Parsing Dynamics of Protein Backbone NH and Side-Chain Methyl Groups using Molecular Dynamics Simulations.

Experimental NMR spectroscopy and theoretical molecular dynamics (MD) simulations provide complementary insights into protein conformational dynamics and hence into biological function. The present work describes an extensive set of backbone NH and side-chain methyl group generalized order parameters for the Escherichia coli ribonuclease HI (RNH) enzyme derived from 2-µs microsecond MD simulations using the OPLS4 and AMBER-FF19SB force fields. The simulated generalized order parameters are compared with values derived from NMR 15N and 13CH2D spin relaxation measurements. The squares of the generalized order parameters, S2 for the N-H bond vector and Saxis2 for the methyl group symmetry axis, characterize the equilibrium distribution of vector orientations in a molecular frame of reference. Optimal agreement between simulated and experimental results was obtained by averaging S2 or Saxis2 calculated by dividing the simulated trajectories into 50 ns blocks (∼five times the rotational diffusion correlation time for RNH). With this procedure, the median absolute deviations (MAD) between experimental and simulated values of S2 and Saxis2 are 0.030 (NH) and 0.061 (CH3) for OPLS4 and 0.041 (NH) and 0.078 (CH3) for AMBER-FF19SB. The MAD between OPLS4 and AMBER-FF19SB are 0.021 (NH) and 0.072 (CH3). The generalized order parameters for the methyl group symmetry axis can be decomposed into contributions from backbone fluctuations, between-rotamer dihedral angle transitions, and within-rotamer dihedral angle fluctuations. Analysis of the simulation trajectories shows that (i) backbone and side chain conformational fluctuations exhibit little correlation and that (ii) fluctuations within rotamers are limited and highly uniform with values that depend on the number of dihedral angles considered. Low values of Saxis2, indicative of enhanced side-chain flexibility, result from between-rotamer transitions that can be enhanced by increased local backbone flexibility.

Deuterium spin relaxation of fractionally deuterated ribonuclease H using paired 475 and 950 MHz NMR spectrometers.

Deuterium (2H) spin relaxation of 13CH2D methyl groups has been widely applied to investigate picosecond-to-nanosecond conformational dynamics in proteins by solution-state NMR spectroscopy. The B0 dependence of the 2H spin relaxation rates is represented by a linear relationship between the spectral density function at three discrete frequencies J(0), J(ωD) and J(2ωD). In this study, the linear relation between 2H relaxation rates at B0 fields separated by a factor of two and the interpolation of rates at intermediate frequencies are combined for a more robust approach for spectral density mapping. The general usefulness of the approach is demonstrated on a fractionally deuterated (55%) and alternate 13C-12C labeled sample of E. coli RNase H. Deuterium relaxation rate constants (R1, R1ρ, RQ, RAP) were measured for 57 well-resolved 13CH2D moieties in RNase H at 1H frequencies of 475 MHz, 500 MHz, 900 MHz, and 950 MHz. The spectral density mapping of the 475/950 MHz data combination was performed independently and jointly to validate the expected relationship between data recorded at B0 fields separated by a factor of two. The final analysis was performed by jointly analyzing 475/950 MHz rates with 700 MHz rates interpolated from 500/900 MHz data to yield six J(ωD) values for each methyl peak. The J(ω) profile for each peak was fit to the original (τM, Sf2, τf) or extended model-free function (τM, Sf2, Ss2, τf, τs) to obtain optimized dynamic parameters.

Breaking down walls: Continuous potential models for internal motions in NMR spin relaxation.

Simple physical models for restricted diffusion in a potential, which provide important insights for NMR spin relaxation, usually are based on free diffusion within rigid boundaries or diffusion in relatively simple continuous potential energy surfaces. The diffusion-in-a-cone model is an example of the former and diffusion in an N-fold cosine potential is an example of the latter. The present work models restricted diffusion for arbitrary potential energy functions on the surface of a cone or a sphere, by expanding the potentials in Fourier or spherical harmonic series, respectively. The results exhibit simple relationships between generalized order parameters and effective correlation times, critical for analysis of experimental spin relaxation data, and illustrate the transition from diffusive-like to jump-like behavior in multi-well potentials.

