Mechanistic Models of Chemical Exchange Induced Relaxation in Protein NMR

Unraveling motion in proteins by combining NMR relaxometry and molecular dynamics simulations: A case study on ubiquitin

Nuclear magnetic resonance (NMR) relaxation experiments shine light onto the dynamics of molecular systems in the picosecond to millisecond timescales. As these methods cannot provide an atomically resolved view of the motion of atoms, functional groups, or domains giving rise to such signals, relaxation techniques have been combined with molecular dynamics (MD) simulations to obtain mechanistic descriptions and gain insights into the functional role of side chain or domain motion. In this work, we present a comparison of five computational methods that permit the joint analysis of MD simulations and NMR relaxation experiments. We discuss their relative strengths and areas of applicability and demonstrate how they may be utilized to interpret the dynamics in MD simulations with the small protein ubiquitin as a test system. We focus on the aliphatic side chains given the rigidity of the backbone of this protein. We find encouraging agreement between experiment, Markov state models built in the χ1/χ2 rotamer space of isoleucine residues, explicit rotamer jump models, and a decomposition of the motion using ROMANCE. These methods allow us to ascribe the dynamics to specific rotamer jumps. Simulations with eight different combinations of force field and water model highlight how the different metrics may be employed to pinpoint force field deficiencies. Furthermore, the presented comparison offers a perspective on the utility of NMR relaxation to serve as validation data for the prediction of kinetics by state-of-the-art biomolecular force fields.

Conformational Dynamics in Proteins: Entangled Slow Fluctuations and Nonequilibrium Reaction Events

Proteins exhibit conformational fluctuations and changes over various time scales, ranging from rapid picosecond-scale local atomic motions to slower microsecond-scale global conformational transformations. In the presence of these intricate fluctuations, chemical reactions occur and functions emerge. These conformational fluctuations of proteins are not merely stochastic random motions but possess distinct spatiotemporal characteristics. Moreover, chemical reactions do not always proceed along a single reaction coordinate in a quasi-equilibrium manner. Therefore, it is essential to understand spatiotemporal conformational fluctuations of proteins and the conformational change processes associated with reactions. In this Perspective, we shed light on the complex dynamics of proteins and their role in enzyme catalysis by presenting recent results regarding dynamic couplings and disorder in the conformational dynamics of proteins and rare but rapid enzymatic reaction events obtained from molecular dynamics simulations.

Biomolecular NMR spectroscopy in the era of artificial intelligence

Biomolecular nuclear magnetic resonance (NMR) spectroscopy and artificial intelligence (AI) have a burgeoning synergy. Deep learning-based structural predictors have forever changed structural biology, yet these tools currently face limitations in accurately characterizing protein dynamics, allostery, and conformational heterogeneity. We begin by highlighting the unique abilities of biomolecular NMR spectroscopy to complement AI-based structural predictions toward addressing these knowledge gaps. We then highlight the direct integration of deep learning approaches into biomolecular NMR methods. AI-based tools can dramatically improve the acquisition and analysis of NMR spectra, enhancing the accuracy and reliability of NMR measurements, thus streamlining experimental processes. Additionally, deep learning enables the development of novel types of NMR experiments that were previously unattainable, expanding the scope and potential of biomolecular NMR spectroscopy. Ultimately, a combination of AI and NMR promises to further revolutionize structural biology on several levels, advance our understanding of complex biomolecular systems, and accelerate drug discovery efforts.

Vendi sampling for molecular simulations: Diversity as a force for faster convergence and better exploration

Molecular dynamics (MD) is the method of choice for understanding the structure, function, and interactions of molecules. However, MD simulations are limited by the strong metastability of many molecules, which traps them in a single conformation basin for an extended amount of time. Enhanced sampling techniques, such as metadynamics and replica exchange, have been developed to overcome this limitation and accelerate the exploration of complex free energy landscapes. In this paper, we propose Vendi Sampling, a replica-based algorithm for increasing the efficiency and efficacy of the exploration of molecular conformation spaces. In Vendi sampling, replicas are simulated in parallel and coupled via a global statistical measure, the Vendi Score, to enhance diversity. Vendi sampling allows for the recovery of unbiased sampling statistics and dramatically improves sampling efficiency. We demonstrate the effectiveness of Vendi sampling in improving molecular dynamics simulations by showing significant improvements in coverage and mixing between metastable states and convergence of free energy estimates for four common benchmarks, including Alanine Dipeptide and Chignolin.

