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THIRST AND VOLUME, ELECTROLYTE HOMEOSTASIS
1Department of Mechanical Engineering, 3Cancer Institute, 2Department of Biological Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada 89154
Submitted 3 April 2003 ; accepted in final form 12 August 2003
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ABSTRACT |
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 TOP ABSTRACT MODELRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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tight epithelium; active solute transport; Michaelis-Menten equation; coupled-water transport
Amphibian skin is a multilayered and a very "tight" epithelium (Ref. 21; Fig. 1). The outermost layer of the skin is the highly permeable stratum corneum (the cornified cells), followed by the stratum granulosum and the stratum spinosum. The innermost layer is the stratum germinativum that faces the basal lamina. The cells in the stratum granulosum, the stratum spinosum, and the stratum germinativum communicate with each other via the gap junctions and are held together by the desmosomes. Tight junctions exist in the stratum granulosum. Amphibian skin is also a heterocellular epithelium. Two types of cells, principal cells (majority) and mitochondria-rich cells (MR cells, minority), are present in the stratum granulosum and stratum spinosum. The MR cells constitute a highly specialized pathway for chloride transport across skin epithelium (epidermis) (21). They are few in number. In toads, the MR cells only occupy <2% of the epithelial cell volume. Their apical membrane area adds up to <1% of the total epidermal surface area (21). Both principal and MR cells have Na+-K+-ATPase on their basolateral cell membranes and the ability to actively transport sodium with the presence of ATP (20, 30).
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In living toads, the driving force for water reabsorption is believed to be the osmotic gradient across skin. Recently, Sullivan et al. (35) observed that the toad, Bufo punctatus, was able to rehydrate more rapidly from a 50 mM NaCl solution than from deionized water despite there being a reduced osmotic gradient for water uptake from the salt solution. A similar phenomenon was observed by Hillyard and Larsen (13) in experiments using Bufo marinus. They dehydrated toads 10-15% of their standard weights and allowed them to rehydrate from either deionized water or from 10 to 120 mM NaCl solutions. They found that nearly fully immersed toads could rehydrate from 50 mM NaCl at a rate that is 
1.45 times that from deionized water. In addition, the rehydration rate from 10 mM NaCl bathing solution was comparable to that from 50 mM NaCl bathing solution. However, water uptake from 100 mM sucrose and 50 mM Na gluconate was reduced relative to deionized water by a fraction predicted from the osmotic gradient. To explain the mechanism for enhanced water gain from dilute salt solutions, we postulate that there is water transport coupled with active solute transport (specifically, active Na+/Cl- transport) provided that both Na+ and Cl- are present in the bathing solutions.
Active solute transport can serve as a driving force for water uptake and is the main driving force in nearly isotonic transport in leaky epithelia such as the proximal tubule epithelium in kidneys (40) and the intestine (22). This active solute transport also exists in tight epithelia of the frog skin (16, 18, 28, 39). Nielsen (28) simultaneously measured the short-circuit current (Na+ flux) and the transepithelial water flux across isolated frog skin, Rana esculenta, bathed with identical Ringer on either side. He found a linear correlation between transepithelial Na+ transport and the water movement, which corresponded to 160 ± 15 molecules of water after each Na+ across the skin.
Figure 2 shows a simplified model for water and solute transport across amphibian skin. Na+ transport across the skin occurs through two pathways, transcellular and paracellular. In transcellular transport, Na+ first enters the cytoplasm via epithelial sodium channels (ENaCs) on the apical cell membrane, and then Na+ in cytoplasm is actively pumped into the intercellular space by Na+-K+-ATPase at the basolateral cell membranes of both principal and MR cells (21). In paracellular transport, sodium/chloride can traverse the tight junction area between cells due to concentration gradients and solvent drag (21). Water transport is also through two pathways. The transcellular water transport is suggested to occur through water channels at the apical and basolateral cell membranes (36) and the paracellular water transport is through the tight junction area.
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There have been numerous studies on epithelial NaCl and water transport across amphibian epidermis since the revolutionary work by Ussing and colleagues (18, 37-39). These studies are summarized in Refs. 15-17, 20, 21, 24, 29, 32-34.
