Andrew J. Baldwin & Lewis E. Kay

Transcription

1 An R 1ρ expression for a spin in chemical exchange between two sites with unequal transverse relaxation rates Andrew J. Baldwin & Lewis E. Kay Journal of Biomolecular NMR ISSN Volume Number J Biomol NMR (13) :11-18 DOI.7/s

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3 J Biomol NMR (13) :11 18 DOI.7/s ARTICLE An R 1q expression for a spin in chemical exchange between two sites with unequal transverse relaxation rates Andrew J. Baldwin Lewis E. Kay Received: 14 August 1 / Accepted: 6 December 1 / Published online: 3 January 13 Ó Springer Science+Business Media Dordrecht 13 Abstract An analytical expression is derived for the rotating frame relaxation rate, R 1q, of a spin exchanging between two sites with different transverse relaxation times. A number of limiting cases are examined, with the equation reducing to formulae derived previously under the assumption of equivalent relaxation rates at each site. The measurement of a pair off-resonance R 1q values, with the carrier displaced equally on either side of the observed correlation, forms the basis of one of the approaches for obtaining signs of chemical shift differences, Dx, of exchanging nuclei. The results presented here establish that this method is relatively insensitive to differential transverse relaxation rates between the exchaning states, greatly simplifying the calculation of optimal parameters in R 1q based experiments that are used for measurement of signs of Dx. Keywords Relaxation dispersion NMR Invisible excited states Protein conformational exchange Spinlock Rotating frame relaxation Differential transverse relaxation A. J. Baldwin (&) L. E. Kay Departments of Molecular Genetics, Biochemistry and Chemistry, University of Toronto, Toronto, ON MS 1A8, Canada andrew.baldwin@chem.ox.ac.uk Present Address: A. J. Baldwin Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, UK L. E. Kay Hospital for Sick Children, Program in Molecular Structure and Function, University Avenue, Toronto, ON MG 1X8, Canada Introduction The structures spontaneously adopted by proteins dictate their function. The interactions that stabilize these structures are individually weak and at biologically relevant temperatures are easily rearranged by thermal motion. Consequently, biological molecules are inherently dynamic (Karplus and Kuriyan ) and are best understood as ensembles of interconverting conformers rather than in terms of single static structures (Lange et al. 8; Lindorff-Larsen et al. ). Moreover, it is increasingly apparent that conformational fluctuations play crucial roles in enabling the functions of protein molecules (Karplus and Kuriyan ). Observing and characterizing such motions however remains challenging. Solution NMR spectroscopy is particularly well suited to characterizing such molecular dynamics over a broad spectrum of time-scales (Ishima and Torchia ; Mittermaier and Kay 6; Palmer et al. 1996). Among the different methodologies, paramagnetic relaxation enhancement (Iwahara and Clore 6), R 1q (Palmer and Massi 6) and Carr-Purcell-Meiboom-Gill (CPMG) (Hansen et al. 8; Korzhnev and Kay 8; Palmer et al. 1) experiments are particularly useful for the study of exchange processes involving the inter-conversion between dominant (ground) and sparsely populated (conformationally excited ) states. Insight into biological processes as diverse as molecular recognition, ligand binding, enzyme catalysis and protein folding, has been achieved using these techniques (Boehr et al. 6; Fraser et al. 9; Henzler-Wildman and Kern 7; Ishima et al. 1999; Korzhnev et al. 4, ; Sugase et al. 7; Tang et al. 7). A rich theoretical basis for understanding PRE, R 1q and CPMG-based experiments as they pertain to systems undergoing chemical exchange has emerged over many years. Notably, analytical expressions exist that describe the

