Chemical exchange in biomacromolecules: Past, present, and future

Abstract

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, mol ecular recognition, catalysis, and allostery by proteins and nucleic acids are highlighted.

1. Introduction

Time-dependent modulations of nuclear spin resonance frequencies give rise to chemical exchange phenomena in NMR spectroscopy [1]. Chemical exchange affects resonance positions, intensities, and linewidths in NMR spectra; conversely, the importance of chemical exchange for interrogating kinetic or other time-dependent processes has been recognized for decades [2,3]. Use of chemical exchange phenomena as probes of dynamic processes in biological macromolecules, such as proteins and nucleic acids, was for many years hindered by the complexity of ¹H spectra [4] and the insensitivity of natural abundance ¹³C or ¹⁵N spectra [5]. The former obstacle has been alleviated by methods for selective protonation of specific sites in otherwise deuterated molecules and the latter by methods for uniform or selective isotopic enrichment with ¹³C or ¹⁵N isotopes (in many applications both approaches are employed, vide infra). Isotopic enrichment with ¹³C or ¹⁵N became feasible first and chemical exchange line broadening was quantified in initial studies by multidimensional NMR spectroscopy of protein dynamics as an excess contribution to the ¹⁵N transverse relaxation rate constant [6]:

$$R_2=R_2^0+R_{\mathit{ex}}$$ (1)

in which R₂ is the observed free-precession transverse relaxation rate constant, $R_2^0$ is the relaxation rate constant in the absence of chemical exchange (dominated by dipole–dipole, chemical shift anisotropy, and quadrupolar relaxation mechanisms in biological macromolecules [1]), and R_ex is the chemical exchange relaxation contribution. Eq. (1) assumes that R_ex is independent of $R_2^0$, a situation that applies if the kinetic rate constants for chemical exchange are large compared to the difference in values of $R_2^0$ for the exchanging chemical species (vide infra). In addition, the value of R_ex, and hence R₂, may depend on radiofrequency (rf) fields applied during the NMR experiment, most commonly as spin-echo trains, spin-lock pulses, or adiabatic sweeps [7]. Qualitative identification of sites of exchange broadening using Eq. (1) remains an important part of many investigations, allowing rationalization of weak or missing resonance signals that hinder assignments and structure determination and identifying contiguous regions of higher flexibility that may be linked to function [8-11].

Experimental methods directly aimed at quantifying chemical exchange, generally by making use of the rf field dependence of R_ex, enable detailed characterization of structures, kinetics, and equilibria of interconverting species, even with populations less than a few percent in solution. These studies originated with applications to ¹³C and ¹⁵N labeled proteins [12-14]; more recently, methods have been developed that allow characterization of exchange broadening of ¹H resonances, making use of the large magnetogyric ratio of the ¹H spin, while minimizing the complex effects of ¹H spin interactions [15-18].

This Perspective briefly reviews these past and present achievements and suggests areas of future interest in quantitative investigation of chemical exchange phenomena in proteins and other biological macromolecules. The Perspective highlights experimental methods for solution NMR spectroscopy that have been developed recently and illustrates applications of these methods with some examples drawn from my own laboratory. A major theme of the Perspective is the development and application of spin-locking methods for probing a wide range of kinetic time scales. Methods for investigations of biomacromolecular conformational dynamics in solid-state NMR are rapidly developing and have many parallels with solution NMR methods (for a recent review, see [19]).

2. Theory

Chemical exchange in NMR spectroscopy is treated theoretically using the Bloch–McConnell equations for Bloch magnetization components (M_x, M_y, M_z) or more generally the Stochastic Liouville Equation (SLE) for arbitrary elements of the density operator [20]. These equations can be formally written as:

$$\frac{d}{\mathit{dt}}\mathit{\rho }(t)=(-i\mathit{L}-\widehat{\mathbf{\Gamma }}+\mathrm{\Xi })\mathit{\rho }(t)$$ (2)

in which ρ(t) = [ρ₁(t), ρ₂(t), …, ρ_N(t)]^T is the density matrix in the Liouville representation with dimension (M · N) × 1 (understood as the difference from the equilibrium density operator); M is the dimension of the spin space; N is the number of exchanging chemical states; ρ_k(t) is the density operator for the kth chemical state; L and Γ̂ are the Liouvillian and relaxation superoperators, respectively, defined as:

$$\mathit{L}=\left[\begin{array}{cccc}{\mathit{L}}_1& 0& \cdots & 0\\ 0& {\mathit{L}}_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\mathit{L}}_N\end{array}\right]$$ (3)

$$\widehat{\mathbf{\Gamma }}=\left[\begin{array}{cccc}{\widehat{\mathbf{\Gamma }}}_1& 0& \cdots & 0\\ 0& {\widehat{\mathbf{\Gamma }}}_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\widehat{\mathbf{\Gamma }}}_N\end{array}\right]$$ (4)

L_k and Γ̂_k are the M × M Liouvillian and relaxation superoperators in the spin space for the kth chemical state; Ξ = K ⊗ E; K is the N × N matrix of kinetic rate constants; E is the M × M identity matrix; K_ij = k_ji (i ≠ j); $K_{\mathit{ii}}=-{\sum }_{j=1}^N{k}_{\mathit{ij}}$; and k_ij is the (pseudo-first-order) rate constant for transition from chemical state i to j. The dimensionality of the Bloch–McConnell and Stochastic Liouville Equations grows rapidly with either the size of the spin system or number of exchanging states; consequently, exact closed-form solutions are available for few situations. Modern computers permit numerical solutions of the SLE for arbitrary cases and software programs are available for data analysis [21,22]. Nonetheless, approximate analytical solutions for limiting cases, such as fast kinetic exchange or highly skewed state populations, continue to be useful for both qualitative understanding and data analysis. For example, when N = 2 and the site populations satisfy p₁ ≫ p₂, i.e. one chemical state is dominant, the perturbation approach of Trott and Palmer gives the free-precession relaxation rate constant as $R_2=R_{21}^0+R_{\mathit{ex}}$ in which:

$$R_{\mathit{ex}}=\frac{k_{12}\left\{\mathrm{\Delta }{\omega }^2+\mathrm{\Delta }{R}_2^0\left(k_{21}+\mathrm{\Delta }{R}_2^0\right)\right\}}{{\left(k_{21}+\mathrm{\Delta }{R}_2^0\right)}^2+\mathrm{\Delta }{\omega }^2}$$ (5)

and $\mathrm{\Delta }{R}_2^0=R_2^0-R_{21}^0$ [23]. In this case, in contrast to Eq. (1), the exchange broadening is not independent of the exchange-free relaxation rate constants $R_{2i}^0$ (and this result can be generalized to N-state exchange).

3. ZZ-exchange spectroscopy

The first heteronuclear experiments specifically intended for characterizing chemical exchange in biological macromolecules were based on concepts from ¹H EXSY and ¹H–X (X = ¹³C or ¹⁵N) HMQC or HSQC heteronuclear correlation experiments. Like ¹H EXSY [24], these so-called ZZ-exchange experiments are appropriate for slow chemical exchange in which resolved resonances are observed for the individual chemical states. Initial experiments by Montelione and Wagner used 2I_zS_z longitudinal two-spin order or S_z magnetization (I = ¹H, S = ¹³C or ¹⁵N) during the mixing period, rather than I_z magnetization as in ¹H EXSY [25]. Subsequent developments by Wüthrich and coworkers [26] and Kay and coworkers [27] led to a widely used pulse sequence for heteronuclear ZZ-exchange spectroscopy using S_z magnetization during the mixing time and HSQC-style frequency labeling. This pulse sequence is shown in Fig. 1a. An example of ZZ-exchange data for the two-state monomer–dimer equilibrium for a protein construct comprising extracellular domains 1 and 2 of mouse E-cadherin is shown in Fig. 2 [28]. These data were analyzed using a composite peak intensity ratio defined by [29]:

$$\mathrm{\Pi }(T)=\frac{I_{\mathit{ij}}(T)I_{\mathit{ji}}(T)}{I_{\mathit{ii}}(T)I_{\mathit{jj}}(T)-I_{\mathit{ij}}(T)I_{\mathit{ji}}(T)}=k_{12}{k}_{21}{T}^2$$ (6)

in which I_ij(T) is the intensity of the crosspeak (i ≠ j) or diagonal peak (i = j) in the spectrum recorded with a mixing time T. This expression both simplifies data analysis and is less sensitive to differences in relaxation rates for (monomer and dimer) exchanging species. To independently determine k₁₂ and k₂₁, the equilibrium constant K_D is determined from fully relaxed correlation spectra; for accurate results, the effects of differential relaxation during coherence transfer periods must be considered [7,30].

