A novel view of domain flexibility in E. coli adenylate kinase based on structural mode-coupling (15)N NMR relaxation

Protein tolerance to random circular permutation correlates with thermostability and local energetics of residue-residue contacts

Adenylate kinase (AK) orthologs with a range of thermostabilities were subjected to random circular permutation, and deep mutational scanning was used to evaluate where new protein termini were nondisruptive to activity. The fraction of circularly permuted variants that retained function in each library correlated with AK thermostability. In addition, analysis of the positional tolerance to new termini, which increase local conformational flexibility, showed that bonds were either functionally sensitive to cleavage across all homologs, differentially sensitive, or uniformly tolerant. The mobile AMP-binding domain, which displays the highest calculated contact energies, presented the greatest tolerance to new termini across all AKs. In contrast, retention of function in the lid and core domains was more dependent upon AK melting temperature. These results show that family permutation profiling identifies primary structure that has been selected by evolution for dynamics that are critical to activity within an enzyme family. These findings also illustrate how deep mutational scanning can be applied to protein homologs in parallel to differentiate how topology, stability, and local energetics govern mutational tolerance.

Fokker-Planck formalism in magnetic resonance simulations

This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, stationary flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author's experiences with writing and using the corresponding modules of the Spinach library - the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use.

The Structure of a Thermophilic Kinase Shapes Fitness upon Random Circular Permutation

Proteins can be engineered for synthetic biology through circular permutation, a sequence rearrangement in which native protein termini become linked and new termini are created elsewhere through backbone fission. However, it remains challenging to anticipate a protein's functional tolerance to circular permutation. Here, we describe new transposons for creating libraries of randomly circularly permuted proteins that minimize peptide additions at their termini, and we use transposase mutagenesis to study the tolerance of a thermophilic adenylate kinase (AK) to circular permutation. We find that libraries expressing permuted AKs with either short or long peptides amended to their N-terminus yield distinct sets of active variants and present evidence that this trend arises because permuted protein expression varies across libraries. Mapping all sites that tolerate backbone cleavage onto AK structure reveals that the largest contiguous regions of sequence that lack cleavage sites are proximal to the phosphotransfer site. A comparison of our results with a range of structure-derived parameters further showed that retention of function correlates to the strongest extent with the distance to the phosphotransfer site, amino acid variability in an AK family sequence alignment, and residue-level deviations in superimposed AK structures. Our work illustrates how permuted protein libraries can be created with minimal peptide additions using transposase mutagenesis, and it reveals a challenge of maintaining consistent expression across permuted variants in a library that minimizes peptide additions. Furthermore, these findings provide a basis for interpreting responses of thermophilic phosphotransferases to circular permutation by calibrating how different structure-derived parameters relate to retention of function in a cellular selection.

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.

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.

Minimum free energy path of ligand-induced transition in adenylate kinase

Large-scale conformational changes in proteins involve barrier-crossing transitions on the complex free energy surfaces of high-dimensional space. Such rare events cannot be efficiently captured by conventional molecular dynamics simulations. Here we show that, by combining the on-the-fly string method and the multi-state Bennett acceptance ratio (MBAR) method, the free energy profile of a conformational transition pathway in Escherichia coli adenylate kinase can be characterized in a high-dimensional space. The minimum free energy paths of the conformational transitions in adenylate kinase were explored by the on-the-fly string method in 20-dimensional space spanned by the 20 largest-amplitude principal modes, and the free energy and various kinds of average physical quantities along the pathways were successfully evaluated by the MBAR method. The influence of ligand binding on the pathways was characterized in terms of rigid-body motions of the lid-shaped ATP-binding domain (LID) and the AMP-binding (AMPbd) domains. It was found that the LID domain was able to partially close without the ligand, while the closure of the AMPbd domain required the ligand binding. The transition state ensemble of the ligand bound form was identified as those structures characterized by highly specific binding of the ligand to the AMPbd domain, and was validated by unrestrained MD simulations. It was also found that complete closure of the LID domain required the dehydration of solvents around the P-loop. These findings suggest that the interplay of the two different types of domain motion is an essential feature in the conformational transition of the enzyme.

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.

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.

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.

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).

