Ternary systems water — impermeable solute (“salts”) — permeable solute have been widely used in cryopreservation of biological cells where the permeable solute is the cryoprotective agent (CPA). The cell membrane is considered “leaky”’ to CPA so the osmotic process is described by a system of two coupled non-linear equations: the osmotic equation (1.1) that drives water against the gradient of total osmotic pressure of both solutes, and the diffusion equation (1.2), which drives the permeable solute towards its gradient across membrane. In case of the “classical” system one permeable solute + impermeable entities (“salts”), the third equation (Boyle-van’t Hoff law (1.3) that states the constant amount of the impermeable components inside the cell (3) is added:
dWdt=-LpRTYf-Y+Zf-Z
(1.1)
dXdt=PsAYf-Y
(1.2)
WZ=WisoZiso=W0Z0=WfZf=const
(1.3)
where W is the cell water volume, t is the time, Lp is the hydraulic conductivity, R is the gas constant, T is the temperature, A is the surface area, Y and Z are the osmolalities of permeable and impermeable solutes respectively, XWY is the intracellular amount (“number of osmoles”) of the permeable solute, Ps is the solute membrane permeability. Subscript “iso” refers to the isosmotic values, “0” is the values at time zero, and “f” refers to the final (equilibrium, extracellular) values. The water permeability can be considered as LpRT.
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