Mechanism for the formation of density gradients through semipermeable membranes

We describe and theoretically analyze here a phenomenon which can take place in a system with two different compartments, each containing the same chemicals, which undergo reactions on the surface of both sides of the membrane which separates the two compartments, in the case where the membrane permeabilities to the various chemicals are different and diffusion is fast. There are two main reasons of interest for this kind of system. First, if the overall system is isolated, starting from the case where the initial concentrations of the chemicals are the same in the two phases, one observes the formation of a transient concentration difference. This difference eventually vanishes, although it might last for a long time, depending upon the value of the relevant parameters. The second reason of interest is that, in the case of an open system, one can achieve a steady-state value of the concentration of some chemicals in the smaller compartment which is higher than that in the external one. These results may prove important, inter alia, to understand the behavior of lipid vesicles in water, a topic which is important for studies on the origin of life as well as for possible future applications.