A Mathematical Model of Solute Coupled Water Transport in Toad Intestine Incorporating Recirculation of the Actively Transported Solute

Abbreviations used in this paper: (Superscripts: compartments, membranes, and pathways) a, apical cell membrane; bm, interspace basement membrane; c, cell; i, inside (serosal) bath; lis, lateral intercellular space; lm, lateral cell membrane lining lis; o, outside (mucosal) bath; s, serosal cell membrane; tm, tight junction membrane; α, compartment (α = o, c, lis, or i); m, membrane (m = a, s, lm, tm, or bm); (Subscripts: solutes, water) ND, non-diffusible solute (cell or bath); S, driving (diffusible) solute; T, paracellular tracer; V, volume (water); (Intensive variables) C_S^α< ψονψεντρατιον οφ S in α mM; C_ND^α< ψονψεντρατιον οφ ND in α mM; TON, concentration of net transportate mM; D^cell, cell density; M_ND^cell, amount of ND per cell; V^cell, cell volume; p^α, hydrostatic pressure of α; (Membrane parameters) P_S^m, permeability to S in m; ^P_Sm,diff,permeability to S of pure diffusion pore in tm or bm; n, binding sites of pump dimensionless; L^m, hydraulic permeability of m; μ^mμ^cell< ρελατιωε ψομπλιανψε ψονσταντ οφ m dimensionless; σ^m, reflection coefficient to S of convection-diffusion pore tm or bm dimensionless; (fluxes) J_S^m, flux of S across m; J_S^pump, pumped flux of S across lm; J_S^pump,max, saturated pump flux across lm; J_S^m,diff, flux of S in pure diffusion pore of tm or bm; J_S^para,IN, paracellular unidirectional influx of S; J_S^para,OUT, paracellular unidirectional outflux of S; J_V^m, water flux across m; J_V, transepithelial water flux.

The assumptions underlying this equation are listed by Patlak et al. 1963. Most significantly, the partition coefficient is unity.

Not only is the intersection with y-axis (Fick diffusion), but also the water flux, J_V, _REV, at which J_S = 0, proportional with the solute permeability, i.e., J_V, _REV = −(P_S/[1 − σ])·log_e[C⁽¹⁾/C⁽²⁾].

The driving solute in small intestine is the charged sodium ion. As explained in Discussion, in this first version of the mathematical model, we have replaced the sodium ion with a diffusible non-electrolyte in order to simplify and generalize the treatment. In comparing our model with experimental data we shall nevertheless draw on measured electrolyte fluxes for identification of physiologically relevant areas of parameter space.

Similar fluxes were obtained by Frizzell and Schultz 1972, and they refer to in vitro studies. The fluxes of Na⁺ measured in vivo (Curran and Solomon, 1958) are larger by a factor of two when compared to in vitro.

Note that the diffusive component of S discussed here is that of convection-diffusion pores. The diffusion permeability of interspace channels with no water permeability is zero in the model's reference state (Table).

Note that despite the transjunctional and the translateral water flux is governed by a similar driving force, is the ratio of these fluxes smaller than the ratio of the hydraulic conductances of the two pathways (L^tm/L^tl). This follows from the necessary condition that the reflection coefficient of the tight-junction convection-diffusion channel is less than unity.

In Nedergaard et al. 1999 Cs⁺ was used as a paracellular marker. If tight-junction and interspace basement membrane pores are water filled and the two cations move through these pores with no specific interaction with pore walls, the selectivity would be given by the ratio of their diffusion coefficients in water, D_Cs/D_Na ≈ 1.5.

It is not contradictory that the net flux of S is directed from lis to outer bath (see Fig. 15) while the paracellular flux-ratio of S is greater than one. We may here think of the paracellular fluxes as being determined with isotopes different from the isotope pumped into lis. Generally, a molecule produced or consumed within a membrane will not influence the ratio of unidirectional tracer fluxes flowing across the membrane (Ussing 1952).