Dynamical Models of Chemical Exchange in Nuclear Magnetic Resonance Spectroscopy

Chemical exchange line-broadening is an important phenomenon in nuclear magnetic resonance (NMR) spectroscopy, in which a nuclear spin experiences more than one magnetic environment as a result of chemical or conformational changes of a molecule. The dynamic process of chemical exchange strongly affects the sensitivity and resolution of NMR experiments, and increasingly provides a powerful probe of the inter-conversion between chemical and conformational states of proteins, nucleic acids, and other biological macromolecules. A simple and often used theoretical description of chemical exchange in NMR spectroscopy is based on an idealized two-state jump model (the random-phase or telegraph signal). However, chemical exchange can also be represented as a barrier-crossing event that can be modeled using chemical reaction rate theory. The time scale of crossing is determined by the barrier height, the temperature, and the dissipation modeled as collisional or frictional damping. This tutorial explores the connection between the NMR theory of chemical exchange line-broadening and strong-collision models for chemical kinetics in statistical mechanics. Theoretical modeling and numerical simulation are used to map the rate of barrier-crossing dynamics of a particle on a potential energy surface to the chemical exchange relaxation rate constant. By developing explicit models for the exchange dynamics, the tutorial aims to elucidate the underlying dynamical processes that give rise to the rich phenomenology of chemical exchange observed in NMR spectroscopy. Software for generating and analyzing the numerical simulations is provided in the form of Python and Fortran source codes.

Motional clustering in supra-τc conformational exchange influences NOE cross-relaxation rate

Biomolecular spin relaxation processes, such as the NOE, are commonly modeled by rotational τ_c-tumbling combined with fast motions on the sub-τ_c timescale. Motions on the supra-τ_c timescale, in contrast, are considered to be completely decorrelated to the molecular tumbling and therefore invisible. Here, we show how supra-τ_c dynamics can nonetheless influence the NOE build-up between methyl groups. This effect arises because supra-τ_c motions can cluster the fast-motion ensembles into discrete states, affecting distance averaging as well as the fast-motion order parameter and hence the cross-relaxation rate. We present a computational approach to estimate methyl-methyl cross-relaxation rates from extensive (>100×τ_c) all-atom molecular dynamics (MD) trajectories on the example of the 723-residue protein Malate Synthase G. The approach uses Markov state models (MSMs) to resolve transitions between metastable states and thus to discriminate between sub-τ_c and supra-τ_c conformational exchange. We find that supra-τ_c exchange typically increases NOESY cross-peak intensities. The methods described in this work extend the theory of modeling sub-μs dynamics in spin relaxation and thus contribute to a quantitative estimation of NOE cross-relaxation rates from MD simulations, eventually leading to increased precision in structural and functional studies of large proteins.

Transient exposure of a buried phosphorylation site in an autoinhibited protein

Autoinhibition is a mechanism used to regulate protein function, often by making functional sites inaccessible through the interaction with a cis-acting inhibitory domain. Such autoinhibitory domains often display a substantial degree of structural disorder when unbound, and only become structured in the inhibited state. These conformational dynamics make it difficult to study the structural origin of regulation, including effects of regulatory post-translational modifications. Here, we study the autoinhibition of the Dbl Homology domain in the protein Vav1 by the so-called acidic inhibitory domain. We use molecular simulations to study the process by which a mostly unstructured inhibitory domain folds upon binding and how transient exposure of a key buried tyrosine residue makes it accessible for phosphorylation. We show that the inhibitory domain, which forms a helix in the bound and inhibited stated, samples helical structures already before binding and that binding occurs via a molten-globule-like intermediate state. Together, our results shed light on key interactions that enable the inhibitory domain to sample a finely tuned equilibrium between an inhibited and a kinase-accessible state.