Electron-nuclear interactions as probes of domain motion in proteins

Long range interactions between nuclear spins and paramagnetic ions can serve as a sensitive monitor of internal motion of various parts of proteins, including functional loops and separate domains. In the case of interdomain motion, the interactions between the ion and NMR-observable nuclei are modulated in direction and magnitude mainly by a combination of overall and interdomain motions. The effects on observable parameters such as paramagnetic relaxation enhancement (PRE) and pseudocontact shift (PCS) can, in principle, be used to characterize motion. These parameters are frequently used for the purpose of structural refinements. However, their use to probe actual domain motions is less common and is lacking a proper theoretical treatment from a motional perspective. In this work, a suitable spin Hamiltonian is incorporated in a two body diffusion model to produce the time correlation function for the nuclear spin-paramagnetic ion interactions. Simulated observables for nuclei in different positions with respect to the paramagnetic ion are produced. Based on these simulations, it demonstrated that both the PRE and the PCS can be very sensitive probes of domain motion. Results for different nuclei within the protein sense different aspects of the motions. Some are more sensitive to the amplitude of the internal motion, others are more sensitive to overall diffusion rates, allowing separation of these contributions. Experimentally, the interaction strength can also be tuned by substitution of different paramagnetic ions or by varying magnetic field strength (in the case of lanthanides) to allow the use of more detailed diffusion models without reducing the reliability of data fitting.

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.

Dynamics in Thermotoga neapolitana adenylate kinase: 15N relaxation and hydrogen-deuterium exchange studies of a hyperthermophilic enzyme highly active at 30 degrees C

Backbone conformational dynamics of Thermotoga neapolitana adenylate kinase in the free form (TNAK) and inhibitor-bound form (TNAK*Ap5A) were investigated at 30 degrees C using (15)N NMR relaxation measurements and NMR monitored hydrogen-deuterium exchange. With kinetic parameters identical to those of Escherichia coli AK (ECAK) at 30 degrees C, TNAK is a unique hyperthermophilic enzyme. These catalytic properties make TNAK an interesting and novel model to study the interplay between protein rigidity, stability, and activity. Comparison of fast time scale dynamics (picosecond to nanosecond) in the open and closed states of TNAK and ECAK at 30 degrees C reveals a uniformly higher rigidity across all domains of TNAK. Within this framework of a rigid TNAK structure, several residues located in the AMP-binding domain and in the core-lid hinge regions display high picosecond to nanosecond time scale flexibility. Together with the recent comparison of ECAK dynamics with those of hyperthermophilic Aquifex aeolicus AK (AAAK), our results provide strong evidence for the role of picosecond to nanosecond time scale fluctuations in both stability and activity. In the slow time scales, TNAK's increased rigidity is not uniform but localized in the AMP-binding and lid domains. The core domain amides of ECAK and TNAK in the open and closed states show comparable protection against exchange. Significantly, the hinges framing the lid domain show similar exchange data in ECAK and TNAK open and closed forms. Our NMR relaxation and hydrogen-deuterium exchange studies therefore suggest that TNAK maintains high activity at 30 degrees C by localizing flexibility to the hinge regions that are key to facilitating conformational changes.

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.

Theoretical study of the effective Chemical Shielding Anisotropy (CSA) in peptide backbone, rating the impact of CSAs on the cross-correlated relaxations in L-alanyl-L-alanine

The dependence of the effective chemical shielding anisotropy (effective CSA, Deltasigma(eff)) on the phi and psi peptide backbone torsion angles was calculated in the l-alanyl-l-alanine (LALA) peptide using the DFT method. The effects of backbone conformation, molecular charge including the cation, zwitterion, and anion forms of the LALA peptide, and the scaling taking into account the length of the dipolar vector were calculated for the effective CSAs in order to assess their structural behaviors and to predict their magnitudes which can be probed for the beta-sheet and alpha-helix backbone conformations via measurement of the cross-correlated relaxation rates (CCR rates). Twenty different CSA-DD cross-correlation mechanisms involving the amide nitrogen and carbonyl carbon chemical shielding tensors and the C(alpha)H(alpha) (alpha-carbon group), NH(N) (amide group), C(alpha)H(N), NH(alpha), C'H(alpha), and C'H(N) (alpha = alpha1, alpha2) dipolar vectors were investigated. The X-C(alpha)H(alpha) (X = N, C'; alpha = alpha1, alpha2) cross-correlations, which had already been studied experimentally, exhibited overall best performance of the calculated effective CSAs in the LALA molecule; they spanned the largest range of values upon variation of the psi and phi torsions and depended dominantly on only one of the two backbone torsion angles. The X-NH(N) (X = N, C') cross-correlations, which had been also probed experimentally, depended on both backbone torsions, which makes their structural assignment more difficult. The N-NH(alpha2) and N-C'H(alpha1) cross-correlations were found to be promising for the determination of various backbone conformations due to the large calculated range of the scaled effective CSA values and due to their predominant dependence on the psi and phi torsions, respectively. The 20 calculated dependencies of effective CSAs on the two backbone torsion angles can facilitate the structural interpretation of CCR rates.

