Local Osmosis and Isotonic Transport

Abstract

Osmotically driven water flow, u (cm/s), between two solutions of identical osmolarity, c_o (300 mM in mammals), has a theoretical isotonic maximum given by u = j/c_o, where j (moles/cm²/s) is the rate of salt transport. In many experimental studies, transport was found to be indistinguishable from isotonic. The purpose of this work is to investigate the conditions for u to approach isotonic. A necessary condition is that the membrane salt/water permeability ratio, ε, must be small: typical physiological values are ε = 10⁻³ to 10⁻⁵, so ε is generally small but this is not sufficient to guarantee near-isotonic transport. If we consider the simplest model of two series membranes, which secrete a tear or drop of sweat (i.e., there are no externally-imposed boundary conditions on the secretion), diffusion is negligible and the predicted osmolarities are: basal = c_o, intracellular ≈ (1 + ε)c_o, secretion ≈ (1 + 2ε)c_o, and u ≈ (1 − 2ε)j/c_o. Note that this model is also appropriate when the transported solution is experimentally collected. Thus, in the absence of external boundary conditions, transport is experimentally indistinguishable from isotonic. However, if external boundary conditions set salt concentrations to c_o on both sides of the epithelium, then fluid transport depends on distributed osmotic gradients in lateral spaces. If lateral spaces are too short and wide, diffusion dominates convection, reduces osmotic gradients and fluid flow is significantly less than isotonic. Moreover, because apical and basolateral membrane water fluxes are linked by the intracellular osmolarity, water flow is maximum when the total water permeability of basolateral membranes equals that of apical membranes. In the context of the renal proximal tubule, data suggest it is transporting at near optimal conditions. Nevertheless, typical physiological values suggest the newly filtered fluid is reabsorbed at a rate u ≈ 0.86 j/c_o, so a hypertonic solution is being reabsorbed. The osmolarity of the filtrate c_F (M) will therefore diminish with distance from the site of filtration (the glomerulus) until the solution being transported is isotonic with the filtrate, u = j/c_F.With this steady-state condition, the distributed model becomes approximately equivalent to two membranes in series. The osmolarities are now: c_F ≈ (1 − 2ε)j/c_o, intracellular ≈ (1 − ε)c_o, lateral spaces ≈ c_o, and u ≈(1 + 2ε)j/c_o. The change in c_F is predicted to occur with a length constant of about 0.3 cm. Thus, membrane transport tends to adjust transmembrane osmotic gradients toward εc_o, which induces water flow that is isotonic to within order ε. These findings provide a plausible hypothesis on how the proximal tubule or other epithelia appear to transport an isotonic solution.

Appendix

In the text, we have presented results from perturbation expansions of the equations describing Fig. 2, 3, 4 and 7. For each of these models, the perturbation approach is quite similar, so in this Appendix we will go through the expansion for Fig. 2A to demonstrate the approach.

The concentrations and fluid flow in Equations 7–9 are expanded in a series in ε.

$$ \eqalign{ & C_{{\rm{i}}} = C^{{{\rm{(0)}}}}_{{\rm{i}}} + \varepsilon C^{{{\rm{(1)}}}}_{{\rm{i}}} {\rm{ + }} \ldots \cr & C_{{\rm{s}}} = C^{{{\rm{(0)}}}}_{{\rm{s}}} + \varepsilon C^{{{\rm{(1)}}}}_{{\rm{s}}} + \ldots \cr & U = U^{{(0)}} + \varepsilon \,U^{{(1)}} + \ldots } $$

((A1))

Equation A1 is inserted into Eqs. 7–9 and terms of like powers in ε are collected to define a series of problems to be solved. For the order (0) problem, we obtain

$$ \eqalign{ & {{{\rm{d}}C^{{{\rm{(0)}}}}_{{\rm{i}}} (y)} \over {{\rm{d}}y}} = 0 \cr & 1 - C^{{{\rm{(0)}}}}_{{\rm{i}}} (0) = 0 \cr & C^{{{\rm{(0)}}}}_{{\rm{s}}} - C^{{{\rm{(0)}}}}_{{\rm{i}}} = 1 \cr & U^{{(0)}} C^{{{\rm{(0)}}}}_{{\rm{s}}} = 1 } $$

((A2))

The solutions to Eq. A2 are:

$$ {\eqalign{ & C^{{{\rm{(0)}}}}_{{\rm{i}}} = 1 \cr & C^{{{\rm{(0)}}}}_{{\rm{s}}} = 1 \cr & U^{{{\rm{(0)}}}} C^{{{\rm{(0)}}}}_{{\rm{s}}} = 1 }} $$

((A3))

The order (1) problems are:

$$ \eqalign{ & {{{\rm{d}}C^{{{\rm{(0)}}}}_{{\rm{i}}} (y)} \over {{\rm{d}}y}} = k_{{\rm{i}}} (U^{{{\rm{(0)}}}} C^{{{\rm{(0)}}}}_{{\rm{i}}} - 1) \cr & C^{{{\rm{(1)}}}}_{{\rm{i}}} (0) = U^{{{\rm{(0)}}}} \cr & C^{{{\rm{(1)}}}}_{{\rm{s}}} - C^{{{\rm{(1)}}}}_{{\rm{i}}} (0) = \alpha U^{{(0)}} \cr & U^{{(0)}} C^{{{\rm{(1)}}}}_{{\rm{s}}} + U^{{(1)}} C^{{{\rm{(1)}}}}_{{\rm{s}}} = 0 } $$

((A4))

If the order (0) solutions in Eq. A3 are inserted into Eq. A4, the results are:

$$ \eqalign{ & C^{{(1)}}_{{\rm{i}}} = 1 \cr & C^{{(1)}}_{{\rm{s}}} = 1 + \alpha \cr & U^{{(1)}} = - (1 + \alpha ) } $$

((A5))

Thus to within order (ε²) the solutions are given by:

$$ \eqalign{ & C_{{\rm{i}}} = 1 + \varepsilon + O(\varepsilon ^{2} ) \cr & C_{{\rm{s}}} = 1 + \varepsilon (1 + \alpha ) + O(\varepsilon ^{2} ) \cr & U = 1 - \varepsilon (1 + \alpha ) + O(\varepsilon ^{2} ) }$$

((A6))

In this simple model, the concentrations and flows are constant to within order (ε²). In the more complicated models of extracellular clefts, the values of C ⁽¹⁾_e (y) and U ⁽⁰⁾_e (y) depend on y. Nevertheless, the more complicated models are analyzed in the same manner, so the expansions will not be presented.