Applications of constraints to general nonlinear least squares calculations. second order reversible consecutive reactions

Abstract

It is often advantageous to formulate least squares problems so that theoretical constraints on the parameters can be determined directly from the experimental observations and do not have to be incorporated into the model equations. An example occurs in kinetics problems dealing with reversible reactions, where microscopic reversibility requires that the ratio of the forward and reverse rate constants, k 1 / k 2 , must equal the equilibrium constants K eq. In previous methods of treating second order reversible reactions, k 1 and k 2 are automatically constrained so that k 1 / k 2 = k eq. An alternative approach is to omit the constraint and then find the best k 1 and k 2 which described the data. A comparison of k 1 / k 2 with K eq serves as an additional criteria for establishing the uniqueness of the proposed model. Successful implementation of this technique usually requires subsets of observations, corresponding to different experimental conditions, and a mechanism for manipulating constraints which are imposed on the parameters and which relate the parameters of one data subset to those of another subset. In the present paper, a new method of organizing a general least squares program is presented which allows full exploitation of multiple subsets of data from different experiments and permits a general system of linear and nonlinear constraints to operate on the least squares parameters. An important advantage for a general least squares program is that linear constraints are not explicitly recognized when the function and its derivatives are evaluated. Consequenlty, linear constraints can be added, deleted or modified without any changes in programming. Nonlinear constraints are imposed as “observations” and, once coded in the program, can be deleted or relaxed through the program's input without further programming changes. The technique is applied to two types of second order reversible reactions where traditional least squares methods are not easily employed.