Characterizing micro-to-millisecond chemical exchange in nucleic acids using offresonance R_1ρ relaxation dispersion

Abstract

This review describes off-resonance R_1ρ relaxation dispersion NMR methods for characterizing microsecond-to-millisecond chemical exchange in uniformly ¹³C/¹⁵N labeled nucleic acids in solution. The review opens with a historical account of key developments that formed the basis for modern R_1ρ techniques used to study chemical exchange in biomolecules. A vector model is then used to describe the R_1ρ relaxation dispersion experiment, and how the exchange contribution to relaxation varies with the amplitude and frequency offset of an applied spin locking field, as well as the population, exchange rate, and differences in chemical shifts of two exchanging species. Mathematical treatment of chemical exchange based on the Bloch-McConnell equations is then presented and used to examine relaxation dispersion profiles for more complex exchange scenarios including three-state exchange. Pulse sequences that employ selective Hartmann-Hahn cross-polarization transfers to excite individual ¹³C or ¹⁵N spins are then described for measuring off-resonance R_1ρ(¹³C) and R_1ρ(¹⁵N) in uniformly ¹³C/¹⁵N labeled DNA and RNA samples prepared using commercially available ¹³C/¹⁵N labeled nucleotide triphosphates. Approaches for analyzing R_1ρ data measured at a single static magnetic field to extract a full set of exchange parameters are then presented that rely on numerical integration of the Bloch-McConnell equations or the use of algebraic expressions. Methods for determining structures of nucleic acid excited states are then reviewed that rely on mutations and chemical modifications to bias conformational equilibria, as well as structurebased approaches to calculate chemical shifts. Applications of the methodology to the study of DNA and RNA conformational dynamics are reviewed and the biological significance of the exchange processes is briefly discussed.

Graphical abstract

1. Introduction

The fundamental importance of conformational flexibility to the biological function of nucleic acids was evident at the birth of structural biology when it was immediately apparent that the two strands in DNA had to come apart to allow access to genetic information[1]. However only over the past decade, thanks in part to developments in solution-state NMR[2–6], has the true nature of nucleic acid flexibility and its roles in replication, gene expression, and regulation come to light[3, 4, 7–9]. Like proteins, the structures of nucleic acids undergo complex fluctuations over timescales spanning more than twelve orders of magnitude from picoseconds to seconds[10–12].

Here, we review methods based on the measurement of R_1ρ(¹³C) and R_1ρ(¹⁵N) relaxation dispersion (RD)[13, 14] for characterizing microsecond-to-millisecond chemical exchange in uniformly ¹³C/¹⁵N labeled nucleic acids in solution. Motions on this timescale are of particular interest because they coincide with the timescales for breaking base pairs (BPs, see glossary for abbreviations), the basic building block of nucleic acid structure. These motions can lead to formation of high-energy non-native conformations referred to as “excited states” (ESs)[15] that correspond to local minima in the free energy landscape. ESs have structural and therefore functional properties that can differ substantially from the native “ground state” (GS) conformation, enabling them to serve as molecular switches or targets for drug discovery. R_1ρ RD experiments have revealed a wide variety of such motions in DNA and RNA, which were previously unknown, and are now believed to play important roles in gene expression and regulation[9, 11, 16–18].

R_1ρ is one of several RD experiments that can be used to characterize microsecond-tomillisecond chemical exchange. R_1ρ experiments offer some advantages relative to Carr Purcell-Meiboom-Gill (CPMG) experiments[19, 20], particularly for nucleic acid applications given the scarcity of ideal ¹⁵N spin probes[9, 21]. These advantages include enhanced sensitivity to faster microsecond timescale motions[22], ability to robustly determine a complete set of exchange parameters using experiments performed at a single static magnetic field[23], and perhaps most importantly, applicability to uniformly ¹³C/¹⁵N labeled samples[13, 21], which can be readily prepared using standard biochemical protocols. However, relative to CPMG, offresonance R_1ρ experiments can be more difficult to implement, and perhaps for this reason, have not been as widely applied to the study of chemical exchange in biomolecules. The goal of this review is to provide readers with the conceptual foundations needed to use R_1ρ RD experiments for characterizing chemical exchange. While the focus is on nucleic acids, many topics covered should be of utility to those interested in studying proteins, and to those who are using other RD methods to characterize chemical exchange in nucleic acids. The reader is also referred to other excellent reviews on R_1ρ RD methods, with specific application to proteins[5, 13]. It should be noted that in recent years, there have also been significant developments in the application of Chemical Exchange Saturation Transfer (CEST) experiments for characterizing slow millisecond timescale chemical exchange in proteins[24–27], that have been extended to nucleic acids[28–32] and that afford advantages similar to those of R_1ρ.

The review is organized in the following manner: Section 2 provides a historical account of key developments that led to R_1ρ RD methods described here. Section 3 uses a vector model to describe dephasing of the magnetization during the R_1ρ experiment as a conceptual framework for understanding basic aspects of the experiment, with a focus on two-state exchange. This is followed by mathematical treatment of chemical exchange using the Bloch-McConnell equations in Section 4. Section 5 reviews pulse sequences for measuring offresonance R_1ρ(¹³C) and R_1ρ(¹⁵N) RD in uniformly labeled nucleic acids while Section 6 is devoted to data analysis methods to obtain the exchange parameters. Section 7 reviews methods for determining structures of nucleic acid ESs, and finally Section 8 reviews some example applications to DNA and RNA. Since we have carried out many off-resonance R_1ρ studies of chemical exchange in nucleic acids, most of the applications reviewed in Sections 7 and 8 come from our group. However, to provide a more balanced perspective of what is now a rapidly evolving field applying a range of NMR RD methods to the study of chemical exchange in nucleic acids, we also refer to earlier on-resonance R_1ρ studies which set the stage for the off resonance R_1ρ applications, and also make references to CPMG and CEST studies of nucleic acids when appropriate.

2. Brief historical account of relaxation dispersion in NMR with a focus on R_1ρ and nucleic acids

The theoretical and experimental underpinnings of NMR RD techniques were conceived soon after the observation of NMR in condensed matter systems independently by Bloch, Hansen, and Packard at Stanford University[33], and by Purcell, Torrey, and Pound at Harvard University[34]. Examining the historical progression from these initial studies helps one to highlight the motivation behind these developments, which frequently was unrelated to measuring dynamics, as well as key challenges that had to be overcome to make modern biological applications possible. We will find that, while the basics of RD experiments were conceived for applications to small organic molecules, the desire to characterize chemical exchange in biomolecules drove key developments in both theory and experiment that made it possible to robustly and quantitatively extract exchange parameters of interest. While the focus will be on R_1ρ, we shall also make references to CPMG and CEST RD experiments to help place these developments into a broader context. A timeline showing some of the milestones to the development of RD NMR is provided in Fig. 1. While we have tried our best to provide an accurate portrayal of this history, this is not intended to be an exhaustive compilation of the many important contributions to the development of RD NMR, and we apologize for any omissions.

Figure 1.

Time diagram showing key developments in RD NMR techniques for characterizing chemical exchange in biomolecules with emphasis placed on R_1ρ studies and their application to nucleic acids.

2.1. 1930–1960: Genesis of NMR relaxation dispersion methods

The R_1ρ experiment involves measuring exchange contributions (R_ex) to nuclear spin relaxation in the rotating frame. The phenomenon of spin relaxation and its experimental measurement was well established for electrons even before the observation of NMR by Bloch and Purcell. The Dutch physicist Cornelis Gorter first reported electron spin-lattice and spinspin relaxation in the mid 1930s[35, 36]. Gorter found that the electron magnetic moments of paramagnetic substances such as the sulfate salts of iron and chromium needed some time to return back to equilibrium after changing the external magnetic field[36]. Gorter, who is credited with coining the term “NMR”[37], is also well-known for his unsuccessful attempt to observe NMR, a failure which ironically could be attributed to his poor choice of lithium fluoride as a sample given its long spin-lattice relaxation time[38].

Theoretical estimates for nuclear spin-lattice relaxation times were available as early as the 1930s[39]. Some of the first quantitative experimental measurements were performed in 1948, when Pound et al. measured spin-lattice (T₁) and spin-spin (T₂) relaxation times in condensed matter[40]. Pound also derived expressions relating T₁ and T₂ to the rotational correlation time, thus establishing a link between relaxation and dynamics[40]. Later efforts were focused on improving the methods used to measure T₁ and T₂[41–43]. While attempting to develop methods for measuring T₁, Erwin Hahn “accidently” discovered the spin-echo which would prove invaluable to measure T₂ and form the basis for the very first RD experiments[44]. A few years later, Carr, Purcell, Meiboom, and Gill[19, 20] extended the spin-echo into a series of 180° pulses to suppress contributions of the static field inhomogeneity and of translational diffusion, resulting in the development of the CPMG experiment. As an unintended consequence, the train of 180° pulses also provided a means to suppress R_ex to a variable extent, thus forming the basis for the CPMG RD experiment (see below).

The phenomena of chemical exchange and many aspects of RD experiments can be described without recourse to a quantum description. The origins of the Bloch-McConnell (B-M) equations[45], which constitute the theoretical foundation for NMR RD experiments, can be traced back to the phenomenological Bloch equations, which were introduced in 1946, that use classical physics to describe the motion of macroscopic nuclear magnetization. Bloch realized that bulk magnetization resulting from the application of a longitudinal magnetic field (B₀) could be forced to undergo precession by application of a transverse magnetic field. He developed equations to describe this dynamic behavior that also treat the influence of T₁ and T₂ relaxation on the magnetization[46]. Bloch theorized that forced precession would induce an electric signal (“nuclear induction”) in an appropriately placed coil, and subsequently verified this prediction using a water sample[33]. Gutowsky, Slichter, Meiboom and others[47–51] later extended the Bloch equations and developed expressions that took into account the influence of chemical exchange on the behavior of magnetization in the absence of transverse magnetic fields. These expressions helped explain why line shapes changed when altering temperature or pH. In the late 1950s, McConnell recognized that chemical exchange provides mechanisms for directly transferring components of magnetization between exchanging states (i.e., M_ji→ M_ki; i = x, y or z; for states j and k), and explicitly introduced exchange terms into the Bloch equations to take this transfer into account, resulting in the development of the B-M equations[45]. Hahn independently came to a similar realization even earlier in 1952 while accounting for the influence of scalar couplings and chemical shifts during spin-echoes[52].

As is the case for CPMG, the R_1ρ experiment was not conceived for the purpose of measuring chemical exchange. In the mid 1950s, while developing a theory to account for the magnetic behavior of solids under the influence of strong transverse radio frequency (RF) fields, Redfield discovered a phenomenon that he referred to as “rotary saturation” or “saturation in the rotating frame”[53]. Redfield realized that one could spin-lock magnetization along an effective field direction through application of an RF field. Inspired by Rabi’s picture of the rotating frame[54], Redfield recognized that spin-locked magnetization could be tilted through application of a second RF field perpendicular to the effective field, in a manner analogous to how Bloch tilted magnetization away from the B₀ field[33]. The subsequent recovery of the magnetization along the effective field following removal of the second RF field was described by a relaxation time T_1ρ, which Redfield recognized to be sensitive to spin-spin interactions in the solid. Redfield measured T_1ρ for metals such as aluminium and copper as a function of the strength of the applied transverse RF field, thus performing the very first R_1ρ RD measurements. R_1ρ RD measurements in liquids were reported some four years later by Solomon in 1959[55] who tested theoretical expressions relating R_1ρ to the rotational correlation time that were developed earlier by Winter et al.[56]. Solomon performed these R_1ρ measurements in a manner similar to the pulsed methods used routinely nowadays. An adiabatic half passage was used to tilt the magnetization to the transverse plane, following which the spin-locking field was applied for a given time and switched off to measure the signal in the transverse plane. The variation of the signal amplitude with the time for which the spin-locking field was applied was used to obtain R_1ρ. An on-resonance RD profile was reported showing the dependence of R_1ρ(¹H) on the applied spin-lock amplitude for the protons of formamide (Fig. 2A).

Figure 2.

Historical progression of the development of R_1ρ RD techniques for applications to nucleic acids. A) The on-resonance R_1ρ(¹H) RD profile measured for formamide by Solomon et al.[55] was the first reported measurement of R_1ρ in liquids. Reproduced with permission from Bibliotheque nationale de France (BNF).B) The chair-chair inter-conversion in cyclohexane was one of the earliest conformational transitions characterized by RD methods[62, 63].C) R_1ρ(¹⁹F)measurements on N-trifluoroacetyl phenylalanine in exchange between free and enzyme-bound states by Sykes et al.[73] constituted the first application of R_1ρ RD in the context of biological macromolecules. D) Some of the earliest applications of R_1ρ in studies of conformational exchange in biomolecules targeted ¹H nuclei (red circles) in nucleic acid duplexes in the early 1990s[109–111]. E) On-resonance R_1ρ(¹⁵N) RD profile for backbone amide ¹⁵N of C38 in bovine pancreatic trypsin inhibitor was one of the first reported characterizations of conformational exchange in isotopically labeled proteins using R_1ρ[122]. Reprinted by permission from [122]. F) Secondary structure transition in U6RNA involving flipping out of a bulge nucleotide; this was one of the first validated models for ESs in nucleic acids[135]. G) First structure of an ES by Kay et al.[153] for a folding intermediate of the G48V mutant of the Fyn SH3 domain obtained using chemical shifts measured using CPMG RD and structurebased chemical shift calculations. Reprinted by permission from [153]. H) Expressions for R_1ρ outside the fast exchange limit by Palmer et al.[23] allowed extensions of the R_1ρ method by demonstrating the feasibility of extracting all exchange parameters at a single magnetic field. RD profiles were shown to be centered at the ES chemical shift. Simulated off-resonance profiles (for p_ES = 0.048, k_ex = 1500 s⁻¹, Δω = 2400 rad s⁻¹, R_1,GS = R_1,ES = 1.5 s⁻¹, R_2,GS = R_2,ES = 11 s⁻¹ and ω₁/2π = 1000 Hz) are shown using the asymmetric population expression in Palmer et al.[23] (dotted line) and fast exchange expression (dashed line). The exact numerical solution is shown as a solid black line. Reproduced with permission from [23]. I) Off-resonance R_1ρ(¹⁵N) RD profile for the amide ¹⁵N of Q11 in a G48M mutant of the Fyn-SH3 domain; this was the first experimental measurement of off-resonance R_1ρ using low spin-lock amplitudes and selective excitation experiments[163]. This study also represented the first instance wherein a complete set of exchange parameters was extracted using R_1ρ measurements. Reprinted with permission from [163]. J) Excited state Hoogsteen BPs in duplex DNA were the first ESs to be structurally characterized in nucleic acids using R_1ρ RD.Rates and populations were obtained using off-resonance R_1ρ experiments as described by Al-Hashimi et al.[115].

2.2. 1960–1970s: Applications to small molecules

In the early 1960s, Meiboom and colleagues realized that RD experiments could be used to characterize chemical exchange. Using on-resonance R_1ρ(¹H)[57] and CPMG(¹H)[58, 59] experiments, they estimated the rates of proton transfer in water. These experiments did not rely on the modulation of the chemical shift by the exchange process, but rather, on the modulation of the proton resonance frequency due to changes in scalar couplings and quadrupolar interactions as the protons exchanged between water molecules containing ¹⁶O and ¹⁷O. During the same period, approaches based on saturation transfer, which later were to form the basis for the CEST experiment, were used to investigate the kinetics of proton exchange between 2-hydroxy acetophenone and salicylaldehyde in 1963[24] and internal proton-transfer in acetylacetone in 1964[60].

RD measurements probing exchange contributions due to the modulation of chemical shifts soon followed with applications to small organic compounds. These landmark studies of conformational exchange in small molecules formed the basis for the modern RD applications to biomolecules. In an early application, Allerhand and Gutowsky used CPMG(¹H) experiments to characterize the hindered rotation of amide groups in dimethylcarbamyl chloride and N,N dimethyltrichloroacetamide[61] as well as chair-to-chair inter-conversion in cyclohexane[62] (Fig. 2B). Analogous R_1ρ RD studies were conducted in the early 1970s by Morgan, Strange and colleagues[63]. The exchange rate (250–2000 s⁻¹) and difference in chemical shift between the exchanging axial and equatorial protons (20–25 Hz) deduced for the chair-to-chair interconversion of cyclohexane (Fig. 2B), assuming two-state exchange between equally populated states, were in good agreement with the values measured previously using CPMG[62]. Morgan, Strange and colleagues also demonstrated the feasibility of using on-resonance R_1ρ(³¹P) and R_1ρ(¹H) to measure J-couplings and T₂ for halides of phosphorus[64] and hydrogen[65, 66], and also examined the impact of RF and B₀ field inhomogeneity on the accuracy of R_1ρ measurements[67]. Other early applications included the determination of the activation energies for the rotation of amide groups in urea derivatives, based on the temperature dependence of R_1ρ[68]. This period also saw applications of saturation transfer to characterize conformational exchange in small molecules[69, 70], kinetics of proton exchange[71], and reaction rates[72].

2.3. 1970s–2000s: Applications to biomolecules

Applications of RD methods to biomolecules began to emerge in the early 1970s with initial studies focusing on measuring RD data on small substrates to characterize enzymesubstrate binding kinetics. In one of the earliest applications of R_1ρ RD to the study of chemical exchange in biomolecules, Sykes[73] in 1969 measured on-resonance R_1ρ(¹⁹F) RD on the trifluoroacetyl-phenylalanine substrate and thereby characterized its binding kinetics to the enzyme chymotrypsin (Fig. 2C). Not only was this the first measurement of R_1ρ RD involving biological macromolecules, it was also the first measurement of exchange contributions to R_1ρ arising from chemical shift modulation. It was shown that the rate constants for binding as well as the chemical shifts of the bound trifluoroacetyl-phenylalanine could be deduced from R_1ρ RD measurements. In analogy to R_1ρ, the first biological application of CPMG RD involved characterizing the kinetics of N-fluoroacetyl-D-tryptophan binding to α-chymotrypsin using ¹⁹F as a probe[74]. During the same period, ¹H CEST experiments were also applied to study the kinetics of trimethoprim binding to dihydrofolate reductase[75]. Many modern applications have continued to use RD NMR with great success to probe the kinetics of intermolecular association[76–79]. This period also saw the application of R_1ρ[80–82] and CPMG[83–85] experiments to measure T₂ for the purpose of characterizing fast picosecond-to-nanosecond timescale motions in proteins and nucleic acids.

During the 1970s and 80s, ¹H RD was used to study chemical exchange due to conformational changes in the biomolecules themselves, with initial applications to protein side chains and oligonucleotides. In one of the earliest applications, Williams et al. used ¹H CEST to measure the temperature-dependent kinetics of rotation of a tyrosine residue about the C^β-C^γ bond in ferrocytochrome C, and obtain the activation energy for the process[86]. R_1ρ(¹H) RD experiments were also used to characterize backbone and side-chain conformational exchange in peptides[87–90]. This period also witnessed the introduction of ROESY experiments that relied on transverse cross relaxation in the presence of a spin-locking field for transferring magnetization between spins[91, 92]. The experiment was later extended to include spin-locking fields far off-resonance[93–95] and has been used to study conformational changes[96], reaction rates[97] and base opening[98] in the context of nucleic acids, in addition to investigating their hydration dynamics[99, 100]. The 1980s also saw the application of imino proton exchange methods by Gueron, Leroy, and co-workers, to study base pair opening in DNA duplexes[101, 102] and tetrads[103].

By the early 1990s, RD NMR had emerged as a technique for probing micro-to-millisecond timescale motions, bridging the gap between picosecond-to-nanosecond timescale motions accessible by T₁, T₂, and NOE measurements[104–106] and millisecond timescale motions detectable by T₂ measurements and line-shape analysis. Given the advances in hardware[107, 108], the stage was set for applications to measure chemical exchange in biomolecules, initially using ¹H probes and later ¹³C and ¹⁵N probes as well.

It may come as a surprise that early applications of RD to study conformational exchange in large biomolecules used R_1ρ rather than CPMG, and that the first target molecules were DNA duplexes rather than proteins. Ikuta[109], James[110] and Frenkiel[111] measured onresonance R_1ρ(¹H) RD for cytosine-H5, guanine-H8 and adenine-H2 and reported micro-tomillisecond timescale exchange in the DNA double helix (Fig. 2D). These early experiments did not provide the chemical shifts of the exchanging species, and as a consequence, it was not possible to deduce the identity of the ES. We now know that these very first observations of chemical exchange in DNA most likely represented transitions between Watson-Crick and Hoogsteen BPs, based on off-resonance R_1ρ measurements performed a quarter century later[17, 112–117] (see Fig 2J). Analogous Hoogsteen transitions were characterized in G-G mismatches during the mid 1990s by Peck and colleagues[118] using R_1ρ(¹H) RD measurements targeting base guanine-H8.

Following the introduction of methods for preparing ¹³C/¹⁵N labeled proteins in the late 1980s and 1990s (reviewed in Bolton et al.[119]) and the development of 2D methods for measuring heteronuclear spin-lattice R₁ (= 1/T₁) and spin-spin R₂ (= 1/T₂) relaxation rate constants[104, 120, 121], it became feasible to carry out detailed residue specific studies of conformational exchange in biomolecules. In an early application, Wuthrich et al.[122] used a 2D R_1ρ(¹⁵N) RD experiment developed by Wagner et al.[123] to probe the kinetics of backbone conformational changes in bovine pancreatic trypsin inhibitor (BPTI) (Fig. 2E). During the same period, Kay, Torchia, and Bax pioneered the 2D CPMG(¹⁵N) experiment[104] to suppress chemical exchange contributions and allow the accurate measurement of residue specific T₂ values in proteins for studies of picosecond-to-nanosecond dynamics. Early applications of CPMG to measure T₂ frequently revealed micro-to-millisecond exchange contributions, but these were often not investigated further to obtain details regarding the underlying exchange process[124, 125]. This changed in 1994, when Arseniev et al. performed the first 2D ¹⁵N CPMG RD measurement on a protein[126] by varying the delay between the hard 180° pulses. Millisecond timescale backbone motions in a fragment of bacteriorhodopsin were characterized and tentatively assigned to the bending of the transmembrane alpha helices. Many studies appeared thereafter during the 1990s applying R_1ρ and CPMG to study conformational exchange in a variety of proteins (reviewed by Loria et al.[22]). Other early applications included the use of CPMG(¹H) RD to investigate the dynamics of water molecules as they exchanged between the bulk and the hydration layer of proteins[127].

The development of methods for preparing ¹³C/¹⁵N labeled RNA[128] and DNA[129] during the late 1980s and early 1990s also opened the door for the application of heteronuclear 2D R_1ρ experiments to nucleic acids, initially with a focus on measuring on-resonance R_1ρ(¹³C). Using on-resonance adenine sugar R_1ρ(C1’) RD measurements, Lancelot et al. in 1997[130] and Chattopadhyaya et al. in 2000[131] reported microsecond timescale exchange in a DNA duplex. While the nature of the exchange process was unknown, it again likely represented Watson Crick to Hoogsteen transitions[17, 112–117]. During the same period, Pardi and Hoogstraten performed some of the first R_1ρ RD measurements on RNA, reporting microsecond timescale exchange in the lead-dependent ribozyme[132] and hammerhead ribozyme[133] based on aromatic base R_1ρ(¹³C) measurements. From these measurements, k_ex as well as the product p_GSp_ESΔω² could be determined, where k_ex is the exchange rate (units s⁻¹), Δω is the difference in chemical shift between the exchanging species (units rad s⁻¹), and p_GS and p_ES are the populations of the GS and ES, respectively. This period also saw application of CPMG(¹³C) RD to RNA by Marion et al., who estimated exchange contributions to transverse relaxation using random fractionally ¹³C labeled samples to suppress homonuclear C-C scalar couplings[134].

This was followed by many studies employing R_1ρ(¹³C) experiments to examine chemical exchange in RNA and DNA during the 2000s. For example, Butcher et al. characterized widespread chemical exchange in the U6-RNA stem loop containing a single nucleotide bulge[135]. The exchange was hypothesized to arise from the flipping out of a bulge nucleotide in a manner dependent on protonation of a flanking A-C mismatch (Fig. 2F). This was validated based on pH-dependent chemical shift measurements, providing a rare example in which structural features of nucleic acid ESs could be deduced and tested. Other studies exposed exchange on a microsecond timescale involving changes in sugar pucker in the GAAA tetraloop[136] as well as changes in exchange dynamics in RNA upon binding to proteins[137]. Subsequent studies by Varani et al. also revealed micro-to-millisecond timescale exchange related to motions of loop residues in RNA and G-C BPs in duplex DNA[138–140].

