An R(1ρ) expression for a spin in chemical exchange between two sites with unequal transverse relaxation rates

Increasing the accuracy of exchange parameters reporting on slow dynamics by performing CEST experiments with 'high' B1 fields

Over the last decade chemical exchange saturation transfer (CEST) NMR methods have emerged as powerful tools to characterize biomolecular conformational dynamics occurring between a visible major state and 'invisible' minor states. The ability of the CEST experiment to detect these minor states, and provide precise exchange parameters, hinges on using appropriate B1 field strengths during the saturation period. Typically, a pair of B1 fields with ω1 (=2πB1) values around the exchange rate kex are chosen. Here we show that the transverse relaxation rate of the minor state resonance (R2,B) also plays a crucial role in determining the B1 fields that lead to the most informative datasets. Using [Formula: see text]  ≥ kex, to guide the choice of B1, instead of kex, leads to data wherefrom substantially more accurate exchange parameters can be derived. The need for higher B1 fields, guided by K, is demonstrated by studying the conformational exchange in two mutants of the 71 residue FF domain with kex ∼ 11 s-1 and ∼ 72 s-1, respectively. In both cases analysis of CEST datasets recorded using B1 field values guided by kex lead to imprecise exchange parameters, whereas using B1 values guided by K resulted in precise site-specific exchange parameters. The conclusions presented here will be valuable while using CEST to study slow processes at sites with large intrinsic relaxation rates, including carbonyl sites in small to medium sized proteins, amide 15N sites in large proteins and when the minor state dips are broadened due to exchange among the minor states.

Nuclear magnetic resonance/single molecule fluorescence combinations to study dynamic protein systems

Many proteins require different structural states or conformations for function, and intrinsically disordered proteins, i.e. proteins without stable three-dimensional structure, are certainly an extreme. Single molecule fluorescence and nuclear magnetic resonance (NMR) spectroscopy are both exceptionally well suited to decipher and describe these states and their interconversion. Different time scales, from picoseconds to several milliseconds, can be addressed by both techniques. The length scales probed and the sample requirements (e.g. concentration, molecular weight, sample complexity) are, however, vastly different, making NMR and single molecule fluorescence an excellent combination for integrated studies. Here, we review recently undertaken approaches for the combined use of NMR and single molecule fluorescence to study protein dynamics.

Microsecond Motion of the Bacterial Transporter EmrE in Lipid Bilayers

The bacterial transporter EmrE is a homo-dimeric membrane protein that effluxes cationic polyaromatic substrates against the concentration gradient by coupling to proton transport. As the archetype of the small multidrug resistance family of transporters, EmrE structure and dynamics provide atomic insights into the mechanism of transport by this family of proteins. We recently determined high-resolution structures of EmrE in complex with a cationic substrate, tetra(4-fluorophenyl)phosphonium (F4-TPP+), using solid-state NMR spectroscopy and an S64V-EmrE mutant. The substrate-bound protein exhibits distinct structures at acidic and basic pH, reflecting changes upon binding or release of a proton from residue E14, respectively. To obtain insight into the protein dynamics that mediate substrate transport, here we measure 15N rotating-frame spin-lattice relaxation (R1ρ) rates of F4-TPP+-bound S64V-EmrE in lipid bilayers under magic-angle spinning (MAS). Using perdeuterated and back-exchanged protein and 1H-detected 15N spin-lock experiments under 55 kHz MAS, we measured 15N R1ρ rates site-specifically. Many residues show spin-lock field-dependent 15N R1ρ relaxation rates. This relaxation dispersion indicates the presence of backbone motions at a rate of about 6000 s-1 at 280 K for the protein at both acidic and basic pH. This motional rate is 3 orders of magnitude faster than the alternating access rate but is within the range estimated for substrate binding. We propose that these microsecond motions may allow EmrE to sample different conformations to facilitate substrate binding and release from the transport pore.

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.