4 1 J Biomol NMR (13) :11 18 effective R 1q or R relaxation rate as a function of the exchange parameters, radio frequency (RF) field strengths and shift offsets (Allerhand et al. 1966; Carver and Richards 197; Deverell et al. 197; Jen 1978; Luz and Meiboom 1963; Miloushev and Palmer ; Palmer et al. 1, ; Palmer and Massi 6; Trott and Palmer ). While explicit use of these equations in fits of experimental data has now given way to numerical approaches involving fast computers, analytical expressions remain useful for providing physical insight that various parameters have on the measured relaxation rates. Most recently, Palmer and coworkers and Ishima and Torchia have derived analytical expressions for CPMG and R 1q experiments that extend the relations that were originally presented many years ago (Ishima and Torchia 1999; Miloushev and Palmer ; Trott and Palmer ). The majority of the derived formulae for transverse relaxation in the presence of CPMG or R 1q fieldsarebasedonthe assumption that the spins in question exchange between states with the same intrinsic transverse relaxation rates, DR =, although Allerhand and Thiele recognized very early on that large differences in transverse relaxation rates of exchanging spins can certainly influence the CPMG profile (Allerhand and Thiele 1966). More recently, Ishima and Torchia have explored through simulations how CPMG dispersion profiles are affected in the case where DR = (IshimaandTorchia6). In addition, Miloushev and Palmer have presented an expression to account for relaxation differences in R 1q experiments that is valid in certain limits (Miloushev and Palmer ). There are clearly many cases where the assumption DR = is reasonable. However, recent advances in NMR methodology have opened the possibilities of studying biological molecules of ever-increasing size (Fiaux et al. ; Sprangers and Kay 7), leading to examples where differential relaxation becomes an important issue. This is the case for the 6 kda ab-crystallin oligomeric ensemble (Baldwin et al. 11a, b), where we have recently characterized a disorder (ground state) to order (excited state) transition with DR * s -1 (Baldwin et al. 1), illustrated schematically in Fig. 1. In an effort to understand how such differences affect measured R 1q relaxation rates we have derived an expression for R 1q in the limit DR = that is valid over a large range of DR values (see below). We show that differential transverse relaxation can significantly influence R 1q rates but that remarkably measurements of signs of chemical shift differences, based on comparison of R 1q values obtained with spinlock fields on either sides of the ground state correlation (Korzhnev et al. 3; Trott and Palmer ), are much less affected by DR =. Results and discussion We consider here the case of two-site conformational exchange involving the interconversion of a molecule A B Fig. 1 Schematic of a two-site exchange process involving the interconversion between protein conformers whereby a fragment is rigidly bound or unattached to the core of the protein. In the case where the protein is small (molecular mass of kda, a) it may be that differential transverse relaxation in the two states can be neglected in the analysis of exchange data (although this must be rigorously established). In the case of a high molecular weight complex, such as ab crystallin (aggregate molecular mass of close to 6 kda, illustrated schematically in b) DR can be large (& s -1 ), complicating analysis of dispersion data between ground (G) and excited (E) states, G k GE E and focus k EG on a single spin reporter. In what follows, the fractional populations of the two states are given by p G = k EG /k ex and p E = 1 - p G = k GE /k ex, the overall exchange rate is k ex = k EG? k GE, the spin of interest in each of the two states evolves with frequencies x G and x E (Dx = x E -x G, rad/s) in the absence of chemical exchange and the difference in intrinsic