## Abstract

A numerical model of the rat distal tubule was developed to simulate water and solute transport in this nephron segment. This model incorporates the following: *1*) Na-Cl cotransporter, K-Cl cotransporter, Na channel, K channel, and Cl channel in the luminal membrane; *2*) Na-K-ATPase, K channel, and Cl channel in the basolateral membrane; and *3*) conductances for Na, K, and Cl in the paracellular pathway. Transport rates were calculated using kinetic equations. Axial heterogeneity was represented by partitioning the model into two subsegments with different sets of model parameters. Model equations derived from the principles of mass conservation and electrical neutrality were solved numerically. Values of the model parameters were adjusted to minimize a penalty function that was devised to quantify the difference between model predictions and experimental results. The developed model could simulate the water and solute transport of the distal tubule in the normal state, as well as in conditions including thiazide or amiloride application and various levels of sodium load and tubular flow rate.

- transporters
- ion channels
- water and mineral metabolism

distal tubule is defined as the portion of the nephron between the macula densa region and the first confluence with another tubule. This part of the nephron is the site where most of potassium secretion takes place, and normally as much as 70% of potassium in the final urine is derived from this segment (44). Furthermore, the distal tubule reabsorbs ∼10% of filtered sodium (45) and is highly permeable to water in the presence of antidiuretic hormone (11), concentrating the initially hyposmotic tubular fluid to isosmolality.

Transport properties of the distal tubule have been studied extensively in the rat using the micropuncture and in vivo microperfusion techniques. Those studies have revealed that *1*) the early part (which corresponds to distal convoluted tubule) and the late part (which is comprised of connecting tubule and initial collecting tubule) have different morphological and physiological characteristics;*2*) the early part reabsorbs sodium mostly through the thiazide-sensitive Na-Cl cotransporter in the luminal cell membrane and has low permeability to water; *3*) the late part is rich in amiloride-sensitive Na channel, has highly negative luminal electrical potential that drives potassium secretion, and is highly permeable to water if antidiuretic hormone is present; *4*) total sodium reabsorption is load dependent; and *5*) total potassium secretion is flow dependent.

Recently, an increasing number of molecules involved in the transport of water and solutes in the distal tubule have been identified by cDNA cloning. These molecules include amiloride-sensitive Na channel (8, 9, 41), thiazide-sensitive Na-Cl cotransporter (22), ROMK K channel (32, 70), and water channel (21). Such progress in molecular biology promotes detailed analysis of transport kinetics by providing sufficient amounts of these molecules for flux measurements. Accordingly, information regarding their transport mechanisms is growing. One of the most effective usages of these data would be to use them to predict the macroscopic transport in the renal tubule with the help of numerical models (65). Numerical models suitable for this purpose had been successfully constructed for proximal tubule (63, 64), thin descending limb of Henle (46, 57), thick ascending limb of Henle (20), and cortical collecting tubule (55). However, to our knowledge, models to simulate distal tubule have not been developed. In this study, we report a numerical model that simulates water and solute transport in the distal tubule of the rat. This model was constructed in the theoretical framework similar to the one adopted in the previous models (55, 64) except that electrodiffusive mass transfer in the tubular lumen was taken into account to accurately predict the profile of luminal electrical potential. With appropriate values of the model parameters, the model could simulate the transport properties of the distal tubule in the basic state, as well as in experimental conditions such as thiazide or amiloride application and various levels of sodium load and tubular flow rate.

### Glossary

*R*- Gas constant, J ⋅ mmol
^{−1}⋅ K^{−1}or mmHg ⋅ cm^{3}⋅ mmol^{−1}⋅ K^{−1} *F*- Faraday constant, C/mmol
*R*_{1}- Outer radius of the distal tubule, cm
*R*_{2}- Inner radius of the distal tubule, cm
*x*- Axial coordinate along the tubule, cm
*TL*- Total length of the distal tubule, cm
*N*- Number of sections
*w*- Width of each section, cm (
*w*=*TL*/*N*) - m
- Superscript used to denote luminal compartment
- c
- Superscript used to denote cellular compartment
- s
- Superscript used to denote serosal compartment
- Imp
- Impermeant solute
*D*_{α}- Diffusion coefficient of solute α, cm
^{2}/s - Concentration of solute
*k*in compartment*p*, mmol/cm^{3} - ψ
^{p} - Electrical potential of compartment
*p*, V - F
_{v} - Tubular flow rate, cm
^{3}/s - Volume flux from compartment
*p*to compartment*q*, cm/s (All the transepithelial fluxes were expressed as transport rate per tubular outer surface area.) *J*_{v}- Transepithelial volume flux, cm/s (
*J*_{v}= + ) - Flux of solute
*k*from compartment*p*to compartment*q*, mmol ⋅ cm^{−2}⋅ s^{−1} *J*_{k}- Transepithelial flux of solute
*k*, mmol ⋅ cm^{−2}⋅ s^{−1}(*J*_{k}= + ) *z*_{k}- Valence of solute
*k* - Valence of impermeant solute in compartment
*p* - Discretized model variables are coded by supplying an additional superscript indicating the section number. For example,
stands for the concentration of intracellular sodium in the
*i*-th section. Superscripts 0 and*N*+ 1 indicate the entrance and exit of the distal tubule, respectively. Thus, and stand for potassium concentrations of fluid entering and leaving the distal tubule, respectively.