Optimizing frequency sampling in CEST experiments

For the past decade chemical exchange saturation transfer (CEST) experiments have been successfully applied to study exchange processes in biomolecules involving sparsely populated, transiently formed conformers. Initial implementations focused on extensive sampling of the CEST frequency domain, requiring significant measurement times. Here we show that the lengthy sampling schemes often used are not optimal and that reduced frequency sampling schedules can be developed without a priori knowledge of the exchange parameters, that only depend on the chosen B₁ field, and, to a lesser extent, on the intrinsic transverse relaxation rates of ground state spins. The reduced sampling approach described here can be used synergistically with other methods for reducing measurement times such as those that excite multiple frequencies in the CEST dimension simultaneously, or make use of non-uniform sampling of indirectly detected time domains, to further decrease measurement times. The proposed approach is validated by analysis of simulated and experimental datasets.

RING NMR dynamics: software for analysis of multiple NMR relaxation experiments

Molecular motions are fundamental to the existence of life, and NMR spectroscopy remains one of the most useful and powerful methods to measure their rates and molecular characteristics. Multiple experimental methods are available for measuring the NMR relaxation properties and these can require different methods for extracting model parameters. We present here a new software application, RING NMR Dynamics, that is designed to support analysis of multiple relaxation types. The initial release of RING NMR Dynamics supports the analysis of exponential decay experiments such as T₁ and T₂, as well as CEST and R₂ and R_1ρ relaxation dispersion. The software runs on multiple operating systems in both a command line mode and a user-friendly GUI that allows visualizing and simulating relaxation data. Interaction with another program, NMRFx Analyst, allows drilling down from the derived relaxation parameters to the raw spectral data.

Protein Dynamics revealed by NMR Relaxation Methods

Structural biology often focuses primarily on three-dimensional structures of biological macromolecules, deposited in the Protein Data Bank (PDB). This resource is a remarkable entity for the world-wide scientific and medical communities, as well as the general public, as it is a growing translation into three-dimensional space of the vast information in genomic databases, e.g. GENBANK. There is, however, significantly more to understanding biological function than the three-dimensional coordinate space for ground-state structures of biomolecules. The vast array of biomolecules experiences natural dynamics, interconversion between multiple conformational states, and molecular recognition and allosteric events that play out on timescales ranging from picoseconds to seconds. This wide range of timescales demands ingenious and sophisticated experimental tools to sample and interpret these motions, thus enabling clearer insight into functional annotation of the PDB. NMR spectroscopy is unique in its ability to sample this range of timescales at atomic resolution and in physiologically relevant conditions using spin relaxation methods. The field is constantly expanding to provide new creative experiments, to yield more detailed coverage of timescales, and to broaden the power of interpretation and analysis methods. This review highlights the current state of the methodology and examines the extension of analysis tools for more complex experiments and dynamic models. The future for understanding protein dynamics is bright, and these extended tools bring greater compatibility with developments in computational molecular dynamics, all of which will further our understanding of biological molecular functions. These facets place NMR as a key component in integrated structural biology.

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

The rotating-frame spin relaxation rate constant, R_1ρ, is a powerful probe of macromolecular chemical and conformational dynamics in relaxation dispersion, CEST, and DEST NMR experiments. The R_1ρ 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.

Exchange saturation transfer and associated NMR techniques for studies of protein interactions involving high-molecular-weight systems

A brief overview of theoretical and experimental aspects of the Dark state Exchange Saturation Transfer (DEST) and lifetime line broadening ([Formula: see text]) NMR methodologies is presented from a physico-chemical perspective. We describe how the field-dependence of [Formula: see text] can be used for determining the exchange regime on the transverse spin relaxation time-scale. Some limitations of DEST/[Formula: see text] methodology in applications to molecular systems with intermediate molecular weights are discussed, and the means of overcoming these limitations via the use of closely related exchange NMR techniques is presented. Finally, several applications of DEST/[Formula: see text] methodology are described from a methodological viewpoint, with an emphasis on providing examples of how kinetic and relaxation parameters of exchange can be reliably extracted from the experimental data in each particular case.

Quantifying Microsecond Exchange in Large Protein Complexes with Accelerated Relaxation Dispersion Experiments in the Solid State

Solid state NMR is a powerful method to obtain information on the structure and dynamics of protein complexes that, due to solubility and size limitations, cannot be achieved by other methods. Here, we present an approach that allows the quantification of microsecond conformational exchange in large protein complexes by using a paramagnetic agent to accelerate ¹⁵N R_1ρ relaxation dispersion measurements and overcome sensitivity limitations. The method is validated on crystalline GB1 and then applied to a >300 kDa precipitated complex of GB1 with full length human immunoglobulin G (IgG). The addition of a paramagnetic agent increased the signal to noise ratio per time unit by a factor of 5, which allowed full relaxation dispersion curves to be recorded on a sample containing less than 50 μg of labelled material in 5 and 10 days on 850 and 700 MHz spectrometers, respectively. We discover a similar exchange process across the β-sheet in GB1 in crystals and in complex with IgG. However, the slow motion observed for a number of residues in the α-helix of crystalline GB1 is not detected in the complex.