The classical two-membrane theory for amphibian skin epithelium or KJU model proposed by Koefoed-Johnsen and Ussing (18) separated the surface of one epithelial cell into apical and basolateral membranes with Na+ pumps at the basolateral membrane (Fig. 2). The compartment model for transepithelial transport was first proposed by Curran and MacIntosh (3) to explain isotonic fluid transport across leaky epithelia in the absence of external osmotic gradients. Since then the original compartment model has been refined by many researchers to explain the fluid transport across leaky epithelia such as in the proximal tubule of the kidney, the ileum, and the small intestine where the transport is nearly isotonic (4, 5, 22, 23, 34, 40). However, there is no quantitative compartment model that is applied to the tight epithelial nonisotonic transport like that of the amphibian skin where there is a passive osmotic gradient in addition to solute-coupled water flux.
In this study we propose a two-barrier functional compartment model to examine the volume flux across amphibian skin driven by both the active solute transport and the osmotic gradient. The classical two-membrane model for one epithelial cell has been modified to describe water and solute transport across amphibian skin consisting of various types of epithelial cells. The amphibian skin is modeled as a well-stirred compartment bounded by an apical barrier and a tissue barrier (Fig. 3). The compartment represents the intercellular spaces between cells in the stratum granulosum. The apical barrier includes both transcellular and paracellular (tight junction) pathways for water and solute transport. The transcellular component of the apical barrier is hypothesized to have the ability to actively transport solutes through Na+-K+-ATPase. We first postulated that the actively transported solute flux satisfies the Michaelis-Menten equation (Fig. 4), i.e., the flux is nearly saturated when the apical bathing salt solution is 100 mosmol/kgH2O. In addition, we first estimated the permeability properties of the tissue barrier comprising stratum spinosum, the stratum germinativum, the basal lamina, and the dermis, based on the knowledge for the endothelial barrier forming the microvessel wall (1, 7, 8). Our model predicts a transepithelial volume flux across amphibian skin that is in good agreement with the experimental measurements in Hillyard and Larsen (13). Furthermore, the volume flux coupled with the active transport is nearly constant when the osmolality of the apical bathing solution is >100 mosmol/kgH2O. The molar ratio between the transported solutes and the coupled water molecules is 1:156, around the value reported in Nielsen (28). In summary, we proposed a two-barrier compartment model that can quantitatively predict various experimental measurements in Hillyard and Larsen (13) and Nielsen (28) for water and solute transport across tight epithelium of the amphibian skin. This model can also be easily modified to explain transport phenomena in other types of epithelia.
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Glossary
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MODEL |
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TOPABSTRACTMODELRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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The apical barrier is a heterogeneous barrier comprising the cell barrier and the TJ barrier. It represents the stratum corneum, the principal/MR cells in the stratum granulosum, and the TJ between the cells. The resistance of the stratum corneum is neglected because it is highly permeable to water and solutes as small as Na+ or Cl-. The cell barrier represents both the principal and MR cells and the difference between principal and MR cells is neglected. In our idealized model, the cell barrier has the ability to actively transport solutes. The solute flux by active transport is denoted by N, with the unit of nanomoles per second per square centimeter. The TJ barrier represents the TJ area between the cells in the stratum granulosum. Estimations for sodium and chloride permeability of this barrier are available in Larsen (21). In our model, the contribution of the TJ or the paracellular pathway to water transport is carefully examined (please see later sections).
The tissue barrier is in series with the apical barrier. It represents the stratum spinosum, the stratum germinativum, the basal lamina, and the dermis, i.e., the layers after the stratum granulosum. We first assume that the dermis is much more permeable to water and solute than other layers and its resistance to water and solute transport is neglected. The permeability of the tissue barrier to water and solutes is estimated later in Parameters.
In this model we hypothesize that the main resistance to water and solute transport comes from the apical barrier. The outflow from the tissue barrier mixes together at the exit to the tissue space. The osmolality of the exit flow can be different from that in the tissue.
Mathematical formulation. Formulas denote the hydrostatic pressure and the osmolality in the lateral intercellular space pI and CI, respectively. The volume flux across the cell barrier on unit surface area, JVC, from the apical side to intercellular space, is
 | (1) |
Here LPC is the hydraulic conductivity of the cell barrier and C is its reflection coefficient for salt solutes. CL is the osmolality of the apical bathing solution and pL is the hydrostatic pressure in the apical side. R is universal gas constant and T is temperature. Similarly, the volume flux across the TJ barrier on unit surface area, JVTJ, from the apical bathing solution to intercellular space, is
 | (2) |
LPTJ is the hydraulic conductivity of the TJ barrier, and TJ is its reflection coefficient for salt solutes.
