Correspondence to: Erik Hviid Larsen, August Krogh Institute, Universitetsparken 13, DK-2100 Copenhagen Ø, Denmark. Fax:45-3532-1567 E-mail:EHLarsen{at}aki.ku.dk.
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Abstract |
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A mathematical model of an absorbing leaky epithelium is developed for analysis of solute coupled water transport. The non-charged driving solute diffuses into cells and is pumped from cells into the lateral intercellular space (lis). All membranes contain water channels with the solute passing those of tight junction and interspace basement membrane by convection-diffusion. With solute permeability of paracellular pathway large relative to paracellular water flow, the paracellular flux ratio of the solute (influx/outflux) is small (24) in agreement with experiments. The virtual solute concentration of fluid emerging from lis is then significantly larger than the concentration in lis. Thus, in absence of external driving forces the model generates isotonic transport provided a component of the solute flux emerging downstream lis is taken up by cells through the serosal membrane and pumped back into lis, i.e., the solute would have to be recirculated. With input variables from toad intestine (Nedergaard, S., E.H. Larsen, and H.H. Ussing, J. Membr. Biol. 168:241251), computations predict that 6080% of the pumped flux stems from serosal bath in agreement with the experimental estimate of the recirculation flux. Robust solutions are obtained with realistic concentrations and pressures of lis, and with the following features. Rate of fluid absorption is governed by the solute permeability of mucosal membrane. Maximum fluid flow is governed by density of pumps on lis-membranes. Energetic efficiency increases with hydraulic conductance of the pathway carrying water from mucosal solution into lis. Uphill water transport is accomplished, but with high hydraulic conductance of cell membranes strength of transport is obscured by water flow through cells. Anomalous solvent drag occurs when back flux of water through cells exceeds inward water flux between cells. Molecules moving along the paracellular pathway are driven by a translateral flow of water, i.e., the model generates pseudo-solvent drag. The associated flux-ratio equation is derived.
Key Words: isotonic transport, convection-diffusion, uphill water transport, anomalous solvent drag, pseudo-solvent drag
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INTRODUCTION |
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Water transport by leaky vertebrate epithelia like small intestine (
Description of the Recirculation Model
The model contains three different pore types in cellular, tight-junction and interspace basement membranes: A diffusion pore (channel) is a water impermeable, but solute permeable pore, a convection-diffusion pore is a water and solute permeable pore, and a water pore (channel) is a water permeable, but solute impermeable pore (i.e., a pore with reflection coefficient of unity for the solute).
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METHODS |
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Compartments and Membrane Pathways
The physical outline of the model is shown in Fig 1, and the name and definition of the associated variables are listed in the Abbreviations. The outer and inner border of the epithelium are exposed to a mucosal (o)1 and a serosal (i) compartment that are well stirred. The cellular (cell) and lateral intercellular space (lis) constitute two well-stirred intraepithelial compartments. The model contains the following five membranes, apical cell membrane (a), serosal cell membrane (s), lateral cell membrane (lm), tight junction membrane (tm), and interspace basement membrane (bm). The following solutes (all non-electrolytes) are considered: A diffusible solute, S, and non-diffusible solutes, ND, of the cellular and of the two external compartments. The diffusible solute, S, is supposed to pass all membranes, but by various mechanisms. An active, saturating pump is located in the membrane lining the lateral intercellular space. The tight junction and the interspace basement membrane contain both pure diffusion channels and convection-diffusion channels through which S is translocated by both convection and diffusion. Only in these membranes is water interacting with solute. The three other membranes contain water channels, and the serosal and apical membrane diffusion channels. By having both convection-diffusion and pure diffusion channels in tight junction membrane, paracellular convection fluxes can be analyzed for the case of convection-diffusion and pure diffusion, respectively, across tight junction.
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Equations
In this section are presented the equations, which constitute the mathematical problem. There are four diffusion fluxes given by Fick's Law:
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(1) |
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(2) |
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(3) |
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(4) |
The pump flux of the driving species is given by:
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(5) |
In all computations, n = 3 and KSpump = 10 mM.
In our treatment of coupling between solute and water flows in convection-diffusion channels we have avoided linearized forms of the Kedem-Katchalsky equation (
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(6) |
where JS is the solute flux, JV the water-volume flux, C the solute concentration, and the reflection coefficient; JV(1 -
) is the convection velocity of the solute. This equation follows from the Smoluchowski equation (
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(7) |
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(8) |
The water flux through the five membranes are calculated from:
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(9) |
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(10) |
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(11) |
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(12) |
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(13) |
It follows that water channels of membranes a, s, and lm are not permeable to the solute, i.e., formally, a,
s, and
lm are all unity. Requirement of mass conservation of diffusible solute and water, respectively, result in four continuity equations:
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(14) |
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(15) |
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(16) |
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(17) |
Inserting Equation 1Equation 2Equation 3Equation 4Equation 5 and Equation 7Equation 8Equation 9Equation 10Equation 11Equation 12 HREF="#FD13">Equation 13 into the conservation equations results in four equations for the pressures and concentrations in the two intraepithelial compartments. In fact, the fluxes given by Equation 1 5 and 713 are to be considered as auxiliary variables defining the various mechanisms by which solute transport takes place between compartments. The primary variables are the concentration of diffusible solutes, CScell and CSlis, that are conserved through Equation 14 and Equation 15, and the pressures, pcell and plis, that serve to ensure mass conservation through Equation 16 and Equation 17. It is important to note that the concentration of non-diffusible solute, CNDcell, only indirectly contributes to the conservation equations since it is not actively involved in the flux balance. As a consequence, the resulting system of equations contains five unknowns but only four equations. However, introducing an auxiliary pressure variable,
