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.
 |
INTRODUCTION |
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 · K1 or
mmHg · cm3 · mmol1 · K1
|
| F |
Faraday constant, C/mmol
|
| R1 |
Outer radius of the distal tubule, cm
|
| R2 |
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 |
Diffusion coefficient of solute  , cm2/s
|
| C pk |
Concentration of solute k in compartment p,
mmol/cm3
|
 p |
Electrical potential of compartment p, V
|
| Fv |
Tubular flow rate, cm3/s
|
| J pqv |
Volume flux from compartment p to compartment q,
cm/s (All the transepithelial fluxes were expressed as transport
rate per tubular outer surface area.)
|
| Jv |
Transepithelial volume flux, cm/s
(Jv = J mcv +
J msv)
|
| J pqk |
Flux of solute k from compartment p to compartment
q,
mmol · cm2 · s1
|
| Jk |
Transepithelial flux of solute k,
mmol · cm2 · s1
(Jk = J mck +
J msk)
|
| zk |
Valence of solute k
|
| zpImp |
Valence of impermeant solute in compartment p
|
|
Discretized model variables are coded by supplying an additional
superscript indicating the section number. For example,
Cc,iNa 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, Cm,0K and Cm,N + 1K 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. 1A). Three
compartments were discriminated: luminal, cellular, and serosal
compartments (Fig. 1B). 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.
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Fig. 1.
Schematic diagram of model tubule. A: outline of the model.
Distal tubule was modeled as a cylinder made of a sheet of epithelial
cells. In the lumen, tubular fluid flows from the proximal end to the
distal end as indicated by arrows. In the derivation of discretized
model equations, tubule was divided into N sections of equal
width. B: magnified view. There are three distinct
compartments; luminal, cellular, and serosal. Solute concentrations and
electrical potential within each compartment as well as tubular flow
rate in the luminal compartment are taken into account in the model.
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Fig. 2.
Transport mechanisms of model tubule. In the luminal cell membrane,
there are Na-Cl cotransporter, K-Cl cotransporter, Na channel, K
channel, and Cl channel. In the basolateral cell membrane, there are
Na-K-ATPase, K channel, and Cl channel. In the paracellular pathway,
which faces luminal and basolateral compartments, there are
conductances for sodium, potassium, and chloride. Axial heterogeneity
of the distal tubule was represented by changing the model parameters
in the early and the late parts of the model tubule. Transport rates
(in units of pmol/min) in the basic state through individual transport
devices in the early (A) or late (B) parts of the
distal tubule are shown by numbers beside arrows that indicate the
direction of transport.
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|
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. 1A). 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
CmNa, CmK,
CmCl, CmImp,
m, CcNa,
CcK, CcCl,
CcImp, c, and Fv.
In the discretized form, they are
Cm,iNa (i = 1, ... , N + 1),
Cm,iK
(i = 1, ... , N + 1),
Cm,iCl
(i = 1, ... , N + 1),
Cm,iImp (i = 1, ... , N + 1),
m,i (i = 0, ... ,
N + 1), Cc,iNa
(i = 0, ... , N + 1),
Cc,iK
(i = 0, ... , N + 1),
Cc,iCl (i = 0, ... , N + 1),
Cc,iImp
(i = 0, ... , N + 1),
c,i (i = 0, ... ,
N + 1), and Fiv
(i = 1, ... , N + 1). Total number of model
variables in the discretized form amounts to 11N + 17. Note
that Cm,0Na, Cm,0K, Cm,0Cl, Cm,0Imp,
and F0v (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 Table
1. Equations 1 to 4
represent conservation of water and solutes in the cellular
compartment. Equation 5 represents electrical neutrality
within the cellular compartment. Equations 6 to
ME-10 represent conservation of water and solutes in the luminal compartment. Equation 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
(Osmq 
Osmp) (Eq. 1). We have assumed reflection coefficients of all the solutes
to be 1.0, and we ignored the difference between osmolarity and
osmolality (Eq. 2). Equation 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 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. 4 is described in APPENDIX 1. Rate of ATP hydrolysis by Na-K-ATPase is assumed to obey Eq. 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. 6); potassium through K-Cl
cotransporter and K channel (Eq. 7); and chloride through Na-Cl cotransporter, K-Cl cotransporter, and Cl channel (Eq. 8). Across the basolateral cell membrane, sodium is transported
through Na-K-ATPase (Eq. 9); potassium through Na-K-ATPase
and K channel (Eq. 10); and chloride through Cl channel
(Eq. 11). Across the paracellular pathway, sodium, potassium
and chloride are transported through selective conductances (Eqs.
12 to 14). Transepithelial transport is comprised of
transcellular transport and paracellular transport (Eqs. 15
to 18). Axial flow of solute in the luminal compartment is the sum of the convective term
(FvCm) and the electrodiffusive term (Eq. 19).
Discretization and Solution of the Model Equations Without
Electrodiffusive Terms
Without the electrodiffusive terms, axial flow of solute (Eq. 19) is reduced to
If Eq. 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. 7 to
10
and because
d(
zF)/dx = 0 from the above consideration
|
(ME-11`)
|
We substituted Eq. 11` for Eq. 11 in
the following derivation of difference equations. Equation 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 Table
3. Applying Eqs. 1 to
5 and 11` to discretized model variables at the
entrance of the tubule (i = 0), Eqs. 1 to
6 are derived. Solving these equations by Newton-Raphson
method, Cc,0Na,
Cc,0K, Cc,0Cl,
Cc,0Imp, c,0, and
m,0 can be determined. Initial values for the iteration
were as follows (6): CcNa, 12 mM;
CcK, 170 mM; CcCl, 15 mM; CcImp, 108 mM; and c,
90 mV.