Note
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The total volume flux across the apical barrier is
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The volume flux across the tissue barrier on unit surface area, JVT, is
 | (4) |
LPT is the hydraulic conductivity of the tissue barrier and T is its reflection coefficient to salt solutes. CT is the osmolality of the well-stirred tissue compartment. pT is the hydrostatic pressure in the tissue side, which is assumed to be the same as that in the apical side pL. Both pL and pT are set to be zero in our model.
Under steady state, the flux into and out of the intercellular space is equal.
 | (5) |
Equation 5 can be rewritten to provide a constraint between pI and CI.
 | (6) |
Applying Eq. 6, the total volume flux, JV, can be expressed only in terms of CI.
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JVC and JVTJ can be also expressed only in term of CI.
The solute flux across the cell barrier on unit surface area, JSC, is
 | (8) |
Here PC is the solute permeability of the cell barrier. L is defined as
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N is the actively transported solute flux. We assume that N satisfies the Michaelis-Menten equation (6)
 | (10) |
Nmax is the maximal flux; K is the Michaelis-Menten constant, which depends on the concentrations of salt solutions and the function of Na+-K+-ATPase.
The solute flux across the TJ barrier on unit surface area, JSTJ, is
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PTJ is the solute permeability of the TJ barrier.
The solute flux across the tissue barrier on unit surface area is
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PT is the solute permeability of the tissue barrier. T is defined as
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Under steady state, the solute flux into and out of the lateral intercellular space is equal.
 | (14) |
Equation 14 provides another constraint for pI and CI.
If the apical bathing solution osmolality CL and the tissue osmolality CT are given and the parameters for both the apical barrier and tissue barrier, LPC, PC, C, LPTJ, PTJ, TJ, LPT, PT, and T, are all known, there are only two unknowns left, CI and pI. There are also two constraints, Eqs. 5 and 14. To solve this problem, we first substitute the hydrostatic pressure pI in terms of CI, using Eq. 6, in Eqs. 1, 2, and 4. In this way, the volume fluxes, JVC, JVTJ, and JVT, are expressed only in terms of CI, as in Eq. 7. We then put the revised forms for JV's in Eqs. 8, 11, and 12 to express all JS's only in terms of CI. The conservation condition for solute flux, Eq. 14, provides an implicit constraint on CI. The meaningful solution for CI is found by using a commercial software MathCAD. After CI is determined, the hydrostatic pressure pI can be determined using Eq. 6. The volume fluxes across the cell barrier, the TJ barrier, and the tissue barrier can be determined using Eqs. 1, 2, and 4. Finally, the solute fluxes, JSC, JSTJ, and JST, can be determined by using Eqs. 8, 11, and 12.
One important variable of interest is the osmolality of the transported solution. In very leaky epithelia such as kidney proximal tubule and intestine, transepithelial transport is nearly isotonic and the osmolality of the transported solution is close to that for serum in the interstitial tissue space (22, 23, 40). For water transport across amphibian skin (tight epithelium), the osmolality of the transported solution is likely to be different from the serum osmolality provided that toads can rehydrate from deionized water where there is no reabsorptive solute flux (13). The osmolality of the transported solution is defined as
 | (15) |
If CE is different from CT, it may induce a perturbation on CT. However, the volume flux across amphibian skin, in our case, is small compared with total blood/lymph flux to skin. This perturbation is negligible and the assumption that CT is constant is valid.
Parameters. The parameter values used in our model are listed in Table 1. Room temperature (25°C) is used in all calculations. CT is the tissue osmolality and its value is set to be 250 mosmol/kgH2O. This value is a typical one for the blood/lymph osmolality of toads based on the experimental measurements (13). CL is the osmolality of the apical bathing solution. On the basis of the concentrations of NaCl bathing solutions (0-120 mM) used in Hillyard and Larsen (13), we choose CL from 0 to 250 mosmol/kgH2O.
