Elucidating Functional Dynamics by R _1ρ and R ₂ Relaxation Dispersion NMR Spectroscopy

Abstract

NMR spectroscopy is the method of choice to measure protein and nucleic acid dynamics on a variety of timescales. Picosecond to nanosecond dynamics can be precisely probed by quantifying R ₁ and R ₂ relaxation rates and heteronuclear NOE values, whereas residual dipolar couplings (RDCs) are sensitive to motion on a wide range of timescales from submicrosecond to milliseconds. Even slower dynamics can be assessed by hydrogen exchange experiments. In a biochemical context, relaxation dispersion NMR spectroscopy is particularly valuable, because it reports on the biologically important timescale from micro- to milliseconds, encompassing the conformational rearrangements of ligand binding, enzymatic reactions, and base pair transitions. From relaxation dispersion measurements, it is possible to obtain structural, kinetic, and thermodynamic information about energetically excited conformational minor states beyond the ground state structure. Here, we review the two methods of R _1ρ and R ₂ relaxation dispersion, focusing on recent developments in pulse sequence design and data processing techniques, as well as applications of the methods to resolve protein–protein interactions.

1 Relaxation Dispersion

Accurate function of biomolecules such as proteins, carbohydrates, and nucleic acids is central to all cellular processes. Although the function of many proteins and nucleic acids would seem to be explained by a single static structure, many biomacromolecules such as enzymes are extremely dynamic. This dynamical character means that the proteins in fact sample many conformations to exert their physiological function in solution. The static ground state structure of a protein can be obtained from X-ray crystallography, NMR spectroscopy, or cryo-electron microscopy; however, the dynamics beyond this ground state structure remain hidden. Because these “invisible” transient dynamic conformations cannot be observed directly, specialized methods are needed that can detect the energetically excited minor states in solution.

Relaxation dispersion NMR spectroscopy is the method of choice to detect and quantify such functionally relevant motion in biomolecules on a timescale from microseconds to milliseconds. Indeed, relaxation dispersion has provided unprecedented insight, with an astonishing level of detail, into the mechanisms of protein binding [1, 2], enzymatic activity [3,4,5,6,7,8,9], base pair exchange [10, 11], post-translational modifications [12], protein folding [13, 14], protein intermediate structures [15], intrinsically disordered proteins [1, 2, 16, 17], and transcriptional activators [18]. This great degree of detail arises (i) because relaxation dispersion provides site-specific information at atomic resolution; and (ii) because the extracted information has such high diversity. The method allows the extraction of structural (chemical shift difference between the major and minor states, Δω), kinetic (lifetime of the excited states, 1/k _ex), and thermodynamic (relative populations of the states, p _i) information about the excited states. Although none of these parameters is directly measurable, the information can be obtained by quantifying the broadening of NMR resonance lines, which is caused by the conformational exchange process.

Relaxation dispersion can be measured by two main methods: R ₂ relaxation dispersion and R _1ρ relaxation dispersion. In the R ₂ relaxation dispersion experiment, the effective transverse relaxation rate is measured as a function of the delay between two successive refocusing pulses in a Carr–Purcell–Meiboom–Gill (CPMG) pulse sequence [19,20,21]. In the R _1ρ relaxation dispersion experiment, the effective rotating-frame relaxation rate is probed as a function of either the amplitude or the offset of an applied spin-lock radiofrequency (RF) field [22,23,24,25,26,27,28,29]. The relaxation rates resulting from either method can then be plotted as a function of the applied field (Fig. 7.1). In the R _1ρ relaxation dispersion experiment, the relaxation rates can also be plotted as a function of the spin-lock offset. In either case, fitting of the data points of the relaxation dispersion profile to a theoretical model enables the relevant structural, kinetic, and thermodynamic parameters to be extracted.

Fig. 7.1

Schematic relaxation dispersion profile. In the presence of chemical exchange (shown), broad signals (large effective R ₂) are obtained at weak applied fields, whereas strong fields refocus the magnetization leading to sharp lines (small effective R ₂). In the absence of exchange (not shown), the relaxation rates are equal at all applied fields and a flat profile is obtained

In this chapter, we briefly review the two relaxation dispersion methods and then discuss several recent theoretical and practical developments. To this end, we focus on protein–ligand interactions and the automation of relaxation dispersion measurements and processing.

2 Accessible Information

The kind of information that can be extracted from relaxation dispersion depends on the exchange regime in question. Assuming the simplest model to describe the exchange between two conformational states A and B in a biomolecule, a two-state exchange model is:

$${\text{A}}\mathop \rightleftarrows \limits_{{k_{\text{BA}} }}^{{k_{\text{AB}} }} {\text{B}} .$$

(7.1)

If the exchange between A and B is slow on the chemical shift timescale (slow exchange regime), fitting of the relaxation rates as a function of the applied field strength to a theoretical model yields the chemical shift difference Δω between the two states, the exchange rate of the process k _ex = k _AB + k _BA, and the populations of the major (p _A) and minor state (p _B) [30]. If the exchange process is fast on the chemical shift timescale (fast exchange regime), unfortunately it is not possible to separate p _A, p _B, and Δω [29, 31]; in this case, only k _ex, $R_{2} {^{0}}$ (the intrinsic transverse relaxation rate in absence of exchange), and the site of the exchange can be derived and interpreted.

3 R _1ρ Relaxation Dispersion

3.1 General Aspects

Spin-lock-based relaxation dispersion experiments are called R _1ρ relaxation dispersion experiments because, during the relaxation block of the pulse sequence, a magnetization decays at the rotating-frame relaxation rate R _1ρ . In proteins, relaxation of the ¹⁵N amide nucleus is most frequently studied [20, 32,33,34]; however, applications of relaxation dispersion to ¹H and ¹³C nuclei have also been described in the literature [35, 36]. For simplicity, here we focus on the ¹⁵N nucleus.

Whether it is possible to resolve a given exchange process by R _1ρ relaxation dispersion depends on the exchange rate k _ex relative to the applied field strength [29]. That is, the exchange rate k _ex should be of the same order as the experimentally applicable effective field amplitudes:

$$\omega_{\text{e}} = \sqrt {\omega_{1}^{2} +\Omega ^{2} } ,$$

(7.2)

where ω _e is the effective field amplitude (in units of rad s⁻¹). ω ₁ and Ω denote the RF field amplitude and the offset of a given resonance from the spin-lock carrier frequency, respectively. In other words, the experimentally applicable field strength directly determines the time window of the dynamical process that can be studied by R _1ρ relaxation dispersion. In general, higher effective fields can be used in spin-lock (R _1ρ relaxation dispersion) experiments than in CPMG (R ₂ relaxation dispersion) experiments; however, spin-lock experiments present other challenges such as accurate field calibration [37], sample heating, and offset effects that are not critical issues with CPMG-type experiments. The spectrometer hardware (amplifier and probe) may impose additional constraints.

