Protein dynamics in the solid-state from 2H NMR lineshape analysis. III. MOMD in the presence of Magic Angle Spinning.

We report on a new approach to the analysis of dynamic NMR lineshapes from polycrystalline (i.e., macroscopically disordered) samples in the presence of Magic Angle Spinning (MAS). This is an application of the Stochastic Liouville Equation developed by Freed and co-workers for treating restricted (i.e., microscopically ordered) motions. The 2H nucleus in an internally-mobile C-CD3 moiety serves as a prototype probe. The acronym is 2H/MOMD/MAS, where MOMD stands for "microscopic-order-macroscopic-disorder." The key elements describing internal motions - their type, the local spatial restrictions, and related features of local geometry - are treated in MOMD generally, within their rigorous three-dimensional tensorial requirements. Based on this representation a single physically well-defined model of local motion has the capability of reproducing experimental spectra. There exist other methods for analyzing dynamic 2H/MAS spectra which advocate simple motional modes. Yet, to reproduce satisfactorily the experimental lineshapes, one has either to use unusual parameter values, or combine several simple motional modes. The multi-simple-mode reasoning assumes independence of the constituent modes, features ambiguity as different simple modes may be used, renders inter-system comparison difficult as the overall models differ, and makes possible model-improvement only by adding yet another simple mode, i.e., changing the overall model. 2H/MOMD/MAS is free of such limitations and inherently provides a clear physical interpretation. These features are illustrated. The advantage of 2H/MOMD/MAS in dealing with sensitive but hardly investigated slow-motional lineshapes is demonstrated by applying it to actual experimental data. The results differ from those obtained previously with a two-site exchange scheme that yielded unusual parameters.

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 [ß3-L3-ß4] 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 ß3 and ß4 (the preceding L2 loop and the remote chain-segment [(α2-helix)-(α2/ß5-turn)-(ß5-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/ß5-turn)-(ß5-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.

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.

The eigenmode perspective of NMR spin relaxation in proteins.

We developed in recent years the two-body (protein and probe) coupled-rotator slowly relaxing local structure (SRLS) approach for elucidating protein dynamics from NMR spin relaxation. So far we used as descriptors the set of physical parameters that enter the SRLS model. They include the global (protein-related) diffusion tensor, D1, the local (probe-related) diffusion tensor, D2, and the local coupling∕ordering potential, u. As common in analyzes based on mesoscopic dynamic models, these parameters have been determined with data-fitting techniques. In this study, we describe structural dynamics in terms of the eigenmodes comprising the SRLS time correlation functions (TCFs) generated by using the best-fit parameters as input to the Smoluchowski equation. An eigenmode is a weighted exponential with decay constant given by an eigenvalue of the Smoluchowski operator, and weighting factor determined by the corresponding eigenvector. Obviously, both quantities depend on the SRLS parameters as determined by the SRLS model. Unlike the set of best-fit parameters, the eigenmodes represent patterns of motion of the probe-protein system. The following new information is obtained for the typical probe, the (15)N-(1)H bond. Two eigenmodes, associated with the protein and the probe, dominate when the time scale separation is large (i.e., D2 >> D1), the tensorial properties are simple, and the local potential is either very strong or very weak. When the potential exceeds these limits while the remaining conditions are preserved, new eigenmodes arise. The multi-exponentiality of the TCFs is associated in this case with the restricted nature of the local motion. When the time scale separation is no longer large, the rotational degrees of freedom of the protein and the probe become statistically dependent (coupled dynamically). The multi-exponentiality of the TCFs is associated in this case with the restricted nature of both the local and the global motion. The effects of local diffusion axiality, potential strength, and extent of mode-coupling on the eigenmode setup are investigated. We detect largely global motional or largely local motional eigenmodes. In addition, we detect mixed eigenmodes associated with correlated∕prograde or anti-correlated∕retrograde rotations of the global (D1) and local (D2) motional modes. The eigenmode paradigm is applied to N-H bond dynamics in the ß-sheet residue K19, and the α-helix residue A34, of the third immunoglobulin-binding domain of streptococcal protein G. The largest contribution to the SRLS TCFs is made by mixed anti-correlated D1 and D2 eigenmodes. The next largest contribution is made by D1-dominated eigenmodes. Eigenmodes dominated by the local motion contribute appreciably to A34 and marginally to K19. Correlated D1 and D2 eigenmodes contribute exclusively to K19 and do not contribute above 1% to A34. The differences between K19 and A34 are delineated and rationalized in terms of the best-fit SRLS parameters and mode-mixing. It may be concluded that eigenmode analysis is complementary and supplementary to data-fitting-based analysis.