Protein dynamics from a NMR perspective: networks of coupled rotators and fractional Brownian dynamics

Nuclear magnetic resonance (NMR) has proven to be the most valuable tool for investigating internal dynamics of proteins. In this perspective, the interpretation of NMR relaxation data eventually relies on a model of the motions. In this article, we propose to compare two radically different approaches that aim at describing internal dynamics in proteins. It is shown that the correlation functions predicted by a network of coupled rotators can be interpreted in terms of a heuristic approach based on fractional Brownian dynamics for each of the vectors in the network. Our results are interpreted in terms of the probability distributions of relaxation modes in both processes, the median of which turns out to be the relevant quantity for the comparison of both models.

An improved picture of methyl dynamics in proteins from slowly relaxing local structure analysis of 2H spin relaxation

Protein dynamics is intimately related to biological function. Core dynamics is usually studied with 2H spin relaxation of the 13CDH2 group, analyzed traditionally with the model-free (MF) approach. We showed recently that MF is oversimplified in several respects. This includes the assumption that the local motion of the dynamic probe and the global motion of the protein are decoupled, the local geometry is simple, and the local ordering is axially symmetric. Because of these simplifications MF has yielded a puzzling picture where the methyl rotation axis is moving rapidly with amplitudes ranging from nearly complete disorder to nearly complete order in tightly packed protein cores. Our conclusions emerged from applying to methyl dynamics in proteins the slowly relaxing local structure (SRLS) approach of Polimeno and Freed (Polimeno, A.; Freed, J. H. J. Phys. Chem. 1995, 99, 10995-11006.), which can be considered the generalization of MF, with all the simplifications mentioned above removed. The SRLS picture derived here for the B1 immunoglobulin binding domain of peptostreptococcal protein L, studied over the temperature range of 15-45 degrees C, is fundamentally different from the MF picture. Thus, methyl dynamics is characterized structurally by rhombic local potentials with varying symmetries and dynamically by tenfold slower rates of local motion. On average, potential rhombicity decreases, mode-coupling increases, and the rate of local motion increases with increasing temperature. The average activation energy for local motion is 2.0 +/- 0.2 kcal/mol. Mode-coupling affects the analysis even at 15 degrees C. The accuracy of the results is improved by including in the experimental data set relaxation rates associated with rank 2 coherences.

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

(15)N-(1)H spin relaxation is a powerful method for deriving information on protein dynamics. The traditional method of data analysis is model-free (MF), where the global and local N-H motions are independent and the local geometry is simplified. The common MF analysis consists of fitting single-field data. The results are typically field-dependent, and multifield data cannot be fit with standard fitting schemes. Cases where known functional dynamics has not been detected by MF were identified by us and others. Recently we applied to spin relaxation in proteins the slowly relaxing local structure (SRLS) approach, which accounts rigorously for mode mixing and general features of local geometry. SRLS was shown to yield MF in appropriate asymptotic limits. We found that the experimental spectral density corresponds quite well to the SRLS spectral density. The MF formulas are often used outside of their validity ranges, allowing small data sets to be force-fitted with good statistics but inaccurate best-fit parameters. This paper focuses on the mechanism of force-fitting and its implications. It is shown that MF analysis force-fits the experimental data because mode mixing, the rhombic symmetry of the local ordering and general features of local geometry are not accounted for. Combined multifield multitemperature data analyzed with the MF approach may lead to the detection of incorrect phenomena, and conformational entropy derived from MF order parameters may be highly inaccurate. On the other hand, fitting to more appropriate models can yield consistent physically insightful information. This requires that the complexity of the theoretical spectral densities matches the integrity of the experimental data. As shown herein, the SRLS spectral densities comply with this requirement.