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Nmax is the maximum of the actively transported solute flux used in Michaelis-Menten equation. To fit the experimental results in Hillyard and Larsen (13), Nmax = 4.0 nmol · s-1 · cm-2 (see RESULTS). K, the Michaelis-Menten constant, is estimated by fitting the experimental results in Hillyard and Larsen (13). Figure 4 shows the dependence of N/Nmax on K. K of 20 mosmol/kgH2O is used in our calculation.
The resistance of each barrier in the compartment model to the transepithelial water transport is not completely known. We hypothesize that the apical barrier offers most of the resistance to water transport. In this way, the hydraulic conductivity of the apical barrier (LPL) should be close to the transepithelial hydraulic conductivity (LP, see Eq. 16). The transepithelial hydraulic conductivity, LP = 54.2x10-7 cm·s-1·Atm-1 (7.1x10-9 cm · s-1 · mmHg-1), for isolated frog skin was measured in Nielsen (28), in the presence of antidiuretic hormone (AVT), a hormone that can increase the transepithelial hydraulic conductivity. This measured value is adopted for LPC, the hydraulic conductivity of the cell barrier for the toad skin during rehydration. There is no measured value for the hydraulic conductivity of the TJ barrier, LPTJ. Because amphibian skin is a tight epithelium, we assume that LPTJ is much smaller than LPC. In fact, in our calculation, LPTJ is set to be one-tenth of LPC. This ratio will be further examined in the DISCUSSION.
The salt permeability of the cell barrier, PC, is set to be 2.4 x10-8 cm/s, an averaged value for measured inner and outer membrane permeability of amphibian principal cells (21). The TJ solute permeability, PTJ, has a value of 5 x10-8 cm/s, a reported value for sodium permeability of the junctional membrane in Larsen (21).
The hydraulic conductivity and solute permeability of the tissue barrier have never been reported. We thus construct a simple model here to provide an estimate for them. We hypothesize that water and solute transfer across the tissue barrier via the intercellular space of epithelia. However, the detailed dimensions of the intercellular space for toad skin are not available. The best known ultrastructure of the intercellular space is that between endothelial cells forming capillary walls. Adamson and Michel (1) showed that the interendothelial cleft appears to be a small slit around endothelial cells. The gap width of the cleft is 20 nm and the depth is 0.5 µm (thickness of the wall) in frog mesenteric capillary. This value has been used in a series of research papers (7-11) to successfully predict water and solute transport across capillary walls. In this study we shall use a slit geometry to provide an estimate for the hydraulic conductivity and salt permeability of the tissue barrier.
From Fig. 1, we can estimate that, on average, the cells are cylindrical with radii r from 5 to 10 µm; the depth of the paracellular pathway varies from 50 to 100 µm. Furthermore, we assume that the gap width W of the paracellular pathway in the tissue barrier is 50-100 nm, larger than that for interendothelial space in frog mesenteric capillary walls. Using a slit model for the paracellular pathway (26), the hydraulic conductivity and the solute permeability of the tissue barrier, LPT and PT, are calculated as
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Here Ds is the salt (Na+ or Cl-) diffusion coefficient in bulk flow at 25°C, Ds = 1.4x10-5 cm2/s (8). µ is solution viscosity at 25°C. µ = 1.01 centipoise (11). The calculated LPT varies from 1.4 to 45x10-7 cm · s-1 · mmHg-1 and the PT varies from 0.7 to 5.6 x10-5 cm/s, which are of the same magnitudes of those for frog mesenteric capillary wall, respectively. In this study, we choose LPT = 4x10-7 cm · s-1 · mmHg-1 and PT = 1.1x10-5 cm/s. Compared with permeability of the apical barrier, the tissue barrier provides much less resistance to water and solute transport.
The tissue barrier is in series with the apical barrier. The total resistance of amphibian skin (inverse of the transepithelial hydraulic conductivity) can be approximated by the sum of the resistance from the apical barrier and from the tissue barrier.
 | (16) |
Using LPL = LPC + LPTJ = 7.8 x10-9 cm · s-1 · mmHg-1 (Table 1) for hydraulic conductivity across the apical barrier, and LPT = 4.0x10-7 cm · s-1 · mmHg-1 for that across the tissue barrier, the hydraulic conductivity across the skin LP = 7.65x10-9 cm · s-1 · mmHg-1. This value is almost the same as that reported in Nielsen (28).