2.4. 2000s-present: Detailed characterization of chemical exchange in biomolecules

The 2000s witnessed key advances in methodology that improved the ability to characterize chemical exchange using NMR RD methods. CPMG(¹³C and ¹⁵N) experiments could not be used to study processes slower than ~2 ms due to difficulties associated with deconvoluting in-phase and anti-phase relaxation contributions during the relaxation period when using long interval periods between 180° pulses[141]. This drawback was addressed by the development of relaxation-compensated CPMG experiments by Palmer and Loria in 1999[142] that permitted the usage of 180° pulse trains with larger intervals by averaging the contributions of in-phase and anti-phase relaxation during the relaxation period. Another limitation was that CPMG data could only be used to determine the magnitude of Δω, but not its sign. This prevented determination of the ES chemical shifts that later proved critical for their structural elucidation. This problem was addressed in 2002 when Kay et al.[143] introduced HSQC/HMQC methods for determining the sign of Δω, permitting for the first time complete characterization of an exchange process using CPMG based methods, for a mutant of T4 lysozyme. Other developments in CPMG experiments (reviewed in Palmer et al.[5]) soon followed, including the use of constant time relaxation periods[144, 145], double/multiple quantum based methods[146–148], extension to different types of spins in proteins[149, 150], and TROSY[151] based methods to extend applications to larger proteins[142, 152]. The first structural characterization of an ES was reported in 2004, when in a landmark study, Kay et al. reported the structure of a protein folding intermediate of a mutant Fyn SH3 domain, using chemical shifts obtained from CPMG(¹⁵N) RD measurements[153] (Fig. 2G).

During the same time period, advances were also being made in the R_1ρ experiment with specific application to proteins. Early R_1ρ RD experiments[123, 154, 155] employed hard pulses to align the magnetization of all spins along their effective fields. This limited the experiment to relatively high spin-lock amplitudes (>1000 Hz), consequently leaving millisecond motions largely undetected. In addition, algebraic expressions relating R_1ρ RD to the exchange parameters (p_ES, k_ex, Δω, R₁ and R₂) were only available for the fast exchange limit[63, 87, 156]. While this allowed estimation of k_ex, resolving p_ES and Δω was not possible; rather only the product p_ESp_GSΔω² could be determined. Finally, methods for suppressing dipole-dipole CSA cross correlated relaxation (DD/CSA)[157] had yet to be developed, and discrepancies were being reported for relaxation rates measured using R_1ρ and CPMG[158, 159].

In the early 2000s, advances were made that addressed these limitations. In a milestone study, Palmer et al. in 2002[23] provided expressions for the exchange contribution to R_1ρ outside of the fast exchange limit (Fig. 2H). These expressions showed that a complete thermodynamic and kinetic characterization of an exchange process including the sign of Δω was feasible based on off-resonance R_1ρ RD data measured at a single static magnetic field. In 2005, Kay et al. introduced an R_1ρ RD experiment that employed a selective excitation scheme developed by Bodenhausen and co-workers[160–162] to investigate exchange contributions at individual ¹⁵N spins in a 1D manner[163]. In addition to permitting facile alignment of the magnetization along the effective field for a wide variety of offsets and spin-lock amplitudes, contributions from cross-correlated dipole-dipole/CSA relaxation[157] and evolution under J_NH scalar couplings could be optimally suppressed with the use of continuous-wave ¹H irradiation during the spin-lock period. This culminated in the first extraction of all exchange parameters using R_1ρ measurements at a single magnetic field by Kay et al. in 2005[163] for a mutant of the Fyn SH3 domain, thus demonstrating the utility of the expressions developed earlier by Palmer et al.[23] (Fig. 2I).

These advances initially made for proteins were later integrated in 2009[21] into a pulse sequence employing low spin-lock amplitudes optimized for measuring R_1ρ(¹³C) for base (C6/C8) and sugar (C1′) spins in nucleic acids. This study demonstrated the feasibility of performing off-resonance R_1ρ RD experiments on uniformly ¹³C/¹⁵N nucleic acids, which could be prepared using commercially available ¹³C/¹⁵N labeled nucleotide triphosphates. This made it possible for the first time to extract all exchange parameters in nucleic acids, including the sign of Δω. By permitting the use of low spin-lock amplitudes, the experiment also made it possible to access slow millisecond timescale exchange in nucleic acids, given the challenges of applying CPMG(¹³C) to uniformly ¹³C/¹⁵N labeled samples. This R_1ρ experiment permitted more detailed studies of chemical exchange in nucleic acids, and when combined with methods for modulating conformational exchange using mutagenesis and chemical modifications (Section 7), it became possible to visualize the structural identity of the nucleic acid ESs, first in DNA[17] (Fig. 2J) and soon after in RNA[16]. The experiment was later extended to target base imino[112, 164] and amino[165] ¹⁵N spins as well as sugar ¹³C spins[117, 166], culminating in 3D structure determination of an ES in the DNA double helix, specifically the Hoogsteen BPs mentioned earlier[116, 117].

Five years following the application of low spin-lock off-resonance R_1ρ to nucleic acids, and building on efforts that resurrected the CEST experiment to characterize chemical exchange in proteins[167–169], Zhang et al.[28, 170] demonstrated the utility of ¹³C CEST for characterizing slow millisecond timescale chemical exchange in uniformly ¹³C/¹⁵N labeled RNA samples, as well as allowing direct access to structural and dynamic properties of ESs such as residual dipolar couplings (RDCs)[171, 172]. Like R_1ρ, the CEST experiment also allows the determination of all exchange parameters and has been widely applied in studies of nucleic acid dynamics[29–32]. Concurrently, significant advances were made in preparing RNA samples with selective labeling schemes, which further extended the domain of applicability of CPMG to nucleic acids[18, 136, 173–177]. With the R_1ρ, CEST, and CPMG methodologies in place, one is now in a much better position to characterize motions in DNA and RNA occurring on the microto-millisecond timescale and to examine their roles in gene expression and regulation.

Perhaps the most important lesson that can be drawn from this brief historical account is that RD methods have their roots in magnetic resonance studies that were not intended to characterize chemical exchange, but rather, flow from human curiosity. This underscores the importance of fostering basic science research in magnetic resonance even if clear applications are not obvious in the short run. Indeed, it is very likely the case that there are other NMR phenomena such as Hahn’s echo discovered many decades ago, that can be adapted to modern applications to biological problems. Our account also illustrates how important it was following the initial discovery of the RD methods to tightly integrate developments in hardware, sample labeling, theory, and pulse sequence design to make possible the modern quantitative applications to biomolecules.

3. Description of R_1ρ relaxation dispersion using a vector model

RD experiments for studying equilibrium chemical exchange can be understood without recourse to a rigorous quantum mechanical description. To understand the R_1ρ RD experiment, one needs to understand how exchange between a GS and an ES leads to dephasing of the observable bulk magnetization, and to what extent dephasing varies with the amplitude (ω₁, units rad s⁻¹) and offset frequency (Ω, units rad s⁻¹) of an applied spin-locking field, as well as the exchange parameters of interest, namely the fractional population of the GS (p_GS) and ES (p_ES), the exchange rate (k_ex, units s⁻¹), and the difference between the chemical shifts of the GS and ES (Δω = ω_0,ES - ω_0,GS and $\Delta \overline{\omega }={\overline{\omega }}_{0,ES}-{\overline{\omega }}_{0,GS}$, where ω_0,ES and ${\overline{\omega }}_{0,ES}$, and ω_0,GS and ${\overline{\omega }}_{0,GS}$ are the Larmor frequencies, of the spin in the ES and GS, in units of rad s⁻¹ and ppm respectively). In this section, we use a vector model to describe how two-state exchange leads to dephasing of the magnetization during an R_1ρ RD experiment. The goal is to help the reader visualize and thereby better understand the R_1ρ experiment. A mathematical treatment applicable to n-state exchange will be presented in Section 4.1. This section will ignore the influence of scalar couplings, the treatment of which requires a quantum-mechanical description based on the Liouville-von Neumann equation[178–180] or average Hamiltonian theory[181].

3.1. Free precession chemical exchange

We first consider the simplest case and examine how chemical exchange leads to dephasing of the magnetization under free precession, i.e., in the absence of a spin-locking field. We will find that similar mechanisms lead to dephasing of the magnetization in the presence of a spin-locking field during an R_1ρ experiment. We consider conformational exchange between a major GS and a sparsely populated ES (Fig. 3A). We will ignore the effects of longitudinal and transverse relaxation, and consider a spin that has different chemical shifts in the GS and ES (ω_0,GS ≠ ω_0,ES).

Figure 3.

Thermodynamic and kinetic characteristics of a system undergoing two-state chemical exchange. A) Free energy diagram for two-state exchange. B) Time evolution of a molecule exchanging between a dominant GS and sparsely populated and short-lived ES. C) Distribution of τ_GS and τ_ES, the dwell times for the GS and ES.

To understand how chemical exchange impacts the NMR spectrum, we first need to understand the stochastic nature of the exchange process as this ultimately dictates how it will dephase bulk magnetization. Two-state exchange can be described by forward (k₁) and backward (k₋₁) rate constants (units s⁻¹):

$$\begin{array}{c}{k}_1\\ GS\rightleftharpoons ES\\ k_{-1}\end{array}$$ (1)

with the exchange rate k_ex = k₁ + k₋₁ (units s⁻¹), which can be understood to be the rate constant at which the equilibrium state is restored following a perturbation that pushes the system away from equilibrium. Assuming transition-state theory, the rate constants are related to the free energies of activation according to the Eyring equation:

$$k_1=\frac{\kappa k_{B}T}{h}\text{exp}\left(\frac{-\Delta G_1^{‡}}{RT}\right)$$ (2)

$$k_{-1}=\frac{\kappa k_{B}T}{h}\text{exp}\left(\frac{-\Delta G_{-1}^{‡}}{RT}\right)$$ (3)

where κ is the transmission coefficient describing the fraction of barrier crossing events that lead to the formation of the products from the reactants, R is the universal gas constant (units J K⁻¹ mol⁻¹), k_B is the Boltzmann constant (units J K⁻¹), T is the temperature (units K), h is Planck’s constant (units J s), and $\Delta G_1^{‡}$ and $\Delta G_{-1}^{‡}$ are free energies of activation (units J mol⁻¹) for the forward and backward reactions, respectively.

Importantly, molecules do not spend a fixed amount of time in either the GS or ES before interconverting (Fig. 3B). Rather, they spend variable amounts of time. A molecule may spend a short time visiting the ES on one occasion but spend a longer time on a second visit. Why is that the case? At the microscopic level, not all molecules are equal; rather they have free energies that follow a Boltzmann distribution $\text{exp}\left(\frac{-\Delta G}{RT}\right)$ where ΔG is the free energy of a molecule relative to a reference state. The probability that a molecule has the energy required to cross a barrier ΔG^‡ will be proportional to $\text{exp}\left(\frac{-\Delta G^{‡}}{RT}\right)$. As a consequence, the dwell time (τ) molecules spend in conformation i follows an exponential distribution $\text{exp}\left(\frac{-\tau }{<{\tau }_i>}\right)<{\tau }_i{>}^{-1}$ where <τ_i>is the average dwell time or lifetime of the i^th state (Fig. 3C). Therefore, while we will often simply refer to the lifetime of the GS or ES, this actually represents an average over what in reality is an exponential distribution of dwell times. The lifetimes of the GS <τ_GS>and ES <τ_ES> are given by reciprocals of k₁ and k₋₁, respectively:

$$<{\tau }_{GS}>=1/k_1$$ (4)

$$<{\tau }_{ES}>=1/k_{-1}$$ (5)

We first consider a hypothetical scenario where a molecule exists in two distinct conformations GS and ES that do not interconvert. The transverse magnetizations M_GS and M_ES are initially aligned along the x-axis after application of a 90°_y pulse and will experience an effective field (B_eff, bold refers to a vector) equal to B₀ aligned along the z-axis of the laboratory frame. Consequently, M_GS and M_ES will precess around B₀ with frequencies ω_0,GS and ω_0,ES, respectively, and the NMR spectrum consists of two lines centered at frequencies ω_0,GS and ω0,_ES.

It is instructive to examine the behavior of the magnetization in a rotating frame, as this will also prove important when describing the dephasing of the magnetization during an R_1ρ experiment (Section 3.2). In a rotating frame precessing at a frequency ω_rf (units rad s⁻¹), M_GS and M_ES precess with offset frequencies Ω_GS and Ω_ES (units rad s⁻¹) respectively, around the z axis:

$${\Omega }_{GS}={\omega }_{0,GS}-{\omega }_{rf}$$ (6)

$${\Omega }_{ES}={\omega }_{0,ES}-{\omega }_{rf}$$ (7)

The magnitude of the effective fields experienced by the GS (B_eff,GS) and the ES (B_eff,ES) is reduced from B₀ to ΔB_GS and ΔB_ES (units Tesla), respectively:

$$B_{eff,GS}=\Delta B_{GS}=\frac{{\Omega }_{GS}}{\gamma }$$ (8)

$$B_{eff,ES}=\Delta B_{ES}=\frac{{\Omega }_{ES}}{\gamma }$$ (9)

where γ is the gyromagnetic ratio (units rad s⁻¹ Tesla⁻¹).

We now examine how chemical exchange affects M_GS and M_ES, and the NMR spectrum. The exchange contribution will vary depending on the NMR chemical shift timescale, defined as the ratio between k_ex and Δω (k_ex/Δω). Under the slow exchange limit (k_ex << Δω), the GS and ES are long-lived and two discrete resonances are observed in the NMR spectrum centered at ω_0,GS and ω_0,ES (Fig. 4A). However, compared to the case in which the GS and ES do not interconvert, the resonances are broadened by the exchange process. This is because exchange leads to the dephasing of the magnetization.

Figure 4.

Chemical exchange under free precession. A) NMR spectra for a system undergoing two-state GS-ES chemical exchange simulated using the B-M equations. Columns correspond to different values of p_GS while rows correspond to different values of k_ex. Simulations were performed assuming $\Delta \overline{\omega }{\text{ (}}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz and R_2,GS = R_2,ES = 10 s⁻¹. B) Magnetization of GS (blue) and ES (red) spins in the rotating frame of the GS at the slow exchange limit. Black arrows represent the passage of time while colored arrows represent exchange events. Dots surrounded by circles denote vectors perpendicular to the plane of the figure. C) GS-ES chemical exchange leads to dephasing with time of the bulk magnetization corresponding to the GS (black arrow). The black dot near the x-axis corresponds to the receiver phase. D) Exponential decay due to chemical exchange of the bulk GS magnetization along the x axis (M_GSx) as a function of time. E) The magnetization of the GS (blue) and ES (red) spins in the rotating frame of the population-weighted average (AVG) resonance in the fast exchange limit.

To visualize dephasing of M_GS, initially aligned along the x-axis, we move into a rotating frame precessing at a frequency of ω_0,GS. In this frame, the magnetization of the GS spins is stationary (Ω_GS = 0) while that of the ES spins precesses with frequency Δω (Fig. 4B). The magnetization belonging to a GS spin will be stationary for the duration of the dwell time τ_GS. When the molecule transforms into the ES, the magnetization associated with the spin will start to precess with frequency Ω_ES = Δω for duration τ_ES, leading to a change in phase angle given by τ_ESΔω (Fig. 4B). The precession then stops again for some duration τ_GS when the molecule switches back to the GS, and so on. Depending on the exchange rate, many such transitions may occur during acquisition. As spins within the same molecule or across molecules will spend varying amounts of time in the GS or ES, the GS spins will be associated with different phase angles leading to dephasing of the bulk GS magnetization along the x-axis (M_GSx) (Fig. 4C). This causes M_GSx to decrease exponentially over time with the rate constant given by R_ex,GS (Fig. 4D). Why is the decay exponential? This again goes back to the exponential distribution of the dwell times τ_GS and τ_ES (Fig. 5), which causes the probability that the magnetization of the different spins are aligned to decrease exponentially with time. Chemical exchange leads to a line-broadening contribution given by R_ex,GS/⫪ to the GS resonance in addition to the linebroadening due to intrinsic transverse relaxation (R_2,GS/⫪). Put alternatively, R_ex,GS denotes the contribution to transverse relaxation of the GS resonance due to chemical exchange, and is dependent on the exchange parameters p_ES, k_ex and Δω.

Figure 5.

The mono-exponential decay of magnetization due to chemical exchange under free precession conditions is a consequence of the exponential dwell time distributions of the GS and ES. (A and B) Normalized bulk x-magnetization M_GSx (black dots) of the GS as a function of time for a system undergoing GS-ES exchange, simulated using the vector model. An exponential fit to the magnetization is shown in red. Panel A is simulated assuming exponential probability distributions for the GS and ES dwell times (< τ_GS > = 0.33 s and < τ_ES > = 0.14 s), while panel B is simulated assuming that the ES and GS dwell times follow a standard normal distribution. Expressions for the probability distributions of τ_GS and τ_ES are given in the inset. Simulations assumed the following exchange parameters: $\Delta \overline{\omega }{\text{ (}}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, and R_1,GS= R_1,ES = R_2,GS = R_2,ES = 0 s⁻¹ for panels A and B.

In an analogous manner, the dephasing of M_ES can be visualized by moving into a rotating frame precessing at ω_0,ES. In this frame, the ES magnetization is stationary (Ω_ES = 0) but will much more quickly transform into the GS and start to precess with frequency Ω_GS = − Δω, typically for a longer time relative to the GS when it transforms to the ES. As a result, the ES bulk magnetization experiences a greater degree of dephasing and the ES resonance experiences a greater degree of line-broadening as compared to the GS resonance, i.e., R_ex,ES > R_ex,GS.

Algebraic expressions for R_ex,GS and R_ex,ES at the slow exchange limit (k_ex << Δω) can be obtained by solving the B-M equations under free precession conditions (Section 4.1)[13, 14, 22]:

$$R_{ex,GS}=p_{ES}{k}_{ex}=k_1$$ (10)

$$R_{ex,ES}=p_{GS}{k}_{ex}=k_{-1}$$ (11)

$$R_{ex,GS}+R_{ex,ES}=k_{ex}$$ (12)

In the slow exchange limit, R_ex,ES > R_ex,GS and are equal to the backward and forward rate constants, respectively. In addition, increasing k_ex increases R_ex because this increases the frequency of transitions between the GS and ES. Furthermore, R_ex is independent of Δω (valid for k_ex/Δω ~< 0.2, Fig. 6) but does depend on p_ES. For k_ex/Δω ~< 0.2, changing Δω for a fixed k_ex does not significantly change the NMR chemical shift timescale and therefore the R_ex contribution. However, it should be noted that the NMR timescale can vary in the presence of a spin-locking field during the R_1ρ experiment[13, 182]. Consequently, although R_ex is independent of Δω in general under slow exchange (Fig. 6), this will not necessarily be the case during the R_1ρ experiment (see Section 6.2). The observed transverse relaxation rate (R_2,obs) is given by the sum of contributions due to intrinsic relaxation (R_2,GS and R_2,ES) and due to chemical exchange (R_ex,GS and R_ex,ES):

$$R_{2,obs,GS}=R_{2,GS}+R_{ex,GS}=R_{2,GS}+p_{ES}{k}_{ex}=R_{2,GS}+k_1$$ (13)

$$R_{2,obs,ES}=R_{2,ES}+R_{ex,ES}=R_{2,ES}+p_{GS}{k}_{ex}=R_{2,ES}+k_{-1}$$ (14)

Figure 6.

Line-width at half maximum (LW_1/2, units Hz) for the dominant GS resonance as a function of $\Delta \overline{\omega }$ for a system undergoing GS-ES exchange under free precession, simulated using the B-M equations (k_ex = 1500 in red and k_ex = 2750 in black). Simulations assumed the following exchange parameters: p_ES = 0.05, γ(¹H)B₀/2π = 700 MHz, R_1,GS= R_1,ES = R_2,GS = R_2,ES = 0.0 s⁻¹, while $\Delta \overline{\omega }{\text{ (}}^{13}\text{C)}$ was varied linearly between 0.1 and 40 ppm in 100 equally spaced increments. Solid lines denote (R₂+p_ESk_ex)/π while dotted lines denote a Δω value where (k_ex/Δω) ~ 0.2.

As k_ex increases, R_ex continues to increase in the slow exchange regime, which can be defined as k_ex/Δω < 0.2. Under slow exchange, the observed resonance frequencies for the GS and ES are to a good approximation given by ω_0,GS and ω_0,ES, respectively. Increasing k_ex further moves the system into the intermediate exchange regime, which can be defined as 0.2 < k_ex/Δω < 2 (Fig. 4A). The broadened GS and ES resonances now begin to shift toward one another until they coalesce when k_ex ~ Δω, at a frequency (ω_0,OBS) distinct from the populationweighted average of ω_0,GS and ω_0,ES (ω_0,AVG = p_GSω_0,GS + p_ESω_0,ES). At coalescence, the line broadening is maximum. This is because the GS-ES transition frequency is high, while <τ_GS> and <τ_ES> are sufficiently long to cause substantial dephasing of the magnetization. The exchange contribution can again be visualized by moving into a rotating frame precessing at the frequency ω_0,OBS. In this frame, both the GS and ES magnetizations undergo precession, resulting in dephasing of the bulk magnetization corresponding to the observed resonance. In contrast to the slow exchange limit, B-M simulations indicate that R_ex depends on Δω in the intermediate exchange regime[14, 22].

As k_ex increases further, the broadened resonance begins to shift and sharpen, and the system moves into the fast exchange regime (k_ex/Δω > 2) (Fig. 4A). Under these conditions, the observed resonance frequency ω_0,OBS is to a good approximation given by the populationweighted average ω_0,AVG. The exchange contribution can again be visualized by moving into a rotating frame precessing at frequency ω_0,AVG (Fig. 4E), in which exchange results in dephasing of the bulk magnetization corresponding to the AVG resonance. Expressions for the exchange contribution (R_ex,AVG) to the observed relaxation rate (R_2,obs,AVE) at the fast exchange limit (k_ex >> Δω) can be obtained by solving the B-M equations (Section 4.1) and are given by[13]:

$$R_{2,obs,AVG}=R_{2,AVG}+R_{ex,AVG}=\left(p_{GS}{R}_{2,GS}+p_{ES}{R}_{2,ES}\right)+p_{GS}{p}_{ES}\Delta {\omega }^2/k_{ex}$$ (15)

As in slow exchange, R_ex depends on p_ES under fast exchange. However, in contrast to slow exchange, decreasing k_ex increases R_ex under fast exchange, as spins spend longer periods of time in the GS and/or ES and conversely, increasing k_ex decreases R_ex as <τ_GS> and/or <τ_ES> decrease. Moreover, R_ex is now proportional to Δω². Note that the appearance of a single resonance in the NMR spectrum does not imply that the system is in fast exchange; the system could be in intermediate exchange or even under slow exchange since the larger R_ex contribution to a minor ES resonance could render it undetectable[22, 183].

3.2. R_1ρ in the absence of chemical exchange

While NMR line shapes carry rich information about chemical exchange[184], they cannot be used to reliably determine all exchange parameters of interest. The R_1ρ experiment, and RD experiments in general, provide an alternative means to extract this information. These experiments modulate the effective field experienced by a given spin, thereby enhancing the sensitivity to the exchange parameters.

The R_1ρ experiment entails the application of a continuous RF spin-locking B₁ field along the x or y-axis, transverse to the static B₀ field. The purpose of the spin-locking field is to control B_eff,GS and B_eff,ES, and therefore, the extent to which the bulk magnetization is dephased due to chemical exchange. In this section, we will ignore chemical exchange and first examine how the magnetization is spin-locked, how changing the spin-lock parameters alters the effective field, and lay out the theoretical basis for describing the R_1ρ experiment using a vector model.