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 ß-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 Rex 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 Rex in MF-based 15N relaxation analysis. Then, we show that forgoing the usage of Rex, SRLS analysis of the very same experimental data provides interesting new information.

The N-Terminal Domain of Aß40-Amyloid Fibril: The MOMD Perspective of its Dynamic Structure from NMR Lineshape Analysis.

We have developed the stochastic microscopic-order-macroscopic-disorder (MOMD) approach for elucidating dynamic structures in the solid-state from 2H NMR lineshapes. In MOMD, the probe experiences an effective/collective motional mode. The latter is described by a potential, u, which represents the local spatial-restrictions, a local-motional diffusion tensor, R, and key features of local geometry. Previously we applied MOMD to the well-structured core domain of the 3-fold-symmetric twisted polymorph of the Aß40-amyloid fibril. Here, we apply it to the N-terminal domain of this fibril. We find that the dynamic structures of the two domains are largely similar but differ in the magnitude and complexity of the key physical parameters. This interpretation differs from previous multisimple-mode (MSM) interpretations of the same experimental data. MSM used for the two domains different combinations of simple motional modes taken to be independent. For the core domain, MOMD and MSM disagree on the character of the dynamic structure. For the N-terminal domain, they even disagree on whether this chain segment is structurally ordered (MOMD finds that it is), and whether it undergoes a phase transition at 260 K where bulklike water located in the fibril matrix freezes (MOMD finds that it does not). These are major differences associated with an important system. While the MOMD description is a physically sound one, there are drawbacks in the MSM descriptions. The results obtained in this study promote our understanding of the dynamic structure of protein aggregates. Thus, they contribute to the effort to pharmacologically control neurodegenerative disorders believed to be caused by such aggregates.

Structural Dynamics by NMR in the Solid State: II. The MOMD Perspective of the Dynamic Structure of Metal-Organic Frameworks Comprising Several Mobile Components.

We describe the application of the microscopic-order-macroscopic-disorder (MOMD) approach, developed for the analysis of dynamic 2H NMR lineshapes in the solid state, to unravel interactions among the constituents of metal-organic frameworks (MOFs) that comprise mobile components. MOMD was applied recently to University of Windsor Dynamic Material (UWDM) MOFs with one mobile crown ether per cavity. In this work, we study UWDM-9-d4, which comprises a mobile 2H-labeled phenyl-ring residue along with an isotopically unlabeled 24C8 crown ether. We also study UiO-68-d4, which is structurally similar to UWDM-9-d4 but lacks the crown ether. The physical picture consists of the NMR probe─the C-D bonds of the phenyl-d4 rotor─diffusing locally (diffusion tensor R) in the presence of a local ordering potential, u. For UiO-68-d4, we find it sufficient to expand u in terms of four real Wigner functions, D0|K|L, overall 2-3 kT in magnitude, with R∥ relatively fast, and R⊥ in the (2.8-5.0) × 102 s-1 range. For UWDM-9-d4, u requires only two terms 2-3 kT in magnitude and slower rate constants R∥ and R⊥. In the more crowded macrocycle-containing UWDM-9-d4 cavity, phenyl-d4 dynamics is more isotropic and is described by a simpler ordering potential. This is ascribed to cooperative phenyl-ring/macrocycle motion, which yields a dynamic structure more uniform in character. The experimental 2H spectra used here were analyzed previously with a multi-simple-mode (MSM) approach where several independent simple motional modes are combined. Where possible, similar features have been identified and used to compare the two approaches.

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, Peq = exp(-u) (u, local potential), of bond-vectors in proteins provide information on structural flexibility. The related conformational entropy, Sk = -∫Peq(ln Peq)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 Peq and Sk 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 ß2/α1 and α2/ß5 more flexible. By comparison, RBD dimerization renders L4 more rigid, and the α-helices, the ß-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.

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, Peq = exp(-u) (u, local potential), and associated conformational entropy, Sk = -∫Peq (ln Peq) dΩ â”€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) Peq 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 Peq 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.

NMR studies of a channel protein without membranes: structure and dynamics of water-solubilized KcsA.