Water and salt traverse the apical cell membrane in the stratum granulosum via specific channels. Sodium enters into the cytoplasm via the epithelial sodium channels (ENaCs) and is pumped out into the intercellular space by Na+-K+-ATPase, whereas water transfers via exclusive water channels (aquaporins) (36). In this study we assume ENaCs are nearly impermeable to water and water channels are impermeable to sodium ion. The reflection coefficient for solute of the cell barrier C is set to be 0.95. This value has been reported for KCl in Larsen (21) for both MR cells and principal cells. We also assume a typical TJ reflection coefficient TJ of 0.80. The high values chosen for C and TJ allow the apical barrier tight enough to build the osmotic gradient for water transport. In the calculation, we try to vary C from 0.95 to 0.99 and vary TJ from 0.40 to 0.95. It is found that the transepithelial volume flux is not sensitive to these values. The reflection coefficient T for the tissue barrier is set to be 0.01, the measured value for frog mesenteric capillaries (26), because the tissue barrier is as permeable as the capillary wall to solutes as small as Na+ or Cl-.
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RESULTS |
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TOPABSTRACTMODELRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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200 g and the estimated reabsorption area is 185 cm2 by a cylinder model for the toad body, the volume flow across toad skin can be estimated as 1.7 x 10-5 cm/s. The second characteristic value is the measured transepithelial volume flux across isolated frog skin (Rana esculenta) in Nielsen (28), 2.2 x 10-6 cm/s, in the presence of AVT. Under the same condition, the actively transported solute flux was measured as 0.765 nmol · s-1 · cm-2. Another characteristic value reported by Nielsen (28) is that there is a linear correlation between actively transported solute flux and the volume flux, which corresponds to that 160 ± 15 molecules of water follow each Na+ across the skin.
On the basis of the aforementioned experimental results, we can determine K and Nmax in the Michaelis-Menten equation that is assumed for the active solute transport across the apical barrier. Figure 4 shows the relation between dimensionless N/Nmax and K. We choose K = 20 mosmol/kgH2O and plot in Fig. 5 the volume flux across the skin JV as a function of N at various apical bathing solution concentration CL. These lines are nearly parallel to each other with the slope 2.8-3.0 nl/nmol. JV shown here is the combined result from the active solute transport and the osmotic gradient across the skin except at CL = 250 mosmol/kgH2O when the osmotic gradient disappears. The larger the N, the higher the JV. The slope of the line for CL = 250 mosmol/kgH2O, 2.8 nl/nmol, corresponds to 156 molecules of water that follow each solute across the skin. This ratio is in the range of 160 ± 15 reported in Nielsen (28). Therefore, by properly choosing K, we can fit the experimental data. To satisfy the observed JV of 1.7 x10-5 cm/s in Hillyard and Larsen (13) at CL = 100 mosmol/kgH2O (50 mM NaCl solution), from Fig. 5, N = 3.5, Nmax should be 4 nmol/cm2. This value is also comparable to that measured by Larsen (21). When N = 0.765 nmol · cm-2 · s-1, corresponding JV in Fig. 5 is 2.2 x 10-6 cm/s under isotonic condition at CL = 250 mosmol/kgH2O. It is the measured value across the frog skin in Nielsen (28).
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Figure 6 shows the model predictions and the experimental results for the volume flux JV as a function of osmolality of the apical bathing solution CL. The results are expressed as the ratio to the volume flux when CL = 0 (bathing solution is deionized water). Lines are model predictions and symbols are experimental data. The top curve shows the result when N satisfies Michaelis-Menten equation with K = 20 mosmol/kgH2O and Nmax = 4 nmol · s-1·cm-2. The bottom curve is when N = 0, representing that the driving force is an osmotic gradient only. We can see that under both cases the model predictions are in good agreement with the experimental data.
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The volume flux in the presence of the active solute transport (top curve) is always larger than that in the absence of the active solute transport (bottom curve). The difference between these two curves, namely, the volume flux coupled with the active solute transport, is of the same order of magnitude as the volume flux from the deionized water due to the osmotic gradient.