Characterizing micro-to-millisecond chemical exchange in nucleic acids using off-resonance R1ρ relaxation dispersion

This review describes off-resonance R1ρ relaxation dispersion NMR methods for characterizing microsecond-to-millisecond chemical exchange in uniformly 13C/15N labeled nucleic acids in solution. The review opens with a historical account of key developments that formed the basis for modern R1ρ techniques used to study chemical exchange in biomolecules. A vector model is then used to describe the R1ρ relaxation dispersion experiment, and how the exchange contribution to relaxation varies with the amplitude and frequency offset of an applied spin-locking field, as well as the population, exchange rate, and differences in chemical shifts of two exchanging species. Mathematical treatment of chemical exchange based on the Bloch-McConnell equations is then presented and used to examine relaxation dispersion profiles for more complex exchange scenarios including three-state exchange. Pulse sequences that employ selective Hartmann-Hahn cross-polarization transfers to excite individual 13C or 15N spins are then described for measuring off-resonance R1ρ(13C) and R1ρ(15N) in uniformly 13C/15N labeled DNA and RNA samples prepared using commercially available 13C/15N labeled nucleotide triphosphates. Approaches for analyzing R1ρ data measured at a single static magnetic field to extract a full set of exchange parameters are then presented that rely on numerical integration of the Bloch-McConnell equations or the use of algebraic expressions. Methods for determining structures of nucleic acid excited states are then reviewed that rely on mutations and chemical modifications to bias conformational equilibria, as well as structure-based approaches to calculate chemical shifts. Applications of the methodology to the study of DNA and RNA conformational dynamics are reviewed and the biological significance of the exchange processes is briefly discussed.

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 R_1ρ, 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.

Unusual Proton Transfer Kinetics in Water at the Temperature of Maximum Density

Water exhibits numerous anomalous properties, many of which remain poorly understood. One of its intriguing behaviors is that it exhibits a temperature of maximum density (TMD) at 4 °C. We provide here new experimental evidence for hitherto unknown abrupt changes in proton transfer kinetics at the TMD. In particular, we show that the lifetime of OH^{-} ions has a maximum at this temperature, in contrast to hydronium ions. Furthermore, base-catalyzed proton transfer shows a sharp local minimum at this temperature, and activation energies change abruptly as well. The measured lifetimes agree with earlier theoretical predictions as the temperature approaches the TMD. Similar results are also found for heavy water at its own TMD. These findings point to a high propensity of forming fourfold coordinated OH^{-} solvation complexes at the TMD, underlining the asymmetry between hydroxide and hydronium transport. These results could help to further elucidate the unusual properties of water and related liquids.

Concerted millisecond timescale dynamics in the intrinsically disordered carboxyl terminus of γ-tubulin induced by mutation of a conserved tyrosine residue

Tubulins are an ancient family of eukaryotic proteins characterized by an amino-terminal globular domain and disordered carboxyl terminus. These carboxyl termini play important roles in modulating the behavior of microtubules in living cells. However, the atomic-level basis of their function is not well understood. These regions contain multiple acidic residues and their overall charges are modulated in vivo by post-translational modifications, for example, phosphorylation. In this study, we describe an application of NMR and computer Monte Carlo simulations to investigate how the modification of local charge alters the conformational sampling of the γ-tubulin carboxyl terminus. We compared the dynamics of two 39-residue polypeptides corresponding to the carboxyl-terminus of yeast γ-tubulin. One polypeptide comprised the wild-type amino acid sequence while the second contained a Y > D mutation at Y11 in the polypeptide (Y445 in the full protein). This mutation introduces additional negative charge at a site that is phosphorylated in vivo and produces a phenotype with perturbed microtubule function. NMR relaxation measurements show that the Y11D mutation produces dramatic changes in the millisecond-timescale motions of the entire polypeptide. This observation is supported by Monte Carlo simulations that-similar to NMR-predict the WT γ-CT is largely unstructured and that the substitution of Tyr 11 with Asp causes the sampling of extended conformations that are unique to the Y11D polypeptide.