Predicting internal protein dynamics from structures using coupled networks of hindered rotators

Internal motions in proteins, such as oscillations of internuclear vectors u(N(i)H(i) (N)) of amide bonds about their equilibrium position, can be characterized by a local order parameter. This dynamic parameter can be determined experimentally by measuring the longitudinal and transverse relaxation rates of (15)N(i) nuclei by suitable NMR methods. In this paper, it is shown that local variations of order parameters S(ii) (2) can be predicted from the knowledge of the structure. To this effect, the diffusive motion of the internuclear vector u(N(i)H(i) (N)) is described in a potential that takes into account the deviations of the angles theta(ij) between u(N(i)H(i) (N)) and neighboring vectors u(N(j)H(j) (N)) from their average value and similarly of deviations of the angles subtended between u(N(i)H(i) (N)) and u(X(j)Y(j)), where X(j) and Y(j) are heavy atoms in the vicinity of the u(N(i)H(i) (N)) vector under investigation. It is shown how the concept of vicinity can be defined by a simple cutoff threshold, i.e., by neglecting vectors u(X(j)Y(j)) with distances d(N(i),X(j))>7.5 A. The local order parameters S(ii) (2) can be predicted from the structure using a limited set of coordinates of heavy atoms. The inclusion of a larger number of heavy atoms does not improve the predictions. Applications to calmodulin, calbindin, and interleukin 4 illustrate the success and limitations of the predictions.

An overview of recent developments in the interpretation and prediction of fast internal protein dynamics

During the past decades, NMR spectroscopy has emerged as a unique tool for the study of protein dynamics. Indeed, relaxation studies on isotopically labeled proteins can provide information on the overall motions as well as the internal fast, sub-nanosecond, dynamics. Therefore, the interpretation and the prediction of spin relaxation rates in proteins are important issues that have motivated numerous theoretical and methodological developments, including the description of overall dynamics and its possible coupling to internal mobility, the introduction of models of internal dynamics, the determination of correlation functions from experimental data, and the relationship between relaxation and thermodynamical quantities. A brief account of recent developments that have proven useful in this domain are presented.

A model of interdomain mobility in a multidomain protein

Domain mobility plays an essential role in the biological function of multidomain systems. The characteristic times of domain motions fall into the interval from nano- to milliseconds, amenable to NMR studies. Proper analysis of NMR relaxation data for these systems in solution has to account for interdomain motions, in addition to the overall tumbling and local intradomain dynamics. Here we propose a model of interdomain mobility in a multidomain protein, which considers domain reorientations as exchange/interconversion between two distinct conformational states of the molecule, combined with fully anisotropic overall tumbling. Analysis of 15N-relaxation data for Lys48-linked diubiquitin at pH 4.5 and 6.8 showed that this model adequately fits the experimental data and allows characterization of both structural and motional properties of diubiquitin, thus providing information about the relative orientation of ubiquitin domains in both interconverting states. The analysis revealed that the two domains reorient on a time scale of 9-30 ns, with the amplitudes sufficient for allowing a protein ligand access to the binding sites sequestered at the interface in the closed conformation. The analysis of a possible mechanism controlling the equilibrium between the interconverting states in diubiquitin points toward protonation of His68, which results in three different charged states of the molecule, with zero, +e, and +2e net charge. Only two of the three states are noticeably populated at pH 4.5 or 6.8, which assures applicability of the two-state model to diubiquitin at these conditions. We also compare our model with the "extended model-free" approach and discuss possible future developments of the model.

Methyl Dynamics in Proteins from NMR Slowly Relaxing Local Structure Spin Relaxation Analysis: A New Perspective