Let us first examine how B₁ applied along the x-axis affects B_eff experienced by a given spin. B₁ is a linearly oscillating field with amplitude 2B₁ (units Tesla) and frequency ω_rf (units rad s⁻¹; positive/negative values refer to clockwise/anti-clockwise rotations, respectively), both of which are specified by the user (Fig. 7A):

$${\mathit{B}}_1=2B_1\text{cos}\left({\omega }_{\text{rf}}\text{t}\right)\widehat{\mathit{x}}$$ (16)

The linearly oscillating B₁ field can be decomposed into two rotating fields B₁⁺ and B₁⁻ each with an amplitude B₁ rotating in opposite directions in the transverse plane with frequencies ω_rf (clockwise) and −ω_rf (counterclockwise) (Fig. 7A):

$${\mathit{B}}_1={\mathit{B}}_1^{+}+{\mathit{B}}_1^{-}$$ (17)

$${\mathit{B}}_1^{+}=B_1\text{cos}\left({\omega }_{\text{rf}}\text{t}\right)\widehat{\mathit{x}}-B_1\text{sin}\left({\omega }_{\text{rf}}\text{t}\right)\widehat{\mathit{y}}$$ (18)

$${\mathit{B}}_1^{-}=B_1\text{cos}\left({\omega }_{\text{rf}}\text{t}\right)\widehat{\mathit{x}}+B_1\text{sin}\left({\omega }_{\text{rf}}\text{t}\right)\widehat{\mathit{y}}$$ (19)

The treatment of B₁ is simplified by working in a frame rotating with frequency +ω_rf (Fig. 7B). In this frame, the vector B₁⁺ is aligned along the x-axis and has a length B₁, while B₁⁻ rotates at 2ω_rf and can be ignored. The magnetization precesses with an offset frequency Ω around the zaxis and the spin experiences a reduced field ΔB along the z-axis as described in Section 3.1. B_eff experienced by an ensemble of spins with an offset Ω is given by the vector sum:

$${\mathit{B}}_{\mathit{e}\mathit{f}\mathit{f}}={\mathit{B}}_1+\Delta \mathit{B}$$ (20)

It is common practice to express B_eff, B₁, and ΔB in terms of the precession frequency ω (units rad s⁻¹) of the magnetization around the effective field:

$$\omega =\gamma \mathit{B}$$ (21)

$$\omega =\gamma B$$ (22)

Using ω is useful because dephasing of the magnetization due to chemical exchange during the R_1ρ experiment is governed by the frequencies of precession. For the remainder of the review, we will replace the vector B and its magnitude B with ω and ω, respectively:

$${\omega }_1=\gamma {\mathit{B}}_1$$ (23)

$${\omega }_1=\gamma B_1$$ (24)

$$\Omega ={\omega }_0-{\omega }_{rf}=\gamma \mathit{\Delta }\mathit{B}$$ (25)

$$\Omega ={\omega }_0-{\omega }_{\text{rf}}=\gamma \Delta B$$ (26)

$${\omega }_{eff}=\Omega +{\omega }_1$$ (27)

$${\omega }_{\text{eff}}={\left({\Omega }^2+{\omega }_1^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$ (28)

Figure 7.

Influence of the spin-locking field during an R_1ρ experiment in the absence of chemical exchange. A) In the lab frame (left), the spin-locking field B₁ (black arrow) can be decomposed into two fields $B_1^{+}$ and $B_1^{-}$ rotating in opposite directions (green and cyan arrows). The short black arrow represents the passage of time. A black dot surrounded by a circle represents a vector perpendicular to the plane of the figure. Black dots near the x-axes correspond to the receiver phase. ω_rf is the angular frequency of the spin-lock. Variation of the spin-locking field amplitude as a function of time is shown on the right. B) Evolution of $B_1^{+}$ and $B_1^{-}$ in the rotating frame of $B_1^{+}$. The static magnetic field is reduced to ΔB. C) Effective fields under on- and off-resonance conditions in the rotating frame of the spin-lock.

The applied ω₁ field will change both the orientation and the magnitude of the effective field ω_eff experienced by a spin in a manner dependent on both Ω and ω₁ as shown in Fig. 7C. For example, when ω₁ is applied on resonance (ω_rf = ω₀), Ω = 0, ω_eff = ω₁ and ω_eff is aligned along the x-axis (Fig. 7C). If on the other hand ω₁ is applied off-resonance (ω_rf ≠ ω₀), Ω ≠ 0, ω_eff = ω₁ + Ω, and ω_eff is no longer in the transverse plane but is tilted away from the z-axis by an angle θ (units rad) that depends on both ω₁ and Ω (Fig. 7C):

$$\theta =ta{n}^{-1}\left(\frac{{\omega }_1}{\Omega }\right)$$ (29)

Figure 8 shows additional examples of how changing ω₁ or Ω (through changes in ω_rf) changes both the amplitude and orientation of ω_eff. In principle, all possible orientations and amplitudes of ω_eff can be realized experimentally by a suitable choice of ω₁ and Ω. As R_1ρ experiments are most sensitive to an exchange process when k_ex ~ ω_eff, the experimentally accessible effective fields define the detection limit of the experiment, as elaborated in Section 4.2.1.

Figure 8.

Modulation of the effective field by variation of the spin-lock amplitude (ω₁) and offset frequency (Ω) in an R_1ρ experiment in the absence of chemical exchange. Influence of changing ω₁ (A) and Ω (B) on the effective field ω_eff in the rotating frame.

During the R_1ρ experiment, ω₁ is applied immediately after the magnetization of interest has been aligned along z′, the direction of ω_eff, using an appropriate preparatory pulse as described in Section 5.2 (Fig. 9). In the absence of ω₁, the magnetization precesses around B₀ with frequency ω₀. However, in the presence of ω₁, precession is eliminated, and the magnetization is ‘spin-locked’ along ω_eff, although its magnitude is subject to decay due to relaxation (Fig. 9). It is important to note that the magnetization remains spin-locked along ω_eff as long as R₂ and R₁ are both smaller than ω₁ and Ω. If those conditions are not met, the magnetization can veer off and deviate from ω_eff (Fig. 10).

Figure 9.

Schematic representation of the R_1ρ experiment in the absence of chemical exchange. The equilibrium magnetization tilted along z′, the direction of ω_eff,OBS, is immediately spin-locked by the application of an RF field (green arrow), after which it decays exponentially due to relaxation with a rate constant R_1ρ. A cross within a circle denotes a vector perpendicular to the plane of the figure. All vector diagrams are in the rotating frame where the spin-locking field appears stationary.

Figure 10.

Time course of the evolution of normalized magnetization during an R_1ρ experiment in the absence of chemical exchange, as a function of the spin-lock amplitude ω₁, the offset Ω and the relaxation rates, simulated using the B-M equations. A) R₁ = 2.5 s⁻¹ and R₂ = 22.5 s⁻¹; B) R₁ = 25 s⁻¹ and R₂ = 225 s⁻¹. The time course for the position of the tip of the net magnetization vector M (with its base at the origin) is denoted as a solid black line, with green and red dots denoting the positions of the vector tip at the start and end (T_max = 0.015s) of the relaxation period, respectively. The effective field direction is denoted using a dashed red line.

The R_1ρ experiment measures the rate of relaxation of the magnetization that is spinlocked along ω_eff (Fig. 9). In the absence of chemical exchange, the relaxation will be governed by R₁ and R₂[13]. To obtain expressions relating R_1ρ to R₁ and R₂, we use the Bloch equations[46], which model the behavior of the magnetization in the rotating frame of the spin locking field:

$$\frac{d}{dt}\left(\begin{array}{c}{M}_x\\ M_y\\ M_z\end{array}\right)=\left(\begin{array}{ccc}-R_2& -\Omega & 0\\ \Omega & -R_2& -{\omega }_1\\ 0& {\omega }_1& -R_1\end{array}\right)\left(\begin{array}{c}{M}_x\\ M_y\\ M_z\end{array}\right)+R_1\left(\begin{array}{c}0\\ 0\\ M_{z,eq}\end{array}\right)$$ (30)

where M_x, M_y and M_z are the x, y and z components of the bulk magnetization respectively and M_z,eq refers to the equilibrium z magnetization. The diagonal and M_z,eq terms describe the influence of relaxation while off-diagonal terms describe interchange between M_x and M_y, and M_y and M_z, due to precession around Ω/z-axis and ω₁/x-axis, respectively, in the absence of chemical exchange and spin-locking field. The Bloch equations assume that the interactions between spins are independent of, and significantly weaker than, the interactions between the spins and the external field, i.e., τ_c << T₂, where τ_c is the rotational correlation time[53]. This condition is readily satisfied for biomacromolecules in solution where τ_c ~ 5–30 ns and T₂ ~ 5 – 200 ms.

Moving into a doubly rotating frame in which the z-axis is aligned along z′ (Fig. 9) simplifies the description of the magnetization and allows derivation of a key expression for R_1ρ[123, 185–188]. This is achieved through rotation of the coordinate frame (in the rotating frame of the spin-locking field) around the y-axis by the angle θ. This second rotation has the effect of nullifying rotations about ω_eff at a frequency of ω_eff. The Bloch equations in the doubly rotating frame are given by:

$$\frac{d}{dt}\left(\begin{array}{c}{M}_x^{\prime }\\ M_y^{\prime }\\ M_z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}-R_{2}co{s}^2(\theta )-R_{1}si{n}^2(\theta )& -{\left({\Omega }^2+{\text{ω}}_1^2\right)}^{1/2}& \left(R_1-R_2\right)sin(\theta )\cos (\theta )\\ {\left({\Omega }^2+{\text{ω}}_1^2\right)}^{1/2}& -R_2& 0\\ \left(R_1-R_2\right)sin(\theta )\cos (\theta )& 0& -R_{2}co{s}^2(\theta )-R_{1}si{n}^2(\theta )\end{array}\right)\left(\begin{array}{c}{M}_x^{\prime }\\ M_y^{\prime }\\ M_z^{\prime }\end{array}\right)+R_1\left(\begin{array}{c}-sin(\theta )M_{z,eq}\\ 0\\ cos(\theta )M_{z,eq}\end{array}\right)$$ (31)

where M_x′, M_y′ and M_z′ are the components of the magnetization in the doubly rotated frame. M_x′ and M_y′ are perpendicular to ω_eff, while M_z′ is oriented along ω_eff. Although algebraic expressions for M_z′ are complicated in the general case, one can make the simplifying assumption[187] ω₁ ≫ (R₁ − R₂)sin(θ)cos(θ) which is valid under most experimental conditions. In general, rapid interconversion between the M_x′ and M_y′ components of the magnetization due to the ${\left({\Omega }^2+{\text{ω}}_1^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ term averages the cross-relaxation term (R₁ − R₂)sin(θ)cos(θ) between M_x′ and M_z′ to zero[187]. The evolution of M_x′ and M_y′, and M_z′ are uncoupled and M_z′ decays mono-exponentially with a rate constant R_1ρ given by a weighted sum of R₁ and R₂:

$$M_Z^{\prime }=M_Z^{\prime }(0)\text{exp}\left(-R_{1\rho }t\right)$$ (32)

$$R_{1\rho }=R_{1}co{s}^2\theta +R_{2}si{n}^2\theta$$ (33)

Thus, rotating the coordinate frame introduces contributions to decay along ω_eff due to longitudinal and transverse relaxation to variable extents[13], depending on the orientation (tilt angle) of ω_eff. The relative weights of R₁ (cos² θ) and R₂ (sin² θ) can be understood as corresponding to their fractional contributions to the length of a unit vector oriented along ω_eff. The above expression shows that in the absence of chemical exchange, R_1ρ does not provide any information beyond that which can be obtained from measuring R₁ and R₂.

The above expressions also lay the basis for the vector model that will be used to describe the R_1ρ experiment in subsequent sections. Setting R₁ and R₂ to 0, the B-M equations in the doubly rotating frame reduce to:

$$\frac{d{M}_x^{\prime }}{dt}=-{\left({\Omega }^2+{\text{ω}}_1^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{M}_y^{\prime }$$ (34)

$$\frac{d{M}_y^{\prime }}{dt}={\left({\Omega }^2+{\text{ω}}_1^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{M}_x^{\prime }$$ (35)

$$\frac{d{M}_z^{\prime }}{dt}=0$$ (36)

From above equations, in the absence of relaxation M_x′ and M_y′ vary sinusoidally with time, while M_z′ stays constant. Therefore, if the magnetization is initially aligned along ω_eff, M_x′(0) = 0 and M_y′(0) = 0, M_z′ does not change with time while M_x′ and M_y′ remain zero; the magnetization is always aligned along ω_eff and is spin-locked. In contrast, the magnetization tilted away from ω_eff (M_x′(0) ≠ 0 or M_y′(0) ≠ 0) will precess around ω_eff on the surface of a cone with a constant amplitude of M_z′(0) and with precession frequency ${\text{ω}}_{\text{eff}}={\left({\Omega }^2+{\text{ω}}_1^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$. Precession of the GS and ES magnetizations around their respective effective fields forms the basis for the vector model, which will be used to describe the R_ex contribution to R_1ρ.

3.3. Dephasing of magnetization during R_1ρ

We now examine how chemical exchange leads to dephasing of the magnetization in the rotating frame during an R_1ρ experiment. For Δω ≠ 0, the GS and ES will be associated with effective fields ω_eff,GS and ω_eff,ES respectively, in the rotating frame of ω₁ (Fig. 11A):

$${\omega }_{\mathit{e}\mathit{f}\mathit{f},\mathit{G}\mathit{S}}={\omega }_1+{\Omega }_{\mathit{G}\mathit{S}}$$ (37)

$${\text{ω}}_{eff,GS}={\left({\text{ω}}_1^2+{\Omega }_{GS}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$ (38)

$${\omega }_{\mathit{e}\mathit{f}\mathit{f},\mathit{E}\mathit{S}}={\omega }_1+{\Omega }_{\mathit{E}\mathit{S}}$$ (39)

$${\text{ω}}_{eff,ES}={\left({\text{ω}}_1^2+{\Omega }_{ES}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$ (40)

where the offsets Ω_GS and Ω_ES are defined in Section 3.1. We see that in the presence of 2-state chemical exchange between GS and ES, one must consider two non-parallel effective fields ω_eff,GS and ω_eff,ES that have different amplitudes, and around which the GS and ES magnetization vectors precess (Fig 11A). For n-state chemical exchange with n unique chemical shifts, there would be one effective field for each of the n states (Fig. 11A). As described below, the dephasing of the magnetization during R_1ρ is analogous to that in free precession, the major difference being that the axes of rotation for the GS and ES magnetization are different.

Figure 11.

R_1ρ in the presence of chemical exchange. A) ω_eff,GS (blue) and ω_eff,ES (red) in the rotating frame where the spin-locking field ω₁ appears stationary, for two-state and n-state exchange. B) Alignment of the net magnetization at the start of an R_1ρ experiment for a system undergoing GS-ES exchange under fast and slow exchange regimes. C) Rotation of the magnetization of GS/ES spins around their respective effective fields leads to dephasing of the bulk magnetization along ω_eff,OBS during the R_1ρ experiment. Shown is a representative example under conditions of fast exchange.

For intermediate and fast exchange, it is useful to define an average effective field ω_eff,AVG corresponding to ω_0,AVG:

$${\omega }_{\mathit{e}\mathit{f}\mathit{f},\mathit{A}\mathit{V}\mathit{G}}={\omega }_1+{\Omega }_{\mathit{A}\mathit{V}\mathit{G}}$$ (41)

$${\Omega }_{\mathit{A}\mathit{V}\mathit{G}}={\Omega }_{AVG}\widehat{\mathit{z}}$$ (42)

$${\Omega }_{AVG}={\text{ω}}_{0,AVG}-{\text{ω}}_{rf}=\left(p_{GS}{\text{ω}}_{0,GS}+p_{ES}{\text{ω}}_{0,ES}\right)-{\text{ω}}_{rf}$$ (43)

In the selective 1D R_1ρ experiments reviewed here in Section 5.2, both M_GS and M_ES are initially aligned along an effective field ω_eff,OBS before the application of the spin-locking field, which depends on ω₁ and Ω = ω_0,OBS – ω_rf, where ω_OBS corresponds to the GS resonance frequency (ω_0,OBS = ω_0,GS) in the case of slow exchange or to the average resonance frequency (ω_0,OBS = ω_0,AVG) in the case of fast exchange (Fig. 11B). In general, simulations indicate that for intermediate to fast exchange (0.1 < k_ex/Δω < 5) and low p_ES (< 10%), alignment along ω_eff,AVG is a valid approximation and does not appreciably influence the RD profiles. However, deviations can arise particularly for high p_ES (> 10%)(Section 6.1).

Now let us examine how the magnetization is dephased due to exchange in the rotating frame in the presence of the spin-locking field. M_GS and M_ES are initially aligned along ω_eff,OBS and precess around ω_eff,GS and ω_eff,ES with frequencies ω_eff,GS and ω_eff,ES, respectively (Fig. 11C). When a GS spin switches to the ES, its associated magnetization starts to precess around ω_eff,ES with frequency ω_eff,ES. When the molecule switches back to the GS, the magnetization associated with the spin starts again to precess around ω_eff,GS with frequency ω_eff,GS and so on. Once again, because molecules spend varying amounts of time in the GS precessing around ω_eff,GS or in the ES precessing around ω_eff,ES, exchange causes dephasing of the bulk magnetization (Fig. 11C) in a manner analogous to free precession in the absence of a spinlocking field (Fig. 4C). Here, ω_eff,GS and ω_eff,ES acts as ‘dephasers’ that scramble or randomize the orientation of M_GS and M_ES. Unlike free precession, in which the magnetization remains transverse, the exchange in this case results in magnetization that is arranged in a 3D “bouquet”, which fans out exponentially with time (Fig. 11C). As in free precession, the exchange causes the projection of the bulk magnetization onto the vector ω_eff,OBS to decay exponentially with time resulting in an R_ex contribution to R_1ρ. The R_1ρ experiment entails measuring the exponential decay of the magnetization along ω_eff,OBS as a function of time, from which R_ex can be deduced. The R_2ρ experiment measures the decay of M_x′/M_y′ orthogonal to ω_eff,OBS[13, 182].

The exponential decay of the magnetization associated with a single 3D bouquet provides a single R_1ρ value that is not sufficient to extract all exchange parameters of interest. Thus, additional data is needed. This is accomplished by changing the scramblers ω_eff,GS and ω_eff,ES by changing the spin-lock parameters ω₁ and Ω (via changes in ω_rf). This changes the extent of dephasing of the magnetization, the resulting magnetization bouquets and the associated R_ex contributions, in a manner that depends on the exchange parameters, providing a means for their determination. Experiments in which the carrier is on-resonance and ω₁ is varied are referred to as “on-resonance” R_1ρ experiments, while those that vary both ω₁ and Ω are referred to as “off-resonance” R_1ρ experiments. The validity of the vector description of the R_1ρ experiment can be verified through comparison with numerical solutions to the B-M equations (Fig. 12).

Figure 12.

Comparison of off-resonance R_1ρ RD profiles as obtained from B-M simulations and the vector model, in the absence of relaxation, for a wide variety of exchange parameters. Rows correspond to variations in k_ex while columns correspond to variations in p_ES. Spin-lock amplitudes are color coded while solid vertical green lines correspond to an offset Ω = –Δω; $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz. 10,000 spins were used for the vector model simulations. The initial alignment of the magnetization for the B-M and vector simulations was performed as described in Section 6.1. In the vector model simulations, the magnetization of the GS and ES spins initially aligned along ω_eff,OBS was allowed to precess about ω_eff,GS and ω_eff,ES with angular velocities ω_eff,GS and ω_eff,ES, respectively. The dwell times of the spins in the GS and ES were sampled from exponential probability distributions as described in Section 3.1, following which they were allowed to exchange with each other, while retaining the same orientation of the magnetization in 3D space prior to exchange. The sum of the magnetization of all the spins is projected along ω_eff,OBS as a function of time to obtain R_1ρ, which is used to obtain R_ex, as described in Section 3.2 – 3.4.

It is interesting to ask whether the net magnetization resulting from the vector sum over the bouquet remains perfectly aligned along ω_eff,OBS. The answer depends on the timescale for chemical exchange as defined under free precession conditions. For fast exchange, the net magnetization as well as M_GS and M_ES are aligned along ω_eff,AVG (Fig. 13A). In contrast, under slow exchange conditions, M_ES can deviate from its initial alignment along ω_eff,GS towards ω_eff,ES while M_GS remains along ω_eff,GS, causing the net magnetization to deviate towards ω_eff,AVG (Fig. 13B). This deviation of the net magnetization is proportional to p_ES and Δω. Even when the net magnetization is not along ω_eff,GS under slow exchange, the decay of the component along ω_eff,GS can still be mono-exponential (Fig. 13B). Note that the above statements are only valid under conditions when the relaxation rates R₁ and R₂ are smaller than ω₁ and Ω, as discussed in Section 3.2.

Figure 13.

Time course of the evolution of the normalized GS, ES and net magnetization during an off-resonance R_1ρ experiment, simulated using the B-M equations. Time course for the positions of the tips of the GS, ES and net magnetization vectors (each with their base at the origin), denoted as solid blue, red and black lines respectively, with green dots in each case denoting the positions of the vector tips at the start of the relaxation period, and violet dots denoting their positions at the end of the relaxation period, respectively. Directions for the GS,ES and AVG effective fields are denoted using blue, red and green dashed lines, respectively. Also shown as insets are the variations of the normalized projections of the net magnetization along ω_eff,OBS (M) as a function of time (black dots), along an exponential fit of the same (red line). Simulations were performed using k_ex = 20000 s⁻¹ for panel A, k_ex = 20 s⁻¹ for panels B and C. The other parameters used for all simulations were p_ES = 0.1, $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, R_1,GS = R_1,ES = 2.5 s⁻¹, R_2,GS = R_2,ES = 22.5 s⁻¹ and T_max = 0.25 s. The initial alignment of the magnetization for the B-M and vector simulations was achieved as described in Section 6.1.

3.4. On-resonance R_1ρ

While off-resonance R_1ρ experiments provide the best means for characterizing exchange as they afford the maximal amount of data, on-resonance experiments are often performed initially, typically to screen spins for the presence of chemical exchange. We review this simpler on-resonance R_1ρ experiment and then later move to the more general offresonance R_1ρ experiments.

In the on-resonance experiment, Ω = 0, ω_eff = ω₁, and R_ex simply adds to R₂ in a manner analogous to free precession (Section 3.1)[23, 63, 87, 156, 189]:

$$R_{1\rho }=R_2+R_{ex}\left({\omega }_1\right)$$ (44)

where R₂ is assumed to be the population-weighted average of R_2,GS and R_2,ES, i.e., R₂ = p_GSR_2,GS + p_ESR_2,ES, which is valid for |R_2,GS − R_2,ES| << k_ex[13, 190]. It is also assumed that R_ex and R₂ are not correlated, which is a reasonable approximation given that they report on dynamics occurring on different timescales (i.e., τ_c << 1/k_ex)[13, 45]. It is also assumed that τ_c << T₂ as described in Section 3.2, k_ex < ω_GS,0 and k_ex < ω_ES,0[191], and that the decay of the magnetization is mono-exponential; deviations from mono-exponential behavior can arise under slow exchange conditions with high p_ES[23](Section 4.2).

Setting R₁ = R₂ = 0, R_1ρ = R_ex, a vector description can be used to model how R_ex varies with ω₁. Increasing ω₁ decreases the angle φ between ω_eff,GS and ω_eff,ES (Fig. 14A). Intuitively, this should reduce dephasing of the magnetization and R_ex since rotations around axes that have similar orientations should be less effective in scrambling magnetization compared to the case when the axes have very different orientations. Indeed, vector simulations show that for hypothetically fixed ω_eff,GS and ω_eff,ES, R_ex is maximum for φ = π/2 while it is zero for φ = 0 (Fig. 15A). It is important to note that in an actual on-resonance R_1ρ experiment, changing ω₁ also changes ω_eff,GS and ω_eff,ES (Fig. 14A). Simulations show that when fixing φ, R_ex increases with ω_eff,ES until it approaches k_ex after which it plateaus, while showing little variation with ω_eff,GS (Fig. 15B). Simulations also suggest that overall, the decrease in φ outweighs the increase in ω_eff,ES to cause a net decrease in R_ex with increasing ω₁ under on-resonance conditions. This can be appreciated from the diminished spread of the magnetization bouquet depicted using a Sanson Flamsteed projection[192] in Fig. 14A and Supplementary Movie 1. At very high ω₁, the measured value of R_1ρ approaches R₂ and this illustrates how by changing ω₁ it is feasible to resolve the R₂ contribution to R_1ρ (Fig. 14B).

Figure 14.

Dependence of on-resonance R_1ρ on exchange and spin-lock parameters.A) Δω and ω₁ modulate R_ex via changes in φ, the angle between ω_eff,GS and ω_eff,ES. Also shown are positions of the magnetization vectors corresponding to GS (blue circles) and ES (red circles) obtained using vector model simulations in the form of Sanson-Flamsteed projections[192]. ω_eff,GS and ω_eff,ES are indicated by black and red stars respectively. Simulations were performed with the following parameters - p_ES = 0.01, k_ex = 20,000 s⁻¹, $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, ω₁/2π = 1000 Hz when on-resonance with the AVG state. Positions of the spins for a larger $\Delta \overline{\omega }{(}^{13}\text{C)}=10\text{\hspace{0.17em}ppm}$ (right) and ω₁/2π = 3000 Hz (left) are also shown. Simulations employed 10,000 spins with a relaxation delay of 0.012s.B) Variation of R₂ + R_ex with ω₁ under on-resonance conditions as a function of exchange parameters, as obtained using B-M simulations. Rows and columns correspond to the indicated values of k_ex and p_ES, respectively. Dashed line denotes the value of R₂. The other exchange parameters used were $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, R_1,GS= R_1,ES = 2.5 s⁻¹ and R_2,GS = R_2,ES = 22.5 s⁻¹. The initial alignment of the magnetization during the B-M simulations was performed as described in Section 6.1.

Figure 15.

