The Tj and Tg relaxation parameters were determined in accordance with the procedures outlined earlier. The T^ and T2 relaxation experiment pulse schemes
are extensions o f the basic ^^N-^H HSQC experiment. The ^^N-^H HSQC experiment is adapted to allow the measurement o f relaxation parameters by the introduction o f an extra relaxation delay prior to the q delay (see ’^N-^H HSQC sequence given above), during which time the magnetisation is either longitudinal (Tj experiments) or transverse (T2 experiments). The intensity o f a given signal in the 2-D spectrum is
consequently dependent on both the duration o f the extra relaxation delay and the relaxation rate o f the corresponding ^^N nucleus. By performing a series o f 2-D experiments where the relaxation delay is successively incremented, it is possible to measure and plot the intensity o f each signal in the spectrum obtained as a function o f the relaxation delay. Therefore, a curve-fitting procedure allows the estimation o f the relaxation rates R1 and R2 (l/T j and I /T2, respectively) for each signal observed
The Tj and T^ pulse sequences contain modifications o f the sensitivity- enhanced HSQC experiment (Zhang et al., 1994), and were essentially identical to those used previously by Farrow et al. (Farrow et al., 1994). The Tj experiments included multiple Carr-Purcell-Meiboom-Gill spin-echo pulse schemes during the relaxation delay in order to compensate for magnetic field inhomogeneity and consequent Tg* relaxation effects. Both Tj and T^ experiments were performed with sweep widths o f 10000 Hz and 2400 Hz in the ^H and dimensions, respectively; 1024 complex t^ data points and 160 complex tz data points were recorded. For nine successive T^ experiments performed at 17 °C on the SH3-P1 protein, the relaxation delay was set as follows: 0.01005, 0.06031, 0.13068, 0.23121, 0.34178, 0.48251, 0.74387, 1.06554 and 1.50785 seconds. In the eight successive Tg experiments performed at 17 °C on the SH3-P1 protein, the relaxation delay was set to: 0.0, 0.016832, 0.033664, 0.050496, 0.067328, 0.08416, 0.117824 and 0.151488 seconds. All other T2
experiments were performed at 32 °C, using the same set o f eight values for the relaxation delay. Since these experiments required data acquisition over long periods (~24 hours) at relatively high temperatures, a short duration ^^N-^H HSQC experiment (-3 0 minutes) was recorded before and after each set o f Tj experiments, allowing the integrity o f the protein sample to be closely monitored.
For the determinations o f both the R1 and R2 relaxation parameters, data points corresponding to the signal intensity at a given site in the spectra were subjected to a least-squares fitting procedure using the following two-parameter model o f an exponential decay:
I (t) = Iq exp (-tR^ Equation NMR-8
- where I is the signal intensity as a function o f the relaxation delay x and R^ is the relaxation rate, with i denoting the type o f relaxation experiment. The uncertainty in the intensity measurement was taken to be the noise level in the spectra.
The relaxation rate parameters R1 and R2 determined for the SH3-P1 protein at 17°C were used to estimate its overall molecular rotational correlation time, Xm- In practice, the ratio R2/R1 can be used to estimate X^. The calculation o f X^ was performed according to published procedures, adopting the assumptions that (1) there are no internal molecular dynamics, and (2) there are no slow (millisecond time scale) conformational averaging effects in the protein (Mandel et al., 1996). In
The use o f NMR relaxation parameters to yield information on molecular dynamics is relatively well-established. The method was introduced in publications reporting heteronuclear NMR experiments which yielded insight into protein dynamics and mobility^ Relaxation data are interpreted using the Lipari-Szabo model-free a p p r o a c h e d . The model-free
formalism describes the internal motions o f a molecule using only two parameters: a generalised order parameter, S, and an effective correlation time. T e . For most N -H bond vectors in proteins
it can be assumed that the overall and internal motions are independent, since they occur on significantly different time-scales (i.e. nanoseconds compared with picoseconds, respectively). In the method employed, the determination o f the overall rotational correlation time T m is
independent o f S, such that local differences in protein dynamics do not greatly affect the value obtained for T m - The contribution o f T e during the determination o f T m is small and is neglected
here. Thus, the model-free formalism allows the measured parameters R1 and R2 to be used to determine T m , to describe the overall motion o f the protein. This protocol has been used
previously^’^ and is outlined further below.
