Simple membrane-based model of the Min oscillator

Min proteins in E. coli bacteria organize into a dynamic pattern oscillating between the two cell poles. This process identifies the middle of the cell and enables symmetric cell division. In an experimental model system consisting of a flat membrane with effectively infinite supply of proteins and energy source, the Min proteins assemble into travelling waves. Here we propose a simple one-dimensional model of the Min dynamics that, unlike the existing models, reproduces the sharp decrease of Min concentration when the majority of protein detaches from the membrane, and even the narrow MinE maximum immediately preceding the detachment. The proposed model thus provides a possible mechanism for the formation of the MinE ring known from cells. The model is restricted to one dimension, with protein interactions described by chemical kinetics allowing at most bimolecular reactions, and explicitly considering only three, membrane-bound, species. The bulk solution above the membrane is approximated as being well-mixed, with constant concentrations of all species. Unlike other models, our proposal does not require autocatalytic binding of MinD to the membrane. Instead, it is assumed that two MinE molecules are necessary to induce the dissociation of the MinD dimer and its subsequent detachment from the membrane. We investigate which reaction schemes lead to unstable homogeneous steady states and limit cycle oscillations, and how diffusion affects their stability. The suggested model qualitatively describes the shape of the Min waves observed on flat membranes, and agrees with the experimental dependence of the wave period on the MinE concentration. These results highlight the importance of MinE presence on the membrane without being bound to MinD, and of the reactions of Min proteins on the membrane.