When using the ¹⁵N amide nucleus as a probe to study the conformational dynamics of a protein, spin-locking fields as high as ω ₁/2π ≈ 6 kHz can be applied [38]. Such strong fields refocus rapidly exchanging magnetization, and thus, very fast chemical exchange processes can be resolved. The spin-lock field can be applied near-resonance (Ω ≃ 0), on-resonance (Ω = 0), or off-resonance (Ω ≠ 0). The effective relaxation rate depends on both ω ₁ and Ω. Thus, either ω ₁ or Ω, or both, can be varied between experiments to obtain spin-lock power (on-resonance R _1ρ [28, 39]) or spin-lock offset (off-resonance R _1ρ [40, 41])-dependent relaxation dispersion profiles.

Early ¹⁵N R _1ρ relaxation dispersion experiments were carried out in a relatively conventional manner; in other words, data were obtained as a series of 2D ¹H–¹⁵N experiments in which a near-resonance spin-lock relaxation building block was included [27, 28, 40, 42]. However, the dependence of the relaxation rate R _1ρ on the offset from the spin-lock carrier frequency Ω posed challenges. In particular, because the amide resonances in a protein have different ¹⁵N chemical shifts, all resonances experience the effect of the spin lock at different offsets. Thus, these experiments were called “near-resonance” experiments, rather than on-resonance experiments. The offset dependence imposes the limitation that weak spin-locking fields cannot be used owing to the resulting tilt θ of the effective field toward the z-axis when ω ₁ becomes small relative to the offset Ω:

$$\tan \theta = \frac{{\omega_{1} }}{\Omega }.$$

(7.3)

A highly tilted effective field maximizes the contribution of R ₁ and minimizes the contribution of R ₂. This is not useful, however, because R ₁ does not report on the exchange process in question: Only the transverse relaxation rate R ₂ contains an exchange contribution [43]:

$$R_{2} = \overline{{R_{2}^{0} }} + R_{\text{ex}} ,$$

(7.4)

where the line above $R_{2} {^{0}}$ indicates that in the case of fast exchange, the relaxation rate will be population-averaged.

In 2005, Kay and coworkers suggested that it would be more beneficial to probe only one of the amide resonances of a protein at a time [25]. An equivalent strategy was also used by Al-Hashimi and coworkers to probe ¹³C spins in nucleic acids [22]. This strategy eliminates the problem of the differential off-resonance effects of the spins of a protein because only one spin is studied at a time, which is particularly advantageous for off-resonance experiments [25]. Moreover, if desired, the spin lock can be applied completely on-resonance (Ω = 0) in all experiments [44]. To probe a single spin at a time, the pulse sequences utilize Hartmann–Hahn cross-polarization (CP) transfer for selective coherence transfer. Although the number of spectra to be obtained is markedly increased in this scheme, experimental time can be somewhat conserved by recording the experiments as a series of 1D NMR spectra.

When the spin lock is applied on-resonance, accurate spin locking can be achieved even with very weak ω ₁ fields. This extends the time window of dynamics that can be studied by R _1ρ relaxation dispersion for the study of slow exchange processes, because this window depends on the value of ω _e relative to k _ex. Moreover, in this approach, ¹H decoupling during the duration of the spin lock is simplified, thereby avoiding potential problems arising from J-coupling and cross-correlation between dipole–dipole interactions and chemical shift anisotropy whenever ¹⁵N transverse coherence is present [25].

3.2 Pulse Sequence of the R _1ρ Relaxation Dispersion Experiment

The pulse sequence of a typical R _1ρ relaxation dispersion experiment is shown in Fig. 7.2. Equilibrium proton magnetization is excited and transferred to the scalar-coupled amide ¹⁵N spin by Hartmann–Hahn cross-polarization [45]. To this end, RF fields of matched frequency are applied on both the ¹H and the ¹⁵N channels. The selectivity of this transfer depends on the amplitude of the matched applied RF fields, which is termed ω _CP. In general, it is possible to select a single amide resonance, that is a single cross-peak in a ¹H¹⁵N–heteronuclear single-quantum coherence (HSQC) spectrum using this technique with very weak matched CP fields, which may be as weak as [46]:

$$\frac{{\omega_{\text{CP}} }}{2\pi } = \frac{\sqrt 3 }{4}J.$$

(7.5)

If the resonance in question is sufficiently isolated in the ¹H¹⁵N–HSQC spectrum, slightly stronger matched CP fields may be applied. By using the program SIMPSON [47], it is possible to simulate the selectivity of this transfer. For example, matched CP fields of ω _CP/2π = 93 Hz ≈ |J _HN| cause coherence transfer in an approximate range of ±100 Hz around the target resonance at a static field of 16.4 T [44]. After CP, the resulting ¹⁵N in-phase coherence is stored on the z-axis and a z-filter is applied to dephase all resonances except the single resonance of interest.

Fig. 7.2

R _1ρ relaxation dispersion. Shown is a pulse sequence to probe micro- to millisecond dynamics by ¹⁵N R _1ρ relaxation dispersion [22, 44]. Pulses are applied with x phase, unless noted otherwise in the diagram. The filled rectangles denote non-selective 90° pulses. A WATERGATE scheme is used for water suppression [85]. The open rectangles represent pulses with variable tip angles as described previously [25]. ¹⁵N decoupling during acquisition is achieved by the WALTZ-16 sequence [86]. The block in square brackets can be replaced by a selective excitation pulse (E-BURP-1 shape, 5 ms). Phase cycle: φ ₁ = 8(y), 8(−y); φ ₂ = x, −x; φ ₅ = 4(x), 4(−x); φ ₆ = 2(x), 2(−x); φ _rec = x, −x, −x, −x, x, x −x, −x, x, x, −x, x, −x, −x, x; for more details, see ref. [44]. Gradients: G ₁ (1 ms, 3 G/cm), G ₂ (1 ms, 40 G/cm), G ₃ (1 ms, 15 G/cm), G ₄ (0.5 ms, 27.5 G/cm), G ₅ (1 ms, 10 G/cm). Abbreviations CP cross-polarization, cw continuous wave irradiation, SL spin lock

While the coherence of interest is still stored on the z-axis, an equilibration time (τ _eq) on the order of 5 ms is employed. In some circumstances, the initial population of each state may differ from the expected value if this equilibration period is omitted. Consider a given spin that exchanges between two conformational states A and B, where A is the major state (population p _A ≃ 0.99) and B is the minor state (population p _B ≃ 0.01). For the given spin, the separation in chemical shift (Δω _AB) between states A and B is relatively large. Thus, when the spin of interest is subjected to selective CP, only the spins of the molecules in which the spin is in state A will feel the effect of selective CP. In other words, only spins in state A transfer their coherence by CP from ¹H to ¹⁵N. The subsequent z-filter dephases all other resonances, including the spins in state B. Thus, the relative populations of A and B will have been perturbed to p _A = 1.0 and p _B = 0. The additional equilibration delay τ _eq makes sure that, at the beginning of the relaxation period (spin lock), the respective populations p _A and p _B are restored to their appropriate equilibrium values. This is important because the populations appear in the theoretical equations that describe R _1ρ . Because of this dependence, accurate values of p _A and p _B are critical to ensure a good agreement between theory and experiment. For small values of Δω _AB, the influence of τ _eq may be limited because spins in state B will be affected by selective CP; however, for Δω _AB values that fall outside the range of the CP coherence transfer profile, the experiment is expected to perform better when a sufficient equilibration delay τ _eq is used.