NMR spin relaxation of (2)H nuclei in (13)CH(2)D groups is a powerful method for studying side-chain motion in proteins. The analysis is typically carried out with the original model-free (MF) approach adapted to methyl dynamics. The latter is described in terms of axial local motions around, and of, the methyl averaging axis, mutually decoupled and independent of the global motion of the protein. Methyl motion is characterized primarily by the axial squared order parameter, S(axis)2, associated with fluctuations of the methyl averaging axis. This view is shown to be oversimplified by applying to typical experimental data the slowly relaxing local structure (SRLS) approach of Polimeno and Freed (Adv. Chem. Phys. 1993, 83, 89) which can be considered the generalization of the MF approach. Neglecting mode coupling and the asymmetry of the local ordering and treating approximately features of local geometry imply inaccurate values of S(axis)2, hence of the residual configurational entropy derived from it. S(axis)2, interpreted as amplitude of motion, was found to range from near disorder to almost complete order. Contrary to this picture, we find with the SRLS approach a moderate distribution in the magnitude of asymmetric local ordering and significant variation in its symmetry. The latter important property can be associated implicitly with the contribution of side-chain rotamer jumps. This is consistent with experimental residual dipolar coupling studies and theoretical work based on molecular dynamics simulations and molecular mechanics considerations. Configurational entropy is obtained in the SRLS approach directly from experimentally determined asymmetric potentials. Inconsistency between order parameters from 2H relaxation and from eta(HC-HH) cross-correlation and increase in order parameters with increasing temperature were observed with the MF approach. These discrepancies are reconciled, and physically tenable temperature dependence is obtained with the SRLS approach.

Activation Energy of Catalysis-Related Domain Motion in E. coli Adenylate Kinase

Adenylate kinase from E. coli (AKeco), folded into domains CORE, AMPbd, and LID, catalyzes the reaction AMP + ATP <--> 2ADP. Previous X-ray crystallography and optical solution methods showed that the domains AMPbd and LID, and the conserved P-loop, execute large-amplitude catalysis-related motions. We used (15)N NMR spin relaxation methods to find that the simplified model-free (MF) analysis does not, whereas our general Slowly Relaxing Local Structure analysis does, detect catalytic domain motion. SRLS set for the first time the correlation time for domain motion at tau(L)perpendicular = 8.2 ns, to be compared with tau(m) = 15.1 ns for global tumbling. These results were obtained at 303 K. Herein we conduct a temperature-dependent investigation of tau(L)perpendicular and tau(m) in the range of 288-310 K. We found that the activation energy for global tumbling is Ea = 16.9 +/- 0.5 kJ/mol, the hydrodynamic volume of hydrated AKeco is 65.6 +/- 2.1 nm3, its radius is 2.50 +/- 0.03 nm, and the number of hydration layers is 1.77. The average tau(L)perpendicular value decreases from 11 ns at 288 K to 4 ns at 310 K, with activation energies of 29.7 +/- 3.3, 32.1 +/- 4.3, and 30.4 +/- 4.3 kJ/mol for the domains AMPbd and LID, and the catalytic P-loop, respectively. These values are two-to-three times smaller than typical activation energies of enzymatic reactions. Hence kinase catalysis appears not to be controlled by domain motion in the ligand-free enzyme. However, the latter process clearly facilitates important mechanical aspects such as steric recognition and capturing of the AMP and ATP substrates, their proper positioning for phosphorylation, and the release of the ADP product.

Interdomain mobility in di-ubiquitin revealed by NMR

Domain orientation and dynamics can play an essential role in the function of multidomain proteins. Lys48-linked polyubiquitin chains, the principal signal for proteasomal protein degradation, adopt a closed conformation at physiological conditions, in which the functionally important residues Leu8, Ile44, and Val70 are sequestered at the interdomain interface. This interface must open in order for these groups to become available for interactions with various chain-recognition factors. Knowledge of the mechanism of domain motion leading to the opening of the interdomain interface in polyubiqutin is, therefore, essential for the understanding of the processes controlling molecular recognition events in polyubiquitin signaling. Here we use NMR to characterize the interdomain dynamics that open the interface in a di-ubiquitin chain. This process occurs via domain reorientations on a 10-ns time scale and with the amplitudes that are sufficient for making functionally important hydrophobic residues in polyubiquitin available for direct interactions with various ubiquitin-binding factors. The analysis revealed the structures of the interconverting conformational states of di-ubiquitin and the rates and amplitudes of this process at near-physiological and acidic pH. The proposed mechanism of domain reorientation is quite general and could serve as a paradigm of interdomain mobility in other multidomain systems.

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.