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(18) |
results in a unique system consisting of four equations to be solved for the four dependent variables, CScell, CSlis, plis, and p*,cell. Although the actual values of pcell and CNDcell are not needed for the mathematical solution of the problem, they are of interest for discussing the physical processes involved. An additional equation for determining these variables can be established if the cell volume is known. The cell volume per unit area of epithelium, Vcell, is related to CNDcell from the equation,
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(19) |
in which the independent variables, Dcell and MNDcell, are the number of cells per unit area of epithelium and the amount of non-diffusible solute per cell, respectively. If the cell was rigid both Vcell and CNDcell would be constants, independent of the properties of mucosal and serosal bathing media. As discussed in
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(20a) |
Here, µa, µs, and µlm are compliance constants (Pa-1) of the respective plasma membrane domains and cellthe cell volume at pressure equilibrium between all compartments. Thus, knowing the cell volume at pressure equilibrium would in principle allow us to determine pcell, CNDcell, and Vcell by solving Equation 18Equation 19. Unfortunately, neither compliance constants nor equilibrium conditions are known and we abstain from estimating these. It should be stressed, however, that with the introduction of the auxiliary variable, p*,cell, Equation 1 HREF="#FD2">Equation 2Equation 3Equation 4Equation 5 and Equation 7Equation 8 HREF="#FD9">Equation 9Equation 10Equation 11Equation 12Equation 13Equation 14 HREF="#FD15">Equation 15Equation 16Equation 17Equation 18, constitute a unique system, which does not depend on any of these quantities, and which can be solved for all remaining dependent variables. An approximate way to proceed is to assume that the relative change in cell volume is so small that it can be neglected. This would imply that the cell volume displaced by, e.g., a pressure induced expansion of the lateral space, by the associated increasing pcell causes a quantitative similar displacement of volume of the infinitely large external compartments. With this assumption Equation 20a leads to,
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(20b) |
where µcell = µa + µlm + µs. Thus, according to this model the hydrostatic pressure of the cell, pcell, adjusts itself to a value between the ambient pressures weighted relative to local compliance constants. With p*,cell being obtained together with the other primary variables, CScell, CSlis, plis, and the fluxes, as explained below (Computing Strategy), subsequently the three remaining unknowns, pcell, CNDcell, and Vcell, are computed from Equation 18, Equation 19, and Equation 20b.
Paracellular unidirectional fluxes.
The model contains equations for handling unidirectional paracellular fluxes of the driving species, S, as well as of solute(s) that cannot pass the cell (paracellular tracers, T). Unidirectional convection-diffusion fluxes of S across the tight junction membrane and the interspace basement membrane, respectively, are given by (always positive):
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(21a) |
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(21b) |
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(22a) |
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(22b) |
Equations for calculating unidirectional fluxes of S in diffusion channels of the interspace-membranes are:
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(23a) |
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(23b) |
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(24a) |
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(24b) |
From the above fluxes, the two paracellular unidirectional fluxes of S, denoted JSpara,IN and JSpara,OUT, respectively, are calculated by:
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(25a) |
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(25b) |
These equations cover the case of steady state unidirectional fluxes of a three-compartment two-barrier system (
Sign conventions Fluxes of solute and water are positive in direction from outer to inner solution and from cell to lateral intercellular space.
Computing Strategy
We built two versions of the model: Solution of Equation 1Equation 2 HREF="#FD3">Equation 3Equation 4Equation 5, Equation 7Equation 8Equation 9 HREF="#FD10">Equation 10Equation 11Equation 12Equation 13Equation 14Equation 15 HREF="#FD16">Equation 16Equation 17, Equation 19, Equation 20aEquation 20b, and Equation 25a,Equation 25b) yields the value of the 20 dependent variables, but allows the net tonicity, TON, of the absorbed fluid to be a derived dependent variable:
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(26) |
Thus, in this version of the model (version 1), dependent on choices of independent variables, the net tonicity may vary from hypotonic over isotonic to hypertonic. To study the model under conditions of strictly isotonic transport, in the other version of the model (version 2), TON, was included as an independent variable, and Equation 2 and Equation 26 were used for deriving an expression for PSs:
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(27) |
In version 2 of the model, therefore, with TON set to isotoni-city (300 mM),PSs was included as a dependent variable, defining the recirculation flux across the basolateral membrane that would make the net absorbate truly isotonic. The physiological implication of this procedure would seem to be that the epithelium actively regulates the recirculation pathway to achieve isotonicity. While this is a reasonable or at least a possible hypothesis, it is not a necessary implication of the mathematical procedure. Since in this paper we wish to study isotonic transport, the condition of isotonicity has to be included in the set of equations to assure that the identified mathematical solutions are in the isotonic area of parameter space. For the computations, we have chosen to make PSs a dependent variable. This, however, is only of historical interest once relevant mathematical solutions (reference states) have been identified. In real life it may as well be another parameter (for instance PSa) or a combination of two or more parameters, which is/are being regulated. Nevertheless, when isotonicity is assumed (version 2) it is important to remember that changing one parameter (see Fig 4 Fig 5 Fig 6 Fig 7 Fig 8) is going to affect the values of all the derived variables (including PSs).
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Equation 1Equation 2Equation 3Equation 4Equation 5, Equation 7 HREF="#FD8">Equation 8Equation 9Equation 10Equation 11Equation 12Equation 13 HREF="#FD14">Equation 14Equation 15Equation 16Equation 17Equation 18, and Equation 27 constitute a set of strongly coupled, non-linear equations for 17 (version 1) or 18 (version 2) dependent variables, respectively. A convenient way of solving this system is to use the iterative Newton-Raphson method, which is a computationally efficient and robust method for non-linear equations. With this method linearization is introduced locally by forming the Jacobian matrix by differentiating the governing equations with respect to the dependent variables.
