Dynamics of ¹³C uniformly enriched cell wall polysaccharide of Streptococcus mitis J22 studied by ¹³C relaxation rates ¹³C R₁, R_1rho and ¹³C-{¹H} NOE

We have studied the dynamics of the motion of a complex polysaccharide having seven sugar residues in the repeating subunit and which is a receptor for lectin interaction in the coaggregation of oral bacteria. Measurements of the longitudinal and the rotating frame relaxation rates and the heteronuclear nuclear Overhauser effects were carried out on a uniformly ¹³C enriched sample using pulse sequences chosen to minimize the effects of ¹³C-¹³C coupling and cross relaxation. T₁ and T_1rho measurements both showed single exponential decay for the anomeric carbon atom resonances of the polysaccharide. The results show the polymer to be highly flexible with a hinge at the 1->6 linked galactofuranoside residue. Since there is no generally accepted scheme for interpreting polysaccharide dynamics, several different methods of data analysis were used including a reduced spectral density function method as well as several different methods in which a series of isotropically decaying rotational correlation functions are assumed. The different analyses all show that there are differing amounts of internal motion in the different residues of the polysaccharide. One possible interpretation of the data, which uses an extended version of the model-free treatment indicates that picosecond motion is exhibited to a similar degree by all the residues in addition to a slower motion on the nanosecond time scale whose amplitude is greatest in the hinge region around the 1->6 linked galactofuranoside residue in the polysaccharide.

While the question of the conformation and dynamics of complex polysaccharides has received considerable attention, a number of questions remain concerning the nature and extent of conformational exchange. The notion of flexibility is generally recognized to be important at least for certain oligosaccharide linkages, (Rutherford et al., 1993; Poppe and van Halbeek, 1992; Maler et al., 1996; Xu et al., 1996a,b). Although it is accepted that exchange can occur among different conformations of the glycosidic linkage, the kinetics of that exchange remains uncertain. NMR relaxation data provide an experimental approach to this question. Directly detected ¹³C T₁ data have been reported for natural abundance sucrose by McCain and Markley (1986) which were interpreted in terms of motion on the picosecond time scale attributed to sugar puckering. An alternative approach in which ¹³C T₁ was measured for a series of homologous oligosaccharides was used to show that motion of a polysaccharide can be viewed in terms of a persistence length of some 10 to 15 sugar units. Polymers longer than this show the same values of T₁ for the central residues regardless of chain length (Benesi and Brant, 1985; Brant et al., 1995). These directly detected ¹³C relaxation experiments generally require sufficiently high concentrations of sample that the viscosity of the solution introduces some concentration dependence to the measured relaxation rates.

Indirect detection offers an improvement in the sensitivity of relaxation rate measurements. In relaxation experiments on the small glycoprotein, ribonuclease B, Rutherford et al., (1993) reported T₁ and T₂ for natural abundance ¹³C. The data showed limited signal to noise at long relaxation delays but were interpreted with the 'model-free' formalism of Lipari and Szabo (1982a,b) to show that the oligosaccharide which is covalently tethered to the protein exhibits internal motion on a time scale of approximately 200 picoseconds. Poppe et al (1994) in experiments on the ganglioside GD1a inserted in a micelle, observed motions of the oligosaccharide on a time scale of about 300 ps superimposed on a tumbling time of 2.8 ns for the micelle. Hricovini and Torri (1995) have reported T₁, T₂, and ¹H NOE data for a pentasaccharide from heparin which was interpreted by the model-free formalism to show a complicated internal motion on the time scale of 15-50 ps.

The recent introduction of isotopic enrichment into proteins with the stable spin 1/2 isotopes, ¹³C and ¹⁵N, has stimulated interest in relaxation rate measurements to study protein dynamics (Peng and Wagner, 1992; Clore et al., 1990b; Kay et al., 1989; Stone et al., 1992). The improvements in the signal to noise ratios brought about by indirect detection and isotopic enrichment have made possible very accurate measurements of the ¹⁵N relaxation rates and very sophisticated theoretical treatments which give detailed insight into the internal motions of proteins. While enrichment of proteins with ¹³C is also common, measurement of relaxation rates in uniformly highly enriched proteins presents some technical challenges in the design of the NMR experiments, (Yamazaki et al., 1994).

