Materials and Methods
3.4. Analysis of Data.
3.4.1 - Calculating signal-to-noise (S/N) ratios
S/N ratios for amide cross peaks in [^H, ^^N]-HSQC and [^H, ^^N]-TROSY spectra were measured as follows. For each selected amide cross peak, a signal intensity value was extracted using ANSIG scripts written in-house (Pfuhl et al. 1999). For spectra of ubiquitin, NH cross peaks were assigned in a residue specific fashion. In the case of PaDDAH, selected NH cross peaks were assigned numerically. The signal intensity values and corresponding peak ‘assignments’ were output as ASCII text files. Noise values were calculated using Plot2 (Boucher 2002) from a region of the spectrum devoid of cross peaks.
S/N ratios were initially analysed for all selected cross peaks. The mean and standard deviation of this data set are strongly biased by the cross peaks from extremely flexible NH groups, such as those at the polypeptide termini, which had extraordinarily high S/N values. The data was, therefore, re-analysed using a standard deviation filter to omit any cross peak with a S/N value outside the original mean plus or minus 1.5 times the original standard deviation. These latter statistics have been presented throughout.
3 .4 .2 - Derivation of^^NR], ^^NR2, H} heteronuclear-NOE and estimation o f Tc ^^N Ti and T2 experiments were acquired with incremented relaxation delay
spectrum was recorded for each relaxation delay period (Tables 3.3 and 3.4). Ti and T2 spectra could therefore be processed as pseudo 3D-spectra with the third
dimension comprising 2D correlation planes with incremented relaxation delays. For each set of relaxation data a reference 2D [^H, ^^N]-correlation spectrum was recorded; either [^H, ^^N]-HSQC or [^H, ^^N]-TROSY experiments depending on the sample being studied. The relaxation spectra were processed with an artificial third dimension consisting of a single point per relaxation delay increment using nmrPipe. In the analysis of relaxation data from ubiquitin samples, residue specific assignment of amide cross peaks in the 2D NH reference spectrum were available. For PaDDAH, selected NH cross peaks were assigned numerically. Only well-resolved cross peaks were selected for relaxation analysis. For each selected amide cross peak in the reference spectrum a signal intensity value was extracted from each of the 2D planes in the pseudo 3D relaxation spectrum using ANSIG scripts written in-house (Pfijhl et al. 1999). These values and corresponding peak assignments were output as text files. To measure the (^H)-^^N heteronuclear NOE, 2D NH correlation spectra were recorded in the presence and absence of proton saturation (see Tables 3.3 and 3.4). The data sets could there be processed as a pseudo 3D experiments. Thus 3D spectra had two or four planes in the third dimension depending on the number of repeated 2D spectra. Signal intensity values were extracted as described for Ti and T2
spectra and output as text files. Signal intensity values for all relaxation experiments were corrected for noise, which was calculated using Plot2 (Boucher 2002) from a region of the spectrum devoid of cross peaks.
For each selected NH cross peak in the Ti or T2 spectrum, the decay of magnetisation
was fit to Equation 4.2 using Mathematical (Wolfram 2002). The cross peak signal intensity and relaxation delay time data sets were fit against Equation 4.2 (see Chapter 4, Section 4.2.1) using a two parameter (lo, the intial signal intensity, and /^i, the relaxation rate constant where i = 1 or 2) least squares regression minimisation (Levenberg-Marquardt method; as described in Kay et al. 1992; Peng and Wagner 1994). A standard deviation for each experimental signal was calculated from the difference between two separate experiments with identical relaxation delay values or, in the absence of repeated data points, the spectral noise value. The uncertainty in the fit for lo and R\ was determined using a Monte Carlo simulation. For each Monte Carlo iteration, a random signal intensity for each relaxation delay time was generated
from a normal distribution about the experimental mean signal intensity for that relaxation delay. Each simulated data set was separately used to optimise Equation 4.2 (see Chapter 4, Section 4.2.1), using the protocol described above for the initial fit. The relaxation rates and standard deviations stated in text correspond to the mean relaxation rate and standard deviation of the distribution of 200 such Monte Carlo iterations.