transverse spin relaxation rates is given by DR. As has been described in the literature (Anet and Basus 1978; Skrynnikov et al. ), chemical exchange leads to a shift in the position of the observed resonance from x G to x OBS where in the limit that p G p E, k EG k GE Dx x ex ¼ x OBS x G ¼ ðk EG þ DR Þ þðdxþ : ð1þ In a typical R 1q experiment that is of interest here, magnetization initially along the z-axis is rotated into the x z plane by a pulse along the y axis with flip angle h flip ¼ tan 1 ðx 1 =d OBS Þ; d OBS ¼ x OBS x RF where x RF is the carrier frequency. The resulting spin-locked magnetization subsequently evolves for a defined period of time under the influence of an RF pulse of strength x 1 applied along the x-axis at an offset of d OBS from the observed ground state frequency. At the completion of this interval magnetization is subsequently restored to the z-axis by a y pulse with flip angle h flip. The decay rate of the signal, R 1q can be analyzed to determine the properties of the underlying conformational exchange (Palmer and Massi 6). The evolution of

5 J Biomol NMR (13) : magnetization during the spin-lock period is given by a set of coupled linear differential equations (McConnell 198), k) equation that results, Ak? B =, contains 76 terms, with k =-B/A = R 1q. M G 1 x R G My G k GE d G k EG M G 1 x d G R G d Mz G k GE x 1 k EG M x dt B Mx E ¼ 1 R G 1 k G y GE k EG C k GE R My E A k M G z EG d G Dx k GE d G þ Dx R E Mz E k C M E B x C EG x 1 A@ M E A k GE x 1 R E 1 k y EG Mz E ƒ! þ M ðþ where M~ O ¼ ; ; R G 1 MG O ; ; ; T, RE 1 ME O T is transpose, R G/E 1 and R G/E are longitudinal and transverse relaxation rates for the spin in states G, E and M G/E are thermal equilibrium magnetization values, proportional to p G and p E. In what follows we assume R E 1 ¼ RG 1 ¼ R 1 and R E 6¼ RG. Note that the assumption of equivalent R 1 values is reasonable even if states G and E differ significantly in correlation time or in intrinsic dynamics since R 1 rates are in general small for the spin probes of dynamics typically used in studies of biomolecules. Following the lead of Trott and Palmer () an expression for the R 1q rate when R E 6¼ RG is derived by evaluating, det j ^R k ^Ej ¼ ð3þ where ^E is the identity matrix and ^R is the square matrix in Eq. (). Of the six eigenvalues of ^R, four are complex, and of the two that are real only one is small in magnitude, corresponding to the R 1q value measured experimentally (Trott and Palmer ). Evaluating the characteristic polynomial that derives from Eq. (3), with the substitutions DR ¼ R E RG and DR ¼ RG R 1 generates 3 non-zero terms of the form CðDxÞ i d j G xk 1 kl EG km GE ðdrþn ðdr Þ o k p where C is a number and the sum of all superscripts is equal to 6. In general it is not possible to obtain an analytical solution for the roots (eigenvalues) of a polynomial of order 6, however, as discussed by Trott and Palmer () only the leading terms of the polynomial must be retained to calculate an accurate expression for R 1q. Because in many practical cases both the eigenvalue of interest (k, corresponding to - R 1q ) and DR are small compared to the other elements in each of the 3 terms (Dx, d G, x 1, k EG, k GE, DR )itis possible to include only those terms in the characteristic polynomial for which n þ p 1 without unduly sacrificing accuracy. This follows directly from the fact that for higher powers of DR and k the terms become small. The linear (in After a lengthy derivation, and with the following definitions, d j ¼ x j x RF ; j fg; Eg d ¼ p G d G þ p E d E ¼ d G þ p E Dx X G ¼ x 1 þ d G X E ¼ x 1 þ d E X ¼ x 1 þ d h ¼ tan 1 x 1 =d we arrive at the central result of the paper, R 1q ¼ C R1 R 1 cos h þðc R R G þ R exþ sin h ð4þ ðþ where sin h ¼ x 1 = X and cos h ¼ d = X. Note that h ¼ h flip only in the limit x G þ p E Dx x OBS. The coefficients C R1, C R and R ex include explicit contributions from DR and are given by C R1 ¼ FP þ FP 1 þ DR ðf 3 F Þ tan h D P þ DR F 3 sin h C R ¼ DP csc h F P cot h F P 1 þ DR F ð6þ D P þ DR F 3 sin h R ex ¼ FP 1 k ex þ DR F 1 D P þ DR F 3 sin h These are, in turn, recast in terms of three expressions that are independent of DR, present in the derivation by Trott and Palmer (), F P 1 ¼ p Gp E Dx F P ¼ k ex þ x 1 þ d G d E d D P ¼ k ex þ X G X E X ð7þ and an additional three terms that are all multiplied by DR in C R1,C R and R ex,