We have recently reported the preparation of a cell wall polysaccharide from Streptococcus mitis J22 which is uniformly ¹³C labeled (Gitti et al., 1994). We have carried out measurements of NOE and ³J_CH for this polymer and have reported molecular modeling studies which establish that it is a rather flexible polysaccharide, (Xu and Bush, accompanying paper). In this paper we report a method for measuring very accurate ¹³C relaxation rates for the seven distinct anomeric carbon atoms in the repeating subunit of this polymer along with several different schemes for interpreting the data in terms of polymer dynamics on the picosecond to nanosecond time scale.

The polysaccharide sample from Streptococcus mitis J22 was biosynthetically enriched with ¹³C to a level of approximately 96% according to the procedure described previously, (Gitti et al., 1994). The sample was dissolved in 99.96% D₂O at a concentration of 7-10 mg/ml at neutral pH without any buffer. The structure of the polysaccharide is, (Abeygunawardana et al., 1990):
NMR experiments were carried on a GE-Omega 500 PSG system controlled by Sun Sparc workstation with reverse-detection through an RPT probe at 24.0°. The pulse sequences in Figure 1 for measuring ¹³C T₁, T_1rho and ¹³C-{¹H} NOE were based on the scheme for relaxation rate measurements on uniformly isotope-labeled biopolymers by Yamazaki et al. (1994). The ¹³C carrier frequency was set within the anomeric carbon region (104-94 ppm) and all ¹³C pulses were selective for the anomeric carbons with a low power level so that the other carbon atoms, such as C2, were not excited. Therefore, the cross relaxation terms for carbons in uniformly isotope-enriched sample did not contribute to the relaxation decay of anomeric carbons and Hartman-Hahn effect was avoided during the carbon spin-lock time for T_1rho measurements. The proton carrier frequency was 500.1321140 MHz with a sweep width of 2564.1 Hz. The proton dimension contained 256 complex data points. %Proton 90 degree pulse was 10.5 ms. The carbon frequency was set in the middle of anomeric carbon chemical shift region (125.770591 MHz) with a spectral width of 4000.0 Hz. The 90° pulse for carbon was 155 ms. Carbon decoupling during acquisition was carried out with WALTZ-16 with field strength of 1612.9 Hz. The ¹J_CH coupling during carbon constant time evolution was removed with a continuous low power proton pulse. Proton decoupling during the relaxation delay period for R₁ and R_1rho measurements was done with a series of 180° pulses which eliminate the cross correlation between ¹H-¹³C dipolar interaction and ¹³C CSA, or cross correlation between ¹³C-¹³C and ¹H-¹³C dipolar interaction. The pre-saturation of the proton resonances for ¹³C-{¹H} NOE measurement was carried out with a series of proton 120° pulses. The carbon 90° pulse of phase phi₃ in the relaxation measurements of T₁ places carbon magnetization along +z and -z on alternate scans to cancel the effect of the carbon longitudinal magnetization at equilibrium (Sklenar et al., 1987). The delay preceding the relaxation delay in the R_1rho measurement aligned magnetization with an offset from the carrier frequency exactly along the orientation of effective B₁ field and the delay after the relaxation delay returned all the magnetization to the horizontal plane. The data matrices contained 32 complex data blocks and 64 scans per block in the R₁ and NOE data and 128 scans per block in the R_1rho data. The delays for the R₁ measurement varied between 10 and 400ms and those for the R_1rho measurement varied between 10 and 120 ms. The carbon carrier frequency was set at varying values close to individual anomeric carbon resonances for the R_1rho measurements and the relaxation rate measurements were repeated by varying carbon carrier frequency, power level and relaxation delay to avoid artifacts and to assess experimental reproducibility. NOE measurements were done by acquiring two spectra with and without proton presaturation during pulse delay (2.5 s).