Next, the target magnetization is flipped back to the transverse plane, where the optional delay ζ separates a target on-resonance coherence (¹⁵N chemical shift Ω = 0) and a CP-excited nearby unwanted coherence (¹⁵N chemical shift Ω ≠ 0). If the target coherence is sufficiently isolated in the ¹H¹⁵N–HSQC spectrum or CP selectivity is sufficient, a value of ζ = 0 may be set: Because ζ is on the order of milliseconds, setting ζ to 0 will maximize sensitivity. The subsequent gradient G₃ dephases the unwanted off-resonance coherences, whereas the target coherence is stored on the z-axis. Whenever ¹⁵N transverse coherence is present, ¹H decoupling is achieved by applying on-resonance continuous wave fields [25].

As a result, only the single target desired coherence is retained at the beginning of the relaxation period in which the spin lock of variable power and/or offset is applied. After the relaxation block, another equilibration delay is employed and coherence is transferred back from ¹⁵N to ¹H for detection as a 1D experiment.

The effective rotating-frame relaxation rate is then obtained according to:

$$R_{{1\uprho}} \text{ } = \text{ } - \frac{1}{T}\ln \left( {\frac{I}{{I_{0} }}} \right),$$

(7.6)

where T is the duration of the spin lock, typically 30–50 ms; I is the peak intensity in an experiment in which a spin lock is applied; and I ₀ is a reference intensity from an experiment in which the spin-lock block is omitted (by setting T = 0). Note that, because of the dependence of R _1ρ on ω ₁, the experimentally applied B ₁ fields must be rigorously calibrated and their actual values should be measured directly [37].

3.3 Automation of R _1ρ Relaxation Dispersion Measurements and Data Processing

Performing the R _1ρ relaxation dispersion experiment separately for one site at a time, as described in the previous section, has many advantages because very weak ω ₁ fields can be used. However, this approach means that a large number of free-induction decays (FIDs) must be recorded, processed, and integrated before the relaxation rates can be calculated and fitted to a theoretical model. The number of FIDs is n × (m + r), where n is the number of probed resonances; m is the number of varied parameters, that is, spin-lock power ω ₁ or offset Ω (i.e., m is the number of R _1ρ relaxation rates (data points) in the final relaxation dispersion profile); and r is the number of reference data recorded. Thus, even for a small protein such as ubiquitin (76 residues; ~72 resolved cross-peaks in the ¹H¹⁵N–HSQC spectrum), approximately 1600 (n = 72; m = 20; r = 2) FIDs would have to be recorded, processed, and peak-picked before the R _1ρ relaxation rates can be calculated and compared with a theoretical model of the exchange process. It is very likely that many researchers have shied away from this approach to relaxation dispersion owing to its apparently labor-intensive procedure.

It is, however, relatively straightforward to make the acquisition and processing of the data more convenient by considering the following points:

• 1. Not all amino acids in a protein report on conformational exchange.

• 2. R _1ρ relaxation dispersion-positive residues can be easily identified.

• 3. Acquisition and processing can be automated.

Based on these three considerations, the acquisition and processing of R _1ρ relaxation dispersion data have been recently automated through the development of the Amaterasu (automated R _1ρ analysis utility) pipeline (Fig. 7.3) [24].

Fig. 7.3

Workflow of Amaterasu. Gray boxes, experimental steps; white boxes, steps automated by the pipeline

3.3.1 Experimental Setup

A dedicated spreadsheet included in the Amaterasu package (http://www.moleng.kyoto-u.ac.jp/~moleng_01/amaterasu) calculates all of the acquisition parameters that are required to set up the experiment. The acquisition pulse program is implemented in such a way that the entire dataset, comprising n residues × (m + r) residue-specific FIDs, is acquired in a single run (i.e., a single acquisition dataset). This type of implementation is termed a “pseudo-2D” experiment and means that only a single experiment needs to be set up at the spectrometer. The acquisition of all residues with all power/offset values is handled entirely by the pulse program.

First, a conventional ¹H¹⁵N–HSQC spectrum is acquired. The spectrum is semi-automatically peak picked in a spectrometer acquisition program, such as Bruker TopSpin. The chemical shifts of all resonances of interest are then copied to the Amaterasu spreadsheet. At this stage, resonances such as tryptophan side chain peaks can be discarded if desired. After the basic pulse lengths and powers (¹H and ¹⁵N hard pulses) are entered, the spreadsheet automatically calculates all parameters necessary to perform the R _1ρ relaxation dispersion experiment, including CP, spin lock, water-flip back, decoupling pulses, and the optional delay ζ. The chemical shifts ω _H, ω _N, the delay ζ, and the spin-lock power or offset values are passed to the pulse program as simple text files. Once these text files are created, the spreadsheet can be closed.

3.3.2 Screening

In almost all known cases, not all residues of a protein will report on conformational exchange on the micro- to millisecond timescale; therefore, it is not recommended that relaxation dispersion profiles are acquired for all residues of a protein, because a great deal of spectrometer time will be spent on acquiring essentially meaningless flat profiles. Instead, an efficient spectroscopist will first identify the dynamic residues of the protein by performing a quick screening experiment in which only two spectra are recorded for all resonances. Each resonance is probed by a weak and a strong spin-lock field ω ₁. For ubiquitin, for example, the ~72 cross-peaks of its ¹H¹⁵N–HSQC spectrum would be picked and two spectra would be acquired for each: one with ω ₁/2π ≃ 50 Hz and one with ω ₁/2π ≃ 3000 Hz. If the residue in question does not exhibit chemical exchange, the two spectra will show peaks of equal intensity. Conversely, if the resonance does exhibit micro- to millisecond conformational dynamics, a lower peak intensity will be observed in the spectrum corresponding to the weak spin-lock experiment. In summary, a screening experiment will collect only the first and the last data point of a relaxation dispersion profile (Fig. 7.1), with the exception that no reference spectrum is acquired, and thus, the relaxation rate R _1ρ is not actually calculated.

If the ratio of these two intensities indicates the presence of chemical exchange, Amaterasu will select the resonance for acquisition of a full relaxation dispersion profile. Unpromising resonances can be discarded at this stage to save acquisition time, except for special cases in which a flat profile is particularly desired (e.g., negative control experiments).

