Nonlinear-Least-Squares Analysis of Slow-Motion EPR Spectra in One and Two Dimensions Using a Modified Levenberg–Marquardt Algorithm

The application of the ‘‘model trust region’’ modification of the instrumentation. Two of the most important advances inLevenberg‐Marquardt minimization algorithm to the analysis of clude the introduction of two-dimensional Fourier-transform one-dimensional CW EPR and multidimensional Fourier-trans- methods (1‐4) , and the extension of both continuous-wave form (FT) EPR spectra especially in the slow-motion regime is ( 5, 6, 7) and pulsed ( 8, 9) EPR methods to high frequencies described. The dynamic parameters describing the slow motion requiring superconducting magnets. By analogy with similar are obtained from least-squares fitting of model calculations based developments in NMR, these enhancements have greatly on the stochastic Liouville equation (SLE) to experimental spectra. increased the amount of structural and dynamic information The trust-region approach is inherently more efficient than the that is available from spectra of spin labels and intrinsic standard Levenberg‐Marquardt algorithm, and the efficiency of paramagnetic species. the procedure may be further increased by a separation-of-variOne important application of EPR, and magnetic resoables method in which a subset of fitting parameters is independently minimized at each iteration, thus reducing the number of nance in general, is to study the molecular dynamics of an parameters to be fitted by nonlinear least squares. A particularly appropriate label in isotropic fluids, ordered phases such useful application of this method occurs in the fitting of multicom- as liquid crystals or biological membranes, on surfaces, or ponent spectra, for which it is possible to obtain the relative popu- attached to macromolecules. Many details of the dynamics lation of each component by the separation-of-variables method. are discernible in the slow-motion regime ( 10) , where the These advantages, combined with recent improvements in the characteristic time scale of the motion is on the order of the computational methods used to solve the SLE, have led to an inverse spectral bandwidth. However, the analysis of sloworder-of-magnitude reduction in computing time, and have made motion spectra is complicated by the fact that the relationship it possible to carry out interactive, real-time fitting on a laboratory between the spectrum and the physical parameters of interest workstation with a graphical interface. Examples of fits to experi- is rather indirect. The partial averaging of EPR spectra by mental data will be given, including multicomponent CW EPR molecular motion or spin dynamics can produce very comspectra as well as two- and three-dimensional FT EPR spectra. Emphasis is placed on the analytic information available from the plicated and irregular lineshapes requiring detailed spectral partial derivatives utilized in the algorithm, and how it may be simulation to extract the desired information. used to estimate the condition and uniqueness of the fit, as well The increase in the number of spectral dimensions and as to estimate confidence limits for the parameters in certain cases. the resolution now available in EPR has been accompanied q 1996 Academic Press, Inc. by a corresponding increase in the computational demands of spectral simulation. The computational requirements are particularly stringent for slow-motion problems, which typi