Time-Lagged Independent Component Analysis of Random Walks and Protein Dynamics

Time-lagged independent component analysis (tICA) is a widely used dimension reduction method for the analysis of molecular dynamics (MD) trajectories and has proven particularly useful for the construction of protein dynamics Markov models. It identifies those “slow” collective degrees of freedom onto which the projections of a given trajectory show maximal autocorrelation for a given lag time. Here we ask how much information on the actual protein dynamics and, in particular, the free energy landscape that governs these dynamics the tICA-projections of MD-trajectories contain, as opposed to noise due to the inherently stochastic nature of each trajectory. To answer this question, we have analyzed the tICA-projections of high dimensional random walks using a combination of analytical and numerical methods. We find that the projections resemble cosine functions and strongly depend on the lag time, exhibiting strikingly complex behavior. In particular, and contrary to previous studies of principal component projections, the projections change noncontinuously with increasing lag time. The tICA-projections of selected 1 μs protein trajectories and those of random walks are strikingly similar, particularly for larger proteins, suggesting that these trajectories contain only little information on the energy landscape that governs the actual protein dynamics. Further the tICA-projections of random walks show clusters very similar to those observed for the protein trajectories, suggesting that clusters in the tICA-projections of protein trajectories do not necessarily reflect local minima in the free energy landscape. We also conclude that, in addition to the previous finding that certain ensemble properties of nonconverged protein trajectories resemble those of random walks; this is also true for their time correlations.

Microscopic insights into dynamic disorder in the isomerization dynamics of the protein BPTI

Understanding the dynamic disorder behind a process, i.e., the dynamic effect of fluctuations that occur on a timescale slower or comparable with the timescale of the process, is essential for elucidating the dynamics and kinetics of complicated molecular processes in biomolecules and liquids. Despite numerous theoretical studies of single-molecule kinetics, our microscopic understanding of dynamic disorder remains limited. In the present study, we investigate the microscopic aspects of dynamic disorder in the isomerization dynamics of the Cys14-Cys38 disulfide bond in the protein bovine pancreatic trypsin inhibitor, which has been observed by nuclear magnetic resonance. We use a theoretical model with a stochastic transition rate coefficient, which is calculated from the 1-ms-long time molecular dynamics trajectory obtained by Shaw et al. [Science 330, 341-346 (2010)]. The isomerization dynamics are expressed by the transitions between coarse-grained states consisting of internal states, i.e., conformational sub-states. In this description, the rate for the transition from the coarse-grained states is stochastically modulated due to fluctuations between internal states. We examine the survival probability for the conformational transitions from a coarse-grained state using a theoretical model, which is a good approximation to the directly calculated survival probability. The dynamic disorder changes from a slow modulation limit to a fast modulation limit depending on the aspects of the coarse-grained states. Our analysis of the rate modulations behind the survival probability, in relation to the fluctuations between internal states, reveals the microscopic origin of dynamic disorder.

Discovery of a hidden transient state in all bromodomain families

Bromodomains (BDs) are small protein modules that interact with acetylated marks in histones. These posttranslational modifications are pivotal to regulate gene expression, making BDs promising targets to treat several diseases. While the general structure of BDs is well known, their dynamical features and their interplay with other macromolecules are poorly understood, hampering the rational design of potent and selective inhibitors. Here, we combine extensive molecular dynamics simulations, Markov state modeling, and available structural data to reveal a transiently formed state that is conserved across all BD families. It involves the breaking of two backbone hydrogen bonds that anchor the ZA-loop with the αA helix, opening a cryptic pocket that partially occludes the one associated to histone binding. By analyzing more than 1,900 experimental structures, we unveil just two adopting the hidden state, explaining why it has been previously unnoticed and providing direct structural evidence for its existence. Our results suggest that this state is an allosteric regulatory switch for BDs, potentially related to a recently unveiled BD-DNA-binding mode.