The NMR data were processed with Felix 2.30 (Biosym) on a Silicon Graphics Indigo workstation. The proton dimension was apodized with a 90° shifted sine-bell function and then zero-filled to 1024 points before Fourier transformation. The data along the carbon dimension were apodized with a 90° shifted sine-bell function and zero-filled to 128 points before Fourier transformation. The peak height was read from each properly phased cross peak. R₁ and R_1rho data were fitted to a two-parameter exponential function shown by equations (1) and (2) with software based on nonlinear least square data fitting using the Levenberg-Marquardt algorithm, (Press et al., 1989).
¹³C-{¹H} NOE values were obtained from the peak ratios of spectra with and without proton saturation. There was little difference in the relaxation rate values when either peak height or peak volume was used.

The experimental R₁, R_1rho and ¹³C-{¹H} NOE data were analyzed in two different ways. In the first method, the reduced spectral density function was used according to methods described by Farrow et al., (1995), by Ishima and Nagyama (1995) and by Lefevre et al., (1996). An analysis was also done using the ``model-free'' formalism which assumes that the rotational correlation function is composed of a small number of exponentially decaying components (Lipari and Szabo, 1982a; 1982b; Clore et al., 1990a,b). The data fitting for the model-free formalism was carried out with a conjugate gradient method on the following merit function, chi², (Press et al., 1989).
In eq. 3, k₁, k₂ and k₃ are coefficients adjusted to facilitate analysis of the experimental data. The parameters describing the molecular motion in the model-free analysis were adjusted and chi² minimized with k₁=1.0, k₂=1.0 and k₃=0.0 . The calculated NOE was monitored to assure good agreement with experimental values. This procedure was followed to avoid trapping by barriers in this non-linear fitting.

Our analysis is based on theoretical calculation of the R₁, R_1rho and ¹³C-{¹H} NOE data for the anomeric ¹³C resonances of the polysaccharide following Clore et al., (1990a) and Peng and Wagner (1992). The relaxation rate R₁, which we have measured for the anomeric ¹³C resonances includes all the autorelaxation terms (rho_C1,H1 and rho__C1,C2), but does not contain the cross relaxation term sigma_C1,C2 because carbon excitation was selective for only the anomeric carbons, and does not contain sigma_{H1, C1} due to proton saturation during the relaxation delay ( Yamazaki et al., 1994). Therefore the dipolar contributions of H1 and of C2 to the relaxation of the anomeric carbon atom are formally identical. Chemical shift anisotropy (CSA) is included in the last term of the R₁ and R_1rho expressions. It is important to recognize that R₁ contains J(0) as a result of ¹³C,¹³C autorelaxation, and this term, although small compared with other relaxation terms in R₁, increases with increasing rotational correlation time.

For the case discussed here, rho_C = R₁ in eq. (6).

In our application, in the equations 4 and 5 for R₁ and R_1rho, the terms in the summation over i and j include C1 for j and both H1 and C2 for i. The values of the constants in equations (4), (5) and (6) are the following: h=6.626 x 10-27 erg s, gamma_H = 2.6752 x 10⁴ gauss^-1s^-1, gamma_C=6.728 x 10³ gauss^-1s^-1, r_CH = 1.09 x 10^-8 cm, r_CC=1.515 x 10^-8 cm. The constant d² = 0.1 (gamma_i gamma_j h/2 pi)².

omega_e² = omega₁² + delta² where omega₁ = 2 pi H₁ and H₁ is the proton B₁ spin lock field strength in Hz. delta is the offset from the spin lock carrier frequency. delta is the chemical shift anisotropy of ¹³C (CSA) which is taken as 50.0 ppm (Bovey, 1980; Hricovini and Torri, 1995). The spin locking tilt angle is beta and sin beta =omega₁ \omega_e.

The uncertainties in spectral density function analysis and model-free formalism as obtained from fitting experimental data were estimated with Monte Carlo simulation. The measured experimental values and their error ranges were assumed as the mean and standard deviation of a Gaussian distribution. The relaxation rates were calculated from each set of 50,000 randomly generated parameters in the spectral density function analysis or model-free formalism. If the calculated relaxation rates fell within 99% of the above Gaussian distribution curve, the value of dynamics parameter was taken into calculating the statistical standard deviation of these parameters.