Deciphering the Dynamic Interaction Profile of an Intrinsically Disordered Protein by NMR Exchange Spectroscopy

Intrinsically disordered proteins (IDPs) display a large number of interaction modes including folding-upon-binding, binding without major structural transitions, or binding through highly dynamic, so-called fuzzy, complexes. The vast majority of experimental information about IDP binding modes have been inferred from crystal structures of proteins in complex with short peptides of IDPs. However, crystal structures provide a mainly static view of the complexes and do not give information about the conformational dynamics experienced by the IDP in the bound state. Knowledge of the dynamics of IDP complexes is of fundamental importance to understand how IDPs engage in highly specific interactions without concomitantly high binding affinity. Here, we combine rotating-frame R_1ρ, Carr-Purcell-Meiboom Gill relaxation dispersion as well as chemical exchange saturation transfer to decipher the dynamic interaction profile of an IDP in complex with its partner. We apply the approach to the dynamic signaling complex formed between the mitogen-activated protein kinase (MAPK) p38α and the intrinsically disordered regulatory domain of the MAPK kinase MKK4. Our study demonstrates that MKK4 employs a subtle combination of interaction modes in order to bind to p38α, leading to a complex displaying significantly different dynamics across the bound regions.

Protein folding by NMR

Protein folding is a highly complex process proceeding through a number of disordered and partially folded nonnative states with various degrees of structural organization. These transiently and sparsely populated species on the protein folding energy landscape play crucial roles in driving folding toward the native conformation, yet some of these nonnative states may also serve as precursors for protein misfolding and aggregation associated with a range of devastating diseases, including neuro-degeneration, diabetes and cancer. Therefore, in vivo protein folding is often reshaped co- and post-translationally through interactions with the ribosome, molecular chaperones and/or other cellular components. Owing to developments in instrumentation and methodology, solution NMR spectroscopy has emerged as the central experimental approach for the detailed characterization of the complex protein folding processes in vitro and in vivo. NMR relaxation dispersion and saturation transfer methods provide the means for a detailed characterization of protein folding kinetics and thermodynamics under native-like conditions, as well as modeling high-resolution structures of weakly populated short-lived conformational states on the protein folding energy landscape. Continuing development of isotope labeling strategies and NMR methods to probe high molecular weight protein assemblies, along with advances of in-cell NMR, have recently allowed protein folding to be studied in the context of ribosome-nascent chain complexes and molecular chaperones, and even inside living cells. Here we review solution NMR approaches to investigate the protein folding energy landscape, and discuss selected applications of NMR methodology to studying protein folding in vitro and in vivo. Together, these examples highlight a vast potential of solution NMR in providing atomistic insights into molecular mechanisms of protein folding and homeostasis in health and disease.

General expressions for R1ρ relaxation for N-site chemical exchange and the special case of linear chains

Exploration of dynamic processes in proteins and nucleic acids by spin-locking NMR experiments has been facilitated by the development of theoretical expressions for the R_1ρ relaxation rate constant covering a variety of kinetic situations. Herein, we present a generalized approximation to the chemical exchange, R_ex, component of R_1ρ for arbitrary kinetic schemes, assuming the presence of a dominant major site population, derived from the negative reciprocal trace of the inverse Bloch-McConnell evolution matrix. This approximation is equivalent to first-order truncation of the characteristic polynomial derived from the Bloch-McConnell evolution matrix. For three- and four-site chemical exchange, the first-order approximations are sufficient to distinguish different kinetic schemes. We also introduce an approach to calculate R_1ρ for linear N-site schemes, using the matrix determinant lemma to reduce the corresponding 3N×3N Bloch-McConnell evolution matrix to a 3×3 matrix. The first- and second order-expansions of the determinant of this 3×3 matrix are closely related to previously derived equations for two-site exchange. The second-order approximations for linear N-site schemes can be used to obtain more accurate approximations for non-linear N-site schemes, such as triangular three-site or star four-site topologies. The expressions presented herein provide powerful means for the estimation of R_ex contributions for both low (CEST-limit) and high (R_1ρ-limit) radiofrequency field strengths, provided that the population of one state is dominant. The general nature of the new expressions allows for consideration of complex kinetic situations in the analysis of NMR spin relaxation data.