Presentation of Results
All variables of the computer program are in SI units. For making comparison with experimental results from literature easy, as listed in the Abbreviations above, computed results given in the main text are in units common to physiological papers. However, in the physiological literature hydraulic conductance can be found in many different units. In tables and figures of the present paper is the hydraulic conductance of membrane m, Lm, given in units of (cm·s-1)/(N·cm-2), which is related to the osmotic permeability, Pf, by:
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(28a) |
With the molar volume of water, VW = 18 cm3/mole, R = 8.31(N·102 cm)·mole-1·K-1 and T = 293 K, the osmotic water permeability, Pf, given in conventional unit of cm·s-1, and the above mentioned Lm are related by:
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(28b) |
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RESULTS |
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Basic Analysis of the Problem
Two barriers delimit the lateral intercellular space of the present model: The tight junctions and the basal interspace membrane. In both membranes are the solute fluxes governed by the non-linear Hertz equation. A pump in the lateral membrane pumps solute into the lis and, dependent on properties of the delimiting membrane, may drive fluid absorption under equilibrium conditions. Before incorporating recirculation into this model, we will consider the simpler model of the lis, the two delimiting membranes, and the pump, and ask two questions: (a) What is the tonicity of the transportate? (b) What is the expected flux-ratio of a paracellular marker that traverses this system?
A single convection-diffusion barrier: The Hertz equation.
We will start our analysis by considering the Hertz equation for a single convection-diffusion barrier and in the absence of solute pumping. With concentrations of the solute in compartment (1) and (2) denoted, C(1) and C(2), the Hertz equation reads:
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(29) |
where JS is the convection-diffusion flux through water filled pores with reflection coefficient, , solute permeability, PS, and water flux, JV. The unidirectional tracer fluxes (both positive) are then given by:
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(30a) |
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(30b) |
with a ratio of:
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(30c) |
Fig 2, left, depicts the net flux, Equation 29, and the two unidirectional fluxes, Equation 30a and Equation 30b, as function of JV·(1 - )/PS. Fig 2 (right) shows the flux-ratio, Equation 30c, as function of JV·(1 -
)/PS. It is clear that the flux-ratio is a measure of JV·(1 -
)/PS, and that this quantity defines the degree to which convection dominates over diffusion. A solute flux-ratio in range of e, corresponding to measurements (
)/PS
1 (assuming nearly the same concentration at the two sides of the membrane), i.e., with a flux of S which contains a significant diffusion component.
From Equation 30aEquation 30bEquation 30c it is easily shown that, lim JS = PS·(C(1) - C(2)) for JV 0, which is pure diffusion.3 In the alternative regime, pure convection, for JV
±
, lim (JS/JV) = C(1)·(1 -
) and C(2)·(1 -
), respectively. Whereas these features appear trivial mathematically, the physiological implications are not as trivial. As shown graphically in the insert of right hand panel of Fig 2, in the regime where JS and JV are both positive and the flux of S contains a significant diffusion component, the concentration of the transportate, JS/JV, may obtain values much above C(1)·(1 -
). Since for C(1) > C(2), JS/JV is monotonic decreasing, JS/JV > C(1)·(1 -
) for all finite values of JV. Thus, with the generally low experimental ratio of paracellular unidirectional fluxes the concentration of the transportate would be significantly larger than C(1)·(1 -
). If we allow the (1)-side to represent the lis (which due to solute pumping fulfils Clis > Ci), there is one value of
, for which the fluid leaving the lis through the basement membrane would be isotonic. At higher values of
, the absorbate would be hypotonic, at lower values, hypertonic. However, the value for
at which isotonicity is obtained would be different for different values of JV. Further, with a high value of
the absolute fluid flow across the membrane for a certain hydrostatic pressure difference between lis and the inner compartment would decrease, Equation 13. As we will see, solute recirculation allows isotonicity for a wide range of JV while maintaining a low
and hence a large fluid transport with modest plis.
Two serial convection-diffusion barriers.
The above analysis applies formally only to a single convection-diffusion barrier described by Hertz equation, in our case either the tight junctions or the basement interspace membrane. The case of two serial barriers delimiting a middle compartment (lis) can be solved by applying Hertz equation for each of the two barriers, and eliminating the concentration in between (Clis) to get the solute and water flux through the entire system. This was done by
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(31) |
where C is the concentration in both inner and outer bath (assumed to be the same), j is the (finite) rate of solute pumping into the middle compartment, 1 and
2 are the reflection coefficients for the first and second barrier, respectively, and
1 = JV(1 -
1)/P1 and
2 = JV(1 -
2)/P2, where P1 and P2 are the solute permeabilities of the first and second barrier, respectively. Considering the transported fluid,
1 >
2 defines water transport from the (1) to the (2) side, as one would expect. With this assumption, they showed that JV < j/(C·(
1 -
2)). They also showed that for all positive finite values of JV, JS/JV > (1 -
2)·C. Thus, the problem of obtaining an isotonic fluid is the same for the two-membrane system as in the case of a single membrane.
Using the above strategy, Equation 30a and Equation 30b, see also Equation 21aEquation 21b, Equation 22aEquation 22b, and Equation 25aEquation 25b, can be used to derive an expression for the ratio of unidirectional fluxes across two serially arranged convection-diffusion barriers. The result is simply:
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(32) |
where now C(1) and C(2) are the concentrations at either side of the middle compartment (in Equation 31 we assumed C(1) = C(2) = C). Thus for the case of a two-barrier system, the ratio of paracellular unidirectional fluxes is still a measure of the relative contribution of convection and diffusion.
The Recirculation Model
In the following we will consider the case of solute recirculation. If all membranes have non-zero hydraulic conductance, i.e., La, Ls, Ltm, Lbm, Llm > 0, under transepithelial equilibrium conditions two pathways are available for transport of fluid across the epithelium. The one is the paracellular route with a water permeability governed by the hydraulic conductance of two membranes, Ltm and Lbm. The other is the translateral route with its water permeability governed by the hydraulic conductance of three membranes, La, Llm, and Lbm. All membranes having a non-zero hydraulic conductance is here treated as the general case that will be dealt with in details immediately below. For this case, tight junction permeable solutes are transported by convection-diffusion across both tight junction and interspace basement membrane obeying principles outlined above in our basic analysis of two serial convection-diffusion barriers. In a subsequent section we discuss features of the model with eliminated tight junction hydraulic conductance, Ltm 0, implying that solutes diffuse across tight junction with water entering the lateral coupling compartment via cells.
The Reference State
Choice of independent variables.