3.3.3 Processing

The software package Amaterasu performs the entire processing procedure from the raw FIDs (in Bruker spectrometer format) to model fitting of the final R _1ρ relaxation dispersion profile in a fully automated manner. First, it splits the pseudo-2D dataset into the respective 1D data. Second, it reads all acquisition parameters and, for each FID, conducts apodization, zero filling, 1D Fourier transform (FT), and automatic phase correction. All processing steps are carried out using the Python library nmrglue [48]. For phase correction, the entropy of the spectrum is minimized as described by Chen and coworkers [49]. Third, Amaterasu performs peak picking on all spectra. Because the chemical shift of each given resonance is known a priori, the program can easily recognize the correct peak based on the known chemical shift. It is also possible to plot all spectra to check for the presence of artifacts arising from peak overlap or erroneous phases. After all of the peaks are integrated, the resulting peak intensities as a function of ω ₁ and/or Ω are passed to the relaxation dispersion data-fitting program GLOVE [50]. The output files of GLOVE will then contain the kinetic and thermodynamic parameters that govern the exchange process, along with the fitted relaxation dispersion profile. For a small protein (~100 amino acids), Amaterasu completes its pipeline, from reading the raw FIDs to the final output of the relaxation dispersion profiles, in less than 1 min on a standard desktop PC (e.g., Mac mini, 2015). Examples of relaxation dispersion profiles obtained with Amaterasu are shown in Fig. 7.4.

Fig. 7.4

¹⁵N R _1ρ relaxation dispersion profiles of amide resonances of the ubiquitin-associated (UBA) domain of the autophagy receptor protein p62 as obtained with Amaterasu and GLOVE [24, 50]. Left, on-resonance R _1ρ relaxation dispersion experiment; right, off-resonance R _1ρ relaxation dispersion experiment. Reproduced with permission from ref. [24]

4 R ₂ Relaxation Dispersion

4.1 General Aspects

Application of a spin-lock pulse is not the only method to keep a coherence from dephasing during a fixed relaxation delay T. A Carr–Purcell–Meiboom–Gill (CPMG) pulse sequence [19,20,21] refocuses a given coherence by repetitive execution of the spin-echo sequence (τ—180°—2τ—180°—τ) _n . Here, n is an integer and the spacing between the two successive 180° pulses, 2τ, is called τ _CP. In general, chemical exchange can be detected by R ₂ relaxation dispersion if the exchange rate k _ex is of the same order as 1/τ _CP. As with spin-lock experiments, CPMG experiments may cause sample heating at high values of n, which may also interfere with theoretical treatments of the experiment because the pulse lengths of the refocusing pulses are assumed to be negligible in these models [41, 43].

In principle, the evolution of both in-phase and anti-phase coherences during the CPMG time must be considered. It has been shown, however, that it is beneficial to average the differential contributions of in-phase and anti-phase coherences to R ₂ in the experiment by including a relaxation-compensating (RC) element in the pulse sequence (Fig. 7.5). The RC element interconverts in-phase and anti-phase coherences in the middle of the relaxation period, which eliminates any artifacts that may arise from different relaxation rates of in-phase and anti-phase coherences of a spin. For simplicity, the sequence in Fig. 7.5 shows only the pulse sequence during the relaxation time T _CPMG including the RC element. In a full sequence, the experiment begins from equilibrium proton magnetization using an INEPT element to obtain the anti-phase ¹⁵N coherence –2H _z N _y . This coherence is then continuously rephased by the CPMG spin-echo sequence while exhibiting transverse relaxation and imprinting the exchange contribution R _ex onto the intrinsic transverse relaxation rate R ₂. In the middle of the pulse sequence, the anti-phase coherence –2H _z N _y encounters the RC element and is thus converted to the in-phase coherence N_x; moreover, the RC element suppresses cross-correlation arising from chemical shift anisotropy and dipole–dipole interaction. After passing the RC element, the in-phase coherence N_x relaxes for the remainder of T _CPMG, thereby averaging the effect of in-phase and anti-phase coherence relaxation during T _CPMG. ¹⁵N frequency labeling during t ₁ and acquisition during t ₂ conclude the experiment.

Fig. 7.5

R ₂ relaxation dispersion. Shown are pulse sequence elements of a CPMG relaxation dispersion experiment for evolution through the relaxation block of length T _CPMG. The sequence starts from an initial coherence described by −2H _z N _y , which can be obtained by an INEPT sequence. Open rectangles indicate 180° pulses. The relaxation compensation (RC) element interchanges anti-phase to in-phase coherence. J is the amide ¹ J _HN-coupling

In the ¹⁵N CPMG experiment, problems can arise from ¹ J _HN-coupling, offset effects, and pulse imperfections, which can lead to erroneous estimation of the effective transverse relaxation rate $R_{2} {^{\text{eff}}}$. These problems can be largely alleviated by applying continuous wave decoupling on the ¹H channel during T _CPMG [51, 52].

4.2 Quantifying Protein–Ligand Interactions by R₂ Relaxation Dispersion

Interestingly, relaxation dispersion has not only been used to study the dynamics of a single protein, but it can also probe interactions between a protein and other molecules [31].

When the interaction between a protein and a ligand is in intermediate or slow exchange on the NMR timescale, then the observed chemical shifts do not appear as population averages between the free and bound states in the experiment. In this situation, therefore, other methods that assume fast exchange conditions, such as chemical shift titration or transferred R ₁ and R ₂ experiments [53,54,55,56], are not applicable. For example, for a binding process in the slow exchange regime, the NMR signal of a protein or a ligand does not move during a titration experiment; accordingly, it is not possible to simply obtain a titration curve (chemical shift difference as a function of protein–ligand ratio) to extract the dissociation constant K _D.

In recent years, multiple NMR methods have been developed to study the kinetic rates of slow and intermediate exchange systems. In principle, R _1ρ dispersion [28], R ₂ dispersion [20, 32, 57, 58], ZZ-exchange [59,60,61], CEST [62, 63], and DEST [64] techniques can all be applied to this end; each of these methods has the potential to derive the kinetic rates of interconversion between the states, even though one of the states is not directly measurable because of its low population and a large transverse relaxation rate (due to the additive contribution of R _ex).

4.2.1 Theory

Consider the interaction between a protein (A) and a ligand (B):

$${\text{A + B}}\mathop \rightleftarrows \limits_{{k_{\text{off}} }}^{{[{\text{B}}]k_{\text{on}} }} {\text{AB}}$$

(7.7)

It has been shown that it is possible to determine [B]k _on and k _off by relaxation dispersion in an analogous way to determination of the kinetic rates of conformational exchange in a single molecule. Note, however, that [B] is unknown and must be determined. Although it is possible to determine K _D from isothermal titration calorimetry (ITC), to measure k _off by NMR, and then to calculate k _on from K _D = k _off/k _on [65], here we discuss the determination of k _on and k _off without any prior information on K _D by using R ₂ relaxation dispersion [1].