Vector model simulations to illustrate the dependence of R_ex on φ, ω_eff,GS and ω_eff,ES during an on-resonance R_1ρ experiment for a system undergoing two-state GS-ES exchange. A) Variation of R_ex with φ for a fixed ω_eff,GS and ω_eff,ES, under fast and slow exchange conditions. Simulations were performed using ω_eff,GS = 1000*2π rad s⁻¹ and ω_eff,ES = 1131*2π rad s⁻¹ when on-resonance with the GS under slow exchange conditions. ω_eff,GS = 1000*2π rad s⁻¹ and ω_eff,ES = 1128*2π rad s⁻¹ under fast exchange conditions while maintaining the AVG state onresonance by suitably adjusting the offset of the GS and ES. Similar trends are observed for alternative values of the precession frequencies (data not shown). B) Heat maps of R_ex as a function of ω_eff,GS and ω_eff,ES for a fixed φ under fast and slow exchange conditions. Horizontal dotted lines (blue) correspond to the condition ω_eff,ES = k_ex. Orientations of the effective fields were kept fixed at those corresponding to application of ω₁/2π = 1000 Hz on resonance. Similar trends are also observed for R_1ρ, and for both R_ex and R_1ρ under off-resonance conditions, as long as the magnetization decay is mono-exponential (data not shown). For panels A and B, simulations were performed using 10,000 spins with p_ES = 0.01, $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, R_1,GS= R_1,ES = 0.0 s⁻¹, R_2,GS = R_2,ES = 0.0 s⁻¹. The relaxation delays used were 3 s and 0.6 s for k_ex = 200 and 20000 s⁻¹, respectively. For both panels, the initial alignment of magnetization and its projection to calculate R_ex, was performed as described in Section 6.1.

The above framework also helps us rationalize why R_ex increases with Δω; as Δω increases so does φ, thus increasing R_ex (Fig. 14A, Supplementary Movie 2). Algebraic expressions (Section 6.2) show that R_ex is proportional to Δω² in the presence of a spin-locking field. How R_ex varies with ω₁ will also depend on other exchange parameters. Fig. 14B shows on-resonance RD profiles for various values of p_ES and k_ex. Corresponding profiles for a wider range of k_ex and p_ES are shown in Fig. 16. From these RD profiles, we see that R_ex increases with p_ES. For p_ES << p_GS, R_ex will be maximum at a given ω₁ when $k_{\text{ex}}={\left(\Delta {\omega }^2+{\omega }_1^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ and will decrease for other k_ex values, as can be appreciated from algebraic expressions in Section 6.2.

Figure 16.

Variation of R₂ + R_ex as a function of ω₁ under on-resonance conditions as a function of exchange parameters, obtained using B-M simulations. Rows and columns correspond to the indicated values of k_ex and p_ES respectively. Panels with dotted lines correspond to exchange scenarios where the decay of the magnetization is not mono-exponential. Other parameters were: $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, R_1,GS = R_1,ES = 2.5 s ⁻¹ and R_2,GS = R_2,ES = 22.5 s⁻¹. The initial alignment of the magnetization was performed as described in Section 6.1.

3.5. Off-resonance R_1ρ

Although the variation of R_ex with ω₁ depends on all exchange parameters, it is not possible to determine each of them individually from on-resonance R_1ρ RD profiles measured at a single magnetic field[13]. Rather, inspection of algebraic equations (see Section 6.2) reveals that in general only k_ex and p_GSp_ESΔω² can be obtained under fast exchange (Eqn. 66) while p_GSp_ESΔω²k_ex and (Δω² + k_ex²) can be determined under slow exchange when p_ES << p_GS (Eqn. 65). Changing ω_rf, the carrier frequency of the spin-locking field, provides a means of obtaining additional data that can address this gap. As will be elaborated upon in Section 4.2, the variation of R_ex with ω_rf during an off-resonance R_1ρ experiment features a symmetric peak centered at a position that corresponds to ω_0,ES[23]. This makes it possible to determine Δω including its sign, thereby enabling one to resolve the p_GSp_ESΔω² term obtained in the onresonance experiment, while generally also improving the ability to characterize all other exchange parameters.

Under fast exchange conditions, changing Ω_AVG in an off-resonance R_1ρ experiment makes it possible to sample additional configurations of ω_eff,GS and ω_eff,ES relative to the onresonance R_1ρ experiment. In principle, ω_eff,GS and ω_eff,ES are each specified by two parameters (length and angle with respect to the z-axis) that can vary independently to assume all allowed configurations. However, experimentally, since both the GS and ES always sense the same ω₁, not all parameters can be varied independently. Rather, once ω₁ and Ω_AVG are specified, all four remaining parameters (two lengths and two angles) are fixed, due to the relations:

$${\Omega }_{GS}={\Omega }_{AVG}-p_{ES}\Delta \omega$$ (45)

$${\Omega }_{ES}={\Omega }_{AVG}-p_{GS}\Delta \omega$$ (46)

Consequently, specifying ω₁ and Ω_AVG determines ω_eff,GS and ω_eff,ES, and the angle φ between them:

$$\phi ={tan}^{-1}\left(\frac{{\omega }_1}{{\Omega }_{GS}}\right)-{tan}^{-1}\left(\frac{{\omega }_1}{{\Omega }_{GS}+\Delta \omega }\right)$$ (47)

In an analogous manner, specifying Ω_GS in addition to ω₁ during an off-resonance R_1ρ experiment under slow exchange conditions also fixes ω_eff,GS and ω_eff,ES.

A general expression for R_1ρ applicable for off-resonance conditions is given by[23]:

$$R_{1\rho }=R_1{cos}^2\theta +R_2{sin}^2\theta +R_{ex}\left({\omega }_1,\Omega \right){sin}^2\theta$$ (48)

where θ is the angle between ω_eff,OBS and the z-axis, which corresponds to ω_eff,AVG for intermediate and fast exchange, and ω_eff,GS for slow exchange (Fig. 11B). Similar assumptions apply as for on-resonance R_1ρ (Section 3.4). The expression is analogous to that for R_1ρ in the absence of exchange, except for the addition of the R_ex term. Note that because R_ex adds to transverse relaxation, it is also weighted by sin²θ. Thus, R_ex will depend on the extent of dephasing of the total magnetization along the effective field as measured by R_1ρ, which in turn depends on both the angle φ and ω_eff,ES as described in Section 3.4, and also on sin²θ, which measures the fractional contribution to R_1ρ from dephasing in the transverse plane.

As mentioned earlier, a key feature of the off-resonance experiment is that R_ex is maximum when on-resonance with the ES, providing a means of determining Δω and ω_0,ES. We can use the vector model to understand this important point. We focus on the case in which ω_eff,ES > k_ex, typically valid for k_ex < ~300 s⁻¹ usually corresponding to the slow exchange regime. Under these conditions, the variation of R_1ρ with Ω_GS is dominated by changes in φ, while the dependency on ω_eff,ES is weak, as R_1ρ has already reached saturation relative to ω_eff,ES (Fig. 15B).

Let us first consider the case in which ω₁ is fixed and the carrier is gradually moved far off-resonance, i.e., |Ω_GS| >> 0. This causes ω_eff,GS and ω_eff,ES to gradually tilt away from the transverse plane, decreasing φ while θ→π for Ω_GS << 0 and θ→0 for Ω_GS >> 0. As the angle φ decreases, the decrease in R_1ρ due to reduced scrambling of the magnetization outweighs the decrease in sin²θ, leading to a net decrease in R_ex (Fig. 17). There comes a point at which ω_eff,GS and ω_eff,ES are aligned nearly parallel to ω₀ (θ→0) and R_1ρ is dominated by R₁ rather than by R₂ + R_ex. This provides a means for resolving the R₁ contribution to R_1ρ. Furthermore, this also increases the uncertainty in R_ex as one moves off-resonance. In practice, Ω is chosen such that |Ω| < 3.5ω₁ and θ < 15°[21].

Figure 17.

Variation of R_1ρ, φ and R_ex with Ω_GS during an off-resonance R_1ρ experiment under slow-exchange conditions with ω_eff,ES < k_ex. Representative diagrams showing ω_eff,GS (blue vector) and ω_eff,ES (red vector) at selected offsets are also shown. The exchange parameters used were: p_ES = 0.01, k_ex = 200 s⁻¹, $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, R_1,GS= R_1,ES = 0.0 s⁻¹, R_2,GS = R_2,ES = 0.0 s⁻¹, ω₁/2π = 600 Hz along with a relaxation delay of 3.0 s. The initial alignment of the magnetization during the B-M simulation was performed as described in Section 6.1.

As we move closer to being on-resonance with the GS or ES, both ω_eff,GS and ω_eff,ES decrease in amplitude and are increasingly tilted toward the transverse plane, tending towards θ→π/2 and consequently increasing sin²θ. This increase in sin²θ is outweighed by the increase in R_1ρ due to the concurrent increase in φ, resulting in a net increase in R_ex (which, in the absence of relaxation, is equal to R_1ρ/sin²θ) (Fig. 17). Again, intuitively, the maximum dephasing and therefore R_1ρ should be achieved when the angle φ is maximum. This is indeed the case; R_1ρ is maximum when placing the carrier right in the middle between ω_0,GS and ω_0,ES (ω_rf = ½ (ω_0,GS + ω_0,ES), Ω_GS = ½(ω_0,GS − ω_0,ES)) (Fig. 17). However, this is not the offset that maximizes R_ex. Rather, as noted by Palmer[23], the maximum R_ex is always achieved when placing the offset on-resonance with ω_0,ES as this maintains a large φ angle while also minimizing θ and therefore sin²θ. This results in a smaller sin²θ term and a larger R_1ρ value, consequently maximizing R_ex (Fig. 17). Put in different words, the trade-off between maximizing φ and consequently R_1ρ, in combination with minimizing θ to reduce sin²θ is best satisfied when onresonance with the ES. The underlying basis for this rule lies in the fact that the orientations of the effective fields are coupled to the angle φ in R_1ρ experiments. Thus, under slow exchange, off-resonance RD profiles are centered at −Δω, while under fast exchange conditions they are centered at -p_GSΔω ~ -Δω when p_ES << p_GS, as described by Palmer et al.[23, 179, 182]. Under intermediate exchange, they are centered at −(Δω-ψ), where ψ is the observed resonance frequency in the presence of exchange, assuming ω_0,GS = 0 and ω_0,ES = Δω.

It should be apparent that off-resonance R_1ρ not only provides a means of characterizing the exchange parameters associated with R_ex and micro-to-millisecond exchange, but, as an added bonus, it simultaneously allows one to determine R₁ and R₂, which can be used to characterize fast picosecond-to-nanosecond dynamics.

4. Illustrative examples of R_1ρ relaxation dispersion profiles

In this section, we review the B-M equations, which are first-order differential equations describing the evolution of the magnetization in the presence of chemical exchange, relaxation, and B₀ and B₁ magnetic fields. We then use the B-M equations to simulate off-resonance R_1ρ RD profiles for a wide range of exchange scenarios, including three-state exchange with various topologies, and for cases in which R_2,GS and R_2,ES are unequal. Theoretical simulations will be accompanied in some cases with experimental examples. The goal is to develop a deeper understanding regarding how off-resonance R_1ρ RD profiles vary as a function of the exchange parameters and to extend description of the R_1ρ experiment to three-state exchange.

4.1. Description of chemical exchange using Bloch-McConnell equations

We first review the B-M equations for a two-state system in the absence of chemical exchange, and later move to systems with two-state chemical exchange. In the absence of chemical exchange, M_GS and M_ES can be treated independently, evolving according to their respective relaxation matrices. For a two-state system without chemical exchange, the Bloch equations take the following form in the rotating frame of ω₁:

$$\frac{d}{dt}\left(\begin{array}{c}{M}_{GSx}\\ M_{ESx}\\ M_{GSy}\\ M_{ESy}\\ M_{GSz}\\ M_{ESz}\end{array}\right)=\left(\begin{array}{cccccc}-R_{2,GS}& 0& -{\Omega }_{GS}& 0& 0& 0\\ 0& -R_{2,ES}& 0& -{\Omega }_{ES}& 0& 0\\ {\Omega }_{GS}& 0& -R_{2,GS}& 0& -{\omega }_1& 0\\ 0& {\Omega }_{ES}& 0& -R_{2,ES}& 0& -{\omega }_1\\ 0& 0& {\omega }_1& 0& -R_{1,GS}& 0\\ 0& 0& 0& {\omega }_1& 0& -R_{1,ES}\end{array}\right)\left(\begin{array}{c}{M}_{GSx}\\ M_{ESx}\\ M_{GSy}\\ M_{ESy}\\ M_{GSz}\\ M_{ESz}\end{array}\right)+\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ R_{1,GS}{M}_{z,eq,GS}\\ R_{1,ES}{M}_{z,eq,ES}\end{array}\right)$$ (49)

As in the case for the Bloch equations for a single state (Section 3.2), the diagonal elements of the relaxation matrix and M_z,eq terms describe the influence of R₁ and R₂ on M_GS and M_ES. The off-diagonal elements in the relaxation matrix describe the interchange between the x and y components (M_GSx→M_GSy and M_ESx→M_ESy) and between the y and z components (M_GSy→M_GSz and M_ESy→M_ESz) components due to precession about the z and x-axes under Ω and ω₁, respectively. The equations are valid for τ_c << T₂ (see Section 3.2).

The Bloch equations can be modified to accommodate chemical exchange, by adding exchange terms describing the magnetization transfer between the states at rates equal to the rate constants for chemical exchange[45]. They can be written in matrix form as follows:

$$\frac{d}{dt}\overrightarrow{M(t)}=K\overrightarrow{M(t)}+\overrightarrow{\mathit{B}}$$ (50)

where

$$\overrightarrow{\mathit{M}(t)}=\left(\begin{array}{l}{M}_{GSx}\hfill \\ M_{ESx}\hfill \\ M_{GSy}\hfill \\ M_{ESy}\hfill \\ M_{GSz}\hfill \\ M_{ESz}\hfill \end{array}\right)$$ (51)

$$K=\left(\begin{array}{cccccc}-R_{2,GS}& 0& -{\Omega }_{GS}& 0& 0& 0\\ 0& -R_{2,ES}& 0& -{\Omega }_{ES}& 0& 0\\ {\Omega }_{GS}& 0& -R_{2,GS}& 0& -{\omega }_1& 0\\ 0& {\Omega }_{ES}& 0& -R_{2,ES}& 0& -{\omega }_1\\ 0& 0& {\omega }_1& 0& -R_{1,GS}& 0\\ 0& 0& 0& {\omega }_1& 0& -R_{1,ES}\end{array}\right)+\left(\begin{array}{cccccc}-k_1& k_{-1}& 0& 0& 0& 0\\ k_1& -k_{-1}& 0& 0& 0& 0\\ 0& 0& -k_1& k_{-1}& 0& 0\\ 0& 0& k_1& -k_{-1}& 0& 0\\ 0& 0& 0& 0& -k_1& k_{-1}\\ 0& 0& 0& 0& k_1& -k_{-1}\end{array}\right)$$ (52)

and

$$\overrightarrow{\mathit{B}}=\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ R_{1,GS}{M}_{z,eq,GS}\\ R_{1,ES}{M}_{Z,eq,ES}\end{array}\right)$$ (53)

The diagonal terms in the matrix of rate constants correspond to the decrease of M_GS and M_ES due to exchange events in which spins leave the GS (M_GS→M_ES) or the ES (M_ES→M_GS). In contrast, the off-diagonal terms correspond to the increase of M_GS and M_ES due to exchange events in which spins arrive at the GS (M_ES→M_GS) or the ES (M_GS→M_ES). As molecules in the GS transform into the ES with a rate of k₁[GS], where [GS] is the concentration of the GS species, the associated components of the magnetization of the GS (M_GSi, i = x, y or z) decrease with rates of k₁M_GSi, as M_GSi is proportional to [GS]. Analogous considerations are also applicable for the case of n-state exchange. The assumptions under which the above equations are valid are given in Section 3.4. It should be noted that the B-M equations can be used to treat the behavior of the magnetization under free precession conditions by setting ω₁ = 0.

The solution to a first-order linear differential equation such as $\frac{d}{dt}\overrightarrow{M(t)}=K\overrightarrow{M(t)}+\overrightarrow{B}$ has the form:

$$\overrightarrow{\mathit{M}(\mathit{t})}=\sum _{i=1}^6\text{exp}\left({\lambda }_{i}t\right){\overrightarrow{\mathit{l}}}_{\mathit{i}}+\overrightarrow{\mathit{a}}$$ (54)

where λ_i and ${\overrightarrow{l}}_i$ are the i^th eigenvalue and eigenvector of matrix K respectively, and $\overrightarrow{\mathit{a}}$ is the stationary solution[23, 193]. For realistic values of exchange parameters, considering the presence of an inhomogeneous spin-locking field, the decay of the magnetization in most cases is mono-exponential and dominated by the least negative eigenvalue, denoted as R_1ρ[23, 163]. Indeed, even if the spin-locking field is homogeneous, with the exception of exchange scenarios with low k_ex and high p_ES, the decay of the magnetization is mono-exponential, as described in Section 4.2. A non-monoexponential decay of the magnetization is typically a consequence of oscillatory motions of a component of the magnetization that is not parallel to the effective field. The effects of such oscillatory motions are diminished by inhomogeneous spin-locking fields because of differences in the frequencies of oscillation of the magnetization in different parts of the sample. An analogous solution is also valid for the case of n-state exchange with arbitrary topological complexity as described by Palmer[193, 194]. Thus, the variation of R_1ρ with ω₁ and Ω can be used to characterize chemical exchange, including n > 2 state exchange with arbitrary topologies, so long as the decay of magnetization is mono-exponential.

4.2. Off-resonance R_1ρ relaxation dispersion profiles for two and three-state exchange

Understanding how off-resonance R_1ρ RD profiles vary with exchange parameters is essential when analyzing R_1ρ data. Here, we explore these variations by numerically integrating the B-M equations. The initial alignment of magnetization assumed in these simulations was performed as described in Section 6.1.

4.2.1. Two-state exchange

Fig. 18 shows representative off-resonance RD profiles for two-state GS⇌ES exchange as a function of p_ES and k_ex. These profiles show the dependence of R_2,eff = R₂ + R_ex on Ω and ω₁. Over this range of p_ES and k_ex, the RD profiles feature a symmetric peak centered near −Δω. The height of the peak relative to the baseline (which is at R₂) is equal to R_ex.

Figure 18.

Changes in R₂ + R_ex as a function of k_ex and p_ES for two-state exchange during an off-resonance R_1ρ experiment, simulated using the B-M equations. Horizontal comparisons show changes in k_ex while vertical comparisons show changes in p_ES. Spin-lock amplitudes are denoted using different colors. Solid gray vertical lines correspond to an offset of –Δω, while the yellow star denotes exchange parameters typical of Watson-Crick to Hoogsteen exchange in B-DNA[115]. Other exchange parameters used are: $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, R_1,GS = R_1,ES = 2.5 s ⁻¹ and R_2,GS = R_2,ES =22.5 s⁻¹. The initial alignment of the magnetization was performed as described in Section 6.1.

Increasing p_ES increases the peak height (R_ex) in addition to its width, and higher spin-lock amplitudes are needed to suppress larger R_ex contributions (Fig. 18). For a given ω₁ and Ω, algebraic expressions in Section 6.2 (Eqn. 65) show that R_ex increases until k_ex approaches ${\left({\Omega }_{ES}^2+{\omega }_1^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ (for p_ES << p_GS) where it reaches a maximum (Fig. 19), and is maximally dependent on ω₁. Representative experimental RD profiles for two-state exchange are shown in Fig. 20 for nucleic acid systems with different exchange regimes.

Figure 19.

Variation of R₂ + R_ex with k_ex during an R_1ρ experiment when on-resonance with the ES (p_ES = 0.01, $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, R_1,GS = R_1,ES = 2.5 s ⁻¹ and R_2,GS = R_2,ES =22.5 s⁻¹). Spin-lock amplitudes are color-coded. Vertical lines denote the indicated values of k_ex. The initial alignment of the magnetization during the B-M simulations was performed as described in Section 6.1.

Figure 20.

Representative experimental off-resonance R_1ρ RD profiles for two-state exchange under fast and intermediate exchange. The profiles were measured for U31-C1′ and C24-C1′ in HIV-1 TAR at 25 °C and 35 °C, respectively[166]. Spin-lock amplitudes are color-coded. Error bars represent experimental uncertainties determined by a Monte-Carlo scheme (Section 6.3). Solid lines denote global fits to the RD data using the B-M equations[166]. The initial alignment of the magnetization during the B-M fitting was performed as described in Section 6.1. Gray vertical lines correspond to the offset value when on resonance with the ES. Reproduced with permission from [166].

Simulated RD profiles for a wider range of p_ES and k_ex are shown in Fig. 21. The RD profiles are flat for extremes of very slow (≤ 0.2 s⁻¹) or fast (≥ 100,000 s⁻¹) k_ex and for low p_ES (≤ 0.01 %) (Fig. 21). In line with this, exchange processes in nucleic acids that have been experimentally characterized using off-resonance R_1ρ RD have p_ES between 0.02 % and 14 %, and k_ex between ~400 s⁻¹ to 50,000 s⁻¹[16, 164, 195]. In general, for fast exchange with k_ex on the order of 20,000 s⁻¹, the RD profiles are broad with small R_ex, making it more difficult to define the peak center and therefore Δω, and higher spin-lock amplitudes are needed to suppress the smaller R_ex contributions. However, the profiles continue to remain centered at −Δω. This is in contrast to the fast exchange equations, which predict that the profiles should be centered at 0[196] (Eqn. 66, Section 6.2).

Figure 21.

Variation of R₂ + R_ex during off-resonance R_1ρ experiments, for a wide range of k_ex and p_ES values for two-state exchange, simulated using the B-M equations. Horizontal comparisons show changes in k_ex while vertical comparisons show changes in p_ES. Different spin-lock amplitudes are denoted by colors. Dotted lines correspond to cases where the decay of the magnetization is not mono-exponential. Solid gray vertical lines correspond to an offset of –Δω, while the yellow star denotes a typical Watson-Crick to Hoogsteen exchange scenario in DNA[115]. Simulations assumed the following parameters: $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, R_1,GS = R_1,ES = 2.5 s ⁻¹ and R_2,GS = R_2,ES = = 22.5 s⁻¹. The initial alignment of the magnetization was performed as described in Section 6.1.

Additional simulations were performed to assess more rigorously the detection limits of the off-resonance R_1ρ RD experiment for two-state exchange. While the results vary depending on the exchange parameters and measurement uncertainty, some general remarks can be made. For typical values of $\Delta \overline{\omega }$, the slowest exchange process that can be reliably detected has k_ex ~ 10 s⁻¹ although even k_ex ~ 1 s⁻¹ can be reliably characterized for high p_ES (Fig. 22). The upper limit for k_ex is ~50,000 s⁻¹ although this can vary depending on the exchange parameters and measurement uncertainty. The lower limit for p_ES is ~0.02 % for a ¹⁵N spin with very large $\Delta \overline{\omega }\left(30\text{\hspace{0.17em}ppm}\right)$ associated with deprotonation[197], but is more typically ~0.1 % for a ¹³C spin with $\Delta \overline{\omega }=3\text{\hspace{0.17em}ppm}$ (Fig. 22). As expected, doubling Δω allows ~4-fold smaller p_ES values to be characterized given that R_ex is proportional to p_ESΔω² (Section 6.2).

Figure 22.

Exploring the limits of the sensitivity for off-resonance R_1ρ experiments using B-M simulations for R_1ρ(¹⁵N)(left) and R_1ρ(¹³C) (right), at γ(¹H)B₀/2π = 700 MHz. For each $\Delta \overline{\omega }$ value, detectability of R_1ρ was assessed for 10,000 (p_ES, k_ex) combinations. 100 p_ES values ranging between 0.005 % to 20 % were chosen such that they were equally spaced on a logarithmic scale. For each of these p_ES values, 100 k_ex values ranging from 1 to 400000 s⁻¹ were chosen such that they were equally spaced on a logarithmic scale. For each (p_ES, k_ex) combination, offresonance R_1ρ RD Profiles were simulated using the B-M equations, assuming spin-locking amplitudes ω₁/2π = 100, 250, 500, 1000 and 2000 Hz, each with 24 offset points Ω evenly spaced such that −3.5*ω₁ < Ω < +3.5*ω₁, without experimental error. A rate R_ex of at least 5 Hz above the baseline R₂ was assumed to be the threshold of detectability. The above curves were obtained by drawing a line through the lowest detectable p_ES for every k_ex value; thus all combinations of k_ex and p_ES to the right of the curves are detectable. Similar limits are observed when profiles are simulated with 5% error and re-fit to the Bloch-McConnell equations, and considered to be detectable when the fitted exchange parameters deviate from the simulated exchange parameters by less than 2-fold. For each panel, the $\Delta \overline{\omega }$ values are color-coded. For all simulations, R_1,GS = R_1,ES = 2.5 s⁻¹ and R_2,GS = R_2,ES = 22.5 s⁻¹. The initial alignment of the magnetization during the B-M simulations was performed as described in Section 6.1.