## METHODS

### Overview of the Model

Distal tubule was modeled as a rigid cylinder made of a single layer of epithelial cells (Fig. 1
*A*). Three compartments were discriminated: luminal, cellular, and serosal compartments (Fig. 1
*B*). The luminal compartment is the space in which tubular fluid flows. The cellular compartment is the cytosolic space of epithelial cells. The serosal compartment is the well-stirred bath surrounding the tubule. Transports of water, sodium, potassium, and chloride among these compartments were considered. Transport between the luminal and cellular compartments is governed by the Na-Cl cotransporter, K-Cl cotransporter, Na channel, K channel, and Cl channel (Fig. 2, *A* and*B*). Transport between the cellular and serosal compartments is governed by Na-K-ATPase, K channel, and Cl channel. Paracellular pathway represents the (direct) transport between the luminal and serosal compartments, and it is comprised of conductive pathways for sodium, potassium, and chloride. Solute fluxes through ion channels were calculated using the Goldman-Hodgkin-Katz current equation (26, 31, 33). Solute fluxes through Na-K-ATPase (69), Na-Cl cotransporter, and K-Cl cotransporter were calculated using appropriate kinetic models. We assumed water flux was proportional to the osmotic difference across the membrane.

A system of differential equations was postulated on the basis of principles of mass conservation and electrical neutrality. To transform these differential equations to difference equations suitable for numerical solution, the model tubule was divided into *N*sections with equal width (Fig. 1
*A*). Then, the continuous model variables (solute concentrations, electrical potential, and tubular flow rate) in the differential equations were replaced by discrete model variables approximating the continuous counterparts at the center of each section. Two sets of difference equations were derived depending on the treatment of axial electrodiffusive mass transfer in the luminal space. In the first set, electrodiffusive terms were neglected as in the previous models (55, 64), and the derived equations could be solved efficiently in terms of computational time and computer memory. In the second set, electrodiffusive terms were taken into account. When each solution of these sets of equations were compared, we found that electrodiffusive terms have to be taken into account to accurately estimate the profile of luminal electrical potential in the distal tubule, which is comprised of heterogeneous subsegments.

A computer program that was developed for this project solved difference equations by the Newton-Raphson method (50) and determined the transepithelial fluxes of water and solutes as well as profiles of solute concentrations and electrical potential in the luminal and cellular compartments. Values of the model parameters were adjusted to minimize a penalty function that was devised to quantify the difference between model predictions and experimental measurements. The definition of the penalty function reflected the results of free-flow micropuncture experiments of the rat. With the use of Powell’s direction set method in multidimensions (50), we could find a set of model parameters that minimizes the penalty function.

### Model Variables

Model variables are comprised of solute concentrations, electrical potential and tubular fluid flow rate in the luminal compartment; and solute concentrations and electrical potential in the cellular compartment. In the continuous form, they are
,
,
,
, ψ^{m},
,
,
,
, ψ^{c}, and F_{v}. In the discretized form, they are
(*i* = 1, … , *N* + 1),
(*i* = 1, … , *N* + 1),
(*i* = 1, … , *N* + 1),
(*i* = 1, … , *N* + 1), ψ^{m,i} (*i* = 0, … ,*N* + 1),
(*i* = 0, … , *N* + 1),
(*i* = 0, … , *N* + 1),
(*i* = 0, … , *N* + 1),
(*i* = 0, … , *N* + 1), ψ^{c,i} (*i* = 0, … ,*N* + 1), and
(*i* = 1, … , *N* + 1). Total number of model variables in the discretized form amounts to 11*N* + 17. Note that
,
,
,
, and
(luminal solute concentrations and flow rate at the entrance of the tubule) are given as boundary conditions and are not model variables.

### Model Equations

Equations describing the steady state of the model are listed in Table1. *Equations ME-1
* to *
ME-4
*represent conservation of water and solutes in the cellular compartment. *Equation ME-5
* represents electrical neutrality within the cellular compartment. *Equations ME-6
* to*ME-10* represent conservation of water and solutes in the luminal compartment. *Equation ME-11
* implies electrical neutrality within the mucosal compartment.

All these equations could be expressed in terms of the model variables using the equations listed in Table 2. Flux of water across the membrane was assumed to be proportional to the difference in osmolality across the membrane (Osm^{q} − Osm^{p}) (*Eq.EOE-1
*). We have assumed reflection coefficients of all the solutes to be 1.0, and we ignored the difference between osmolarity and osmolality (*Eq. EOE-2
*). *Equation EOE-3
* is the Goldman-Hodgkin-Katz current equation (26, 31, 33) which was applied to all the conductive pathways (ion channels and paracellular pathways) in the present study. *Equation EOE-4
* is a kinetic equation used for the electroneutral cation and anion cotransporter with the stoichiometry of 1:1 (Na-Cl cotransporters and K-Cl cotransporters). Derivation of *Eq. EOE-4
* is described in
1. Rate of ATP hydrolysis by Na-K-ATPase is assumed to obey *Eq.EOE-5
* (69). As one molecule of ATP is hydrolyzed, three molecules of sodium are transported out of the cell and two molecules of potassium are transported into the cell.

Across the luminal cell membrane, sodium is transported through Na-Cl cotransporter and Na channel (*Eq. EOE-6
*); potassium through K-Cl cotransporter and K channel (*Eq. EOE-7
*); and chloride through Na-Cl cotransporter, K-Cl cotransporter, and Cl channel (*Eq.EOE-8
*). Across the basolateral cell membrane, sodium is transported through Na-K-ATPase (*Eq. EOE-9
*); potassium through Na-K-ATPase and K channel (*Eq. EOE-10
*); and chloride through Cl channel (*Eq. EOE-11
*). Across the paracellular pathway, sodium, potassium and chloride are transported through selective conductances (*Eqs.EOE-12
* to *
EOE-14
*). Transepithelial transport is comprised of transcellular transport and paracellular transport (*Eqs. EOE-15
*to *
EOE-18
*). Axial flow of solute α in the luminal compartment is the sum of the convective term (F_{v}
) and the electrodiffusive term (*Eq. EOE-19
*).

