Communication Nonlinear-least-squares analysis of slow motional regime EPR spectra q Khaled Khairy a,b, * , David Budil a , Piotr Fajer b a Department of Chemistry and Chemical Biology, Northeastern University, Boston, MA 02115, USA b National High Magnetic Field Laboratory and Inst. Molecular Biophysics, Tallahassee, FL 32310, USA Received 27 March 2006; revised 26 July 2006 Available online 24 August 2006 Abstract A comparison between the full Newton-type optimization NL2SNO, the Levenberg–Marquardt method with the model-trust region modification, and the simplex algorithm is made in the context of the iterative fitting of EPR spectra. EPR lineshape simulations are based on the stochastic Liouville equation (SLE), with an anisotropic diffusion tensor and an anisotropic restraining potential describing the motional amplitude of the spin label. The simplex algorithm was found to be the most reliable, and an approach—incorporating both NL2SNO as well as the downhill simplex methods—is proposed as a strategy-of-choice. Ó 2006 Elsevier Inc. All rights reserved. Keywords: EPR; Spectrum fitting; Nonlinear optimization; Levenberg–Marquardt; Simplex 1. Introduction The emergence of applications in the study of biological membranes and structural biology, for both NMR and EPR, have significantly increased the demand for robust, accurate, and efficient programs for the analysis of experi- mental data. A suitable minimization scheme can be used to elicit both structural and dynamic information from the spectroscopic lineshapes. Iterative analysis of spectra is based on: (a) a suitable physical model to simulate the theoretical spectrum, (b) an algorithm that improves a given set of model parameters, based on a suitable criterion such as minimizing v 2 [1], and (c) a measure for the goodness of fit. In this report, the simulation of EPR spectra involves the solution of the stochastic Liouville equation (SLE) of motion [2]. The SLE is of special importance in the slow- motion regime of the spin label and provides a versatile means for implementing detailed dynamical models that in- clude fully anisotropic rates and amplitudes of motion [3]. SLE-based simulations are CPU-intensive, and—in all but the simplest cases—remain a time-consuming step in the analysis of experimental spectra. This is especially true at high magnetic fields [4,5], which necessitates finding an effi- cient and robust strategy that reduces the number of itera- tions needed for an acceptable fit. Some of the more well-known parameter optimization algorithms are the downhill simplex [6], Powell [7], evolu- tionary Monte Carlo [8], and conjugate gradients methods [9], as well as Newton-type methods such as Gauss–New- ton [1], Levenberg–Marquardt [10,11], and Levenberg– Marquardt with the model-trust-region modification [12]. A number of these have been applied to various curve fit- ting problems involving EPR spectra: simplex in powder patterns and orientational distribution of rigid samples [13,14], Levenberg–Marquardt for dynamically averaged line shapes [15], and simulations in the slow motional regime [16]. At the end of an optimization it is desirable to be able to test for the goodness of fit, a number of such tests is available, for example the reduced v 2 = v 2 /(n  p), www.elsevier.com/locate/jmr Journal of Magnetic Resonance 183 (2006) 152–159 1090-7807/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmr.2006.07.022 q This work was supported by National Science Foundation Grants MCB960094 (D.E.B.) and MCB0346650 (P.G.F.), and an IHRP grant from the National High Magnetic Field Laboratory (P.G.F.). * Corresponding author. Present address: Max Planck Institute of Molecular Cell Biology and Genetics, Dresden 01307, Germany. Fax: +49 351 2102020. E-mail address: khairy@mpi-cbg.de (K. Khairy).