With the results of our basic analysis in mind, the standard input variables were selected for obtaining a paracellular flux-ratio and absolute fluxes approximating those of Na+ in toad small intestine, and an intracellular concentration of S, which amounts to less than 10% of its the mucosal concentration. In experiments with glucose in mucosal bath
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Before entering the analysis, regarding the choice of independent variables two points should be discussed. With glucose in bathing solutions, the ratio of transepithelial unidirectional Na+ fluxes (considering in both directions the sum of cellular and paracellular fluxes) is similar for toad and rat small intestine (22.5). The fluxes of rat, however, are about three times larger than those of toad. For example, the transepithelial unidirectional influx in rat ileum is ~3,000 pmol·s-1·cm-2 (·cm2, which is fairly large for a leaky epithelium.
·cm2. This calculation made use of their experimental estimate of the paracellular unidirectional influx under transepithelial equilibrium conditions, JNAin,PARA = 1361 ± 250 pmol·s-1·cm-2, by assuming that the flux was due to diffusion, only. But as this component most likely is due to convection-diffusion, it is expected to be larger than the pure diffusion flux. Thus, the value of 204
·cm2 is expected to be an underestimate. In the calculations immediately below we wish to simulate the toad preparation, for which we have estimates of the recirculation flux. Therefore, the somewhat low tight junction permeability of PStm = 1.5·10-6 cm·s-1 has been chosen together with an interspace basement membrane permeability, PSbm = 3.5·10-5 cm·s-1. This latter permeability would correspond to a resistance of 23
·cm2, which is similar to the experimental estimate of 22
·cm2 of toad gallbladder (
Features of the reference state. With non-zero water permeability of all membranes and similar hydrostatic pressure (1 atm) and concentrations (300 mM) of external compartments, the hydrostatic pressure of the cell, pcell = 1.0043 atm, while the sum of concentrations of diffusible and non-diffusible solutes of the cell is 324.6 mM (Fig 3). This latter value amounts to a measurable hyperosmolality (8.2%) of the cellular compartment. With trans-lateral water flux the major determinant of the steady state cellular hyperosmolality is the hydraulic water permeability of the plasma membranes, which here is assumed to be fairly low. It is not a surprising result that the osmotic concentration of the cell drops to near that of the external solutions if either La or Ls, or both, are increased. For example, the total osmotic pressure of the cells would be ~308.5 mM if La and Ls are increased by a factor of five, and this results in minor changes, only, of all other dependent variables of the reference state (not shown). If the hydraulic conductance of the lateral membrane is set to zero (Llm = 0), the translateral water flux is abolished so that the total osmotic pressure of the cell drops to 300.16 mM with a hydrostatic pressure of, pcell = 1.0044 atm. With all other independent variables kept at their reference values, neither does this maneuver result in any significant change of the reference state's dependent variables indicated in Fig 3 (calculations not shown). We return to the question about the significance of the plasma membranes' water permeability in the discussing of uphill water transport.
At steady state the flux of water from the outer solution into the lateral intercellular space is accomplished by the concentration of S of the lateral intercellular space (336.9 mM) being larger than that of the outer solution (300 mM), that is, = 1.12. Likewise, the excess hydrostatic pressure of lis, which provides the driving force for moving water into the inner compartment, is in range of plausible values, plis - pi = 6.5 cm H2O. This particular mathematical solution is governed by, PSbm= 3.5·10-5 cm·s-1. Thus, with
= 4.845·10-6·(1 - 10-5)/3.5·10-5 = 0.138, the flux of S through water pores of the interspace basement membrane contains a significant diffusion component (see Fig 2). As a result, the virtual concentration of the transportate emerging downstream the lateral space,
= 2836/4.845 = 583.3 mM, is significantly larger than the steady state concentration of the lateral intercellular space fluid (336.9 mM)6. With the restriction of net isotonicity of the absorbate,
= 300 mM, our mathematical solution gives a stationary uptake of S across the serosal cell membrane, JSs = -1561 pmol·s-1·cm-2. This means that 66% of the molecules pumped through the lateral membrane is derived from the inner compartment (
= 1561/2373 = 0.658), which is in accord with the estimate obtained in experiments with toad small intestine (Table 2).
Robustness of mathematical solutions.
With only five dependent parameters measured at this stage (Table 2), the model is of course underdetermined, and it follows that the reference state of Fig 3 is only one of many that could be constructed to be consistent with the data. However, if the mathematical model provides faithful solutions to our physiological problem the qualitative behavior of the model should not be sensitive to exact values of variables of the input list. In other words, in the neighborhood of each independent variable listed in Table 1, also new sets of mathematical solutions should be compatible with the reference state. We have investigated this by examining how paracellular flux-ratio, CSlis, plis, and recirculation flux vary with imposed variations of tight junction reflection coefficient, interspace hydraulic conductivities, and solute permeabilities. In doing this we used version 2 of the model, so that serosal membrane solute permeability and hence recirculation flux would all the time be adjusted for achieving net isotonicity of the absorbate. Effects of varying tight junction properties are depicted in Fig 4 Fig 5 Fig 6, which will be discussed together. As mentioned above (Fig 2), increasing the ratio, JV(1 - )/Ps, leads to increase of the flux-ratio of solutes passing water permeable pores. Because an increase of Ltm, or a decrease of
tm, result in an increase of JVtm, for relatively small values of Ltm the paracellular flux-ratio becomes a monotonic increasing function of Ltm (Fig 4 A), and a monotonic decreasing function of
tm (Fig 5 A). Decreasing PStm has the effect of reducing the diffusion component of JStm and, thus, increasing the flux-ratio (Fig 6 A). The force driving water across tight junction is given by the difference in effective osmotic pressure between lateral intercellular space and outer (mucosal) compartment (Equation 12). Thus, decreasing hydraulic water permeability (Fig 4 B), or reflection coefficient (Fig 5 B), result in new steady states with increased CSlis. Mathematical solutions providing physiologically relevant flux-ratios also provide plausible values of CSlis, i.e.,
< 1.30. In the above ranges of the three independent variables, also interspace hydrostatic pressure (Fig 4 C, 5 C, and 6 C) and recirculation flux (Fig 4 D, 5 D, and 6 D) exhibit shallow monotonic variation about the reference state. In conclusion, by letting each of the three independent variables, Ltm,
tm, and PStm span almost a decade, mathematical solutions are obtained which are qualitatively similar to the reference state.
