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Comparison of ¹³CH and ¹⁵NH backbone dynamics in protein GB1

Department of Biochemistry, Molecular Biology & Biophysics, University of Minnesota, Minneapolis, Minnesota 55455, USA

Reprint requests to: Kevin H. Mayo, Department of Biochemistry, Molecular Biology & Biophysics, 6-155 Jackson Hall, University of Minnesota, 321 Church Street, Minneapolis, MN 55455, USA; e-mail: mayox001{at}tc.umn.edu; fax: 612-624-5121.

(RECEIVED August 14, 2002; FINAL REVISION February 3, 2003; ACCEPTED February 12, 2003)

Article and publication are at http://www.proteinscience.org/cgi/doi/10.1110/ps.0228703.

	Abstract

TOP Abstract Introduction Results Discussion Materials and methods References

 
This study presents a site-resolved experimental view of backbone CH and NH internal motions in the 56-residue immunoglobulin-binding domain of streptococcal protein G, GB1. Using ¹³CH and ¹⁵NH NMR relaxation data [T₁, T₂, and NOE] acquired at three resonance frequencies (¹H frequencies of 500, 600, and 800 MHz), spectral density functions were calculated as F() = 2J() to provide a model-independent way to visualize and analyze internal motional correlation time distributions for backbone groups in GB1. Line broadening in F() curves indicates the presence of nanosecond time scale internal motions (0.8 to 5 nsec) for all CH and NH groups. Deconvolution of F() curves effectively separates overall tumbling and internal motional correlation time distributions to yield more accurate order parameters than determined by using standard model free approaches. Compared to NH groups, CH internal motions are more broadly distributed on the nanosecond time scale, and larger CH order parameters are related to correlated bond rotations for CH fluctuations. Motional parameters for NH groups are more structurally correlated, with NH order parameters, for example, being larger for residues in more structured regions of ß-sheet and helix and generally smaller for residues in the loop and turns. This is most likely related to the observation that NH order parameters are correlated to hydrogen bonding. This study contributes to the general understanding of protein dynamics and exemplifies an alternative and easier way to analyze NMR relaxation data.

Keywords: NMR; relaxation; spectral density; correlation times

	Introduction

TOP Abstract Introduction Results Discussion Materials and methods References

 
The 56 residue immunoglobulin-binding domain of streptococcal protein G (GB1) is an ideal model system with which to investigate protein dynamics by using NMR relaxation. It is a small, yet highly stable (Alexander et al. 1992) (T_m of 87 °C at pH 5.4), well-structured protein (Gronenborn et al. 1991) that is folded as a four-stranded ß-sheet (residues 2–8, 13–20, 42–46, and 51–56), on top of which lies an -helix running from residues 22 to 37 (Fig. 1). GB1 already has been extensively studied in terms of its thermodynamic stability. Over the predenaturation temperature range between 5 and 30°C, the calorimetrically determined free energy of unfolding for GB1 (Alexander et al. 1992) varies little and, on raising the temperature further, falls gradually to zero by 87°C. In this regard, GB1 behaves thermodynamically like a larger protein and, therefore, may be considered representative of any number of proteins.

View larger version (163K): [in this window] [in a new window]   Figure 1. Overall fold for GB1. The overall fold of GB1 is illustrated as given by Gronenborn et al. (1991).

Once NMR relaxation parameters have been acquired, most researchers analyze their data using the Lipari-Szabo (1982a,b) "model free" approach. The major limitation with using this method for analysis is that it is only valid for isotropic overall tumbling with internal rotational jumps between n equivalent states. In actual proteins, a multitude of internal motional correlation times are required to fully describe various bond rotational fluctuations and correlated motions. A novel method was recently developed to avoid many of the problems associated with the standard model free approaches (Idiyatullin et al. 2001). This new approach yields a motional correlation time distribution without assuming the nature of the molecular motions or the number of motional modes, that is, Lorentzians, involved in the dynamics processes. Here, we have used ¹³C and ¹⁵N NMR relaxation data and this new model free approach to examine ¹³CH and ¹⁵NH internal motional modes in protein GB1.

