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Microsecond timescale backbone conformational dynamics in ubiquitin studied with NMR R₁ relaxation experiments

Department of Biochemistry and Molecular Biophysics, Columbia University, New York, New York 10032, USA

Reprint requests to: Arthur G. Palmer III, Department of Biochemistry and Molecular Biophysics, Columbia University, 630 West 168th Street, New York, NY 10032, USA; e-mail:agp6{at}columbia.edu; fax: (212) 305-6949.

(RECEIVED September 24, 2004; FINAL REVISION September 24, 2004; ACCEPTED November 19, 2004)

	Abstract

TOP Abstract Introduction Results and Discussion Materials and methods References

 
NMR spin relaxation experiments are used to characterize the dynamics of the backbone of ubiquitin. Chemical exchange processes affecting residues Ile 23, Asn 25, Thr 55, and Val 70 are characterized using on- and off-resonance rotating-frame ¹⁵N R₁ relaxation experiments to have a kinetic exchange rate constant of 25,000 sec^–1 at 280 K. The exchange process affecting residues 23, 25, and 55 appears to result from disruption of N-cap hydrogen bonds of the -helix and possibly from repacking of the side chain of Ile 23. Chemical exchange processes affecting other residues on the surface of ubiquitin are identified using ¹H-¹⁵N multiple quantum relaxation experiments. These residues are located near or at the regions known to interact with various enzymes of the ubiquitin-dependent protein degradation pathway.

Keywords: protein dynamics; chemical exchange; ¹⁵N spin relaxation; multiple-quantum relaxation

Article and publication are at http://www.proteinscience.org/cgi/doi/10.1110/ps.041139505.

	Introduction

TOP Abstract Introduction Results and Discussion Materials and methods References

 
Ubiquitin, one of the most highly conserved proteins in eukaryotes, is the signature component of the ubiquitin-dependent protein degradation pathway (Hochstrasser 1996; Hershko and Ciechanover 1998). Briefly, ubiquitin is activated by adenylation of its C terminus and linked covalently to an ubiquitin-activating enzyme, E1. Next, ubiquitin is transferred from E1 to an ubiquitin-conjugating enzyme, E2. Finally, ubiquitin is attached covalently to the -amino group of a lysine residue of the target protein. This last step often involves an ubiquitin–protein ligase, E3. Polyubiquitinated proteins subsequently are targeted to the 26S proteasome for degradation. Deubiquitinating enzymes serve a regulatory role by removing ubiquitin from target proteins.

Investigations by site-directed mutagenesis and NMR spectroscopy have identified amino acid residues in ubiquitin that have crucial roles in interactions with E1, E2, and deubiquitinating enzymes. These residues include the four arginines at positions 42, 54, 72, and 74. In particular, Arg 54 and Arg 72 are involved in the initial binding of free ubiquitin to E1 (Burch and Haas 1994). In addition, Leu 8, Ile 44 and Val 70 form a hydrophobic patch on the surface of ubiquitin that is essential for binding to E1 (Beal et al. 1996; Haas and Siepmann 1997). Many of the residues important for interactions with E1 are also involved in binding to E2 enzymes (Miura et al. 1999; Hamilton et al. 2000, 2001) and to deubiquitinating enzymes (Wilkinson et al. 1999). Residues 7–9, 40–49, and 70–76 are most strongly implicated in the interaction with E2 enzymes and deubiquitinating enzymes.

Conformational flexibility within the interaction surfaces of proteins may facilitate protein–protein interactions through induced fit mechanisms and may allow recognition of more than one binding partner (James et al. 2003). Protein conformational dynamics on microsecond to millisecond timescales are manifest as chemical exchange line-broadening in NMR spectroscopy, and a number of techniques have been developed to characterize enhanced transverse relaxation of nuclear spin resonances in proteins (Palmer 2004).