6 14 J Biomol NMR (13) :11 18 F 1 ¼ p E X G þ k ex þ DR p G k ex F ¼ k ex þ x 1 k ex þ DR p G F 3 ¼ 3p E k ex þ p G k ex þ x 1 k ex þ DR þ DR p E k ex X G! X G x 1 ð8þ Equation () is complicated and it is useful to consider certain limiting cases. Assuming for the moment that DR = it is straightforward to show that R DR! 1q ¼ R 1 cos h þ R G þ RDR! ex sin h: ð13þ From the expression for R DR! ex in Eq. (1) it is clear that so long as DR Dx kex \\ þ X E k ex kex þ X G k ex þ X E k ex þ x 1 Dx ð14þ differential transverse relaxation makes little contribution R 1q ¼ R þ kex þ x 1 þ d G d E d ðr 1 R Þcos h þ p G p E Dx ðk ex þ R 1 R Þsin h k ex þ X G X E X ð9þ (R E ¼ RG ¼ R ) which is identical to Eq. (11) of Trott and Palmer () (not withstanding a small typographical error in their expression). In the limit that DR/k ex 1 and d G p E Dx it follows that R P 1q ¼ R 1 cos h þðr þ R P ex Þ sin h where R P ex ¼ p Gp E Dx k ex X E X G þ k X ex ðþ ffi p Gp E Dx k ex X E þ ; ð11þ k ex that is Eq. (1) of Trott and Palmer (). Equation () can be simplified in the limit when DR is small by performing a Taylors series expansion about DR =, retaining only the linear terms and enforcing d G p E Dx, p G p E. It can be shown that under these conditions C DR! R1 ¼ 1 þ p EDx kex þ tan h þ DR N X E kex þ tan h X E C DR! R ¼ 1 p EDx kex þ DR N X E kex þ X E R DR! ex N ¼ p E ¼ p EDx k ex k ex þ X E 1 þ DR k ex k ex þ X G Dx k ex þ x 1 k ex þ X E! 1 3 cos h kex k ex þ x 1 Dx! kex þ X k E ex!! ð1þ Substitution of the expressions in Eq. (1) into Eq. () with the requirement that DR/k ex 1 and retaining only the leading terms gives to R 1q, while as k ex increases the effects of DR become more pronounced (see below). Values of R 1q as a function of x 1 have been calculated with (i) Eq. (11) of Trott and Palmer (), modified as they describe with R 1 and R in this expression replaced by population weighted averages, (ii) with Eq. () or (iii) numerically for exchange parameters indicated in the legend to Fig. a, b. When DR = (Fig.a) excellent agreement is obtained with the numerically computed values. The slight deviations observed for small x 1 values are in keeping with the assumptions used in the derivation of analytical expressions based on linearization of the characteristic polynomial of the matrix in Eq. (). Significant deviations are noted for the Trott and Palmer expression when DR is large (R G = 6s-1, R E =s-1 ), as expected, while rates calculated with Eq. () are in good agreement with those obtained numerically, Fig. b. These differences decrease as x 1 increases but do not go to zero in the present case as this would require in addition that DR /k ex 1, a condition that is not fulfilled in this example (see legend to Fig. ). Of note, similar calculations, but as a function of k ex, show that the differences between the expressions do converge to the exchange free rate in the tilted frame in the limit that k ex?? with R 1 and R values given by population weighted averages. We have also performed an extensive grid search in parameter space to ascertain the accuracy of Eq. () more generally. Numerical simulations for R G RE s1,.1 % B p E B %, 1. rad/s B Dx B 1, rad/s, s -1 Bk ex B, s -1, rad/s B x 1 B, rad/s,.1 rad/s B d G B, rad/s show that, so long as ðd G þ x 1 Þ: [ Aðp E DxÞ with A =.1, errors are less than s -1.Figurec plots the deviations in R 1q values calculated from Eq. ()orfromtheformula oftrottand Palmer for a number of R E values as a function of A.Asthevalue

