Solution of the Haldane equation for substrate inhibition enzyme kinetics using the decomposition method

The Haldane equation has been widely used to describe substrate inhibition kinetics and biodegradation of inhibitory substrates. However, the differential form of the Haldane equation does not have an explicit closed form solution. In this study, we present an explicit solution to the Haldane equation as a recursive series using the decomposition method. We have divided the time interval into several subintervals and have used the first few terms of the series over these subintervals. These low-order solutions provided accurate solutions of the substrate concentration in the Haldane equation. Accuracies on the order of 10−4 were reached when only three terms were used in the series solution and the subinterval was 1% of the total time interval. This is adequate for most practical applications as most experimental measurements of substrate concentrations are seldom characterized by accuracies greater than 10−3. Also, results of any desired accuracy can be obtained by decreasing the subinterval size or increasing the number of terms in the solution. The algebraic nature of this solution and its accuracy make the decomposition method an attractive alternative to numerical approaches such as differential equation evaluation and root-solving techniques currently used to compute substrate concentration in the Haldane equation.