Previous NMR investigations have identified chemical exchange processes that affect a variety of sites within ubiquitin. At room temperature, ¹⁵N chemical exchange broadening is most evident for residues Ile 23 and Asn 25, while the resonance for Glu 24 is extremely weak in HSQC spectra (Schneider et al. 1992; Tjandra et al. 1995; Fushman and Cowburn 1998; de Alba et al. 1999; Carlomagno et al. 2000; Meiler et al. 2001; Tolman et al. 2001). Recent ¹⁵N R₁ relaxation measurements in supercooled water at T = 260 K have established the presence of a chemical exchange process affecting Val 70 with an exchange rate constant (vide infra) of k_ex = 7500 ± 1600 sec^–1 (Mills and Szyperski 2002). Relaxation of zero-quantum (ZQ) and double-quantum (DQ) coherences was used by Bodenhausen and coworkers to identify exchange processes affecting residues Ile 23 and Asn 25 (Dittmer and Bodenhausen 2004; Wist et al. 2004). CPMG measurements reported exchange rate constants of k_ex = 4200 sec^–1 for Ile 23 at 296 K using ¹H-¹⁵N ZQ/DQ relaxation (Dittmer and Bodenhausen 2004) and k_ex = 1064 sec^–1 for Asn 25 at 300 K using ¹⁵N-¹³C' ZQ/DQ relaxation (Wist et al. 2004). Novel TROSY-based ZQ and DQ experiments were used by Majumdar and Ghose (2004) to identify Glu 24, Asn 25, Glu 51, and Asp 52 as a continuous surface-exposed patch likely to be subject to correlated motions.

This article reports the results of on- and off-resonance ¹⁵N R₁ (Massi et al. 2004) and ¹H-¹⁵N ZQ/DQ (Kloibert and Konrat 2000) relaxation experiments. Newly developed decoupling schemes were applied to allow the use of weak radiofrequency fields in the R₁ experiments (Massi et al. 2004). R₁ relaxation dispersion at 280 K is used to quantify chemical exchange processes affecting residues Ile 23, and Asn 25, located in the N terminus of the -helix; Thr 55, in the loop between -strand 4 and the short second 3–10 helix; and Val 70, in -strand 5. Exchange processes affecting additional residues located on the surface of ubiquitin are identified using ZQ/DQ relaxation.

	Results and Discussion

TOP Abstract Introduction Results and Discussion Materials and methods References

 
R₁ relaxation dispersion curves
Figure 1 shows the conformational exchange contribution 0) to the transverse relaxation rate constant for ¹⁵N of ubiquitin at temperatures of 280 K and 298 K, estimated using a Hahn spin-echo experiment (Wang et al. 2001b; Wang and Palmer 2003) (see Materials and Methods). At T = 298 K, only residue Asn 25, located in the -helix, has a value of R_ex (_e 0) that is clearly significantly greater than zero. When _xy values, measured at 298 K, are plotted against R₂ values for ubiquitin, both Ile 23 and Asn 25 are identified as subject to chemical exchange linebroadening (data not shown). However, as shown in Figure 2, R_ex(_e) dispersion curves for these residues are independent of _e, indicating that the exchange kinetics, k_ex are too fast, relative to _N, to be refocused even at the highest values of _e used in the off-resonance R₁ experiment. R_ex(_e) was calculated for each residue from measured values of R₁, R₁, and _xy relaxation rate constants (see Materials and Methods). When the temperature is decreased to 280 K, the exchange contribution for Asn 25 is increased, and significant exchange contributions are observed for three additional residues: Ile 23 at the N terminus of the -helix, Thr 55 in the turn region between -strand 4 and second short 3–10 helix, and Val 70 in -strand 5 of the C-terminal region. The positions of the residues that are observed to undergo chemical exchange at 280 K are mapped onto the three-dimensional structure of ubiquitin in Figure 3A.

View larger version (24K): [in this window] [in a new window]   Figure 1. Conformational exchange contribution to transverse relaxation, Rex, for ubiquitin at T = 298 K (open circles) and T = 280 K (filled circles) at B0 = 11.7 T.

View larger version (11K): [in this window] [in a new window]   Figure 2. Conformational exchange contribution to transverse relaxation, Rex as a function of the effective field, e, at a static magnetic field of 11.7 T and T = 298 K for Ile 23 (circles) and Asn 25 (squares) in ubiquitin.

