Simple membrane-based model of the Min oscillator
New Journal of Physics
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
... 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. necessary for the oscillations. Although many details of the interaction of Min proteins with the membrane and among themselves are known [12, 13] , the exact mechanism of the oscillations is not yet fully understood. An important step towards the understanding of the Min dynamics was the observation of dynamic Min patterns on artificially created supported lipid bilayers    , on structured membrane surfaces , on unsupported lipid membranes  , and within microcompartments enclosing a small volume  . These synthetic systems allow easier control of all relevant parameters, such as protein concentration, membrane composition and volume geometry, and so enable quantitative studies of their effect on the Min pattern dynamics. The models of Min oscillations are usually based on the reaction-diffusion mechanism. The steady state of the reaction system is unstable, which leads to inhomogeneous oscillating protein distribution. The early models employed effective reaction terms [20, 21] or included effective processes such as aggregation current describing the tendency of MinD to aggregate on the membrane    . Although the introduction of sufficient nonlinearity in the rate equations in these models results in instability and complex dynamics, it does not always allow a direct interpretation in terms of biochemical reactions. A clear interpretation is possible with models containing only terms describing simple reaction steps. While some models contain cubic terms corresponding to non-realistic trimolecular reactions  , other models employ only at most bimolecular reactions [16,     . Several other models attempt to explain the Min dynamics by relying on the observed MinD polymerization into filaments , often combined with preferential nucleation at the cell poles    . There are two classes of models that are relatively simple, contain only chemical reaction terms, make minimum assumptions beyond firmly established facts, and exhibit Min dynamics similar to those in cells or flat membranes. In the first of them, the finite rate of conversion of MinD-ADP, which cannot bind to the membrane, to the MinD-ATP in the bulk solution plays an important role [25, 28] . In the other class of models, the existence of MinE not attached to MinD on the membrane is important, and the reactions in the bulk are not considered (i.e. are assumed to proceed fast) [16, 27] . Here we propose a model related to the second type. Contrary to most other models, the model we suggest does not require autocatalytic binding of MinD to the membrane. Instead, it assumes that two MinE molecules are necessary to induce MinD detachment from the membrane. This assumption is based on MinD being a dimer with two binding sites for MinE and two bound ATP molecules. The model is very simple, not far from (in terms of complexity) the minimal chemical model that can exhibit instability and periodic oscillations  . It includes only reactions on the membrane and explicitly contains only three, membrane-bound, species. In comparison to other models, it provides so far the best qualitative description of the shapes of the observed invitro concentration profiles of MinE on lipid bilayers, and reproduces the experimental dependence of the wave period on the MinE concentration. We first study the dynamical behaviour and stability of this model considered as a well-mixed system, neglecting diffusion. We explore modifications of the model concerning the way of attachment of MinE to the membrane and the necessity of two MinE molecules for MinD membrane detachment. Then we add diffusion and observe its effects on the wave shape and propagation. Finally, we briefly consider this model in closed geometry with a finite pool of MinD and MinE molecules.