The dilute polysaccharide used in these experiments eliminates problems which arise from dependence of relaxation rates on concentration which has been encountered in some previous polysaccharide studies. Nevertheless, the signal to noise ratio for the relaxation data is excellent with a ¹³C enriched sample when compared to natural abundance polysaccharides. The data illustrated in Figure 2 show little scatter, even at long delay times and the fit to a single exponential decay is excellent. Table 1 and Figure 3 and 4 show that there are small differences in the T₁ and NOE data among the signals for the seven anomeric carbon atoms.

Table 1 Experimental data of 13C relaxation rates

residue	R1 s-1a   R1r s-1a	  NOE a	R1r/R1
a	1.22	   10.33	1.08	8.47
b	1.28	   11.43	1.05	8.93
g	1.32	   10.75	1.03	8.14
c	1.11	   7.97		1.26	7.18
d	1.10	   5.26		1.20	4.78
e	1.19	   7.06		1.14	5.93
f	1.16	   9.17		1.19	7.90

a. The experimental error of the measurements is about 10% 
of the values as indicated by the error bar on Figures 3 and 4

One method for analysis of relaxation data, which requires the determination of additional relaxation rates, is the full spectral density function method of Peng and Wagner (1992). With three sets of relaxation rates as reported in this work, an alternative reduced spectral density function analysis is possible, (Lefevre et al., 1996; Ishima and Nagayama, 1995; Farrow et al., 1995). Equations (4-7) can be simplified with the assumption of the form of eq. (8) for J(omega).
The first and second terms in (8) represent contributions to J(omega) from overall tumbling and from internal motion respectively, (Farrow et al., 1995). With the inclusion of eq. (8), the equations (4-7) can be reduced to eqs. (9)-(11) in which three new frequencies, omega_p, omega_q and omega_r are defined.
The new forms of Equations (9)-(11) are recast with appropriate gyromagnetic ratios of ¹H and ¹³C.
Equations (12-14) are analogous to eq. (11) of Farrow et al., (1995) with ¹³C substituted for ¹⁵N. In order to simplify eq. (5) to accommodate the reduced spectral density function analysis, it is necessary to introduce further simplifications into the treatment of the R_1rho data. Because the offset of the anomeric carbon resonance frequency from the carrier is small compared to the B₁ field strength, the rotating frame is tilted close to the x,y plane. Thereforesin²(beta) = 1.0 and sin⁴(beta/2) + cos⁴(beta/2) = 0.5 in eq. (5) and omega_e is set =0. The validity of an additional assumption, that relaxation of C1 by C2 can be neglected, will be discussed below. In this particular system the ¹³C-¹³C auto-relaxation is only a few percent of the total relaxation. With these approximations, eq. (5) may be written as,
For this situation, eq. (15) becomes the same as the formula for T₂ relaxation. Similarly Equations (4) and (7) can be recast using eqs. (12-14) into eqs. (16) and (17) respectively.
Eqs. (15-17) involve the spectral density sampled at omega_C, at 0 and at three frequencies close to and above omega_H. The situation discussed here for ¹³C differs somewhat from that of ¹⁵N relaxation treated by Farrow et al., (1995) in that the three frequencies differ more substantially from omega_H as a result of the difference in sign and magnitude of the magnetogyric ratios of the heteronuclei. Thus, two different approaches are taken to reduce the five spectral density function values in Equations (15)-(17) to three needed for our analysis. It is necessary to assume a description of the shape of the decay of the spectral density function at and above the proton frequency, omega_H. Although the exact values of spectral density function will be greatly influenced by the nature of the assumptions, we will focus on the correlation of the internal motion on the nano-second scale with the spectral density function values at zero frequency. These values are less sensitive to the details of the assumptions as will be seen in the following. In the first case we assume that J(omega_H) is essentially constant near omega_H. Then J(1.116omega_H) and J(1.563omega_H) may be replaced with J (1.058omega_H). The calculated J(0), J(omega_C), J(1.058omega_H), J(1.116omega_H) and J(1.563omega_H) derived by this method are listed in Table 2.