If the exchange process—that is, the association–dissociation equilibrium—occurs on the timescale of milliseconds with a comparably large chemical shift change Δω, we can assume that the effective transverse relaxation rate R ₂ of the free and the bound resonances will increase by an additional relaxation rate R _ex, as described by Eq. 7.4. $R_{2} {^{\text{eff}}}$ can be obtained from the R ₂ relaxation dispersion experiment, thereby allowing extraction of the rate k _ex of the association–dissociation process. In addition, structural information in form of the chemical shift difference Δω between the free and bound states can be obtained. $R_{2} {^{\text{eff}}}$ can be accurately described by numerical solution of the Bloch–McConnell equations [66, 67]. Alternatively, we can assume a simple two-state exchange model and describe $R_{2} {^{\text{eff}}}$ as follows [68]:

$$\begin{array}{*{20}l} {R_{2}^{\text{eff}} = R_{2}^{0} + \frac{1}{2}\left\{ {k_{\text{ex}} - \frac{1}{{\tau_{\text{cp}} }}\cosh^{ - 1} \left[ {D_{ + } \cosh \left( {\eta_{ + } } \right) - D_{ - } \cos \left( {\eta_{ - } } \right)} \right]} \right\}} \hfill \\ {D_{ \pm } = \frac{1}{2}\left[ { \pm 1 + \frac{{\Psi + 2\Delta \omega^{2} }}{{\sqrt {\Psi ^{2} + \xi^{2} } }}} \right]} \hfill \\ {\eta_{ \pm } = \tau_{\text{cp}} \sqrt {\frac{1}{2}\left( { \pm\Psi + \sqrt {\Psi ^{2} + \xi^{2} } } \right)} } \hfill \\ {\Psi = k_{\text{ex}}^{ 2} -\Delta \omega^{2} } \hfill \\ {\xi = 2\Delta \omega \left( { [ {\text{B]}}k_{\text{on}} - k_{\text{off}} } \right)} \hfill \\ {k_{\text{ex}} = [ {\text{B]}}k_{\text{on}} + k_{\text{off}} ,} \hfill \\ \end{array}$$

(7.8)

where $R_{2} {^{0}}$ is the intrinsic relaxation rate in the absence of exchange and is assumed to be identical for the free and bound states; and τ _CP is the delay between two successive 180° pulses in the CPMG pulse train. τ _CP is thus a known, preset parameter when the experiment is performed. If the interaction between protein and ligand is in the intermediate or slow exchange regime, it is possible to separate the parameter k _ex into [B]k _on and k _off.

In general, the concentration of unbound ligand—namely, the concentration of free B—is described by:

$$[{\text{B}}] = \frac{1}{2}\left\{ { - K_{\text{D}} - [{\text{A}}]_{0} + [{\text{B}}]_{0} + \sqrt {\left( {K_{\text{D}} + [{\text{A}}]_{0} - [{\text{B}}]_{0} } \right)^{2} + 4[{\text{B}}]_{0} K_{\text{D}} } } \right\},$$

(7.9)

where [A]₀ and [B]₀ are the total concentrations of A and B, respectively. To separate [B]k _on in order to calculate k _on, we notice that [B] depends on [A]₀ and [B]₀. As a result, multiple samples with varying total amounts of [A]₀ and/or [B]₀ can be prepared, and global fits of the R ₂ relaxation dispersion profiles can be performed for the various samples. In most cases, it is wise to keep one of the concentrations constant, ideally the concentration [A]₀ of the isotope-labeled protein, to obtain relaxation dispersion profiles with similar sensitivity from different measurements. Thus, it is most feasible to vary the total concentration of the unlabeled ligand [B]₀. As described in detail in ref. [31], it is necessary to include the total concentrations [A]₀ and [B]₀ as variable fitting parameters in the fitting procedure under realistic experimental conditions; otherwise, a 5% mismatch of the concentration can lead to a twofold error in the final K _D value. Moreover, the NMR samples for these experiments must be prepared with extreme care—preferably by dilution from single respective concentrated stock solutions—to obtain accurate concentration ratios. Taken together, relaxation dispersion is measured for multiple samples with multiple concentration ratios using a modified version of Eq. 7.9, in which the parameter a is introduced as follows:

$$[{\text{B}}] = \frac{1}{2}\left\{ { - K_{\text{D}} - [{\text{A}}]_{0} + a[{\text{B}}]_{0} + \sqrt {\left( {K_{\text{D}} + [{\text{A}}]_{0} - a[{\text{B}}]_{0} } \right)^{2} + 4a[{\text{B}}]_{0} K_{\text{D}} } } \right\}.$$

(7.10)

4.2.2 Example 1: Interaction Between the pKID Domain of CREB and the KIX Domain of CBP/p300

The method to obtain K _D by R ₂ relaxation dispersion was originally applied to study the phosphorylated kinase-inducible domain (pKID), an intrinsically disordered protein that forms part of the transcription factor CREB. pKID binds to the KIX domain of CBP/p300 [1]. To study the affinity of the two proteins, a [¹⁵N]-labeled sample of pKID was prepared and relaxation dispersion profiles were obtained for samples with concentration ratios (KIX/pKID) of 0.95, 1.00, 1.05, and 1.10 at two distinct static magnetic fields of 500 and 800 MHz (Fig. 7.6). The induced fit model was used to fit all relaxation dispersion data (4 concentration ratios, 2 B ₀ values) as follows:

$${\text{A + B}}\mathop \rightleftarrows \limits_{{k_{\text{off}} }}^{{ [ {\text{B]}}k_{\text{on}} }} \underline{\text{A:B}} \mathop \rightleftarrows \limits_{{k_{ - 1} }}^{{k_{ 1} }} {\text{A:B,}}$$

(7.11)

where the underline indicates that the conformation of the state A:B is different from the final bound state A:B. For each residue, the chemical shift differences were defined as global parameters. The folding (k ₁) and unfolding (k ₋₁) rates were considered as global parameters for all residues falling within the same cluster, that is, residues that are in close proximity in the same secondary structure element and thus presumably fold/unfold in a cooperative manner. As a result, it was possible to obtain site-specific K _D values for the binding of pKID to KIX. Strikingly, the mean K _D value obtained from the site-specific analysis showed excellent agreement with the global, macroscopic dissociation constant obtained by ITC [69]. Based on the relaxation dispersion data combined with chemical shift titrations, it was concluded that pKID engages with KIX, while it is unfolded through the formation of non-specific encounter complexes; upon KIX binding, pKID adopts a partially folded intermediate, which subsequently evolves toward a fully folded bound state. In other words, this model consists of four distinct states: free pKID, a non-specific encounter complex, a bound-folding intermediate, and the final bound-folded state. Note that this model assumes that the exchange between the free state and the encounter complex is too fast to be traceable by R ₂ relaxation dispersion. In summary, relaxation dispersion deciphered the coupled folding and binding mechanism of an intrinsically disordered protein.

Fig. 7.6

Protein–ligand ratio-dependent R ₂ relaxation dispersion profiles. The relaxation dispersion profiles were obtained at 800 MHz (filled circles) and 500 MHz (open circles). The data (reproduced with permission from [1]) show Arg¹²⁴ of 1 mM pKID in the presence of 0.95, 1.00, 1.05, and 1.10 mM KIX. A global fit of the relaxation dispersion data provided the values of k _on, k _off, k ₁, k ₋₁, and K _D, as well as chemical shift differences between the distinct states

4.2.3 Example 2: Interaction Between the Transactivation Domain of c-Myb and KIX

The transactivation domain of c-Myb, a transcription factor, also binds the KIX domain. Similar to pKID, the transactivation domain of c-Myb is an intrinsically disordered protein [16]. Again, the mechanism of binding has been resolved in great molecular detail by R ₂ relaxation dispersion. In essence, it was deduced that the N-terminal region of c-Myb binds to KIX while already in a principally folded conformation. Conversely, the C-terminal region of c-Myb folds after binding to KIX, and again this binding is explained by an induced fit mechanism.