It is as expected that interspace and cellular hydrostatic pressures are the only dependent variables that are significantly influenced by changes of the hydraulic permeability of interspace basement membrane (Fig 7). Thus, while plis - pbath varies from 48.2 to 4.05 cm H2O (Fig 7 C), pcell - pbath varies in parallel from 32.1 to 2.70 cm H2O (not shown). The corresponding variation of CNDcell is trivial, from 305.4 to 304.5 mM (not shown). The interspace hydrostatic pressure also increases with increasing interspace basement membrane reflection coefficient, bm (not shown). This is because the effective osmotic force directed from inner compartment to interspace increases with the reflection coefficient (Equation 13). With virtually unchanged volume flow the increase of plis compensates for this increase of the counteracting osmotic force. As discussed above, the concentration of the fluid emerging from lateral intercellular space is much sensitive to the solute permeability of the basement membrane and, pari passu, so is the recirculation flux (Fig 8 D). However, it is worth noting that no dramatic effects are observed by varying PSbm within an order of magnitude. At any of the chosen values of this parameter is isotonic transport achieved by recirculating a significant proportion of the pumped solute flux.
Significance of solute recirculation. The reference state was found with two independent requirements. (a) Permeabilities were selected for reproducing solute fluxes compatible with measured Na+ fluxes in toad intestine. (b) We asked for a mathematical solution containing an isotonic net transportate (version 2 with Equation 27 and TON = 300 mM). Therefore, the fairly large recirculation flux given by the model (~0.7), which is shown above to be a robust result, is a derived quantity. The experimental analysis resulted in an estimate of similar magnitude (Table 2). Thus, the computed result indirectly verifies the hypothesis of the experimental paper that recirculation is of significance for maintaining isotonic transport. For exploring this feature in more detail we computed the tonicity of the transportate as a function of the recirculation flux. This was achieved with version 1 (containing Equation 26) by varying serosal permeability (PSs) about the value listed in Table 1. From the results shown in Fig 9 A it can be seen that even with tonicities of the net transportate ranging within ± 15% of the tonicity of the bathing solutions is there a very significant recirculation flux. This range is covered by values of PSs that are within x0.67 and x1.5 the standard value (Fig 9 B). Since the tonicity of the transportate of toad intestine was not measured, the exact value of the permeability of the recirculation pathway cannot be inferred from these computations. For investigating the explanatory range of the model, in computations requiring use of version 2 we decided to set TON = 300 mM. But as the analysis above showed that neither is the behavior of the model dependent on exact values of input variables nor is it of critical importance whether we run the model under assumption of truly isotonic or near isotonic transport, results obtained with the above choice are expected to apply more generally for epithelia that may generate a near isotonic transportate.
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Rate Limiting Steps of Isotonic Volume Flow
Varying apical membrane solute permeability
In the computations shown in Fig 10 the translateral flux of S was varied by varying solute permeability of the apical cell membrane. The stationary flux of the isotonic transportate is indicated by the net solute flux, JSnet, and the water flux, JV, with, = 300 mM (model version 2). It can be seen that in the recirculation model is the transepithelial paracellular flux of water, JV, driven by the transcellular active solute flux with S being derived from mucosal solution (Fig 10 A). The paracellular flux ratios are within the physiological range with the relative recirculation flux (
) being virtually unaffected (Fig 10 B). In Fig 10 C is shown that the large variation of the fluxes of water and solutes in the model are accomplished with little variation of cell volume. The associated interspace parameters (Fig 10 D) exhibit somewhat larger variations, but both are probably being kept within realizable values.
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Significance of hydraulic conductance
Thus, in the recirculation model, the rate of active solute transport is governing the transport of water through the epithelium. The coupling of the solute and water fluxes takes place in the lateral interspace, with direction of water flow determined by the relative values of the two reflection coefficients, that is, tm >
bm directs absorption. It would be of interest, therefore, to analyze effects of varying the hydraulic conductance of the epithelium on transepithelial solute and water flows (Fig 11). In these computations, with their relative magnitudes maintained the five hydraulic conductances were increased, while all other independent variables were kept at their reference state values (Table 1). By increasing the hydraulic conductance, Clis and the virtual concentration of the fluid emerging from lateral space,
, both approach the concentration of bathing solutions (Fig 11 A). Concomitantly the solute flux of S emerging from the lateral space, JSbm and, pari passu, the recirculation flux, -JSs (Fig 11 B), are diminished. Thus, the response to an increase of the hydraulic conductance is a diminished driving force for the flux of water with the volume flow kept within a narrow range about its reference state of 4.2 nl·s-1·cm-2 (Fig 11 C). It follows, that the paracellular flux ratio of the driving solute is also kept within a narrow range (Fig 11 C). The physiological significance of this is that solute transport becomes energetically more effective as indicated by the increase of
(Fig 11 D). With the individual hydraulic conductances of x10 their reference values, the fluxes entering the calculation are: The flux into the interspace from cells via the pump and from the outer solution via tight junction is, 1176.2 and 523.9 pmol·s-1·cm-2, respectively. With cellular reuptake of 338.6 pmol·s-1·cm-2, we thus obtain,
= (1176.2 + 523.9 338.6)/1176.2 = 1.16. The general conclusion is that for a solute pump driving a transepithelial water flow, the thermodynamic efficiency can be higher than that of the pump itself. It is noted that the above extra solute flux of 185.3 pmol·s-1·cm-2 is due to the pump-driven paracellular convection-diffusion flux being larger than the recirculation flux. For the sodium pump this would imply that more than 3 Na+ are transported across the epithelium per ATP molecule split by the Na+/K+-ATPase. This interesting feature of the model shall be explored in more detail below.
