Simulation of Time-Dependent Enzyme Kinetics by Multistage Picard Iterative Method

Abstract

The time-dependent behaviour of reactant concentrations in a basic enzyme--substrate system is analysed using a modified formulation of the Picard Iterative Method (PIM). In this approach, the classical PIM is reformulated into a multistage framework, where the solution is approximated over successive time subintervals, resulting in a hybrid numerical-analytical scheme known as the Multistage Picard Iterative Method (MPIM). Numerical comparisons show that the standard PIM suffers from increasing error accumulation and reduced accuracy as time progresses, while the Pade-enhanced PIM improves convergence only within a limited time range. In contrast, the MPIM maintains excellent agreement with the classical fourth-order Runge-Kutta (RK4) method throughout the entire simulation interval, producing significantly smaller absolute errors and enhanced numerical stability. These results confirm that the multistage reformulation effectively suppresses error propagation and provides a reliable and accurate approach for modelling enzyme--substrate dynamics and solving nonlinear biochemical systems over extended time domains.