View larger version (33K): [in this window] [in a new window]   Figure 3. Cartoon representation of ubiquitin that highlights the positions of the exchanging residues Ile 23, Asn 25, Thr 55, Val 70 (A). Regions characterized by different secondary structure are represented with different colors: helices ,red; -strands, yellow; coil and turns, blue. The boxed region is shown as an expansion in B. Each residue is labeled at the backbone N atom. The dashed lines indicate hydrogen bonds.

1

View larger version (21K): [in this window] [in a new window]   Figure 4. Conformational exchange contribution to transverse relaxation, Rex, as a function of the effective field, e at T = 280 K. Circles and squares represent data collected at a static magnetic field of 11.7 T and 14.1 T, respectively. Lines are the results of simultaneously fitting the data at 11.7 T (circles) and 14.1 T (dashed), using equation 2.

1

1

1

Previous work in supercooled water at 260 K has already identified Val 70 as an exchanging residue with a rate constant k_ex equal to 7500 ± 1600 sec^–1 (Mills and Szyperski 2002). In the present study this motional process is determined to have a value of k_ex = 26,000 ± 4000 sec^–1 at 280 K. The data at two temperatures allow the activation barrier for the exchange process to be estimated as 36 kJ/mol.

Multiple quantum experiments
Amide ¹H-¹⁵N differential relaxation rates of ZQ and DQ coherences, R_MQ, of ubiquitin measured at 280 K are depicted in Figure 5. R_MQ is proportional to both _N and _H. Therefore, this experiment can detect exchange processes that are not detected by the R₁ experiment, provided that _H is sufficiently large. The residues characterized by the highest values of R_MQ—Thr 9, Ile 23, Ile 30, Lys 33, Leu 43, Phe 45, Thr 55, and Val 70—are located in the loop regions, on the -helix at the N and C termini, and on -strands 3 and 5. Exchange broadening for residues Ile 23, Leu 43, Phe 45, and Thr 55 have been identified previously in ZQ/DQ CPMG experiments (Dittmer and Bodenhausen 2004). Asn 25, which shows the highest _N from the R₁ experiment, has R_MQ 0, indicating that the ¹H chemical shift for Asn 25 does not change as a result of exchange. Using R₁ and R_MQ measurements, the ratio of _H /_N can be evaluated for a given residue (as described in Materials and Methods). Relative values of _H/_N are 1.81 ± 0.03, 2.0 ± 0.2, and 0.78 ± 0.05 for residues Ile 23, Thr 55, and Val 70, respectively. Values of _H/_N in the range of 1–3 suggest that the exchange process results from dihedral angle and/or hydrogen bonding fluctuations, rather than variations in ring current shifts (Ishima et al. 1998; Wishart and Case 2001).

View larger version (25K): [in this window] [in a new window]   Figure 5. Amide 1H-15N differential relaxation rate constants for ZQ and DQ coherences, RMQ, for ubiquitin measured at 11.7 T and T = 280 K.

Correlations between structural changes and _N

The values of _H/_N obtained from residues Ile 23, Thr 55, and Val 70 suggest that changes in hydrogen bonding and dihedral angles are the most likely sources of conformational dynamics. Residues Ile 23 and Asn 25 are located at the N terminus of the -helix in ubiquitin and form capping interactions with Thr 22 and the loop comprising residues Glu 51–Leu 56 (Fig. 3B). Disruption of the helix capping interactions provides one possible mechanism for chemical exchange broadening. Hydrogen bonding provides a significant contribution to the structure dependent ¹⁵N chemical shifts in proteins (Xu and Case 2002). Xu and Case and Oldfield and coworkers (deDios et al. 1993; Xu and Case 2002) have shown that the hydrogen bond status of the amide group itself (direct effect) and the carbonyl group of the preceding residue (indirect effect) both contribute to the ¹⁵N chemical shielding. Model calculations suggest that removal of the Ile 23 (NH)–Arg 54 (CO) hydrogen bond results in a -1.5 ppm change in the ¹⁵N chemical shift of Ile 23 due to the direct hydrogen bonding effect and a –4.2 ppm change in the ¹⁵N chemical shift of Thr 55 due to the indirect hydrogen bonding effect. Assuming that the disruption of the Ile 23–Arg 54 hydrogen bond accounts for the relaxation dispersion observed for Thr 55, a chemical shift change of that magnitude would correspond to a minor state population p_B 0.02.