Protein Allostery at Atomic Resolution

Protein allostery is a phenomenon involving the long range coupling between two distal sites in a protein. In order to elucidate allostery at atomic resoluion on the ligand-binding WW domain of the enzyme Pin1, multistate structures were calculated from exact nuclear Overhauser effect (eNOE). In its free form, the protein undergoes a microsecond exchange between two states, one of which is predisposed to interact with its parent catalytic domain. In presence of the positive allosteric ligand, the equilibrium between the two states is shifted towards domain-domain interaction, suggesting a population shift model. In contrast, the allostery-suppressing ligand decouples the side-chain arrangement at the inter-domain interface thereby reducing the inter-domain interaction. As such, this mechanism is an example of dynamic allostery. The presented distinct modes of action highlight the power of the interplay between dynamics and function in the biological activity of proteins.

Quantitative comparison between sub-millisecond time resolution single-molecule FRET measurements and 10-second molecular simulations of a biosensor protein

Molecular Dynamics (MD) simulations seek to provide atomic-level insights into conformationally dynamic biological systems at experimentally relevant time resolutions, such as those afforded by single-molecule fluorescence measurements. However, limitations in the time scales of MD simulations and the time resolution of single-molecule measurements have challenged efforts to obtain overlapping temporal regimes required for close quantitative comparisons. Achieving such overlap has the potential to provide novel theories, hypotheses, and interpretations that can inform idealized experimental designs that maximize the detection of the desired reaction coordinate. Here, we report MD simulations at time scales overlapping with in vitro single-molecule Förster (fluorescence) resonance energy transfer (smFRET) measurements of the amino acid binding protein LIV-BP^SS at sub-millisecond resolution. Computationally efficient all-atom structure-based simulations, calibrated against explicit solvent simulations, were employed for sampling multiple cycles of LIV-BP^SS clamshell-like conformational changes on the time scale of seconds, examining the relationship between these events and those observed by smFRET. The MD simulations agree with the smFRET measurements and provide valuable information on local dynamics of fluorophores at their sites of attachment on LIV-BP^SS and the correlations between fluorophore motions and large-scale conformational changes between LIV-BP^SS domains. We further utilize the MD simulations to inform the interpretation of smFRET data, including Förster radius (R₀) and fluorophore orientation factor (κ²) determinations. The approach we describe can be readily extended to distinct biochemical systems, allowing for the interpretation of any FRET system conjugated to protein or ribonucleoprotein complexes, including those with more conformational processes, as well as those implementing multi-color smFRET.

Förster (fluorescence) resonance energy transfer (FRET) has been used extensively by biophysicists as a molecular-scale ruler that yields fundamental structural and kinetic insights into transient processes including complex formation and conformational rearrangements required for biological function. FRET techniques require the identification of informative fluorophore labeling sites, spaced at defined distances to inform on a reaction coordinate of interest and consideration of noise sources that have the potential to obscure quantitative interpretations. Here, we describe an approach to leverage advancements in computationally efficient all-atom structure-based molecular dynamics simulations in which structural dynamics observed via FRET can be interpreted in full atomistic detail on commensurate time scales. We demonstrate the potential of this approach using a model FRET system, the amino acid binding protein LIV-BP^SS conjugated to self-healing organic fluorophores. LIV-BP^SS exhibits large scale, sub-millisecond clamshell-like conformational changes between open and closed conformations associated with ligand unbinding and binding, respectively. Our findings inform on the molecular basis of the dynamics observed by smFRET and on strategies to optimize fluorophore labeling sites, the manner of fluorophore attachment, and fluorophore composition.

Local frustration determines loop opening during the catalytic cycle of an oxidoreductase

Local structural frustration, the existence of mutually exclusive competing interactions, may explain why some proteins are dynamic while others are rigid. Frustration is thought to underpin biomolecular recognition and the flexibility of protein-binding sites. Here, we show how a small chemical modification, the oxidation of two cysteine thiols to a disulfide bond, during the catalytic cycle of the N-terminal domain of the key bacterial oxidoreductase DsbD (nDsbD), introduces frustration ultimately influencing protein function. In oxidized nDsbD, local frustration disrupts the packing of the protective cap-loop region against the active site allowing loop opening. By contrast, in reduced nDsbD the cap loop is rigid, always protecting the active-site thiols from the oxidizing environment of the periplasm. Our results point toward an intricate coupling between the dynamics of the active-site cysteines and of the cap loop which modulates the association reactions of nDsbD with its partners resulting in optimized protein function.

