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Fitting of NMR line shapes using LineShapeKin or any other curve-fitting software yields what is called 'best-fit parameters' for Kd, Koff and frequency of B state, w0(B). The software attempts to calculate the errors of the parameter values to give us an idea of how relaible the fitting result is. It is generally known that the best fitting results for line shape analysis are obtained in fast-intermediate or slow-intermediate exchange modes (when we have significant peak broadening and shifting). Other usual requirements for the 'best experimental data' include high signal/noise ratio, many titration points and small spacing between these points in vicinity of stoichiometric point. Despite existence of these recommendations it is still very difficult to access how faithful the particular fitting results are utilizing specific experimental HSQC titration series.

Direct parameter mapping (DPM) as an approach to analyze how stable will be result of fitting the given experimental data with the specific mathematical model. Direct parameter mapping, introduced for analysis of relaxation dispersion data in {Kovrigin, 2006 #3950}, performs calculation of the mathematical model over an exhaustive space of the experimental parameters. Sum of squared deviations (SSD) of each test calculation from experimental data is recorded for every parameter space point and compared to the sum of squared residuals achieved in the fitting procedure that yielded the 'best-fit' parameters. Now one can select *alternative * parameter sets that produce SSD from the experimental data as small as or comparable to 'best-fit' SSD.

This procedure is NOT a substitute for least squares fitting but instead it produces complementary information. Inspecting the surface of the SSD function in the parameter space one can easily see which parameters are reliably determined while which ones are instead correlated and will produce similar fit quality from a broad range of combinations. The least squares routine may stop at any point within the range of these parameter combinations as they produce similar SSD to experimental data. It is possible to obtain information on variability of the results by starting least square minimization routine from a grid of starting conditions (this is implemented in LineShapeKin). Additional problem with fitting results is that typical errors of parameter values reported by minimization protocols *do not* represent the range of variability due to *correlation *as was clearly demonstrated in {Kovrigin, 2006 #3950}.

Direct Paramter Mapping procedure exhaustively tests a large range of parameter combinations to measure the correlations between the fitting parameters using the dataset with its *specific* signal/noise, line shapes and titration series spacing. Any changes in these parameters from experimental system to experimental system will change relative degree of correlation of different fitting parameters. It is not reasonably possible to take *all *these experimental conditions into account to develop a simple analytical measure of robustness of fitting results. However, one may obtain an adequate idea of parameter correlations in every specific case using Direct Parameter Mapping analysis.

In NMRline shape analysis we have at least three parameters to fit: equilibrium dissociation constant, Kd, kinetic off-rate constant, Koff and position of the resonance corresponding to state B, w0(B). Plotting SSD to the experimental data for every combination of these parameters will generate a 4-dimensional SSD surface (exactly the one the minimization routine descents along to the SSD minimum located at the 'best-fit' parameter combination). It is difficult to visualize this result and instead one can project 4D space into a set of 2D planes to facilitate the analysis.

The goal of the analysis is to identify 'equivalent' parameter sets, which produce SSDs similar to the 'best-fit' parameters set and visualize any trends in mutual correlations of the parameters. The approach developed in {Kovrigin, 2006 #3950} was to map equivalent parameter sets onto 2D planes corresponding to mutual correlations of pairs of parameters. Technically, choose two parameter, say Kd and Koff, to analyse correlations of. We search SSD matrix for any parameter sets that produce SSD smaller than a set limit, say experimental 'best-fit' SSD. We disregard value of the third parameter in this case and plot dots for every such a parameter set. Entire list of equivalent parameter plotted produces a group of specific shape that directly reflect mutual correlation of the two fitting variables considered.

- DPM analysis is performed on a three-parameter fitting model. Two parameter model may be accomodated by using a dummy third parameter. More than three-parameter models will have to have other parameters fixed to specific values. This three-parameter limitation may be lifted by necessary modifications to the code of the DPM module.

- Before you start with DPM analysis you must have successfully completed fitting runs (project_name='single' or 'individual') and have your Scale Factor determined and entered in 'scale_factor.txt' in the residue-specific dataset.

- Get the m-files from 6.Fitting_Kd_Koff_w0B.DPM/
6.Fitting_Kd_Koff_w0B

- The model employed in DPM by default is three-parameter (Kd, Koff, and w0(B)) model. You need to run it first in a
*single*or*individual*fitting mode (project_name='single' or 'individual') to ensure the spectral data may be adequately fitted.- NOTE: If your fits are NOT good you will inevitably see extensive correlation between fitting parameters telling your that there is a bunch of other parameter sets that will reproduce your data this bad. Work on extracting better region from spectrum for fitting or slicing the peaks on the shoulder if a problem of overlap occurs .

- NOTE: If your fits are NOT good you will inevitably see extensive correlation between fitting parameters telling your that there is a bunch of other parameter sets that will reproduce your data this bad. Work on extracting better region from spectrum for fitting or slicing the peaks on the shoulder if a problem of overlap occurs .
- After you ensured good fit results you should enter the
*final*best-fit values of w0(B) into the w0_B array in**setup.m**

- You are ready to run DPM calculations (project_name= 'DPM'). The calculations are performed in two subsequent sessions. In the first run (DPM_mode='computation') the program computes an matrix of SSD values corresponding to every grid point in the 3D parameter space and saves this matrix on a disk.

to be finished.