Interestingly, under slow exchange (k_ex/Δω < 0.1) with high p_ES (> 1 %), the simulated RD profiles are asymmetric, particularly for low ω₁ (Fig. 21). These asymmetries arise due to gradual changes in the alignment of M_ES from its initial state along ω_eff,GS toward ω_eff,ES during the relaxation period. For high p_ES, this veering off of M_ES also leads to oscillations in the magnetization decay[23, 163](p_ES ≥ 1%, k_ex ≤ 20 s⁻¹) (Fig. 13C). These contributions can be suppressed by avoiding an equilibration delay following selective excitation of the GS so as to minimize M_ES during the relaxation period (Section 5.2)[28].

The RD profiles in Fig. 21 also show that 10-fold changes in p_ES or k_ex can result in very large changes in the RD profiles. For example, large and significant RD (p_ES = 10 %, k_ex = 20000 s⁻¹) can become undetectable (p_ES = 10 %, k_ex = 100000 s⁻¹). One cannot therefore simply relate a change in magnitude in the RD profile to a corresponding change in exchange parameters. A related point is that there is a danger of over-interpreting the absence of RD as evidence for the absence of an ES. Thus, it is advisable to vary experimental conditions in order to bring exchange processes within the detection limits of the experiment.

4.2.2. Three-state exchange: Star-like topology without minor exchange

So far we have considered two-state exchange involving a single ES. However, several R_1ρ studies report two detectable ESs in nucleic acids[117, 195, 197]. As detailed by Palmer[193], the way in which a second ES impacts the RD profile will depend on the kinetic topology. We first consider the topology ES2⇌GS⇌ES1 in which direct exchange between ES1 and ES2 (ES1⇌ES2) is assumed to be sufficiently slow as to have a negligible effect on R_1ρ (Fig. 23). The shape of such topologies in which multiple ESs exchange with the GS, but not amongst each other, resembles a star, and therefore these topologies are called “star-like”. We examine how inclusion of ES2⇌GS exchange impacts the two-state RD profiles observed for GS⇌ES1 exchange. To aid in visualizing the two ES contributions, the simulations assume that the ESs have Δω values with opposite signs, i.e., Δω₁Δω₂ < 0, where Δω₁ = ω_0,ES1 - ω_0,GS and Δω₂ = ω_0,ES2 - ω_0,GS, ω_0,ES1 and ω_0,ES2 being the Larmor frequencies of ES1 and ES2, respectively.

Figure 23.

Off-resonance R_1ρ RD profiles for three-state exchange simulated using B-M equations. A) Off-resonance R_1ρ RD profiles for three-state exchange in a star-like topology with p_ES1 = 1 %, p_ES2 = 2 %, k_ex,1 = 2,000 s⁻¹, $\Delta {\overline{\omega }}_1$(¹³C) = 5 ppm ¹³C, $\Delta {\overline{\omega }}_2$(¹³C) = −7 ppm, γ(¹H)B₀/2π = 700 MHz, R_1,GS = R_1,ES = 2.5 s^-1 and R_2,GS = R_2,ES = 22.5 s^-1, with k_ex,2 = 1,000 s^-1 (left) and kex,2 = 10,000 s⁻¹ (right). Dotted lines correspond to RD profiles for individual GS-ES1 and GS-ES2 exchange, while the solid lines correspond to the RD profiles for the three-state exchange process. Solid blue and green lines correspond to offsets of -Δω₁ and -Δω₂ respectively. B) ESs exchanging in a three-state star-like topology with similar chemical shifts ($\Delta {\overline{\omega }}_1$(¹³C) = $\Delta {\overline{\omega }}_2$(¹³C)= 3 ppm, γ(¹H)B₀/2π = 700 MHz) but different exchange rates and populations (p_ES1 = 3 %, p_ES2 = 5 %, k_ex,1 = 15000 s^-1, k_ex,2 = 1000 s^-1, R_1,GS = R_1,ES = 2.5 s^-1 and R_2,GS = R_2,ES = 22.5 s^-1) can be resolved using off-resonance R_1ρ RD (fitted parameters: p_ES1 = 2.5 ± 0.4 %, p_ES2 = 5.0 ± 0.7 %, k_ex,1= 14092 ± 1605 s⁻¹, k_ex,2= 926 ± 220 s⁻¹, $\Delta {\overline{\omega }}_1{(}^{13}\text{C)}=3.2\pm 0.3\text{\hspace{0.17em}ppm}$ and $\Delta {\overline{\omega }}_2{(}^{13}\text{C)}=3.1\pm 0.7\text{\hspace{0.17em}ppm}$. The B-M profiles were corrupted by noise with 5 % errors in R_1ρ prior to fitting. A vertical green line denotes an offset of -Δω₁. In both panels, spin-lock amplitudes are color coded. The direction of the initial alignment of the magnetization during B-M simulations was determined as described in Section 6.1, while taking into consideration the ES with the highest population.

As shown in Fig. 23A, the RD profiles for star-like three-state exchange are approximately given by the sum of two RD profiles for each of the two two-state exchange processes, with the three-state exchange resulting in a larger R_ex compared to either of the two separate two-state processes[193]. Two separate RD peaks are observed centered at −Δω₁ and −Δω₂. Similar behavior would be applicable for the case of having three or more ESs in a startopology, in which each ES would be associated with its own RD profile. It should be noted that two ESs in a star-like topology can be resolved even if they have identical Δω values, as long as they have sufficiently different k_ex and/or p_ES values (Fig. 23B).

An example of a star-like topology has been reported for DNA duplexes in the context of Watson-Crick to Hoogsteen (A(anti)-T(anti) to A(syn)-T(anti) and G(anti)-C(anti) to G(syn)-C⁺(anti)) transitions in B-DNA. The RD profiles measured for the sugar C3′ show two discrete peaks[117] (Fig. 24). Here, the C3′ spins sense Watson-Crick to Hoogsteen transitions of the constituent BP and its neighbor. The RD profiles could be satisfactorily fitted to three-state exchange with a star-like exchange topology as expected, given that the Watson-Crick to Hoogsteen transitions at neighboring BPs are independent of one another and can occur with different rates[117].

Figure 24.

Representative experimental off-resonance R_1ρ RD profiles for a system with threestate exchange in a star-like topology. Measurements were performed on C15-C3′ in A₆-DNA at pH 6.8, 25 °C, as described by Al-Hashimi et al.[117]. Spin-lock amplitudes are color-coded. Error bars represent experimental uncertainties determined by a Monte-Carlo scheme (Section 6.3). Solid lines denote a global fit to the RD data[117]. The initial alignment of the magnetization during the B-M fitting was performed as described in Section 6.1, while taking into consideration the ES with the highest population. Vertical black lines correspond to offsets of -Δω₁ and -Δω₂. Reprinted by permission from [117].

4.2.3. Three-state exchange: Triangular and linear topologies with minor exchange

We now consider how the off-resonance R_1ρ RD profiles vary for three-state exchange in the presence of direct exchange between the two ESs in a triangular topology. By definition exchange processes not involving the dominant GS are referred to as ‘minor’[193], thus the exchange rate between the two ESs is denoted by k_ex,minor. As expected, when k_ex,minor is slow relative to k_ex,1 describing GS⇌ES1 and k_ex,2 describing ES2⇌GS exchange, the RD profiles are identical to those obtained for the star topology (Fig. 25). The contribution from exchange between the two ESs increases as k_ex,minor approaches k_ex,1 or k_ex,2. This has the effect of ‘averaging’ the ES1 and ES2 chemical shifts[195], resulting in an asymmetric RD profile in which the positions of RD peaks of the two ESs shift toward one another. Further increasing k_ex,minor results in coalescence of the two ES RD peaks at a position determined by a populationweighted average over the Δω values of ES1 and ES2 (Fig. 25). This is analogous to what happens in free precession chemical exchange when two resonances coalesce in going from slow to fast exchange (Fig. 4A)[22].

Figure 25.

Off-resonance R_1ρ RD profiles for three-state exchange in the presence of direct “minor” exchange between the ES1 and ES2, simulated using B-M equations. Variation of offresonance R_1ρ RD profiles for three-state exchange with triangular and linear exchange topologies as a function of k_ex,minor and k_ex,1. Vertical blue and green lines correspond to offsets Δω₁ and -Δω₂, respectively. The RD profile in gray corresponds to three-state exchange in a star-like topology (p_ES1 = 1 %, p_ES2 = 2 %, k_ex,1= 2000 s⁻¹, k_ex,2 = 1000 s⁻¹, $\Delta {\overline{\omega }}_1{(}^{13}\text{C)}=5\text{\hspace{0.17em}ppm}$,$\Delta {\overline{\omega }}_2{(}^{13}\text{C)}=-7\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz, R_1,GS = R_1,ES = 2.5 s⁻¹ and R_2,GS = R_2,ES = 22.5 s⁻¹). Exchange parameters (other than k_ex values) for the linear exchange topologies were the same as for the star-like topology. Different spin-lock amplitudes are indicated by colors. The direction of initial alignment of the magnetization during B-M simulations was determined as described in Section 6.1, while taking into consideration the ES with the highest population.

Direct exchange between ESs has been reported experimentally for G-T/U mismatches exchanging between the wobble GS and two Watson-Crick-like ESs formed by tautomerization (ES1) or ionization (ES2) of the bases[165, 195, 197]. Under neutral pH, the RD profiles show a single ES (ES1), which was shown to be comprised of two distinct tautomeric species that exist in rapid equilibrium G^enol-T/U⇌G-T^enol/U^enol leading to a single coalesced RD peak[195]. Interestingly, a second ES (ES2) comprising anionic Watson-Crick G-T⁻/U⁻ was observed at higher pH, leading to deviations in the RD profiles from the perfect symmetry expected for two-state exchange. Some RD profiles showed a statistically significant improvement when fitted to a three-state model with minor exchange in a triangular rather than a star-like topology[195] (Fig. 26). It should be noted that fitting RD profiles for a three-state system undergoing minor exchange without taking it into account can result in errors in the estimates of Δω₁ and Δω₂.

Figure 26.

Representative experimental off-resonance R_1ρ RD profiles showing three-state exchange with “minor” exchange i.e., direct exchange between the two excited states[195]. Shown in solid lines are three-state fits to the experimental off-resonance R_1ρ RD profiles (hpGT-GGC, pH 8.0, 10 °C) with and without direct “minor” exchange between the two excited states[195], with the inclusion of “minor” exchange fitting the data better. Spin-lock amplitudes are color-coded. Error bars represent experimental uncertainty determined based on a Monte-Carlo error propagation scheme (Section 6.3). The direction of initial alignment of the magnetization during B-M fitting was determined as described in Section 6.1, while taking into consideration the ES with the highest population. Reprinted by permission from [195].

When the exchange is slow between the GS and one of the two ESs, the system can be described by a ‘linear’ topology GS⇌ES1⇌ES2[193]. Here, the detectability of ES2 can be determined by the kinetics of the GS⇌ES1 exchange. Once again, two discrete ESs peaks are observed or a single coalesced peak, depending on the exchange parameters including the relative values of k_ex,1 and k_ex,minor (Fig. 25).

As the appearance of three-state RD profiles can vary greatly with exchange parameters, the authors’ group provides an interactive tool (https://github.com/alhashimilab/InteractiveProfiles), which allows a user to vary the exchange parameters for a three-state system and simulate the effect on the RD profile.

4.2.4. Impact of unequal R₂ for ground and excited states

The above simulations assume equal R₁ and R₂ values for the GS and ES, i.e., ΔR₁ = R_1,ES – R_1,GS = 0 and ΔR₂ = R_2,ES – R_2,GS = 0. Deviations from this assumption can arise, particularly for R₂. For example, transitions toward ESs can entail the flipping in or out of nucleotide bases[16] that can be accompanied by large changes in ps-ns dynamics and therefore R₂[7, 198–200]. Especially for large RNAs, this can amount to ΔR₂ for C6/C8 and C1′ as large as ~100 s⁻¹[201]. Secondly, nucleic acid ESs can feature ionized or tautomeric bases[195, 197], which can have different dipolar contributions to relaxation, profoundly affecting R₂. Finally, melting/annealing of duplexes as well as dimerization of hairpins can involve large changes in overall molecular tumbling that can also lead to large ΔR₂[190]. It is therefore instructive to understand how differences in ΔR₂ impact the RD profiles.

As noted by Kay et al., a non-zero ΔR₂ has characteristic effects on the RD profile[187, 202]. For large ΔR₂ ~ 100 s⁻¹, the RD peak is no longer symmetric, and R_2,eff values are larger for offsets that correspond to having the ES near resonance[187] (Fig. 27A). The magnitude of the asymmetry depends primarily on ΔR₂/k_ex; in the limit of ΔR₂/k_ex = 0, the RD profiles are symmetric but they become increasingly more asymmetric with increasing ΔR₂/k_ex[202]. The asymmetry due to unequal R₂ is in general larger for smaller $\Delta \overline{\omega }$ values (Fig. 27B). Even in the limit of $\Delta \overline{\omega }=0\text{ ppm}$, an ES can be detected provided it has a sufficiently large ΔR₂; for p_ES > 10 %, ΔR₂ as small as 20 – 30 s⁻¹ can be detected even when $\Delta \overline{\omega }=0\text{ ppm}$ [187]. The off-resonance RD profiles in such cases are symmetrically centered at the carrier frequency but are ‘inverted’ (Fig. 27C) relative to the canonical two-state RD profiles (Section 4.2.1).

Figure 27.

Impact of unequal R_2,GS and R_2,ES on off-resonance R_1ρ RD profiles for two-state exchange simulated using B-M equations. Variation of off-resonance R_1ρ RD profiles with ΔR₂ as function of k_ex for A) $\Delta \overline{\omega }{(}^{13}\text{C)}=15\text{\hspace{0.17em}ppm}$, B) $\Delta \overline{\omega }{(}^{13}\text{C)}=3\text{\hspace{0.17em}ppm}$ and C) $\Delta \overline{\omega }{(}^{13}\text{C)}=0\text{\hspace{0.17em}ppm}$. Different ΔR₂ values are color coded. Exchange parameters are specified in inset. For all simulations γ(¹H)B₀/2π = 700 MHz and R_1,GS = R_1,ES = 2.5 s⁻¹. Direction of initial alignment of the magnetization during B-M simulations was determined as described in Section 6.1. While fitting the mono-exponential decay, the magnetization for the first 0.05 s was not considered, in order to exclude the fast initial decay, as proposed by Kay et al.[187].

5. Experiments for measuring off-resonance R_1ρ relaxation dispersion in uniformly ¹³C/¹⁵N labeled nucleic acids

5.1. Challenges and solutions for measuring off-resonance R_1ρ

1D R_1ρ experiments employing selective excitation schemes reviewed here for measuring R_1ρ(¹⁵N) and R_1ρ(¹³C) in nucleic acids are straightforward adaptations of experiments originally developed to measure R_1ρ(¹⁵N) and R_1ρ(¹³C) in proteins[154, 203, 204]. Before we describe the 1D R_1ρ experiments, we review three key challenges that were addressed to allow optimal measurements of R_1ρ in biomolecules.

5.1.1. Spin-lock amplitude limitations associated with alignment of magnetization along the effective field

So far we have considered the behavior of a single spin during the R_1ρ experiment. In practice, one is interested in measuring R_1ρ simultaneously for a large number of spins in a biomolecule. This presents a challenge, as the different spins in a molecule generally have different Larmor frequencies and will therefore be associated with different effective fields. Aligning all spins along their respective effective fields is not straightforward, particularly when using low spin-lock amplitudes, for which differences in the effective field orientations are most significant. Early heteronuclear 2D R_1ρ experiments employed high spin-lock amplitudes (>1000 Hz) and/or high offsets and used hard pulses to align the magnetization of all spins along a common effective field[154, 203, 204]. As R_1ρ experiments are sensitive to exchange processes with k_ex ~ ω_eff, this limited sensitivity to motions on the fast microsecond timescale. The misalignment of spins that are far off-resonance, and gradual decay of the magnetization component perpendicular to ω_eff,OBS due to the inhomogeneity of the spin-locking field results in oscillations during the relaxation period and in losses in sensitivity[205, 206]. Alternative magnetization alignment schemes were subsequently developed that used adiabatic ramps with amplitude, phase or frequency modulation to optimally align as many spins as possible along their respective effective fields[181, 205, 207–209]. While these schemes improved magnetization alignment, they were still limited to high spin-lock amplitudes, since low amplitudes require long adiabatic sweeps to optimally align the magnetization of individual spins along a broad range of effective field orientations, resulting in large sensitivity losses due to relaxation[181, 208].

5.1.2. Suppressing cross-correlated relaxation and heteronuclear scalar couplings

It has long been recognized that cross correlation between dipolar and chemical shift anisotropy (DD_NH/CSA_N) contributions to relaxation can affect ¹⁵N T₁ and T₂ measurements in macromolecules[157, 210, 211]. Cross-correlated relaxation converts transverse or longitudinal spin operators of order one to order two (S_x -> 2S_xI_z and S_z -> 2S_zI_z where S = ¹⁵N and I = ¹H), thus introducing an alternative mode of relaxation in addition to auto-relaxation. For nucleic acids, in addition to sizeable DD_NH/CSA_N contributions to imino ¹⁵N relaxation, suppressing DD_CH/CSA_C is also important for ¹³C relaxation measurements in which the DD_CH/CSA_C contributions can be significant[201].

During the 1990s, it became apparent that various schemes used to suppress DD_NH/CSA_N interactions in ¹⁵N RD experiments, such as appropriately spaced 180° ¹H pulses and ¹H decoupling elements[122, 205, 207, 209, 212, 213] were leading to systematic discrepancies in T₂ values measured using R_1ρ and CPMG[158, 159]. This was later attributed to the evolution of heteronuclear ¹J_NH couplings[180, 181] during the R_1ρ experiment, as a result of simultaneously pulsing on¹⁵N and ¹H for spin-locking and decoupling respectively. This led to the development of new approaches for more effectively suppressing both CSA_N/DD_NH and ¹J_NH scalar coupling evolution in R_1ρ experiments that employ appropriately spaced ¹H pulses[181] or continuous wave ¹H irradiation with periodic phase inversion[180]. Although the 2D experiment developed by Palmer et al.[181] could be used to measured R_1ρ(¹⁵N) RD using low spin-lock amplitudes (150–1000 Hz), in practice, the use of adiabatic pulses for magnetization alignment resulted in significant losses in sensitivity, precluding collection of a large set of off-resonance R_1ρ RD data for obtaining a complete set of exchange parameters.

5.1.3. Homonuclear scalar couplings complicate application of CPMG(¹³C) in uniformly labeled nucleic acids

For proteins, CPMG(¹⁵N) RD experiments provide a wealth of information regarding chemical exchange[22]. For nucleic acids, suitable imino ¹⁵N probes are only available for half the residues (guanine and uridine/thymine) and rarely for flexible sites of interest due to rapid exchange of imino protons with the solvent. For this reason, there has always been interest in using ¹³C spins as probes of nucleic acid dynamics[134]. However, extensive homonuclear C-C scalar (J_CC) coupling networks in base and sugar moieties in uniformly ¹³C labeled nucleic acid samples complicate the application of CPMG RD experiments as hard 180° pulses can excite many coupled ¹³C spins simultaneously[134, 136]. To address this problem, much effort has been directed toward the production of site-labeled[136, 177] or fractionally labeled[134] nucleotide-triphosphates (NTPs) through biomass production for use in enzymatic synthesis. Unfortunately, the site-labeled precursors are not commercially available. Interestingly, even though methods for site-specific incorporation of ¹³C labels using phosphoramidite chemistry were available as early as the 1990s[214], it is only recently[18, 173, 174, 176, 215] that these methods have been used to enable CPMG studies of chemical exchange in nucleic acids. However, as is the case with site-labeled NTPs, these amidites are not commercially available. Furthermore, obtaining a large number of chemical shifts for ES characterization can become prohibitive with site-labeled samples, owing to the necessity for synthesizing separate samples with different labeling schemes.

5.1.4. Selective excitation addresses challenges in R_1ρ relaxation dispersion studies of nucleic acids

The 1D R_1ρ(¹⁵N) experiment introduced by Kay et al. in 2005[163] addresses many of the above-mentioned limitations. The experiment employs selective excitation schemes pioneered by Bodenhausen and colleagues[160–162] to excite signals corresponding to individual spins of interest. Rather than measuring R_1ρ simultaneously for multiple spins at once, off-resonance R_1ρ RD profiles are measured one at a time for specific spins in a molecule. This bypassed the need to simultaneously align multiple spins along their effective fields altogether since each spin is aligned one at a time. By obviating the need for adiabatic ramps, much lower spin-lock amplitudes (~25 Hz for ¹⁵N) could be employed, broadening the sensitivity of R_1ρ to millisecond timescale motions[163]. In addition, DD_NH/CSA_N (or DD_CH/CSA_C) and heteronuclear J_NH (or J_CH) scalar coupling evolution could be effectively suppressed by the application of an on-resonance ¹H continuous wave field[163, 211] during the spin-lock period. In addition, J_CN and J_CC scalar couplings could be more readily suppressed in R_1ρ experiments by application of selective spin locking fields during the relaxation period. One downside is as the extra time needed to collect separate RD profiles for individual spins in a given molecule, relative to 2D R_1ρ experiments.

A straightforward modification of the original 1D R_1ρ(¹⁵N) experiment allows R_1ρ(¹⁵N) and R_1ρ(¹³C) data to be measured in uniformly ¹³C/¹⁵N labeled nucleic acids[21, 112], and will be described in Section 5.2. These R_1ρ experiments have been used to measure RD for a variety of sugar and base ¹³C/¹⁵N probes to obtain insights into chemical exchange over a broad range of timescales from micro to milliseconds in uniformly ¹³C/¹⁵N labeled nucleic acids ranging in size from 24–56 nucleotides[16, 21, 112, 117, 164–166].

5.2. Selective 1D pulse sequences for measuring R_1ρ in uniformly ¹³C/¹⁵N labeled nucleic acids

Pulse sequences for measuring R_1ρ(¹⁵N) and R_1ρ(¹³C) in uniformly labeled nucleic acids in a 1D manner are shown in Fig. 28A. The experiments employ selective cross polarization (CP)[160–162] to excite individual ¹⁵N or ¹³C spins of interest through the selective transfer of magnetization from directly bonded ¹H spins. This is accomplished by applying a 90°_x pulse to create transverse I_y (I = ¹H) magnetization followed by spin-locking in the rotating frame by application of a long and selective 90° RF pulse (B_I) along the y-axis. Simultaneously, a long RF pulse (B_S) is applied along x on the S spin (S = ¹³C or ¹⁵N) to satisfy the Hartmann-Hahn matching condition γ_IB_I = γ_SB_S. The spins are then allowed to exchange magnetization in a manner dependent on the magnitude and duration of the RF fields, leading to generation of transverse S_x magnetization. In theory, complete transfer of magnetization from spin I to spin S takes place when the duration of the CP transfer (τ_CP) is set to (2k-1)/J_IS and the amplitude of the RF pulses (ω_CP = γ_IB_I = γ_SB_S) is set to ${\omega }_{\text{CP}}/2\pi ={\left(\frac{\left(4n^2-{(2k-1)}^2\right)}{{(2k-1)}^2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{J}_{IS}/4$ (valid for 2(ω_CP/2π) ~ J_IS), where n and k are positive integers such that n ≥ k and J_IS is the scalar coupling between spins I and S. In practice, smaller values of τ_cp (= 1/J_IS) are preferred so as to minimize sensitivity losses due to relaxation. Furthermore, as demonstrated by Bodenhausen et al.[216] lower values of ${\omega }_{\text{CP}}/2\pi \left(=3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{J}_{IS}/4\right)$ are also preferred because the transfer is more selective and less sensitive to RF field inhomogeneities and mismatches in the Hartmann-Hahn matching condition. The values of ω_CP and τ_CP are adjusted from their initial settings (Table 1) to maximize the signal intensity at the end of the R_1ρ experiment, when setting the relaxation delay T_relax = 0. For AX and non-equivalent AX₂ spin systems (A = ¹³C/¹⁵N, X = ¹H), the CP transfer generates A_x magnetization from longitudinal X_z. For equivalent AX₂ spin systems, the transfer generates A_x + 4X_1ZX_2ZA_X coherences. In the latter case, some magnetization will be lost due to competing pathways arising from evolution of the three-spin coherences[217].

Figure 28.