The heteronuclear NOE value for each selected NH cross peak was the ratio of noise-corrected signal intensities from 2D {^H}-^^N heteronuclear correlation spectra with and without proton saturation. If repeated experiments were conducted the ratio was derived using the mean of the saturated and non-saturated signal intensities. Uncertainties in the estimation of the NOE values were propagated from the experimental reproducibility of saturated and non-saturated signal intensity values. In the absence of repeated experiments, the error was calculated from the spectral noise.
The isotropic rotational correlation time, Xc, for a protein was derived from a RilRi
data set which had been filtered for internal mobility and chemical exchange contributions to R\, Ri and the {^H}-^^N heteronuclear NOE values (described in Chapter 4, Section 4.2.1). A Monte Carlo procedure using the experimental mean
R2IR1 ratio and standard deviation of the filtered data set was used to generate a simulated distribution of 1000 R2lR\ ratios. Xc values were determined (see Chapter 4, Section 4.2.1) for each simulated R2IR1 ratio (see Chapter 4, Equation 4.4). The values of Xc stated in the text refer to the mean and standard deviation of the resultant distribution of artificial Xc values.
3 .4.3- Calculating proton transverse relaxation rates
The signal intensity in a selected region in the ID spectrum of a 1-1 spin echo experiment was extracted using nmrPipe scripts written in-house and output as ASCII text files. Signals in the amide proton region of the ID proton spectrum were selected. Five identical experiments were recorded per sample and the proton T2 calculated in
each case using Equation 5.1 (see Chapter 5, Section 5.4.1). T2 values stated in the
3 .4 .4 - Calculating translational diffusion rates
Multiple ID PSG diffusion experiments were recorded per sample with incremented gradient strengths between 6.5 and 51.7 G cm '\ The exact gradient strengths employed varied between experiments. The mean log(signal intensity) and standard deviation per gradient strength were fit against Equation 5.5 (see Chapter 5, Section 5.4.2) \Vi Mathematical using a two parameter (lo, the intial signal intensity, and the translation diffusion coefficient) least squares regression minimisation (Levenberg-Marquardt method). The errors in the fit for Iq and Dz were determined using a Monte Carlo simulation. For each Monte Carlo iteration, a random signal intensity for each gradient strength was generated from a normal distribution about the experimental mean signal intensity for that gradient strength. The simulated data set was then used to optimise Equation 5.5 (see Chapter 5, Section 5.4.2), using the protocol described above for the initial fit. The values and standard deviations of Dz
stated in text correspond to the mean Dz and standard deviation of the distribution of 1000 Monte Carlo iterations.
3.4 .5 - Estimating the dissociation constants o f WT PaDDAH from analytical SEC
The elution volume of each peak was measured using the software packages accompanying the chromatography systems used (see Section 3.2.10). 8 separate SEC experiments were run at each sample concentration. Apparent molecular weight (app. MW) values were derived from the peak elution volumes using Equation 6.1 (see Chapter 6, Section 6.2.3). The experimental data set (comprising an average app. MW and standard deviation for each of the six sample concentrations used) was used to fit against Equation 6.12 (see Chapter 6, Section 6.2.4) m Mathematica® using a three parameter (Kd, the homodimer dissociation constant, MWm, the app. MW of a monomeric species and MWd, the app. MW of a homodimer) least squares regression minimisation (Levenberg-Marquardt method). Loading protein concentrations were adjusted for dilution on the column using the rationale of Manning and co-workers (Manning et al. 1996). The dilution factor for WT PaDDAH data was derived from the ratio of the average elution peak width at half-height (in ml) and the sample loading volume (always 100 pi).
In instances where MWm and MWd were defined, the error of the fit for Kd was
app. MW for each sample concentration was generated from a normal distribution about the experimental mean app. MW at that concentration (using the standard error of each experimental data set). The simulated data set was used to optimise Equation 6.12 (see Chapter 6, Section 6.2.4), using the protocol described above for the initial fit. The values and standard deviations of Kd given in text correspond to the mean and standard deviation of the distribution of values generated from 200 Monte Carlo iterations.