positive sign of the ROE effect also allows one immediately ROESY to distinguish cross relaxation from chemical exchange effects, which are always negative. This latter application has proven 6,7 Ad Bax and Stephan Grzesiek critical, for example, in studying hydration of proteins. National Institutes of Health, Bethesda, MD, USA Other important applications lie in the area of macromolec- ular structure determination, where the rotating frame Over- hauser enhancement, owing to its positive sign, tends to be far less susceptible to spin diffusion effects than its longitudi- 1 Introduction 1 nal counterpart.8,9 Although for macromolecules, the inherent 2 Principles of Rotating Frame Overhauser Spectroscopy 1 sensitivity of the ROESY experiment is less than for NOESY, 3 Comparison With Laboratory Frame NOE 2 numerous useful applications to proteins and nucleic acids 4 Experimental Realization 4 have been reported. In addition to the nonselective ROESY 5 Minimization of J Contributions 5 experiment, a number of very interesting selective variants 6 Examples of ROESY Applications 7 have been developed that can selectively probe small sub- 7 Different Areas of Application 7 sets of the Redfield cross-relaxation matrix. These elegant 8 Related Articles 10 ROESY variants are particularly powerful for probing cross 9 References 10 correlation,10– 12 and they have been treated and reviewed in great detail by Bull.13 The discussion below is restricted to the more commonly used nonselective ROESY experiments and their various applications. 1 INTRODUCTION Molecular motion in liquids results in a time dependence 2 PRINCIPLES OF ROTATING FRAME of the dipolar coupling between two spins, which gives OVERHAUSER SPECTROSCOPY rise to transitions between the nuclear spin states. These transitions constitute the basis of the nuclear Overhauser The state of a spin system is most conveniently described by enhancement (NOE) effect, which has long been one of the the density matrix ρˆ. For a system of two spin- 1 nuclei, A and 1 2 NMR cornerstones for the study of molecular structure. As X, ρˆ isasimple4× 4 matrix. Its diagonal elements, ρ11, ρ22, discussed in NOESY, the transition rates are determined by ρ33,andρ44 correspond to the populations of energy levels −6 the square of the dipolar coupling, i.e., by r ,wherer is the 1, ... , 4. An off-diagonal element such as ρ12 represents internuclear distance, by the angular Larmor frequency ω0, the coherence or transition between levels 1 and 2, etc. In and by the decay constant τ c of the autocorrelation function order to describe relaxation phenomena, the Hamiltonian is of the time-dependent dipolar coupling D(t). In conventional ˆ usually split into a spin-dependent part HS and a spin–lattice homonuclear NOE experiments, the change in z magnetization coupling Hˆ . The spin-dependent component Hˆ consists of of a spin B is monitored after the z magnetization of a spin SL S the Zeeman coupling to the main external magnetic field (Hˆ ) A in its immediate vicinity has been perturbed. Provided τ 0 c plus other, much smaller, purely spin-dependent components is known, the rate at which the longitudinal magnetization Hˆ of spin B changes after perturbing the z magnetization of A ( 1) such as chemical shift, J coupling, and coupling of provides a measure of r.Forω τ < 1.12, the z magnetization the spins to radiofrequency (rf) fields. The spin–lattice part 0 c Hˆ of spin B will increase to a value higher than its Boltzmann SL describes the coupling of the spins to the external heat equilibrium value when saturating the resonance of A (positive bath, which is a stochastic process. Transformation into the Hˆ NOE), whereas the opposite holds for ω0τ c > 1.12. For many interaction frame makes the Zeeman term 0 disappear in interesting molecules of intermediate size (500–2000 Da), the explicit form from the Hamiltonian, and hides it as a time product ω0τ c is frequently found to be close to 1, resulting in dependence of the operators and the density matrix. In this very small NOE effects. interaction frame and under a number of approximations, the As outlined above and described in detail elsewhere,1 – 4 the time dependence of the density matrix ρˆ is then described by NOE is a cross-relaxation effect between longitudinal com- ponents of nuclear spin magnetization. In 1983, Bothner-By dρ(t)ˆ =−i[Hˆ , ρˆ] − Rˆρˆ (1) and co-workers developed a different type of NOE experi- dt 1 ment, which measures cross relaxation between magnetiza- tion components that are perpendicular to the static magnetic Rˆ is the Redfield relaxation matrix, which describes the field.5 The original name of this experiment is cross relaxation return to equilibrium of the spin system. The relation appropriate for minimolecules