In addition to the hydrogen bond contribution, ¹⁵N chemical shifts exhibit a strong dependence on local backbone and side-chain conformation. Although there is no clear correlation between ¹⁵N chemical shifts and / angles, such as for ¹³C chemical shifts (Wishart and Case 2001; Xu and Case 2002), trends have been observed for variation in ¹⁵N chemical shift with side chain ₁ dihedral angles for particular amino acid types. For the case of Ile, an average upfield shift of 4.5 ppm is expected for a change in ₁ from –60° to +60° (Fig. 6A). Examination of the local structure of Ile 23 in ubiquitin reveals that such a ₁ isomerization for the completely buried side chain could not proceed without additional conformational changes of the backbone in the vicinity and would likely disrupt the Ile 23–Arg 54 hydrogen bond. Assuming a –4.5 ppm change in chemical shift associated with the ₁ transition and a –1.5 ppm change calculated for disruption of the hydrogen bond as a potential mechanism of exchange broadening for Ile 23 would correspond, again, to a minor state population p_B 0.02. The estimate p_B 0.02 indicates that the forward rate constant for transitions from major to minor states, k₁ is 500 sec^–1.

View larger version (32K): [in this window] [in a new window]   Figure 6. Dependence of the 15N chemical shift on side chain 1 dihedral angle. 15N secondary chemical shifts are plotted as a function of side chain 1 torsional angle for all isoleucine (A) and asparagine (B) residues in the RefDB chemical shift database (Zhang et al. 2003).

Glu 24, which is broadened extensively and essentially unobservable, and Asn 25 also form N-cap interactions. Based on the minor site population p_B = 0.02 derived above and the value of _ex reported in Table 1, a value of _N ~9 ppm is estimated for Asn 25, and an even larger value would be necessary to account for the extreme broadening of Glu 24. The dependence of the ¹⁵N chemical shifts on ₁ dihedral angles is clearly residue-type specific. For example, ₁ isomerization for asparagine residues does not perturb the ¹⁵N chemical shift average, although considerable variation about zero is possible (Fig. 6B). Disruption of the N-cap hydrogen bonds would contribute only ~2 ppm to the observed chemical shift changes. The secondary chemical shift for Asn 25 is only 3 ppm. Thus, the large values of _N for Glu 24 and Asn 25 must result from multiple sources, including changes in dihedral angles and redistribution of electrostatic and hydrogen bonding interactions. Model calculations, database analysis, and comparison to random coil shifts cannot uniquely identify the multiple mechanisms of exchange broadening for these residues.

Interestingly, a site population for the minor species of p_B = 0.02 for Val 70, estimated from the secondary chemical shift, is in good agreement with the site populations estimated for residues Ile 23 and Thr 55.