	Results

TOP Abstract Introduction Results Discussion Materials and methods References

 
¹³CH and ¹⁵NH relaxation data were acquired at three frequencies (¹H frequencies of 500, 600, and 800 MHz). These results are exemplified in Figure 2 with ¹³C and ¹⁵N NMR relaxation data (T₁, T₂, and {¹H}-¹³C NOE) at 500 and 800 MHz. Using these relaxation data, F() curves were calculated and motional correlation time distributions are plotted in Figure 3 for residues M1, T11, A26, A34, D40, and A48 in protein GB1. Similarly appearing curves were obtained for all ¹³CH and ¹⁵NH groups of residues throughout the protein. Two observations common to these distributions can be made: (1) F() curves for ¹³CH are generally shifted to lower frequency compared to those for ¹⁵NH. This is primarily due to increased viscosity of the ¹³C-enriched protein in D₂O, which results in somewhat slower overall tumbling motions. (2) The maximum of F(), F()_max, for ¹³CH is usually greater than for ¹⁵NH. As will be discussed below, this results partly from a shift of the ¹³CH internal motional correlation time distribution closer to that of overall tumbling and partly from correlated bond rotations that influence ¹³CH fluctuations more than they do those of ¹⁵NH.

View larger version (58K): [in this window] [in a new window]   Figure 2. R1, R2, and NOE data for protein GB1. 13C and 15N NMR relaxation data [T1, T2, {1H}- 15N NOE, and {1H}-13C NOE] on random fractionally 13C-enriched protein GB1 and from a separate sample, uniformly 15N-enriched protein GB1 are plotted versus the GB1 residue number for data acquired at 1H spectrometer frequencies of 500 MHz (open circles) and 800 MHz (filled squares). Secondary structure elements for the protein are given between panels. The temperature for all measurements was 5°C.

View larger version (26K): [in this window] [in a new window]   Figure 3. Plots of F() for six residues in GB1. Because is inversely related to the correlation time, F() gives the distribution of motional correlation times for CH and NH fluctuations over the nanosecond to picosecond range. F() motional correlation time distributions are exemplified with six CH and NH groups: M1 (ß-strand 1), T11 (turn 1), A26 (-helix), A34 (-helix), D40 (loop), and A48 (turn 2) through the structure of GB1. Solid lines give F() curves for 15NH, whereas dotted lines give F() curves for 13CH. F() curves were truncated at 100 psec because internal motions occurring on the picosecond time scale are not accurately defined.

Contributions from multiple Lorentzians was assessed and semiquantified by taking the line width of F() curves at half-height, 1/2 F()_max, a value that has been termed _LR (Idiyatullin et al. 2001). A single Lorentzian resulting from overall tumbling of a spherical molecule has an _LR value of 13.93. GB1, however, is ellipsoid in shape with a D_||/D_z ratio of 1.8. Because of this, _LR depends on the orientation of a particular C-H and N—H vector within the molecular frame of GB1. Using the program TENSOR II (Daragan and Mayo 1997) and the coordinates for GB1 averaged over 60 NMR-derived structures from the PDB database [accession code: GB1], values of _LR for C-H and N—H vectors in GB1 have been calculated and found to fall generally in the range from about 14 to 15 (Fig. 4A, B). Some vectors are oriented more parallel to the long axis of the molecule, yielding smaller values of _LR, whereas others are oriented more perpendicular to this axis, resulting in larger values of _LR. Experimentally determined values for _LR are plotted versus residue number in Figure 4C. In most cases, the observed _LR is larger than expected for overall tumbling alone, indicating that the shape of the molecule plays a minor role in this broadening phenomenon. Moreover, the magnitudes of _LR demonstrate that there are significant contributions to F() from internal motions occurring on the nanosecond time scale.

View larger version (49K): [in this window] [in a new window]   Figure 4. RL plotted versus residue number from GB1. Calculated values for RL are plotted for CH (A) and NH (B) vectors in GB1 and for experimentally determined values of RL for 13CH (open squares) and for 15NH (filled circles) (C). Values of RL for C—H and N—H vectors in GB1 were calculated using the program TENSOR II (Daragan and Mayo 1997) and the coordinates for GB1 averaged over 60 NMR-derived structures from the PDB (accession code GB1). Experimentally determined values for RL are plotted for 5°C. For calculated values of RL, the scale is more expanded to illustrate the effect through the sequence. Secondary structure elements in GB1 are shown between panels.

