Bistability and hysteresis in self-assembling micelle systems: phenomenology and deterministic dynamics

We analyse a simplified dynamical model that emulates the overall process of self-assembly of amphiphilic molecules into micelles under non-equilibrium conditions. The study is motivated by a review of experimental evidence in the literature for the occurrence of bistability and hysteresis in diverse self-assembling molecular systems. Singularity theory conditions and classifications for bifurcations are used to map the bifurcation structure of the model. The analysis predicts that self-assembling micellar systems may exhibit bistability and hysteresis. It also provides a map of the boundaries of multiplicity, which effectively defines the role of the model parameters in producing and maintaining hysteretic behaviour. Anomalous hysteresis is identified in the model, and the implications of this for the design and engineering of self-assembling systems are discussed with reference to experimental data for the autocatalytic production of caprylate micelles. We also investigate the occurrence of oscillatory behaviour. In a model having a single outflow or sink rate f for amphiphile and micelles it is shown that Hopf bifurcations to limit cycles are effectively trapped at f = -infinity. In a more realistic model, where the sink rates for amphiphile and micelles are finitely coupled or uncoupled, limit cycles can move into the physical parameter space and interact with the hysteretic region. The results of this study suggest that the hysteresis loop in non-equilibrium self-assembling systems could be used as a switch.