Fig. 1.

Pulse sequence diagrams for (a) in-phase HSQC-detected [27] and (b) ¹⁵N TROSY-selected [31] ZZ-exchange experiments; the sequence in (a) is suitable for ¹³C or ¹⁵N spins (although water flip-back water suppression could be added in the case of ¹⁵N spins in similar fashion as in (b)). Narrow and wide bars represent 90° and 180° pulses, respectively; short narrow open bars represent water-selective 90° pulses; and wide open bars represent crafted 180° pulses that leave the water magnetization unperturbed [1]. All pulse phases are x unless indicated otherwise. The delays are Δ = 1/(4J_XH) and τ = 1/(8J_XH). Gradients are used to suppress unwanted coherences and pulse imperfections. (a) Decoupling during the relaxation delay uses a train of ¹H 180° pulses. Decoupling during acquisition is achieved with the GARP sequence [161]. The phase cycle is ϕ₁ = x, −x; ϕ₂ = x,x,y,y,−x,−x,−y,−y; and receiver = x,−x,−x,x. Frequency discrimination is obtained by shifting the phase of the receiver and ϕ₁ according to the States-TPPI protocol [162]. (b) The phase cycle is ϕ₁ = 4(x,−x,−y,y), ϕ₂ = 2(135°,315°,45°,225°), 2(315°,135°,225°,45°), ϕ₃ = 4(x,−x,−y,y), ϕ₄ = 4(x,−x,y,−y), ϕ_{5 - 5} = 2(y,y,y,y,−y,−y,−y,−y), ϕ₆ = 2(−y,−y,−y,−y,y,y,y,y), ϕ₇ = 4(−x,x,−y,y), ϕ₈ = 2(x,x,x,x,−x,−x,−x,−x), ϕ_rec = (x,−x,y,−y,x,−x,−y,y,−x,x,−y,y,−x,x,y,−y). Frequency discrimination is obtained by shifting the phase of the receiver and ϕ₄ according to the States-TPPI protocol [162].

Fig. 2.

ZZ-exchange characterization of a monomer–dimer equilibrium for E-cadherin domains 1 and 2. (a) Composite peak intensity ratio Π(T) for ¹⁵N ZZ-exchange measurements for residues Ile38 and Asp90 of a protein construct consisting of wild-type mouse E-cadherin extracellular domains 1 and 2 (EC1–EC2). The inset shows ¹⁵N ZZ-exchange spectra of residue Ile38 at several mixing times. The solid lines are best-fits to the experimental data to a modification of Eq. (5) for a monomer–dimer equilibrium, yielding k₁₂ = k_on = (1.0 ± 0.1) × 10⁴ M⁻¹ s⁻¹ and k₂₁ = k_off = 0.8 ± 0.1 s⁻¹ at 299 K. (b) Crystal structure of E-cadherin EC1–EC2 domains (Protein Data Bank (PDB) ID code 2qvf) showing the strand-swapped dimer interface of wild type protein. Reprinted from Y. Li, N. Altorelli, F. Bahna, B. Honig, L. Shapiro, A.G. Palmer, Mechanism of E-cadherin dimerization probed by NMR relaxation dispersion, Proc. Natl. Acad. Sci. USA. 110 (2013) 16462–16467.

For proteins that are ¹⁵N labeled and deuterated at non-exchangeable sites, ¹⁵N TROSY-selected [31] and TROSY-detected [32] ZZ-exchange experiment have been described. In this and other TROSY-based experiments [33], TROSY-selected methods measure the relaxation rate constant of the TROSY resonance (e.g. I^βS_z or I^βS^+/−) and detect using a two-dimensional TROSY sequence. In contrast, TROSY-detected methods measure the relaxation rate constant of in-phase magnetization and detect using a TROSY sequence. TROSY-selected experiments have the advantage of reduced intrinsic relaxation rate constants (vide infra), although fast amide proton solvent exchange compromises sensitivity. The pulse sequence for the TROSY-selected experiment is shown in Fig. 1b. This pulse sequence has been employed by Henzler-Widman and coworkers as part of an investigation of the small multi-drug resistance transporter EmrE in bicelles, showing that the protein exchanges between asymmetric dimeric conformations associated with inward and outward facing states of the transporter in membranes [34].

Although ZZ-exchange spectroscopy is now a well-established technique, the method is limited in two ways: (i) the chemical shifts of resonances must be resolved in the various chemical states and (ii) exchange kinetics must be faster than the relaxation rates of the magnetizations or coherences present during the mixing time of the pulse sequence. In some instances, resonances for certain nuclear spin types may be better resolved than others. For example, the X-nucleus frequency labeling periods shown in Fig. 1 can be replaced with ¹H spin evolution periods or the experiments extended into a third frequency dimension [35]. Alternatively, if the macromolecule of interest is weakly aligned in liquid crystalline media, a difference in resonance frequencies for a selected multiplet component may be observed, even in the absence of a chemical shift difference, owing to differences in residual dipolar couplings (RDCs) in different chemical states [36,37]. An example is shown in Fig. 3 for the small molecule N,N-dimethyltrichloroacetamide (DMTCA), which undergoes hindered rotation exchanging the two methyl groups. A ZZ-exchange experiment with a ¹H inversion pulse during the t₁ frequency-labeling interval shows the conventional pattern of auto- and cross-peaks (Fig. 3b). If decoupling is omitted during t₁, the multiplet patterns in each state are determined by the sum of the scalar and residual dipolar couplings (Fig. 3a). If ¹³C chemical shift evolution during t₁ is suppressed intentionally, ZZ-exchange cross peaks are still observable for the upfield and downfield multiplet components reflecting differences in RDCs for the two methyl positions (Fig. 3c). Singlet states, which have relaxation rates many times smaller than conventional z-magnetization or longitudinal two-spin order, have been shown to allow detection of very slow chemical exchange processes [38]. Routine applications to proteins or other biological macromolecules await appropriate isotopic labeling and pulse sequence schemes for selective excitation of singlet states in proteins and other biological macromolecules.

Fig. 3.

ZZ-exchange spectra for DMTCA aligned in poly-γ-benzyl-l-glutamate (PBLG) using a mixing time of 75 ms and T = 281.3 K. Auto-peaks are shown in red and blue boxes for the two methyl ¹³C spins in DMTCA; cross peak patterns are indicated by dotted lines. (a) Coupled ZZ-exchange experiment in which cross peaks develop between individual multiplet component pairs reflecting both conformational differences in chemical shifts and RDCs. (b) Conventional ¹H decoupled ZZ-exchange experiment reflecting only differences in chemical shifts. (c) The case in which Δω = 0 is simulated by adding 180° ¹H and ¹³C pulses in the midpoint of the t₁ frequency labeling delay. (d) Expansion of the boxed region in (c) shows that cross peaks are observed owing to difference in RDCs for the two exchanging methyl groups even in the absence of chemical shift differences.