LC-Photo-CIDNP hyperpolarization of biomolecules bearing a quasi-isolated spin pair: Magnetic-Field dependence via a rapid-shuttling device.

Liquid-state low-concentration photochemically induced dynamic nuclear polarization (LC-photo-CIDNP) is an emerging technology tailored to enhance the sensitivity of NMR spectroscopy via LED- or laser-mediated optical irradiation. LC-photo-CIDNP is particularly useful to detect solvent-exposed aromatic residues (Trp, Tyr), either in isolation or within polypeptides and proteins. This study investigates the magnetic-field dependence of the LC-photo-CIDNP of Trp-α-13C-ß,ß,2,4,5,6,7-d7, a Trp isotopolog bearing a quasi-isolated 1Hα-13Cαspin pair (QISP). We employed a new rapid-shuttling side-illumination field-cycling device that enables ultra-fast (90-120 ms) vertical movements of NMR samples within the bore of a superconducting magnet. Thus, LC-photo-CIDNP hyperpolarization occurs at low field, while hyperpolarized signals are detected at high field (700 MHz). Resonance lineshapes were excellent, and the effect of several fields (1.18-7.08 T range) on hyperpolarization efficiency could be readily explored. Remarkably, unprecedented LC-photo-CIDNP enhancements ε ≅ 1,200 were obtained at 50 MHz (1.18 T), suggesting exciting avenues to hypersensitive LED-enhanced NMR in liquids at low field.

Site-based description of R1ρ relaxation in local reference frames.

Nuclear magnetic spin relaxation in the presence of an applied radiofrequency field depends critically on chemical exchange processes that transfer nuclear spins between chemical or conformational environments with distinct resonance frequencies. Characterization of chemical exchange processes in R1ρ relaxation dispersion, CEST, and DEST experiments provides powerful insights into chemical and conformational kinetics of biological macromolecules. The present work reformulates expressions for magnetization evolution and the R1ρ relaxation rate constant by focussing on the orientations of the tilted rotating frames of reference for magnetization components in different sites, by treating the spin-locking field strength as a perturbation to free-precession evolution, and by applying the Homotopy Analysis and Laplace transform methods to approximate solutions to the Bloch-McConnell equations. The results provide an expression for R1ρ that is invariant to the topology of the kinetic scheme, an approximate equation for evolution of spin-locked z-magnetization, and an approach for effective simplification of chemical exchange topologies for 2- and N-site chemical exchange processes. The theoretical approach also provides an accurate approximation for relaxation during a constant-amplitude radiofrequency field in the absence of exchange.

Comparisons of Ribonuclease HI Homologs and Mutants Uncover a Multistate Model for Substrate Recognition.

Ribonuclease HI (RNHI) nonspecifically cleaves the RNA strand of RNA:DNA hybrid duplexes in a myriad of biological processes. Several RNHI homologs contain an extended domain, termed the handle region, which is critical to substrate binding. Nuclear magnetic resonance (NMR) spectroscopy and molecular dynamics (MD) simulations have suggested a kinetic model in which the handle region can exist in open (substrate-binding competent) or closed (substrate-binding incompetent) states in homologs containing arginine or lysine at position 88 (using sequence numbering of E. coli RNHI), while the handle region populates states intermediate between the open and closed conformers in homologs with asparagine at residue 88 [Stafford, K. A., et al., PLoS Comput. Biol. 2013, 9, 1-10]. NMR parameters characterizing handle region dynamics are highly correlated with enzymatic activity for RNHI homologs with two-state (open/closed) handle regions [Martin, J. A., et al., Biochemistry 2020, 59, 3201-3205]. The work presented herein shows that homologs containing asparagine 88 display distinct structural features compared with their counterparts containing arginine or lysine 88. Comparisons of RNHI homologs and site-directed mutants with asparagine 88 support a kinetic model for handle region dynamics that includes 12 unique transitions between eight conformations. Overall, these findings present an example of the structure-function relationships of enzymes and spotlight the use of NMR spectroscopy and MD simulations in uncovering fine details of conformational preferences.