### Discretization and Solution of the Model Equations Without Electrodiffusive Terms

Without the electrodiffusive terms, axial flow of solute α (*Eq.EOE-19
*) is reduced to
If *Eq. ME-11
* is taken into account, then it follows that axial electrical current flow is constantly zero (and its derivative is also zero) along the tubule
Therefore, transepithelial electrical current is also zero, which can be deduced as follows. From *Eqs. ME-7
* to*
ME-10
*
and because d(∑_{α}
*z*
_{α}F_{α})/d*x*= 0 from the above consideration
Equation ME-11`We substituted *Eq. ME-11`
* for *Eq. ME-11
* in the following derivation of difference equations. *EquationME-11`
* implies that, in the model without electrodiffusion, luminal electrical potential is determined as the potential invoking no (local) transepithelial electrical current.

Discretized model equations are listed in Table3. Applying *Eqs. ME-1
* to*
ME-5
* and *
ME-11`
* to discretized model variables at the entrance of the tubule (*i* = 0), *Eqs. EDE1-1
* to*
EDE1-6
* are derived. Solving these equations by Newton-Raphson method,
,
,
,
, ψ^{c,0}, and ψ^{m,0} can be determined. Initial values for the iteration were as follows (6):
, 12 mM;
, 170 mM;
, 15 mM;
, 108 mM; and ψ^{c}, −90 mV.

*Equations EDE1-7
* to *
EDE1-17
* are derived from *Eqs.ME-1
* to *
ME-10
* and *
ME-11`
*. By sequentially solving these equations with *i* = 1 to *N* + 1, values for all the discretized model variables can be determined.

### Discretization and Solution of the Model Equations With Electrodiffusive Terms

In the derivation of the difference equations from the model equations taking into account electrodiffusive terms, additional (discretized) model variables estimating the derivative of luminal solute concentrations (
/d*x*) and luminal electrical potential (dψ^{i}/d*x*) have been introduced. Discretized model equations are listed in Table4. *Equations EDE2-1
* to *
EDE2-5
*and *Eq. EDE2-11
* are from *Eqs. ME-1
* to *
ME-5
* and*
ME-11. Equations EDE2-6
* to *
EDE2-10
* are difference equations derived from *Eqs. ME-6
* to *
ME-10.* In these equations,
can be expressed in terms of model variables using *Eq. EDE2-17,* which corresponds to *Eq. EOE-19
* in Table 2. Relationships between
and
/d*x* are as in*Eq. DE2-12.* Similar relations also apply to ψ^{m,i} and dψ^{i}/d*x*(*Eq. DE2-13*). *Equations EDE2-14
* and *
EDE2-16
*represent the boundary conditons. In micropuncture experiments, tubular fluid leaving the distal tubule is usually collected into a collecting pipette that can be regarded as electrically insulated. We assumed that in the steady state, solute concentrations and electrical potential are homogeneous in the collecting pipette including its orifice (*Eqs.EDE2-14
* and *
EDE2-15
*). We also assumed that the system providing the tubular fluid to the distal tubule (upstream nephron segment or perfusion pipette) can be regarded, as a first approximation, as being electrically insulated and that there is no net axial electrical current at the beginning of the distal tubule (*Eq.EDE2-16
*). This system of equations was solved with the Newton-Raphson method. In those calculations, solution of the equations without electrodiffusive terms was used as the initial values for iteration.

### Constants

Constants used in the present study are listed in Table5. Temperature was assumed to be 37°C.*R*
_{1} and *R*
_{2} were from de Mello-Aires et al. (15). Total length of the model distal tubule (*TL*) was from Good and Wright (28). Distal tubule was divided into two parts: early and late distal tubules. Lengths of the early and late distal tubules were 0.1 cm (42) and 0.13 cm, respectively. Number of sections (*N* ) was chosen to be 23, and width of each section (*w*) was 0.01 cm. Diffusion coefficients of solutes at infinite dilution were from Hille (31) and Cussler (14). (Diffusion coefficient of impermeant solute was that of urea.) Values for serosal sodium and potassium concentrations were from Malnic et al. (45), and those for chloride were from Kunau et al. (39). Concentration of impermeant solute in the serosal compartment was calculated from the reported osmolality of the rat plasma (305.3 mosmol/kg) (59)
Because most of the impermeant solutes in the mucosal compartment can be assumed to be urea, the valence (
) was considered to be 0. Valence of impermeant solute in the serosal compartment was calculated from solute concentrations based on the principle of electrical neutrality.

### Penalty Function

Transport characteristics of the model tubule are completely specified by a set of model parameters. Ideally, values of these parameters should be determined by experimental measurements. However, since direct evaluations are not available for most of the parameters, we attempted to find the values with which the model simulates the real distal tubule best. Specifically, we devised a penalty function that gives larger values as the model prediction deviates from experimental results and systematically searched for the parameter values which minimize this function.

In this report, we confined our studies to use the results of free-flow micropuncture experiments in the rat, since experimental conditions were relatively uniform in those studies. We summarized their results as follows. *1*) In the basic state, the distal tubule reabsorbs water, sodium, and chloride; secretes potassium; and has negative luminal electrical potential. *2*) The distal tubule has thiazide-sensitive Na-Cl cotransporters. Application of thiazide reduces sodium reabsorption but has a negligible effect on transepithelial potential difference. *3*) The distal tubule has amiloride-sensitive sodium channels. Application of amiloride depolarizes luminal electrical potential but has a minimal effect on total sodium reabsorption. *4*) Sodium reabsorption is load dependent. As the rate of sodium entry increases, sodium reabsorption increases. *5*) Potassium secretion is flow dependent. As the initial tubular flow rate increases, potassium secretion increases.