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Significance of solute-pump density
Under transepithelial equilibrium conditions is the transepithelial water flux governed by the apical permeability of the driving solute implying that volume flow saturates with saturation of the lateral solute pumps (Fig 10 C). Furthermore, increasing the hydraulic conductance of the epithelium increases the energetic efficiency of coupling between solute and water fluxes with very minor effect on the transepithelial water flux (Fig 11C and Fig D). Among vertebrate (kidney) epithelia isotonic volume reabsorption is varying more than one order of magnitude from ~1 nl·s-1·cm-2 in amphibian kidney (e.g.,
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In the computations shown in the inset of Fig 12 the volume flow across the epithelium was maintained at about the same value (~60 nl·cm-2·s-1), while the ratio of transjunctional and translateral hydraulic conductance, Ltm/Ltl, was varied (translateral hydraulic conductance, Ltl = La · Llis/[La + Llis]). Large energetic efficiencies of transepithelial active solute transport is being obtained with large ratio of Ltm/Ltl, and with an unmeasurable small osmotic concentration difference between bath and lateral intercellular space (303.2 mM CSlis - CSbath
301.5 mM, not shown). An efficiency of
1.5 would be in accord with mammalian proximal tubule (see Discussion). It is seen that this high efficiency of the model is associated with a transjunctional hydraulic conductance that is at least an order of magnitude larger than the hydraulic conductance of the translateral pathway.7 Even under these conditions can the recirculation flux be significant (for Ltm/Ltl = 37.5 is
= 1.57, and
= 0.11). If
tm is increased, e.g., to 0.85, the recirculation flux is being decreased to a small value (see Fig 5). But as the tight-junction convection-diffusion flux also decreases, the energetic efficiency becomes smaller (calculations not shown). The general conclusion is that in a compartment model like the one presented based on simple convection-diffusion theory, an energetic efficiency of solute transport larger than that of the pump itself requires a relatively large flow of water through convection-diffusion pores of tight junction.
Effect of varying ambient osmolality
Since the osmolality of the net transportate (TON in Equation 27) can be set to that of the bathing solutions it follows that the model can produce an isotonic absorbate whatever ambient osmolality is chosen. Thus, the model directly reproduces the finding that rabbit gallbladder transports fluid with a tonicity approximating that of the bathing solution used (
In the computations shown in Table 3 the concentration of the external compartments was changed in a symmetrical way using version 2 of the model (CSo=CSi=) with the input parameters adjusted so that computed volume flow and energetic efficiency of solute transport are similar to those of rabbit gallbladder, i.e., JV = 1114 nl·s-1·cm-2 (
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As the cellular solute concentration decreases with decreasing ambient concentration (not shown), the pumped flux of S is being reduced (column 5), and so is the concentration of the lateral intercellular space (column 6). As a result, the driving force for transporting water from the outer bath into the lateral intercellular space remains virtually constant. Therefore, the transepithelial volume flow does not change significantly upon dilution of the bathing solutions (column 7). This is not confirmed by experiments with gallbladder, which showed a significantly increased volume flow with decreasing ambient osmolality (
Water Transport in Presence of Transepithelial Osmotic Gradients
Uphill water transport
Increasing the mucosal osmolarity by adding a non-permeable solute to the outer bath results in diminished steady state water fluxes across the two apical barriers, and eventually these fluxes become "backward". Thus, at some point net water flow reverses, switching the epithelium from net absorption to net secretion, which makes adjustment of the osmolarity of transport via recirculation meaningless. Therefore, in performing these strength of transport calculations we used version 1 of the model, in which mucosal osmolarity was raised under conditions of unchanged recirculation permeability. It can be seen from the computed results in Fig 13 A, with input variables of the reference state the two apical fluxes, JVtm and JVa, reverse at an external hyperosmolality of ~30 and 55 mM, respectively. Fig 13 B depicts the dependence of the net flux of water on luminal osmolarity (JVnet = JVa+JVtm= JVs+JVbm). The computations showed that, JVnet = 0.000 nl·cm-2·s-1, if the concentration of the non-diffusible osmolyte was increased to, NDo = 31.487 mM. Thus, JVa+JVtm = JVs+JVbm
0 for an external osmotic concentration of, CSo+
NDo = 331.487 mM. Following
bm
0, and in experiments conducted with CSo=CSi=CSbath, and po=pi=pbath, it follows (Equation 9Equation 10, Equation 12, and Equation 13) that JVnet = 0 is fulfilled for:
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(33) |
The strength of transport, therefore, is a function of both paracellular and cellular transport parameters including the hydraulic conductance of the epithelium. If modelling is confined to the paracellular pathway, net volume flow ceases when, JVtm = JVbm = 0, implying that, JStm=PStm(CSbath-CSlis), JSbm=PSbm(CSlis-CSbath), and plis=pbath. Thus, with mass conservation of lateral space (Equation 15), at zero paracellular volume flow Equation 33 reduces to:
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(34) |
This equation shows, for a lateral intercellular space-model is the strength of transport depending, simply, on the ratio of the active solute flux into the lateral space and the sum of ion permeabilities governing the dissipative solute fluxes out of this space. It is noted that Equation 34 does not contain the hydraulic conductance of the pathway. These were the conclusions of
For the set of computations shown in Fig 13, the solute concentration of lis at which volume flow just ceases is, CSlis = 354.857 mM with plis = 1.00137 atm. Thus, the driving force for volume flow in tight junction is outward resulting in a water flux of, JVtm = -0.250 nl·s-1·cm-2 (Equation 11) that is numerically equal to the influx of water across the apical membrane. This example illustrates a major result given by modelling both cellular and paracellular pathways for volume and ion flows, that is, intraepithelial water fluxes may prevail despite the transepithelial net water flux is zero. The associated paracellular flux-ratio of the driving solute, S, together with the flux-ratio of the paracellular tracer, T, are shown in Fig 13 C. If the two molecules share convection-diffusion pores of the delimiting membranes of lis with similar selectivity, i.e., , and with CSo=CSi, CTo=CTi, according to Equation 32 the relationship between the ratios of paracellular unidirectional fluxes would be:
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(35) |
which is the result of model computations (Fig 13 C, = 1.5)8. At the external concentration at which net water flux is zero, the ratio of the paracellular tracer-flux is less than unity. This follows from the fact mentioned that the condition, JVnet = 0, is accomplished with non-zero water flows at the apical and basal boundaries of the epithelium, and that a flux-ratio less than unity is obtained when, JVtm
(Equation 32, CSo=CSi). In the present computations (Fig 13) is JVnet = 0.000 for JVtm = -0.250, and JVbm = 1.017 nl·cm-2·s-1, respectively, thus
= 0.95. The general conclusion is that with circular water flows across the outer and inner boundaries of the epithelium a paracellular flux-ratio less than unity does not necessarily imply that the water flux across both delimiting membranes of the lateral space is outward.