4. CPMG and R_1ρ relaxation dispersion

The greatest degree of development in the use of chemical exchange phenomena in biological macromolecules has occurred for the technique of relaxation dispersion, in which relaxation rate constants are measured as functions of applied rf fields. These experiments are utilized when exchange kinetics are too fast or populations of the higher-energy states are too low to observe resolved resonances for the individual chemical states. In these situations a single resonance is observed, reflecting the population-averaged chemical shift for fast exchange, or the major highly populated species for slow exchange. Two principle experimental approaches exist: in one, a windowed Carr–Purcell–Meiboom–Gill (CPMG) train of 180° rf pulses is applied during the relaxation delay; in the other, a windowless rf field (which may be amplitude or phase modulated) is utilized [7,39]. In the CPMG approach, the dispersion of transverse relaxation rate constants, R₂, is measured as a function of the inter-pulse delay, τ_cp. In the other approach, dispersion of R_1ρ relaxation rate constants is measured as a function of the effective field in the rotating frame, by varying the rf amplitude or frequency; related techniques based on R_2ρ relaxation also have been described [40,41]. Although many analytical solutions have been derived for the exchange contribution to relaxation in CPMG [42-45] and R_1ρ experiments [23,39,46-49], for the present discussion, only approximations for two-site exchange in which the equilibrium site populations of the two chemical states satisfy p₁ ≫ p₂ are given to illustrate the two experiments [45,46]:

$$R_{\mathit{ex}}=\frac{p_1{p}_2\mathrm{\Delta }{\omega }^2}{k_{\mathit{ex}}}\left[\frac{k_{\mathit{ex}}^2}{k_{\mathit{ex}}^2+{\omega }_e^2}\right]$$ (7)

in which k_ex = k₁ + k₂, Δω = Ω₂ − Ω₁; Ω_k is the resonance offset frequency in the kth state;

$$\begin{array}{lll}{\omega }_e^2& =& {[144/{\tau }_{\mathit{cp}}^4+p_2^2\mathrm{\Delta }{\omega }^4]}^{1/2}(\mathrm{CPMG})\\ {\omega }_e^2& =& {\omega }_{1,\mathit{eff}}^2{\omega }_{2,\mathit{eff}}^2/{\omega }_{\mathit{eff}}^2({\mathrm{R}}_{1\rho })\end{array}$$ (8)

${\omega }_{\mathit{eff}}={({\overline{\Omega }}^2+{\omega }_1^2)}^{1/2}$ is the effective field in the rotating frame for the population-average resonance offset frequency, Ω̄; ${\omega }_{k,\mathit{eff}}={\left({\Omega }_k^2+{\omega }_1^2\right)}^{1/2}$ is the corresponding effective field for the kth chemical state; and ω₁ is the amplitude of the spin-lock rf field. The dependence of R_ex on kinetic parameters is illustrated in Fig. 4. The total transverse relaxation rate constant for the CPMG experiment is obtained by substituting Eq. (7) into Eq. (1). The observed relaxation rate constant in the tilted reference frame of the R_1ρ experiment is obtained as:

$$R_{1\rho }=R_1{\mathit{cos}}^2\theta +(R_2^0+R_{\mathit{ex}}){\sin }^2\theta$$ (9)

in which tan θ = ω₁/Ω̄ The expressions in Eqs. (1) and (7)-(9) assume that transverse relaxation rate constants $R_{2k}^0$ are identical in both chemical states. The effects of violations of this assumption have been discussed and are small if $|\mathrm{\Delta }{R}_2^0|\ll k_{\mathit{ex}}$ [49,50]. The limit in which $R_2^0$ for the sparsely populated state is extremely large forms the basis for the Dark-state Exchange Saturation Transfer (DEST) experiment (vide infra) [51]. In the fast exchange limit (k_ex > Δω), CPMG and R_1ρ experiments yield very similar dispersion profiles and allow extraction of two parameters, p₁p₂Δω² and k_ex by fitting to the experimental data. When exchange is intermediate-to-slow (k_ex < Δω), both experiments allow determination of p₁p₂ separately from Δω. In addition, the R_1ρ relaxation rate constant depends on the resonance offset of spins in the sparsely populated state, because ${\omega }_e^2\to {\Omega }_2^2+{\omega }_1^2$; thus, the R_1ρ approach permits direct determination of the frequencies of the otherwise unobservable resonances. The CPMG experiment depends only on |Δω|; however, when exchange is not in the fast limit, peak shifts in HMQC/HSQC spectra provide information on the absolute sign of chemical shift differences to augment CPMG results [52]. The relative merits of these different approaches for determination of the sign of Δω have been discussed [53,54].

Fig. 4.

Normalized relaxation dispersion for (a) CPMG and (b) R_1ρ experiments. Calculations were performed for a minor site population of p₂ = 0.05. In (a), the black curve shows the dispersion profile obtained in fast exchange (Δω/k_ex = 0.28) and is independent of scaling of Δω for different static magnetic fields (and thus, as shown by Eq. (7), the full dispersion curve scales quadratically with the applied static magnetic field). The green and red curves show the profiles obtained for slow exchange; the static magnetic field for the green curve (Δω/k_ex = 3.7) is 1.33 times larger than for the red curve (Δω/k_ex = 2.8), as would be obtained using 600 and 800 MHz NMR spectrometers. The reduced dispersion amplitude for the green curve results, again using Eq. (7), with a full dispersion curve that scales less than quadratically with the applied static magnetic field. In (b), ω₁/2π = 150 Hz, and Δω = 3 ppm, assuming a static magnetic field strength of 14.1 T (600 MHz). The green and blue curves show results for k_ex = 2000 s⁻¹ in which the resonance offset from the population-averaged position is shifted (green) in the direction of the minor state resonance or (blue) away from the minor state resonance. The maximum in the green curve occurs when the spin-lock rf field is on-resonance with peak position in the minor chemical state. The red and black curves show that when k_ex = 20,000 s⁻¹ is increased towards the fast-exchange limit, the differences in (red, towards minor resonance) and (black, away from minor resonance) sweep directions is reduced.

Some initial pessimism about the viability of CPMG methods for studies of biological macromolecules [55] was eliminated by the development of the relaxation-compensated CPMG pulse sequence [12]. The key insight of this experiment was that the effects of scalar J-coupling evolution during the variable inter-pulse spacing in the CPMG pulse train could be averaged by interchanging in-phase and anti-phase spin operators in an IS spin system at the midpoint of the relaxation period. This approach has been applied to other spin systems subsequently, proving its generality [56,57]. An alternative approach has been developed recently, in which ¹H continuous-wave decoupling during the CPMG sequence is used to prevent evolution of the scalar coupling altogether [58]. This latter approach is particularly important for systems subject to rapid solvent exchange and has been applied to lysine NH3 groups in proteins [59]. Recoupling artifacts arising from matching between ¹H decoupling pulses and X-nucleus spin-lock pulses in R_1ρ experiments initially restricted spin-lock field strengths to values much larger than one-bond scalar coupling constants, but redesign of pulse sequences has ameliorated these limitations [60,61]. Modern cryogenically cooled NMR probes also increase the maximum strength of spin-lock fields that can be employed, enabling investigations of faster kinetic processes [62]. Experiments for multiple-quantum, rather than single quantum, coherences also have been described [63-72]. Multiple-quantum relaxation rate constants may be more easily measured than single quantum relaxation rate constants in some instances [70,71] and in general provide information on whether the resonance frequency fluctuations for the spins involved in the multiple quantum coherence are (anti-) correlated [63,73].

Other developments have increased the robustness of relaxation dispersion experiments. Millet and coworkers showed that the static magnetic field dependence of chemical exchange is important in defining the chemical exchange time scale [74]; global fitting of relaxation dispersion curves recorded at two or more static magnetic fields has proven to be very effective. The values of the relaxation rate constant in the limits of very small (when the full exchange broadening is obtained) and very large (when the exchange broadening is eliminated) effective fields also are important constraints on data analysis. Hahn-echo experiments, in which only one or two spin-echo sequences are utilized, approximate the free-precession limit rate constant [75,76] and a very robust ¹⁵N TROSY-selected Hahn-echo sequence has been developed [77]. The large field limit, when all exchange broadening is suppressed, has been approximated by a number of methods, including field-dependence of relaxation rate constants [78], relaxation interference rate constants [79], linear combinations of single- and multiple-quantum relaxation rate constants in spin-locking experiments [69], and joint analysis of CPMG and R_1ρ data [80]. An elegant pulse sequence for the latter has been published recently [81], indicating the continued developments in this area.