5 Fitting of the Relaxation Rates to a Theoretical Model

The theoretical model used for fitting depends on the exchange regime and the number of exchange sites. A vast amount of work has been done in the development of analytical equations to describe relaxation dispersion (Table 7.1). These equations can be used to fit the experimental relaxation dispersion profiles. Fitting can be performed by implementing a selected equation in commercial or free packages such as Mathematica, MatLab, SciLab, or similar software. Alternatively, a relaxation dispersion-dedicated free software package such as GLOVE (http://www.scripps.edu/wright/?page_id=17) can be used. In the following, we illustrate both theoretical and practical aspects of fitting of relaxation dispersion data by GLOVE.

Table 7.1 Reports of analytical equations to approximate relaxation dispersion

5.1 Least-Squares Fitting in GLOVE

GLOVE is a dedicated relaxation dispersion data-fitting software, although it also supports several other NMR experiments such as R ₁, R ₂, and CLEANEX-PM [70]. It solves nonlinear least square problems by using the algorithm of Levenberg–Marquardt. GLOVE is written in C++, which assures rapid fitting of the data. To fit the relaxation dispersion profiles, the GLOVE software attempts to minimize the statistical variable χ ² in an iterative manner. χ ² is given as a function of the experimental and calculated effective R ₂ rates:

$$\chi^{2} = \sum\limits_{i = 1}^{N} {\left( {\frac{{R_{ 2}^{{i , {\text{exp }}}} - R_{ 2}^{{i , {\text{calc }}}} }}{{\sigma^{i} }}} \right)}^{2} .$$

(7.12)

where the variables $R_{2} {^{{i , {\text{exp}}}}}$ and $R_{2} {^{{i,{\text{calc}}}}}$ denote the experimental and calculated values of $R_{2} {^{\text{eff}}}$, respectively; and the parameter σ ⁱ represents the experimental error.

To minimize the function iteratively, initial parameters must be provided. In GLOVE, there are five distinct methods to provide the initial parameters. Importantly, it is possible to run multiple methods or to repeat the same method. The program stores the parameter set corresponding to the lowest obtained value of χ ² and replaces this set if a better fit (a lower value of χ ²) is found.

The method named ONE is a minimization routine that starts from the lowest limit or an optionally specified value. As soon as the minimization routine finds a local minimum, the fitting process stops. ONEEX is equivalent to ONE, with the exception that the fitting process does not stop until both global and local parameters (optimized separately) have converged into local minima. When global fitting of parameters is desired, it is advisable to use ONEEX even though the fit converges more slowly. The methods GRID, RANDOM, and MCMIN stop when they reach the same condition as described for ONE; however, they explore the parameter space much more rapidly at the earlier stages of the fitting procedure.

As indicated by its name, GRID represents a grid search. Initially, the global parameters are fixed (to a given grid point) and the local parameters of each dataset are varied using a grid search algorithm. Next, the fix on the global parameters is removed, and then, all parameters including global parameters are optimized. The procedure is iterated to explore all grid points of the parameter space of the global parameters.

Lastly, the methods RANDOM and MCMIN represent a random search and minimization by a Monte Carlo algorithm, respectively. RANDOM uses a random number generator to choose a random initial value from a range of parameters, and χ ² is subsequently minimized by other iterative methods as specified by the user. Thus, RANDOM can be used to quickly find good starting values to determine the global minimum from the entire range of specified parameters. Conversely, MCMIN achieves a more accurate determination of the global minimum after initial establishment of the fit parameter set provided by RANDOM and other methods. This Monte Carlo implementation in GLOVE is described in detail in the next section.

5.2 Monte Carlo Minimization Algorithm in GLOVE

Monte Carlo minimization is a popular method in molecular simulations. For example, it has been used to find the global energy minimum during the folding of a peptide structure by randomly varying the dihedral angles in order to overcome large energy barriers [71]. In a protein-folding context, the Metropolis criterion is used to compare the newly energy-minimized structure with a previous conformation [72, 73]. In the minimization context of GLOVE, the initial parameter values are the currently available best-fit parameters to which positive or negative random numbers, which follow a Gaussian distribution, are added. This enables the parameter set to overcome the barriers of a local minimum. The newly obtained parameter set can then be minimized by using conventional least-squares methods as described above. If the new χ ² is smaller than the previous value, the Monte Carlo step is accepted.

Monte Carlo minimization (MCMIN) runs until no better parameter set is found for a total number of iterations as specified by the user; in general, this number is higher than 5. Moreover, if a lower value of χ ² is obtained, the iteration counter is reset to 0. The magnitude of the Monte Carlo trial steps can be controlled by the user by specifying a scaling factor in the input file to GLOVE. For efficient minimization of χ ², it is important to choose this factor judiciously. In particular, a too small scaling factor would not overcome the barriers of a local minimum efficiently. Conversely, if the scaling factor is too large, resulting in a large Monte Carlo step, the new set of parameters may be completely different from the current best parameter set, which may lead to a marked increase in the value of χ ². An efficient protocol to determine the global minimum most accurately and rapidly is therefore to run MCMIN multiple times with successively smaller scaling factors. Because the fitting parameters are far from the best-fit solution at the early stages of fitting, the MCMIN scaling factor should be set to a comparably large value. The parameters can be varied very finely by using a small scaling factor in the final stages of the fitting in order to determine the global minimum most accurately.

5.3 Two-State Exchange

For simplicity, we describe the most common model used to fit relaxation dispersion data; however, many additional published equations, including two- and three-state exchanges, are available within GLOVE. A relaxation dispersion profile, which reports on a two-state exchange process (Eq. 7.1), is described by the equation of Carver and Richards [68] for all exchange regimes under realistic experimental conditions. The Carver and Richards equation calculates the effective transverse relaxation constant $R_{2} {^{\text{eff}}}$ as:

$$\begin{aligned} R_{2}^{\text{eff}} & = \frac{1}{2}\left\{ {R_{{ 2 {\text{A}}}}^{0} + R_{{ 2 {\text{B}}}}^{0} + k_{\text{AB}} + k_{\text{BA}} - \frac{1}{{\tau_{\text{CP}} }}\cosh^{ - 1} \left[ {D_{ + } \cosh \left( {\eta_{ + } } \right) - D_{ - } \cos \left( {\eta_{ - } } \right)} \right]} \right\} \\ D_{ \pm } & = \frac{1}{2}\left[ { \pm 1 + \frac{{\Psi + 2\Delta \omega^{2} }}{{\sqrt {\Psi ^{2} + \xi^{2} } }}} \right] \\ \eta_{ \pm } & = \tau_{\text{CP}} \sqrt {\frac{1}{2}\left( { \pm\Psi + \sqrt {\Psi ^{2} + \xi^{2} } } \right)} \\\Psi & = \left( {R_{{ 2 {\text{A}}}}^{0} - R_{{ 2 {\text{B}}}}^{0} + k_{\text{AB}} - k_{\text{BA}} } \right)^{2} -\Delta \omega^{2} + 4k_{\text{AB}} k_{\text{BA}} \\ \xi & = 2\Delta \omega \left( {R_{{ 2 {\text{A}}}}^{0} - R_{{ 2 {\text{B}}}}^{0} + k_{\text{AB}} - k_{\text{BA}} } \right). \\ \end{aligned}$$