A) Pulse sequences for measuring off-resonance R_1ρ(¹³C) and R_1ρ(¹⁵N) in nucleic acids with selective excitation and low spin-lock amplitudes. Hard90° pulses are denoted using filled narrow rectangles, while open narrow rectangles (excluding the purge elements in the ¹³C sequence) denote hard pulses with flip angles φ. The flip angle φ of the hard pulses φ₃ and φ₄ is equal to arctan(ω₁/Ω), where ω₁ and Ω are the spin-locking amplitude and offset for ¹³C or ¹⁵N. The flip angle of the hard pulse on the proton channel in the ¹⁵N R_1ρ sequence is equal to tan⁻¹(ω_1H/Ω_H) where ω_1H is the amplitude of the proton spin-locking field and Ω_H is the frequency offset of the signal of interest from water. Wide open rectangles denote periods of continuous RF irradiation. Reprinted with permission from [9].(B) The different spins that can be targeted for R_1ρ RD measurements in uniformly ¹³C/¹⁵N enriched nucleic acids using the pulse sequences in part (A) are highlighted by red circles. The ¹H spins that can be targeted using the 1D R_1ρ pulse sequence developed by Petzold et al.[222] are also shown using red circles. Spectral crowding in RNA between the C3′ and C2′ sugar carbons prevents measurement of R_1ρ(C3′ and C2′) in RNA[166] unless one uses selectively labeled samples, while relaxation contributions from germinal protons complicates R_1ρ(C5′) measurements in both DNA and RNA[166] and R_1ρ(C2′) measurements in DNA unless selective deuteration is used[131].

Table 1. Optimal settings for 1D off-resonance R_1ρ experiments using the pulse sequences shown in Fig. 21A.

Parameter	Optimal Value
ωCP/2πa	85 Hzb, c
τCP	~1/4 |1JAH|d, ~3/4 |1JAH|e
Τeq	~3/kex
ω1/2πf	G-N1/U-N3	> ~50 Hz
	A-C8/G-C8	> ~45 Hz
	A-C2	> ~40 Hz
	C1′/C3′/C4′	> ~150 Hz
	C-C6/C-C5/U-C6/U-C5/T-C6	> ~200 Hz
	A-NH2/G-NH2/C-NH2	> ~70 Hz
ω1H/2π	~10 kHz for 13C and ~5 – 8 kHz for 15Ng
ζ	π/2δh

^a(ω_CP/2π) >3^½/4 |¹J_AH|, A = ¹³C/¹⁵N

^bR_1ρ(¹⁵N) measurements on imino ¹⁵N (N1/N3). In theory, ω_CP/2π should be chosen to be equal to 3^½J_IS/4, the smallest value that can lead to complete magnetization transfer between spins, while also being selective and less sensitive to RF field inhomogeneities and mismatches in the Hartmann-Hahn matching condition[216]. However, in practice, a value of ω_CP/2π (= 85 Hz) larger than 3^½J_IS/4 ~ 39 Hz is typically employed so as to minimize losses of the magnetization due to conformational exchange[163].

^cR_1ρ(¹³C) measurements on aromatic and sugar ¹³C nuclei (C2/C5/C6/C8/C1′/C3′/C4′)

^dFor AX (e.g. N1–H1 and C8–H8) and non-equivalent AX₂ (e.g. amino NH₂ group in WatsonCrick G-C BPs) spin systems[161, 217]

^eFor equivalent AX₂ spin systems (e.g. amino NH₂ group in G-T/U wobbles)[161].

^fShould be greater than 3 times ²J_NN or ¹J_CC[21, 28]. The largest value that can be used depends on probe hardware limitations.

^gShould be > ω_eff(max)

^hδ is the offset in Hz of the undesired resonance in the ¹³C/¹⁵N dimension from the resonance of interest

The transverse S_x magnetization is then rotated to S_z through application of a hard 90° pulse. This is a followed by a delay τ_eq ~ 3/k_ex to allow equilibration of p_GS and p_ES (Fig. 28A). This is necessary as the ES, since it has a chemical shift different from the GS, may not have been excited by the selective Hartmann-Hahn transfer. A second hard pulse with a tip angle θ, i.e., the tilt angle of ω_eff,OBS, is then used to align the S_z magnetization of both the GS and ES along ω_eff,OBS (Fig. 11B). A purge element is used to suppress magnetization from spins that have similar ¹H but different ¹³C or ¹⁵N Larmor frequencies via the application of a delay (ζ), that allows for dephasing of spins precessing with a frequency different from that of the target[163]. The ζ delay and purge element can be placed either before[21] or right after the spin-lock period[112, 163] (Fig. 28A).

The ¹³C/¹⁵N magnetization is then spin-locked during the relaxation period (T_relax) along ω_eff,OBS by the application of a spin-locking field to the spin of interest. Here, care is taken to minimize Hartmann-Hahn transfers between pairs of scalar coupled spins such as C5–C6 (in pyrimidines), C1′-C2′, and C4/6-C8 (in purines) to typically < 1% of the maximum transfer efficiency[21]. This is accomplished by ensuring that the difference in the effective field strengths experienced by two coupled spins (S1 and S2) is much greater than the coupling constant between them. The Hartmann-Hahn transfer efficiencies (A_HAHA) between spins S1 and S2 can be computed using[92]:

$$A_{HAHA}={\left(1+{\left(\frac{{\omega }_{eff,S1}-{\omega }_{eff,s2}}{0.5J_n\left(1+cos\left({\theta }_{S1}-{\theta }_{S2}\right)\right)}\right)}^2\right)}^{-1}$$ (55)

where ω_eff and θ are the effective fields and tip angles of the two spins, and J_n is the scalar coupling between them. The offsets that minimize unwanted Hartmann-Hahn transfers (A_HAHA < 1%) in R_1ρ(¹³C) RD experiments are listed for a variety of ¹³C spins in Table 2. In general, extraneous Hartmann-Hahn magnetization transfers are less relevant for ¹⁵N spins compared to ¹³C, owing to the smaller size of the J_n couplings (< 10 Hz)[218].

Table 2. Offsets for R_1ρ(¹³C) measurements in nucleic acids that are susceptible to Hartmann-Hahn transfers during application of a spin-lock.

Magnetization transfer between nuclei that are coupled to each other can serve as an extraneous mode of relaxation during the application of spin-locking pulses in an R_1ρ experiment. Offset values that are susceptible to such magnetization transfer via attainment of a Hartmann-Hahn matching condition (efficiency A_HAHA > 1%, Eqn. 55) for a pair of spins S1 and S2 were computed assuming γ(¹H)B₀/2π = 700 MHz and spin-lock amplitudes between 50–3600 Hz. J_n is the multiple-bond scalar coupling constant between the spins under consideration while $\Delta {\overline{\omega }}_{\text{S}2-\text{S}1,\text{avg}}$ is the average difference in chemical shift between them. J_n couplings were obtained by van Buuren et al.[218].

S1	S2	Nucleic Acid	Jn(Hz)	$\Delta {\overline{\omega }}_{\text{S2}-\text{S1,avg}}(\text{ppm})$	Offsets (Hz)
C4′	C5′	RNA	−40	−17	(−1700, −1300)
C4′	C3′	RNA	−40	−9	(−900, −700)
C4′	C5′	DNA	−40	−20	(−1900, −1600)
C4′	C3′	DNA	−40	−9	(−900, −700)
C3′	C4′	DNA	−40	9	(700, 900)
C3′	C2′	DNA	−40	−38	(−3500, −3100)
C1′	C2′	RNA	−42	−15	(−1450, −1150)
C1′	C2′	DNA	−42	−46	(−4200, −3900)
C/U-C6	C/U-C5	RNA/DNA	−68	−44	(−4100, −3600)
C/U-C5	C/U-C6	RNA/DNA	−68	44	(3600, 4100)
C/U-C5	C/U-C4	RNA/DNA	−60	70	(5800, 6400)

Selective irradiation of the ¹³C or ¹⁵N spins with a spin-locking field (Fig. 28A) during the relaxation period also serves to suppress evolution under J_CC, J_NC, and J_NN. These scalar coupling constants also set a lower limit on the spin-lock amplitude that can be used and, consequently, the slowest process that can be probed using R_1ρ when working with isotopically labeled samples. Typically, ω₁ is set to be approximately three times larger than the largest relevant scalar coupling (Table 1). In contrast, the highest spin-lock amplitude that can be used is determined by the probe hardware, and this in turn determines the fastest process that can be probed using off-resonance R_1ρ (Table 1). Spin-lock amplitudes as high as ~6.4 kHz for ¹⁵N can be used with a cryo-probe (QCI Bruker) as compared to 4 kHz for a room-temperature probe, allowing measurement of exchange processes with exchange rates as high as 40000 s⁻¹[219].

During the relaxation period, evolution under ¹J_NH (or ¹J_CH) and DD_NH/CSA_N (or DD_CH/CSA_C) are efficiently suppressed using a strong ¹H continuous-wave field (ω_1H) applied on the resonance of interest[211, 220] (Fig. 28A). ω_1H is chosen to be larger than the maximum effective (¹³C/¹⁵N) spin-locking field (ω_1H > ω_eff(max)) to avoid unwanted C-H or N-H polarization transfers during the spin-lock period through accidentally fulfilling a Hartmann-Hahn matching condition[163] (Table 1). The amino NH₂ spin system is also subject to dipole-dipole (DD_NHA/DD_NHB) cross-correlated relaxation, which can result in the generation of doubly antiphase magnetization (4I_1zI_2zS_x) from in-phase magnetization during the relaxation period. The DD_NHA/DD_NHB cross-correlated relaxation contribution scales as (τ_c/2)(3cos²θ − 1)[221], where θ is the angle between the two N-H bonds. The angular term is small (~ −0.125) for amino –NH₂ groups in which the angle between the bonds is ~120°. Thus, the DD_NHA/DD_NHB contribution for a 10 BP duplex is estimated to be ~0.5 s⁻¹ and can be safely ignored[165]. In an analogous manner, extraneous contributions from J couplings and cross relaxation must also be taken into account when performing ¹H RD experiments, for which pulse sequences based on selective 1D excitation have also been reported[222, 223]. As the higher γ value for ¹H allows one to achieve higher spin-lock amplitudes, R_1ρ(¹H) holds great promise for characterizing faster exchange processes in nucleic acids in the future, potentially without the need for isotopic enrichment.

At the end of the relaxation period, the component of the net magnetization along ω_eff,OBS is tilted back along z by the application of a hard pulse, and is transferred back from ¹³C or ¹⁵N to ¹H using CP for subsequent detection (Fig. 28A). Consistent temperature throughout the experiment is maintained by a heat-compensation element[21, 163] that is applied far offresonance; this consists of ¹H and ¹³C or ¹⁵N spin-locking fields applied for a given amount of time (T_max – T_relax), where T_max is the longest relaxation time and T_relax is the relaxation time in a given scan. T_max is typically set to a value that results in ~30% intensity loss relative to the value measured at T_relax = 0 for the resonance of interest, while also taking into account probe hardware limitations that may arise for longer T_max. The water signal is suppressed using either low amplitude presaturation during inter-scan delays with additional purge elements for ¹³C experiments or with a WATERGATE element[224] before acquisition for ¹⁵N experiments (Fig. 28A). Decoupling pulses are applied to ¹³C or ¹⁵N during acquisition. The spins in nucleic acids that can be targeted using these 1D R_1ρ experiments are shown in Fig. 28B.

The experiment yields a 1D ¹H NMR spectrum for the resonance of interest and in some cases additional nearby resonances, especially if their chemical shifts overlap in both ¹H and ¹³C/¹⁵N dimensions (Fig. 29). A total of ~10 T_relax delays are typically measured and the intensity of the peak of interest is fitted to a mono-exponential decay I = I₀ exp(−R_1ρT_relax) using a nonlinear fitting algorithm to obtain the value of R_1ρ (Fig. 29). On-resonance R_1ρ data is typically collected initially by setting Ω = 0 and decay curves are measured at ~ 20 ω₁/2π values ranging between 50 – 2000 Hz and 50 – 3600 for ¹⁵N and ¹³C, respectively. Off-resonance R_1ρ data are then measured by varying both Ω/2π and ω₁/2π (typically three or more values spanning the ranges described above). The maximum value Ω_max/2π is generally set to ~3 – 4 × ω₁/2π (Section 3.5). Typically, a total of ~ 24 values of Ω/2π are chosen with equal spacing to span ± Ω_max/2π symmetrically about Ω/2π = 0. In cases where there is an a priori estimate of ω_0,ES, offsets can be chosen accordingly to sample points where R_ex would be non-zero. Note that it may be important to collect R_1ρ data for multiple experimental sample conditions, as the appearance of the profiles is sensitive to the conditions used (Section 4.2, Section 7.2.3).

Figure 29.

Flowchart describing the various steps for measuring and analyzing R_1ρ RD data using the pulse sequences shown in Fig. 28A.

5.3. Calibrating the spin-lock amplitude

When setting up the R_1ρ experiment it is important to calibrate the ¹³C and ¹⁵N spin-lock amplitudes[22, 180]. This involves finding the optimal power in dB (decibels) or W (watts) for the spin-lock pulse of a given frequency. In general, spin-lock amplitude calibration is done when setting up experiments on a new probe and need not be repeated when switching samples or changing temperature. The calibration is performed using a modified version of the R_1ρ pulse sequence in which the pulses with phases ϕ ₃ and ϕ ₄ flanking the spin-lock element that tilt magnetization along ω_eff,OBS have been removed (Fig. 28A). This can be done by omitting the pulses from the pulse program or by simply setting the pulse duration to zero. The rationale is to detect the precession of net magnetization around the x or y axis due to the spin-locking field, and use this information to infer its amplitude. The following protocol can be used for calibration:

1. Select a sharp and well-resolved resonance for calibration that exhibits minimal or no chemical exchange to avoid complications from exchange processes. For experiments with nucleic acids, it is preferable to perform ¹³C calibration using A-C8 or G-C8 spins as they permit the use of spin-lock amplitudes as low as ~50 Hz (Table 1).

2. Compile a table containing ω₁/2π values (in Hz) to be calibrated and associated power levels P (in dB). For ¹³C/¹⁵N calibration, we choose 24 ω₁/2π values between 50 and 3600 Hz and 50 and 2000 Hz respectively. The power levels P for each spin-locking field can initially be estimated based on the calibration of rectangular 90° pulse lengths for ¹⁵N or ¹³C.

3. Record 1D experiments for an array of spin-lock durations T_relax under on-resonance conditions for each ω₁ value. Usually, 33 durations are used in increments of π/(2ω₁) from 0 to 8*2π/ω₁ s[21]. For example, a spin-lock amplitude ω₁/2π of 100 Hz would be incremented 33 times from 0–80 ms in increments of 2.5 ms. This ensures that there are 4 points for one rotation of the net magnetization about the spin-locking field, enabling extraction of the frequency of rotation, in accordance with the Nyquist sampling theorem.

4. The resultant peak intensity should be a sinusoidal function of ω_1,realT_relax where ω_1,real is the real spin-lock frequency corresponding to the power P (Fig. 30A). Calculate ω_1,real by fitting the peak intensities to the equation I(T_relax) = I₀exp(−R · T_relax)cos(ω_1,realT_relax + ϕ), where I is the peak intensity, I₀ is the intensity when T_relax = 0, R is the relaxation rate and ϕ is a phase correction factor.

5. Repeat Step 2 and 3 for all ω₁ in the table, and obtain ω_1,real for all P values.

6. Fit P and ω_1,real to a logarithmic equation P = alog₁₀(ω_1,real) + b. Ideally this fit should give R² > 0.99. If the data points deviate from linearity, polynomial fitting can be adopted instead. An example of an ideal calibration curve is shown in Fig. 30B.

7. Use the equation given by the fit in step 6 to back-calculate P for a desired ω₁ within the range of amplitudes used for calibration.

Figure 30.

Calibration of the spin-lock amplitude in R_1ρ experiments. A) Nutation curve showing variation of signal intensity (black dots) as a function of time during an R_1ρ calibration experiment. The red line is a fit of the intensity to the equation in the inset. B) Variation of the actual spin-lock frequency ω_1,real, with the power P of the spin-lock field, obtained by fitting nutation curves in part (A) (black dots). The red line denotes a fit to the equation in the inset.

6. Data Analysis

Two general approaches can be used to analyze off-resonance R_1ρ data to extract the exchange parameters (Fig. 29). The most general approach involves numerically integrating the B-M equations[45] to find exchange parameters that best satisfy the measured R_1ρ data. A second approach to fit R_1ρ data involves using algebraic expressions that make various assumptions and which are valid for different ranges of exchange conditions. The two approaches are often used hand in hand, with the algebraic expressions often being used to obtain initial guesses for the B-M fitting as well as for developing an intuitive understanding for how the RD profiles vary with spin-lock and exchange parameters.

6.1. Numerical integration of Bloch-McConnell equations

In the authors’ group, we use an in-house python program called “Bloch-McConnell Numerical Simulator” or BMNS (available from https://github.com/alhashimilab/BMNS) to numerically integrate the B-M equations and thereby simulate or fit R_1ρ data. The analysis described below is valid when the magnetization decay is dominated by a single exponential decay[13, 23]. In uncommon cases where the decay is not mono-exponential, one needs to directly fit it by integrating the B-M equations, without extracting R_1ρ. BMNS takes the following parameters as input:

1. List of Ω, ω₁, R_1ρ and the uncertainty of R_1ρ (see Section 6.3).

2. Initial guesses for the exchange parameters (p_ES1, p_ES2, k_ex1, k_ex2, Δω₁, Δω₂, R_2,GS, R_2,ES1, R_2,ES2, R_1,GS, R_1,ES1, R_1,ES2) for a given exchange topology as well as their ranges. For example, for three-state exchange, one has to specify whether the topology is star-like, linear, or triangular by appropriately setting individual rate constants to zero (Section 4.2). The initial guesses are typically either assigned arbitrarily and/or obtained from fitting the R_1ρ data to an appropriate algebraic expression (Section 6.2). The ranges for the exchange parameters are typically chosen to be large to avoid biases. In general, fitting the RD data is carried out assuming R_1,GS = R_1,ES and R_2,GS = R_2,ES unless there is reason to suspect otherwise. Note that deviations from the latter assumption can result in significant errors in the fitted p_ES and k_ex values even when |R_2,GS- R_2,ES| << k_ex[190].

3. The initial alignment of the magnetization. For slow exchange, the magnetization is initially aligned along ω_eff,GS (GS alignment), while more typically, for fast and intermediate exchange, the magnetization is aligned along ω_eff,AVG (AVG alignment). If not known a priori, the mode of alignment can be determined during the fitting procedure based on the value of k_ex/|Δω|; with k_ex/|Δω| ≤ 1 and k_ex /|Δω| > 1 corresponding to GS and AVG alignments, respectively. We recommend fitting RD profiles assuming both possible alignments for completeness. For low p_ES, the choice of alignment has little effect on fitting R_1ρ data. However, for intermediate to fast exchange (0.1 < k_ex /Δω < 5), with high p_ES (> 10%), a line shape analysis shows that the observed peak is actually neither at ω_0,GS nor at ω_0,AVG (Fig. 31A). Thus, calculation of effective fields assuming GS or AVG alignments leads to asymmetries in the RD profile around Ω = 0, that are especially pronounced at low ω₁, when deviations in Ω can have a large influence on the effective field orientations (Fig. 31B, 31C). This artifact can be corrected by selecting an initial magnetization alignment based on line shape simulations, and then adjusting the effective fields to take this into account (Eqn. 37, 39). This also enables determination of the effective fields and alignment directions for any arbitrary set of exchange parameters. Temperaturedependent R_1ρ measurements can also be used to deduce the alignment scheme when both schemes equally fit the data well, as described in Section 6.4.3. Each of the above inputs can be specified for individual R_1ρ data sets measured on multiple spins, which can also be fit globally using shared parameters.

Figure 31.

Asymmetries near 0 offset observed in off-resonance R_1ρ RD profiles due to improper alignment of the initial magnetization under intermediate exchange conditions for two-state exchange, obtained using B-M simulations. A) Line shape (LS) simulation using the B-M equations in free precession conditions in the absence of a spin-locking field, for a system in intermediate exchange (p_ES = 10 %, k_ex = 3000 s⁻¹, $\Delta \overline{\omega }{(}^{13}\text{C)}=5\text{\hspace{0.17em}ppm}$, γ(¹H)B₀/2π = 700 MHz,R_1,GS = R_1,ES = 0 s⁻¹, R_2,GS = R_2,ES = 16 s⁻¹). B) Directions of the effective fields (ω_eff,GS: blue, ω_eff,ES: red, ω_eff,AVG: green, ω_eff,OBS: black dotted lines) and initial alignment of the bulk magnetization (GS: blue, ES: red arrows) at the start of the R_1ρ experiment (ω₁/2π = 100 Hz, Ω = −300 Hz) for the system in (A) where the magnetization is initially aligned along the effective field of the ground state (GS Align, left), population weighted average (AVG Align, middle) and the observed (OBS) resonance obtained from a lineshape simulation (LS Align, right) . (C) Offresonance R_1ρ RD profiles for the system in (A) with R_1,GS = R_1,ES = 2.5 s⁻¹, when the magnetization is initially aligned along ω_eff,GS (GS Align, left), ω_eff,AVG (AVG align, middle) and ω_eff,OBS (LS Align, right). Note that in all cases, the magnetization at the end of the relaxation delay (0.05 s) is projected parallel to the initial alignment direction to calculate R_1ρ. The different spin-lock amplitudes are color coded. Also shown in insets are representative decays of the normalized magnetization (black dots) along the initial alignment direction during the relaxation decay, along with exponential fits (red line).

Fitting of the R_1ρ data is then carried out through least-squares minimization of the function F that quantifies the agreement between the experimentally measured R_1ρ data $\left(R_{1\rho (i)}^{exp}\right)$ and values calculated $\left(R_{1\rho (i)}^{Calc}\right)$ using the B-M equations for a given set of exchange parameters and a user-specific exchange topology:

$$F=\frac{{\chi }^2}{2}$$ (56)

$${\chi }^2={\sum }_{i=1}^N{\left(\frac{R_{1\rho (i)}^{Calc}-R_{1\rho (i)}^{exp}}{{\sigma }_{exp(i)}}\right)}^2$$ (57)

σ_exp(i) is the experimental uncertainty in R_1ρ (estimated using Monte-Carlo or bootstrap approaches outlined in Section 6.3) and the summation i is over the total number of R_1ρ data points N (total number of combinations of spin-lock amplitudes and offsets).

The initial guesses for the exchange parameters are used as input to predict the R_1ρ data by using the B-M equations (Section 4.1). This involves numerically integrating the B-M equations to determine M_GS and M_ES as a function of time, and then fitting the projection of the net magnetization along ω_eff,OBS (depending on the initial alignment of the magnetization) to determine R_1ρ. The χ² value is then computed and a second iteration is attempted choosing different values for the exchange parameters from the user defined range, in a manner that minimizes the χ², using least-squares minimization[225]. BMNS uses a Trust Region Reflective algorithm for this purpose. This process is repeated iteratively until a local minimum is found. To minimize computational cost and also avoid being trapped in local minima in the multidimensional parameter space during least-squares minimization, an adaptive memory programming framework[226] for global space optimization can be used. Here, once a local minimum is found, the search function tunnels out of this local minimum and resumes optimization from a new starting point. Alternatively, a brute force grid search over the multidimensional parameter space can also be performed, although it is computationally expensive.

The goodness of the fit is evaluated by normalizing the computed χ² values to take into account the degrees of freedom to obtain a reduced χ² (χ²red), so as to enable comparisons between different exchange models:

$${\chi }^{2}red=\frac{{\chi }^2}{N-K}$$ (58)

where N is the total number of R_1ρ data points and K the number of fitted parameters for a given exchange model. In general, a value χ²red < 1.5 is considered to be reasonable though care should be taken if a fit gives χ²red < 1 as this may indicate over-fitting of the data. Convergence of the fitted exchange parameters when starting with different initial guesses can also be used to assess the robustness of the fit. The different exchange models (two-state versus three-state star-like/triangular/linear) and fitting modes (individual versus global) used in the B-M fitting are evaluated for statistical significance using Akaike information (AIC) and Bayesian information (BIC) criteria[227, 228] (Fig. 29). They quantify the likelihood or probability of one model relative to another for fitting a given data set, by taking into account the goodness of fit and the complexity or degrees of freedom of the models:

$$AICweight=\frac{\text{exp}(-0.5\Delta AIC)}{1+\text{exp}(-0.5\Delta AIC)}$$ (59)

$$BICweight=\frac{\text{exp}(-0.5\Delta BIC)}{1+\text{exp}(-0.5\Delta BIC)}$$ (60)

where ΔAIC and ΔBIC denote the differences in AIC and BIC values between the models. The AIC and BIC values themselves are defined as follows:

$$AIC=\begin{array}{c}\begin{array}{ccc}Nln\left(\frac{RSS}{N}\right)+2K,& & where\frac{N}{K}\ge 40\end{array}\\ Nln\left(\frac{RSS}{N}\right)+2K+\frac{2K(K+1)}{N-K-1},where\frac{N}{K}<40\end{array}$$ (61)

$$BIC=Nln\left(\frac{RSS}{N}\right)+Kln(N)$$ (62)

where $RSS={\Sigma }_{i=1}^N{\left(R_{1\rho (i)}^{Calc}-R_{1\rho (i)}^{exp}\right)}^2$ is the residual sum of squares for a given exchange model. Degeneracies can arise near under particular exchange scenarios that can make it difficult to reliably determine all exchange parameters. Near the fast exchange limit (k_ex/Δω > 10), it becomes difficult to reliably separate Δω from p_ES as R_ex becomes proportional to p_ESΔω²[13, 196]. Under slow exchange conditions, there are also degeneracies in resolving p_ES and k_ex as R_ex becomes proportional to p_ESk_ex[229, 230].