The external concentration at which net water flow reverses (Fig 13 B) is remarkably different from that observed in studies of rat jejunum (see Discussion). Model analysis of gallbladder by tm = 0.80, the net water flow reverses at an external concentration of 445 mM, i.e., CNDo = 145 mM (computations not shown). If the paracellular tracer permeabilities are decreased correspondingly, the ratio of paracellular tracer fluxes is 1.41 at a luminal concentration of 450 mM, which compares with a ratio of 1.36 obtained in 134Cs+-flux studies of toad small intestine with 150 mM urea added to the luminal perfusion solution (
Anomalous solvent drag
Equation 33 indicates that the hydraulic conductance of the cellular membranes constitutes another membrane variable of significance for the epithelium's capacity for uphill water transport. Also here we will demonstrate the feature by a minimum of changes of the input variables. For maintaining the strength of transport along the paracellular pathway its parameters were kept at the values indicated in the section above (see also legend of Fig 14), but with the water permeability of the apical and the serosal membrane increased by a factor of 10. Fig 14A and Fig B, shows that the high cellular water permeability now results in reversed net water flow already at an external concentration of ~310 mM. However, there is a substantial inward flow of water between cells (Fig 14 A). This results in a range of external osmolarities in which net water flux is outward, but, nevertheless, the ratio of the paracellular tracer-flux is larger than unity (, Fig 14 C). Furthermore, by inspection of Fig 14 it can be seen that there is a range of external osmotic concentrations (345355 mM) of which the water flux across tight junction is outward, but the flux-ratio of the paracellular marker molecule is above unity. As mentioned above, such a behavior follows from Equation 32, that is, for luminal concentrations below 355 mM is the argument of this equation positive and the flux ratio >1. The combination of an outward net water flux and a paracellular flux-ratio above unity was first observed in so-called leaky frog skin, and it was denoted anomalous solvent drag (
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Pseudo-Solvent Drag
If tight junction water permeability, but not solute permeability, is zero, water enters the lateral space via cells and paracellular markers enter the space via tight junctions. For this limiting case convection occurs in the interspace basement membrane, only. With starting point in our reference state we achieved this in two steps. Firstly, the hydraulic and diffusion permeability of tight junction convection-diffusion pores were decreased by a factor of 106 with the diffusion permeability of the pure diffusion channels of tight junction set to, PStm,diff = 1.5·10-6 cm·s-1 (driving solute) and PTtm,diff = 2.25·10-6 cm·s-1 (paracellular marker), respectively, i.e., similar to those of the convection-diffusion pore of the reference state (Table 1). Since water now passes the cells of low hydraulic conductance, at steady state the osmotic concentration of the lateral space is significantly increased (to 408 mM) and so is the recirculation flux ( = 0.87, computations not shown). For obtaining more realistic values of these variables it is necessary to increase the hydraulic conductance of the translateral route. We will illustrate important features of the pseudo-solvent drag mechanism by arbitrarily increasing the values of all three plasma membrane hydraulic conductances (La, Ls, and Llis) by a factor of five, which brought CSlis and
closer to the values of the reference state (see Fig 15). The net solute flux of 756 pmol·s-1·cm-2 and the paracellular influx of 433 pmol·s-1·cm-2 are within the range of fluxes measured experimentally (Table 2). With vanishing small tight junction water permeability and pure diffusion governing tight junction flux of S, all sets of mathematical solutions, necessarily CSlis>CSo, would be characterized by net diffusion loss of the driving species from lis to the outer compartment, in casu, JStm = -47.0 pmol·s-1·cm-2 (Fig 15). Thus, also across the apical border is there recirculation of the driving species. However, with eliminated tight junction water permeability, and with the above mentioned choice of independent variables, the paracellular flux ratio of the driving species and of paracellular marker are significantly smaller than those estimated experimentally,
= 1.15 and
= 1.10, respectively.9 The paracellular flux-ratios can be raised by decreasing the interspace basement membrane solute permeability by a factor of more than five, or by further increasing the hydraulic conductance of serosal membrane such as to obtain an even larger recirculation flux of water across the inner border of the epithelium (computations not shown, but see Equation 36a). But this adds nothing to the major conclusion, that is, as none of the maneuvers proposed seem justified, it is indicated that with eliminated tight junction water permeability and with physiologically relevant choice of independent variables the model cannot possibly reproduce all aspects of measured cation fluxes in toad small intestine.
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With diffusion pores of tight junction and convection-diffusion pores of the interspace basement membrane, according to Equation 25a and Equation 25b the paracellular flux-ratio would be independent of the permeability coefficient of tight junction:
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(36a) |
The relationship between the flux-ratios of two paracellular markers, S and T, with CSo=CSi and CTo=CTi, would then be:
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(36b) |
Since the reflection coefficient of the interspace basement membrane is small, within a certain range of molecular size we can write, , i.e., with little error the exponent of Equation 36b can be replaced by the ratio of the two molecules' diffusion coefficient in water. This analysis indicates that by proper choice of paracellular marker, a significant paracellular flux-ratio can be generated in an epithelium with water entering the paracellular space via cells, only. In a study of guinea-pig gallbladder,
0.2·10-5 cm2·s-1. Thus, with the paracellular flux-ratio of the driving species being 1.15 (Fig 15) according to Equation 36b, that of inulin would be 2.5. While this calculation presupposes that also the second species (here T = inulin) can permeate tight junction, the result is independent of the selectivity of tight junction.