Table 2 Reduced spectral density function analysis (Method 1)

(s/rad)		J(0) 		J(wC) 		J(1.058wH)
						or  J(1.116wH)
						or  J(1.563wH)
Residues	x 10-9 		x 10-10 	x 10-12
a		1.06±0.01	1.84±0.13	2.29±1.87
b		1.18±0.01	1.95±0.08	1.50±2.06
g		1.09±0.01	2.03±0.07	0.93±0.98
c		0.79±0.01	1.56±0.07	6.76±2.17
d		0.48±0.01	1.59±0.07	5.15±2.08
e		0.68±0.01	1.75±0.08	3.90±1.81
f		0.92±0.01	1.68±0.07	5.16±1.77

A second approach may provide a better description of the decay of the shape of J(omega) at or beyond the proton frequency. The assumption that J(omega) decays like 1 / omega² is equivalent to the assumption that a single exponentially decaying rotational correlation function dominates in the region of omega_H. Under this assumption J(1.116 omega_H) and J(1.563omega_H) can be estimated from the relation J(epsilon omega_H) = (1.058/epsilon)² × J(1.058omega_H) where epsilon =1.116 and 1.563. The calculated J(0), J(omega_C), J(1.058omega_H), J(1.116omega_H) and J(1.563omega_H) derived according to this method are listed in Table 3 and plotted in Figure 4

Table 3 Reduced spectral density function analysis (Method 2)

(s/rad)		J(0)		J(wC)		J(1.058wH) 	J(1.116wH)	J(1.563wH) 
Residues	x 10-9 		x 10-10		x 10-12		x 10-12		x 10-12
a		1.05±0.01	1.79±0.08	4.99±4.58	4.49±4.12	2.29±2.10
b		1.18±0.01	1.92±0.13	3.27±4.06	2.94±3.65	1.50±1.86
g		1.09±0.01	2.00±0.08	2.02±1.60	1.82±1.44	0.93±0.73
c		0.79±0.01	1.41±0.12	14.75±4.24	13.26±3.82	6.76±1.94
d		0.47±0.01	1.47±0.11	11.25±4.52	10.11±4.07	5.15±2.07
e		0.67±0.01	1.67±0.09	8.52±4.93	7.66±4.43	3.90±2.26
f		0.91±0.01	1.56±0.06	11.27±4.20	10.13±3.78	5.16±1.92

The results of the two treatments summarized in Tables 2 and 3 are quite different for J(omega) in the region of the proton frequency, and show a slight difference for the carbon frequency but very similar values for J(0). The uncertainties in spectral density function values at high frequency (at and beyond the proton frequency) are much larger than those at low or zero frequency. The values of J(0) derived from the reduced spectral function analysis are reliable and informative concerning the internal motion on the nano or sub-nano second time scale. Figure 5 illustrates graphically the data of Table 3 showing that J(0) varies among the residues of the polymer subunit in a similar manner to that of R_1rho.

One may raise the objection to the analysis described above that it is not so appropriate for ¹³C as it is for ¹⁵N relaxation for which it was originally derived. It is possible to present an alternative interpretation of our data using the treatment of Lipari and Szabo (1982a,b) which provides a more concrete description of the internal motion including both time and amplitudes. This model, which is known as the 'model-free' treatment, assumes a series of exponentially decaying isotropic rotational correlation functions. There are several different versions of this treatment available, which differ in the number of exponentially decaying components and in assumptions about their rates. In the simplest formulation (Lipari and Szabo, 1982a,b), an overall tumbling described by tau_R is modulated by a very fast internal rotation tau_e. In order to survey the range of overall tumbling times (tau_R) which best fit the relaxation data for each individual residue, we minimized the merit function shown by eq. 3 with the simple isotropic diffusion Lipari-Szabo model described by eq. (18) for each individual anomeric carbon resonance.
In eq. (18), tau = tau_R tau_e /(tau_R +tau_e). Values of S², tau_R and tau_e were derived using eq. 3 to fit R₁ and R_1rho with an iterative adjustment to fit the NOE data. The estimate of tau_e in Table 4 is crude since the data fit is not very sensitive to this parameter. While the fit of Table 4 to the data is satisfactory, the use of different tau_R values for different residues is not fully consistent with the theory in which a single molecular tumbling rate is modulated by faster internal motion. The overall tumbling times tau_R for residues a, b, g and f are similar, but those for residues d, e and c are substantially faster.

