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 worldwide 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 co-ordinate 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 insights 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.
The structural biology of proteins and nucleic acids has undergone many revolutions in the past few decades, and there are now multiple techniques, including both solution and solid-state NMR in addition to X-ray crystallography and cryoEM to determine accurate three-dimensional structures of these biomolecules. Current trends are to combine methods (hybrid methodology) [1–3] to reveal ever more complex structures and the nature of assemblies into larger multicomponent structures. Equally important to the geometrical structure are the motions within these molecules that take place across a very wide timescale (Figure 1). The motions are coupled to biological function, and developing an understanding of these motions and the states that are interconverting will be critical to advancing beyond a rigid ‘lego block’ assembly of molecular complexes. The ability to detect the presence of motions, to assign timescales to the motions, to gain insight into exchange rate and populations of interconverting states [4,5], and to determine structural features [chemical shifts, PRE, RDCs (residual dipolar couplings)] [6,7] of the low-population (invisible) states is unique to NMR spectroscopy.
Correlation of molecular motions with NMR observables.
In the NMR experiment, the time behavior of spins (/nuclei/resonances) within the structural components is modulated by the dynamic processes and can be extracted through experiments that are tailored to specific time regimes. The general principle is to create an initial state of the spin magnetization, allow it to evolve in the presence of the dynamic/exchange process, and, at the end of the evolution, detect the change in the magnetization. This is related to the general principle of multidimensional NMR spectroscopy [8,9], and it enables the encoding of an exchange process onto the observable NMR signal of different nuclei within the biomolecule. The development of methodologies has been the ingenuity to create different initial states and the control and selection of the processes that are effective during the evolution. The ability to sample separate and independent time regimes is extremely powerful, and in recent years, the toolbox of methods has grown increasingly sophisticated, while, at the same time, becoming more accessible to the broad community of structural biologists. Previous reviews [10–15] have provided careful descriptions of individual experiments and the associated tools to interpret the data.
It is convenient to classify/separate the intramolecular motions into structural categories of (i) local motions of bond vectors and torsion angles, (ii) segmental motions (e.g. loops or secondary structural elements), and (iii) domain motions (e.g. subdomains of a larger molecule connected by linkers exhibiting flexibility to varying extents). The current NMR tool box of dynamics methods provides insights into each of these motional regimes. The methods are developed around the timescales of the motion and will be described in this framework.
At the present stage of development, key issues involve the interpretation of dynamics data into physically meaningful models that can be correlated with biological function. The use of the common two-site exchange model may be intrinsically illogical in relation to complex biological molecules and the insights provided by increasingly long molecular dynamics trajectories; however, mathematical complexities have until recently imposed this predominant limitation. Here, we provide a review of the current methodologies and representative accomplishments, based on the different time regimes that may be evaluated, and discuss the pitfalls and solutions to interpretations of relaxation dispersion experiments.