S²_CH and S²_NH values, plotted in Figure 5, fall approximately between 0.5 and 0.8. S²_NH values appear to be more structurally correlated, with generally lower values being observed for residues located at the N-terminus, the C-terminal part of the helix, and within turns 1 and 2 and the loop. This may have been anticipated because a computational analysis on peptide dynamics using the "internally restricted correlated rotation" (IRCR) model (Daragan and Mayo 1996b) indicates that S²_NH is more sensitive to structural differences than is S²_CH. Moreover, analysis of these data indicates that S²_NH is correlated to hydrogen bonding. In GB1, NHs of 36 out of 56 residues are involved in hydrogen bonds. An average of these S²_NH values is 0.76 ± 0.04, whereas the average S²_NH for all other nonhydrogen bonded NHs, aside from those of the N- and C-terminal residues, is 0.62 ± 0.05. Even when omitting turn and loop residues from the calculation, this difference remains. Another observation is that S²_CH values, for any given residue, tend to be larger than S²_NH values. As will be discussed later, the reason for this is related to more complicated CH bond motions involving correlated , , (side-chain) bond rotations.

View larger version (43K): [in this window] [in a new window]   Figure 5. 13CH and 15NH order parameters versus residue number. Order parameters were obtained using the F() deconvolution approach as described in the Materials and Methods section. Open circles are for 13CH and filled circles are for 15NH. Secondary structure elements in GB1 are shown between panels.

View larger version (51K): [in this window] [in a new window]   Figure 6. Fi()max and iR values plotted versus the residue number. iR (A) and Fi()max (B) values are plotted for 13CH (open circles) and 15NH (filled squares) versus the GB1 residue number. iR represents the inverse of the frequency at Fi()max. Secondary structure elements in GB1 are shown between panels. The insert exemplifies the function Fi() for 15NH for two residues M1 and A34 that results from deconvolution of F() into Fo() and Fi().

	Discussion

TOP Abstract Introduction Results Discussion Materials and methods References

Although a number of ¹⁵N NMR relaxation-based studies on protein dynamics have reported the presence of nanosecond time scale internal motions for a few residues (e.g., Clore et al. 1991; Barchi et al. 1994; Mandel et al. 1996; Coles et al. 1999; Berglund et al. 2000; Hidehito et al. 2000; Seewald et al. 2000; Zajicek et al. 2000; Baber et al. 2001; Canet et al. 2001), most protein dynamics studies do not, or perhaps cannot, accurately derive internal motional correlation times, primarily due to the paucity of NMR relaxation data and limitations of the motional model(s) used. In fact, two of these dynamics investigations (Barchi et al. 1994; Seewald et al. 2000) were performed on the same protein GB1, and nanosecond time scale internal motions were reported for only a handful of NHs, and even these were different in each study. Moreover, when internal motions are analyzed, discussion is limited most often to the picosecond time scale. However, even on an 800-MHz NMR spectrometer, the accessible frequency range is limited to about 160 psec and picosecond time scale internal motions (less than about 100 psec) cannot be accurately determined when overall tumbling is much slower than this limiting value, and order parameters are not at their lower limit as for fast internal rotations of methyl groups in side chains (Daragan and Mayo 1996a, 1997, 1998).

Given the present findings with protein GB1, it is plausible that all groups in all proteins undergo nanosecond time scale internal motions. This proposal is consistent with the concept forwarded by Frauenfelder et al. (1991) that there exists a hierarchy of internal motions in proteins. This hierarchy ranges from picoseconds to nanoseconds to microseconds to milliseconds and slower. NMR relaxation studies, which are most sensitive to motions occurring on the nanosecond to picosecond time scales, on small GXG tripeptides devoid of folded structure revealed that internal motions for backbone CH and NH groups fall in the 70- to 100-psec range (Mikhailov et al. 1999), whereas in protein GB1, internal motions for backbone groups fall in the subnanosecond to 5-nsec range, as well as in the picosecond range. Internal motions on the picosecond time scale were mostly omitted from this discussion primarily because these faster time scale motions are least accurately determined as mentioned above. Nevertheless, because spectral density functions for GB1 trail into the picosecond range, it is most probable that picosecond time scale internal motions (less than about 100 psec) contribute about 10% or less to the spectral density function. In any event, because of the prevalence of nanosecond time scale internal motions, caution should be exercised by investigators who only use the Lipari-Szabo model free approach to analyze NMR relaxation data. When overall tumbling and internal motions occur on the same time scale, multiple Lorentzians overlap, and this leads to overestimated order parameters and underestimated overall tumbling correlation times when using the Lipari-Szabo approach or any approach that uses fewer Lorentzians than the number of motional modes present. Using the F() approach avoids this problem.