Large amplitude conformational change in proteins explored with a plastic network model: adenylate kinase

The plastic network model (PNM) is used to generate a conformational change pathway for Escherichia coli adenylate kinase based on two crystal structures, namely that of an open and a closed conformer. In this model, the energy basins corresponding to known conformers are connected at their lowest common energies. The results are used to evaluate and analyze the minimal energy pathways between these basins. The open to closed transition analysis provides an identification of hinges that is in agreement with the existing definitions based on the available X-ray structures. The elastic energy distribution and the C(alpha) pseudo-dihedral variation provide similar information on these hinges. The ensemble of the 45 published structures for this protein and closely related proteins is shown to always be within 3.0 A of the pathway, which corresponds to a conformational change between two end structures that differ by a C(alpha)-atom root-mean-squared deviation of 7.1A.

Molecular cloning and characterization of a novel adenylate kinase 3 gene from Clonorchis sinensis

Adenylate kinase (AK) is a ubiquitous enzyme that contributes to the homeostasis of adenine nucleotides in living cells. AK catalyzes reversible high energy phosphoryl transfer reactions between ATP (or GTP) and AMP to generate ADP (or GDP). From a Clonorchis sinensis adult worm cDNA library, we isolated a cDNA clone encoding a novel AK3 isozyme. The 956 bp cDNA encodes a putative protein of 228 amino acids with a predicted molecular mass of 26.2 kDa. The recombinant CsAK3 protein produced in Escherichia coli can be refolded into a functional protein with AK3 activity. The optimum pH and temperature for the enzyme are 8.5 and 40 degrees C, respectively. The calculated activation energy is 56.04 kJ mol-1. The Km of the CsAK3 for AMP and GTP are 118 microM and 359 microM, respectively. CsAK3 is inhibited by Ap5A (>70% inhibition by 2.0 mM AP5A). Ap5A may be a potential lead compound acting on C. sinensis in which AK3 as a drug target.

Isotropic Reorientational Eigenmode Dynamics Complements NMR Relaxation Measurements for RNA

(13)C NMR relaxation measurements alone are often not sufficient to describe the motions of small RNA molecules in solution. In the case where the global tumbling time of the RNA is on the same time scale as its internal motions, standard Lipari-Szabo analysis becomes inadequate, and other methods must be used to describe the dynamics. Here, molecular dynamics simulations of the iron-responsive element (IRE) RNA hairpin are analyzed using isotropic reorientational eigenmode dynamics (iRED) to provide a picture of the motions of the RNA. The results show that indeed there is no separability of global and internal motions, and thus the order parameters determined from experimental data cannot be quantitatively accurate. iRED analysis also identifies correlated motions, providing a new picture of the dynamics of the IRE loop.

Gated electron transfers and electron pathways in azurin: a NMR dynamic study at multiple fields and temperatures

Dynamic properties of electron transfer pathways in a small blue copper cupredoxin are explored using an extensive 15N NMR relaxation study of reduced Pseudomonas aeruginosa azurin at four magnetic fields (500-900 MHz) and at two temperatures chosen well below the melting point of the protein. Following a careful model-free analysis, several protein regions with different dynamic regimes are identified. Nanosecond time-scale mobility characterizes various residues of the hydrophobic surface patch believed to mark the natural entry point for electrons, notably the surface-exposed copper-ligand His117. These findings are consistent with a gated electron transfer process according to the "dynamic docking" model. Residues 47-49 along intramolecular pathways of electrons show rigidity that is remarkably conserved when increasing the temperature. Three different conformational exchange processes were observed in the millisecond range, one near the only disulfide bridge in the molecule and two near the copper ion. The latter two processes are consistent with previous data such as crystal structures at various pH values and NMR relaxation dispersion experiments; they may indicate an additional gated electron transfer mechanism at slower time-scales.

A simple model for NMR relaxation in the presence of internal motions with dynamical coupling

In this paper some effects of dynamical coupling between two interaction-carrying vectors on the internal auto- and cross-correlation functions are investigated in the limit of small amplitude motions. A linearized Langevin approach allows the derivation of explicit expressions for these correlation functions and for the corresponding order parameters.

