrinsically disordered proteins from residual dipolar couplings

We present a method based on the maximum entropy principle that can re-weight an ensemble of protein structures based on data from residual dipolar couplings (RDCs).