6.2. Algebraic equations

Algebraic expressions that make various approximations can also be used to fit the variations of off-resonance R_1ρ with ω₁ and Ω to extract the desired exchange parameters (reviewed in Massi et al.[13]). They give a better understanding of the dependence of R_1ρ on the exchange parameters relative to the B-M equations. These expressions assume that the decay of the magnetization is mono-exponential. The most versatile expression for two-site chemical exchange is given by the following equation derived from a Laguerre approximation[231] and more recently via perturbation theory applied to an n-site BM evolution matrix[194]:

$$\frac{\begin{array}{l}{R}_{1\rho }=R_1{cos}^2\theta +R_2{sin}^2\theta +\hfill \\ {sin}^2\theta p_{GS}{p}_{ES}{k}_{ex}\Delta {\omega }^2\hfill \end{array}}{\left\{{\omega }_{eff,GS}^2{\omega }_{eff,ES}^2/{\omega }_{eff,AVG}^2+k_{ex}^2-{sin}^2\theta p_{GS}{p}_{ES}\Delta {\omega }^2\left(1+\frac{2k_{ex}^2\left(p_{GS}{\omega }_{eff,GS}^2+p_{ES}{\omega }_{eff,ES}^2\right)}{{\omega }_{eff,GS}^2{\omega }_{eff,ES}^2+{\omega }_{eff,AVG}^2{k}_{ex}^2}\right)\right\}}$$ (63)

$$\theta ={tan}^{-1}\left(\frac{{\omega }_1}{{\Omega }_{AVG}}\right)$$ (64)

The Laguerre equation is valid under conditions of fast exchange, and deviations can arise for slow to intermediate exchange when p_ES > 30% especially for low spin-lock amplitudes (< 400 Hz)[9, 231]. This results from the fact that the Laguerre equation assumes AVG alignment.

Simpler expressions have also been derived that entail the use of alternative approximations[13]. An expression for R_1ρ valid for very asymmetric populations (p_ES << p_GS) for arbitrary chemical shifts is given below:

$$R_{1\rho }=R_1{cos}^2\theta +R_2{sin}^2\theta +\frac{{sin}^2\theta p_{GS}{p}_{ES}{k}_{ex}\Delta {\omega }^2}{{\Omega }_{ES}^2+k_{ex}^2+{\omega }_1^2}$$ (65)

while an alternative expression that is valid only in the fast exchange limit (k_ex >> Δω) for arbitrary populations is given by[63, 189]:

$$R_{1\rho }=R_1{cos}^2\theta +R_2{sin}^2\theta +\frac{{sin}^2\theta p_{GS}{p}_{ES}{k}_{ex}\Delta {\omega }^2}{k_{ex}^2+{\omega }_1^2+{\Omega }_{AVG}^2}$$ (66)

The two equations above can be used to fit off-resonance data; however, deviations can arise under conditions when the assumptions inherent to the equations are not satisfied.

For three-state exchange without direct “minor” exchange between ES1 and ES2 (ES2⇌GS⇌ES1), one can use the following equation[193]:

$$R_{1\rho }=R_1{cos}^2\theta +R_2{sin}^2\theta +{sin}^2\theta \left(\frac{k_1\Delta {\omega }_1^2}{{\Omega }_{ES1}^2+{\omega }_1^2+k_{-1}^2}+\frac{k_2\Delta {\omega }_2^2}{{\Omega }_{ES2}^2+{\omega }_1^2+k_{-2}^2}\right)$$ (67)

where k₁ and k_-1 are the GS⇌ES1 forward and reverse rate constants, and k₂ and k_-2 are the GS⇌ES2 forward and reverse rate constants, respectively. From the equation, it can be appreciated that the exchange contribution for a star-like three-state topology is essentially the sum of exchange contributions due to the individual two-state processes as described in Section 4.2.2. Expressions for three-state exchange with significant minor exchange have also been reported[193]. These equations have been generalized to n-site exchange for star-like, complex linear, and non-linear topologies by Palmer et al.[194]. All the above equations assume that R₂ results from a population-average over all ES and GS species. Algebraic expressions for R_1ρ for cases where R₂ for the GS and ES are not equal have also been reported in the literature[187, 202].

When fitting R_1ρ data using the above equations, one starts with broad initial estimates for the exchange parameters and optimizes initial guesses to achieve agreement with the experimental data. However, biased starting estimates coupled with conventional local optimization methods can lead to being trapped in local minima. In such cases, the use of simulated annealing and basin-hopping schemes[226] can prove beneficial for finding global minima and for exploring the search space more broadly.

χ²red values can be used to assess the quality of fit as described in Section 6.1. A poor fit may suggest deviations from the assumed exchange model or from the range of validityof the algebraic equations used. Once again, statistical tests (e.g. AIC and BIC weights)[227, 228] can be used to assess the likelihood of one model or exchange topology over the other.

6.3. Estimating uncertainties in exchange parameters

Uncertainty in the fitted exchange parameters can be estimated using the standard error of the fit, as well as bootstrap and Monte Carlo based approaches[196]. A parent set of R_1ρ rates (Ω, ω₁, R_1ρ and σ) is used for all methods. The standard error of a fit can be obtained as the square root of the diagonal elements of the covariance matrix of the fitted parameters.

In the bootstrap approach:

1. A large number (typically 500–1000) child data sets of the same size as the parent data set are generated by randomly sampling R_1ρ data points with replacement from the parent data set. Here, an individual data point can be either excluded or chosen multiple times in a given child data set.

2. Each child data set and the parent data set are fit to the Laguerre[231] or B-M equations to obtain the five exchange parameters: p_ES, k_ex, Δω, R₁ and R₂.

3. The uncertainty in the exchange parameters is then determined by calculating the standard deviation between the fitted parameters of the child data sets and those of the parent data set.

In the Monte Carlo approach:

1. The parent data set is fit to the Laguerre equation[231] or B-M equations[45] to obtain the exchange parameters.

2. 1000 child data sets are generated using the exchange parameters and the Laguerre or B-M equations, and the associated Ω and ω₁ values of the parent data set. Each child data set is then corrupted by noise according to the R_1ρ error from the parent data set and re-fitted using the Laguerre equation or B-M integration.

3. The uncertainty in the exchange parameters is then determined by calculating the standard deviation between the fitted parameters of the child data sets and those of the parent data set.

Similar methods can also be used for determination of errors in R_1ρ for each ω₁ and Ω combination while fitting the mono-exponential decays.

6.4. Extracting exchange parameters

6.4.1. Deducing the total number of excited states

Early on, it is important to deduce the total number of ESs sensed by the R_1ρ measurements as this influences the models used in fitting the data. Deviations from symmetric RD profiles expected for two-state exchange, and the appearance of more than one RD peak is an indication of additional ESs as described for three-state exchange in Figure 24. In addition, different spins in a molecule may sense different ESs. To gauge such cases, the R_1ρ data measured for the spins are initially subjected to B-M fitting individually. A plot of k_ex versus p_ES for the different spins is then examined to identify the total number of distinct ESs. A representative k_ex vs. p_ES plot for HIV-1 TAR shows clusters at two distinct locations consistent with two ESs (Fig. 32). Whether or not global fitting of the data with a common p_ES and k_ex is statistically justified also provides a means of gauging the total number of ESs.

Figure 32.

Deducing the number of ESs sensed by R_1ρ RD measurements. A plot of p_ES vs. k_ex as obtained from R_1ρ measurements on individual spins in a given molecule displays clusters corresponding to kinetic and/or thermodynamically distinct ESs. Shown is a representative p_ES vs. k_ex plot obtained from ¹³C R_1ρ RD (C6/C8/C4′/C1′ spins) measurements on HIV-1 TAR (secondary structure shown on the left) at pH 6.4, 25 °C[166], indicating the presence of two distinct ESs (ES1 and ES2). The zoomed in view of the ES2 region is provided in the inset. Errors were estimated using a Monte-Carlo approach while fitting the individual RD profiles, as explained in Section 6.3.

6.4.2. Deducing the chemical shifts of the excited states

The chemical shift of the ES (ω_0,ES) can be calculated from the Δω value obtained from fitting the RD data and peak position (ω_0,OBS) measured from a 2D HSQC spectrum. For intermediate-slow exchange with k_ex /Δω < 1, the initial alignment of the magnetization is along ω_eff,GS (Section 6.1) and ω_0,ES is given by:

$${\omega }_{0,ES}={\omega }_{0,GS}+\Delta \omega$$ (68)

For intermediate-fast exchange k_ex /Δω > 1 for which the initial alignment of the magnetization is along ω_eff,AVG (Section 6.1), ω_0,ES can be calculated using:

$${\omega }_{0,ES}={\omega }_{0,AVG}+p_{GS}\Delta \omega$$ (69)

Note that for special exchange scenarios such as intermediate to fast exchange (0.1 < k_ex /Δω < 5) with high p_ES > 10% (Section 6.1), ω_0,OBS does not correspond to either ω_0,AVG nor ω_0,GS. Here, line shape simulations under free precession assuming the fitted exchange parameters with ω_0,GS = 0 and ω_0,ES = Δω are used to estimate the shift (ψ) in resonance frequency due to the exchange and obtain ω_0,ES:

$${\omega }_{0,ES}={\omega }_{0,OBS}-\Psi +\Delta \omega$$ (70)

6.4.3. Obtaining kinetic and thermodynamic parameters

Kinetic and thermodynamic parameters that provide important insights into the energetics of the ES and transition state (TS) can be derived from the exchange parameters[15, 17]. The free energy difference between the ES and GS (ΔG_ES-GS) at a given temperature can be obtained from the measured p_ES and p_GS using the following equation:

$$\Delta G_{ES-GS}(T)=-\text{R}Tln\left(\frac{p_{ES}}{p_{GS}}\right)$$ (71)

The activation barrier for the forward $\left(\Delta G_1^{‡}(T)\right)$ or reverse $\left(\Delta G_{-1}^{‡}(T)\right)$ reaction and the corresponding enthalpies ($\Delta H_1^{‡}$ and $\Delta H_{-1}^{‡}$) can be obtained from temperature-dependent R_1ρ measurements using the following equation derived by Russu et al.[232] that is free of statistical compensation effects and is valid if the specific heat capacity can be assumed to be constant:

$$\text{ln}\left(\frac{k_i(T)}{T}\right)=ln\left(\frac{k_B\kappa }{h}\right)-\frac{\Delta G_i^{‡}\left(T_{hm}\right)}{R{T}_{hm}}-\frac{\Delta H_i^{‡}}{R}\left(\frac{1}{T}-\frac{1}{T_{hm}}\right)$$ (72)

$$T_{hm}=n{\left({\sum }_{i=1}^n\left(\frac{1}{T_i}\right)\right)}^{-1}$$ (73)

where i = 1 or −1 and T_hm is the harmonic mean of the experimental temperatures used. The transmission coefficient κ is often assumed to be equal to 1. This assumes that the frequency of crossing the activation barrier can be approximated by $\frac{k_{B}T}{h}$, which is valid for the case of a bond vibration. However, as noted by Kay et al., this frequency may be different in the context of conformational changes in biomolecules[233]. Indeed, taking this into account by adjusting the value of κ to be different from 1 can have a large impact on the obtained free energies of activation. For example, assuming κ = 3.3 * 10⁻⁶ as determined by Kay et al. for transitions involving formation of ESs in proteins[233] can change the free energies of activation for Watson-Crick to Hoogsteen transitions by as much as 7 kcal/mol (or 29 kJ/mol) (Fig. 33). In contrast, the enthalpies for the forward and backward reactions do not depend on κ and therefore can be reliably extracted from temperature-dependent rate constants obtained from RD measurements. The free energies and enthalpies obtained as described above can be used to obtain the entropy ($\Delta S_1^{‡}$ and $\Delta S_{-1}^{‡}$) of activation for the forward and reverse reactions:

$$T_{hm}\Delta S_i^{‡}=\Delta H_i^{‡}-\Delta G_i^{‡}\left(T_{hm}\right)\#$$ (74)

While performing R_1ρ RD measurements as a function of temperature, accurate estimates for the sample temperature can be obtained by recording spectra of 99.8% methanol-d₄ (Cambridge Isotope Laboratories, Inc.) as described by Berger et al.[234]. The enthalpy (ΔH_ES–GS) and entropy (ΔS_ES–GS) change for the GS-ES transition can also be calculated as follows:

$$\Delta H_{ES-GS}=\Delta H_{-1}^{‡}-\Delta H_1^{‡}\#$$ (75)

$$\Delta S_{ES-GS}=\Delta S_{-1}^{‡}-\Delta S_1^{‡}\#$$ (76)

Figure 33.

Thermodynamics and kinetics of GS-ES exchange characterized using temperature dependent R_1ρ RD measurements. Free energy diagram (1 kcal/mol = 4.184 kJ/mol) for the Watson-Crick to Hoogsteen transition (κ = 1, left and κ = 3.3 * 10⁻⁶, right) for A16-T9 in A₆-DNA at pH 5.4, 25 °C obtained using off-resonance R_1ρ RD, as reported by Al-Hashimi et al.[17]. The secondary structure of A₆-DNA is shown in inset.

The temperature dependence from a van’t Hoff analysis can also be used to discriminate between GS versus AVG alignment during B-M fitting[235], in cases where the initial magnetization alignment is ambiguous, and both schemes fit the data equally well. A van’t Hoff plot of ln(k₁/T) (or k_-1) versus (1/T – 1/T_hm)(K⁻¹) (Eqn. 72 and 73) is expected to exhibit linearity if the correct magnetization alignment scheme is assumed, provided that it does not change as a function of temperature. Such an analysis was used to select AVG over GS alignment while fitting RD profiles by Al-Hashimi et al.[235].

In addition to obtaining insights into the thermodynamics of transition states and their kinetics, RD methods can also be used for structural characterization of the TS via ϕ value analysis[115, 236] or measurement of kinetic isotope effects for exchange processes[237].

7. Determining structures of excited states in nucleic acids

The biological significance of an exchange process cannot be fully appreciated without determining the structure of the ES. RD experiments provide structural information regarding the ES in the form of chemical shifts. Here, we review how chemical shifts, combined with methods for stabilizing or destabilizing ESs using mutagenesis and chemical modifications, as well as computational approaches for predicting chemical shifts and nucleic acid structures, can be used to determine structures for nucleic acid ESs at various degrees of resolution. These approaches are generally applicable, and can be used in conjunction with CPMG and CEST experiments as well.

7.1. Inferring base pairing and secondary structure from chemical shift fingerprints

Most if not all ESs in nucleic acids characterized to date involve changes in base pairing or secondary structure. The need to break one or two BPs during the transition explains why they occur on the slow micro-to-millisecond timescale. The set of chemical shifts measured for a given ES constitutes a ‘fingerprint’, which can be used to infer various aspects of its base pairing and secondary structure. Key to this approach are recent improvements in our understanding of chemical shift-structure relationships in nucleic acids (reviewed in Chi et al.[238]). The ¹³C chemical shifts primarily depend on the base and sugar conformation[239–248], whereas the ¹⁵N chemical shifts primarily report on base pairing and hydrogen bonding[240, 249–255], respectively. Below, we review how these dependencies of chemical shifts on structure give rise to chemical shift fingerprints that can aid structural elucidation of nucleic acid ESs.

Nucleic acids are primarily composed of canonical Watson-Crick A-T/U and G-C BPs in A-form RNA and B-form DNA helices, respectively. G-C and A-T/U Watson-Crick BPs that are sandwiched between other Watson-Crick BPs have characteristic ¹³C and ¹⁵N chemical shifts that fall within a narrow spectral region[9], owing to their well defined and regular geometry characterized by canonical values for the backbone torsion angles (anti χ, α~300°, β~180°, γ~60°, ε~210°, ζ~270°) and sugar pucker (C3′-endo in RNA & primarily C2′-endo in DNA) (Fig. 34A). These chemical shift fingerprints can readily be used to infer whether transitions toward an ES entail creation or loss of Watson-Crick BPs.

Figure 34.

Chemical shift/structure relationships in nucleic acids can aid structural elucidation of ESs. A) Scatter plot of ¹³C chemical shifts obtained from the BMRB for different secondary structure contexts in RNA (Watson-Crick BPs in A-form helices: black; nucleotides in bulges, internal or apical loops: gold; syn bases: blue). B) Distribution of ¹H and ¹⁵N chemical shifts for U-N3 and G-N1 as a function of the BP type (G-C Watson-Crick BPs: red; U-A Watson-Crick BPs: blue; G-U wobbles: purple; U-G and U-U wobbles: green; sheared G-A mispairs: orange, with the base whose shifts are shown being colored), as obtained from the BMRB. C) Correlation between the C1′ and C4′ chemical shifts in RNA as obtained from a survey of the BMRB for nucleotides in different structural contexts (helical: blue; non-helical: orange; flanking,i.e., helical BP neighboring loops or bulges: green). Black lines correspond to the upper boundaries for C3′-endo sugar chemical shifts as obtained based on the average and standard deviation of the chemical shifts for helical nucleotides.(A) and (B) were reprinted with permission from [9] while (C) was reproduced with permission from [166].

On the other hand, nucleotides not involved in Watson-Crick pairing have chemical shifts that differ from canonical Watson-Crick BPs for at least a subset of ¹³C and ¹⁵N spins. For example, Hoogsteen BPs such as G(syn)-C⁺(anti), A(syn)-T/U(anti), and G(syn)-G(anti), G(syn)A⁺(anti) and G(anti)-A(syn) have $\Delta {\overline{\omega }}_{\text{C}1}$ and $\Delta {\overline{\omega }}_{\text{C}8}$ values for the syn nucleotide, that are shifted downfield by 1 – 4 ppm relative to Watson-Crick BPs[17, 235]. Furthermore, many mismatches such as A(anti)⁺-C(anti), A(anti)⁺-G(syn), G(syn)-C⁺(anti), and C(anti)-C(anti)⁺ have protonated bases with $\Delta {\overline{\omega }}_{\text{A}-\text{C}2}~-8\text{\hspace{0.17em}ppm}$, $\Delta {\overline{\omega }}_{\text{A}-\text{C}8}~-4\text{\hspace{0.17em}ppm}$, $\Delta {\overline{\omega }}_{\text{C}-\text{C}6}~-2.5\text{\hspace{0.17em}ppm}$, and $\Delta {\overline{\omega }}_{\text{C}-\text{N}4}~-10\text{\hspace{0.17em}ppm}$ relative to their non-protonated counterparts[17, 235, 253, 256]. Rare tautomeric and anionic forms of the bases which occur for example in Watson-Crick-like mismatches such as G^enol-T/U, GT^enol/U^enol and G-T⁻/U⁻[257], feature $\Delta {\overline{\omega }}_{\text{G}-\text{N}1}$ and $\Delta {\overline{\omega }}_{\text{T/U}-\text{N}3}~50\text{\hspace{0.17em}ppm}$, and $\Delta {\overline{\omega }}_{\text{G}-\text{C}8}$ and $\Delta {\overline{\omega }}_{\text{T/U}-\text{C}6}~2\text{\hspace{0.17em}ppm}$ relative to bases in their canonical tautomeric and ionic forms[165, 195, 197]. The imino ¹⁵N chemical shifts are also sensitive to changes in base pairing, with upfield shifts in T/UN3 and G-N1 occurring upon transitions of Watson-Crick A-T/U or G-C BPs to mismatches such as T/U-G, T/U-T/U and G-T/U[9] (Fig. 34B). Unpaired nucleotides in DNA and RNA also have characteristic chemical shifts relative to their duplex counterparts, particularly for the sugar ¹³C (C1′, C3′, and C4′) due to changes in sugar pucker[166, 239, 244–247] away from canonical C2′endo and C3′-endo geometries respectively (see below) as well as 1D6D8 angles. This may also be accompanied by changes in the base ¹³C chemical shifts (C6/C8) due to changes in stacking and χ angle[239–243].

In general, C3′-endo sugars in RNA helices have downfield shifted ω_C1′ and upfield shifted ω_C4′ relative to C2′-endo sugars, which typically occur in non-helical residues such as in bulges, and which tend to have upfield shifted ω_C1′ and downfield shifted ω_C4′[166, 240, 242, 244, 247] (Fig. 34C). In RNA, changes in sugar pucker are frequently accompanied by coupled changes in χ angle[258] that also influence ω_C1′. In some cases, these changes can be compensatory leading to small changes in ω_C1′, and consequently, to a lack of observable RD[166]. This shows how the absence of detectable RD for a particular spin cannot be taken as evidence for the absence of chemical exchange. This also underscores the importance of characterizing the conformation of the ES using as many chemical shift probes as possible, in addition to highlighting potential pitfalls of inferring structural changes based on a limited set of RD measurements. In DNA, C3′-endo sugars have $\Delta {\overline{\omega }}_{\text{C}3\text{′}}~-7\text{\hspace{0.17em}ppm}$ and $\Delta {\overline{\omega }}_{\text{C}4\text{′}}~-2.5\text{\hspace{0.17em}ppm}$ relative to C2′-endo sugars[239, 240, 245, 246].

R_1ρ(¹H) experiments[222] make it possible to measure ¹H chemical shifts that provide information regarding the ES structure that is complementary to ¹³C and ¹⁵N chemical shifts. Proton chemical shifts are sensitive to ring-currents and magnetic anisotropy effects and therefore stacking and χ angles[238]. When combined with empirical tools to predict ¹H chemical shifts based on structure[259–261] and sequence[262, 263], it is likely that proton chemical shifts will be extensively used in the future to help determine the structures of nucleic acid ESs.

Based on the above chemical shift-structure relationships, chemical shift fingerprints (Fig. 35A) can be used to obtain insights into structural features of ESs, and guide the development of structural models for the ES, which can range from localized changes in pairing at an individual BP, to more concerted changes in base pairing that lead to changes in secondary structure as is often observed in RNA[16, 264]. Inferring possible structures for such RNA ESs is greatly aided by secondary structure prediction programs such as Mfold[265] and MC-Fold[266], which can be used to generate a list of putative ES secondary structures ranked according to their predicted free energies[16, 264]. While the lowest energy structures typically correspond to the GS, RNA ESs frequently appear as higher energy structures that form through reshuffling of at least one BP. These putative ES structures are examined for their ability to explain the observed chemical shift fingerprints at sites where conformational changes occur, as well as the absence of RD at sites where little or no conformational change is expected.

Figure 35.

Probing base pairing in nucleic acid ESs using mutations or chemical modifications designed to stabilize the ES relative to the GS. A) Secondary structure of HIV-1 TAR RNA with the $\Delta \overline{\omega }$ values obtained from off-resonance R_1ρ RD experiments[16, 166, 264], color-coded for ES1 and ES2. White circles correspond to sites where no RD was observed. B) Kinetic network of chemical exchange for HIV-1 TAR with ES1 and ES2[16, 264]. Regions undergoing structural changes when transforming into the ES are indicated in red. The exchange rates and populations shown were obtained using NMR RD measurements[16, 264]. Mutations used for stabilizing the ESs are indicated in green. C) Secondary structure of the G28U mutant of HIV-1 TAR in the GS and ES2 conformations. The mutation site is highlighted in red. D)Correlation between $\Delta {\overline{\omega }}_{\text{mutant}}$, the chemical shift difference between the indicated mutant and unmodified HIV-1 TAR and $\Delta {\overline{\omega }}_{\text{RD}}$, the change in chemical shift obtained using RD measurements on unmodified HIV-1 TAR, monitoring the GS to ES2 exchange[16, 264]. Vertical line of points at $\Delta {\overline{\omega }}_{\text{RD}}=0$ correspond to resonances where RD was flat and $\Delta {\overline{\omega }}_{\text{RD}}$ was assumed to be equal to zero. E) m¹A and m¹G disrupt Watson-Crick base pairing and create an energetic bias towards the formation of Hoogsteen BPs in DNA. In panels (B) and (C) Watson-Crick BPs and G-U mismatches are indicated using thick lines while thin lines correspond to all other mismatches.