Structural and dynamic studies on ligand-free adenylate kinase from Mycobacterium tuberculosis revealed a closed conformation that can be related to the reduced catalytic activity

Tuberculosis is the leading cause of death worldwide from a single infectious disease. Search of new therapeutic tools requires the discovery and biochemical characterization of new potential targets among the bacterial proteins essential for the survival and virulence. Among them are the nucleoside monophosphate kinases, involved in the nucleotide biosynthesis. In this work, we determined the solution structure of adenylate kinase (AK) from Mycobacterium tuberculosis (AKmt), a protein of 181 residues that was found to be essential for bacterial survival. The structure was calculated by a simulated annealing protocol and energy minimization using experimental restraints, collected by nuclear magnetic resonance spectroscopy. The final, well-defined 20 NMR structures show an average root-mean-square deviation of 0.77 A for the backbone atoms in regular secondary structure segments. The protein has a central CORE domain, composed of a five-stranded parallel beta-sheet surrounded by seven alpha-helices, and two peripheral domains, AMPbd and LID. As compared to other crystallographic structures of free form AKs, AKmt is more compact, with the AMP(bd) domain closer to the CORE of the protein. Analysis of the (15)N relaxation data enabled us to obtain the global rotational correlation time (9.19 ns) and the generalized order parameters (S(2)) of amide vectors along the polypeptide sequence. The protein exhibits restricted movements on a picosecond to nanosecond time scale in the secondary structural regions with amplitudes characterized by an average S(2)() value of 0.87. The loops beta1/alpha1, beta2/alpha2, alpha2/alpha3, alpha3/alpha4, alpha4/beta3, beta3/alpha5, alpha6/alpha7 (LID), alpha7/alpha8, and beta5/alpha9 exhibit rapid fluctuations with enhanced amplitudes. These structural and dynamic features of AKmt may be related to its low catalytic activity that is 10-fold lower than in their eukaryote counterparts.

Structures of thermophilic and mesophilic adenylate kinases from the genus Methanococcus

The crystal structures of adenylate kinases from the thermophile Methanococcus thermolithotrophicus and the mesophile Methanococcus voltae have been solved to resolutions of 2.8A and 2.5A, respectively. The structures of the enzymes are similar to that of the adenylate kinase from archaeal Sulfolobus acidocaldarius in many respects such as the extended central beta-sheets, the short LID domain, and the trimeric state. The analysis of unligated and AMP-bound subunits of M.voltae suggests that movements of two mobile domains are not independent of each other. The methanococcal structures are examined with respect to their lack of the "invariant" Lys residue within the phosphate-binding loop, and two Arg residues in the LID domain are proposed as substituting residues based on their conservation among archaeal adenylate kinases and mobility within the structures. Since S.acidocaldarius adenylate kinase has the invariant Lys residue as well as the two Arg residues, its phosphate-binding loop is examined and compared with those of other adenylate kinases. On the basis of the comparison and other available biochemical data, the unusual conformation of the Lys residue in S.acidocaldarius adenylate kinase is explained. Despite possessing 78% sequence identity, the methanococcal enzymes exhibit significantly different thermal stabilities. To study the determinants of thermostability, several structural features including salt-links, hydrogen bonds, packing density, surface to volume ratio and buried surface area are compared between the enzymes. From their difference in apolar buried surface area, hydrophobic interaction is proposed to be a basis for the disparate thermostabilities, and the corresponding free energy difference is also estimated. Results of previous mutational studies are interpreted in terms of the crystal structures, and support the importance of hydrophobic interactions in thermostability.

End-to-end correlation for a C-12 hydrocarbon chain

The 19F nuclear spin-lattice relaxation rate constants were measured as a function of magnetic field strength for 1,12-diaminododecane labeled at one end with a nitroxide radical and at the other with a trifluoromethyl group. The magnetic relaxation dispersion profile (MRD) reports the spectral density function appropriate to the end-to-end correlation function for the doubly labeled molecule. After extrapolation to zero concentration to eliminate the intermolecular relaxation contribution to relaxation, the resulting intramolecular MRD profile was compared with several model approaches. The rotational model for the spectral density functions as included in the Solomon-Bloembergen-Morgan equations does not describe the data well. The earlier model of Freed for nuclear spin relaxation induced by a freely diffusing paramagnetic co-solute is not rigorous for this case because the paramagnet is tethered to the observed nuclear spin and only a restricted space in the immediate vicinity of the nuclear spin is accessible for pseudo-translational diffusion of one end of the molecule with respect to the other. A generalization of the Torrey model for magnetic relaxation by translational diffusion developed by Nevzorov and Freed, which includes the effect of restrictions imposed by the finite length of the chain, describes the experiment within experimental errors. A simple modification of the Hwang-Freed model that does not specifically include the dynamical effects of the finite tether also provides a good approximation to the data when the tether chain is sufficiently long.