emulated by locked spins, or between its elements and the spectral densities is described CAMELSPIN, but it is more commonly known as ROESY, in detail in Relaxation Theory: Density Matrix Formu- for rotating-frame Overhauser enhancement spectroscopy. As lation. A product R1213ρ13, for example, describes the loss implied in the original acronym of this experiment, the trans- per unit time of the density matrix element ρ12 caused by verse cross relaxation is positive for all values of the rotational the matrix element ρ13. It is important to realize, however, correlation time, i.e., it is the same as that found for small that relaxation is usually a slow process relative to evolution molecules in the NOESY experiment. The original motivation under the static Hamiltonian. Consequently, cross relaxation for this experiment was to study the structure of intermedi- can only occur if the off-diagonal elements ρ12 and ρ13 ate size molecules, which have near-zero longitudinal NOE oscillate at nearly the same frequencies in the laboratory frame, effects. However, as pointed out in the original report,5 the i.e., if the corresponding resonances overlap. In this case, the Encyclopedia of Magnetic Resonance, Online © 2007 John Wiley & Sons, Ltd. This article is © 2007 John Wiley & Sons, Ltd. This article was published in the Encyclopedia of Magnetic Resonance in 2007 by John Wiley & Sons, Ltd. DOI: 10.1002/9780470034590.emrstm0473 2 ROESY cross relaxation between the 1–2 and the 1–3 transitions is (a) termed a secular process. If their frequencies are different, 0.65 the exchange of magnetization is a nonsecular process, and can be neglected. ROESY and its selective variants all rely on 0.60 ˆ modifying the spin Hamiltonian H1 in such a manner that cross max 0.55 relaxation can be probed between transitions that would differ in frequency in the absence of radiofrequency irradiation. In ROE 0.50 the standard ROESY experiment, this is accomplished simply by the application of a strong monochromatic rf field, which 0.45 locks all magnetization that is parallel to it and dephases any perpendicular components. Neglecting off-resonance effects, 0.40 the cross-relaxation rate for magnetization components locked along the rf field is therefore identical to the cross relaxation 0.1 1 10 between the transverse magnetization components of two w0tc exactly overlapping resonances (identical spins). 1.0 (b) Following Bothner-By et al.,5 the time dependence of the spin locked magnetization components mA and mB of two homonuclear spins A and B is described by 0.5 AB R dmA(t) =− − / AB R2AmA(t) RABmB(t) (2a) L dt R 0.0 dm (t) B =−R m (t) − R m (t) (2b) dt AB A 2B B where RAB is the cross-relaxation rate of interest, and R2A –0.5 and R2B are the autorelaxation rates, describing the loss of 0.1 1 10 magnetization, which also includes processes other than cross w0tc relaxation between spins A and B. For convenience, we assume below that R2A = R2B = R2, which results in a particularly Figure 1 (a) Maximum obtainable ROE enhancement ROEmax in simple solution of equations (2a) and (2b). If two experiments the isolated two-spin pair approximation, assuming that the spins are relaxed exclusively by homonuclear dipolar interactions. (b) The ratios are done, one with mA(0) = mB(0) = 1 and one with mA(0) of the cross relaxation rates in the laboratory frame, RL , and in the = −mB(0) = 1 (for example, by selectively inverting the AB magnetization of spin B immediately prior to a nonselective rotating frame, RAB, as functions of ω0τ c ◦ 90 x pulse, which is followed by a y-axis spin lock), the time dependence of mA(t)andmB(t) in these two cases is given The maximum Overhauser enhancement obtainable for this by isolated two-spin pair, using the ROE mixing time of equation (4), is shown graphically in Figure 1(a). In practice, the mA(t) = mB(t) = exp[−(R2 + RAB)t](3a)isolated spin pair approximation is usually not valid, and frequently R is much larger than R . In this case, it follows =− = − − 2 AB mA(t) mB(t) exp[ (R2 RAB)t] (3b) from equation (4) that the maximum ROE effect is obtained for The difference between equations (3a) and (3b) represents the ROE, which has a maximum value for t ≈ 1/R2 (8) 1 R + R t = ln 2 AB (4) which means that the highest sensitivity is obtained for a 2R R − R AB 2 AB mixing period on the order of the transverse relaxation time T 2, as measured, for example, by a selective spin echo experiment. In the isolated spin pair approximation, R2 and RAB are given by γ 4h2(µ /4π)2 3 COMPARISON WITH LABORATORY FRAME NOE R = 0 [5J(0) + 9J(ω ) + 6J(2ω )](5) 2 80π 2r6 0 0 In the isolated spin pair approximation and assuming rigid 4 2 2 body motion, comparison of the ROE and NOE results is γ h (µ0/4π) RAB = [4J(0) + 6J(ω0)](6)straightforward.