Mathematical modelling of enzymatically cross-linked polymer-phenol conjugates using deterministic and stochastic methods

In this study, two different modelling approaches, namely, a deterministic and a stochastic one, are developed to model the enzymatic cross-linking of polymer-phenol conjugates. A comprehensive kinetic mechanism is postulated to describe the elementary reactions in the cross-linking of polymer-phenol chains in the presence of the horseradish peroxidase (HRP)-H2O2 initiation system. In the first approach, a moments-based model is derived to account for the conservation of all molecular species and leading moments of the number chain length distribution (NCLD) in the reactive system. In the second approach, a stochastic Monte Carlo kinetic model is formulated to follow the time evolution of a sample of cross-linkable polymer chains and calculate the weight chain length distribution (WCLD). From the numerical solutions of both models, the dynamic evolution of the concentrations of all the reactive species, the gelation onset time, the sol and gel mass fractions as well as the number and weight average molecular weights of the cross-linkable polymer chains are calculated. The two derived models are validated using experimental kinetic measurements on the enzymatic cross-linking of tyramine-modified hyaluronic acid and carboxymethyl-chitin. It is shown that both models can accurately predict the gelation onset time of the two cross-linkable systems over a wide range of variations in HRP and H2O2 concentrations. Finally, the MC model predictions on the weight average number of polymer chains in the cross-linked molecules are compared to Flory's analytical solution on the tetra-functional cross-linking of polymer chains of uniform length.