Fast dynamics (ps–ns)
Examination of the fast time regime, representing local motions of the protein backbone or side-chain rotational motions, was revolutionized in the late 1980s and early 1990s [16–19] with the development of two-dimensional pulse schemes and inverse detection approaches for making relaxation time experiments. These techniques dramatically reduced problems with low sensitivity and low resolution of the one-dimensional direct-detected 13C/15N relaxation experiments. The approach established the framework for using two- and three-dimensional spectroscopy to sample relaxation phenomena, and the experiments are now rather mature and established . The standard set of experiments, longitudinal relaxation rate (R1), transverse relaxation rate (R2), and heteronuclear steady state NOE (ssNOE), have become popular and routine experiments to measure the protein backbone dynamics at the ps–ns timescale are readily available. Interpretation of the R1, R2, ssNOE data involves the evaluation of various motional models and approaches, such as anisotropic model-free analysis [17,21] and spectral density mapping [22,23]. These tools, available in software packages such as RELAX [24–26], ModelFree , and others , provide robust analyses of the standard experimental data. Insights into the local flexibility/motions of the protein backbone are readily obtained in these analyses and, while often expressed as simply a plot of the ssNOE versus residue number, more complete analysis [29,30] reveals the overall tumbling time, τc, for the molecule, the local fast correlation time, τf, or the frequencies of the local motions (ns–ps), and a measure of the amplitudes of the local motions of each residue expressed as the order parameter S2 for the NH vector. An early, significant application of this technique suggested correlation across a hierarchy of timescales within mesophilic and thermophilic variants of adenylate kinases , which was interpreted as the fast local dynamics in hinge regions facilitating the large and collective motions to produce a catalytically competent state. Similarly, conformational entropy of the protein backbone was demonstrated to be the only/primary driving force for the biological activity/functions of catabolite activator protein . Insights into intrinsically unfolded proteins are also available from such experiments for multiple field strengths, ModelFree analyses, and spectral density mapping . These examples illustrate the power of revealing functional consequences of dynamics that is not discernible from the static three-dimensional structure, regardless of the technique used to determine the structure. Later, following the discovery of methyl-TROSY effects  and the invention of straightforward, specific methyl-labeling schemes , various relaxation experiments were developed to measure methyl side-chain dynamics [36–41]. The use of methyl probes provides greatly improved sensitivity and resolution, which enables the investigation of very challenging systems using solution NMR spectroscopy and expands our understanding of the dynamics in large protein complexes. The application of this new technique has led to several important discoveries in biological systems, such as the molecular recognition through conformational entropy , the dynamic behaviors of the 20S proteasome , and the allosteric signals transmitted through the hydrophobic core of protein kinase A  (Figure 2).
Ligand binding and the corresponding changes in fast dynamics of methyl groups in the catalytic core of protein kinase A.
Intermediate dynamics (μs–ms)
In the μs–ms timescale, there are many important biological functions coupled to molecular motions, such as catalysis, ligand binding, and allostery (Figure 1). Motions in this timescale modulate the transverse relaxation times of nuclei, some of which can be seen directly in the linewidth of resonances. More generally, motions in this time regime, particularly when the dynamics is an exchange between two states where one state is present in a relatively low population, are detected via changes in R2 or the rotating-frame relaxation rate (R1ρ), as a function of the averaging process created by the pulse sequence, referred to as relaxation dispersion [45,46]. Often the spectrum of the system of interest is dominated by the ground-state structure and exhibits only the spectral features of that state. The low-population state is not observed directly and is only revealed as modulations of magnetization corresponding to the high-population state; hence, these tools are often described as ‘seeing the invisible’. Among the different relaxation dispersion techniques, Carr-Purcell–Meiboom-Gill (CPMG) spin-echo experiments and R1ρ spin-lock experiments are the most classical and popular experiments for probing the conformational dynamics of macromolecules at the μs–ms timescale. These methods were adapted from measuring chemical exchange phenomena in small molecules [47–51] to incorporation in multidimensional, inverse-detected experiments to investigate biological dynamic processes in macromolecules with atomic resolution [52–55]. Utilizing a two-site exchange model of the dynamic process, analytic descriptions have been derived [56–59] that yield a description of the populations of the two sites (pa and pb), the exchange rate between the two sites (kex), and the chemical shift difference between the two sites (Δω). The chemical shifts reveal insights into the structure of the minor, invisible state [60,61]. Moreover, due to the simple implementation and the accurate analytic descriptions, the application of the CPMG technique has led to many accomplishments in understanding various biological functions, such as the roles of conformational dynamics in the active site of the ribonuclease binase , the influence of conformational exchange in the catalytic steps in dihydrofolate reductase , the revelation of invisible and excited states of proteins [60,61], the correlation of conformational exchange rate with the turnover rate of prolyl cis–trans isomerase cyclophilin A , and initiation and regulation of allostery along reaction pathways [64,65]. In these examples, the extracted chemical shifts of the minor states, obtained from Δω in the CPMG analysis, are frequently used to infer structural information about the minor state. The other types of experiments, R1ρ spin-lock experiments, are used in both solution-state NMR and solid-state NMR to probe fast conformational dynamics in soluble proteins  and membrane proteins . In these applications, the R1ρ experiments are generally of higher sensitivity compared with constant-time CPMG experiments and have the potential to probe faster exchange events [68,69].