On the nanosecond time scale, motional fluctuations for CHs are slower than they are for NHs. Slower internal motions for backbone CHs suggest more complicated, concerted types of motions. This interpretation is supported by results on partially folded peptides where NH backbone motions could be described by motions about a single bond, that is, the C-N -bond, whereas CH motions could only be explained fully by considering concerted , , and bond rotations (Ramirez et al. 1998; Idiyatullin et al. 2000). Furthermore, this idea is consistent with the observed larger order parameters for CHs compared to those for NHs. The average value for S² in ß-sheet segments is 0.7 for ¹³CH and 0.6 for ¹⁵NH, whereas the average value for S² for the helix is 0.65 for ¹³CH as well as for ¹⁵NH. Although most protein dynamics studies interpret S² values in terms of restricted motional amplitudes, angular variance is only one contribution to the order parameter. Another is the presence of correlated bond rotations, which can have a significant influence on S² (Daragan and Mayo 1996b, 1997). When considering both motional amplitudes and correlated motions, order parameters for CH and NH backbone bond motions can be expressed (Daragan and Mayo 1997) as:

(1a)

(1b)

(1c)

(1d)

c is the rotational correlation coefficient for _i and _i backbone rotations; and c_' is the rotational correlation coefficient for _i-1 and _i backbone rotations, and R² is the average squared amplitude of and rotations [R² = ]. For a short peptide segment, _i and _i bonds are illustrated in Figure 7. Notice that order parameters defined in this way (equation 1a–d) are structure dependent. For model peptides conformed in -helix and ß-strand (extended) structures, Daragan and Mayo (1996b) have estimated typical ranges for c_' and c correlation coefficients:

View larger version (11K): [in this window] [in a new window]   Figure 7. Sketch of peptide backbone. A tripeptide backbone is illustrated to show and backbone dihedral angles as discussed in the text.

(2a)

(2b)

(2c)

(2d)

Molecular dynamics calculations on protein GB1 (E. Ermakova, I. Nesmelova, V.A. Daragan, D. Idiyatullin, and K.H. Mayo, unpubl.) show that for most residues, c is closer to zero, whereas c_' is largely negative. In any event, the question here is how do these correlation coefficients affect the order parameter. For ¹⁵NH motions, correlated backbone motions will always act to minimize S²_NH because c_' is always negative. On the other hand, for ¹³CH motions, c is negative and c_' acts to increase S². Because it is reasonable to assume that R²_CH has about the same value as R²_NH (Daragan and Mayo 1997), the observation that S²_CH is greater than S²_NH, especially for ß-sheet residues, supports the proposition that CH fluctuations are related to correlated bond rotations.

	Materials and methods

TOP Abstract Introduction Results Discussion Materials and methods References

 
Protein production
The 56-residue protein GB1 was produced as a recombinant protein as described by Barchi et al. (1994). Escherichia coli containing the expression system for GB1 were grown on M9 minimal media containing glucose as the carbon source, 20% of which was uniformly ¹³C-enriched glucose. The ¹³C-enriched protein was purified by HPLC using a linear acetonitrile/water gradient, and purity was checked by MALDI-TOF mass spectrometry and analytical HPLC on a C18 Bondclone (Phenomenex) column. For NMR measurements, freeze-dried samples were dissolved in D₂O in 20 mM potassium phosphate. Protein concentration, determined from the dry weight of freeze-dried samples, was 10 mg/mL. The pH was adjusted to pH 5.25 by adding microliter quantities of NaOD or DCl.

NMR relaxation experiments
With ¹³C- and ¹⁵N-enriched GB1, spin-lattice (T₁), rotating frame spin-lattice (T₁) relaxation times and heteronuclear NOEs were measured at three Larmor precession frequencies (¹H frequencies of 500, 600, and 800 MHz) on Varian Inova 500, 600, and 800 NMR spectrometers equipped with triple-resonance probes. The temperature was held at 5°C, with calibration being performed by using the chemical shifts of resonances from methanol.

¹³C and ¹⁵N T₁ and T₁ relaxation times were measured by using the HSQCSE pulse sequence (Yamazaki et al. 1994; Farrow et al. 1995), which employs pulsed field gradients for the coherence transfer pathway whereby magnetization passes from ¹H to ¹³C (or ¹⁵N) and back again to ¹H for observation. To attenuate crosscorrelation between dipolar and chemical-shift anisotropy mechanisms during the relaxation period, an even number of 180° ¹H pulses with alternating phase were applied every 5 msec (Kay et al. 1992; Palmer III et al. 1992). Spectra were recorded by using seven different relaxation delays with values from 10 msec up to 2T_i, where T_i is the respective relaxation time. For all T₁ and T₁ experiments, the recycle period was 1.7 sec, and all relaxation curves followed single exponential decay. To prevent ¹³C-¹³C scalar and dipolar coupling effects on relaxation parameters (Yamazaki et al. 1994), GB1 was randomly ¹³C-enriched to only 20%. In this case, residual effects on T₁ values from neighboring ¹³C groups was less than 5% and negligible for ¹³C-{¹H}NOEs. This was estimated by comparing relaxation parameters measured using 100% ¹³C-enriched GB1 and 20% ¹³C-enriched GB1. For T₁ relaxation experiments, the carrier for ¹³C was set at 59 ppm and the frequency of the field lock was 2.6 kHz.