Definition of the penalty function incorporating these characteristics is presented in Table 6. We chose 13 representative experimental measurements (*E*
_{1} to*E*
_{13}). *E*
_{1} to*E*
_{5} were from experiments conducted in the basic state. Total water reabsorption rate (*E*
_{1}) and total sodium reabsorption rate (*E*
_{2}) were from Malnic et al. (45). Total potassium secretion rate (*E*
_{3}) was from Malnic et al. (44). Luminal electrical potential in the late distal tubule (*E*
_{5}) was from Hayslett et al. (30). Luminal electrical potential in the early distal tubule (*E*
_{4}) was based on the experimental observation that luminal potential in the early distal tubule was depolarized by 23.3 mV relative to that in the late distal tubule (5). Model predictions corresponding to these values (*S*
_{1} to*S*
_{5}) were calculated with the boundary condition listed in Table 6. Values of this boundary condition were determined from reports of free-flow micropuncture experiments (44, 45). Simulated values of luminal potential in the early (*S*
_{4}) and late (*S*
_{5}) distal tubules were the values of luminal potential at the midpositions of these subsegments (ψ^{m,5} and ψ^{m,16}).

*E*
_{6} to *E*
_{8} are from experiments conducted in the presence of thiazide. *E*
_{6}represents total sodium reabsorption rate. *E*
_{7} and*E*
_{8} represent luminal potential in the early and late distal tubules, respectively. These values were based on the results of Costanzo and Windhager (13), which showed that thiazide reduced sodium reabsorption by 43% compared with the basic state and did not affect luminal electrical potential. Model predictions (*S*
_{6} to *S*
_{8}) corresponding to*E*
_{6} to *E*
_{8} were calculated similarly as in the basic state, except that
(both in the early and late distal tubules) was reduced by 90%.

*E*
_{9} to *E*
_{11} are related to the distal tubular function when amiloride was applied.*E*
_{9} represents total sodium reabsorption rate. This value was set equal to that of the basic state (*E*
_{2}) based on the report of Duarte et al. (16) in which amiloride had essentially no effect on total sodium reabsorption.*E*
_{10} and *E*
_{11} are luminal potential in the early and late distal tubules, respectively, and the values for these were based on the report by Barratt (4) that amiloride depolarized luminal potential by 1.5 and 20.5 mV in the early and late distal tubules, respectively. Model predictions (*S*
_{9} to *S*
_{11}) corresponding to *E*
_{9} to *E*
_{11} were calculated with the same boundary conditions as in the basic state, except that
(both in the early and late distal tubules) was reduced by 99%.

*E*
_{12} is from the measurement (37) of total sodium reabsorption rate when sodium load was increased to 2,160 pmol/min. Model prediction (*S*
_{12}) corresponding to*E*
_{12} was calculated using the boundary condition that simulates the one observed when the experiment was conducted (37).

*E*
_{13} is from the measurement of total potassium secretion rate when tubular flow rate was experimentally increased to 38.1 nl/min (38). *S*
_{12} is the simulated value obtained with the boundary condition resembling the one observed in the experiment (38).

### Parameter Search

There were 40 model parameters in total. The minimization algorithm used in this study (Powell’s direction set method) is an iterative procedure and requires a set of initial (presumptive) values for the parameters to start with. These initial parameters, listed in Table7, were deduced from (manual) calculations that were based on the information relevant to the basic state (Tables6
*B* and 12). Apical hydraulic conductivity (
) in the late distal tubule had a value (7.48 × 10^{−8}cm ⋅ s^{−1} ⋅ mmHg^{−1}) similar to those reported for transepithelial hydraulic conductivity of the nondiuretic rat (6.47 × 10^{−8}cm ⋅ s^{−1} ⋅ mmHg^{−1}, Ref. 54; and 10.03 × 10^{−8}cm ⋅ s^{−1} ⋅ mmHg^{−1}, Ref. 58). Intrinsic dissociation constants of Na-Cl cotransporter (
and
) were selected to account for the measured kinetic constants of this transporter. With the listed values, the kinetic equation (*Eq. EOE-4
* in Table 2) exhibits effective affinities (53) of 25.0 and 13.6 mM for sodium and chloride, respectively. These values are identical to the ones reported with the thiazide-sensitive Na-Cl cotransporter expressed in*Xenopus* oocytes (22). Intrinsic dissociation constants for K-Cl cotransporter (
and
) were also selected to account for the kinetic constants reported for the recently cloned K-Cl cotransporter (25) that is expressed in the kidney. The kinetic equation predicts effective affinities of 25.0 and 50.0 mM for potassium and chloride, respectively. These values are identical to the experimental results (25). Maximal rate of ATP hydrolysis (*J*
_{a,max}) was 317 pmol ⋅ min^{−1} ⋅ tube^{−1}, which was similar to the experimental value (34) for dissected rat distal tubule (257 pmol ⋅ min^{−1} ⋅ tube^{−1}). Value of kinetic parameter of Na-K-ATPase (*K*
_{Na,ATPase}) was that of Na-K-ATPase with rat α1-isoform, which is the predominant isoform in the kidney, reported in a recent study (69). Ion permeabilities through the paracellular pathway (
,
, and
) were calculated from a measurement of transepithelial conductance (2.6 mS/cm^{2}) in the early distal tubule (43). With the listed values, paracellular pathway of the model has conductance of 2.5 mS/cm^{2}.

To calculate initial values for other parameters, we made following presumptions: cytosolic solute concentrations are close to the values reported by Beck et al. (6); luminal potassium is nearly at equilibrium with cytosolic potassium at the end of the distal tubule; the early distal tubule accounts for 80% of total sodium reabsorption; potassium transport rate through K channel is equal to chloride transport rate through Cl channel in the basolateral membrane; potassium secretion rate in the early distal tubule is small (1 pmol/min); apical K-Cl cotransporter and K channel contribute equally to potassium secretion both in the early and late distal tubules; apical Na-Cl cotransporter accounts for 90% of the sodium reabsorption in the early distal tubule; water reabsorption rate in the early distal tubule is 1 nl/min; hydraulic conductivities in the basolateral membrane ( ) are 100-fold larger than that in the luminal membrane ( ); and water flux in the paracellular pathway is negligible.

