Diffusion with memory in two cases of biological interest

The complexity of a biological structure, such as membrane where the transport process may carry solid particles which may obstruct some of the pores, diminishing their size and making the permeability dependent on the local structure of the medium, suggests the introduction of a space-dependent diffusion constant. In this note, the profile concentration of diffusing solutes inside a cell membrane has been calculated on the basis of the Fick diffusion equation modified by introducing a memory formalism (diffusion with memory). This approach has been employed to describe the concentration profile inside the membrane when a sudden change of the concentration in the medium bathing one of its face is applied for a limited interval of time. A further application of the method concerns the so-called concentration boundary layer that occurs at the membrane-aqueous medium interface, where the solute concentration depends, even at considerable depth, on the local structure of the interface. These profiles are compared to some recent experiments concerning the diffusion of ethanol in a layer close to a nephrophane membrane. This approach generalizes the diffusion models based on the Fick equation to more complex systems, where a space-independent diffusion coefficient could be inappropriate to take into account the large variety of diffusion processes in biological systems.