Recently, several new types of experiments [70–73] have been designed to either qualitatively or quantitatively extract the information regarding conformational dynamics at the μs–ms timescale. One of the most sophisticated and powerful experiments is the adiabatic relaxation dispersion experiment [72,74,75] (Figure 3), which has been shown to be able to probe a wide range of conformational dynamics with the exchange rates from 102 to 105 s−1 . This adiabatic relaxation dispersion experiment does not yield analytic solutions for the behavior of the magnetization, and the full power of the experiment must be coupled with a powerful new data analysis tool, geometric approximation. The experimental and data analysis combination will be a new and powerful tool for interrogating the dynamic processes across a wide timescale in biological macromolecules . The geometric approximation methodology also enables the consideration of more complex exchange models to analyze both the adiabatic relaxation dispersion experiments and the traditional CPMG experiments [75,76] (vide infra).
The pulse sequences of adiabatic relaxation dispersion experiments.
Slow dynamics (ms–s)
The slow time regime of ms–s embodies many important biological events, including unfolding events and complex formation between multiple components. This time regime was originally accessible by zz-exchange spectroscopy [77,78], when the population of the two states is sufficient to directly detect both species (e.g. ≥80 : 20). The detectable exchange rate is limited by the T1 of the observed nuclei (nominally 0.1–10 s−1) , and the data can directly identify the exchanging resonances in each state . The presence of low-population states in this exchange regime was previously very difficult to identify and quantitate. However, recently, the use of chemical exchange saturation transfer (CEST  or DEST ) spectroscopy has been shown to be a facile and powerful tool to examine this time regime, which can be traced back to its early application to a small molecule . The spectroscopic methods have evolved to use multidimensional, indirect detection schemes, including non-uniform sampling  (similar to the evolution of CPMG and R1ρ experiments), and there has been rapid development to provide a range of experiments enabling monitoring backbone and side-chain sites, as well as 1H, 15N, and 13C nuclei . These methods are sensitive to exchange processes that occur in the 20–200 s−1 range, and they can reveal ‘invisible’ states with populations as low as 1% . They are very powerful for revealing the kinetic parameters for these moderately slow processes, and the future is bright to detect exchange events with high sensitivity. The ability to obtain chemical shifts and, in favorable cases, RDCs of the low-population state (which could be an unfolded, alternate conformation or ligand/complex-bound state) can lead to structural determination of these invisible states. Analyses of these exchange processes have also been performed primarily using a two-state model with numerical integration/analysis . Extension of this model to more complex topologies has not been reported, and the general problem remains formidable (vide infra).
Accurate data analysis of relaxation dispersion experiments
There is widespread interest in understanding the correlation between biological processes and dynamics in macromolecules using relaxation dispersion experiments [85–87]. The earlier techniques, such as CPMG and R1ρ experiments, together with recent CEST and adiabatic relaxation dispersion techniques can reveal significantly more information for these dynamic processes. However, the insight sought is critically dependent on accurate analyses and reliable interpretation of the experimental data. In general, there are several important factors that determine the accuracy of the fit parameters during data analysis. Often, the focus is either on providing the most ‘accurate solutions’ [57,59,88,89] or on the best sampling schemes for finding the global minima [90–92], where the algorithm searches for the best match (or the lowest deviation) between experimental data and the provided solutions. However, the extents to which the amount of information in a given data set and the potential experimental errors affect the accuracy of the data analysis are often ignored or given little discussion in the literature. Indeed, these last two problems are more intractable. In this section, we demonstrate how new tools or approaches can help to investigate the problems, to provide deeper insights into the problems, as well as to provide possible solutions. Mathematically, it is definitely possible to distinguish between exchange models by investigating the high-resolution, theoretically simulated solution surfaces for a given relaxation dispersion experiment. However, practically, it is virtually impossible to obtain enough data points to represent the complete geometric features of the solution surface of an unknown exchange model by varying experimental conditions. Thus, model selection is one of the most notorious problems in the data analysis of relaxation dispersion experiments, and there is significant impetus to develop robust tools to guide researchers.
