Protein dynamics from NMR: the slowly relaxing local structure analysis compared with model-free analysis

Local Structures in Proteins from Microsecond Molecular Dynamics Simulations: 2. The Role of Symmetry in GTPase Binding and Dimer Formation

The Rho GTPase binding domain of Plexin-B1 (RBD) prevails in solution as dimer. Under appropriate circumstances, it binds the small GTPase Rac1 to yield the complex RBD-Rac1. Here, we study RBD dimerization and complex formation from a symmetry-based perspective using data derived from 1 μs long MD simulations. The quantities investigated are the local potentials, u(MD), prevailing at the N-H sites of the protein. These potentials are statistical in character providing an empirical description of the local structure. To establish more methodical description, a method for approximating them by explicit functions, u(simulated), was developed in the preceding article in this journal issue. These functions are combinations of analytical Wigner functions, DL,K, belonging to the D2h point group. The D2h subgroups Ag and B2u are found to dominate u(simulated); the B1u subgroup contributes in some cases. The Ag (B2u) functions have axial or rhombic symmetry. For the first time, local potentials in proteins can be quantitatively characterized in terms of their strength (rhombicity) evaluated by axial Ag (rhombic Ag and B2u) contributions. Until now, the chain-segment [β3-L3-β4] and to some extent the α2-helix have been associated with GTPase binding. Here, we find that this process causes an increase (decrease) in the potential strength of β3 and β4 (the preceding L2 loop and the remote chain-segment [(α2-helix)-(α2/β5-turn)-(β5-strand)]), suggesting effects of counterbalancing and allostery. There is evidence for the L2 loop being associated with RBD-GTPase binding. Until now only the L4 loop has been associated with RBD dimerization. The latter process is found to cause an increase (decrease) in the potential strength and rhombicity of the L4 loop (the adjacent chain-segment [(α2-helix)-(α2/β5-turn)-(β5-strand)]), suggesting counterbalancing activity. On average, the RBD dimer features stronger local potentials than RBD-Rac1. The novel information inherent in these findings is mesoscopic in character. Prospects of interest include exploring relation to atomistic force-field parameters.

Local Structures in Proteins from Microsecond Molecular Dynamics Simulations: A Symmetry-Based Perspective

We report on a new method for the characterization of local structures in proteins based on extensive molecular dynamics (MD) simulations, here, 1 μs in length. The N-H bond of the Rho GTPase binding domain of plexin-B1 (RBD) serves as a probe and the potential, u(MD), which restricts its internal motion, as a qualifier of the local dynamic structure. u(MD) is derived from the MD trajectory as a function of the polar angles, (θ, φ), which specify the N-H orientation in the protein. u(MD) is statistical in character yielding empirical descriptions. To establish more insightful methodical descriptions, we develop a comprehensive method which approximates u(MD) by combinations of analytical Wigner functions that belong to the D2h point group. These combinations, called u(simulated), make it possible to gain a new perspective of local dynamic structures in proteins based on explicit potentials/free energy surfaces and associated probability densities, entropy, and ordering. A simpler method was developed previously using 100 ns MD simulations. In that case, the traditional "perpendicular N-H ordering" setting centered at Cα-Cα with (θ, φ) = (90, 90) and generally, featuring positive φ, prevailed. u(MD) derived from 1 μs MD simulations is considerably more complex requiring substantial model enhancement. The enhanced method applies to the well-structured sections of the RBD. It only applies partly to its loops where u(MD) extends into the negative-φ region where we detect nonperpendicular N-H ordering. This arrangement requires devising new reference structures and making substantial algorithmic changes, to be performed in future work. Here, we focus on developing the comprehensive method and using it to investigate perpendicular ordering settings. We find that secondary structures (loops) exhibit varying (virtually invariant) potentials with Ag, B2u, and B1u (Ag and B2u) D2h symmetry. Application to RBD dimerization and RBD binding to the GTPase Rac1 is described in the subsequent article. Applications to other probes, proteins, and biological functions, based on explicit local potentials, probability densities, entropy, and ordering, are possible.

Microsecond MD Simulations of the Plexin-B1 RBD: N-H Probability Density as Descriptor of Structural Dynamics, Dimerization-Related Conformational Entropy, and Transient Dimer Asymmetry

Amide-bond equilibrium probability density, P_eq = exp(-u) (u, local potential), and associated conformational entropy, S_k = -∫P_eq (ln P_eq) dΩ ─ln ∫dΩ, are derived for the Rho GTPase binding domain of Plexin-B1 (RBD) as monomer and dimer from 1 μs MD simulations. The objective is to elucidate the effect of dimerization on the dynamic structure of the RBD. Dispersed (peaked) P_eq functions indicate "flexibility" ("rigidity"; the respective concepts are used below in this context). The L1 and L3 loops are throughout highly flexible, the L2 loop and the secondary structure elements are generally rigid, and the L4 loop is flexible in the monomer and rigid in the dimer. Overall, many residues are more flexible in the dimer. These features, and their implications, are discussed. Unexpectedly, we find that monomer unit 1 of the dimer (in short, d1) is unusually flexible, whereas monomer unit 2 (in short, d2) is as rigid as the RBD monomer. This is revealed due to their engagement in slow-to-intermediate conformational exchange detected previously by ¹⁵N relaxation experiments. Such motions occur with rates on the order of 10³-10⁴ s^-1; hence, they cannot be completely sampled over the course of 1 μs simulation. However, the extent to which rigid d2 is affected is small enough to enable physically relevant analysis. The entropy difference between d2 and the monomer yields an entropic contribution of -7 ± 0.7 kJ/mol to the free energy of RBD dimerization. In previous work aimed at similar objectives we used 50-100 ns MD simulations. Those results and the present result differ considerably. In summary, bond-vector P_eq functions derived directly from long MD simulations are useful descriptors of protein structural dynamics and provide accurate conformational entropy. Within the scope of slow conformational exchange, they can be useful, even in the presence of incomplete sampling.

Microsecond MD Simulations of the Plexin-B1 RBD: 2. N-H Probability Densities and Conformational Entropy in Ligand-Free, Rac1-Bound, and Dimer RBD

Orientational probability densities, P_eq = exp(-u) (u, local potential), of bond-vectors in proteins provide information on structural flexibility. The related conformational entropy, S_k = -∫P_eq(ln P_eq)dΩ - ln ∫dΩ, provides the entropic contribution to the free energy of the physical/biological process studied. We have developed a new method for deriving P_eq and S_k from MD simulations, using the N-H bond as probe. Recently we used it to study the dimerization of the Rho GTPase binding domain of Plexin-B1 (RBD). Here we use it to study RBD binding to the small GTPase Rac1. In both cases 1 μs MD simulations have been employed. The RBD has the ubiquitin fold with four mostly long loops. L3 is associated with GTPase binding, L4 with RBD dimerization, L2 participates in interdomain interactions, and L1 has not been associated with function. We find that RBD-Rac1 binding renders L1, L3, and L4 more rigid and the turns β₂/α₁ and α₂/β₅ more flexible. By comparison, RBD dimerization renders L4 more rigid, and the α-helices, the β-strands, and L2 more flexible. The rigidity of L1 in RBDRAC is consistent with L1-L3 contacts seen in previous MD simulations. The analysis of the L3-loop reveals two states of distinct flexibility which we associate with involvement in slow conformational exchange processes differing in their rates. Overall, the N-H bonds make an unfavorable entropic contribution of (5.9 ± 0.9) kJ/mol to the free energy of RBD-Rac1 binding; they were found to make a favorably contribution of (-7.0 ± 0.7) kJ/mol to the free energy of RBD dimerization. In summary, the present study provides a new perspective on the impact of Rac1 binding and dimerization on the flexibility characteristics of the RBD. Further studies are stimulated by the results of this work.