Table 4 Data from individual residue fitting to Lipari-Szabo model
(Equation 18)


Residues tauR (ns)	taue (ns)	   S2		R1(s-1)	R1r(s-1)  NOE
a	4.6 ± 0.05	0.002 ± 0.001	0.42 ± 0.02	1.21	10.3	1.17
b	4.7 ± 0.05	0.001 ± 0.000	0.45 ± 0.02	1.27	11.4	1.16
g	4.4 ± 0.05	0.0006 ± 0.000	0.46 ± 0.02	1.33	10.7	1.15
c	4.1 ± 0.05	0.0008 ± 0.0002	0.36 ± 0.01	1.12	7.97	1.16
d	3.2 ± 0.06	0.0008 ± 0.0002	0.29 ± 0.01	1.10	5.25	1.17
e	3.6 ± 0.06	0.0001 ± 0.0000	0.36 ± 0.02	1.21	7.04	1.15
f	4.4 ± 0.05	0.001 ± 0.000	0.39 ± 0.01	1.16	9.17	1.16

For this reason, we have attempted to fit the data to a single overall tumbling time, tau_R, modulated by internal motions which differ among the residues. The overall tumbling time of residue b, which is the largest from Table 4, is selected and the fitting was accomplished by minimizing the merit function chi² shown in Equation 3. The result of this iterative fitting of an internal motion time tau_e and order parameter for each individual residue according to eq. (18), is shown in Table 5.

Table 5 Data from fitting to Lipari-Szabo model with tauR=4.7ns (Equation 18)


Residue	   taue (ns)	   S2		R1 (s-1)R1r (s-1)NOE
a	0.006 ± 0.002	0.41 ± 0.02	1.22	10.3	1.26
b	0.002 ± 0.001	0.45 ± 0.02	1.28	11.4	1.17
g	0.012 ± 0.003	0.42 ± 0.02	1.32	10.8	1.34
c	0.017 ± 0.003	0.31 ± 0.02	1.11	7.97	1.54
d	0.032 ± 0.002	0.19 ± 0.02	1.10	5.25	2.04
e	0.028 ± 0.003	0.27 ± 0.02	1.19	7.06	1.81
f	0.010 ± 0.003	0.37 ± 0.02	1.16	9.17	1.36

The order parameters of internal motion indicate that the residue d (galactofuranose) has the largest internal motion amplitude and residue b is relatively restricted in its internal motion. All the residues show much smaller order parameters than the values found for rigid biopolymers. The internal motion is on the time scale from one to tens of pico-seconds which is 2-3 orders of magnitude smaller than the overall tumbling time, tau_R. However, the fitting is not satisfactory since the NOE values calculated with the parameters of this model for internal motion are larger than the experimental values.

Although one possible explanation for the unsatisfactory fit of Table 5 could be anisotropic tumbling of the polysaccharide subunit, (Schurr et al. 1994; Hricovini and Tori, 1995), we have some evidence indicating that anisotropy is not significant. In model building based on ³J_CH data and NOE data for the polysaccharide of S. mitis J22, we have observed that the orientation of the C1-H1 vectors of residues a, b, g and f are quite different so that it is likely that these vectors would differ in their orientation with respect to the principal axis of any anisotropic motion, (Xu and Bush, accompanying paper). Nevertheless, the tau_R of these residues are similar and the most pronounced differences are seen for the galactofuranoside, residue d (Table 4). In addition, ¹H NOE studies on an isolated heptasaccharide gave values which are very similar to NOE data for the intact polysaccharide providing evidence that the contribution of anisotropic motion in the polysaccharide is not important, (Xu et al., 1996b). Therefore, we conclude that the differences in tau_R among the residues of the polysaccharide do not arise from anisotropy.

Other possible explanations for the unsatisfactory fit to the model include slow chemical exchange processes and multiple modes of internal motion on different time scales. The present data provide no evidence for slow exchange processes and we were unable to reconcile the data of Table 5 with any reasonable assumptions regarding exchange. But Clore et al. (1990a,b) proposed a model which is an extension of the original simple Lipari-Szabo model and which includes three different types of exponentially decaying rotational correlations. This formalism includes two different internal motions, one fast characterized by tau_f and one slower motion characterized by tau_s, which are superimposed on the overall molecular tumbling. Eq. (19) describes J(omega) for this model and a reduced formalism is given in (eq. 20) which is normally used because the second term in eq. (19) is small as a result of small tau_f.
In eqs. (19) and (20), The results of fitting our data to eq. (20) by minimizing chi² in eq. 3 is listed in Table 6 and a bar plot of the order parameters for the seven residues of the repeating subunit of the polysaccharide is shown in Figure 6.