As for ZZ-exchange experiments, HSQC-detected, TROSY-detected, and TROSY-selected CPMG and R_1ρ pulse sequences have been developed for ¹⁵N spins; versions of the TROSY-detected [82] and TROSY-selected [83] R_1ρ sequences are shown in in Fig. 5 for illustration. Examples of R_1ρ data are shown in Fig. 6 for the protein ubiquitin to illustrate the improvement offered by the TROSY-selected pulse sequence because the exchange-free relaxation rate constant for the TROSY component of the ¹H-¹⁵N doublet satisfies $R_2^{\mathrm{\beta },0}<R_2^0$ [83]. Relaxation dispersion experiments applied to macromolecules in dilute liquid crystalline media also permit measurement of RDCs [36,37] and residual chemical shift anisotropies [84] for sparsely populated states. Relaxation dispersion at multiple static magnetic fields and multiple temperatures, combined with analyses of new X-ray crystal structures and computationally generated ensembles of structures (vide infra) have resulted in detailed understanding of the conformational kinetic processes that lead to line broadening in ubiquitin [13,71,72,85,86].

Fig. 5.

Pulse sequences for (a) TROSY-detected [82] and (b) TROSY-selected [83] R_1ρ experiments. Narrow and wide bars depict 90° and 180° pulses, respectively; short narrow open bars represent water-selective 90° pulses; and wide open bars represent crafted 180° pulses that leave the water magnetization unperturbed [1]. Composite pulses are ${90}_x^{°}{210}_y^{°}{90}_x^{°}$, shown as three closely spaced bars. All pulses are x-phase unless otherwise indicated. The delays are Δ = 1/(4J_NH), τ = 1/(8J_NH), Ξ > Ge, ε = Δ − ζ/2, ζ > Gd. The spin-lock fields are shown as open rectangles; the triangular segments are adiabatic pulse schemes to rotate the magnetization from (to) the z-axis to (from) the orientation of the effective field in the rotating frame [163]. Phase cycles are (a) ϕ₁ = 4y, 4(−y); ϕ₂ = y,x,−y,−x; ϕ₃ = y; and receiver phase = y,−x,−y,x,−y,x,y,−x and (b) ϕ₁ = x,−x; ϕ₂ = x,x,−x,−x; ϕ₃ = y; ϕ₄ = x; ϕ₅ = 4(135°) 4(315°); and receiver = (x,−x,−x,x,−x,x,x,−x). Gradients Ge and Gd are used for coherence selection; other gradients are for artifact suppression. Gradients are rectangular or shaped as indicated (for details see the original publications). Echo/antiecho quadrature detection [164] is achieved by (a) inverting ϕ₃, and the sign of gradient Ge and using ϕ₂ = y,−x,−y,x; and (b) inverting ϕ₃, ϕ₄, and the sign of gradient Ge. The ϕ₁ and receiver phases are inverted for each t₁ increment to shift axial peaks to the edge of the spectrum.

Fig. 6.

¹⁵N R_1ρ relaxation dispersion for ubiquitin. Relaxation dispersion curves for two exchange-broadened residues in ubiquitin, (circles) Asn 25 and (squares) Ile 23, collected using (blue) TROSY-detected or (red) TROSY-selected R_1ρ experiments. Solid lines represent the fits to the experimental data obtained using the chemical exchange parameters of Massi et al. [86] and R₂0 or $R_2^{\mathrm{\beta },0}$ as an adjustable parameter. The $R_2^{\mathrm{\beta },0}$ values obtained in the TROSY-selected R_1ρ experiment are 5.27 ± 0.05 s⁻¹ and 5.62 ± 0.07 s⁻¹ for Ile 23 and Asn 25, as compared to the $R_2^0$ values of 10.8 ± 0.1 s⁻¹ and 10.6 ± 0.1 s⁻¹ obtained in the TROSY-detected R_1ρ experiment. Representative TROSY-selected R_1ρ relaxation dispersion curve for one of the non-exchanging residues, Thr 66, is shown with triangles. Reprinted with permission from T.I. Igumenova, A.G. Palmer, Off-resonance TROSY-selected R_1ρ experiment with improved sensitivity for medium- and high-molecular-weight proteins, J. Am. Chem. Soc. 128 (2006) 8110–8111. Copyright 2006 American Chemical Society.

5. ¹³C and ¹H relaxation dispersion

In proteins, ¹⁵N spins are convenient probes for chemical exchange processes because (i) these spins are rare so that coupling interactions between ¹⁵N spins are weak, (ii) dipole and chemical shift relaxation rates are relatively small and can be reduced further by deuteration and TROSY methods, (iii) isotopic enrichment is straightforward, and (iv) ¹⁵N chemical shifts are affected by many magnetic influences. On the other hand, ¹⁵N spins have drawbacks as probes: (i) fast solvent exchange of directly bonded hydrogen atoms reduces the efficiency of relaxation methods and (ii) the low magnetogyric ratio of ¹⁵N nuclear limits the amplitude of rf fields that can be applied experimentally. Over the past few years, numerous methods have been developed for relaxation dispersion measurements of ¹H and ¹³C spins. These methods rely on schemes for isotopic enrichment that reduce unwanted scalar and dipolar coupling interactions and on pulse sequence designs that minimize systematic interference from residual interactions. Determination of chemical shifts for as many sites and nuclei as possible is important for calculation of structures of sparsely populated states (vide infra). Accordingly, methods have been developed for ¹H^N [87-89], ¹H^α [90-92], ¹³CO [93-95], ¹³C^α [92,94,96], and ¹³C^β [97] spins. Related methods have been developed for aromatic groups, which are relatively rare in proteins but provide additional probes of the hydrophobic core dynamics [98] and side chain ¹H spins [18].

An example of ¹H^N CPMG relaxation dispersion for the monomeric form of mouse E-cadherin extracellular domains 1 and 2 is shown in Fig. 7 [28]. The chemical exchange broadening results from formation of a sparsely populated intermediate, termed the “X-dimer” from X-ray studies, that is structurally distinct from the strand-swapped dimer. Comparisons of the wild-type protein with a mutant that blocks formation of the X-dimer establishes that an X-dimer-like state is an on-pathway intermediate for the strand-swapped dimer described in Fig. 2. Thus, the combination of ZZ-exchange and relaxation dispersion techniques allow full kinetic characterization of dimerization of wild-type E-cadherin in solution.

Fig. 7.

¹H^N CPMG relaxation dispersion data for mouse E-cadherin extracellular domains 1 and 2. (a) Dispersion profiles for the monomer resonances of (a) Ile 7 and (b) Gln 101 for two different total monomer protein concentrations of (red, blue) 374 μM and (purple) 97 μM recorded at (red, purple) 600 and (blue) 800 MHz ¹H frequencies. The solid lines are fits to the data, yielding k_ex = 1890 ± 130 s⁻¹ and p₂ = 0.025 ± 0.003 at the higher concentration at the lower concentration, p₂ = 0.017 ± 0.003. (c) X-dimer interface (drawn from the X-ray crystal structure of the E89A mutant, PDB ID code 3lni) with residues showing relaxation dispersion highlighted as stick representations. The green spheres represent bound calcium ions. Reprinted from Y. Li, N. Altorelli, F. Bahna, B. Honig, L. Shapiro, A.G. Palmer, Mechanism of E-cadherin dimerization probed by NMR relaxation dispersion, Proc. Natl. Acad. Sci. USA 110 (2013) 16462–16467.

