Simulation of pulsatory liposome working using a linear approximation for transmembrane pore dynamics

This paper presents an analytical solution of the differential equations describing the pulsatory liposome dynamics. We consider a unilamellar liposome filled with an aqueous solution of osmotic solute inserted in a hypotonic aqueous medium. Due to the osmosis process the liposome has a cyclic evolution. The lipid vesicle swells to a critical size, at which point a transbilayer pore suddenly appears. Part of the internal solution leaks through this pore. The liposome relaxes and returns to the initial size. The swelling starts again and the liposome goes through a periodical process. The swelling of the liposome is described by a differential equation. The appearance of the pore changes the evolution of the liposome. The internal solution comes out through the pore and the liposome starts its deflation (relaxation). The evolution of the pore has two phases: first, the radius of the pore increases to its maximum value, then the radius decreases until it disappears, and the liposome reaches its initial size. During each cycle, the liposome will release a quantity (a pulse) of the solution from its interior. All the processes which contribute to the liposome relaxing and its coming back to the initial size are described by three differential equations. This system of differential equations can be integrated using numerical methods. The functions – which model our biological engine in three stages, are as follows: R(t) - the liposome radius, r(t) - the pore radius, C(t) - solute concentration, Q(t) - the osmotic solute amount inside the liposome. The graphs representing these functions contain important linear portions, which suggested a solution using analytical methods. Based on some analytical methods, we solve these equations, and their explicit solutions are validated by comparing with numerical results of previous studies.