7.2. Using chemical and environmental perturbations to determine the structure and function of excited states in nucleic acids

Once generated, a putative ES model is subjected to experimental tests. This involves testing predictions about the impact of mutations, modifications or changes in experimental conditions that bias the system toward or away from the ES. These predictions are then tested experimentally using chemical shift or RD measurements. As ESs are typically destabilized relative to the GS by < 4 kcal/mol or 17 kJ/mol, it often proves feasible to invert the GS-ES equilibrium and achieve p_ES > 50 % through single nucleotide or atom substitutions[16, 31, 116, 264, 267, 268]. This can be achieved by stabilizing the ES, destabilizing the GS, or both.

7.2.1. Stabilizing the excited state relative to the ground state by mutations or chemical modifications

If a putative ES model is accurate, one would predict that a mutation or modification stabilizing an ES would cause changes in chemical shifts throughout the RNA at sites with significant RD, resulting in chemical shifts for the mutant that are similar to those measured for the ES using RD measurements in the parent wild-type RNA. Equally important, the mutation/modification should minimally induce changes in chemical shifts for sites that do not show any RD. In addition to verifying the putative ES model, mutations/modifications provide a means by which to verify whether or not an exchange process involving multiple spins is indeed concerted. This is important because similar p_ES and k_ex amongst various spins cannot be taken as definitive evidence for a concerted exchange process as various segments in a molecule may undergo transitions independently, yet have similar exchange parameters by chance. In the latter case, one would expect that mutations/modifications would lead to local stabilization of a given segment without affecting other segments. On the other hand, for a cooperative process, the mutation/modification would affect spins in multiple segments. In this manner, mutations and modifications can provide insights into the cooperativity of conformational transitions.

A general strategy for stabilizing the ES relative to the GS is to convert non-canonical mismatches unique to the ES into more stable canonical Watson-Crick BPs while simultaneously converting canonical Watson-Crick BPs in the GS into non-canonical mismatches[9, 16, 264]. For example, in HIV-1 TAR, an ES could be stabilized by a G28U point substitution mutation, which replaces the G28-A35 BP in the ES with a more stable U28-A35 BP while at the same time replacing the G28-C37 BP in the GS with a less stable U28-C37 BP[264] (Fig. 35B, 35C). 2D NOESY-based distances confirm that the G28U mutant folds into the predicted ES2-like secondary structure[264]. The chemical shifts measured for the G28U mutant were found to be in good agreement with those measured for ES2 in HIV-1 TAR using R_1ρ (Fig. 35D). In addition, only small differences in chemical shifts were observed for several sites that do not show any RD (e.g. U42-N3 and A22-C8)[264]. The fact that a single point substitution mutation in the upper helix changed the conformation of remotely positioned nucleotides in the bulge such as U23, provides independent support for a concerted transition, in which all BPs are reshuffled in going from the GS to ES2.

ES structures can also be stabilized using commercially available chemical modifications. For example, the definitive confirmation that a Hoogsteen BPs in duplex DNA was the ES detected by R_1ρ came with the use of N¹-methyl adenosine (m¹A) and N¹-methyl guanosine (m¹G) modifications, to induce Hoogsteen BPs at specific sites within DNA duplexes[17, 116]. m¹A and m¹G sterically block Watson-Crick type hydrogen-bonding and knock out one hydrogen bond in the Watson-Crick BP without significantly affecting the stability of the Hoogsteen BP (Fig. 35E). The modified nucleotides formed the expected Hoogsteen BPs as verified by 2D NOESY without affecting Watson-Crick base pairing at neighboring sites, which is expected if the Watson-Crick to Hoogsteen transitions occur independently of one another[17, 116]. The chemical shifts of the m¹A and m¹G modified duplexes were in excellent agreement with those measured by RD on the C1′ and C6/C8 atoms, and Density Functional Theory (DFT) calculations[17]. Moreover, the m¹A and m¹G mutants revealed that other resonances should experience large chemical shift perturbations due to formation of Hoogsteen BPs, including upfield shifted ¹⁵N signals in imino guanine-N1 and thymine-N3, due to changes in hydrogen bonding[112], and changes in sugar ¹³C signals (adenine and guanine C3′ and C4′, and thymine and cytosine C3′) indicative of changes in sugar pucker[117]. These predictions were confirmed in subsequent studies in which R_1ρ RD were used to measure the ES chemical shifts at these new sites[112, 117]. Once again, no RD was observed for ¹³C signals (e.g. thymine and cytosine C4′) showing small chemical shift perturbations in the m¹A and m¹G mutants[117].

An ES stabilizing mutant is not expected to reproduce all ES chemical shifts, especially for residues near the mutation site, and in many cases more than one mutant will be required to verify different aspects of the ES structure[16, 264]. Returning to the G28U mutant mimic of ES2 in TAR, differences as large as 1.5 ppm for C1′ and N1 were observed near the site of the mutation[264] (Fig. 35D). As this mutant replaces tandem G-A mismatches in ES2 with a single G-A mismatch and a Watson-Crick BP, it is likely to alter the structure around these sites. Much better agreement for the G-A mismatch chemical shifts was obtained with a second mutant (ES2^uucg) that retains the tandem G-A mismatches[269] (Fig. 35B, D). Likewise, in the case of ES Hoogsteen BPs, the m¹A modification introduces a positive charge to the base, and consequently, the m¹A-C8 chemical shift was downfield-shifted more significantly than in the ES Hoogsteen BP[17, 116]. The A-C8 chemical shifts measured for an A-T Hoogsteen BPs stabilized using a drug were found to be in better agreement with those measured by RD[17, 270].

7.2.2. Destabilizing the excited state relative to the ground state by mutations or chemical modifications

Putative ES models can also be tested by introducing mutations or modifications that are predicted to destabilize the ES relative to the GS, reducing p_ES, or even making the ES invisible to RD detection. Returning to the TAR example, ES2 could be destabilized by replacing the wild-type apical loop with a UUCG apical loop (Fig. 36A). This mutation quenched RD at all sites sensitive to ES2 including sites within the bulge that are distant from the mutation site[264].

Figure 36.

Probing the structure of nucleic acid ESs using mutations or chemical modifications designed to stabilize the GS relative to the ES. A) Replacement of the HIV-1 TAR apical loop with a UUCG tetraloop destabilizes ES2[264]. B) N^6,6-dimethyl adenine (m^6,6A) disrupts base pairing of adenine via the Watson-Crick face in RNA, thereby shifting the GS-ES1 equilibrium in HIV-1 TAR towards the GS[16], in which A35 is unpaired. C) N7-deaza guanine and adenine disrupt Hoogsteen hydrogen bonding and bias A-T and G-C BPs towards the native Watson-Crick conformation[112]. D) Inosine disrupts hydrogen bonding involving the Guanine amino group thereby destabilizing Watson-Crick-like mismatched BPs[165]. In all cases, modifications are highlighted in red. In panels (A) and (B) Watson-Crick BPs and G-U mismatches in the secondary structures are indicated using thick lines while thin lines correspond to all other mismatches.

The chemical modification N⁶,N⁶-dimethyl substituted adenine (m^6,6A), which blocks interactions of adenine at the Watson-Crick face was used to disrupt an A-C mismatch that forms in TAR ES1 (Fig. 35B), favoring instead the conformation where adenine bulges out as observed in the GS[16] (Fig. 36B). As ES1 is in fast exchange and is highly populated (p_ES = 13 %), the m^6,6A mutant showed perturbations in the chemical shifts towards the GS at sites showing RD for ES1[16].

Destabilizing a putative ES structure also provides a powerful strategy to test hydrogen bonding that is unique to the ES. For example, if Hoogsteen BPs in DNA were indeed stabilized by hydrogen bonds involving purine-N7, replacement of N7 with C7 (N7-deaza-purines, c7A and c7G) should significantly destabilize the BP[112] (Fig. 36C). Indeed, N7-deaza-purines quenched Watson-Crick to Hoogsteen RD in DNA selectively at the modified BP without affecting Watson-Crick to Hoogsteen exchange at neighboring sites[112]. As a second example, inosine was used to show that ES Watson-Crick-like G-T mismatches were hydrogen bonded in a Watson-Crick-like fashion[165] (Fig. 36D). Removal of the exocyclic amino group was predicted to destabilize ES Watson-Crick-like mismatches by 1–2 kcal/mol or 4–8 kJ/mol, pushing them beyond the limit of detection of R_1ρ RD experiments, as was verified experimentally[165].

7.2.3. Stabilizing the ground or excited states by changing experimental conditions

Changing pH or temperature provides an alternative means for testing putative ES models. Many ESs feature the formation of BPs that are stabilized by base protonation or deprotonation such as A⁺(anti)-C(anti), A⁺(anti)-G(syn), and G(syn)-C⁺(anti), while examples of deprotonated BPs include Watson-Crick-like G-T⁻/U⁻ mismatches. Insights into the appearance or loss of such BPs upon formation of an ES can be obtained from pH-dependent RD measurements. For example, in the case of TAR ES1, decreasing the pH increased the population of ES1, supporting the interpretation that ES1 contained a protonated A⁺-C BP[16]. The pH dependence of RD measurements was also used to resolve anionic Watson-Crick-like G-T⁻ (in DNA) and G-U⁻ (in RNA) mismatches from their tautomeric counterparts (G^enol-T/U and G-T^enol/U^enol) based on the observed increase in p_ES with increasing pH (Section 8.2). In contrast, the population of the tautomeric ES (G^enol-T/U and G-T^enol/U^enol) was found to be independent of pH, consistent with a tautomeric species. Changes in pH were also used to stabilize the structure of an ES of U6RNA[135] and the Murine Leukemia Virus pseudoknot[267], both of which involved the formation of A-C⁺ BPs.

Changing the temperature can also help to resolve RD contributions from two distinct exchange processes. For example, in the case of HIV-1 TAR, the R_ex contribution from a fast exchange process involving ES1 could be diminished by increasing the temperature, allowing better resolution of a second and slower exchange process involving ES2[264].

Lastly, for the case of exchange scenarios where one is monitoring binding equilibria, it is feasible to bias the equilibrium towards the bound or free conformation by altering the concentrations of the binding partners, so as to obtain their chemical shifts[32, 78, 79].

7.3. Solving high-resolution structures and dynamic ensembles for excited states in nucleic acids

Once validated, high resolution structures or dynamic ensembles can be obtained for a given ES-mutant and be used as high-resolution models for the ES conformation[117]. While this is a potentially general strategy for determining structures of ESs in nucleic acids, a number of precautions have to be taken. First, care must be taken to validate the mutant that stabilizes the ES using a large number of R_1ρ measurements on a variety of spins. Second, the structure or dynamic ensemble of the ES mutant determined using NOEs and/or RDCs should be further validated for its ability to independently predict additional experimental data such as the ES chemical shifts.

The above approach has been applied to determine the first atomic resolution structure of a nucleic acid ES, in particular Hoogsteen BPs in two different DNA duplexes, using m¹A to stabilize the Hoogsteen conformation[117] (Fig. 35E). The m¹A mutant was validated based on the excellent agreement observed between its chemical shifts and those measured for the ES using RD for a variety of base and sugar spins (Fig. 37A). Dynamic structural ensembles (Fig. 37C) were determined for the m¹A-T mutants and the corresponding unmodified duplexes[116] using RDCs and computational molecular dynamics simulations[271–274]. An Automated Fragmentation Quantum Mechanics/Molecular mechanics approach[275] was then used to predict chemical shifts for every conformer in the ensembles of the m¹A-T mutant as well as the unmodified DNA[117]. The computed chemical shifts were averaged over the ensembles and the differences between the GS and ES chemical shifts thus computed (Δω_Ens) were compared to values obtained using RD (Δω_RD). Excellent agreement was observed between the Δω_Ens and Δω_RD values (Fig. 37B), thus validating the generated ensembles. This also made it possible to rationalize the unique chemical shift signatures associated with the ES in terms of structural features[117].

Figure 37.

High-resolution structural characterization of ES Hoogsteen BPs in B-DNA.A) Validation of the ES mutant (m¹A) using an extensive set of chemical shifts. Shown is a representative example of A₆-DNA (secondary structure on the left), with the site of m¹A16 modification to trap Hoogsteen BPs indicated in red. Correlation between $\Delta {\overline{\omega }}_{\text{m}1\text{A}-\text{A}}$ (the chemical shift difference between the Hoogsteen-stabilizing A₆-DNA^m1A16 and unmodified A₆-DNA duplex), and $\Delta {\overline{\omega }}_{\text{RD}}$ values obtained from R_1ρ RD measurements on unmodified A₆-DNA, monitoring the Watson-Crick to Hoogsteen exchange at A16. B) Correlation between $\Delta {\overline{\omega }}_{\text{Ens}}$, the calculated chemical shift difference between the ensembles of the A₆-DNA^m1A16 and A₆-DNA duplexes, and $\Delta {\overline{\omega }}_{\text{RD}}$ (C) Dynamic ensembles for A₆-DNA (green, left) and A₆-DNA^m1A16 (red, right) obtained as described by Al-Hashimi et al.[116]. The average inter-helical kink angle β, obtained from an Euler angle analysis[296, 297] is also indicated. In panels A and B, two RMSD and R values are shown. The first value is calculated with the inclusion of all the resonances for which RD was measured, while the second value is calculated excluding data with flat RD where $\Delta {\overline{\omega }}_{\text{RD}}$ was assumed to be zero. Reprinted by permission from [117].

The above strategy can in principle be extended to RNA, for which there is a growing number of tools for predicting ¹³C/¹⁵N chemical shifts based on 3D structures[259, 276]. In addition, even prior to the above study, methods had been developed to solve ES structures of proteins that involve measuring direct structural restraints, such as residual dipolar coupings (RDCs)[277–279], residual chemical shift anisotropies (RCSAs)[280], as well as paramagnetic relaxation enhancements (PREs)[281]. Building on these applications, CEST approaches have recently been developed to measure RDCs for nucleic acid ESs[170]. These approaches can readily be extended in the future to measure ES RCSAs in nucleic acids[282, 283]. These new types of data could be used in the future in conjunction with the mutation strategy described above to robustly determine structures and dynamic ensembles for nucleic acid ESs.

7.4. Using mutations to assess the biological functions of excited states in nucleic acids

Importantly, GS or ES stabilizing mutants can also be used in functional assays to assess the biological importance of the ES[16, 284]. For example, a mutation that destabilizes an ES in the HIV-1 Stem Loop 1 (SL1) was shown to impede duplex dimerization, implying that the ES plays a role for nucleating the transition[16]. More recently, ES-mutants of HIV-1 TAR were used in cell-based experiments to show that the non-native conformation disrupts transactivation[284]. Key to this assessment was the use of subsequent “rescue mutations” that push the ES-GS equilibrium to favor the GS; such mutations reinforce the fact that changes in conformation, rather than changes in sequence due to the mutations themelves, are responsible for the observed biological response.

8. Examples of applications of 1D selective R_1ρ relaxation dispersion for the analysis of chemical exchange in nucleic acids

In this section, we review examples of applications of R_1ρ RD to the study of chemical exchange in nucleic acids that illustrate the different types of motions characterized to date and their biological implications.

8.1. Watson-Crick to Hoogsteen exchange in DNA and RNA duplexes

As stated throughout this review, application of off-resonance R_1ρ RD methodology resulted in the discovery that in the DNA double-helix, A-T and G-C Watson-Crick BPs continually undergo exchange with their Hoogsteen counterparts. Based on these studies, at any given point in time, ~1% and ~0.1% of A-T and G-C BPs in the DNA double helix are Hoogsteen rather than Watson-Crick BPs. This implies that at any given point in time, there could potentially be tens of millions of Hoogsteen BPs in the human genome. In vivo, DNA experiences supercoiling, torsional stress, under-winding, and stretching as it wraps around histones to form nucleosomes. Hoogsteen BPs could provide mechanisms for stabilizing such stressed structures of highly compacted DNA. Furthermore, in addition to the currently known proteins such as p53 that bind to Hoogsteen BPs, other proteins may also promote or recognize Hoogsteen BPs when binding to DNA[114]. Indeed, drugs have been shown to bind duplex DNA and to induce dramatic changes in the Hoogsteen breathing dynamics[270]. We are today still in the early stages of examining the occurrence and biological significance of Hoogsteen BPs in DNA.

In sharp contrast to B-DNA, transient Hoogsteen BPs were not detected in A-RNA duplexes using R_1ρ RD[235]. It was later shown that Hoogsteen BPs are less stable in A-RNA because syn purine bases are sterically disfavored in the A-form conformation, and because there is a greater energetic cost associated with constricting the backbone to form Hoogsteen type hydrogen bonds in A-RNA relative to that in B-DNA[285]. The markedly different propensities to accommodate Hoogsteen BPs in RNA and DNA duplexes is another important difference between the two nucleic acid polymers, in addition to the presence or absence of the 2′-hydroxyl group. The ability to absorb chemical damage at the Watson-Crick face by forming Hoogsteen BPs makes DNA a better custodian of genetic information and may represent one of the evolutionary pressures that resulted in B-DNA becoming the main carrier of genetic information in higher-order organisms[235]. On the other hand, the inability of A-RNA to accommodate Hoogsteen BPs explains how post-transcriptional modifications such as m¹A and m¹G can markedly influence RNA folding and function[235].

8.2. Exchange between wobble and Watson-Crick-like tautomeric and anionic G-T/U base pairs in DNA and RNA duplexes

Watson-Crick to Hoogsteen exchange converts canonical Watson-Crick G-C and A-T BPs into non-canonical Hoogsteen conformations. R_1ρ studies have also uncovered an inverse process in which non-canonical G-T/U wobble mismatches are converted into canonical Watson-Crick-like BPs stabilized by rare tautomeric and anionic forms of the bases[165, 195, 197] (Fig. 38A). In contrast to Hoogsteen transitions, this exchange process occurs in both BDNA and A-RNA duplexes. By adopting conformations similar to Watson-Crick BPs, these ESs can evade fidelity checkpoints and contribute to errors during replication, transcription, and translation[286, 287]. These rare tautomeric bases were originally proposed as a potential source of replication errors by Watson and Crick in 1954, who outlined analogous tautomeric BPs in the case of the A-C mismatch[288].

Figure 38.

Watson-Crick-like G-T mismatches and their roles in generating spontaneous mutations during DNA replication. A) Kinetic topology for transforming between G-T/U wobble mismatches and their Watson-Crick-like counterparts formed via tautomerization or ionization. The exchange rates shown were obtained using NMR RD measurements[195, 197]. B) Minimal kinetic model for Watson-Crick BP incorporation by DNA polymerases. The incorporation of a G-T mismatch requires in addition, as shown in the lower pathway, a tautomerization or ionization step allowing for the formation of a Watson-Crick-like species.(Bottom left) Histograms showing F_pol, the fidelity of dTTP/^5BrdUTP mis-incorporation errors catalyzed by Alfalfa Mosaic Virus Reverse Transcriptase (AMV RT) as computed using kinetic simulations allowing incorporation of tautomeric (blue, M_ES1), ionic (green, M_ES2) or both (orange, M_ES1 + M_ES2) Watson-Crick-like excited states, and experimentally measured values (gray).(Bottom right) Measured and simulated rates (k_obs) of mis-incorporation for dTTP/^5BrdUTP catalyzed by human DNA polymerase β. Reprinted by permission from [195].

A kinetic model for the generation of errors during DNA replication constructed using the rate constants of the formation of Watson-Crick-like BPs measured by NMR could predict the frequency of G-T mis-incorporation across a wide range of pH, polymerases, and modified mutagenic bases[195] (Fig. 38B). Likewise, the uridine 5-oxyacetic acid (cmo⁵U) modification found in tRNA was recently shown to bias the G-cmo⁵U ensemble toward anionic Watson-Cricklike conformations, potentially explaining how the modification “reprograms” translation[289]. Taken together, these studies suggest that Watson-Crick-like ESs provide a general mechanism to explain the generation of errors and re-writing of information transfer in every step of the central dogma of molecular biology.

8.3. Exchange between alternative secondary structures in RNA

RNA molecules can form ESs that dramatically alter their secondary structure. Fig. 39 shows different types of ESs in RNA that have been characterized to date using off-resonance R_1ρ RD methods[16, 164]. Functions are beginning to emerge for many of these ESs in RNA. For example, as mentioned previously in the context of HIV-1 SL1[16], RNA ESs appear to provide means by which to break down large conformational transitions into multiple kinetically labile microscopic steps[290]. A similar mechanism has been proposed for an ES in P5abc, which is proposed to form an intermediate during tertiary folding[164]. RNA ESs characterized by other RD methods have also been shown to play important roles in gene regulation[31] and in ligand binding[32, 78, 79].

Figure 39.

RNA ESs feature localized changes in secondary structure. Shown are the secondary structures of ES determined for A) bacterial A-site, B) p5abc and C) HIV-SL1 using selective 1D R_1ρ RD measurements. The part of the ES undergoing structural changes on transforming from the GS is indicated in red. Thick lines correspond to Watson-Crick BPs and G-U mismatches, whereas thin lines denote other non-canonical mispairing. The populations of the states and the rate constants for GS-ES exchange, as obtained using R_1ρ RD measurements[9, 16] are also shown.

As they remodel structure and disrupt biological activity, RNA ESs are proposed as opportune targets for the development of RNA-targeted therapeutics[16]. In addition, there is mounting evidence that the mechanism underlying the disease phenotype of many single nucleotide polymorphisms (SNPs) involves changes in RNA ensembles by stabilizing alternative ES conformations[291 – 293]. Future studies of ESs in RNA will undoubtedly clarify their potential roles in biology and disease.

9. Conclusions

Since the earliest studies of chemical exchange in DNA in the late 1980s, the R_1ρ RD methodology has matured significantly during the past four decades to the point that it can now be routinely applied to characterize ESs in DNA and RNA. These advances coupled to development of CEST[28, 31] as well as CPMG[18, 136, 174, 294] methodology have opened the door to a world of previously unknown dynamics in DNA and RNA that play essential roles in gene expression and regulation. Central to these developments are experiments for accurately measuring RD data for a variety of spins in uniformly ¹³C/¹⁵N labeled samples over a range of spin-lock amplitudes, using mutations and chemical modifications to stabilize or destabilize ESs relative to the GS, an improved understanding of chemical-shift/structure relationships in nucleic acids, as well as advances in predicting nucleic acid chemical shifts based on 3D structure. Existing methods can immediately be applied to study the dynamics in higher-order DNA structures (e.g. quadruplexes, triplexes etc.)[295] and in RNAs ranging from simple singlestrands[199] to RNAs with complex tertiary folds[31, 164]. We can also anticipate advances that extend the methodology in the future to allow studies of dynamics in larger RNAs and DNAs and their complexes with proteins, and provide additional structural and dynamic information regarding ESs. We hope that this review will provide a useful resource for these future studies.

Highlights.

• Historical account of NMR RD with a focus on nucleic acid R_1ρ is presented

• A vector model is used to conceptually understand R_1ρ RD

• Approaches for fitting, simulating and visualizing R_1ρ data are described

• Application of R_1ρ RD to study complex three-state exchange scenarios are examined

• Strategies for determining structures of nucleic acid ESs are reviewed

Glossary

A

Adenine

AIC

Akaike Information Criterion

AVG

population-weighted average

BIC

Bayesian Information Criterion

B-M

Bloch-McConnell

BMNS

Bloch-McConnell Numerical Simulator

BPs

base pairs

BPTI

Bovine Pancreatic Trypsin Inhibitor

C

Cytosine

CEST

Chemical Exchange Saturation Transfer

cmo⁵U

Uridine 5-oxyacetic acid

CP

cross polarization

CPMG

Carr-Purcell-Meiboom-Gill

CSA

Chemical Shift Anisotropy

c7A

N7-deaza adenosine

c7G

N7-deaza guanosine

DD/CSA

Dipole-Dipole CSA cross-correlated relaxation

DD/DD

Dipole-dipole cross-correlated relaxation

DNA

Deoxyribonucleic acid

ES

Excited State

ES1/ES2

Excited State 1/Excited State 2

G

Guanine

GS

Ground State

HMQC

Heteronuclear Multiple Quantum Coherence

HSQC

Heteronuclear Single Quantum Coherence

I

Inosine

ms

millisecond

m¹A

N¹-methyl adenosine

m¹G

N¹-methyl guanosine

m^6,6A

N⁶,N⁶-dimethyl adenine

NMR

Nuclear Magnetic Resonance

NOE

Nuclear Overhauser Effect

ns

nanosecond

NTP

nucleotide tri-phosphate

OBS

observed

ps

picosecond

RCSA

Residual Chemical Shift Anisotropy

RD

Relaxation Dispersion

RDC

Residual Dipolar Coupling

RF

Radio Frequency

RNA

Ribonucleic acid

RSS

residual sum of squares

SH3

SRC Homology Domain 3

SL1

Stem Loop 1

T

Thymine

TROSY

Transverse Relaxation Optimized Spectroscopy

TS

Transition State

U

Uracil

ROESY

Rotating frame Overhauser Spectroscopy