When CL is >100 mosmol/kgH2O, the coupled volume flux is nearly constant because the two curves in Fig. 6 are nearly parallel. This indicates the saturation of active solute transport when CL 
100 mosmol/kgH2O (See Fig. 4 at K = 20 mosmol/kgH2O). When CL < 100 mosmol/kgH2O, this model predicts a nearly constant transepithelial water flux from both active and passive transport. This is in agreement with the experimental measurements in Hillyard and Larsen (13) because they suggested that the hydration rate across the toad skin is nearly constant when the apical salt solution was dilute (<100 mosmol/kgH2O).
In Fig. 7A, we plot the intercellular osmolality CI as a function of the apical bathing solution osmolality CL in the presence and the absence of active solute flux N. When N = 0, CI is always less than tissue osmolality CT (250 mosmol/kgH2O) but greater than CL at each CL. This induces an osmotic gradient driving water from the apical side to the tissue side. However, when N is not zero but satisfies the Michaelis-Menten equation with K = 20 mosmol/kgH2O and Nmax = 4 nmol · s-1 · cm-2, CI can be either hypotonic or hypertonic relative to CT depending on CL. When CL is <150 mosmol/kgH2O, CI is hypotonic to CT. When CL is >150 mosmol/kgH2O, CI is hypertonic to CT.
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We plot in Fig. 7B the hydrostatic pressure pI in the intercellular space as a function of the apical bathing solution osmolality CL in the presence and the absence of active solute flux N. When N = 0, pI is negative, but very close to both pressures in the apical and tissue sides that are set to zero. This pressure difference will draw water into the intercellular space from both sides. However, compared with osmotic gradient that drives water from the intercellular space to the tissue side (1,500 mmHg at CL = 20 mosmol/kgH2O), the contribution to JV from the negative hydrostatic pressure difference can be neglected. When N 
0, pI ranges from 22 to 41 mmHg when CL ranges from 20 to 250 mosmol/kgH2O. Again, the volume flux contributed from the hydrostatic pressure difference is negligible compared with the contribution from the osmotic gradient, which is 1,500 mmHg at CL = 250 mosmol/kgH2O.
In Fig. 7C, we plot the transepithelial solute flux JS as a function of CL in the presence and the absence of active solute flux N. We notice that JS is almost completely contributed by N. When N = 0, JS is close to zero at each CL.
Figure 7D shows the osmolality of the transported solution CE = JS/JV as a function of the apical bathing solution osmolality CL in the presence and the absence of active solute flux N. When N = 0, CE is almost zero due to negligible JS (Fig. 7C). When N 0, CE increases with increasing CL. CE is greater than CT = 250 mosmol/kgH2O when CL > 150 mosmol/kgH2O. Higher CE would induce a disturbance in CT at the exit of the tissue barrier. However, compared with the much greater amount of water and solutes in the tissue region, this disturbance can be neglected.
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DISCUSSION |
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TOPABSTRACTMODELRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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The hydraulic conductivity of the apical barrier LPL is the summation of the transcellular permeability LPC and the paracellular permeability LPTJ. In our model, we use the measured value by Nielsen (28) for LPC, 7.1 x 10-9 cm · s-1 · mmHg-1, which was the measurement in the presence of a hormone AVT. In the absence of AVT, it was measured as 9.3 x10-10 cm · s-1 · mmHg-1, which is roughly one-tenth of that in the presence of AVT.
We test the influence of LPC on the volume flux JV, which is shown in Fig. 8. The 10 times higher LPC would increase JV by up to 3.5 times that for the lower LPC in both cases of N = 0 and N 0. We choose the higher LPC to predict the experimental results in Hillyard and Larsen (13), because the toads in the experiments were under dehydration conditions and there should be some hormonal effect (12, 25).
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In Fig. 9, we test the influence of hydraulic conductivity of the TJ barrier LPTJ on JV. Three LPTJ values, 0.02LPC, 0.1LPC, and 0.5LPC, are chosen for the comparison. LPC = 7.1 x10-9 cm · s-1 · mmHg-1. The 25 times change in LPTJ, from 0.02LPC to 0.5LPC, would only induce up to 18% increase in JV. The difference in lines of LPTJ = 0.02LPC and LPTJ = 0.1LPC is negligible.
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The influence of other estimated parameters, LPT, PT, C, TJ, and T, has been discussed correspondingly in the parameter section. Although there is a lack of experimental sources for these parameters, the estimates used in our model are shown to be reasonable.