Steady-state {¹H }-¹³C and {¹H }-¹⁵N NOEs were determined from two spectra recorded with proton broadband irradiation and in the absence of proton saturation (Yamazaki et al. 1994; Farrow et al. 1995). Saturation was achieved by application of 120° ¹H pulses applied every 5 msec (Markley et al. 1971) during repetition times of 3 sec.

All experiments were repeated two times for T₁ and T₁ and four times for NOEs. Average values are reported.

Relaxation data analysis
Because relaxation measurements were performed at high frequency where the influence of chemical shift anisotropy and chemical exchange terms are significant, the square dependence of these terms on the value of the magnetic field was used during the minimization procedure described below. The CSA constants used were -25 ppm for ¹³C (Peng and Wagner 1994) and -170 ppm for ¹⁵N (Tjandra et al. 1996). Moreover, the length of the amide HN bond used in ¹⁵N relaxation studies is the subject of an ongoing debate (Korzhnev et al. 2001). Because this work does not propose to solve the problem of NH bond vibrational corrections, a traditional value of 1.02 Å for the length of the N—H bond was used (Kay et al. 1989), which is widely employed by other authors in field.

We have recently developed an approach to analyzing NMR relaxation data, which yields a correlation time distribution for molecular motion without assuming the nature of the molecular motions or the number of motional modes, that is, Lorentzians (Idiyatullin et al. 2001). We introduced the function F(), which is related to the spectral density function as:

(3)

Although the function J() is almost equivalent to the imaginary part of the dielectric spectrum as described by Debye (1929) and has been used for many years in various branches of science, we are not aware that it has ever been used to interpret NMR relaxation data as is being done here.

F() can be represented as the sum of the peaks corresponding to the motional modes, and the spectral density function J() can be expressed as a sum of Lorentzians:

(4)

If motional correlation times are well separated, positions of peak maxima on the frequency axis, , are equal to the inverse of the correlation times 1/_i and the peak maxima are equal to the weighting coefficients c_i in equation 4. Although values for c_i and _i for i > 2 cannot be determined accurately, F() can be fairly accurately described for any number of Lorentzians. This is due to the fact that during minimization F() remains stable, even though mutual switching of c_i and _i values prevents determination of their specific values. For J(0) = c₀₀ + c₁₁, for example, various combinations of c₀, ₀, c₁, and ₁ give the same value of J(0). In this regard, J(0) can be determined more accurately than can the individual terms in J(0). The same can be stated for F(), that is, many combinations of c_i and _i give the same value of F() over a wide range of frequencies. These properties of F() allow the function to be used easily in the interpretation of NMR relaxation data where there are a large number of motional modes.

The largest peak in a F() curve corresponds to low-frequency motions associated with overall tumbling, and any other peaks or shoulders on the higher frequency side result from the presence of a distribution of internal motional correlation times on the nanosecond and picosecond time scales. The highest frequency peak, that is, picosecond time scale, cannot be determined accurately because the accessible spectrometer frequency range is limited, with a 900-MHz spectrometer, for example, to correlation times in the frequency range up to 2(900 + 225) MHz = 7 GHz (_H + _C), corresponding to a correlation time of 140 psec. The height of the major peak, F()_max, is equal to c₀ in equation 4 when there is no nanosecond time scale internal motions. For molecules tumbling isotropically, the coefficient c₀ can be interpreted as the squared order parameter. However, when nanosecond time scale internal motions are present, Lorentzians for overall tumbling and internal motions overlap and lead to the inequality:

(5)

When _o and _i are not well-separated, the presence of nanosecond time scale internal motions may be reflected in the half-height line width, _LR, of the major peak of F(). In effect, _LR = _R/_L, where _R and _L are the high- and low-frequency positions taken at the half-height of F(), that is, F(_R) = F(_L) = 0.5F()_max. If the spectral density function can be described by a single Lorentzian, then _LR = 13.93.

