Application of a genetic algorithm: near optimal estimation of the rate and equilibrium constants of complex reaction mechanisms Marcel Maeder, Yorck-Michael Neuhold, Graeme Puxty * Department of Chemistry, University of Newcastle, University Drive, Callaghan NSW 2308, Australia Received 10 June 2003; accepted 15 November 2003 Abstract In iterative non-linear least-squares fitting, the reliable estimation of initial parameters that lead to convergence to the global optimum can be difficult. Irrespective of the algorithm used, poor parameter estimates can lead to abortive divergence if initial guesses are far from the true values or in rare cases convergence to a local optimum. For determination of the parameters of complex reaction mechanisms, where often little is known about what value these parameters should take, the task of determining good initial estimates can be time consuming and unreliable. In this contribution, the methodology of applying a genetic algorithm (GA) to the task of determining initial parameter estimates that lie near the global optimum is explained. A generalised genetic algorithm was implemented according to the methodology and the results of its application are also given. The parameter estimates obtained were then used as the starting parameters for a gradient search method, which quickly converged to the global optimum. The genetic algorithm was successfully applied to both simulated kinetic measurements where the reaction mechanism contained one equilibrium constant and two rate constants to be fitted, and to kinetic measurements of the complexation of Cu 2+ by 1,4,8,11-tetraazacyclotetradecane where two equilibrium and two rate constants were fitted. The implementation of the algorithm is such that it can be generally applied to any reaction mechanism that can be expressed by standard chemistry notation. The control parameters of the algorithm can be varied through a simple user interface to account for parameter range and the number of parameters involved. D 2004 Elsevier B.V. All rights reserved. Keywords: Genetic algorithm; Kinetics; Rate constants; Equilibrium constants 1. Introduction The iterative fitting of non-linear parameters to mea- sured data using gradient or direct search methods is a procedure broadly applied across chemistry and science as a whole [1–3]. However, one of the drawbacks of these optimisation methods is their sensitivity to the initial parameters supplied to them. Only if the parameter esti- mates are in the vicinity of the global optimum can reliable and robust convergence be expected. If the initial estimates are far from the global optimum, it is likely a gradient or direct search method will not converge at all or will converge to a local optimum. This begs the question how do we determine reasonable initial estimates for parameters if they are difficult to generate. In this contri- bution, we discuss the basic methodology of applying a genetic algorithm (GA) to the determination of the param- eters of difficult fitting problems, specifically complex reaction mechanisms. We then show an implementation of a GA according to the methodology and give the results of its application to the task of determining good initial parameter estimates for further refinement by a gradient method. We are specifically dealing with the determination of the rate and equilibrium constants that quantitatively describe complex reaction mechanisms from multivariate kinetic spectrophotometric measurements (e.g. the acquisi- tion of a series of absorption spectra, measured as a function of the reaction time). The application of GAs to problems in chemistry and science has only been recent. Interest in evolutionary algorithms began with the work of Rechenberg [4] and Holland [5]. Limiting their early application was the fact that GAs are computationally intensive, often requiring thousands if not millions of evaluations of the objective function used to define the fitness of members of the GA. It is only with recent advances in computing power that the 0169-7439/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemolab.2003.11.006 * Fax: +61-2-49215472. E-mail address: chmm@cc.newcastle.edu.au (G. Puxty). www.elsevier.com/locate/chemolab Chemometrics and Intelligent Laboratory Systems 70 (2004) 193 – 203