With the initial parameter values in Table 7, the penalty function was 33.8 (calculation with electrodiffusive terms). Starting with these values, preliminary attempts to adjust all the model parameters to minimize the penalty function were unsuccessful. The minimization procedure required too much computational time (more than 3 days), and the acquired solutions were mostly unrealistic in terms of predicted solute concentrations and electrical potential in the cytosolic space. Hence, we restricted the parameters for adjustment to the ones representing the maximum transport rates of the luminal transporters ( , , and in the early distal tubule; and , , , , and in the late distal tubule). These parameters are underlined in Table 7. Exclusion of the basolateral parameters ensured the acceptable predictions for cytosolic variables. and in the early distal tubule were also excluded from optimization, since it was obvious from experiments (52) that they should remain very small. Furthermore, one constraint was imposed in the minimization procedure: in the early distal tubule is 10-fold larger than that in the late distal tubule. Without this constraint, the search procedure converged to a much larger value of in the late distal tubule than that of the early distal tubule, which is contradictory to experimental results (12, 19). Accordingly, among the eight parameters adjusted, seven of these were independently varied. Optimized values of these parameters are listed in Table8, with which the penalty function was 0.327. Thus minimization procedure reduced the value of the penalty function by a factor of 103. Other parameters are also included in Table 8 to present the complete set of model parameters used throughout this study.

### Computer Software

A computer program for this project was implemented with C++ programming language (56). Discretized model equations were solved by Newton-Raphson method in multidimensions (50). Systems of linear equations were solved by LU decomposition (50). Minimization of the penalty function was done using direction set (Powell’s) method in multidimensions (50), which do not require derivatives of the target function. All the calculations were conducted in 64-bit double precision. The program was compiled by Metrowerks C/C++ compiler (version 1.8) and run on a personal computer with 180 MHz PowerPC 604e CPU running under MacOS.

## RESULTS

### Comparison Between Calculations With and Without Electrodiffusive Terms

To solve the model equations with our computer program, it took ∼0.4 and 14 s without and with electrodiffusive terms, respectively. This difference consequently affected the time required to calculate the penalty function, and it took 4 and 73 s. Minimization of the penalty function completed in ∼30 min or 12 h depending on whether we neglect those terms. Necessary heap memory was 0.15 megabytes without electrodiffusion and 1.7 megabytes with electrodiffusion. Thus, calculations without electrodiffusive terms could be conducted much more efficiently.

When there is no electromotive force, we can safely drop the electrodiffusive terms in modeling the renal tubule (40). To examine whether similar simplification was acceptable in our case, solutions of the model equations with or without electrodiffusion were compared. Profiles of luminal variables of both models are plotted in Fig.3, and profiles of transepithelal water and solute fluxes are plotted in Fig. 4. As can be seen in Fig. 3, *B*–*D*, predictions of the two models were similar with regard to luminal flow rate and solute concentrations. This was also the case for water and solute fluxes (Fig. 4) and cytosolic variables (data not shown), although a small difference in the transepithelial potassium flux was observed in the late distal tubule (Fig. 4
*C*). However, there was a significant difference in the luminal potential profile (Fig.3
*A*) between these models. The model without electrodiffusion (broken line, Fig. 3
*A*) predicted an unrealistically abrupt change in luminal potential at the junction of the early and late distal tubules (*x* = 0.10 cm), in contrast to the smooth transition discernible in the model with electrodiffusive terms (solid line, Fig.3
*A*).1 This was not unexpected, since in the model without electrodiffusion the luminal potential is determined as the potential with which there is no local transepithelial current (note *Eq. ME-11`
*) and the levels of luminal potential satisfying this condition are quite different between the two subsegments. Thus, in modeling the distal tubule, simplification by dropping the electrodiffusive terms was considered to be unacceptable. In the following part of this report, only the results with electrodiffusive terms will be presented.

### Basic State

The model could simulate total water and solute reabsorptions as well as luminal potential of the distal tubule in the basic state (Table9). More detailed description of the magnitude of transport rate through individual transport devices can be found in Table 10. Profiles of luminal variables and transepithelial fluxes are plotted as continuous lines in Figs. 3 and 4. Most (80%) of water reabsorption took place in the late distal tubule (Table 10; Fig. 6
*A*) due to the 14-fold higher water permeability in the late subsegment of the model tubule than that in the early subsegment. This is consistent with the report by Woodhall and Tisher (66), which showed that the late subsegment is the primary site of water reabsorption in the distal tubule. Initially hyposmotic (184 mosmol/kg) tubular fluid (Table 6
*B*) was concentrated to near isomolality at the proximal half of the late distal tubule. Total sodium reabsorption was 389 pmol/min, of which 84% was in the early distal tubule (Table 10; Fig. 6
*B*). Sodium reabsorption in the early subsegment occurred almost exclusively through the luminal Na-Cl cotransporter, whereas both Na-Cl cotransporter and Na channel contributed in the late subsegment. The low level of luminal sodium concentration and hyperpolarized luminal potential (Fig.5) resulted in substantial sodium secretion through paracellular pathways (46 and 88 pmol/min in the early and late distal tubules, respectively). Potassium was secreted mainly in the late distal tubule (Table 10 and Fig.6
*C*), which is compatible with the report by Stanton et al. (52) that potassium secretion in the early subsegment was negligible. Potassium secretion was mostly through luminal K channel in the present model.2 Intratubular potassium concentration progressively increased along the tubule and reached a plateau at 14.1 mM (Fig. 5), which is close to the equilibrium concentration of 13.8 mM. Values of cytosolic variables (Table 11) were compatible with the experimental results by Beck et al. (6).