Biological implications
An increasing number of investigations have identified sites in ubiquitin that are known to be involved in protein–protein interactions and are subject to chemical exchange linebroadening in NMR experiments (Mills and Szyperski 2002; Majumdar and Ghose 2004). In the present work, R₁ experiments quantify exchange processes with a rate constant of 25,000 sec^–1 at 280 K affecting residues Ile 23, Asn 25, Thr 55, and Val 70. Arg 54 has been found to be important in the interaction of free ubiquitin with E1 (Burch and Haas 1994). Both Ile 23 and Thr 55 are connected to Arg 54, either covalently or through hydrogen bonding, suggesting that a single correlated kinetic process affects all three residues. As noted above, the exchange process may involve the disruption of the hydrogen bond between Ile 23 and Arg 54. Val 70 is located in -strand 5 at the C terminus of ubiquitin, and together with Leu 8 and Ile 44 is part of a hydrophobic patch essential for the recognition of E1. Val 70 also is known to have a role in the interaction with E2. The exchange rate constant for conformational dynamics of Val 70 has been measured previously by Mills and Szyperski at 260 K (Mills and Szyperski 2002); the present results allow an apparent activation energy for the conformational process to be estimated. Residues with large R_MQ relaxation rate constants cover an extended part of the exposed surface of the protein and are clustered together in two main regions. As Figure 7 shows, one of these regions, consisting of residues Thr 9, Ile 23, Leu 43, Phe 45, Thr 55, and Val 70, is contiguous and partly overlapped with the region of ubiq-uitin that is involved in the binding to the ubiquitin activating and conjugating enzymes E1 and E2 (Burch and Haas 1994; Beal et al. 1996; Haas and Siepmann 1997; Miura et al. 1999; Wilkinson et al. 1999; Hamilton et al. 2000, 2001). This is a distinct surface patch from Glu 24, Asn 25, Asp 52, and Glu 51 previously identified by Majumdar and Ghose (2004). The observation and quantification of chemical exchange processes affecting two groups of residues in ubiq-uitin located near or at the region of interactions with E1 and E2 give insight into the conformational dynamics of ubiquitin that may be necessary for recognition of diverse E1 and E2 partners.

View larger version (73K): [in this window] [in a new window]   Figure 7. Amide 1H-15N RMQ for ubiquitin measured at 11.7 T and T = 280 K. Data are represented with different colors on the surface of the protein. Residues with RMQ < 3 are represented as gray, residues with 3 RMQ < 5 are yellow, residues with 5 RMQ < 8 are orange, residues with 8 RMQ < 15 are red, and residues with RMQ 15 are purple. The opaque residues are those known to be directly involved in the interaction with E1 and E2 enzymes. All other residues are depicted as transparent.

1

	Materials and methods

TOP Abstract Introduction Results and Discussion Materials and methods References

 
Theory
The relaxation rate constant, R₁ for decay of the magnetization component parallel to the effective field in the rotating frame, _e = (₁² + ²)^1/2, is a function of the amplitude of the applied rf field, ₁, and of the resonance offset from the spin-lock carrier, The relaxation rate constant in the rotating frame is given by Deverell et al. (1970) and Abragam (1983),

(1)

in which = arctan(₁/) is the tilt angle between the static magnetic field and the effective field in the rotating frame. Thus, ¹⁵N R₂ rate constants can be calculated using measured values of R₁ and R₁ relaxation rate constants. In the presence of a conformational exchange process, R₂ = R₂⁰ + R_ex, in which R₂⁰ is the relaxation rate constant due to relaxation mechanisms other than exchange. The conformational exchange contribution to transverse relaxation, R_ex, depends on the effective field, _e. The kinetic parameters that characterize the exchange process can be obtained from the variation of R_ex as a function of _e. For two-site exchange that is fast on the chemical shift timescale (Deverell et al. 1970),

(2)

in which k_ex is the exchange rate constant (given as the sum of the forward and reverse kinetic rate constants for two-site exchange),

(3)

p_A and p_B are the fractional populations of sites A and B, and _N is the difference between the ¹⁵N chemical shifts of the two sites A and B. Relaxation dispersion is observed for values of _e that are experimentally accessible over the range _e < k_ex to _e > k_ex. If k_ex >>_e, then R_ex is approximately independent of _e.

The difference between the relaxation rate constants for ¹H-¹⁵N ZQ and DQ coherences is given by (Kloibert and Konrat 2000),

(4)

in which R⁰_ZQ and R⁰_DQ are the relaxation rate constants for ZQ and DQ coherences from processes other than chemical exchange, respectively, and _H is the difference between the ¹H chemical shifts of the two sites A and B. Thus, if both R₁ and R_MQ measurements are available for a given residue:

(5)

and R⁰_ZQ - R⁰_DQ is assumed to be small.