Particularly important developments have been reported using methods based on the special properties of methyl groups. Rapid rotation of the methyl group reduces ¹H–¹³C dipole–dipole relaxation of the ¹³C group by a factor of ~1/9. In one set of experiments, the AX₃ methyl spin system is simplified to an approximately AX system by random fractional deuteration and selection by pulse sequence design for ¹³CHD₂ isotopomers. Methods then exist for measuring chemical exchange of ¹³C [99] and ¹H spins [15-17]; as an example, Fig. 8a shows a recently reported pulse sequence for ¹H R_1ρ measurements [17]. A variant of this experiment has been applied to methionine methyl groups in the α₇ ring of the 20S proteasome from T. acidiphilium [16]. The results indicate that relaxation dispersion to be measured in very large molecular species prepared with protonated ¹³CH₃ groups in a highly deuterated background (including stereospecifically deuterating one methyl group in Val and Leu residues) [100]. Fig. 8b shows a related pulse sequence for measurements of ZQ/DQ relaxation rate constants by a methyl-TROSY Hahn-echo sequence that uses ZQ evolution during t₁ and suppresses anti-TROSY multiplet components [70]. Even for large systems, filtration reduces spectral artifacts from methyl groups in flexible or disordered regions.

Fig. 8.

Relaxation measurements for methyl groups. (a) ¹H R_1ρ pulse sequence for ¹³CHD₂ groups [17] and (b) ZQ-TROSY Hahn-echo pulse sequence for ¹³CH₃ groups [70]. Narrow and wide bars depict 90° and 180° pulses respectively; wide open bars represent crafted 180° pulses that leave the water magnetization unperturbed [1]. All pulses are x-phase unless otherwise indicated. Decoupling during acquisition uses WALTZ-16 [165]. (a) The delays are τ_a = 1.67 ms and τ_b = 2.0 ms. For constant time evolution, T_c = 14.3 ms, Δ₁ = τ_b + t₁/2, Δ₂ = T_c − τ_b, and Δ₃ = T_c − t₁/2. For isolated ¹³CHD₂ methyl groups, non-constant time evolution uses Δ₁ = τ_b + t₁/2, Δ₂ = t₁/2, and Δ₃ = τ_b. The spin-lock fields are shown as open rectangles; the triangular segments are adiabatic pulse schemes to rotate the magnetization from (to) the z-axis to (from) the orientation of the effective field in the rotating frame [163]. Phase cycles are ϕ₁ = x,−x; ϕ₂ = x,x,y,y,−x,−x,−y,−y; and receiver phase = x,−x,−x,y,−y. Gradients are for artifact suppression. Frequency discrimination is obtained by shifting the phase of the receiver and ϕ₁ according to the States-TPPI protocol [162]. (b) The delay τ = 1/(2J_CH) ≈ 3.91 ms. Phase cycles are ϕ₁ = x,−x; and receiver phase = x,−x. Gradients G_X and G_SQ are used for coherence selection; other gradients are for artifact suppression. For recording ZQ relaxation, ϕ₁ = x. Multiplet filtration is obtained by adding data sets recorded with (i) Δ₁ = 0, Δ₂ = T/2 + τ/4 + t₁, Δ₃ = 0 and Δ₄ = T/2 + τ/4 and (ii) Δ₁ = τ/4, Δ₂ = T/2 + t₁, Δ₃ = τ/4 and Δ₄ = T/2. The corresponding data sets for echo/antiecho quadrature detection use (iii) Δ₁ = 0, Δ₂ = T/2 + τ/4, Δ₃ = 0 and Δ₄ = T/2 + τ/4 + t₁ and (iv) Δ₁ = T/2, Δ₂ = τ/4, Δ₃ = T/2 + t₁ and Δ₄ = τ/4. Gradient G_X = G_ZQ for data sets (i–iii) and G_DQ for data set (iv). The ϕ₁ and receiver phases are inverted for each t₁ increment to shift axial peaks to the edge of the spectrum. Corresponding values of parameters for measuring DQ relaxation rate constants are given in the original publication.

Fig. 9 illustrates the application of ¹³C relaxation dispersion measurements to characterize a sparsely populated state of the villin headpiece domain HP67 [96], which has also been investigated by ¹⁵N relaxation dispersion [101]. This application used random fractional labeling with ¹³C to minimize ¹³C–¹³C scalar and dipolar interactions. Two different relaxation dispersion approaches were utilized: for ¹³C^α spins, CPMG data were acquired at a single static magnetic field and these data were supplemented by Hahn-echo rate constants recorded at five static magnetic fields. The field dependence of the Hahn-echo data was used both to establish the chemical exchange regime and to determine $R_2^0$ as a constraint for fitting CPMG dispersion curves. For ¹³CH₃ methyl groups, CPMG relaxation dispersion was measured at two static magnetic fields and analyzed globally. HP67 consists of an N-terminal subdomain (residues 10–42) that transiently unfolds at equilibrium under native-like conditions and a highly stable C-terminal subdomain (residues 43–76). Comparisons between p₁p₂ Δω2 and secondary chemical shifts reveal three groupings of residues in the intermediate: (i) residues that are highly disordered, (ii) residues that maintain partial native-like structure, and (iii) residues that make non-native-like interactions. Residues in the second category are spatially located in the interface between the N- and C-terminal domains and residues in the third category are located in the vicinity of His 41, which can become protonated in the intermediate state.

Fig. 9.

Relaxation dispersion characterization of a folding intermediate for the villin headpiece domain HP67. (a) Backbone ¹³C^α CPMG relaxation dispersion profiles for Arg 37 measured at a static magnetic field strength of 14.1 T. The solid lines are fits to the data. The dashed line corresponds to the value of $R_2^0$ used to constrain the data in the limit of infinitely fast pulsing. (b) The corresponding Hahn-echo R₂ dispersion data are plotted versus ${\omega }_c^2$ in the inset; the solid line is a linear fit to the data; the y-intercept determines $R_2^0$. Methyl ¹³C CPMG relaxation dispersion profiles measured at static magnetic field strengths of (○) 14.1 T and (●) 18.8 T for (c) Val20 C^γ2 and (d) Val33 C^γ2. The solid lines are fits to the data. (e) Analysis of ϕ_ex = p1p2Δω2 for ¹³C^α and methyl ¹³C spins in HP67. Values determined from individual fits of R_ex and the global value of k_ex = 3190 ± 180 s⁻¹ are plotted versus |Ω_sec| for (a) ¹³C^α (●,○) and (b) methyl ¹³C (■,□) spins. Green symbols represent data for residues that are assumed to be fully unfolded in the intermediate ensemble; the dashed green line yields a slope of 0.104 ± 0.005, corresponding to a population of (1.09 ± 0.11)% for the intermediate state. The solid line has a slope of 0.0105, corresponding to p₂ = (1.11 ± 0.09)% observed previously for ¹⁵N spin relaxation dispersion [101]. Data points that were excluded from the fit are grouped into two categories: (i) those that lie above and to the left of the fitted line adopt non-native conformations in the intermediate and (ii) those that lie below and to the right of the fitted line maintain residual native-like interactions in the intermediate. Residues in (i) and (ii) are colored with brown and blue gradients, respectively, with the color of the data points becoming lighter as the ratio |Δδ/Ω_sec| deviates from unity, in which |Δδ| was obtained from ϕ_ex assuming p₂ = (1.11 ± 0.09)%. (f) Exchange broadened ¹³C^α and ¹³C methyl groups are mapped onto the structure of HP67. Atoms are colored to correlate with the classifications depicted in (e). Two regions of residual interactions are maintained the intermediate: (i) at the interface of the N- and C-terminal subdomains and (ii) in the vicinity of His41. Reprinted from N.E. O’Connell, M.J. Grey, Y. Tang, P. Kosuri, V.Z. Miloushev, D.P. Raleigh, A.G. Palmer, Partially folded equilibrium intermediate of the villin headpiece HP67 defined by ¹³C relaxation dispersion, J. Biomol. NMR 45 (2009) 85–98, with permission from Springer.