Dynamical Models of Chemical Exchange in Nuclear Magnetic Resonance Spectroscopy.

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

Bootstrap Aggregation for Model Selection in the Model-free Formalism.

The ability to make robust inferences about the dynamics of biological macromolecules using NMR spectroscopy depends heavily on the application of appropriate theoretical models for nuclear spin relaxation. Data analysis for NMR laboratory-frame relaxation experiments typically involves selecting one of several model-free spectral density functions using a bias-corrected fitness test. Here, advances in statistical model selection theory, termed bootstrap aggregation or bagging, are applied to 15N spin relaxation data, developing a multimodel inference solution to the model-free selection problem. The approach is illustrated using data sets recorded at four static magnetic fields for the bZip domain of the S. cerevisiae transcription factor GCN4.

Approximate Representations of Shaped Pulses Using the Homotopy Analysis Method.

The evolution of nuclear spin magnetization during a radiofrequency pulse in the absence of relaxation or coupling interactions can be described by three Euler angles. The Euler angles in turn can be obtained from the solution of a Riccati differential equation; however, analytic solutions exist only for rectangular and chirp pulses. The Homotopy Analysis Method is used to obtain new approximate solutions to the Riccati equation for shaped radiofrequency pulses in NMR spectroscopy. The results of even relatively low orders of approximation are highly accurate and can be calculated very efficiently. The results are extended in a second application of the Homotopy Analysis Method to represent relaxation as a perturbation of the magnetization trajectory calculated in the absence of relaxation. The Homotopy Analysis Method is powerful and flexible and is likely to have other applications in magnetic resonance.

Dimerization of Cadherin-11 involves multi-site coupled unfolding and strand swapping.

Cadherin extracellular domain 1 (EC1) mediates homophilic dimerization in adherens junctions. Conserved Trp2 and Trp4 residues in type II cadherins anchor the EC1 A strand intermolecularly in strand-swapped dimers. Herein, NMR spectroscopy is used to elucidate the roles of Trp2 and Trp4 in Cadherin-11 dimerization. The monomeric state, with the A strand and Trp side chains packed intramolecularly, is in equilibrium with sparsely populated partially and fully A-strand-exposed states, in which Trp2 (and Trp4, respectively) side-chain packing is disrupted. Exchange kinetics between the major state and the partially (fully) A-strand-exposed state is slow-intermediate (intermediate-fast). A separate very fast process exchanges ordered and random-coil BC-loop conformations with populations dependent on A-strand exposure and dimerization status. In addition, very slow processes connect the folded A-strand-exposed conformation to partially unfolded states, which may represent additional domain-swapping intermediates. The dimerization mechanism of type II cadherins is revealed as coupled folding and strand swapping.

Asterix/Gtsf1 links tRNAs and piRNA silencing of retrotransposons.

The Piwi-interacting RNA (piRNA) pathway safeguards genomic integrity by silencing transposable elements (transposons) in the germline. While Piwi is the central piRNA factor, others including Asterix/Gtsf1 have also been demonstrated to be critical for effective silencing. Here, using enhanced crosslinking and immunoprecipitation (eCLIP) with a custom informatic pipeline, we show that Asterix/Gtsf1 specifically binds tRNAs in cellular contexts. We determined the structure of mouse Gtsf1 by NMR spectroscopy and identified the RNA-binding interface on the protein's first zinc finger, which was corroborated by biochemical analysis as well as cryo-EM structures of Gtsf1 in complex with co-purifying tRNA. Consistent with the known dependence of long terminal repeat (LTR) retrotransposons on tRNA primers, we demonstrate that LTR retrotransposons are, in fact, preferentially de-repressed in Asterix mutants. Together, these findings link Asterix/Gtsf1, tRNAs, and LTR retrotransposon silencing and suggest that Asterix exploits tRNA dependence to identify transposon transcripts and promote piRNA silencing.

