Determination of uniqueness of the best-fit parameters of in NMR line shape analysis


Concept of Direct Parameter Mapping

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.


Direct Parameter Mapping implementation for analysis of NMR line shapes

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.





to be finished.