Compartment models. Compartment models are widely used to model "leaky" epithelia for the coupled water transport, such as in the proximal tubule of the kidney, in the ileum, and in the small intestine, where transport is nearly isotonic (3-5, 22, 23, 34, 40). Transport across amphibian skin, the "tight" epithelia, is far from isotonic as shown in Hillyard and Larsen (13). The concentration of the apical bathing solution can be as low as zero (deionized water), whereas the lymph osmolality under subcutaneous skin varies from 220 to 260 mosmol/kgH2O (13). Therefore, unlike the isotonic transport in leaky epithelia, both the osmotic gradient and the active solute transport can act as driving forces for transepithelial water uptake.
We tried to adopt the four compartments-five membranes model by Larsen et al. (22) for the coupled water transport in toad intestine to model the volume flow across the amphibian skin. We failed due to the lack of permeability data for each membrane and being unable to simplify a series of nonlinear equations because there is no isotonic condition in our case.
In the current study, we are able to functionally simplify the skin into three compartments-two barriers structure with the active solute transport mechanism at the apical barrier. All the needed permeability parameters are either from measured data or from simple model estimations. We first proposed to use the Michaelis-Menten equation for the active solute transport by Na+-K+-ATPase, which is a commonly used relationship in this type of molecular event (6). We show that this assumed relationship for the active solute transport of epithelia in amphibian skin can lead us to a good prediction for a variety of experimental observations.
There is an observation in Hillyard and Larsen's experiment (13) showing that the total solute content after rehydration increased by 8% compared with that in the hydrated and dehydrated states, when the apical NaCl solution was 120 mM. There was only slight increase (2%) when the apical solutions were 10 and 50 mM, no increase when it was deionized water. From Figs. 6 and 7C, we notice that when the apical solution CL increases from 100 to 250 mosmol/kgH2O, the solute flux JS increases but the volume flux JV decreases. In this way, there would be a net increase in salt content when CL = 240 mosmol/kgH2O (or 120 mM NaCl) compared with that when CL = 100 mosmol/kgH2O (or 50 mM NaCl) if the toad uptakes the same amount of the volume.
Another observation in Hillyard and Larsen (13) was that adding amiloride (a pyrazine diuretic) to block the epithelial sodium channels (ENaC, Fig. 2) had no effect on the rehydration degree when CL = 20 and 100 mosmol/kgH2O. But the addition of amiloride significantly reduced the degree of rehydration relative to the same toads when CL = 240 mosmol/kgH2O or 120 mM NaCl. Where is the source for the salt that is pumped into the intercellular space to induce an osmotic gradient for water uptake if the apical sodium channel is blocked by amiloride? The sodium recirculation theory proposed in Larsen et al. (22, 23) may be employed in the current model to elucidate the amiloride effect in the future.
So far it is still under debate whether the transepithelial water flow coupled with ion transport is driven by local osmosis or electro-osmosis or molecular water pumping under nonosmotic conditions (34). A recent study by Sanchez et al. (31) showed the role of electro-osmosis in fluid transport across corneal endothelium (a leaky epithelium). Nielsen (28) in his study showed that the coupled water flow to Na+ transport across frog skin epithelium was due to local osmosis. For this reason, we neglect the contribution of the transepithelial potential difference to the transepithelial water flux (electro-osmosis). However, our model can be easily modified to include the electro-osmosis contribution for other types of epithelia where electro-osmosis is important.
In summary, we have developed a new two-barrier compartment model with the active solute transport mechanism for the water flux across amphibian skin (tight epithelia). This model can successfully explain the experimental results in Hillyard and Larsen (13) and Nielsen (28). This model may also be applied in other cases for water transport across tight epithelia, such as bladder and colon epithelia.
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DISCLOSURES |
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FOOTNOTES |
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  S. D. Hillyard, V. Baula, W. Tuttle, N. J. Willumsen, and E. H. Larsen Behavioral and Neural Responses of Toads to Salt Solutions Correlate with Basolateral Membrane Potential of Epidermal Cells of the Skin Chem Senses, October 1, 2007; 32(8): 765 - 773. [Abstract] [Full Text] [PDF]
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, hydration rates from NaCl bathing solution; 
, hydration rates from 50 mM Na gluconate; 
, hydration rates from 100 mM sucrose.