Algebraic expressions for Carr-Purcell-Meiboom-Gill relaxation dispersion for N-site chemical exchange.

The Carr-Purcell-Meiboom-Gill (CPMG) NMR relaxation dispersion experiment measures the effective relaxation rate constant during a train of spin-echo pulse sequence elements as a function of the echo time. The CPMG experiment is a powerful method for characterizing chemical and conformational dynamic processes, termed chemical and conformational exchange, on µs-ms time scales, comparable to the experimentally accessible echo times. Approximate theoretical expressions for the effective relaxation rate constant for N-site chemical exchange have been reported (H. Koss, M. Rance, and A. G. Palmer, Biochemistry 57, 4753-4763 (2018)). Expressions for the effective relaxation rate constant have been improved by using the Cayley-Hamilton theorem to obtain simple and accurate approximations of the average Liouvillian for the CPMG experiment. The improved accuracy of the results allows efficient analyses of experimental data. In addition, the relationship is clarified between the approach of Koss and coworkers and that of Jen (1978).

Quantifying the Relationship between Conformational Dynamics and Enzymatic Activity in Ribonuclease HI Homologues.

Ribonuclease HI (RNHI), a ubiquitous, non-sequence-specific endonuclease, cleaves the RNA strand in RNA/DNA hybrids. RNHI functions in replication and genome maintenance, and retroviral reverse transcriptases contain an essential ribonuclease H domain. Nuclear magnetic resonance (NMR) spectroscopy combined with molecular dynamics (MD) simulations suggests a model in which the extended handle region domain of Escherichia coli RNHI populates (substrate-binding-competent) "open" and (substrate-binding-incompetent) "closed" states, while the thermophilic Thermus thermophilus RNHI mainly populates the closed state at 300 K [Stafford, K. A., Robustelli, P., and Palmer, A. G., III (2013) PLoS Comput. Biol. 9, 1-10]. In addition, an in silico-designed mutant E. coli Val98Ala RNHI was predicted to populate primarily the closed state. The work presented here validates this model and confirms the predicted properties of the designed mutant. MD simulations suggest that the conformational preferences of the handle region correlate with the conformations of Trp85, Thr92, and Val101. NMR residual dipolar coupling constants, three-bond scalar coupling constants, and chemical shifts experimentally define the conformational states of these residues and hence of the handle domain. These NMR parameters correlate with the Michaelis constants for RNHI homologues, confirming the important role of the handle region in the modulation of substrate recognition and illustrating the power of NMR spectroscopy in dissecting the conformational preferences underlying enzyme function.

Compact expressions for R1ρ relaxation for N-site chemical exchange using Schur decomposition.

The rotating-frame spin relaxation rate constant, R1ρ, is a powerful probe of macromolecular chemical and conformational dynamics in relaxation dispersion, CEST, and DEST NMR experiments. The R1ρ relaxation rate constant is given by the absolute value of the largest (least negative) eigenvalue of the Bloch-McConnell evolution matrix; however, estimation of this eigenvalue require inversion of 3 N × 3 N dimensional matrices, in which N is the number of interconverting sites or states for a given nuclear spin in a molecule. The Schur complement is used to reduce the problem of calculating the characteristic polynomial of a 3 N × 3 N matrix to that of calculating the characteristic polynomial of a 3 × 3 matrix. The resulting expressions for N-site chemical exchange are more numerically tractable, because the largest matrix inversion also is of dimension 3 × 3. In addition, the simplifications offered by the Schurr complement conveniently illustrate the effects of fast or slow kinetic steps within an N-site kinetic topology.