NMR spectroscopy and data processing
NMR ¹⁵N relaxation measurements were performed on a 1.25 mM [U-¹⁵ N] sample of ubiquitin (90% H₂O/10% D₂O, 10 mM sodium phosphate buffer (pH 5.8), T = 280 K and 298 K). Relaxation data were collected at static magnetic field strengths of 11.7 T and 14.1 T, using Bruker DRX500 and DRX600 spectrometers. Temperatures were calibrated using a sample of 100% methanol. Relaxation rate constants were determined from series of two-dimensional spectra recorded with different relaxation delays. Intensities of cross-peaks were fitted to mono-exponential or hyperbolic tangent decay functions as appropriate to give spin relaxation rate constants; uncertainties were estimated by jackknife simulations (Mosteller and Tukey 1977). Spectra were processed using nmr-Pipe (Delaglio et al. 1995), Sparky (University of California, San Francisco), and Curvefit (http://www.palmer.hs.columbia.edu).

¹⁵N R₁ and R₂ values were measured at 298 K and 280 K as described previously using inversion recovery and CPMG experiments (Farrow et al. 1994). Maximum relaxation delay values of 800 msec and 200 msec were used for R₁ and R₂, respectively. The interval between ¹⁵N 180° pulses in the CPMG sequence of the R₂ experiment was equal to 1 msec. Twelve spectra, including duplicates, are recorded for each measurement.

R₂⁰ values were determined at 280 K and 298 K from measurements of the rate constant for ¹⁵N-¹H dipole–dipole/¹⁵N chemical shift anisotropy, CSA, cross-correlated transverse relaxation, _xy, by an established method (Kroenke et al. 1998). Five pairs of spectra, each one corresponding to a different relaxation time delay, were recorded; the maximum value of the relaxation delay used was 106.8 msec. _xy rates were calculated by fitting the ratio of the signal intensities N_y2H_zN_y (t) to the function tanh(_xyt). The average value of the ratio = R₂⁰/_xy was obtained from data for residues in ubiquitin that were unaffected by chemical exchange broadening, assuming constant ¹⁵N CSA (Tjandra et al. 1996; Fushman et al. 1998; Wang et al. 2001a). R₂⁰ values for the residues undergoing conformational exchange were calculated as R₂⁰ = _xy using values of obtained at 11.7 T and 14.1 T.

R_ex(_e0) was estimated using a Hahn spin-echo experiment (Wang et al. 2001b; Wang and Palmer 2003). Five spectra were acquired with t = 0 msec, and five spectra with t = 115.8 msec. WALTZ 16 ¹H decoupling was employed during the relaxation period t. The apparent relaxation rate constant, R₂^HE, was calculated by fitting the ratio of the signal intensities I(t)/I(0) to an exponential decay function. R_ex(_e0) was calculated for each residue as R_ex(_e0)=R₂^HE – R₂⁰.

¹⁵N₁ rate constants were measured as previously described (Mulder et al. 1998; Massi et al. 2004). The maximum length of the spin-lock period in all experiments was equal to 160 msec. Twelve spectra, including duplicates, were recorded. On-resonance R₁ experiments were performed at 280 K using different spin-lock field strengths (370 Hz, 1000 Hz, and 1300 Hz at 11.7 T, and 370 Hz and 1000 Hz at 14.1 T). Off-resonance R1 experiments, where the spin-lock rf frequency was set outside the spectral region of interest were performed at 280 K using nine and seven different effective fields, respectively, at 11.7 T and 14.1 T. The different effective fields were obtained using several values of the spin-lock field strength (1270 Hz, 1765 Hz, 1790 Hz, 1830 Hz, and 1870 Hz at 11.7 T; 1000 Hz and 1700 Hz at 14.7 T) and different resonance offsets. The nominal tilt angles used, calculated from the middle of the spectrum, were 69.8°, 52.0°, 51.8°, 47.3°, 46.3°, 40.1°, 35.1°, 30.5°, 26.9°, and 69.5°, 50.5°, 45.5°, 40.3°, 34.3°, 29.5°, and 25.9°, respectively, at 11.7 T and 14.1 T. The tanh/tan adiabatic pulse, used in the off-resonance R₁ experiments, had a duration of 10 msec and the frequency sweep began at –15,000 Hz from the carrier frequency. On- and off-resonance R₁ experiments were performed at 300 K using a static magnetic field strength of 11.7 T. On-resonance R₁ experiments were performed using spin-lock field strengths equal to 250 Hz, 500 Hz, and 1000 Hz. The off-resonance R₁ experiment used a field strength of 1257 Hz and an offset of 2600 Hz.