### Thiazide

The model could simulate the effects of thiazide on total reabsorption of sodium and luminal potential of the distal tubule (Table 9). In the simulation, thiazide reduced total sodium reabsorption by 49% and had only small effects on luminal potential, which is consistent with experimental results. To present the results of the simulation in further detail, fluxes through individual transport mechanisms are listed in Table 10, and profiles of luminal variables and transepithelial fluxes are plotted in Figs. 5 and 6, respectively. As can be seen in Table 10, the reduction in sodium reabsorption occurred largely in the early distal tubule. This reflects the property of the model distal tubule in which ∼90% of thiazide-sensitive Na-Cl cotransporters are in the early distal tubule. In the late distal tubule, in addition to the relative sparsity of the luminal Na-Cl cotransporter, compensatory increase in sodium reabsorption through the luminal Na channel contributed to the smaller reduction in sodium reabsorption. As a result of the inhibition of luminal Na-Cl cotransporters, luminal concentrations of both sodium and chloride tended to increase (Fig. 5), which should have contributed to the thiazide-induced 140% increase in potassium secretion (Table 10) by two mechanisms. In the first place, the consequent increase in osmolality of the tubular fluid hindered water reabsorption and led to higher flow rates throughout distal tubule. Indeed, tubular flow rate at the end of the distal tubule increased by a factor of 1.5 in response to thiazide. Since potassium secretion is a flow-dependent process as shown below, the rise in tubular flow rate should bring about an increase in potassium secretion. In the second place, increased luminal sodium chloride concentration depolarized cytosolic potential and hyperpolarized luminal potential in the late distal tubule, increasing the electromotive force of potassium secretion. This is not an intuitively evident result, since increases in luminal sodium and chloride concentrations would have opposite effects. Increase in luminal sodium would depolarize the apical cell membrane and hyperpolarize luminal potential, and an increase in luminal chloride would induce the opposite effects. Consequently, direction of changes depends on other factors including relative magnitude of the conductances of these ions. Simulation of thiazide application predicted cytosolic depolarization (Table 11) and luminal hyperpolarization. In summary, the main effects of thiazide in the simulation were 49% reduction in total sodium reabsorption due to the inhibition of Na-Cl cotransporter and 140% increase in total potassium secretion due to increased tubular flow rate, depolarized cytosolic potential in the late distal tubule, and hyperpolarized luminal potential.

### Amiloride

The model could simulate the effects of amiloride on total sodium reabsorption and luminal potential of the distal tubule (Table 9). In the simulation, amiloride reduced total sodium reabsorption by only 5.4%, but markedly depolarized luminal potential. The latter effect was more prominent in the late distal tubule, and at its end, luminal potential depolarized by 16.1 mV in response to amiloride. To present the results of the simulation in further detail, fluxes through individual transport mechanisms are listed in Table 10, and profiles of luminal variables and transepithelial fluxes are plotted in Figs.7 and 8, respectively. Although amiloride abolished sodium reabsorption through luminal Na channels (Table 10), compensatory increase in sodium reabsorption through luminal Na-Cl cotransporters and decrease in sodium secretion through paracellular pathways prevented a large change in total sodium reabsorption. Inhibition of (electrogenic) luminal Na channel led to depolarization of luminal potential and hyperpolarization of cytosolic potential, which caused a marked inhibition of potassium secretion by reducing the electromotive force driving potassium secretion. Specifically, luminal potential at the end of distal tubule was −8.9 mV, whereas it was −25.1 mV in the basic state, and average cytosolic potential in the late distal tubule hyperpolarized by 1.6 mV in response to amiloride (Table 11). These changes lowered equilibrium concentration of luminal potassium to 7.7 mM, which is close to the luminal potassium concentration at the end of the distal tubule (7.2 mM). In summary, the main effect of amiloride in the simulation was depolarization of luminal potential, which led to a marked reduction of tubular potassium secretion.

### Sodium Load

The model prediction of total sodium reabsorption under the condition of high sodium load was comparable with experimental results (Table 9). Details of the simulation are presented in Table 10 and Figs.9 and 10. Water reabsorption increased to 11.51 nl/min (Table 10). The late distal tubule appears to contribute to most of the increase (Fig.10
*A*). To simulate high sodium load, we raised initial luminal sodium concentration by 1.2-fold and tubular flow rate by 4.4-fold (Table 6). At this level of flow rate, osmolality at the end of the tubule no longer achieved isosmolality and was significantly hyposmotic (256 mosmol/kg). Sodium reabsorption increased to 687 pmol/min (Table10). Both luminal Na channel and Na-Cl cotransporter contributed to this increase in sodium reabsorption, which was driven by higher luminal sodium concentration in increased sodium load. Changes in other factors including intracellular sodium concentration, luminal potential, and cytosolic potential were in the opposite directions to explain the increased sodium reabsorption. Potassium secretion was markedly increased to 106.3 pmol/min. Hyperpolarized luminal potential, as well as lower luminal potassium concentration, contributed to this increase in potassium secretion. Figure11 presents total sodium reabsorptions in various levels of sodium load. The solid line in Fig. 11 indicates the model prediction, and the solid circles indicate experimental results from Kunau et al. (37) and Costanzo and Windhager (13). The model predicted the positive correlation between sodium load and sodium reabsorption observed in experimental measurements.

### High Flow Rate

The boundary condition of high flow rate was similar to that of high sodium load (Table 6). Accordingly, simulation results in high flow rate were similar to those in high sodium load, thus their details are not presented here. Figure 12 presents the relationship between potassium secretion and initial flow rate; the solid line indicates the model prediction, and the solid circles indicate the experimental results from Kunau et al. (38) and Reineck et al. (51). The model prediction is in good agreement with the experimental observation.