Detection of chemical exchange in methyl groups of macromolecules.

The zero- and double-quantum methyl TROSY Hahn-echo and the methyl 1H-1H dipole-dipole cross-correlation nuclear magnetic resonance experiments enable estimation of multiple quantum chemical exchange broadening in methyl groups in proteins. The two relaxation rate constants are established to be linearly dependent using molecular dynamics simulations and empirical analysis of experimental data. This relationship allows chemical exchange broadening to be recognized as an increase in the Hahn-echo relaxation rate constant. The approach is illustrated by analyzing relaxation data collected at three temperatures for E. coli ribonuclease HI and by analyzing relaxation data collected for different cofactor and substrate complexes of E. coli AlkB.

Chemical Exchange.

The phenomenon of chemical or conformational exchange in NMR spectroscopy has enabled detailed characterization of time-dependent aspects of biomolecular function, including folding, molecular recognition, allostery, and catalysis, on timescales from microsecond to second. Importantly, NMR methods based on a variety of spin relaxation parameters have been developed that provide quantitative information on interconversion kinetics, thermodynamic properties, and structural features of molecular states populated to a fraction of a percent at equilibrium and otherwise unobservable by other NMR approaches. The ongoing development of more sophisticated experimental techniques and the necessity to apply these methods to larger and more complex molecular systems engenders a corresponding need for theoretical advances describing such techniques and facilitating data analysis in applications. This review surveys current aspects of the theory of chemical exchange, as utilized in ZZ-exchange; Hahn and Carr-Purcell-Meiboom-Gill (CPMG) spin-echo; and R1ρ, chemical exchange saturation transfer (CEST), and dark state saturation transfer (DEST) spin-locking experiments. The review emphasizes theoretical results for kinetic topologies with more than two interconverting states, both to obtain compact analytical forms suitable for data analysis and to establish conditions for distinguishability between alternative kinetic schemes.

Analysis of NMR Spin-Relaxation Data Using an Inverse Gaussian Distribution Function.

Spin relaxation in solution-state NMR spectroscopy is a powerful approach to explore the conformational dynamics of biological macromolecules. Probability distribution functions for overall or internal correlation times have been used previously to model spectral density functions central to spin-relaxation theory. Applications to biological macromolecules rely on transverse relaxation rate constants, and when studying nanosecond timescale motions, sampling at ultralow frequencies is often necessary. Consequently, appropriate distribution functions necessitate spectral density functions that are accurate and convergent as frequencies approach zero. In this work, the inverse Gaussian probability distribution function is derived from general properties of spectral density functions at low and high frequencies for macromolecules in solution, using the principle of maximal entropy. This normalized distribution function is first used to calculate the correlation function, followed by the spectral density function. The resulting model-free spectral density functions are finite at a frequency of zero and can be used to describe distributions of either overall or internal correlation times using the model-free ansatz. To validate the approach, 15N spin-relaxation data for the bZip transcription factor domain of the Saccharomyces cerevisiae protein GCN4, in the absence of cognate DNA, were analyzed using the inverse Gaussian probability distribution for intramolecular correlation times. The results extend previous models for the conformational dynamics of the intrinsically disordered, DNA-binding region of the bZip transcription factor domain.

Optimized coherence selection.

A theoretical framework for optimized coherence order selection in regards to cogwheel phase cycling is developed based on analysis of solutions of the associated homogeneous linear Diophantine equation. The empirical Hughes-Carravetta-Levitt conjectures are derived. A general formula is constructed that is guaranteed to provide valid phase cycles. Optimality of this formula is proven for non-sparse sets of coherence levels with relatively prime lower absolute bounds. Smaller phase cycles can be constructed when these conditions do not hold. The resultant formula can be incorporated into NMR spectrometer software to automatically generate cogwheel phase cycles.