Slowly Relaxing Local Structure Analysis of 15N Relaxation from the Proteins p50 and Human Neutrophil Gelatinase-Associated Lipocalin: New Insights into the Dynamic Structure of β-Barrel Proteins

Nuclear magnetic resonance relaxation analysis is a powerful method for studying the internal mobility of proteins. We have developed for analysis the slowly relaxing local structure (SRLS) approach. SRLS is general in its nature in several respects, including the tensorial representation of the physical quantities comprising the dynamic model. By controlling tensor symmetry, a broad range of systems can be treated with physical relevance, typically with data-fitting techniques. In simple limits, SRLS yields the traditional model-free (MF) method. In the present context, MF simplicity means featuring the highest possible tensor symmetry. This renders MF-based data-fitting susceptible to the usage of fit parameters, yielding physically ill-defined results. A typical candidate is the R_ex term, devised to represent ms-μs motions but often invoked by the fitting scheme just to improve the statistics. Here, we consider two such cases using the N-H bond as probe and the proteins p50 and human neutrophil gelatinase-associated lipocalin as paradigm systems. We illustrate the harm caused by the physically unjustified involvement of R_ex in MF-based ¹⁵N relaxation analysis. Then, we show that forgoing the usage of R_ex, SRLS analysis of the very same experimental data provides interesting new information.

Explicit models of motions to analyze NMR relaxation data in proteins

Nuclear Magnetic Resonance (NMR) is a tool of choice to characterize molecular motions. In biological macromolecules, pico- to nano-second motions, in particular, can be probed by nuclear spin relaxation rates which depend on the time fluctuations of the orientations of spin interaction frames. For the past 40 years, relaxation rates have been successfully analyzed using the Model Free (MF) approach which makes no assumption on the nature of motions and reports on the effective amplitude and time-scale of the motions. However, obtaining a mechanistic picture of motions from this type of analysis is difficult at best, unless complemented with molecular dynamics (MD) simulations. In spite of their limited accuracy, such simulations can be used to obtain the information necessary to build explicit models of motions designed to analyze NMR relaxation data. Here, we present how to build such models, suited in particular to describe motions of methyl-bearing protein side-chains and compare them with the MF approach. We show on synthetic data that explicit models of motions are more robust in the presence of rotamer jumps which dominate the relaxation in methyl groups of protein side-chains. We expect this work to motivate the use of explicit models of motion to analyze MD and NMR data.

Detecting anisotropic segmental dynamics in disordered proteins by cross-correlated spin relaxation

Among the numerous contributions of Geoffrey Bodenhausen to NMR spectroscopy, his developments in the field of spin-relaxation methodology and theory will definitely have a long lasting impact. Starting with his seminal contributions to the excitation of multiple-quantum coherences, he and his group thoroughly investigated the intricate relaxation properties of these "forbidden fruits" and developed experimental techniques to reveal the relevance of previously largely ignored cross-correlated relaxation (CCR) effects, as "the essential is invisible to the eyes". Here we consider CCR within the challenging context of intrinsically disordered proteins (IDPs) and emphasize its potential and relevance for the studies of structural dynamics of IDPs in the future years to come. Conventionally, dynamics of globularly folded proteins are modeled and understood as deviations from otherwise rigid structures tumbling in solution. However, with increasing protein flexibility, as observed for IDPs, this apparent dichotomy between structure and dynamics becomes blurred. Although complex dynamics and ensemble averaging might impair the extraction of mechanistic details even further, spin relaxation uniquely encodes a protein's structural memory. Due to significant methodological developments, such as high-dimensional non-uniform sampling techniques, spin relaxation in IDPs can now be monitored in unprecedented resolution. Not embedded within a rigid globular fold, conventional 15 N spin probes might not suffice to capture the inherently local nature of IDP dynamics. To better describe and understand possible segmental motions of IDPs, we propose an experimental approach to detect the signature of anisotropic segmental dynamics by quantifying cross-correlated spin relaxation of individual 15 N 1 H N and 13 C ' 13 C α spin pairs. By adapting Geoffrey Bodenhausen's symmetrical reconversion principle to obtain zero frequency spectral density values, we can define and demonstrate more sensitive means to characterize anisotropic dynamics in IDPs.

Structural Dynamics from NMR Relaxation by SRLS Analysis: Local Geometry, Potential Energy Landscapes, and Spectral Densities

We have developed the two-body coupled-rotator slowly relaxing local structure (SRLS) approach for elucidating protein dynamics by nuclear magnetic resonance (NMR) relaxation. The rotators are represented by diffusion tensors D₁ for overall protein tumbling and D₂ for locally ordered probe motion. D₁ and D₂ are coupled dynamically by a potential, u, typically given by linear combinations of the Wigner functions D₀₀² and (D₀₂² + D_0-2²). Until now, our SRLS analyses provided the tensors, D₁ and D₂, the potential, u, and the geometric link between SRLS and NMR. Here we enhance this description by also examining the SRLS spectral densities obtained by solving the SRLS Smoluchowski equation. In addition, we show that the form of u specified above complies with two NMR-detected potential energy landscapes representing preferential ordering along N-H or C^α-C^α. Pictorial illustrations thereof are provided. The extended SRLS analysis is applied to ¹⁵N-H relaxation from the carbohydrate recognition domain of galectin-3 (Gal3C) in complex with two diastereomeric ligands, S and R. We find that D₂ is isotropic with a principal value, D₂, of 10¹⁰ s^-1 on average, and it is faster in the strands β₃, β₅, and β₈. The potential, u, is strong (∼20 kT); it is slightly rhombic when N-H is the main ordering axis and highly rhombic when C^α-C^α is the main ordering axis. Gal3C-S exhibits primarily preferential ordering along C^α-C^α; Gal3C-R exhibits both types of ordering. The binding-associated polypeptide chain segment of Gal3C-S is homogeneous, whereas that of Gal3C-R is diversified, with regard to D₂ and ordering preference. We associate these features with the previously determined diminished binding constant of Gal3C-R in comparison with Gal3C-S. Thus, the present study enhances the SRLS analysis, in general, and provides new insights into the dynamic structure and binding properties of Gal3C-S and Gal3C-R, in particular.