## DISCUSSION

In summary, we developed a numerical model of the rat distal tubule that could simulate the experimental results in various conditions including the basic state, in the presence of thiazide or amiloride, and with increased sodium load and flow rate. Validity of the present model can be discussed from three aspects: validity of the experimental data to simulate; validity of the assumptions made to formulate the model; and validity of the model calculation.

### Validity of the Experimental Data to Simulate

As an estimate of the validity of experimental data on which we based our numerical model, a summary of collected experimental measurements is presented in Table 12. Values adopted in the definition of the penalty function are underlined in Table 12. Several points deserve mention with regard to the construction of the present model. *1*) Length of distal tubule (Table 12
*C*) increases as the rat grows. The value used in this simulation was typical of rats weighing 200 to 300 g. *2*) There is a controversy about the value of luminal electrical potential in the distal tubule (Table 12, *H* and *I* ). Some researchers (1, 2, 4, 5) reported values significantly more depolarized than the values reported by others (43, 67). The origin of such a difference is assumed to be due to the difference in types of electrode (Ling-Gerard vs. large-tipped) used to impale the tubule (2). We adopted a value from a study that conducted simultaneous electrical recording with both Ling-Gerard and large-tipped electrodes (30).*3*) Intracellular potential of distal tubular cells had been reported to be −67.4 to −92 mV (Table 12
*S*). We speculate that these values tend to underestimate the magnitude of intracellular hyperpolarization. Some measurements (35, 36) were conducted with double-barreled electrodes, and their primary aim was measurement of intracellular ionic activities. Usage of such large-pore electrodes with mammalian cells likely caused depolarization of intracellular potential due to large leak current and cell damage. Even in the case of measurements conducted with Ling-Gerard electrode, “they [potential recordings] declined rapidly, and were rarely stable for longer than a few seconds” (24), which suggests that distal tubular cells were probably damaged by microelectrode impalement. Thus we have determined basolateral parameters to make the intracellular potential of model tubular cells close to the most hyperpolarized value reported (−92 mV). This value is also close to the intracellular potential (−90 to −95 mV) of a mathematical model of the cortical collecting tubule (55), which is a nephron segment functionally similar to the late distal tubule.

### Validity of the Assumptions Made to Formulate the Model

We have considered only fluxes of water, sodium, potassium, and chloride in the present model. Certainly, this is a substantial simplification in view of the fact that distal tubule also transports urea, hydrogen, bicarbonate, and calcium. However, we concentrated on transport of sodium, potassium, and chloride in this study, because in magnitude the solute flux ascribable to them (more than 700 pmol/min) overwhelms fluxes of other solutes [urea, 70 pmol/min (3); hydrogen/bicarbonate, 50 pmol/min (10); and calcium, 4 pmol/min (13)]. Disregarding the transport of hydrogen and bicarbonate, we could exclude, from the model, intercalated cells, which are involved primarily in hydrogen/bicarbonate transport. Consequently, cells in the present model can be regarded as distal convoluted tubular cells in the early subsegment and connecting tubular cells or principal cells in the late subsegment.

We have assumed that the intercellular space has the same hydraulic pressure and solute concentrations as the basolateral space, because almost no information is currently available concerning the transport involved in the intercellular space. This assumption may not be valid under extreme experimental conditions, since massive water reabsorption, induced in diabetes insipidus rats with antidiuretic hormone, is known to be accompanied by expansion of the intercellular space (66), which suggests the presence of a pressure gradient between this space and the basolateral space. However, such an expansion has not been observed in Wistar and Sprague-Dawley rats in the normal state (66).

Existence of the thiazide-sensitive Na-Cl cotransporter in the luminal membrane of the distal tubule is supported by physiological (12, 19) as well as histochemical (47, 49, 68) studies. For example, in in vivo microperfusion experiments Ellison et al. (19) have shown that thiazide diuretics abolished sodium reabsorption in the early distal tubule. However, effects of thiazide on the late distal tubule were less remarkable and were statistically insignificant. This is consistent with the histochemical finding that expression of the thiazide-sensitive Na-Cl cotransporter was considerably weaker in the late distal tubule than in the early distal tubule (47, 49). Such a heterogeneous distribution of Na-Cl cotransporter was represented in our model by enforcing in the early distal tubule to be 10-fold larger than that in the late distal tubule.

There is ample evidence for the existence of the amiloride-sensitive Na channel in the apical membrane of the distal tubule. In physiological experiments (4, 16), luminal amiloride application depolarized the luminal electrical potential, which is explainable by suppression of the (inherently electrogenic) amiloride-sensitive Na channel. Although effects of amiloride on total sodium reabsorption in the distal tubule were not readily discernible in micropuncture experiments (16), in vivo microperfusion experiments revealed that there was a modest but statistically significant reduction of sodium reabsorption when amiloride was included in the perfusate (62). On the other hand, histochemical studies also demonstrated that all three subunits (8, 9,41), which are required to reproduce the fully functional channel, are detectable in the apical membrane throughout the distal tubule (17). In the present model, we have assumed separate values of maximal transport capacities of the apical Na channel ( ) in the early and late subsegments. We varied their values independently while optimizing the model parameters, and the result showed 11-fold larger luminal Na conductance in the late than in the early subsegments (Table 8). Future experiments may discover whether such an axial heterogeneity is actually present. If it were not the case, then we should have to add another condition (for example, enforcing equal value of in the two subsegments) while searching the optimal model parameters.

