Network formation in negative charged membranes by two divalent cations and the catastrophe

The ligands of Ca2+-Cu2+-phosphatidylserine (PS) complexes in membrane networks at the water-oil interface through the symmetry breaking instability and the head groups of PS molecules were changed into a solid-like state. A first step in this transition is described by the following scheme in one unit in which the molar ratio is Ca2+: Cu2+: PS = 1:2:4; [Oh]+2[Oh]*----3[Oh]*, where [Oh]* denotes a little distorted ligand structure [LnM2+...2H2O] from [LnM2+2H2O], where Ln is PS molecules (n = 2 to Cu2+ and 4 to Ca2+). All the ligands are changed to [D4h] by the unit-unit interaction due to the network formation; [Oh]*----[D4h]. The whole system is equivalent to Schlögl's scheme and is given by a cubic state equation for suitable variables transformations: x = -x3 - ux - v, where x corresponds to the concentration of [Oh]*, and u and v are related to rate constants in the first and the second steps, and they also depend on the initial [Oh] and the final [D4h] concentrations. This system is transferred into a new state with a cusp catastrophe.