Structural studies of polytopic membrane proteins are often hampered by the vagaries of these proteins in membrane mimetic environments and by the difficulties in handling them with conventional techniques. Designing and creating water-soluble analogues with preserved native structures offer an attractive alternative. We report here solution NMR studies of WSK3, a water-soluble analogue of the potassium channel KcsA. The WSK3 NMR structure (PDB ID code 2K1E) resembles the KcsA crystal structures, validating the approach. By more stringent comparison criteria, however, the introduction of several charged residues aimed at improving water solubility seems to have led to the possible formations of a few salt bridges and hydrogen bonds not present in the native structure, resulting in slight differences in the structure of WSK3 relative to KcsA. NMR dynamics measurements show that WSK3 is highly flexible in the absence of a lipid environment. Reduced spectral density mapping and model-free analyses reveal dynamic characteristics consistent with an isotropically tumbling tetramer experiencing slow (nanosecond) motions with unusually low local ordering. An altered hydrogen-bond network near the selectivity filter and the pore helix, and the intrinsically dynamic nature of the selectivity filter, support the notion that this region is crucial for slow inactivation. Our results have implications not only for the design of water-soluble analogues of membrane proteins but also for our understanding of the basic determinants of intrinsic protein structure and dynamics.

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 + D0-22). 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 Cα-Cα. 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 ß3, ß5, and ß8. The potential, u, is strong (∼20 kT); it is slightly rhombic when N-H is the main ordering axis and highly rhombic when Cα-Cα is the main ordering axis. Gal3C-S exhibits primarily preferential ordering along Cα-Cα; 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 ß (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, ß. 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 (Rex) 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. Rex 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.

Measurement of bond vector orientations in invisible excited states of proteins.

The focus of structural biology is on studies of the highly populated, ground states of biological molecules; states that are only sparsely and transiently populated are more difficult to probe because they are invisible to most structural methods. Yet, such states can play critical roles in biochemical processes such as ligand binding, enzyme catalysis, and protein folding. A description of these states in terms of structure and dynamics is, therefore, of great importance. Here, we present a method, based on relaxation dispersion NMR spectroscopy of weakly aligned molecules in a magnetic field, that can provide such a description by direct measurement of backbone amide bond vector orientations in transient, low populated states that are not observable directly. Such information, obtained through the measurement of residual dipolar couplings, has until now been restricted to proteins that produce observable spectra. The methodology is applied and validated in a study of the binding of a target peptide to an SH3 domain from the yeast protein Abp1p and subsequently used in an application to protein folding of a mutational variant of the Fyn SH3 domain where (1)H-(15)N dipolar couplings of the invisible unfolded state of the domain are obtained. The approach, which can be used to obtain orientational restraints at other sites in proteins as well, promises to significantly extend the available information necessary for providing a site-specific characterization of structural properties of transient, low populated states that have to this point remained recalcitrant to detailed analysis.

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 Sk ≡ S/k = -∫(Peq ln Peq)dΩ - ln∫dΩ. The quantity Peq = 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, D0KL. 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 Sk. For L = 1 (L = 2), Sk is the same (differs) for parallel and perpendicular ordering. The plots of Sk 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. Sk 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.

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.

Structural Dynamics by NMR in the Solid State: The Unified MOMD Perspective Applied to Organic Frameworks with Interlocked Molecules.

The microscopic-order-macroscopic-disorder (MOMD) approach for NMR lineshape analysis has been applied to the University of Windsor Dynamic Materials (UWDM) of types 1, 2, α-3, ß-3, and 5, which are metal-organic frameworks (MOFs) comprising mobile mechanically interlocked molecules (MIMs). The mobile MIM components are selectively deuterated crown ether macrocycles - 24C6, 22C6, and B24C6. Their motion is described in MOMD by an effective/collective dynamic mode characterized by a diffusion tensor, R, a restricting/ordering potential, u, expanded in the Wigner rotation matrix elements, D0, KL, and features of local geometry. Experimental 2H lineshapes are available over 220 K (on average) and in some cases 320 K. They are reproduced with axial R, u given by the terms D0,02 and D0,|2|2, and established local geometry. For UWDM of types 1, ß-3, and 5, where the macrocycle resides in a relatively loose space, u is in the 1-3 kT, R∥ in the (1.0-2.5) × 106 s-1, and R⊥ in the (0.4-2.5) × 104 s-1 range; the deuterium atom is bonded to a carbon atom with tetrahedral coordination character. For UWDM of types 2 and α-3, where the macrocycle resides in a much tighter space, a substantial change in the symmetry of u and the coordination character of the 2H-bonded carbon are detected at higher temperatures. The activation energies for R∥ and R⊥ are characteristic of each system. The MOMD model is general; effective/collective dynamic modes are treated. The characteristics of motion, ordering, and geometry are physically well-defined; they differ from case to case in extent and symmetry but not in essence. Physical clarity and consistency provide new insights. A previous interpretation of the same experimental data used models consisting of collections of independent simple motions. These models are specific to each case and temperature. Within their scope, generating consistent physical pictures and comparing cases are difficult; possible collective modes are neglected.