SRLS Analysis of 15N-1H NMR Relaxation from the Protein S100A1: Dynamic Structure, Calcium Binding, and Related Changes in Conformational Entropy

We report on amide (N-H) NMR relaxation from the protein S100A1 analyzed with the slowly relaxing local structure (SRLS) approach. S100A1 comprises two calcium-binding "EF-hands" (helix-loop-helix motifs) connected by a linker. The dynamic structure of this protein, in both calcium-free and calcium-bound form, is described as the restricted local N-H motion coupled to isotropic protein tumbling. The restrictions are given by a rhombic potential, u (∼10 kT), the local motion by a diffusion tensor with rate constant D₂ (∼10⁹ s^-1), and principal axis tilted from the N-H bond at angle β (10-20°). This parameter combination provides a physically insightful picture of the dynamic structure of S100A1 from the N-H bond perspective. Calcium binding primarily affects the C-terminal EF-hand, among others slowing down the motion of helices III and IV approximately 10-fold. Overall, it brings about significant changes in the shape of the local potential, u, and the orientation of the local diffusion axis, β. Conformational entropy derived from u makes an unfavorable entropic contribution to the free energy of calcium binding estimated at 8.6 ± 0.5 kJ/mol. The N-terminal EF-hand undergoes moderate changes. These findings provide new insights into the calcium-binding process. The same data were analyzed previously with the extended model-free (EMF) method, which is a simple limit of SRLS. In that interpretation, the protein tumbles anisotropically. Locally, calcium binding increases ordering in the loops of S100A1 and conformational exchange (R_ex) in the helices of its N-terminal EF-hand. These are very unusual features. We show that they most likely stem from problematic data-fitting, oversimplifications inherent in EMF, and experimental imperfections. R_ex is shown to be mainly a fit parameter. By reanalyzing the experimental data with SRLS, which is largely free of these deficiencies, we obtain-as delineated above-physically-relevant structural, kinetic, geometric, and binding information.

Conformational Entropy from Mobile Bond Vectors in Proteins: A Viewpoint that Unifies NMR Relaxation Theory and Molecular Dynamics Simulation Approaches

A new method for determining conformational entropy in proteins is reported. Proteins prevail as conformational ensembles, p ∝ exp(-u). By selecting a bond vector (e.g., N-H) as a conformation representative, molecular dynamics simulations can provide (relative to a reference structure) p as approximate Boltzmann probability density and u as N-H potential of mean force (POMF). The latter is as accurate as implied by the force field but statistical in character; this limits the insights it can provide and its utilization. Conformational entropy is given exclusively by u. Deriving it from POMFs renders it accurate but statistical in character. Previously, we devised explicit (i.e., analytical but not exact) potentials made of Wigner functions, D_0K^L, with L ≤ 4, which closely resemble the corresponding POMFs in form; hence, they also approach the latter in accuracy. Such potentials can be beneficially characterized/compared in terms of composition, symmetry, and associated order parameters. In this study, we develop a method for deriving conformational entropy from them, which also features the benefits specified above. The method developed is applied to the dimerization of the Rho GTPase-binding domain of plexin-B1. Insights into local ordering, entropy compensation, and features of allostery are gained. In previous work, we developed the slowly relaxing local structure (SRLS) approach for the analysis of NMR relaxation from restricted bond vector motion in proteins. SRLS comprises explicit (restricting) potentials of the kind developed here. It also comprises diffusion tensors describing the local motion and related features of local geometry. The complete model fits experimental data. In future work, the explicit potentials developed here will be inserted unchanged in SRLS-based data fitting, thereby improving the picture of structural dynamics. Given that SRLS is unique in featuring potentials that can closely approach the corresponding POMFs in accuracy, the present study is an important step toward generally improving protein dynamics by NMR relaxation.

Conformational Entropy from Restricted Bond-Vector Motion in Proteins: The Symmetry of the Local Restrictions and Relation to NMR Relaxation

Locally mobile bond-vectors contribute to the conformational entropy of the protein, given by S_k ≡ S/k = -∫(P_eq ln P_eq)dΩ - ln∫dΩ. The quantity P_eq = exp(-u)/Z is the orientational probability density, where Z is the partition function and u is the spatially restricting potential exerted by the immediate internal protein surroundings at the site of the motion of the bond-vector. It is appropriate to expand the potential, u, which restricts local rotational reorientation, in the basis set of the real combinations of the Wigner rotation matrix elements, D_0K^L. For small molecules dissolved in anisotropic media, one typically keeps the lowest even L, L = 2, nonpolar potential in axial or rhombic form. For bond-vectors anchored at the protein, the lowest odd L, L = 1, polar potential is to be used in axial or rhombic form. Here, we investigate the effect of the symmetry and polarity of these potentials on S_k. For L = 1 (L = 2), S_k is the same (differs) for parallel and perpendicular ordering. The plots of S_k as a function of the coefficients of the rhombic L = 1 (L = 2) potential exhibit high-symmetry (specific low-symmetry) patterns with parameter-range-dependent sensitivity. Similar statements apply to analogous plots of the potential minima. S_k is also examined as a function of the order parameters defined in terms of u. Graphs displaying these correlations, and applications illustrating their usage, are provided. The features delineated above are generally useful for devising orienting potentials that best suit given physical circumstances. They are particularly useful for bond-vectors acting as NMR relaxation probes in proteins, when their restricted local motion is analyzed with stochastic models featuring Wigner-function-made potentials. The relaxation probes could also be molecules adsorbed at surfaces, inserted into membranes, or interlocked within metal-organic frameworks.

Theoretical and computational framework for the analysis of the relaxation properties of arbitrary spin systems. Application to high-resolution relaxometry

A wide variety of nuclear magnetic resonance experiments rely on the prediction and analysis of relaxation processes. Recently, innovative approaches have been introduced where the sample travels through a broad range of magnetic fields in the course of the experiment, such as dissolution dynamic nuclear polarization or high-resolution relaxometry. Understanding the relaxation properties of nuclear spin systems over orders of magnitude of magnetic fields is essential to rationalize the results of these experiments. For example, during a high-resolution relaxometry experiment, the absence of control of nuclear spin relaxation pathways during the sample transfers and relaxation delays leads to systematic deviations of polarization decays from an ideal mono-exponential decay with the pure longitudinal relaxation rate. These deviations have to be taken into account to describe quantitatively the dynamics of the system. Here, we present computational tools to (1) calculate analytical expressions of relaxation rates for a broad variety of spin systems and (2) use these analytical expressions to correct the deviations arising in high-resolution relaxometry experiments. These tools lead to a better understanding of nuclear spin relaxation, which is required to improve the sensitivity of many pulse sequences, and to better characterize motions in macromolecules.

Local Ordering at the N-H Sites of the Rho GTPase Binding Domain of Plexin-B1: Impact of Dimerization

We have developed a new molecular dynamics (MD) based method for describing analytically local potentials at mobile N-H sites in proteins. Here we apply it to the monomer and dimer of the Rho GTPase binding domain (RBD) of the transmembrane receptor plexin-B1 to gain insight into dimerization, which can compete with Rho GTPase binding. In our method, the local potential is given by linear combinations, u^(D_L,K), of the real combinations of the Wigner rotation matrix elements, D_L,K, with L = 1-4 and appropriate symmetry. The combination that "fits best" the corresponding MD potential of mean force, u^(MD), is the potential we are seeking, u^{(D_L,K - BEST)}. For practical reasons the fitting process involves probability distributions, P_eq ∝ exp(-u), instead of potentials, u. The symmetry of the potential, u^(D_L,K), may be related to the irreducible representations of the D_2h point group. The monomer (dimer) potentials have mostly A_g and B_2u (B_1u and B_2u) symmetry. For the monomer, the associated probability distributions are generally dispersed in space, shallow, and centered at the "reference N-H orientation" (defined in section 3.1. below); for the dimer many are more concentrated, deep and centered away from the "reference N-H orientation". The u^(D_L,K) functions provide a consistent description of the potential energy landscape at protein N-H sites. The L1-loop of the plexin-B1 RBD is not seen in the crystal structure, and many resonances of the L4 loop are missing in the NMR ¹⁵N-¹H HSQC spectrum of the dimer; we suggest reasons for these features. An allosteric signal transmission pathway was reported previously for the monomer. We find that it has shallow N-H potentials at its ends, which become deeper as one proceeds toward the middle, complementing structurally the previously derived dynamic picture. Prospects of this study include correlating u^{(D_L,K - BEST)} with MD force-fields, and using them without further adjustment in NMR relaxation analysis schemes.