The Ti and T2 relaxation times (where 1 /T i and I /T2 correspond to the relaxation
rates R1 and R2, respectively) o f an amide ^^N nucleus are dominated by the dipolar interaction o f the ^^N nucleus with its attached proton and by chemical shift anisotropy (CSA), as described previously^. The terms 1 /T i and I /T2 are given by:
1 /T i = d^ [ /(C O h - COn) + 3 /(C 0 n ) + 6 /(C 0 h + COn) ] + c^/(COn) (1) I /T2 = 0.5d^ [ 4/(0) + /((O h - (O n) + 3 /(C 0 n ) + 6 /(C 0 h ) + 6 /(C 0 h + (O n) ] +
( 1 / 6 ) c 2 [ 3 / ( ( 0 N ) + 4 / ( 0 ) ] ( 2 )
- where the constants (R and c^ are defined as:
d^ = 0.1 h^/{4-K^{ y^NH and c^ = (2/15)
- where J, is the gyromagnetic ratio o f spin i ; (O h and (On are the and ^^N Larmor frequencies, t n h is the intemuclear ^^N distance (1.02Â), Ho is the magnetic field strength, and the parallel and perpendicular components o f the assumed axially symmetrical ^^N chemical shift tensor are represented by C\\ and The latter assumption has been shown to be valid for peptide bonds®.
The term / ( ( O , ) represents the spectral density function, where CO, defines the Larmor
frequency o f nucleus i (The term /(O) describes the spectral density at zero frequency). The spectral density function, fundamental for the theoretical description o f NMR relaxation, allows the motional characteristics o f a system to be expressed in terms o f the power at a frequency CO.
A plot o f the spectral density function against frequency yields a curve whose shape, but not integral, is modulated by the rate o f molecular motion. The spectral density function can be represented using the model-free formalism, employing a rriinimum number o f parameters to describe the tumbling motion o f a macromolecule and the internal motions o f the ^H—^®N bond vector, in the following expression:
accordance with the Lipari-Szabo model-free formalism, the simultaneous use o f both relaxation parameters to determine allows the effects o f internal motion in the protein to be neglected. That is, although both R1 and R2 show a dependency on the order parameter'^ (S^), when the ratio R2/R1 is used to determine the dependency on vanishes. Correction: The protocol for determining is given opposite.
The rotational correlation time, T^, o f a molecule can be used to estimate its size via the Stokes-Einstein relation:
Xm = XIH/ k T Equation NMR-9
- where is the solvent viscosity, V the molecular volume o f the sample, k the Boltzmann constant and T the absolute temperature. Thus, knowing also the density o f the protein, the calculated value o f X„ can be used to estimate the apparent molecular mass o f the protein in solution.
In subsequent experiments, full determinations o f X^ were not used in order to derive insight into the dynamics and apparent molecular sizes o f the proteins studied. Rather, it was considered sufficient to determine only a single relaxation parameter: R2. While this approach is perhaps not ideal, it has the advantage o f significantly reducing the experimental data acquisition time. The use o f R2 values to provide insight into molecular dynamics was considered sufficient since, for most proteins, R2 is related to x^ in an approximately linear manner {Figure 2-7). In particular, an almost linear relationship is seen for the range o f X^ values likely to be encountered in NMR experiments o f biological macromolecules. In contrast, R1 displays a more complex, non-linear relationship with respect to X^, and thus would be more likely to introduce errors if used to describe dynamic characteristics such as molecular tumbling rate, and thereby molecular size (Figure 2-7).