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DISCUSSION |
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This study deals with a simple mathematical description and computer assisted analysis of the recirculation theory for solute-coupled water absorption. This type of transport is in the body found in so-called leaky epithelia of, e.g., (reviews) small intestine (
In this first stage of our analysis, the mathematical description is simplified by considering solute transports in an electroneutral regime, so that a non-electrolyte rather than the charged sodium ion constitutes the driving species. The reason why we expect that such a description gives useful insights into some general aspects of the mechanism of solute-coupled fluid transport is that in many leaky epithelia is the transepithelial electric potential difference close to zero. During physiological activity convection-diffusion constitutes the major, in some cases the only, mechanism of transepithelial paracellular flow of diffusible ions. Furthermore, some ion pathways of luminal and serosal cell membranes are co- and counter transporters working in an electro-neutral fashion (for vertebrate small intestine, reviewed by, e.g.,
Isotonic Fluid Transport in Absence of External Transepithelial Driving Force
With exactly similar solute composition, hydrostatic pressure, and electrical potential, respectively, of the two external compartments, truly isotonic transport occurs when the tonicity of transported fluid is identical with the tonicity of bathing solutions. Previous models dealing with coupling in a well-stirred intraepithelial compartment of solute and water fluxes, account for near-isotonic transport, only. That is to say, extracellular solution's tonicities are identical within experimental error of measurements, which is ~2%. Thus, in one type of modelling (
Cation fluxes of small intestine (
With tonicity of the transported fluid set to that of the external compartments (Equation 27) it follows that the model generates isotonic transport whatever external osmolality is assumed (Table 3). This remarkable feature was demonstrated for rabbit gallbladder by
Our analysis addresses the debated question about the driving force for water movement from mucosal solution to lateral space. In a study with ion sensitive microelectrodes of electrolyte concentrations of the lateral intercellular space fluid
Our computations indicate similar small gradients and predict that even large water flows can be associated with small osmotic concentration difference between mucosal and interspace solutions without compromising the robustness of the system (p. 14; Fig 12; and Table 3).
Transport of Water Uphill
Our model contains two pathways for transepithelial water movement, the paracellular pathway and the cellular pathway. Water flows between cells and from cells to lis are driven by the pump, while water flow through cells is driven by the transepithelial osmotic gradient imposed by the investigator. Thus, the transepithelial osmotic gradient at which net uphill water movement stops is given by the balance between the pump-driven water influx and the water permeability of the cell membranes. With an adverse osmotic gradient across the epithelium, if the pump-driven water flux is larger than the backward flux of water through cells, the epithelium exhibits uphill transport of water (Fig 13).
Paracellular Convection Fluxes
In leaky epithelia, hydrophilic solutes entering the lateral intercellular space via tight junction exhibit net inward fluxes in the absence of external electrochemical driving forces. It is a generally accepted hypothesis that this is accomplished by a transepithelial flow of water that entrains the solutes (
Anomalous Solvent Drag
Anomalous solvent drag refers to the paradox that epithelia generating osmotic water flux in outward direction may exhibit inwardly directed net flux of paracellular marker molecules maintained at transepithelial thermodynamic equilibrium (
Large Isotonic Fluid Flows Associated With High Density of Sodium Pumps
Our computations reproduced
Energetic Cost of Active Sodium Transport Spans Large Range
In tight epithelia like frog skin and vertebrate urinary bladder the sodium pump transports ~18 Na+ per molecule O2 consumed (
Our computations show that recirculation of the driving solute through cells may result in metabolic energy expenditure that is significantly larger than would be predicted from the net active transport (Fig 3; Table 3). Our analysis also revealed conditions at which recirculation is small and paracellular convection-diffusion relatively large so that net absorption of the driving solute would occur with an overall efficiency larger than that of the pump itself (Fig 12; Table 3). Thus, the recirculation model accounts in a simple way for the above-mentioned paradoxical findings.
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Footnotes |
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1 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; , compartment (
= 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) CS
<
o
o
o
S in
mM; CND
<
o
o
o
ND in
mM; TON, concentration of net transportate mM; Dcell, cell density; MNDcell, amount of ND per cell; V cell, cell volume; p
, hydrostatic pressure of
; (Membrane parameters) PSm, permeability to S in m; PSm,diff,permeability to S of pure diffusion pore in tm or bm; n, binding sites of pump dimensionless; Lm, hydraulic permeability of m;
<
oµ
o
o
m dimensionless;
m, reflection coefficient to S of convection-diffusion pore tm or bm dimensionless; (fluxes) JSm, flux of S across m; JSpump, pumped flux of S across lm; JSpump,max, saturated pump flux across lm; JSm,diff, flux of S in pure diffusion pore of tm or bm; JSpara,IN, paracellular unidirectional influx of S; JSpara,OUT, paracellular unidirectional outflux of S; JVm, water flux across m; JV, transepithelial water flux.
2 The assumptions underlying this equation are listed by
3 Not only is the intersection with y-axis (Fick diffusion), but also the water flux, JV, REV, at which JS = 0, proportional with the solute permeability, i.e., JV, REV = -(PS/[1 - ])·loge[C(1)/C(2)].
4 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.
5 Similar fluxes were obtained by
6 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 1).
7 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 (Ltm/Ltl). This follows from the necessary condition that the reflection coefficient of the tight-junction convection-diffusion channel is less than unity.
8 In 1.5.
9 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 (
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Acknowledgements |
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The study was supported by the Danish Natural Science Research Council (11-0971).
Submitted: 14 May 1999
Revised: 31 May 2000
Accepted: 1 June 2000
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References |
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