Local ordering and dynamics in anisotropic media by magnetic resonance: from liquid crystals to proteins

Magnetic resonance methods have been used extensively for over 50 years to elucidate molecular structure and dynamics of liquid crystals (LCs), providing information quite unique in its rigour and extent. The ESR- or NMR-active probe is often a solute molecule reporting on characteristics associated with the surrounding (LC) medium, which exerts the spatial restrictions on the probe. The theoretical approaches developed for LCs are applicable to anisotropic media in general. Of particular interest is the interior space of a globular protein labelled, e.g. with a nitroxide moiety or a ¹⁵N-¹H bond. The ESR or NMR label plays the role of the probe and the internal protein surroundings the role of the anisotropic medium. A general feature of the restricted motions is the local ordering, i.e. the nature, magnitude and symmetry of the spatial restraints exerted at the site of the moving probe. This property is the main theme of the present review article. We outline its treatment in our work from both the theoretical and the experimental points of view, highlighting the new physical insights gained. Our illustrations include studies on thermotropic (nematic and smectic) and lyotropic liquid crystals formed by phospholipids, in addition to studies of proteins.

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.

An NMR Confirmation for Increased Folded State Entropy Following Loop Truncation

Previous studies conducted on flexible loop regions in proteins revealed that the energetic consequences of changing loop length predominantly arise from the entropic cost of ordering a loop during folding. However, in an earlier study of human acylphosphatase (hmAcP) using experimental and computational approaches, we showed that thermodynamic stabilization upon loop truncation can be attributed mainly to the increased entropy of the folded state. Here, using 15N NMR spectroscopy, we studied the effect of loop truncation on hmAcP backbone dynamics on the picosecond-nanosecond timescale with the aim of confirming the effect of folded state entropy on protein stability. NMR-relaxation-derived N-H squared generalized order parameters reveal that loop truncation results in a significant increase in protein conformational flexibility. Comparison of these results with previously acquired all-atom molecular dynamics simulation, analyzed here in terms of squared generalized NMR order parameters, demonstrates general agreement between the two methods. The NMR study not only provides direct evidence for the enhanced conformational entropy of the folded state of hmAcP upon loop truncation but also gives a quantitative measure of the observed effects.

Determination of Protein ps-ns Motions by High-Resolution Relaxometry

Many of the functions of biomacromolecules can be rationalized by the characterization of their conformational energy landscapes: the structures of the dominant states, transitions between states and motions within states. Nuclear magnetic resonance (NMR) spectroscopy is the technique of choice to study internal motions in proteins. The determination of motions on picosecond to nanosecond timescales requires the measurement of nuclear spin relaxation rates at multiple magnetic fields. High sensitivity and resolution are obtained only at high magnetic fields, so that, until recently, site-specific relaxation rates in biomolecules were only measured over a narrow range of high magnetic fields. This limitation was particularly striking for the quantification of motions on nanosecond timescales, close to the correlation time for overall rotational diffusion. High-resolution relaxometry is an emerging technique to investigate picosecond-nanosecond motions of proteins. This approach uses a high-field NMR spectrometer equipped with a sample shuttle device, which allows for the measurement of the relaxation rate constants at low magnetic fields, while preserving the sensitivity and resolution of a high-field NMR spectrometer. The combined analysis of high-resolution relaxometry and standard high-field relaxation data provides a more accurate description of the dynamics of proteins, in particular in the nanosecond range. The purpose of this chapter is to describe how to perform high-resolution relaxometry experiments and how to analyze the rates measured with this technique.

15N-H-Related Conformational Entropy Changes Entailed By Plexin-B1 RBD Dimerization: Combined Molecular Dynamics/NMR Relaxation Approach

We report on a new method for determining function-related conformational entropy changes in proteins. Plexin-B1 RBD dimerization serves as example, and internally mobile N-H bonds serve as probes. S_k (entropy in units of k_BT) is given by -∫(P_eqlnP_eq)dΩ, where P_eq = exp(-u) is the probability density for probe orientation, and u the local potential. Previous slowly relaxing local structure (SRLS) analyses of ¹⁵N-H relaxation in proteins determined linear combinations of D₀₀²(Ω) and (D₀₂²(Ω) + D_0-2²(Ω)) (D_0K^L(Ω) represents a Wigner rotation matrix element in uniaxial local medium) as "best-fit" form of u. SRLS also determined the "best-fit" orientation of the related ordering tensor. On the basis of this information the coefficients (in the linear combination) of the terms specified above are determined with molecular dynamics (MD) simulations. With the explicit expression for u thus in hand, S_k is calculated. We find that in general S_k decreases, i.e., the local order increases, upon plexin-B1 RBD dimerization. The largest decrease in S_k occurs in the helices α₁ and α₂, followed by the α₂/β₆ turn. Only the relatively small peripheral β₂ strand, β₂/α₁ turn, and L3 loop become more disordered. That α-helices dominate ΔS_k = S_k(dimer) - S_k(monomer), a few peripheral outliers partly counterbalance the overall decrease in S_k, and the probability density function, P_eq, has rhombic symmetry given that the underlying potential function, u, has rhombic symmetry, are interesting features. We also derive S² (the proxy of u in the simple "model-free (MF)" limit of SRLS) with MD. Its conversion into a potential requires assumptions and adopting a simple axial form of u. Ensuing ΔS_k(MF) profiles are u-dependent and differ from ΔS_k(SRLS). A method that provides consistent, general, and accurate S_k, atomistic/mesoscopic in nature, has been developed. Its ability to provide new insights in protein research has been illustrated.

Coupling and Decoupling of Rotational and Translational Diffusion of Proteins under Crowding Conditions

Molecular motion of biopolymers in vivo is known to be strongly influenced by the high concentration of organic matter inside cells, usually referred to as crowding conditions. To elucidate the effect of intermolecular interactions on Brownian motion of proteins, we performed (1)H pulsed-field gradient NMR and fluorescence correlation spectroscopy (FCS) experiments combined with small-angle X-ray scattering (SAXS) and viscosity measurements for three proteins, αB-crystalline (αBc), bovine serum albumin, and hen egg-white lysozyme (HEWL) in aqueous solution. Our results demonstrate that long-time translational diffusion quantitatively follows the expected increase of macro-viscosity upon increasing the protein concentration in all cases, while rotational diffusion as assessed by polarized FCS and previous multi-frequency (1)H NMR relaxometry experiments reveals protein-specific behavior spanning the full range between the limiting cases of full decoupling from (αBc) and full coupling to (HEWL) the macro-viscosity. SAXS was used to study the interactions between the proteins in solution, whereby it is shown that the three cases cover the range between a weakly interacting hard-sphere system (αBc) and screened Coulomb repulsion combined with short-range attraction (HEWL). Our results, as well as insights from the recent literature, suggest that the unusual rotational-translational coupling may be due to anisotropic interactions originating from hydrodynamic shape effects combined with high charge and possibly a patchy charge distribution.