It is noteworthy that this strategy is potentially limited, since R2 is not a strictly linear function o f X^ and may be affected by slow dynamics corresponding to conformational exchanges on the millisecond time scale (Clore et al., 1990). Therefore the interpretation o f R2 data alone with respect to molecular dynamics should be performed with caution. In particular, the value o f R2 alone does not factor out the
The term describes the internal molecular motions of a spherical molecule. A value o f S^=l corresponds to a rigid sphere, with S^=0 describing the alternative non-rigid extreme.
assumes that molecular tumbling is isotropic. The effective correlation time resulting from internal motions is described by Te , where 1/T = 1/Tm + 1/Te . However, under conditions where Te< 100 ps and Tm > Ins, and T2 is not significantly shortened by chemical or
conformational exchange, the T1/T2 ratio is essentially independent o f both S and T e . The insignificance o f S and T e when calculating T m means that T m is largely independent o f internal dynamics. Thus, the spectral density function is simplified to:
J W = S 2 T m / ( 1 + C O ^ T m ^ ) ( 4 ) A value for T m can subsequently be determined from the ratio o f Ti and T2, by
substituting equations 1 and 2 with the term for the spectral density function given in equation 4. The term is ‘factored out’ o f the ratio T1/T2 since occurs with almost identical frequency in
both terms. Thus, the experimentally determined values o f Ti and T2 are used determine a value
for T m per residue. The overall value o f T m is taken as the average o f calculated values. (The residues which yield Tm ~ ten-fold different from the mean are excluded from the calculation o f the overall T m . This exclusion is used since it is likely that the ^H— bond vectors in these residues undergo additional forms o f molecular motion, e.g. slow conformational exchange, leading to shortened T2 relaxation times). This describes the protocol used for the determination
o f the rotational correlation time ( T m ) o f the SH3-P1 protein.
Although this approach accounts neither for anisotropic tumbling nor for variable CSA o f different nuclei, the method was considered appropriate for describing the overall molecular dynamics o f the protein since the SH3 domain alone is a globular protein and CSA variations are generally small.
^ferences
1. Kay, L.E., Torchia, D.A. & Bax, A. (1989). Backbone dynamics o f proteins as studied by inverse detected heteronuclear NM R spectroscopy: application to Staphylococcal nuclease, biochemistry^ 28,8972-8979. 2. Clore, G.M., Driscoll, P.C., Wingfield, P.T. et al. (1990). Analysis o f the backbone dynamics o f interleukin-1P using two-dimensional inverse detected heteronuclear NM R spectroscopy.
Biochemistry, 29,7387-7401.
3. Lipari, G. & Szabo, A. (1982). Model-free approach to the interpretation o f nuclear magnetic resonance relaxation in macromolecules. 1. Theory and range o f validity. Journal of the American Chemical Society, 104,
4546-4559.
4. Lipari, G. & Szabo, A. (1982). Model-free approach to the interpretation o f nuclear magnetic resonance relaxation in macromolecules. 2. Analysis o f experimental results. Journal of the American Chemical Society,
104,4559-4570.
5. Farrow, N.A., Muhandiram, R., Singer, A.U. et al. (1994). Backbone dynamics o f a free and a phosphopeptide-complexed Src homology 2 domain studied by ^^N NM R relaxation. Biochemistry, 33,
5984-6803.
6. Pascal, S.M., Yamazaki, T., Singer, A.U. et al. (1995). Structural and dynamic characterisation o f the phosphotyrosine binding region o f a Src homology 2 domain-phosphopeptide complex by NMR relaxation, proton exchange, and chemical shift approaches. Biochemistry, 34,11353-11362.
7. Abragam, A. (1961). Principles of nuclear magnetism,1-599, Clarendon Press, Oxford.
8. Hiyama, Y., Niu, C.-H., Silverton, J.V. et al. (1988). Journal of the American Chemical Society, 110, 2378- 2383.
Figure 2-7: Calculated values of the N M R relaxation parameters R1 and R2, with variedfrom 1-30 ns,for H frequeng ôOOMH;^. Values determined as described in Mandel et al (1996). It was assumed thatfast motions do not contribute; the order parameter was set to 0.85.
rH a 2.5 40 35 2 30 20 15 10 0.5 0 20 25 30 35 0 5 10 15 0 5 10 15 20 25 30 (ns)
dependency on the order parameter, S^. However, S^~0.85 for most residues in the regular secondary structure elements o f a protein. Thus, by assuming that the most o f the signals in the spectra correspond to residues in regular secondary structures, the errors introduced by assuming that R2 values closely relate to molecular dynamics are likely to be small. Thus, using the Lipari-Szabo formalism to relate observed R2 values to Tjn, estimations o f the molecular size o f a molecule can subsequently made using the Stokes-Einstein relation.