Modelling the swelling assay for aquaporin expression

The standard method of assaying the water transporting capability of a putative aquaporin-like entity is to express that entity in a cell of normally low water permeability and to measure the enhancement of swelling when the cell is subjected to hypo-osmotic shock. Because of the heterogeneous nature of cytoplasm, the interplay of advection and diffusion, and the coupling of internal and external media via a semipermeable elastic membrane, even simplified mathematical models can be difficult to resolve. This class of diffusion problem seems to have been but little studied. In this paper, the cell and its surround are at first modelled as perfectly-mixed phases separated by an ideal semipermeable membrane with vanishingly small elastic modulus; and the time course of swelling is evaluated analytically. This time course was found to be non-exponential, but such unexpected behavior should not seriously affect the traditional interpretation of experimental results because its short time limit is linear as in the traditional model; and normally only short time data are available. Next, the simplifications of diffusive equilibrium and of vanishing elastic modulus are examined. It is shown that diffusive equilibrium will be true only when diffusive movement of osmolyte is rather faster than the swelling and that this will probably not be the case for many assays. On the other hand, it should often be possible to neglect the elastic modulus. Finally, a more comprehensive model is formulated for a spherical cell in a hypotonic medium and the swelling behavior described in terms as a moving boundary problem (This type of moving boundary problem is often called a Stefan problem [ http://en.wikipedia.org/wiki/Stefan_problem ]) in which two phases containing diffusive osmolyte are separated by a weakly-elastic ideally-semipermeable membrane, the water flux across which is linear in the osmolality difference across it. This type of behavior was evaluated numerically by finite-difference time-domain techniques and found to be qualitatively similar to that of the perfect-mixing simplification.