A theoretical approach to cluster formation in biological membranes

A theoretical analysis of cluster formation within the lipid matrix of biological membranes is presented. Various models are analysed: (a) one-dimensional monolayer, (b) two-dimensional monolayer and (c) one dimensional bilayer. Furthermore, lipid-protein interactions are considered. The model is based on differential equations for the probabilities ai and bi which characterize the occupation of the lattice site i by the lipids A and B, respectively. These differential equations are an approximation of the Master-equation. Steady states as well as time-dependent variations are analysed. Depending on the interaction energies of the two lipids, different stationary lipid distributions are obtained, including clusters of lipids A or B and alternating structures. The distributions may be dynamically stable or unstable. It is shown that phase transitions within the lipid matrix may be induced by alteration of the composition of the membrane, by changing the interaction energies of the lipids, by variation of the temperature or by lipid-protein interactions. The transitions between different stationary distributions are studied by use of bifurcation diagrams. The analysis of time-dependent states reveals that unstable structures of the membrane may be important for certain time periods. Consideration of the lipid bilayer leads to a great number of possible distributions, which may be symmetric or asymmetric with respect to the outer and inner leaflets of the membrane.