Analysis of a generalized Becker-Doring model of self-reproducing micelles

We analyse a model for the kinetics of formation of caprylate micelles from ethyl caprylate during hydrolysis by aqueous alkali. This chemical process provides a simple route to the formation of bounded cell-like structures under prebiotic conditions. Experimentally, it is observed that there is an extended induction period during which the concentration of micelles remains small; at the end of this period, ethyl caprylate is consumed and micelles are formed very rapidly. The theoretical model studied here is one proposed by Billingham & Coveney in an earlier paper (Billingham & Coveney 1994), based on a generalization of the Becker-Doring equations. It accounts for the formation of micelles under general non-equilibrium conditions. We derive a reduced description of the system which preserves all the properties of the infinite-dimensional Becker-Daring equations. Centre manifold theory is used to extract the nonlinear dynamics which govern the start-up behaviour of the system. From a careful incorporation of the perturbation terms into the centre manifold procedure an estimate of the induction time of the system is found which is in good agreement with experiment. This perturbed reduced system correctly predicts the form of the induction time, which reduces to zero as the amount of added monomer reaches a value corresponding to the critical micelle concentration. This behaviour was absent from previous models. A more general model is also proposed, which incorporates more rate processes than previously considered, while maintaining all the mathematical structure of the Becker-Doring system.