(7.13)

Equation 7.13 resembles Eq. 7.8 but has a more general nature. In Eq. 7.13, Δω denotes the difference in chemical shift between the states A and B and has the units rad s⁻¹; and $R_{{2{\text{A}}}} {^{0}}$ and $R_{{2{\text{B}}}} {^{0}}$ are the transverse relaxation rates of the respective states A and B. Note that Eq. 7.13 considers that these values differ: In principle, the intrinsic transverse relaxation rates may be different; however, in practice, they are usually assumed to be the same. Thus,

$$R_{2} {^{0}} = R_{{2{\text{A}}}} {^{0}} = R_{{2{\text{B}}}} {^{0}} .$$

(7.14)

Importantly, this assumption does not have a large effect on the analysis of the exchange process in the case that the rate of the exchange is larger than the difference between $R_{{2{\text{A}}}} {^{0}}$ and $R_{{2{\text{B}}}} {^{0}}$ (i.e., if k _ex ≫ |$R_{{2{\text{A}}}} {^{0}}$ − $R_{{2{\text{B}}}} {^{0}}$|). Moreover, we reduce the parameter space by defining k _ex as the sum of the reaction rates of the forward and backward reaction. Thus,

$$k_{\text{ex}} = \, k_{\text{AB}} + k_{\text{BA}} .$$

(7.15)

In addition, the product of p _A × p _B is treated as a single fitting parameter a = p _A p _B. After fitting, p _B (the population of the minor state) can then be calculated as:

$$p_{\text{B}} = \frac{{\sqrt {1 - 4a} }}{2}.$$

(7.16)

This further enhances the efficiency and stability of the calculation. In summary, the Carver and Richards equation [68] is implemented in GLOVE as:

$$\begin{aligned} R_{2}^{\text{eff}} & = R_{ 2}^{0} + \frac{1}{2}\left\{ {k_{\text{ex}} - \frac{1}{{\tau_{\text{CP}} }}\cosh^{ - 1} \left[ {D_{ + } \cosh \left( {\eta_{ + } } \right) - D_{ - } \cos \left( {\eta_{ - } } \right)} \right]} \right\} \\ D_{ \pm } & = \frac{1}{2}\left[ { \pm 1 + \frac{{\Psi + 2\Delta \omega^{2} }}{{\sqrt {\Psi ^{2} + \xi^{2} } }}} \right] \\ \eta_{ \pm } & = \tau_{\text{CP}} \sqrt {\frac{1}{2}\left( { \pm\Psi + \sqrt {\Psi ^{2} + \xi^{2} } } \right)} \\\Psi & = k_{\text{ex}}^{2} - \Delta \omega^{2} \\ \xi & = 2\Delta \omega k_{\text{ex}} \sqrt {1 - 4p_{\text{A}} p_{\text{B}} } . \\ \end{aligned}$$

(7.17)

Note that GLOVE calculates the partial derivatives of $R_{2} {^{\text{eff}}}$ (required by the least-squares minimization algorithm) with respect to the fitting parameters analytically instead of numerically, which helps to speed up the calculation.

5.4 Workflow for Processing Relaxation Dispersion Data in GLOVE

In this section, a practical approach to processing relaxation dispersion data using the software package GLOVE is presented. First, the relaxation dispersion spectra are measured. In the case of R _1ρ relaxation dispersion data, Amaterasu can be used, which will automatically pass the relaxation rates and spin-lock powers to GLOVE.

In the case of R ₂ relaxation dispersion data, additional steps are necessary. Initially, all spectra are processed by exactly the same processing parameters, including the functions for solvent suppression, apodization, Fourier transform, zero- and first-order phase correction, and baseline correction. It is advisable to use a low order for the baseline correction function in order not to perturb the intensities of relatively small signals. It is not feasible to apply linear prediction because it may interfere with quantitative analysis of the NMR data. However, it is possible to reduce the acquisition time of relaxation dispersion datasets by nonlinear sampling methods [74]. Next, it is necessary to obtain the integrated peak intensities to calculate the effective transverse relaxation rates. This is done by using the tool pkfit, which is included in the GLOVE package. The experimental error in the peak intensity is calculated from the standard deviation of the amplitudes of the noise in each spectrum and the deviation between peak intensities in duplicated data. Thus, after running pkfit, the output file contains the static magnetic field B ₀ (MHz), the relaxation time T _CPMG (s), and the peak intensities of all resonances as a function of 1/τ _CP (s⁻¹). The delay τ _CP in the pulse sequence is the time between two 180° pulses of the CPMG pulse train. Note that some groups use a different notation by defining τ _CPMG as half the delay between two 180° pulses; in this case, the relaxation rates are plotted as a function of ν_CPMG = 1/(4τ _CPMG) instead of 1/τ _CP. Although the horizontal axis differs in the two representations, it is straightforward to convert ν_CPMG to 1/τ _CP according to:

$$1/\tau_{\text{CP}} = \, 2\nu_{\text{CPMG}} .$$

(7.18)

In complete analogy to Eq. 7.6, the effective $R_{2} {^{\text{eff}}}$ relaxation rates are then calculated according to:

$$R_{2}^{\text{eff}} \left( {\tau_{\text{CP}} } \right) = - \frac{ 1}{{T_{\text{CPMG}} }}\ln \left( {\frac{{I\left( {\tau_{\text{CP}} } \right)}}{{I_{0} }}} \right),$$

(7.19)

where I(τ _CP) is the peak intensity in the spectrum corresponding to a particular value of τ _CP; and I ₀ is the intensity in a reference spectrum in which the relaxation block is omitted (by setting T _CPMG = 0). This calculation is carried out by the tool cpmg2glove in the GLOVE package. Moreover, the experimental error in $R_{2} {^{\text{eff}}}$ is calculated from the spectral RMSD noise according to [75]:

$$\sigma^{i} = \frac{{\varepsilon_{I} }}{{I\left( {\tau_{\text{CP}} } \right) \times T_{\text{CPMG}} }},$$

(7.20)

where ε _I is the RMSD noise of the spectrum. The output of the tool cpmg2glove is the input file for GLOVE and contains all data necessary to perform curve fitting of the relaxation dispersion profile. The file may contain only one measurement or multiple different measurements (e.g., under different values of B ₀, temperatures or ligand concentrations).