7 J Biomol NMR (13) :11 18 R 1 (s -1 ) A Eq. [11], T&P Eq. [] N 1 (Hz) 1 (Hz) B R 1 (Eq. [11],T&P) - R 1 (N),red ( (s -1 ) 3 C 3 E R = s -1 E R = s -1 E R = s -1 E G E R =R R = s A R 1 (Eq. []) - R 1 (N), blue ( (s -1 ) Fig. a, b Plots of R 1q (x 1 ) calculated with (i) Eq. (11) of Trott and Palmer (), modified such that R 1 and R in this expression are replaced by population weighted averages (red), (ii) with Eq. () of the present manuscript (blue) and (iii) by solving numerically for R 1q using Eq. () (black) for DR = (a) and R G ¼ 6s1, R E ¼ s1 (b). Values of k ex = 1, s -1, Dx = 1, rad/s, R 1 =. s -1, d G = rad/s, p B = 3 % have been used. In the limit x 1!1Eq. () reduces to R 1q ¼ R G þ DRpE 1þDR =k ex that differs slightly from a population weighted average of rates, p G R G þ p ER E. Note that an identical expression is obtained in this limit for the transverse relaxation rate of A decreases in the expression above the range of possible d G, x 1 values increases, leading to cases where R 1q becomes sufficiently large that retaining only those terms up to linear in k in the polynomial of Eq. (3) is no longer appropriate so that Eq. () becomes less accurate. As an interesting application of the work described here we consider the measurement of R 1q as a function of offset from the ground state peak to obtain the sign of Dx, and hence the value for the resonance position of the spin in the excited conformer. This information is not available from Carr-Purcell-Meiboom- Gill experiments that are typically recorded to characterize exchanging systems since relaxation dispersion profiles are only sensitive to Dx. In their original derivation of an expression for R 1q outside the fast exchange regime Trott and Palmer () noted that R 1q values are particularly sensitive to the placement of the spin lock field relative to the resonance frequency in the ground state, with a maximum in R ex when the carrier is positioned on resonance with the peak derived from the excited state. Thus, R 1q (d G = Dx) \ R 1q (d G =-Dx). Our laboratory has exploited this result by measuring R 1q values where the spin lock is placed equidistantly upfield and derived from the CPMG experiment as well. c Errors in R 1q values calculated from Eq. () [blue, R 1q (Eq. ())] or Eq. (11) of Trott and Palmer [red, R 1q (Eq. (11), T&P)] for R E ¼ ð6; ; ; s1þ as a function of A, with ðd G þ x 1 Þ: [ Aðp E DxÞ and spanning the range R G RE s1,.1 % B p E B %, 1. rad/s B Dx B 1, rad/s, s -1 Bk ex B, s -1, rad/s B x 1 B, rad/s,.1 rad/s B d G B, rad/s. For A =.1 errors are less than s -1 using Eq. (). Values of R 1q calculated by numerically computing the eigenvalues of the matrix in Eq. (3) are denoted by R 1q (N) downfield of the major state correlation in a pair of experiments. Positions of the carrier and the magnitude of the spinlock field (x 1 ) are optimized on a per-residue basis, taking into account the exchange parameters that have been measured previously, by maximizing the difference in R 1q values for a given spin lock duration, T relax. Using this approach we have developed a suite of off-resonance R 1q experiments with weak spinlock fields to measure the signs of chemical shift differences for 1 H a, 1 H N, N, 13 C a and 13 C-methyl nuclei in proteins that exchange between different conformations on the ms timescale (Auer et al. 9, ; Baldwin and Kay 1; Korzhnev et al. 3, ). Equations (1) and (13) presented above can be used to derive an expression for the difference in R 1q relaxation rates recorded with the spinlock carrier placed downfield ðr þ 1q Þ and upfield ðr 1qÞ of the ground state resonance position, DR 1q ¼ R þ 1q R 1q, that forms the basis for measurement of the signs of chemical shift differences. Under the same set of assumptions that were used in the derivation of Eq. (1) it can be shown that DR DR! 1q 4p E jd OPT jdx 3 k ex sin h ¼ kex þ X OPT þ 4d 1 DR Dx OPT Dx k ex kex þ X OPT þ Dx k ex þ x 1 X kex þ X OPT þ 4d Dx Dx Dx OPT OPT þ k ex!! ðþ

