Mode-Coupling Analysis of 15N CSA−15N-1H Dipolar Cross-Correlation in Proteins. Rhombic Potentials at the N−H Bond

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.

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.

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.

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.

Backbone dynamics of deoxy and carbonmonoxy hemoglobin by NMR/SRLS

The slowly relaxing local structure (SRLS) approach, developed for NMR spin relaxation analysis in proteins, is applied herein to amide ¹⁵N relaxation in deoxy and carbonmonoxy hemoglobin. Experimental data including ¹⁵N T₁, T₂ and ¹⁵N-{¹H} NOE, acquired at 11.7 and 14.1 T, and 29 and 34 °C, are analyzed. The restricted local motion of the N-H bond is described in terms of the principal value (S(0)(2)) and orientation (β(D)) of an axial local ordering tensor, S, and the principal values (R(||)(L) and R(⊥)(L)) and orientation (β(O)) of an axial local diffusion tensor, R(L). The parameters c₀² (the potential coefficient in terms of which S(0)(2) is defined), R(||)(L), β(D), and β(O) are determined by data fitting; R(⊥)(L) is set equal to the global motional rate, R(C), found previously to be (5.2-5.8) × 10⁶ 1/s in the temperature range investigated. The principal axis of S is (nearly) parallel to the C(i-1)(α)-C(i)(α) axis; when the two axes are parallel, β(D) = -101.3° (in the frame used). The principal axis of R(L) is (nearly) parallel to the N-H bond; when the two axes are parallel, β(O) = -101.3°. For "rigid" N-H bonds located in secondary structure elements the best-fit parameters are S(0)(2) = 0.88-0.95 (corresponding to local potentials of 8.6-19.9 k(B)T), R(||)(L) = 10⁹-10¹⁰ 1/s, β(D) = -101.3° ± 2.0°, and β(O) = -101.3° ± 4°. For flexible N-H bonds located in loops the best-fit values are S(0)(2) = 0.75-0.80 (corresponding to local potentials of 4.5-5.5 k(B)T), R(||)(L) = (1.0-6.3) × 10⁸ 1/s, β(D) = -101.3° ± 4.0°, and β(O) = -101.3° ± 10°. These results are important in view of their physical clarity, inherent potential for further interpretation, consistency, and new qualitative insights provided (vide infra).

The physical basis of model-free analysis of NMR relaxation data from proteins and complex fluids

NMR relaxation experiments have provided a wealth of information about molecular motions in macromolecules and ordered fluids. Even though a rigorous theory of spin relaxation is available, the complexity of the investigated systems often makes the interpretation of limited datasets challenging and ambiguous. To allow physically meaningful information to be extracted from the data without commitment to detailed dynamical models, several versions of a model-free (MF) approach to data analysis have been developed. During the past 2 decades, the MF approach has been used in the vast majority of all NMR relaxation studies of internal motions in proteins and other macromolecules, and it has also played an important role in studies of colloidal systems. Although the MF approach has been almost universally adopted, substantial disagreement remains about its physical foundations and range of validity. It is our aim here to clarify these issues. To this end, we first present rigorous derivations of the three well-known MF formulas for the time correlation function relevant for isotropic solutions. These derivations are more general than the original ones, thereby substantially extending the range of validity of the MF approach. We point out several common misconceptions and explain the physical significance of the approximations involved. In particular, we discuss symmetry requirements and the dynamical decoupling approximation that plays a key role in the MF approach. We also derive a new MF formula, applicable to anisotropic fluids and solids, including microcrystalline protein samples. The so-called slowly relaxing local structure (SRLS) model has been advanced as an alternative to the MF approach that does not require dynamical decoupling of internal and global motions. To resolve the existing controversy about the relative merits of the SRLS model and the MF approach, we formulate and solve a planar version of the SRLS model. The analytical solution of this model reveals the unphysical consequences of the symmetrical two-body Smoluchowski equation as applied to protein dynamics, thus refuting the widely held belief that the SRLS model is more accurate than the MF approach. The different results obtained by analyzing data with these two approaches therefore do not indicate the importance of dynamical coupling between internal and global motions. Finally, we explore the two principal mechanisms of dynamical coupling in proteins: torque-mediated and friction-mediated coupling. We argue by way of specific analytically solvable models that torque-mediated coupling (which the SRLS model attempts to capture) is unimportant because the relatively slow internal motions that might couple to the global motion tend to be intermittent (jumplike) in character, whereas friction-mediated coupling (which neither the SRLS model nor the MF approach incorporates) may be important for proteins with unstructured parts or flexibly connected domains.