After fitting of the R _1ρ or R ₂ relaxation dispersion data, GLOVE writes an output file containing a summary of the results and plots of the fitted relaxation dispersion profile in the format Xmgr or Grace. An Xmgr file for each resonance of the dataset is created, but it is possible to combine all of these plots in a single PDF file for convenience. This is achieved by the tool mplot, which is included in the GLOVE package. During the fitting process, GLOVE will write the reduced value of χ ², which is χ ² divided by the number of degrees of freedom of the fitting model, to the standard output (typically, the console), enabling the user to monitor the convergence of the fit in real time. Uncertainties in the resulting fitting parameters can be calculated by standard deviation, Monte Carlo, and/or jackknife methods [76, 77].

5.5 Examples of Relaxation Dispersion Curve Fitting by GLOVE

Here, we illustrate curve fitting of relaxation dispersion data by fitting the R ₂ relaxation dispersion profiles of KIX. In total, 110 R ₂ relaxation dispersion profiles (55 resonances; 2 values of B ₀) were subjected to analysis. KIX exhibits a two-state exchange between its native and a non-native conformation [78]. The relaxation dispersion profiles were obtained at two distinct magnetic fields on a Bruker DRX600 and a Bruker DMX750 spectrometer at a temperature of 25 °C using a [¹⁵N]-labeled sample of KIX (concentration: 0.5 mM) [74]. Spectra were acquired for τ _CP values of 10, 5, 3.33, 2.5, 2.0, 1.66, 1.43, 1.25, 1.0, 0.83, 0.71, 0.63, 0.55, 0.5, 0.4, and 0.33 ms at a relaxation time, T _CPMG, of 40 ms.

The performance of GLOVE was evaluated on an Apple iMac (Intel Core i7 dual core, 3.4 GHz) with the GLOVE executable file compiled by using the Intel C++ compiler [50]. The speed and accuracy of the fit were evaluated by using the directives ONE, ONEEX, and GRID. In addition, the combinations GRID + ONEEX, RANDOM + ONEEX, MCMIN + ONEEX, and RANDOM + MCMIN + ONEEX were tested. A combined workflow of RANDOM + MCMIN + ONEEX was found to yield the best results [50].

The relaxation dispersion profiles of KIX were subjected to a global fit using the Carver and Richards equation [68]. The exchange rate k _ex and the product of the populations p _A p _B were specified as global parameters. The possible range of the parameters was specified as [100–2500] for Δω, [5–4000] for k _ex, and [0.005–0.09] for the product p _A p _B. The initial parameter value of $R_{2} {^{0}}$ was taken from the lowest value of $R_{2} {^{\text{eff}}}$ (the right-most data point in the relaxation dispersion profile). For the minimization routines ONE, ONEEX, and MCMIN, the calculation starts from the lower limits of the parameter range. In the evaluation study, grid sizes from 2 to 20 were evaluated for the method GRID for the parameters Δω, k _ex, p _A p _B [50]. In MCMIN, the scaling factor was sequentially reduced from 0.1 to a final value of 0.001. Note that, because RANDOM and MCMIN rely on random number generators, they provide a different result at each execution. Thus, methods relying only on RANDOM on MCMIN alone were repeated 10 times to evaluate the average and standard deviation of each fit.

The 110 relaxation dispersion profiles of KIX were subjected to a global fit, where the overall lowest value of χ ², reached by many methods, was 1.45047, indicating that it represents the global minimum. At this value of χ ², the global parameters of the exchange rate k _ex converged to 600 ± 5 s⁻¹. Moreover, the product of p _A p _B converged to 0.0343 ± 0.0002. Examples of the corresponding relaxation dispersion profiles are shown in Fig. 7.7. Note that these plots were obtained directly from GLOVE and no further adjustment of the image to obtain publication quality was necessary.

Fig. 7.7

Relaxation dispersion profiles for KIX as fitted by GLOVE. The best-fit curves are shown for a dataset collected at a ¹⁵N frequency of 60.83 MHz (black line) and 76.01 MHz (red line), respectively. Residue numbers are indicated at the top (–HN)

The global minimum was not found by ONE, which means that relaxation dispersion profile fitting has the intrinsic problem of becoming trapped in local minima (Table 7.2). Thus, good initial values, or alternatively multiple fits starting from different initial parameters, are necessary to find the global minimum. ONEEX found a smaller χ ² value; however, it did not locate the global minimum either. The GRID method with a grid size of 11 resulted in a χ ² value of 1.45056, which is very near to the global minimum. Moreover, when ONEEX was followed by GRID, the global minimum was always found. Thus, ONEEX is very useful at the last stages of the fit to converge to the global minimum. Importantly, fitting strategies that used only RANDOM or MCMIN alone did not find the global minimum. Conversely, when applying ONEEX after RANDOM, the global minimum was always found. Likewise, the combination RANDOM + MCMIN + ONEEX always found the global minimum and was computationally faster than other methods.

Table 7.2 Relaxation dispersion curve-fitting methods in GLOVE

6 Outlook

Without a doubt, both R _1ρ and R ₂ relaxation dispersion will continue to contribute to the elucidation of a myriad of functionally important dynamical processes in the life of biomolecules such as proteins and nucleic acids. If sufficiently accurate relaxation dispersion profiles are obtained, standard desktop computers are now fast enough to solve Bloch–McConnell equations numerically without relying on approximate methods (Table 7.1). Application of relaxation dispersion to larger molecules remains challenging because of signal overlap and other complications; however, methods are continuously being developed to tackle these challenges [79,80,81,82].

The great wealth of information provided by relaxation dispersion sometimes overshadows the most important question in this context. What is the excited state? Even if relaxation dispersion can be unambiguously detected for a specific site in a biomolecule, initially it is often unknown what the dynamical process in question is. In other words, it is necessary to “assign” the excited state unambiguously.

If Δω can be successfully derived in terms of both sign and magnitude, the chemical shift of the excited state may provide hints about its structural nature. A combination of relaxation dispersion with chemical shift-based methods such as CS-ROSETTA is particularly useful in this case [83]. Alternatively, the excited state chemical shift determined from Δω may be compared with database reference chemical shifts from known structures or chemical shifts calculated by density functional theory [11]. Although the millisecond timescale exceeds the computational cost of most current laboratories for direct visualization of the dynamics by molecular dynamics (MD) simulations, visualization of possible dynamical processes by steered MD simulations is a potential computationally inexpensive approach [11]. In addition, comparison of the experimental relaxation dispersion-derived kinetic–thermodynamic data obtained from relaxation dispersion profiles acquired at different temperatures with theoretical free-energy calculations seems to be a successful strategy [11]. Lastly, it has been demonstrated that including additional biophysical methods such as single-molecule Förster resonance energy transfer (FRET), time-resolved X-ray crystallography, and principle component analysis (PCA) of conventional MD trajectories can successfully establish dynamical models of protein function over a wide range of timescales [84].