Local Ordering at Mobile Sites in Proteins from Nuclear Magnetic Resonance Relaxation: The Role of Site Symmetry

Restricted motions in proteins (e.g., N-H bond dynamics) are studied effectively with NMR. By analogy with restricted motions in liquid crystals (LC), the local ordering has in the past been primarily represented by potentials comprising the L = 2, |K| = 0, 2 spherical harmonics. However, probes dissolved in LCs experience nonpolar ordering, often referred to as alignment, while protein-anchored probes experience polar ordering, often referred to as orientation. In this study we investigate the role of local (site) symmetry in the context of the polarity of the local ordering. We find that potentials comprising the L = 1, |K| = 0, 1 spherical harmonics represent adequately polar ordering. It is useful to characterize potential symmetry in terms of the irreducible representations of D2h point group, which is already implicit in the definition of the rotational diffusion tensor. Thus, the relevant rhombic L = 1 potentials have B1u and B3u symmetry whereas the relevant rhombic L = 2 potentials have Ag symmetry. A comprehensive scheme where local potentials and corresponding probability density functions (PDFs) are represented in Cartesian and spherical coordinates clarifies how they are affected by polar and nonpolar ordering. The Cartesian coordinates are chosen so that the principal axis of polar axial PDF is pointing along the z-axis, whereas the principal axis of the nonpolar axial PDF is pointing along ±z. Two-term axial potentials with 1 ≤ L ≤ 3 exhibit substantial diversity; they are expected to be useful in NMR-relaxation-data-fitting. It is shown how potential coefficients are reflected in the experimental order parameters. The comprehensive scheme representing local potentials and PDFs is exemplified for the L = 2 case using experimental data from (15)N-labeled plexin-B1 and thioredoxin, (2)H-, and (13)C-labeled benzenehexa-n-alkanoates, and nitroxide-labeled T4 lysozyme. Future prospects for improved ordering analysis based on combined atomistic and mesoscopic approaches are delineated.

Polar Versus Non-polar Local Ordering at Mobile Sites in Proteins: Slowly Relaxing Local Structure Analysis of (15)N Relaxation in the Third Immunoglobulin-Binding Domain of Streptococcal Protein G

We developed recently the slowly relaxing local structure (SRLS) approach for studying restricted motions in proteins by NMR. The spatial restrictions have been described by potentials comprising the traditional L = 2, K = 0, 2 spherical harmonics. However, the latter are associated with non-polar ordering whereas protein-anchored probes experience polar ordering, described by odd-L spherical harmonics. Here we extend the SRLS potential to include the L = 1, K = 0, 1 spherical harmonics and analyze (15)N-(1)H relaxation from the third immunoglobulin-binding domain of streptococcal protein G (GB3) with the polar L = 1 potential (coefficients c0(1) and c1(1)) or the non-polar L = 2 potential (coefficients c0(2) and c2(2)). Strong potentials, with ⟨c0(1)⟩ ∼ 60 for L = 1 and ⟨c0(2)⟩ ∼ 20 for L = 2 (in units of kBT), are detected. In the α-helix of GB3 the coefficients of the rhombic terms are c1(1) ∼ c2(2) ∼ 0; in the preceding (following) chain segment they are ⟨c1(1)⟩ ∼ 6 for L = 1 and ⟨c2(2)⟩ ∼ 14 for L = 2 (⟨c1(1)⟩ ∼ 3 for L = 1 and ⟨c2(2)⟩ ∼ 7 for L = 2). The local diffusion rate, D2, lies in the 5 × 10(9)-1 × 10(11) s(-1) range; it is generally larger for L = 1. The main ordering axis deviates moderately from the N-H bond. Corresponding L = 1 and L = 2 potentials and probability density functions are illustrated for residues A26 of the α-helix, Y3 of the β1-strand, and L12 of the β1/β2 loop; they differ considerably. Polar/orientational ordering is shown to be associated with GB3 binding to its cognate Fab fragment. The polarity of the local ordering is clearly an important factor.

Validation of Molecular Dynamics Simulations of Biomolecules Using NMR Spin Relaxation as Benchmarks: Application to the AMBER99SB Force Field

Biological function of biomolecules is accompanied by a wide range of motional behavior. Accurate modeling of dynamics by molecular dynamics (MD) computer simulations is therefore a useful approach toward the understanding of biomolecular function. NMR spin relaxation measurements provide rigorous benchmarks for assessing important aspects of MD simulations, such as the amount and time scales of conformational space sampling, which are intimately related to the underlying molecular mechanics force field. Until recently, most simulations produced trajectories that exhibited too much dynamics particularly in flexible loop regions. Recent modifications made to the backbone φ and ψ torsion angle potentials of the AMBER and CHARMM force fields indicate that these changes produce more realistic molecular dynamics behavior. To assess the consequences of these changes, we performed a series of 5-20 ns molecular dynamics trajectories of human ubiquitin using the AMBER99 and AMBER99SB force fields for different conditions and water models and compare the results with NMR experimental backbone N-H S(2) order parameters. A quantitative analysis of the trajectories shows significantly improved agreement with experimental NMR data for the AMBER99SB force field as compared to AMBER99. Because NMR spin relaxation data (T1, T2, NOE) reflect the combined effects of spatial and temporal fluctuations of bond vectors, it is found that comparison of experimental and back-calculated NMR spin-relaxation data provides a more objective way of assessing the quality of the trajectory than order parameters alone. Analysis of a key mobile β-hairpin in ubiquitin demonstrates that the dynamics of mobile sites are not only reduced by the modified force field, but the extent of motional correlations between amino acids is also markedly diminished.

