Estimation Procedures for Consecutive First Order Irreversible Reactions

Solutions of the differential equations for a series of consecutive irreversible first order reactions are presented in a general form which allows the use of a single nonliniear regression program to estimate values of the reaction constants for any chain length. The observations may consist of a linear combination of compartments. The number of reactions is treated as a parameter which can also be estimated. The mean transit time through the system of reactions is equal to the sum of the reciprocals of the reaction constants. This provides a useful constraint on the choice of starting values of the reaction constants. When all material is initially in the first compartment, a lower bound on the minimum number of precursors at any point in the chain can be obtained by setting all reaction constants equal and finding the number of reactions which yields a minimum sum of squares of errors. When the actual reaction constants are numerically close to each other, the lower bound will be the minimal number of precursors; otherwise, the lower bound will underestimate the number of precursors. A numerical example is presented.