Geometric approximation, why bother?
To date, analytic tools and solutions have been well established to achieve the following: (i) to analyze the steady-state information of the spin system (such as rotating-frame R1ρ and free-precession R2 relaxation rates) [57,88,93], (ii) to accelerate the numerical analysis of certain data sets (such as adiabatic R1ρ relaxation rates) [94,95], and (iii) to analyze certain data sets under simple exchange models (such as CPMG data in the two-site exchange model) [58,59]. Numerical analysis/integration has been the predominate method to analyze relaxation dispersion data from the conventional experiments (CPMG, R1ρ) with complex exchange models (more than two sites) [96–98] or from other complicated experiments (such as CEST) . However, in the complex cases, such as the numerical analysis of adiabatic relaxation dispersion data, mathematical integration is very computationally expensive and limits the number of the sampling points. Practically, these limitations, together with the potential bias of the local gradients, frequently trap the deterministic algorithm in local minima of the energy surface [99,100] (Figure 4b,d). Recently, geometric approximation was introduced as a new computational concept designed to enable analysis of sophisticated physical phenomena lacking simple analytic descriptions and to accelerate the data analysis . Fortunately, the NMR experiment is uniquely suited to complete simulation of spin behavior. Hence, it is possible to simulate the magnetization response to arbitrary experimental pulse schemes for either simple two-site or any other more complex models [75,76]. The simulation can be quite time-consuming; however, it only needs to be performed once (using large computational resources). The output creates libraries that can be searched and interpolated, using geometric approximation, to find the matching solution for the observed behavior in the corresponding experimental data. Thus, geometric approximation together with stochastic Monte Carlo sampling [75,76] provides a better likelihood to achieve the global minima in an efficient manner, as judged by comparison of best fit results with simulated input data for both numerical integration and geometric approximation (Figure 4a–d). Although it can be challenging to build libraries with manageable sizes for a given high-dimensional solution surface, we have demonstrated novel reduction approaches to deal with adiabatic relaxation dispersion and three-site exchange CPMG [75,76]. Because the method is used to approximate numerical solutions, it has no fundamental restriction on the information that can be analyzed and can broadly treat both steady-state and non-steady-state information, as well as mixtures of the two in complex pulse schemes (such as some non-adiabatic behaviors of spin systems as well as adiabatic R2ρ relaxation rates) [75,101]. Looking forward, the geometric approximation methodology will allow the accurate and efficient analyses of relaxation dispersion profiles induced by any custom-tailored shaped pulses or other pulse schemes, which we anticipate will stimulate the further development of more sophisticated and powerful relaxation dispersion techniques.
Comparison of the data analysis of adiabatic relaxation dispersion experiments using geometric approximation and conventional numerical analysis. Relaxation dispersion input data were simulated from random values of kex (0.1–100 ks−1), Δω (1–8 ppm), pa (55–99%).
Under-interpretation or over-interpretation?
Model selection is one of the most problematic issues for the analysis of relaxation dispersion data. Owing to the limited number of the relaxation rates in a given data set, the limited number of experimental replicates, the systematic experimental errors, and other factors, it is potentially dangerous to use only the χ2 analysis or any other information criterion to select a specific model without careful validations. Furthermore, the use of a simple two-site exchange model is clearly an oversimplification when there are many possible conformational exchange events in macromolecules in solution. The model is the predominant approach for two reasons: (i) it is mathematically and computationally tractable and (ii) if there is sufficiently fast exchange of multiple minor states, then they can be modeled as one ‘average’ second state. However, to provide functional insights into biological processes, it is critical to provide valid assessment of the exchanging states/sites and the accuracy of the extracted parameters. Although it is difficult to address this type of problem rigorously, we can reveal some insights and provide possible solutions based on statistical analyses combined with our geometric approximation methodology.