Protein Dynamics in the Solid State from (2)H NMR Line Shape Analysis. II. MOMD Applied to C-D and C-CD3 Probes

Deuterium line shape analysis from mobile C–D and C–CD₃ groups has emerged as a particularly useful tool for studying dynamics in the solid state. The theoretical models devised so far consist typically of sets of independent dynamic modes. Each such mode is simple and usually case-specific. In this scenario, model improvement entails adding yet another mode (thereby changing the overall model), comparison of different cases is difficult, and ambiguity is unavoidable. We recently developed the microscopic order macroscopic disorder (MOMD) approach as a single-mode alternative. In MOMD, the local spatial restrictions are expressed by an anisotropic potential, the local motion by a diffusion tensor, and the local molecular geometry by relative (magnetic and model-related) tensor orientations, all of adjustable symmetry. This approach provides a consistent method of analysis, thus resolving the issues above. In this study, we apply MOMD to PS-adsorbed LKα14 peptide and dimethylammonium tetraphenylborate (C–CD₃ and N–CD₃ dynamics, respectively), as well as HhaI methyltransferase target DNA and phase III of benzene-6-hexanoate (C–D dynamics). The success with fitting these four disparate cases, as well as the two cases in the previous report, demonstrates the generality of this MOMD-based approach. In this study, C–D and C–CD₃ are both found to execute axial diffusion (rates R_⊥ and R_∥) in the presence of a rhombic potential given by the L = 2 spherical harmonics (coefficients c₀² and c₂²). R_⊥ (R_∥) is in the 10²–10³ (10⁴–10⁵) s^–1 range, and c₀² and c₂² are on the order of 2–3 k_BT. Specific parameter values are determined for each mobile site. The diffusion and quadrupolar tensors are tilted at either 120° (consistent with trans–gauche isomerization) or nearly 110.5° (consistent with methyl exchange). Future prospects include extension of the MOMD formalism to include MAS, and application to ¹⁵N and ¹³C nuclei.

Coupling between overall rotational diffusion and domain motions in proteins and its effect on dielectric spectra

In this work, we formulate a closed-form solution of the model of a semirigid molecule for the case of fluctuating and reorienting molecular electric dipole moment. We illustrate with numeric calculations the impact of protein domain motions on dielectric spectra using the example of the 128 kDa protein dimer of Enzyme I. We demonstrate that the most drastic effect occurs for situations when the characteristic time of protein domain dynamics is comparable to the time of overall molecular rotational diffusion. We suggest that protein domain motions could be a possible explanation for the high-frequency contribution that accompanies the major relaxation dispersion peak in the dielectric spectra of protein aqueous solutions. We propose that the presented computational methodology could be used for the simultaneous analysis of dielectric spectroscopy and nuclear magnetic resonance data. Proteins 2015; 83:1571-1581. © 2015 Wiley Periodicals, Inc.

Protein dynamics in the solid state from 2H NMR line shape analysis: a consistent perspective

Deuterium line shape analysis of CD3 groups has emerged as a particularly useful tool for studying microsecond-millisecond protein motions in the solid state. The models devised so far consist of several independently conceived simple jump-type motions. They are comprised of physical quantities encoded in their simplest form; improvements are only possible by adding yet another simple motion, thereby changing the model. The various treatments developed are case-specific; hence comparison among the different systems is not possible. Here we develop a new methodology for (2)H NMR line shape analysis free of these limitations. It is based on the microscopic-order-macroscopic-disorder (MOMD) approach. In MOMD motions are described by diffusion tensors, spatial restrictions by potentials/ordering tensors, and geometric features by relative tensor orientations. Jump-type motions are recovered in the limit of large orientational potentials. Model improvement is accomplished by monitoring the magnitude, symmetry, and orientation of the various tensors. The generality of MOMD makes possible comparison among different scenarios. CD3 line shapes from the Chicken Villin Headpiece Subdomain and the Streptomyces Subtilisin Inhibitor are used as experimental examples. All of these spectra are reproduced by using rhombic local potentials constrained for simplicity to be given by the L = 2 spherical harmonics, and by axial diffusion tensors. Potential strength and rhombicity are found to be ca. 2-3 k(B)T. The diffusion tensor is tilted at 120° from the C-CD3 axis. The perpendicular (parallel) correlation times for local motion are 0.1-1.0 ms (3.3-30 μs). Activation energies in the 1.1-8.0 kcal/mol range are estimated. Future prospects include extension to the (2)H relaxation limit, application to the (15)N and (13)C NMR nuclei, and accounting for collective motions and anisotropic media.

Investigation of glycofullerene dynamics by NMR spectroscopy

Mannose residues linked to flexible spacers on slowly diffusing glycofullerenes may facilitate efficient rebinding to receptors.

High-field EPR on membrane proteins - crossing the gap to NMR

In this review on advanced EPR spectroscopy, which addresses both the EPR and NMR communities, considerable emphasis is put on delineating the complementarity of NMR and EPR concerning the measurement of molecular interactions in large biomolecules. From these interactions, detailed information can be revealed on structure and dynamics of macromolecules embedded in solution- or solid-state environments. New developments in pulsed microwave and sweepable cryomagnet technology as well as ultrafast electronics for signal data handling and processing have pushed to new horizons the limits of EPR spectroscopy and its multifrequency extensions concerning the sensitivity of detection, the selectivity with respect to interactions, and the resolution in frequency and time domains. One of the most important advances has been the extension of EPR to high magnetic fields and microwave frequencies, very much in analogy to what happens in NMR. This is exemplified by referring to ongoing efforts for signal enhancement in both NMR and EPR double-resonance techniques by exploiting dynamic nuclear or electron spin polarization via unpaired electron spins and their electron-nuclear or electron-electron interactions. Signal and resolution enhancements are particularly spectacular for double-resonance techniques such as ENDOR and PELDOR at high magnetic fields. They provide greatly improved orientational selection for disordered samples that approaches single-crystal resolution at canonical g-tensor orientations - even for molecules with small g-anisotropies. Exchange of experience between the EPR and NMR communities allows for handling polarization and resolution improvement strategies in an optimal manner. Consequently, a dramatic improvement of EPR detection sensitivity could be achieved, even for short-lived paramagnetic reaction intermediates. Unique structural and dynamic information is thus revealed that can hardly be obtained by any other analytical techniques. Micromolar quantities of sample molecules have become sufficient to characterize stable and transient reaction intermediates of complex molecular systems - offering highly interesting applications for chemists, biochemists and molecular biologists. In three case studies, representative examples of advanced EPR spectroscopy are reviewed: (I) High-field PELDOR and ENDOR structure determination of cation-anion radical pairs in reaction centers from photosynthetic purple bacteria and cyanobacteria (Photosystem I); (II) High-field ENDOR and ELDOR-detected NMR spectroscopy on the oxygen-evolving complex of Photosystem II; and (III) High-field electron dipolar spectroscopy on nitroxide spin-labelled bacteriorhodopsin for structure-function studies. An extended conclusion with an outlook to further developments and applications is also presented.

The time correlation function perspective of NMR relaxation in proteins

We applied over a decade ago the two-body coupled-rotator slowly relaxing local structure (SRLS) approach to NMR relaxation in proteins. One rotator is the globally moving protein and the other rotator is the locally moving probe (spin-bearing moiety, typically the (15)N-(1)H bond). So far we applied SRLS to (15)N-H relaxation from seven different proteins within the scope of the commonly used data-fitting paradigm. Here, we solve the SRLS Smoluchowski equation using typical best-fit parameters as input, to obtain the corresponding generic time correlation functions (TCFs). The following new information is obtained. For actual rhombic local ordering and main ordering axis pointing along C(i-1)(α)-C(i)(α), the measurable TCF is dominated by the (K,K') = (-2,2), (2,2), and (0,2) components (K is the order of the rank 2 local ordering tensor), determined largely by the local motion. Global diffusion axiality affects the analysis significantly when the ratio between the parallel and perpendicular components exceeds approximately 1.5. Local diffusion axiality has a large and intricate effect on the analysis. Mode-coupling becomes important when the ratio between the global and local motional rates falls below 0.01. The traditional method of analysis--model-free (MF)--represents a simple limit of SRLS. The conditions under which the MF and SRLS TCFs are the same are specified. The validity ranges of wobble-in-a-cone and rotation on the surface of a cone as local motions are determined. The evolution of the intricate Smoluchowski operator from the simple diffusion operator for a sphere reorienting in isotropic medium is delineated. This highlights the fact that SRLS is an extension of the established stochastic theories for treating restricted motions. This study lays the groundwork for TCF-based comparison between mesoscopic SRLS and atomistic molecular dynamics.