General theoretical/computational tool for interpreting NMR spin relaxation in proteins

We developed in recent years the slowly relaxing local structure (SRLS) approach for analyzing NMR spin relaxation in proteins. SRLS is a two-body coupled rotator model which accounts rigorously for mode-coupling between the global motion of the protein and the local motion of the spin-bearing probe and allows for general properties of the second rank tensors involved. We showed that a general tool of data analysis requires both capabilities. Several important functionalities were missing in our previous implementations of SRLS in data fitting schemes, and in some important cases, the calculations were tedious. Here we present a general implementation which allows for asymmetric local and global diffusion tensors, distinct local ordering and local diffusion frames, and features a rhombic local potential which includes Wigner matrix element terms of ranks 2 and 4. A recently developed hydrodynamics-based approach for calculating global diffusion tensors has been incorporated into the data-fitting scheme. The computational efficiency of the latter has been increased significantly through object-oriented programming within the scope of the C++ programming language, and code parallelization. A convenient graphical user interface is provided. Currently autocorrelated (15)N spin relaxation data can be analyzed effectively. Adaptation to any autocorrelated and cross-correlated relaxation analysis is straightforward. New physical insight is gleaned on largely preserved local structure in solution, even in chain segments which experience slow local motion. Prospects associated with improved dynamic models, and new applications made possible by the current implementation of SRLS, are delineated.

Toward an integrated computational approach to CW-ESR spectra of free radicals

Interpretation of structural properties and dynamic behaviour of molecules in solution is of fundamental importance to understand their stability, chemical reactivity and catalytic action. Information can be gained, in principle, by a variety of spectroscopic techniques, magnetic as well as optical. In particular, continuous wave electron spin resonance (CW-ESR) measurements are highly informative. However, the wealth of structural and dynamic information which can be extracted from ESR spectroscopy is, at present, limited by the necessity of employing computationally efficient models, which are increasingly complex as they need to take into account diverse relaxation processes affecting the spectrum. In this paper, we address the basic theoretical tools needed to predict, essentially ab initio, CW-ESR spectra observables according to the stochastic Liouville equation (SLE) approach, combined with quantum mechanical and hybrid methods for the accurate and efficient computation of structural, spectroscopic and magnetic properties of molecular systems. We shall discuss, on one hand, the quantum mechanical calculation of magnetic observables, via density functional theory (DFT), time-dependent DFT (TD-DFT) and application of the polarizable continuum model (PCM) for the description of environmental effects, including anisotropic environments and systems where different regions are characterized by different dielectric constants. One the other hand, the explicit evaluation of dynamical effects will be discussed based on the numerically exact treatment of the SLE in the presence of several relaxation processes, which has been proven to be a challenging task.

Escherichia coli adenylate kinase dynamics: comparison of elastic network model modes with mode-coupling (15)N-NMR relaxation data

The dynamics of adenylate kinase of Escherichia coli (AKeco) and its complex with the inhibitor AP(5)A, are characterized by correlating the theoretical results obtained with the Gaussian Network Model (GNM) and the anisotropic network model (ANM) with the order parameters and correlation times obtained with Slowly Relaxing Local Structure (SRLS) analysis of (15)N-NMR relaxation data. The AMPbd and LID domains of AKeco execute in solution large amplitude motions associated with the catalytic reaction Mg(+2)*ATP + AMP --> Mg(+2)*ADP + ADP. Two sets of correlation times and order parameters were determined by NMR/SRLS for AKeco, attributed to slow (nanoseconds) motions with correlation time tau( perpendicular) and low order parameters, and fast (picoseconds) motions with correlation time tau( parallel) and high order parameters. The structural connotation of these patterns is examined herein by subjecting AKeco and AKeco*AP(5)A to GNM analysis, which yields the dynamic spectrum in terms of slow and fast modes. The low/high NMR order parameters correlate with the slow/fast modes of the backbone elucidated with GNM. Likewise, tau( parallel) and tau( perpendicular) are associated with fast and slow GNM modes, respectively. Catalysis-related domain motion of AMPbd and LID in AKeco, occurring per NMR with correlation time tau( perpendicular), is associated with the first and second collective slow (global) GNM modes. The ANM-predicted deformations of the unliganded enzyme conform to the functional reconfiguration induced by ligand-binding, indicating the structural disposition (or potential) of the enzyme to bind its substrates. It is shown that NMR/SRLS and GNM/ANM analyses can be advantageously synthesized to provide insights into the molecular mechanisms that control biological function.