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, D0KL, 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.

Evidence for domain motion in proteins affecting global diffusion properties: a nuclear magnetic resonance study.

The rotational diffusion of proteins is an important hydrodynamic property. Compact protein structures were found previously to exhibit hydration layer viscosity, etaloc, higher than the viscosity of bulk water, eta. This implies an apparent activation energy for rotational diffusion higher than the activation energy of water viscosity, Eeta=15.4+/-0.3 kJ/mol. In this study we examine etaloc of internally mobile proteins using 15N spin relaxation methods. We also examine the activation enthalpy, DeltaH#, and activation entropy, DeltaS#, for rotational diffusion. Of particular relevance are internally mobile ligand-free forms and compact ligand-bound forms of multidomain proteins. Adenylate kinase (AKeco) and Ca2+-calmodulin (Ca2+-CaM) are typical examples. For AKeco (Ca2+-CaM) we find that DeltaH# is 14.5+/-0.5 (15.7+/-0.4) kJ/mol. For the complex of AKeco with the inhibitor AP5A (the complex of Ca2+-CaM with the peptide smMLCKp), we find that DeltaH# is 18.1+/-0.7 (18.2+/-0.5) kJ/mol. The internally mobile outer surface protein A has DeltaH#=12.6+/-0.8 kJ/mol, and the compact protein Staphylococcal nuclease has DeltaH#=18.8+/-0.6 kJ/mol. For the internally mobile and compact proteins studied, <|DeltaS(|> equals 62+/-7 J/(mol K) and 44+/-5 J/(mol K), respectively. The fact is that etaloc>eta (DeltaH#>Eeta) for compact proteins was ascribed previously to electrostatic interactions between surface sites and water rigidifying the hydration layer. We find herein that obliteration of these interactions by domain motion leads to etaloc approximately eta, DeltaH# approximately Eeta, and large activation entropy for internally mobile protein structures.

Domain mobility in proteins from NMR/SRLS.

Enhanced internal mobility in proteins is typically functional. Domain motion in enzymes, necessarily related to catalysis, is a prototype in this context. Experimental (15)N spin relaxation data from E. coli adenylate kinase report qualitatively on nanosecond motion experienced by the domains AMPbd and LID. Previous quantitative analysis based on the mode-coupling slowly relaxing local structure approach confirmed nanosecond mobility but yielded unduly small local ordering and local geometry not interpretable directly in terms of the local protein structure. Here, we show that these features ensue from having assumed axial local ordering and highly axial local diffusion. After eliminating these simplified second-rank tensor properties, a physically sound picture, with the local motion interpretable as domain motion, is obtained. Rhombic local ordering, with components given by = 0.471, = -0.952 and = 0.481, and main ordering axis, Y(M), lying along C(alpha)(i-1) - C(alpha)(i), has been determined. The associated rhombic potential is given by axial (rhombic) coefficients of = -3.3 ( = 17.8). The average correlation time for domain motion is 10.4 (6.4) ns at 288 (302) K; the corresponding correlation time for global motion is 20.6 (14.9) ns. The rates for domain motion exhibit noteworthy Arrhenius-type temperature-dependence, yielding activation energies of 63.8 +/- 7.0 (53.0 +/- 9.1) kJ/mol for the AMPbd (LID) domain. The traditional model-free analysis ignores mode-coupling and simplifies tensor properties. Within its scope, the AKeco backbone emerges as largely rigid, approximately = 0.94; the main ordering axis, Z(M), lies along N-H, approximately = 16 (c(2)(2) = 0); and the slow local motional correlation time lies at the low end of the nanosecond time scale.