Fast and accurate fitting of relaxation dispersion data using the flexible software package GLOVE

Abstract

Relaxation dispersion spectroscopy is one of the most widely used techniques for the analysis of protein dynamics. To obtain a detailed understanding of the protein function from the view point of dynamics, it is essential to fit relaxation dispersion data accurately. The grid search method is commonly used for relaxation dispersion curve fits, but it does not always find the global minimum that provides the best-fit parameter set. Also, the fitting quality does not always improve with increase of the grid size although the computational time becomes longer. This is because relaxation dispersion curve fitting suffers from a local minimum problem, which is a general problem in non-linear least squares curve fitting. Therefore, in order to fit relaxation dispersion data rapidly and accurately, we developed a new fitting program called GLOVE that minimizes global and local parameters alternately, and incorporates a Monte-Carlo minimization method that enables fitting parameters to pass through local minima with low computational cost. GLOVE also implements a random search method, which sets up initial parameter values randomly within user-defined ranges. We demonstrate here that the combined use of the three methods can find the global minimum more rapidly and more accurately than grid search alone.

Figure 1

Schematic representation of the Monte-Carlo minimization method implemented in GLOVE. The dashed line arrow represents the Monte-Carlo process that adds random values to the current best fit parameters, enabling the parameters to pass through a local minimum. The reduced χ² value usually increases in this step. The new parameter set is subsequently minimized as represented by the curved solid arrow.

Figure 2

Procedure for the analysis of relaxation dispersion data. The programs included in the GLOVE software package are shown in Courier New font. The main part of the data fitting using the GLOVE program, whose executable command is glove, is shown as the grey background.

Figure 3

Representative ¹⁵N relaxation dispersion profiles for KIX with the best-fit curves. The relaxation dispersion data were collected at ¹⁵N frequencies of 60.83 MHz (black line) and 76.01 MHz (red line). The plots were initially created by GLOVE for individual residues, and merged using mplot into a single PDF file. The numbers followed by “-HN” on the upper left of the plots are the residue number.

Figure 4

Fitting accuracy and speed using the grid search method. (A) The reduced χ² values of the fits using the methods GRID (black) and GRID+ONEEX (red) plotted against the grid size. The inset is an enlarged view of the same plot. The symbols have been omitted for clarity. (B) The computational time for the fits using GRID (black) and GRID+ONEEX (red) plotted against the grid size. The vertical scale is shown on the left-hand side of the plot. The green line represents the total grid size N^total, whose vertical scale is shown on the right-hand side of the plot. N^total is calculated as: $N^{\mathit{total}}=\prod _i{N}_i^{\text{global}}\sum _j\prod _k{N}_{j,k}^{\text{local}}$, where N_i^global and N_j,k^local represents the grid sizes of the i-th global parameter and the k-th local parameter in the j-th dataset, respectively.