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.

Dynamic design: manipulation of millisecond timescale motions on the energy landscape of cyclophilin A

Proteins need to interconvert between many conformations in order to function, many of which are formed transiently, and sparsely populated. Particularly when the lifetimes of these states approach the millisecond timescale, identifying the relevant structures and the mechanism by which they interconvert remains a tremendous challenge. Here we introduce a novel combination of accelerated MD (aMD) simulations and Markov state modelling (MSM) to explore these 'excited' conformational states. Applying this to the highly dynamic protein CypA, a protein involved in immune response and associated with HIV infection, we identify five principally populated conformational states and the atomistic mechanism by which they interconvert. A rational design strategy predicted that the mutant D66A should stabilise the minor conformations and substantially alter the dynamics, whereas the similar mutant H70A should leave the landscape broadly unchanged. These predictions are confirmed using CPMG and R1ρ solution state NMR measurements. By efficiently exploring functionally relevant, but sparsely populated conformations with millisecond lifetimes in silico, our aMD/MSM method has tremendous promise for the design of dynamic protein free energy landscapes for both protein engineering and drug discovery.

Dynamic graphical models of molecular kinetics

Most current molecular dynamics simulation and analysis methods rely on the idea that the molecular system can be represented by a single global state (e.g., a Markov state in a Markov state model [MSM]). In this approach, molecules can be extensively sampled and analyzed when they only possess a few metastable states, such as small- to medium-sized proteins. However, this approach breaks down in frustrated systems and in large protein assemblies, where the number of global metastable states may grow exponentially with the system size. To address this problem, we here introduce dynamic graphical models (DGMs) that describe molecules as assemblies of coupled subsystems, akin to how spins interact in the Ising model. The change of each subsystem state is only governed by the states of itself and its neighbors. DGMs require fewer parameters than MSMs or other global state models; in particular, we do not need to observe all global system configurations to characterize them. Therefore, DGMs can predict previously unobserved molecular configurations. As a proof of concept, we demonstrate that DGMs can faithfully describe molecular thermodynamics and kinetics and predict previously unobserved metastable states for Ising models and protein simulations.

Role of conformational dynamics in the evolution of novel enzyme function

The free energy landscape concept that describes enzymes as an ensemble of differently populated conformational sub-states in dynamic equilibrium is key for evaluating enzyme activity, enantioselectivity, and specificity. Mutations introduced in the enzyme sequence can alter the populations of the pre-existing conformational states, thus strongly modifying the enzyme ability to accommodate alternative substrates, revert its enantiopreferences, and even increase the activity for some residual promiscuous reactions. In this feature article, we present an overview of the current experimental and computational strategies to explore the conformational free energy landscape of enzymes. We provide a series of recent publications that highlight the key role of conformational dynamics for the enzyme evolution towards new functions and substrates, and provide some perspectives on how conformational dynamism should be considered in future computational enzyme design protocols.

A scalable approach to the computation of invariant measures for high-dimensional Markovian systems

The Markovian invariant measure is a central concept in many disciplines. Conventional numerical techniques for data-driven computation of invariant measures rely on estimation and further numerical processing of a transition matrix. Here we show how the quality of data-driven estimation of a transition matrix crucially depends on the validity of the statistical independence assumption for transition probabilities. Moreover, the cost of the invariant measure computation in general scales cubically with the dimension - and is usually unfeasible for realistic high-dimensional systems. We introduce a method relaxing the independence assumption of transition probabilities that scales quadratically in situations with latent variables. Applications of the method are illustrated on the Lorenz-63 system and for the molecular dynamics (MD) simulation data of the α-synuclein protein. We demonstrate how the conventional methodologies do not provide good estimates of the invariant measure based upon the available α-synuclein MD data. Applying the introduced approach to these MD data we detect two robust meta-stable states of α-synuclein and a linear transition between them, involving transient formation of secondary structure, qualitatively consistent with previous purely experimental reports.

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.