6. CEST and DEST

Two very similar techniques have been developed to obtain information on chemically exchanging systems in which the exchange time scale is slow and the populations of the sparsely populated states are too low for reliable quantification by CPMG relaxation dispersion or ZZ-exchange methods alone: Chemical Exchange Saturation Transfer (CEST) [102-104] and DEST (vide supra) [51,105]. CEST was originally developed as a contrast mechanism in magnetic resonance imaging [106] and subsequently adapted to high resolution solution NMR spectroscopy [107]. In both CEST and DEST, the intensity of the resonance signal from the major chemical species is measured during application of a weak rf field. The frequency of the rf field is varied across the spectrum in separate experiments and the (normalized) intensity of the resonance plotted as a function of resonance offset. The experiments utilize the effects on the observed signal when the rf field is close to resonance with the unobservable sparsely populated state(s). The experiments differ primarily in that the CEST experiment relies on conventional exchange broadening by the variation in chemical shift between the major and minor states, while the DEST experiment relies on the difference in transverse relaxation rate constants between rapidly and slowly tumbling species in solution, typically a monomeric protein and a very large assemblage, such as an amyloid fibril.

A pulse sequence for the ¹⁵N CEST/DEST experiment is shown in Fig. 10 [103]. The experiment is essentially a ¹⁵N R₁ relaxation measurement with the addition of the weak rf field applied to the ¹⁵N spins during the relaxation delay, T. The resulting profiles of I(T)/I(0) as a function of resonance offset are analyzed numerically using Eqs. (2)-(4) with ${\mathit{\rho }}_k(t)=[M_{\mathit{kx}}(t),M_{\mathit{ky}}(t),M_{\mathit{kz}}(t)-p_k{M}_z^{\mathit{eq}}]$,

$$\mathbf{K}=\left[\begin{array}{cc}-k_{12}& k_{21}\\ k_{12}& -k_{21}\end{array}\right]-i{\mathit{L}}_k-{\widehat{\mathbf{\Gamma }}}_k=\left[\begin{array}{ccc}-R_{1k}& -{\Omega }_k& {\omega }_1\\ {\Omega }_k& -R_{2k}& 0\\ -{\omega }_1& 0& -R_{2k}\end{array}\right]$$ (10)

and $M_z^{\mathit{eq}}$ is the total equilibrium magnetization. Both CEST and DEST experiments are conventionally described as saturation transfer methods: when resonant with the sparsely populated states, the weak rf field saturates the minor state magnetization and the exchange kinetics transfer this saturation to the magnetization of spins in the major state. Many experiments in NMR spectroscopy can be usefully regarded from multiple theoretical perspectives. In this instance, both experiments can be treated as off-resonance R_1ρ experiments to obtain insight into the expected results. For the CEST experiment, provided that $\mathrm{\Delta }{R}_2^0=|R_{22}^0-R_{21}^0|\ll k_{\mathit{ex}}=k_{12}+k_{21}$ the expected rate constant is given by Eqs. (7) and (8). Baldwin and Kay have extended the results of Eqs. (7) and (8) to the situation in which $|\mathrm{\Delta }{R}_2^0|>0$ [49]; the perturbation approach of Trott and Palmer [23] also can be extended to $|\mathrm{\Delta }{R}_2^0|>0$ and yields a simpler expression (but slightly less accurate as p₂ increases):

$$R_{\mathit{ex}}=k_{12}\frac{\left\{\mathrm{\Delta }{\omega }^2+{(\mathrm{\Delta }{R}_2^0)}^2\right\}{k}_{21}+\mathrm{\Delta }{R}_2^0({\omega }_{1,\mathit{eff}}^2+k_{21}^2)}{k_{21}\left\{{\omega }_{2,\mathit{eff}}^2+k_{21}^2+{\left(\mathrm{\Delta }{R}_2^0\right)}^2\right\}+\mathrm{\Delta }{R}_2^0{\omega }_1^2}$$ (11)

If ω₁ → 0 and Ω₁ → 0, then Eq. (10) reduces to Eq. (5) and if $\mathrm{\Delta }{R}_2^0\to 0$and p_b → 0, then Eq. (10) reduces to Eqs. (7) and (8). As already noted, the perturbation approach also can be applied to obtain relaxation rate constants for N-site exchange. The observed signal in the CEST or DEST experiments is approximately:

$$I(T)/I(0)={\cos }^2\theta \exp (-T·R_{1\rho })$$ (12)

in which the factor of cos²θ results from projection of the longitudinal magnetization onto the tilted reference frame (this factor rapidly approaches unity for weak rf fields off-resonance with respect to the population-average chemical shift). This analysis does not consider the contribution near resonance from evolution of magnetization components orthogonal to the tilted z-axis; for long irradiation times used in CEST and DEST experiments, these components dephase owing to R_2ρ relaxation and ω₁ inhomogeneity. Sample calculations are shown in Fig. 11 for a system with similar kinetic parameters, but $\mathrm{\Delta }{R}_2^0=0$ for application of the CEST experiment and $\mathrm{\Delta }{R}_2^0\gg 0$ for application of the DEST experiment. An additional calculation shows the result of $\mathrm{\Delta }{R}_2^0>0$ in the CEST experiment. In this latter case, the width of the I(T)/I(0) curve at the position of the minor resonance is broadened; as the value of $\mathrm{\Delta }{R}_2^0$ becomes larger and larger, this peak merges into the central peak and CEST becomes equivalent to DEST. The agreement between the exact calculation and the R_1ρ approximation is excellent for both experiments. Thus, a viewpoint is established that provides unified insights into CEST, DEST, and conventional R_1ρ experiments over the full range of spin-locking amplitudes.

Fig. 10.

¹⁵N CEST/DEST pulse sequence [103]. Narrow and wide bars depict 90° and 180° pulses, respectively. All pulses are x-phase unless otherwise indicated. The ¹H transmitter is positioned on the water resonance throughout the sequence except during the relaxation period T, when it is moved to the center of the amide region (8.4 ppm). The ¹⁵N transmitter is placed at 119 ppm except during T, when it is relocated to the desired offset. A coherent decoupling train consisting of 90_x240_y90_x pulses is used for ¹H decoupling during T for CEST experiments; decoupling can be obtained with two composite 180° pulses as in Fig. 5 for DEST experiments. In the CEST experiment, temperature compensation is obtained by applying the ¹H decoupling scheme for a time T_max − T immediately after the completion of acquisition, in which T_max is the maximum exchange time used in the experiment (typically T = T_max or 0). ¹⁵N decoupling during acquisition is achieved with WALTZ-16 [165]. Delays are Δ = 1/(4J_NH), and ε > Gd. The phase cycle is ϕ₁ = {x, −x}, ϕ₂ = {y}, ϕ₃ = {2x,2y,2(−x), 2(−y)}, ϕ₄ = {x}, receiver = {x, −x, −x,x}. Weak bipolar gradients, depicted as solid black bars, are applied during the t₁ period. Unlabeled gradients are used to suppress unwanted coherences and artifacts, whereas Ge and Gd are encoding and decoding gradients, respectively, for echo/antiecho coherence selection, obtained by inverting the signs of ϕ₄ and Ge [164]. Values of ϕ₂ and the receiver phase are inverted between t₁ points to shift axial peaks to the edge of the spectrum. A conventional HSQC-detected R_1ρ pulse sequence is obtained by replacing the ¹H decoupling and ¹⁵N spin-lock during T with the scheme (including the adiabatic sweeps) shown in Fig. 5a.

Fig. 11.