Quantifying protein dynamics in the ps-ns time regime by NMR relaxation

Both ¹⁵N chemical shift anisotropy (CSA) and sufficiently rapid exchange linebroadening transitions exhibit relaxation contributions that are proportional to the square of the magnetic field. Deconvoluting these contributions is further complicated by residue-dependent variations in protein amide ¹⁵N CSA values which have proven difficult to accurately measure. Exploiting recently reported improvements for the implementation of T₁ and T_1ρ experiments, field strength-dependent studies have been carried out on the B3 domain of protein G (GB3) as well as on the immunophilin FKBP12 and a H87V variant of that protein in which the major conformational exchange linebroadening transition is suppressed. By applying a zero frequency spectral density rescaling analysis to the relaxation data collected at magnetic fields from 500 to 900 MHz ¹H, differential residue-specific ¹⁵N CSA values have been obtained for GB3 which correlate with those derived from solid state and liquid crystalline NMR measurements to a level similar to the correlation among those previously reported studies. Application of this analysis protocol to FKBP12 demonstrated an efficient quantitation of both weak exchange linebroadening contributions and differential residue-specific ¹⁵N CSA values. Experimental access to such differential residue-specific ¹⁵N CSA values should significantly facilitate more accurate comparisons with molecular dynamics simulations of protein motion that occurs within the timeframe of global molecular tumbling.

Evaluating the influence of initial magnetization conditions on extracted exchange parameters in NMR relaxation experiments: applications to CPMG and CEST

Transient excursions of native protein states to functionally relevant higher energy conformations often occur on the μs-ms timescale. NMR spectroscopy has emerged as an important tool to probe such processes using techniques such as Carr-Purcell-Meiboom-Gill (CPMG) relaxation dispersion and Chemical Exchange Saturation Transfer (CEST). The extraction of kinetic and structural parameters from these measurements is predicated upon mathematical modeling of the resulting relaxation profiles, which in turn relies on knowledge of the initial magnetization conditions at the start of the CPMG/CEST relaxation elements in these experiments. Most fitting programs simply assume initial magnetization conditions that are given by equilibrium populations, which may be incorrect in certain implementations of experiments. In this study we have quantified the systematic errors in extracted parameters that are generated from analyses of CPMG and CEST experiments using incorrect initial boundary conditions. We find that the errors in exchange rates (k ex ) and populations (p E ) are typically small (<10 %) and thus can be safely ignored in most cases. However, errors become larger and cannot be fully neglected (20-40 %) as k ex falls near the lower limit of each method or when short CPMG/CEST relaxation elements are used in these experiments. The source of the errors can be rationalized and their magnitude given by a simple functional form. Despite the fact that errors tend to be small, it is recommended that the correct boundary conditions be implemented in fitting programs so as to obtain as robust estimates of exchange parameters as possible.

A dynamic look backward and forward

The 2015 Gunther Laukien Prize recognized solution NMR studies of protein dynamics and thermodynamics. This Perspective surveys aspects of the development and application of NMR spin relaxation for investigations of protein flexibility and function over multiple time scales in solution. Methods highlighted include analysis of overall rotational diffusion, theoretical descriptions of R1ρ relaxation, and molecular dynamics simulations to interpret NMR spin relaxation. Applications are illustrated for the zinc-finger domain Xfin-31, the calcium-binding proteins calbindin D9k and calmodulin, and the bZip transcription factor of GCN4.

Speeding-up exchange-mediated saturation transfer experiments by Fourier transform

Protein motions over various time scales are crucial for protein function. NMR relaxation dispersion experiments play a key role in explaining these motions. However, the study of slow conformational changes with lowly populated states remained elusive. The recently developed exchange-mediated saturation transfer experiments allow the detection and characterization of such motions, but require extensive measurement time. Here we show that, by making use of Fourier transform, the total acquisition time required to measure an exchange-mediated saturation transfer profile can be reduced by twofold in case that one applies linear prediction. In addition, we demonstrate that the analytical solution for R1ρ experiments can be used for fitting the exchange-mediated saturation transfer profile. Furthermore, we show that simultaneous analysis of exchange-mediated saturation transfer profiles with two different radio-frequency field strengths is required for accurate and precise characterization of the exchange process and the exchanging states.