The spin-lock field strengths used in the R₁ experiments were calibrated by off-resonance continuous wave decoupling as previously described (Palmer et al. 2001). Reported uncertainties are obtained from curve fitting. Variation in ₁ due to B₁ inhomogeneity is ~10%, as measured by a transient nutation experiment (Guenneugues et al. 1999).

Amide ¹⁵N-¹H differential relaxation of the DQ and ZQ coherences, R_MQ, was measured at 11.7 T as previously described (Kloibert and Konrat 2000; Wang and Palmer 2002). The spin echo delays, T, for the multiple quantum coherence relaxation period were 10, 20, 30, 40, and 50 msec, and five measurements were recorded for each delay. R_MQ, rates were calculated by fitting the ratio of the signal intensities 2N_yH_y(t)/2N_xH_x (t) the function tanh(R_MQ T/2) (Kloibert and Konrat 2000).

Dispersion curve fitting
R₂ rate constants were determined from measured values of R₁ and R₁ using equation 1. Use of R₂ rather than R₁ is preferable because R₂ does not depend on the tilt angle, which varies with field strength and the resonance offset from the spin-lock carrier for any given resonance. The resulting R₂ values, from all experiments at the two static magnetic fields, 11.7 T and 14.1 T, were used together with the R₂⁰ values to simultaneously optimize k_ex, _ex and R₂^opt using the following equations:

(6)

in which & R⁰₂ = R⁰_2opt. As a first approximation

(7)

in which d = (µ₀/4) _HN r^–3_NH , C = _NB₀/ 3µ₀ is the permeability of vacuum; is Planck’s constant divided by 2; _H and N are the gyromagnetic ratios of ¹H and ¹⁵N, respectively; f = B/B₀ r^–3_NH ^–1/3 = 1.02 Å, = –172 ppm is the difference between the principal components of the ¹⁵N CSA tensor, _|| and (Kroenke et al. 1999); J() is the spectral density function evaluated at the frequency , B_o = 11.7 T; and B is either 11.7 T or 14.1 T.

Nonlinear least-squares optimization used the Levenberg-Marquardt algorithm implemented in Mathematica (Wolfram). Uncertainties in fitted parameters were estimated by jackknife simulations.

Structural dependence of ¹⁵N chemical shifts
The structural dependence of ¹⁵N chemical shifts was analyzed using the RefDB database of uniformly referenced chemical shifts (Zhang et al. 2003) as previously described (Grey et al. 2003). Briefly, for each protein entry in the RefDB with a three-dimensional structure determined at high resolution, a structure-dependent secondary chemical shift was calculated for each residue as _N – _rc, where _N is the referenced ¹⁵ N chemical shift and _rc is the sequence-corrected random coil chemical shift (Braun et al. 1994). Structural parameters , , ₁, and ₂ were calculated for each residue based on the atomic coordinates of the corresponding PDB file.

Model chemical shifts calculations were performed using the program SHIFTS (Xu and Case 2001, 2002). To examine the contribution of the Ile 23–Arg 54 hydrogen bond to the ¹⁵N chemical shifts of Ile 23 and Thr 55, a fragment of ubiquitin corresponding to the residues Thr 22–Val 26 (A) and Asp 52–Leu 56 (B) was used. ¹⁵N chemical shifts were calculated for both fragments together (presence of hydrogen bond) and on each fragment separately to mimic the effect of removing the Ile 23–Arg 54 hydrogen bond.

	Acknowledgments

 
This work was supported by NIH grants GM59273 (A.G.P.) and GM08281 (M.J.G.). We thank Clay Bracken (Weill Medical College of Cornell University) for providing the ¹⁵N sample of ubiquitin.

	References

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