Theoretical (a) CEST and (b) DEST profiles. Calculations assumed k_ex = 50 s⁻¹, p₂ = 0.015, Ω₁ = −0.076 ppm, Ω₂ = 5 ppm, and T = 0.48 s. ΔΩ is measured relative to the population-averaged resonance position. In (a) R₁₁ = R₁₂ = 1 s⁻¹, R₂₁ = R₂₂ = 20 s⁻¹, ω₁/2π = 25 Hz. In (b) R₂₂ = 20,000 s⁻¹ and (dashed) ω₁/2π = 150 Hz and (solid) ω₁/2π = 300 Hz. Black lines give the numerical solutions of the Bloch–McConnell equations (Eqs. (2)-(4) and (10)) and red lines give the R_1ρ approximations using (a) Eqs. (7)-(9) and (12) or (b) Eqs. (7)-(9), (11) and (12). The blue dashed line in (a) shows the result of increasing R₂₂ = 200 s⁻¹ in the CEST experiment, calculated using Eqs. (7)-(9), (11), and (12) as for the DEST experiment.

7. Applications to nucleic acids

The above chemical exchange methods developed and applied to proteins have analogues adapted for applications to DNA and RNA nucleic acid molecules; such approaches have been reviewed recently [108] and are only described in brief herein. In contrast to proteins, nucleic acids have more sparsely distributed ¹⁵N spins suitable as probes of chemical exchange, notably the imino nitrogen sites of G and T/U nucleobases. Rapid solvent exchange of imino hydrogens can reduce the sensitivity and accuracy of relaxation measurements, particularly for solvent-exposed conformational states. Thus, to circumvent these concerns, an early application to the lead-dependent ribozyme utilized ¹³C R_1ρ measurements for the isolated C2 sites of A and C8 sites of A and G nucleobases in uniformly ¹³C/¹⁵N enriched molecules [109]. The authors noted the potential difficulties posed by ¹³C–¹³C scalar and dipolar interactions for C6 of pyrimidines and other ¹³C sites in fully labeled RNA molecules. As for proteins, isotopic labeling strategies have been developed to increase the number of isolated ¹³C spins suitable for relaxation dispersion and other relaxation measurements [110-112]. Selective excitation and decoupling schemes can be used to minimize complications of ¹³C–¹³C interactions in natural abundance and isotopically enriched molecules [113]. The ¹H–¹H dipole–dipole interactions of imino ¹H spins can be suppressed approximately by pulse sequence elements that average ROE and NOE interactions in ¹H–¹H ZZ-exchange experiments [114]. Applications of chemical exchange methods to nucleic acids have revealed a host of intramolecular dynamic processes, including loop repacking and rearrangement of secondary structures [108,109]. In notable applications, ¹³C ZZ-exchange or relaxation dispersion methods have been used to characterize transitions between Watson–Crick and Hoogsteen base pairing in DNA oligonucleotides [115,116] and folding equilibria of “bistable” RNA oligonucleotides [117].

8. Computational methods

Computational methods, such as molecular dynamics simulations, have long been used in conjunction with NMR spin relaxation studies of conformational dynamics on ps–ns time scales [118-121]. The continued growth in computational power and methods has begun to have similar impact on NMR investigations of chemical exchange in at least three areas: generation of structural models that may represent minor chemical states, calculation of structures of minor species from chemical shifts and other NMR-observables extracted from relaxation dispersion studies, and direct simulation of chemical exchange processes.

In the first area, structural models have been generated from many approaches, including hand-building hypothetical structures [122], extraction of frames from molecular dynamics (MD) simulations [123], and computation of ensembles of structures using other experimental data, such as residual dipole couplings [85,124]. These approaches might be described as “forward methods”: the structural models are validated or selected by comparing values of Δω (for slow exchange) or p₁p₂ Δω2 (for fast exchange) calculated from the models to experimental data. The second approach is an “inverse method”: given the experimental data, structural models are calculated using the experimental data as restraints on the structure determination algorithm [125]. The first example used the CS-Rosetta program to determine the structure of a folding intermediate for the FF domain from HYPA/FBP11 [126]. The calculation used backbone ¹⁵N, ¹H^N, ¹³C^α, ¹H^α, and ¹³CO relaxation dispersion measurements at static field strengths of 11.7 T and 18.8 T to obtain chemical shifts and amide N–H RDCs of the intermediate. A similar protocol was used subsequently to determine the structure of sparsely populated state of the T4 lysozyme L99A mutant [127]. In an alternative approach [128], the structure of a sparsely populated equilibrium on-pathway folding intermediate of the A39V/N53P/V55L mutant of the Fyn SH3 domain was determined from relaxation dispersion data, including ¹⁵N, ¹H^N, ¹³CO, ¹³Cα, and ¹Hα backbone chemical shifts; ¹⁵N–¹H^N RDCs; and ¹³CO residual chemical shift anisotropies, using restrained MD simulations and CamShift [125]. The C-terminal residues 57–59 are disordered in the intermediate, disrupting β-strand 5. Mutants lacking the C-terminal residues form fibrillar aggregates, suggesting that the intermediate is aggregation-prone and can initiate fibrillation. Importantly, in all three cases, the mutations designed from the calculated structures allow validation of these approaches by stabilizing or mimicking the sparsely populated states. As larger sets of chemical shifts become available from experiments such as CEST, direct calculations of the structures of sparsely populated chemical states will become routine and allow more structural insights into the properties and functions of rare conformations of biological macromolecules.

Parallel developments of powerful computer hardware and efficient simulation algorithms have extended the length of MD simulations from order 10² ps in the early 1990s to a few milliseconds (10⁶ ps) at present. These simulation lengths begin to approach the values of 1/k_ex observed in relaxation dispersion studies and raise the prospect that relaxation dispersion results could be interpreted from simulations, in similar fashion as for faster time scale dynamic processes. In complete analogy to the calculation of time-correlation functions for the dipole–dipole interaction in conventional analyses of laboratory frame relaxation data, the time correlation function for chemical shifts is calculated by using SPARTA+ or other semi-empirical chemical shift computer program to obtain chemical shift values for individual frames of the MD trajectory. The Fourier transform of this correlation function is the spectral density function for the process and is directly related to the relaxation dispersion profile in the fast motional limit (in which a Redfield treatment applies) [129,130]:

$$R_{\mathit{ex}}={\omega }_0^2{\mathit{\int }}_0^{\infty }\langle [\delta (t)-\overline{\delta }][\delta (t+\tau )-\overline{\delta }]\rangle f(\tau )\mathit{d\tau }$$ (13)

in which ω₀ is the Larmor frequency, δ(t) is the calculated chemical shift (in ppm) at time t in the trajectory, angle brackets indicate averaging over the trajectory. The function f(t) = 1, cos(ω₁t), or tri[πt/(2τ_cp)] for free-precession, R_1ρ, or CPMG experiments, respectively, and tri(x) is the triangle wave with extreme values (0, 1), (π, −1), (2π, 1) et cetera.

The most detailed analyses to date have been based on a 1-ms simulation of BPTI [130,131]. Extensive experimental data exist for BPTI on the dynamics of disulfide bond isomerization [122,132], flipping of aromatic rings [133], and exchange of buried water molecules with bulk solvent [134]. Conformations consistent with experimental results are sampled in the simulation, but the statistical sampling of conformations in the simulation does not converge to the experimentally determined populations of the alternative states [130]. Thus, chemical exchange rate constants calculated from the trajectory are in qualitative, but not quantitative agreement with experiment. Additional conformations are observed in the MD simulations that have not been detected experimentally by relaxation dispersion methods. Amide hydrogen exchange in proteins reports on very small populations of conformations competent for exchange with solvent. Recent work has compared amide hydrogen exchange rates for BPTI with predictions from MD simulations and concluded that the populations of such novel conformations are likely overestimated by current computational methods [135]. In related work, the exchange of buried water molecules in BPTI with bulk solvent in the 1-ms simulation has been analyzed using a stochastic point process formalism [136]. Even a 1-ms simulation is short compared to the kinetic rate constants observed experimentally for BPTI, limiting statistical convergence of the MD analysis. However, inexorable advances in computer power (coupled with developments in force fields and water models) will continue to increase the synergy between NMR relaxation and MD simulations. Indeed, 1-ms MD simulations of the 76-residue protein ubiquitin have been reported recently [137].

