Collective Properties of the Exactly Solvable Model of Ion-Channel Assemblies in a Biological Cell Membrane

The behaviour of a system of ion channels formed across the cell membrane is presented. The infinite number of channels with an infinite coupling is introduced first as a reference point for the detailed derivation of the thermal-equilibrium probability distribution and the classification of the phase transitions. Fluctuations in a finite system are discussed next. We propose a new, step-like model of the ion channel switching, for which we provide the analytical results. The relation of this model to experiment is also provided. Finally, the master equation for the finite channel-number membrane is analysed numerically with the help of an exact-diagonalization technique. In particular, the decay-time of a metastable solution is estimated. The results do not agree with those obtained perturbationally, the difference is explained by the proposed frozen-diffusion approach.