Using ¹³C and ¹⁵N relaxation data, F() curves, which are independent of any motional model, were determined over the frequency range 0 < <6.28 x 10⁹ rad/sec by using the Monte Carlo minimization protocol described previously by Idiyatullin et al. (2001) using the program FRELAN (www.nmr-relaxation.com) developed in our lab. Because is inversely related to the correlation time, these plots, in fact, give the distribution of motional correlation times over the nanosecond to picosecond range. These distributions are truncated at about 100 psec because of inaccuracies in determining F() at shorter correlation times, tbat is, higher frequencies, where actual experimental data are lacking. Three main parameters define F(): F()_max, _max, and _LR, where _max is the inverse of the frequency at F()_max. With well-separated correlation time distributions for overall tumbling and internal motion, _max will be close to the value of the correlation time for overall tumbling, ₀. However, if internal motions have correlation times close to ₀, the value _max will be less than the value of ₀.

To obtain parameters related to overall tumbling and internal motion, F() was deconvoluted into two parts:

(6)

where the weighting coefficient, c₀, can be interpreted as the squared order parameter, S², and F₀() contains overall tumbling correlation times:

(7)

and F_i() contains correlation times for internal motions:

(8)

To simplify the deconvolution procedure, it was assumed that F₀() can be described by a single Lorentzian because correlation times in equation 7 are narrowly distributed. In this case, one needs to determine x = _o (x > _max) to minimize F_i() for all < _max under the condition:

(9)

By doing this, correlation times which are close to _o become accentuated and maxima associated with overall tumbling and internal motions become better separated. By deriving F_i() in this way, the internal motional correlation time distribution can be visualized and the associated parameters F_i()_max and _i can be used as general terms to describe internal motions for N—H and C—H vectors.

To calculate F(), one needs to determine the coefficients c_i and correlation times _i for i = 0, 1, . . ., N, with N being as large as possible for any given set of experimental data. For example, with three relaxation parameters (e.g., T₁, T₂, and NOE) acquired at three magnetic field strengths, nine fitting parameters, that is, five Lorentzians, can be determined by taking into account that c_i = 1. Although it is practically impossible to obtain reliable values for c_i and _i for N > 2, linear combinations and other functions of c_i and _i over a given frequency range are very stable and can be determined accurately (LeMaster 1999; Idiyatullin et al. 2001). This is basically the key to obtaining the function F() from experiment.

One way to calculate F() is to use the Monte Carlo procedure and a simple step-by-step algorithm outlined below:

1. Randomly take five values of the internal motional correlation time _i and amplitudes c_i. To cover a range of possible errors and to account for the influence of rotational anisotropy, t₂ should be about 50 to 70% larger than the estimated overall correlation time _o for a given protein molecule. To estimate _o, the empirical equation can be used (Daragan and Mayo 1997):
   
   

   (10)

   
   where n_R is the number of residues, and T is the temperature in Kelvin.
   

2. Using minimization program, find appropriate values c_i and _i that best fit the experimental data, that is, minimize the sum of the nine terms:
   
   

   (11)

   
   R_j are experimental parameters; R_j^calc are calculated parameters, and _j are the experimental errors in determining R_j. If ² < 1, then store the set of c_i and _i. If ² > 1, then repeat steps 1 and 2. By repeating this process n times, one obtains n sets of c_i and _i.
   

3. For each k-th set of c_i and _i, calculate F()_k. The probability P_k of finding the k-th set of c_i and _i is (Andrec et al. 1999):
   
   

   (12)

   
   R_jk^calc are calculated parameters for the k-th set.
   

4. 5n-weighted Lorentzians can now be used to describe F(). The average value of F() and the standard deviation () of F() can be calculated as:
   
   

   (13a)

   
   
   

   (13b)

   
   with p_k = P_k/ P_k.
   

	Acknowledgments

 
This work was supported by a research grant from the National Institutes of Health (NIH, GM-58005) and benefited from use of the high field NMR facility at the University of Minnesota. NMR instrumentation was provided with funds from the NSF (BIR-961477), the University of Minnesota Medical School, and the Minnesota Medical Foundation. We also thank Judy Haseman for preparing ¹³C-and ¹⁵N-enriched samples of protein GB1.

The publication costs of this article were defrayed in part by payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 USC section 1734 solely to indicate this fact.

	References

TOP Abstract Introduction Results Discussion Materials and methods References

 
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