8 - 16 J Biomol NMR (13) :11 18 R1ρ (s-1 ) + A + + [R 1ρ -R1ρ (ΔR =)] (s -1 ) 4 3 i) iv) 3 - ΔR= s -1 ΔR= R 1ρ (s -1 ) - - [R 1ρ -R 1ρ (ΔR =)] (s -1 ) 4 3 ii) v) 3 - ΔR= s -1 ΔR= ΔR 1ρ -ΔR 1ρ (ΔR =) (s -1 ) ΔR 1ρ (s -1 ) 4 3 iii) vi) 3 - ΔR= s -1 ΔR= B (11.7T) (ppb) ϖex [ ϖ ex - ϖ ex (ΔR =) ](11.7T) (ppb) i) iv) ΔR= s -1 ΔR= ϖ ex (18.8T) (ppb) [ ϖ ex - ϖ ex (ΔR =) ](18.8T) (ppb) ii) v) ΔR= s -1 ΔR= k ex (s-1) Δϖex - Δϖ ex (ΔR =) (ppb) Δϖ ex (ppb) 3 iii) 3 vi) ΔR= s -1 ΔR= Fig. 3 a Plots of R þ 1q (i), R 1q (ii) and DR 1q (iii) calculated numerically, as a function of Dx and k ex, with DR = s -1 (blue) or (red). (iv vi) Corresponding differences, Z(DR = s -1 ) - Z(DR = s -1 ), where Z {R þ 1q,R 1q,DR 1q}, as labelled along the vertical axes. b (i iii) Exchange induced frequency shifts (ppm), where d OPT ¼ x OBS x RF;OPT x G x RF;OPT X OPT ¼ x 1;OPT þ ð16þ d OPT and x RF,OPT, x 1,OPT are the optimized position of the carrier and spinlock field strength, respectively. - ex (Dx,k ex ), calculated for static magnetic fields of 11.7 and 18.8T along with D- ex = - ex (11.7T) - - ex (18.8T), DR = s -1 (blue) and DR = s -1 (red). (iv vi) - ex (DR = s -1 ) - - ex (DR =s -1 ) at 11.7T (iv) and 18.8T (v), D- ex (DR = s -1 ) -D- ex (DR = s -1 ) (vi) From Eq. () it is clear that the sign of Dx determines the sign of DR 1q, as expected from the work of Trott and Palmer (); note that it can be proved that the sign of the denominator of the first term is[ and extensive numerical simulations establish that even for DR values as large as s -1 the signs of DR 1q and Dx are the same. Thus, measurement of the sign of DR 1q allows the determination of

9 J Biomol NMR (13) : the sign of Dx and for positive (negative) values of DR 1q the excited state peak is downfield (upfield) of the ground state correlation. It is not immediately transparent from Eq. () how DR = influences this scenario; however because DR is multiplied by an expression that is, in turn, given by the difference of two positive terms there is the possibility for attenuation of the effects of differential relaxation. Indeed, this turns out to be the case, as numerical simulations indicate. Figure 3ai iii plot R þ 1q (i), R 1q (ii) and DR 1q (iii) calculated from Eq. () numerically, as a function of Dx and k ex for a spin exchanging between two states with DR = s -1 (blue) or (red). Shown in Fig. 3aiv vi are the differences between each of these values, Z(DR = s -1 ) - Z(DR = s -1 ), where Z {R þ 1q,R 1q,DR 1q}, as labelled along the vertical axis of each panel. Most notably, DR 1q is rather insensitive to DR, even for very large values. Similar plots for the exchange induced frequency shift (ppm), that can also be used to establish the sign of Dx (Skrynnikov et al. ), are shown in Fig. 3bi iii. Here - ex (Dx,k ex ) values evaluated at 11.7 and 18.8T are shown, along with the difference, D- ex = - ex (11.7T) - - ex (18.8T) for both DR = s -1 (blue) and DR = s -1 (red). In addition absolute values of the differences between - ex = - ex (DR = s -1 ) and - ex (DR = ) are plotted in Fig. 3biv (11.7T) and 3Bv (18.8T), respectively, along with D- ex (DR = s -1 ) - D- ex (DR = ) in Fig. 3bvi. Values of D- ex are significantly more sensitive to DR than are DR 1q rates, as indicated in the Figure. Indeed, in a previous application involving studies of chemical exchange in the 6 kda ab crystallin complex we noted that D- ex values were much smaller than anticipated based on the measured exchange parameters, while large DR 1q values were still obtained from which signs of chemical shift differences could be established (Baldwin and Kay 1). In summary, we have derived an expression for R 1q for a spin exchanging between two sites with DR =. While analytical expressions of the sort presented here can be used for