General method for the derivation and numerical solution of epithelial transport models

Abstract

A general method is presented for the formulation and numerical evaluation of mathematical models describing epithelial transport. The method is based on the principles of conservation of mass, and maintenance of electroneutrality within the cells and bathing solutions. It is therefore independent of the specific membrane transport mechanisms, and can be used to evaluate different models describing arbitrary transport processes (including passive, active and cotransport processes). Detailed numerical methods are presented that allow computation of steady-state and transient responses under open-circuit, current-clamp and voltage-clamp conditions, using a general-purpose laboratory minicomputer. To evaluate the utility of this approach, a specific model is presented that is consistent with the Koefoed-Johnson and Ussing hypothesis of sodium transport in tight epithelia (Acta Physiol. Scand. 42:298-308, 1958). This model considers passive transport of an arbitrary number of permeant solutes, active transport of sodium and potassium, and osmotically induced water transport across the apical and basolateral membranes. Results of the model are compared to published experimental measurements in rabbit urinary bladder epithelium.

Transepithelial glucose transport and Na+/K+ homeostasis in enterocytes: an integrative model

“…7 and 8 we integrate the net flow of each species over the length of the time step to update the concentrations. This iterative calculation of the membrane potentials is well proven for epithelial models (25,43,85). A detailed description of our implementation and validation of the iterative method is given in Iterative Calculation of Membrane Potentials.…”

“…43. At each time step we start with guess values for the membrane potentials and calculate the flows; the estimate for the membrane potential is then refined using the Newton-Raphson method and the flows are recalculated until the net cell current (Eq.…”

“…43. It utilizes two nested Newton-Raphson steps to refine the estimates until the conditions for cell and epithelial electroneutrality are met.…”

Model of ion transport regulation in chloride-secreting airway epithelial cells. Integrated description of electrical, chemical, and fluorescence measurements

“…In analyzing the complex effects of Ca and cAMP agonists on I., transepithelial conductances, and intracellular Ca (Hartmann et al, 1989), it became clear that a mathematical model was necessary to interpret the data and to design experiments to distinguish among competing hypotheses. We have therefore developed a quantitative electrokinetic model of canine tracheal epithelial cells based on the general mathematical formulation developed by Latta et al (1984) and extended by Verkman and Alpern (1987). When formulated with a minimal set of assumptions, the model accurately reproduced a number of important published observations.…”

Fluid dilution and efficiency of Na+transport in a mathematical model of a thick ascending limb cell

“…The cytosol is assumed to be dilute and well stirred. With these assumptions, equations for solute and water conservation are given by Latta et al (27).…”