Table 6 Data from fitting to Clore's model with tauR=4.7ns 
(Equation 20)


residues   Sf2		   Ss2		   taus (ns)	R1 (s-1)R1r (s-1)NOE
a	0.43 ± 0.02	0.96 ± 0.03	1.20 ± 0.35	1.22	10.3	1.15
b	0.46 ± 0.01	0.99 ± 0.07	0.30 ± 0.09	1.28	11.4	1.16
g	0.45 ± 0.01	0.94 ± 0.03	1.10 ± 0.32	1.32	10.8	1.16
c	0.35 ± 0.01	0.87 ± 0.04	1.10 ± 0.32	1.11	7.97	1.19
d	0.29 ± 0.01	0.63 ± 0.05	1.00 ± 0.29	1.10	5.25	1.27
e	0.34 ± 0.01	0.75 ± 0.04	1.10 ± 0.32	1.19	7.06	1.22
f	0.39 ± 0.01	0.93 ± 0.03	1.20 ± 0.35	1.16	9.17	1.16

The rates calculated with eq. (20) in Table 6 fit rather well with the experimental data of Table 1. While the fit does not guarantee that this model correctly describes the motions of the polysaccharide, it does offer a plausible framework for describing possible motions. The order parameters (S²) of internal motion calculated from the product of S_f²s² b but much smaller for residues c, d and e.

We have also used the total error function (eq. 21) (Dellwo and Wand, 1989) to evaluate the quality of data fitting to model-free formalism by Clore et al.,(1990).
In eq. (21), the summation extends over all N residues and k denotes the number of measured variables used in the data fit. The value of E for fitting the results in Table 6 is 0.157, which indicates that this is an excellent data fit to the model of Clore et al. (1990b) compared to the fitting for proteins.

It is clear from the data analysis according to the formalisms of Lipari and Szabo (1982a,b) and of Clore et al. (1990a) that the order parameters generally do not depend strongly on the choice of model. The order parameter S² for residue d is the smallest and for residue b is the largest among the seven residues. Regardless of the choice of model, the residues fall into two groups with residues a, b, g and f showing less motion and residues c, d and e being more mobile.

According to the model of Clore et al., (1990a,b), two internal motions are necessary, one on a picosecond and the other on a nanosecond time scale, in order to properly account for R₁, R_1rho, and ¹³C-{¹H} NOE for all the residues. A possible interpretation of this result is that the fast motion contains sugar puckering motions which have been described in molecular dynamics simulations of saccharides, (Brady, 1990; Hadjuk et al., 1993). But examination of data in the literature shows that the fast motion cannot be completely explained by sugar puckering. In an analysis of relaxation data on sucrose using the treatment of Lipari and Szabo (1982a,b), McCain and Markley (1986) found order parameters S² = 0.89 and a recent study of cyclodextrins by Kowalewski and Widmalm (1994) found values of the order parameter S² = 0.81 - 0.86. These values are substantially larger than those reported in Table 4, Table 5 and Table 6. Therefore it is likely that there are some motions of the glycosidic bonds of the polysaccharide included in the fast motion. Curiously these motions are similar for all the residues of the polysaccharide but they have slightly greater amplitude for residue d, the galactofuranoside.

There is a clear variation of dynamics as a function of individual residue with the most significant deviation of both the slow and the fast internal motion from the average occurring for the residues b, g and d. The motions around residues { b and g are more restricted and the motions of residue d have larger amplitude. It is important to note that this conclusion does not rest only on the 'model-free' analysis and it is also implied by the reduced spectral density analysis of Table 3 and Figure 5. The different dynamics of these residues are presumably the result of structural variation at these two sites. Polysaccharide modeling based on NOESY and ³J_CH data shows that around residues b and g (the antigenic site) the conformational space is very crowded and local conformational exchange is hindered, (Xu and Bush, accompanying paper). The increased dynamic flexibility around residue d may result from the 1->6 linkages and from the puckering of the galactofuranoside ring.