Potassium secretion in the distal tubule is considered to be mostly through K channel in the luminal cell membrane (23). In cortical collecting tubule (48), the luminal K channel was identified to be ROMK inward rectifier (32, 70). Also in the distal tubule, ROMK channel was demonstrated to be present (7), justifying the luminal K channel in the model. Besides the luminal K channel, it is also suggested that there exists K-Cl cotransport in the apical membrane of the distal tubule by in vivo microperfusion experiments (18, 60). Consequently, we incorporated both K channel and K-Cl cotransporter in the luminal membrane. However, optimization procedure (i.e., minimizing the penalty function) resulted in a small value of maximal transport capacity of apical K-Cl cotransporter in the late distal tubule (Table 8). (Maximal transport capacity of the K-Cl cotransporter in the early distal tubule was set to a small fixed value in the present model.)

### Validity of the Model Calculation

Because of the nonlinearity of the model equations, it is difficult to demonstrate the appropriateness of the differencing scheme adopted in this study. As an alternative, we have tested whether the numerical solutions converge to a limit as we shorten the section width. We made four models with different section widths: *1*) 0.033 cm in the early and 0.043 cm in the late distal tubule; *2*) 0.020 cm in the early and 0.022 cm in the late distal tubule; *3*) 0.010 cm in the early and the late distal tubule (reported model); and*4*) 0.005 cm in the early and the late distal tubule. Using the same parameters as listed in Table 8, we solved model equations taking into account electrodiffusive terms. Results are plotted in Fig.13. Solutions can be seen to converge to a limit as we take digitization intervals smaller. The solution with 0.010-cm section width and the one with 0.005-cm section width coincided with less than a 0.5-mV difference, which validates that the sampling interval adopted in the present model formulation was practically small enough.

### Future Extension of the Model

This study is a first attempt to develop a model of the distal tubule, and we have concentrated largely on mathematical formulation and implementation of an efficient program. Accordingly, the model has several restrictions that should be improved in future revisions.*1*) The model was optimized to simulate only results of micropuncture experiments. It would be interesting to test whether this model also simulates the results from in vivo microperfusion experiments (13, 18, 19, 27-29, 60, 61) in which solute concentrations and flow rate had been manipulated more extensively.*2*) The model handles transport of only water, sodium, potassium, and chloride. Other solutes such as hydrogen, bicarbonate, magnesium, and calcium should be incorporated, and this is possible within the mathematical framework and numerical methods used in the present model. *3*) The program lacks flexible user interface and is not ideally easy to use. This problem can be solved by providing an intuitive graphical user interface to the program, which would allow for researchers to test the appropriateness of the model by themselves as well as to use the model to predict water and solute transport in physiological and pathophysiological situations.

## Acknowledgments

We thank Drs. N. Yamashita and K. Cho for critical reading of the manuscript and helpful discussions.

## Appendix

### A Kinetic Model of Cation-Anion Cotransporters

The model transporter is a cation-anion cotransporter with a stoichiometry of 1:1, which was applied to both Na-Cl cotransporter and K-Cl cotransporter. The state diagram of the model is illustrated in Fig. 14, in which following assumptions were made: *1*) the transporter has one cation binding site and one anion binding site; *2*) binding to each site occurs in any order and independently of each other; *3*) affinities of binding sites are symmetrical; *4*) transition of the transporter across the membrane is a much slower process compared with other binding or dissociation processes, and bound and unbound forms of the transporters are essentially in equilibrium on each side; and *5*) rate constants of transition of the transporter across the membrane are symmetrical. The whole transport cycle can be outlined as follows: binding of cation (C) and anion (A) to an unloaded transporter (T) converts the transporter to half loaded states (TC or TA) and then to the fully loaded state (TAC); fully loaded transporter can cross the membrane to the other side (T′A′C′); and after dissociation of both ions, the empty transporter (T′) crosses the membrane back to return to the original state (T). It should be noted that transitions between T and T′ or between TAC and T′A′C′ do not necessarily imply actual translocation of the transporter molecule but likely correspond to some conformational change that forces the binding sites to face the other side of the membrane.

From the *assumption 4* (see above), we can assume that states T, TA, TC, and TAC are near equilibrium, and if we use the same symbol for each kinetic state in Figure 16 and for its concentration, then
Equation EAK-1
Equation EAK-2
Equation EAK-3where α_{1} =*k*
_{1}A/*k*
_{2}, and α_{2}= *k*
_{3}C/*k*
_{4}.

Similarly
Equation EAK-4
Equation EAK-5
Equation EAK-6where α′_{1} =*k*
_{1}A′/*k*
_{2}, and α′_{2} =*k*
_{3}C′/*k*
_{4}.

Sum of the concentrations of all the states are constant (S) Equation EAK-7

Finally, the total number of transporters in one side of the membrane should be constant in steady state
Equation EAK-8These eight *Eqs. EAK-1
* to *
EAK-8
* can be solved for T, TA, TC, TAC, T′, T′A′, T′C′, and T′A′C′, and flux of cation (F_{C}) or anion (F_{A}) can be expressed as
where *K*
_{A} =*k*
_{2}/*k*
_{1} is the intrinsic dissociation constant of anion binding site, and *K*
_{C}= *k*
_{4}/*k*
_{3} is the intrinsic dissociation constant of cation binding site. If we further assume that*k*
_{5} = *k*
_{6}, the above equation reduces to *Eq. EOE-4
* in Table 2, in which
=*k*
_{6}
*S*. We note that maximal transport rate is ½ ×
in this notation.

## Footnotes

Address for reprint requests and other correspondence: H. Chang, Health Service Center, Univ. of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan.

↵1 A practical way to lessen the difference between the two models may be to change parameter values gradually at the junction of the two subsegments rather than abruptly as in the present models. Such an arrangement of parameters is justified, since transition between these subsegments is a gradual process in the case of the rat (42).

↵2 However, ad hoc exploration revealed that models with significant (up to 8 pmol/min) potassium secretion through luminal K-Cl cotransporter could also be constructed by readjusting the parameters, although the penalty function increased to the level of 0.5.

- Copyright © 1999 the American Physiological Society