Visualizing transient dark states by NMR spectroscopy

Myriad biological processes proceed through states that defy characterization by conventional atomic-resolution structural biological methods. The invisibility of these 'dark' states can arise from their transient nature, low equilibrium population, large molecular weight, and/or heterogeneity. Although they are invisible, these dark states underlie a range of processes, acting as encounter complexes between proteins and as intermediates in protein folding and aggregation. New methods have made these states accessible to high-resolution analysis by nuclear magnetic resonance (NMR) spectroscopy, as long as the dark state is in dynamic equilibrium with an NMR-visible species. These methods - paramagnetic NMR, relaxation dispersion, saturation transfer, lifetime line broadening, and hydrogen exchange - allow the exploration of otherwise invisible states in exchange with a visible species over a range of timescales, each taking advantage of some unique property of the dark state to amplify its effect on a particular NMR observable. In this review, we introduce these methods and explore two specific techniques - paramagnetic relaxation enhancement and dark state exchange saturation transfer - in greater detail.

An exact solution for R2,eff in CPMG experiments in the case of two site chemical exchange

The Carr-Purcell-Meiboom-Gill (CPMG) experiment is widely used to quantitatively analyse the effects of chemical exchange on NMR spectra. In a CPMG experiment, the effective transverse relaxation rate, R2,eff, is typically measured as a function of the pulse frequency, νCPMG. Here, an exact expression for how R2,eff varies with νCPMG is derived for the commonly encountered scenario of two-site chemical exchange of in-phase magnetisation. This result, summarised in Appendix A, generalises a frequently used equation derived by Carver and Richards, published in 1972. The expression enables more rapid analysis of CPMG data by both speeding up calculation of R2,eff over numerical methods by a factor of ca. 130, and yields exact derivatives for use in data analysis. Moreover, the derivation provides insight into the physical principles behind the experiment.

Evaluating the uncertainty in exchange parameters determined from off-resonance R1ρ relaxation dispersion for systems in fast exchange

Spin relaxation in the rotating frame (R1ρ) is a powerful NMR technique for characterizing fast microsecond timescale exchange processes directed toward short-lived excited states in biomolecules. At the limit of fast exchange, only k(ex)=k(1)+k(-1) and Φex=p(G)p(E)(Δω)(2) can be determined from R1ρ data limiting the ability to characterize the structure and energetics of the excited state conformation. Here, we use simulations to examine the uncertainty with which exchange parameters can be determined for two state systems in intermediate-to-fast exchange using off-resonance R1ρ relaxation dispersion. R1ρ data computed by solving the Bloch-McConnell equations reveals small but significant asymmetry with respect to offset (R1ρ (ΔΩ)≠R1ρ (-ΔΩ)), which is a hallmark of slow-to-intermediate exchange, even under conditions of fast exchange for free precession chemical exchange line broadening (k(ex)/Δω>10). A grid search analysis combined with bootstrap and Monte-Carlo based statistical approaches for estimating uncertainty in exchange parameters reveals that both the sign and magnitude of Δω can be determined at a useful level of uncertainty for systems in fast exchange (k(ex)/Δω<10) but that this depends on the uncertainty in the R1ρ data and requires a thorough examination of the multidimensional variation of χ(2) as a function of exchange parameters. Results from simulations are complemented by analysis of experimental R1ρ data measured in three nucleic acid systems with exchange processes occurring on the slow (k(ex)/Δω=0.2; pE=∼0.7%), fast (k(ex)/Δω=∼10-16; p(E)=∼13%) and very fast (k(ex)=39,000 s(-1)) chemical shift timescales.

Chemical exchange in biomacromolecules: past, present, and future

The perspective reviews quantitative investigations of chemical exchange phenomena in proteins and other biological macromolecules using NMR spectroscopy, particularly relaxation dispersion methods. The emphasis is on techniques and applications that quantify the populations, interconversion kinetics, and structural features of sparsely populated conformational states in equilibrium with a highly populated ground state. Applications to folding, molecular recognition, catalysis, and allostery by proteins and nucleic acids are highlighted.