Most of the NMR relaxation rate data reported in the recent literature have focused on ¹⁵N with rather less attention on ¹³C. In the latter case, problems due to ¹³C-¹³C interaction arise for uniformly highly enriched samples, (Yamazaki et al., 1994). The use of ¹³C pulses which are selective for the anomeric carbon resonance region of the polysaccharide NMR spectrum along with the pulse sequences proposed by Yamazaki et al., (1994) avoid these problems and give good single exponential decays for T₁ and T_1rho ( Figure 2). We were not able to measure T₂ for our sample but the use of on-resonance T_1rho provided information of the same type which is critical to the successful interpretation of the relaxation rate data in terms of internal motion on the nanosecond time scale. Our analysis of the data with the reduced spectral density function neglected C1-C2 autorelaxation but the effect can be estimated if we assume that J(omega) derived from the model-free treatment is approximately correct. In eqs. (4) and (5) j and i are taken as C1 and C2 and using J(omega) calculated according to either eq. (18) (Table 5) or to eq. (20) (Table 6) we can calculate the expected contributions to both R₁ and to R_1rho resulting from ¹³C-¹³C autorelaxation. The contributions range from 0.03 sec^-1 for residue d to 0.12 sec^-1 for residue b. These effects, which amount to only 1 to 2% of the total R₁ or R_1rho for our example, could become substantial for slower rotational correlation times. The effects of chemical shift anisotropy, which can be significant for ¹⁵N, are less important for the ¹³C case (Jarvet et al., 1996).

Since little is known about the detailed dynamics of polysaccharides, interpretation of relaxation rate data in terms of polysaccharide dynamics is rather uncertain and almost any method chosen is subject to some criticism. The reduced spectral density method requires some questionable assumptions concerning the shape of the J(omega) curve in the region of omega_H. But, independent of those assumptions, the data appear to indicate that J(0) is smaller for the galactofuranoside residue, {\bf d}, than for the other residues in the polymer implying that internal motion, presumably on the picosecond to nanosecond time scale, is greater for that residue. This conclusion is not greatly influenced by the method of data treatment and rests on the observed T_1rho data which show slower rates for residue d. Other methods of interpretation based on the 'model-free' approach of Lipari and Szabo (1982a,b) assume a series of isotropic exponentially decaying rotational correlation functions. Application to this problem requires some explicit assumptions about the nature of these discrete relaxation times. Most previous applications of these methods have been to globular proteins for which the assumption of an overall rotational correlation time, tau_R, modulated by one or more distinct internal motions is physically reasonable. Appropriation of this type of model for use in polysaccharides may be questioned since the meaning of tau_R for such a system is not entirely clear. It is our interpretation that tau_R (4.7 ns) used in the construction of Table 5 and 6 refers to the approximately isotropic motion of a unit about the size of the 'persistence length' discussed by Brant et al. (1995). For this polymer, this unit must contain approximately 10 sugar residues. In spite of some justifiable criticism of our 'model-free' treatment of the data, the results do agree with the reduced spectral density method on the matter of the greater mobility of the residues in the hinge region around the galactofuranoside, d. The 'model-free' treatment further suggests that picosecond motions of the glycosidic linkages are more uniformly distributed along the polysaccharide and that the motions of the hinge region on the time scale of a few nanoseconds are responsible for the great flexibility of the polymer.

It is quite possible there could be internal motions of this polysaccharide on a time scale longer than tau_R which we have failed to detect. In fact there are almost certainly motions of the larger blocks of residues which are responsible for the observation that relaxation rates measured for a series of oligosaccharides become independent of chain length at about 10 to 15 residues (Brant et al., 1995). Such motions, which are subject to a substantial viscous drag by the solvent may be on time scales of µ-sec to ms and might not be detected in our experiments. Slower internal motions of this type have been referred to as slow exchange phenomena, (Clore et al., 1990b). They may be most effectively detected by R_1rho measurements as a function of carrier offset and B₁ field strength (Markus et al., 1996; Akke and Palmer, 1996).