data fitting we much prefer to use (exact) numerical approaches that start from the Bloch-McConnell equations. Nevertheless, formula are valuable as a teaching tool providing physical insight into how the range of exchange parameters can influence measured relaxation rates. The work described here adds to a growing body of theoretical analyses of CPMG and R 1q dispersion experiments and considers the important case of differential transverse relaxation that will become increasingly relevant as applications to high molecular weight biomolecules become more common. Acknowledgments A.J.B. acknowledges the Canadian Institutes of Health Research (CIHR) for a postdoctoral fellowship and the BBSRC and Merton College for David Phillips and Fitzjames fellowships respectively. This work was supported by grants from the CIHR and the Natural Sciences and Engineering Research Council of Canada. L.E.K. holds a Canada Research Chair in Biochemistry. References Allerhand A, Thiele E (1966) Analysis of Carr-Prucell spinecho NMR experiments on multiple spin systems. II. The effect of chemical exchange. J Chem Phys 4:9 916 Allerhand A, Gutowsky HS, Jonas J, Meinzer RA (1966) Nuclear magnetic resonance methods for determining chemical-exchange rates. J Am Chem Soc 88: Anet FA, Basus VJ (1978) Limiting equations for exchanging broadening in -site NMR systems with very unequal populations. J Magn Reson 3: Auer R, Neudecker P, Muhandiram DR, Lundstrom P, Hansen DF, Konrat R, Kay LE (9) Measuring the signs of 1H(alpha) chemical shift differences between ground and excited protein states by off-resonance spin-lock R(1rho) NMR spectroscopy. J Am Chem Soc 131: Auer R, Hansen DF, Neudecker P, Korzhnev DM, Muhandiram DR, Konrat R, Kay LE () Measurement of signs of chemical shift differences between ground and excited protein states: a comparison between H(S/M)QC and R1rho methods. J Biomol NMR 46: 16 Baldwin AJ, Kay LE (1) Measurement of the signs of methyl 13C chemical shift differences between interconverting ground and excited protein states by R1rho: an application to ab crystallin. J Biomol NMR 3:1 1 Baldwin AJ, Hilton GR, Lioe H, Bagneris C, Benesch JL, Kay LE (11a) Quaternary dynamics of alpha B-crystallin as a direct consequence of localised tertiary fluctuations in the C-terminus. J Mol Biol 413:3 3 Baldwin AJ, Lioe H, Robinson CV, Kay LE, Benesch JL (11b) AlphaB-crystallin polydispersity is a consequence of unbiased quaternary dynamics. J Mol Biol 413:97 39 Baldwin AJ, Walsh P, Hansen DF, Hilton GR, Benesch JL, Sharpe S, Kay LE (1) Probing dynamic conformations of the high molecular weight ab-crystallin heat shock protein ensemble by NMR spectroscopy. J Am Chem Soc 134:343 3 Boehr DD, McElheny D, Dyson HJ, Wright PE (6) The dynamic energy landscape of dihydrofolate reductase catalysis. Science 313: Carver JP, Richards RE (197) A general two-site solution for the chemical exchange produced dependence of T upon the Carr- Purcell pulse separation. J Magn Reson 6:89 Deverell C, Morgan RE, Strange JH (197) Studies of chemical exchange by nuclear magnetic relaxation in the rotating frame. Mol Phys 18:3 9 Fiaux J, Bertelsen EB, Horwich AL, Wüthrich K () NMR analysis of a 9K GroEL GroES complex. Nature 418:7 11 Fraser JS, Clarkson MW, Degnan SC, Erion R, Kern D, Alber T (9) Hidden alternative structures of proline isomerase essential for catalysis. Nature 46: Hansen DF, Vallurupalli P, Kay LE (8) Using relaxation dispersion NMR spectroscopy to determine structures of excited, invisible protein states. J Biomol NMR 41:113 1 Henzler-Wildman K, Kern D (7) Dynamic personalities of proteins. Nature 4: Ishima R, Torchia DA (1999) Estimating the time scale of chemical exchange of proteins from measurements of transverse relaxation rates in solution. J Biomol NMR 14: Ishima R, Torchia DA () Protein dynamics from NMR. Nat Struct Biol 7:74 743

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