Analysis of 15N-1H NMR relaxation in proteins by a combined experimental and molecular dynamics simulation approach: picosecond-nanosecond dynamics of the Rho GTPase binding domain of plexin-B1 in the dimeric state indicates allosteric pathways

We investigate picosecond–nanosecond dynamics of the Rho-GTPase Binding Domain (RBD) of plexin-B1, which plays a key role in plexin-mediated cell signaling. Backbone 15N relaxation data of the dimeric RBD are analyzed with the model-free (MF) method, and with the slowly relaxing local structure/molecular dynamics (SRLS-MD) approach. Independent analysis of the MD trajectories, based on the MF paradigm, is also carried out. MF is a widely popular and simple method, SRLS is a general approach, and SRLS-MD is an integrated approach we developed recently. Corresponding parameters from the RBD dimer, a previously studied RBD monomer mutant, and the previously studied complex of the latter with the GTPase Rac1, are compared. The L2, L3, and L4 loops of the plexin-B1 RBD are involved in interactions with other plexin domains, GTPase binding, and RBD dimerization, respectively. Peptide groups in the loops of both the monomeric and dimeric RBD are found to experience weak and moderately asymmetric local ordering centered approximately at the C(i–1)(α)–C(i)(α) axes, and nanosecond backbone motion. Peptide groups in the α-helices and the β-strands of the dimer (the β-strands of the monomer) experience strong and highly asymmetric local ordering centered approximately at the C(i–1)(α)–C(i)(α) axes (N–H bonds). N–H fluctuations occur on the picosecond time scale. An allosteric pathway for GTPase binding, providing new insights into plexin function, is delineated.

Stochastic modeling of flexible biomolecules applied to NMR relaxation. 2. Interpretation of complex dynamics in linear oligosaccharides

A computational stochastic approach is applied to the description of flexible molecules. By combining (i) molecular dynamics simulations, (ii) hydrodynamics approaches, and (iii) a multidimensional diffusive description for internal and global dynamics, it is possible to build an efficient integrated approach to the interpretation of relaxation processes in flexible systems. In particular, the model is applied to the interpretation of nuclear magnetic relaxation measurements of linear oligosaccharides, namely a mannose-containing trisaccharide and the pentasaccharide LNF-1. Experimental data are reproduced with sufficient accuracy without free model parameters.

Stochastic modeling of flexible biomolecules applied to NMR relaxation. I. Internal dynamics of cyclodextrins: γ-cyclodextrin as a case study

In this work, we address the description of the dynamics of cyclodextrins in relation with nuclear magnetic resonance (NMR) relaxation data collected for hydroxymethyl groups. We define an integrated computational approach based on the definition and parametrization of a stochastic equation able to describe the relevant degrees of freedom affecting the NMR observables. The computational protocol merges molecular dynamics simulations and hydrodynamics approaches for the evaluation of most of the molecular parameters entering the stochastic description of the system. We apply the method to the interpretation of the (13)C NMR relaxation of the -CH(2)OH group of cyclodextrins. We use γ-cyclodextrin as a case study. Results are in agreement with quantitative and qualitative analyses performed in the past with simpler models and molecular dynamics simulations. The element of novelty in our approach is in the treatment of the coupling of the relevant internal (glucopyranose ring twisting/tilting and hydroxymethyl group jumps) and global (molecular tumbling) degrees of freedom.

Sensitivity of 2H NMR spectroscopy to motional models: proteins and highly viscous liquids as examples

In order to study to what extent mechanisms of molecular motion can be unambiguously revealed by (2)H NMR spectroscopy, (2)H spectra for proteins (chicken villin protein headpiece HP36, selectively methyl-deuterated at leucine-69, C(δ) D(3)) and binary systems of high viscosity (benzene-d(6) in tricresyl phosphate) have been carefully analyzed as illustrative examples (the spectra are taken from the literature). In the first case, a model of restricted diffusion mediated by jumps between rotameric orientations has been tested against jump- and free diffusion models which describe rotational motion combined with jump dynamics. It has been found that the set of (2)H spectra of methyl-deuterated at leucine-69 chicken villin protein headpiece HP36 can be consistently explained by different motional models as well as by a gaussian distribution of correlation times assuming isotropic rotation (simple brownian diffusion model). The last finding shows that when the possible distribution of correlation times is not very broad one might not be able to distinguish between heterogeneous and homogenous (but more complex) dynamics by analyzing (2)H lineshapes. For benzene-d(6) in tricresyl phosphate, the dynamics is heterogeneous and it has been demonstrated that a gaussian distribution of correlation times reproduces well the experimental lineshapes, while for a Cole-Davidson distribution the agreement is somewhat worse. For inquires into the sensitivity of quadrupolar NMR spectral analysis (by "quadrupolar NMR spectroscopy we understand NMR spectroscopy of nuclei possessing quadrupole moment), the recently presented theoretical approach [Kruk et al., J. Chem. Phys. 135, 224511 (2011)] has been used as it allows simulating quadrupolar spectra for arbitrary motional conditions by employing the stochastic Liouville equation.

SRLS analysis of 15N relaxation from bacteriophage T4 lysozyme: a tensorial perspective that features domain motion

Bacteriophage T4L lysozyme (T4L) comprises two domains connected by a helical linker. Several methods detected ns domain motion associated with the binding of the peptidoglycan substrate. An ESR study of nitroxide-labeled T4L, based on the slowly relaxing local structure (SRLS) approach, detected ns local motion involving the nitroxide and the helix housing it. (15)N−H spin relaxation data from T4L acquired at magnetic fields of 11.7 and 18.8 T, and 298 K, were analyzed previously with the model-free (MF) method. The results did not detect domain motion. SRLS is the generalization of MF. Here, we apply it to the same data analyzed previously with MF. The restricted local N−H motion is described in terms of tilted axial local ordering (S) and local diffusion (D(2)) tensors; dynamical coupling to the global tumbling is accounted for. We find that D(2,⊥) is 1.62 × 10(7) (1.56 × 10(7)) s(−1) for the N-terminal (C-terminal) domain. This dynamic mode represents domain motion. For the linker D(2,⊥) is the same as the rate of global tumbling, given by (1.46 ± 0.04) × 10(7) s(−1). D(2,∥) is 1.3 × 10(9), 1.8 × 10(9) and 5.3 × 10(9) s(−1) for the N-terminal domain, the C-terminal domain, and the linker, respectively. This dynamic mode represents N−H bond vector fluctuations. The principal axis of D(2) is virtually parallel to the N−H bond. The order parameter, S(0)(2), is 0.910 ± 0.046 for most N−H bonds. The principal axis of S is tilted from the C(i−1)(α) −C(i)(α) axis by −2° to 6° for the N-, and C-terminal domains, and by 2.5° for the linker. The tensorial-perspective-based and mode-coupling-based SRLS picture provides new insights into the structural dynamics of bacteriophage T4 lysozyme.