9. Discussion

Investigations of conformational dynamics of biological macromolecules are germane particularly to three broad problems in biology: folding and other disorder–order transitions, molecular recognition and oligomerization, and enzyme catalysis. Emerging themes in many of these applications are: (i) “allosteric pathways” composed of spatially adjacent moieties with similar conformational dynamical properties enable coupling between spatially distal regions of structure and (ii) conformational differences between higher-energy sparsely populated states and the major ground state species enable tuning and regulation of function [138-143]. Chemical exchange phenomena are convenient probes of aspects of each of these phenomena owing to a fortuitous matching between the kinetic time scales of these processes and the (ever-increasing) time scales accessible to NMR experiments as well as to the exquisite sensitivity of the chemical shift (and dipole coupling constants) to chemical environment. Experimental developments in the recent past and foreseeable future provide both access to novel chemical systems, such as the DEST method for monomer-fibril exchange [51] or methyl-TROSY techniques for very large molecular machines [100], and increased accuracy of existing methods [82], allowing ever-finer dissection of dynamic contributions to function. A testament of the power of NMR spin relaxation methods and the subtlety of biological macromolecules is that nearly every application to a novel system reveals some unexpected feature linking conformational flexibility and dynamics to function.

Exchange methods have been used to measure ultra-fast protein folding rates that are otherwise difficult to access experimentally [144]. Relaxation dispersion measurements also allow folding rates to be measured at equilibrium under native-like conditions. These measurements are then important controls for other techniques that require perturbation with denaturant. Relaxation dispersion methods also have been used (vide supra) to characterize sparsely populated conformational states in solution that may represent (on- or off-pathway) folding intermediates [96,104,126,145]. Relaxation dispersion results for wild-type and mutant proteins can be used for ϕ-value analysis [146,147]. The NMR ϕ-value approach has the advantage over more usual methods, such as fluorescence spectroscopy, in that confounding structural perturbations caused by mutation can be identified by examination of chemical shift changes between wild-type and mutant species.

In many applications of relaxation dispersion measurements, sparsely populated states of apo proteins are detected that resemble the conformations observed in complexes with other molecules or otherwise activated states [85,148]. These results naturally leads to the hypothesis that molecular recognition or activation is through a “selected-fit” or “population-shift” mechanism. However, as emphasized by a number of authors [29,149], induced-fit and selected-fit mechanisms both are likely to be feasible in particular systems and the flux through each pathway depends on the various rate constants for steps along each pathway. For example, the following scheme represents the selected-fit mechanism along the upper pathway and the induced-fit mechanism along the lower pathway for simple ligand binding:

in which P and P* are ground-state (binding-incompetent) and activated-state (binding-competent) conformations of the protein, PL is a non-specific encounter complex, and P * L is the final complex. The relative flux through each pathway, when the initial concentration of P * L is negligible, is given by [29]:

$$F=\left(\frac{k_2{k}_{\mathit{ns}}^{\mathit{on}}}{k_1{k}_{\mathit{on}}}\right)\left(\frac{k_{-1}+k_{\mathit{on}}[L]}{k_2+k_{\mathit{off}}^{\mathit{ns}}}\right)=\left(\frac{K}{K_{\mathit{ns}}}\right)\left(\frac{k_2}{k_{\mathit{on}}[L]}\right)$$ (14)

and K = (k₋₁ + k_on[L])/k₁ and $K_{\mathit{ns}}=(k_2+k_{\mathit{off}}^{\mathit{ns}})/(k_{\mathit{off}}^{\mathit{ns}}[L])$. Full characterization of interaction mechanisms thus requires characterization of all the kinetic rate constants in the proposed mechanistic scheme, a goal that no doubt will spur additional experimental developments as well as joint investigations coupling NMR methods with other biophysical computational and experimental approaches. Furthermore, more complex, multi-step interaction mechanisms, such as coupled folding and binding [150], share aspects of both selected- and induced-fit recognition. Intramolecular interactions that compete with intermolecular ones are recognized increasingly as important for autoinhibition and modulation of affinity [151,152]. Investigations by chemical exchange methods have provided insight into the role of transient fluctuations to “activated” states in these processes.

The investigation of enzyme catalysis naturally bifurcates into two areas: contributions to the chemical step of catalysis and contributions to other aspects of enzyme function, including, substrate and cofactor binding, product release, and regulation of activity. Aspects of the latter category involve molecular interactions that share features with investigations of molecular recognition; consequently many of the same experimental approaches and interpretive schemes are utilized. A major insight, emerging from studies of DHFR, is that conformational fluctuations can poise an enzyme to bind the next substrate or cofactor in a multi-step enzymatic cycle [140,153]. The controversy in this area revolves around whether conformational dynamics observable by NMR spectroscopy [154-156] contribute to the rate of the chemical step, i.e. exert an effect on k_cat [157]. An experimental difficulty is that mutations, or other perturbations, that alter conformational dynamics (transitions over the energetic barriers relevant to the conformational change) also may alter subtly the (distinct) activation barrier controlling the chemical step. As discussed theoretically, molecular motions on a free-energy surface do not alter the surface, but could contribute to prefactors (entropic barriers or effective diffusion constants) in models for chemical kinetics [158,159]. Relaxation dispersion methods now are being applied to an ever-increasing number of enzymes; the resulting NMR data, combined with computational methods, promises to resolve long-standing questions in enzyme biophysics.

An important question that arises in all investigations of chemical exchange broadening is how many chemical states are kinetically coupled. Most of the analyses discussed above have assumed two-state-like behavior. Three-state exchange has been identified in a number of systems [122,145,146]. Variation of the sample temperature (or other relevant experimental parameter) alters the relative contribution of the exchange processes to the observed relaxation dispersion data and facilitates dissection of multi-step pathways. The CEST experiment (the weak field R_1ρ limit) can distinguish in favorable cases between two- and three-state exchange [23,103]. Resolution of kinetic parameters for four-state exchange mechanisms is likely to require additional constraints on fitting parameters from other biophysical or biochemical experiments [160]. As a simplification, theoretical analyses of free-precession and R_1ρ relaxation rate constants illustrate that rapid averaging within a subset of sites reduces an N-site problem to exchange between a smaller number of “effective” states [23]. For example, investigations of folding implicitly assume that “random-coil” conformations are exchanging between allowed regions of conformational space on time scales faster than can be resolved by relaxation dispersion methods. As evident from long MD simulations [130,131], additional states not yet detected by NMR relaxation dispersion methods may exist and be functional. Increased precision and accuracy of experimental methods promises to allow detection of additional and even rarer conformational states of biological macromolecules.

10. Conclusion

Detailed investigations of sparsely populated chemical or conformational states of proteins and nucleic acids have established the power of relaxation dispersion and other chemical exchange techniques in NMR spectroscopy. Global analysis of data from multiple nuclear spin types, multiple magnetic fields, and multiple techniques improves the reliability of fitted kinetic model parameters; however, efficiency, that is minimizing total NMR acquisition time while maximizing sensitivity and accuracy, is critically important for applications of NMR spectroscopy to biological macromolecules, which may have limited solubilities and stabilities under NMR-suitable conditions. The pace of development of new techniques is likely to continue unabated and progress can be anticipated in experimental design to optimize information content and efficiency. Fundamental insights have been obtained into the functional roles of conformations other than ground state structures, and often populated at only a few percent in solution, in folding, recognition, and catalysis by biological macromolecules. In all likelihood, only a small number of possible biological mechanisms utilizing the properties of these alternative states have been described so far. Unanticipated insights no doubt will emerge from applications to ever more complex biological systems. If “past is prologue” (The Tempest), then the future is bright for chemical exchange investigations of function of biological macromolecules.