The size-templating matrix effect in vesicle formation I: A microscopic model and analysis

Recently it has been reported that the growth of vesicles is strongly affected by the presence of preformed vesicles. In particular, the final distribution is strongly biased toward the size of the preformed vesicles. This size-templating effect is called the matrix effect. Turbidity measurements have revealed some characteristic features of such systems in which a fast phase of dynamics over which turbidity rises sharply is followed by a much slower phase of kinetics. Experimentally it has been observed that the larger the size of a pre-added vesicle, the higher the turbidity at the end of the fast phase and the longer this phase persists. Additionally, a system without pre-added vesicles has slower, but more sustained, dynamics. We develop a microscopic model based on a novel generalization of the Becker-Doring equations of nucleation, which describes the stepwise growth of vesicles. We assume that a preexisting vesicle provides a catalytic surface on which a vesicle fragment can grow at an accelerated rate. The model involves aggregation and fragmentation rate coefficients, which together determine an equilibrium solution. The model derived is strongly nonlinear and describes the dynamics far from the equilibrium solution. We numerically integrate the model from a variety of initial conditions and use an approximate measure of turbidity to compare our results with experiments previously performed. Comparison with the experiments show good qualitative agreement, validating the modeling assumptions made.