Standard tensorial analysis of local ordering in proteins from residual dipolar couplings

Residual dipolar couplings (RDCs) in proteins arise from independent external medium-related and internal protein-related ordering of the spin-bearing probe. Griesinger et al. developed a method for treating RDCs in proteins. The global ordering is given in the standard manner by a rank 2 tensor specified in a known molecular frame, MF. The local ordering is described by the spherical harmonic ensemble averages, <Y(2m)(θ, φ)>, m = 0, ±1, ±2, also given in MF. From these quantities, a method we call mf-RDC derives the squared generalized order parameter (S(rdc)(2)), the amplitude (direction) of the anisotropic disorder, η (Φ′), and an approximation, (N−H)(eff), to the average probe orientation, i.e., to the local director. (N−H)(eff) is determined through a frame transformation where <Y(20)> is maximized. Φ′ is associated with a subsequent frame transformation where <Y(22) + Y(2−2)> is maximized. The mf-RDC method was applied previously to N−H and C−C(methyl) sites in ubiquitin. In this study, we convert the respective <Y(2m)(θ, φ)>'s into a Saupe tensor, which is diagonalized. This is the standard procedure. It yields the eigenvalues, S(xx), S(yy), and S(zz), and the Principal Axis System (PAS) of the rank 2 local ordering tensor, S(l). S(rdc)(2), η, and Φ′ can be recast as S(xx), S(yy), and S(zz). The mf-RDC frame transformations are not the same as the conventional Wigner rotation. The standard tensorial analysis provides new information. The contribution of local ordering rhombicity to S(rdc)(2) is evaluated. For the α-helix of ubiquitin, the main local ordering axis is assigned as C(i−1)(α) − C(i)(α); for the methyl sites, it is associated with the C−C(methyl) axis, as in mf-RDC. Ordering strength correlates with methyl type. The strength (rhombicity) of S(l) associated with picosecond−nanosecond local motions is reduced moderately (substantially) by nanosecond−millisecond local motions. A scheme for analyzing experimental RDCs based on the standard tensorial perspective, which allows for arbitrary orientation of the local director in the protein and of the PAS of S(l) in the probe, is formulated.

Coupling between internal dynamics and rotational diffusion in the presence of exchange between discrete molecular conformations

We present a general formalism for the computation of orientation correlation functions involving a molecular system undergoing rotational diffusion in the presence of transitions between discrete conformational states. In this formalism, there are no proscriptions on the time scales of conformational rearrangement relative to that for rotational diffusion, and the rotational diffusion tensors of the different states can be completely arbitrary. Although closed-form results are limited to the frequency domain, this is generally useful for many spectroscopic observables as the result allows the computation of the spectral density function. We specialize the results for the computation of the frequency-domain correlation function associated with the NMR relaxation.

SRLS analysis of 15N spin relaxation from E. coli ribonuclease HI: the tensorial perspective

15N–H relaxation parameters from ribonuclease HI (RNase H), acquired in previous work at magnetic fields of 14.1 and 18.8 T, and at 300 K, are analyzed with the mode-coupling slowly relaxing local structure (SRLS) approach. In accordance with standard theoretical treatments of restricted motions, SRLS approaches N-H bond dynamics from a tensorial perspective. As shown previously, a physically adequate description of this phenomenon has to account for the asymmetry of the local spatial restrictions. So far, we used rhombic local ordering tensors; this is straightforward but computationally demanding. Here, we propose substantiating the asymmetry of the local spatial restrictions in terms of tilted axial local ordering (S) and local diffusion (D2) tensors. Although less straightforward, this description provides physically sound structural and dynamic information and is efficient computationally. We find that the local order parameter, S(0)2, is on average 0.89 (0.84, and may be as small as 0.6) for the secondary structure elements (loops). The main local ordering axis deviates from the C(i-1)α-C(i)α axis by less than 6°. At 300 K, D(2,perpendicular) is virtually the same as the global diffusion rate, D1 = 1.8 × 10(7) s(-1). The correlation time 1/6D(2,parallel) ranges from 3-125 (208-344) ps for the secondary structure elements (loops) and is on average 125 ps for the C-terminal segment. The main local diffusion axis deviates from the N-H bond by less than 2° (10°) for the secondary structure elements (loops). An effective data-fitting protocol, which leads in most cases to unambiguous results with limited uncertainty, has been devised. A physically sound and computationally effective methodology for analyzing 15N relaxation in proteins, that provides a new picture of N–H bond structural dynamics in proteins, has been set forth.

NMR order parameters calculated in an expanding reference frame: identifying sites of short- and long-range motion

NMR order parameters are calculated from molecular dynamics computer simulations of ubiquitin and the apo (Ca(2+)-free) state of calbindin D(9k). Calculations are performed in an expanding reference frame so as to discriminate between the effects of short- and long-range motions. This approach reveals that the dominant contributions to the order parameters are short-range. Longer-range contributions are limited to specific sites, many of which have been recognized in previous studies of correlated motions. These sites are identified on the basis of an effective reorientational number, n ( eff ). Not only does this parameter identify sites of short- and long-range motion, it also provides a way of evaluating the separability condition that is key to the Lipari-Szabo model-free method. When analyzed in conjunction with the Prompers-Brüschweiler separability index, the n ( eff ) values indicate that longer-range motions play a more prominent role in apo calbindin than they do in ubiquitin.

Integrated computational approach to the analysis of NMR relaxation in proteins: application to ps-ns main chain 15N-1H and global dynamics of the Rho GTPase binding domain of plexin-B1

An integrated computational methodology for interpreting NMR spin relaxation in proteins has been developed. It combines a two-body coupled-rotator stochastic model with a hydrodynamics-based approach for protein diffusion, together with molecular dynamics based calculations for the evaluation of the coupling potential of mean force. The method is applied to ¹⁵N relaxation of N-H bonds in the Rho GTPase binding (RBD) domain of plexin-B1, which exhibits intricate internal mobility. Bond vector dynamics are characterized by a rhombic local ordering tensor, S, with principal values S₀² and S₂², and an axial local diffusion tensor, D₂, with principal values D(2,||) and D(2,⊥). For α-helices and β-sheets we find that S₀² ~ -0.5 (strong local ordering), -1.2 < S₂² < -0.8 (large S tensor anisotropy), D(2,⊥) ~ D₁ = 1.93 × 10⁷ s⁻¹ (D₁ is the global diffusion rate), and log(D(2,||)/D₁) ~ 4. For α-helices the z-axis of the local ordering frame is parallel to the C(α)-C(α) axis. For β-sheets the z-axes of the S and D₂ tensors are parallel to the N-H bond. For loops and terminal chain segments the local ordering is generally weaker and more isotropic. On average, D(2,⊥) ~ D₁ also, but log(D(2,||)/D₁) is on the order of 1-2. The tensor orientations are diversified. This study sets forth an integrated computational approach for treating NMR relaxation in proteins by combining stochastic modeling and molecular dynamics. The approach developed provides new insights by its application to a protein that experiences complex dynamics.