First, consider the case of under-interpretation of CPMG relaxation dispersion data sets, using input data simulated for two linear three-site exchange models which are then fit using geometric approximation with a two-state model (Figure 5). All dynamic parameters are randomly selected to simulate input data based on three-site McConnell equations. The linear three-site exchange model can be conceptually decomposed into two exchange events: (i) one process is dominant in the relaxation dispersion profiles, existing between the most populated state and the connected minor state and (ii) the other processes act as interfering events (or as a type of ‘experimental noise’). Surprisingly, we find that the fit kex values closely approximate the input kex values of the dominant exchange events, as long as these events are within the detectable CPMG regime in our experimental settings (kex < 2π × 1000 s−1) (Figure 5a,b). The accuracy of the fit Δω values will be compromised more if the interfering exchange event is connected to the minor state B (Figure 5c). On the contrary, no matter where the interfering exchange event is introduced, the fit population values of the state A are always overestimated in our test conditions (Figure 5e,f). These results suggest that we can still reliably extract the kex values of the dominant exchange events within the experimentally detectable window, even if there are other exchange events interfering with or breaking our simple assumption of the two-site exchange model.
Statistical analysis of under-interpretation of CPMG relaxation dispersion data.
In the case of over-interpretation of CPMG relaxation dispersion data, simulated data are created using the simple two-site exchange model in the McConnell equations, and the data are fit using geometric approximation with a linear two-site exchange model and two randomly chosen Δω values, which are fixed during data analysis. We find that the data (300 data sets, each of which has data points at two different magnetic fields and νCPMG ranging from 25 to 1000 Hz) can still be fit moderately well with an average deviation of 8% from the simulated values. In some cases, the data can be fit very well by adjusting kex values and population differences even when the given Δω values are incorrect (Figure 6). Hence, the fits suggest that the data analysis using a three-site model requires a large amount of experimental data and additional validations to avoid misinterpretations in general.
Examples of over-interpretation of CPMG relaxation dispersion data.
Although actual experimental situations are potentially more complicated than the cases tested here, the results (Figures 5 and 6) imply that it may be safer to ‘under-interpret’ (using a two-site model) rather than to ‘over-interpret’ (using a three-site or n-site model) unless there are independent experimental indications of the need for a more complex model. An indication of a more complex model may be the significant deviations of the fit dynamic parameters from other experimental results, such as chemical shift differences and population differences (Figure 5c–f), and it is necessary to have as much relaxation dispersion data as possible, including multiple magnetic field strengths, and possibly variable temperature or other perturbations of the process. Moreover, the derived Δω values can be used to identify the detected chemical exchange events by comparing them with chemical shift perturbation data [5,87,97]; however, the derived Δω values are more sensitive to other interfering exchange events and may not exactly match observed perturbation data, or the perturbation data may not be available (as for an intramolecular process). We demonstrate here that utilization of simulations and geometric approximation provides a new approach and tools to characterize these intractable problems, with the potential to explore and find possible solutions.
The power of the complete toolkit of relaxation experiments covering the ps–ns through the ms–s range is very impressive to annotate the growing structural biology information available to the biological community. The accessibility of methods that can be applied to small isolated proteins [96,102] (or domains) through to large multicomponent biological machines [43,97] illustrates the powerful role for NMR spectroscopy in the future. Importantly, the ability to examine the timescale of 102–105 s−1 via the recent adiabatic relaxation dispersion experiments, coupled with geometric approximation analyses, indicates that a broader examination of allosteric events will be forthcoming. Thus, the contribution of NMR to the broad structural biology field, including the hybrid methodologies, will remain critical and play an increasingly significant role in understanding structure and function.
Protein dynamics can be assessed at atomic resolution over timescales from picoseconds to seconds using NMR relaxation methods.
NMR relaxation methods enable the sampling and description of states that are otherwise invisible and enrich the annotation of static, ground-state structures in the context of integrated structural biology.
Development of methods in the millisecond to second range reveals new insights for slow exchange and binding interactions.
Development of methods sampling the microsecond range (kex ∼102–105 s−1) will open a new area of examination pertinent to allosteric events.
Powerful new data analysis methods enable the consideration of complex models of exchange and help develop insights into under- and over-interpretation of experimental data.
The Authors declare that there are no competing interests associated with the manuscript.