Equations 7 to 17 are derived from Eqs.
1 to 10 and 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 (dCi/dx)
and luminal electrical potential (di/dx)
have been introduced. Discretized model equations are listed in Table
4. Equations 1 to 5
and Eq. 11 are from Eqs. 1 to 5 and 11. Equations 6 to 10 are difference
equations derived from Eqs. 6 to 10. In these
equations, Fi can be expressed
in terms of model variables using Eq. 17, which corresponds to Eq. 19 in Table 2. Relationships between
Cm,i and
dCi/dx are as in
Eq. DE2-12. Similar relations also apply to
m,i and di/dx
(Eq. DE2-13). Equations 14 and 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. 14 and 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. 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 Table
5. Temperature was assumed to be 37°C.
R1 and R2 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
(zmImp) 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 (E1 to
E13). E1 to
E5 were from experiments conducted in the basic
state. Total water reabsorption rate (E1) and
total sodium reabsorption rate (E2) were from
Malnic et al. (45). Total potassium secretion rate
(E3) was from Malnic et al. (44). Luminal
electrical potential in the late distal tubule (E5) was from Hayslett et al. (30). Luminal
electrical potential in the early distal tubule
(E4) 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 (S1 to
S5) 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 (S4) and
late (S5) distal tubules were the values of
luminal potential at the midpositions of these subsegments
(m,5 and m,16).
E6 to E8 are from experiments
conducted in the presence of thiazide. E6
represents total sodium reabsorption rate. E7 and E8 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
(S6 to S8) corresponding to
E6 to E8 were calculated
similarly as in the basic state, except that
J mcNa,max (both in the early
and late distal tubules) was reduced by 90%.
E9 to E11 are related to the
distal tubular function when amiloride was applied.
E9 represents total sodium reabsorption rate. This
value was set equal to that of the basic state
(E2) based on the report of Duarte et al. (16) in
which amiloride had essentially no effect on total sodium reabsorption.
E10 and E11 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 (S9 to S11)
corresponding to E9 to E11 were
calculated with the same boundary conditions as in the basic state,
except that P mcNa (both in the
early and late distal tubules) was reduced by 99%.
E12 is from the measurement (37) of total sodium
reabsorption rate when sodium load was increased to 2,160 pmol/min.
Model prediction (S12) corresponding to
E12 was calculated using the boundary condition
that simulates the one observed when the experiment was conducted (37).
E13 is from the measurement of total potassium
secretion rate when tubular flow rate was experimentally increased to
38.1 nl/min (38). S12 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 Table
7, were deduced from (manual) calculations
that were based on the information relevant to the basic state (Tables 6B and 12). Apical hydraulic conductivity
(Lmcv) in the late distal
tubule had a value (7.48 × 108
cm · s1 · mmHg1)
similar to those reported for transepithelial hydraulic conductivity of
the nondiuretic rat (6.47 × 108
cm · s1 · mmHg1,
Ref. 54; and 10.03 × 108
cm · s1 · mmHg1,
Ref. 58). Intrinsic dissociation constants of Na-Cl cotransporter (KmcNa,NaCl and
KmcCl,NaCl) were selected to
account for the measured kinetic constants of this transporter. With
the listed values, the kinetic equation (Eq. 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 (KmcK,KCl and
KmcCl,KCl) 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 (Ja,max) was 317 pmol · min1 · tube1,
which was similar to the experimental value (34) for dissected rat
distal tubule (257 pmol · min1 · tube1).
Value of kinetic parameter of Na-K-ATPase
(KNa,ATPase) was that of Na-K-ATPase
with rat 1-isoform, which is the predominant isoform in the kidney,
reported in a recent study (69). Ion permeabilities through the
paracellular pathway (PmsNa, PmsK, and
PmsCl) were calculated from a
measurement of transepithelial conductance (2.6 mS/cm2)
in the early distal tubule (43). With the listed values, paracellular pathway of the model has conductance of 2.5 mS/cm2.
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 (Lscv) are 100-fold larger
than that in the luminal membrane
(Lmcv); 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
(J mcNaCl,max, PmcNa, and
PmcCl in the early distal
tubule; and J mcNaCl,max, J mcKCl,max,
PmcNa, PmcK, and
PmcCl in the late distal
tubule). These parameters are underlined in Table 7. Exclusion of the
basolateral parameters ensured the acceptable predictions for cytosolic
variables. J mcKCl,max and
PmcK 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:
J mcNaCl,max 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 J mcNaCl,max 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 Table 8, 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. 4C). However, there was a
significant difference in the luminal potential profile (Fig.
3A) between these models. The model without electrodiffusion (broken line, Fig. 3A) 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.
3A).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. 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.
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Fig. 3.
Comparison of profiles of luminal variables predicted by models with
(solid lines) and without (broken lines) electrodiffusion. Model
parameters listed in Table 8 were used. A: luminal electrical
potential. B: tubular flow rate. C: luminal sodium
concentration. D: luminal potassium concentration. Note the
abrupt change of electrical potential at the junction of the early and
late distal tubules (x = 0.10 cm) in the solution without
electrodiffusive terms.
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Fig. 4.
Comparison of profiles of water and solute fluxes predicted by models
with (solid lines) and without (broken lines) electrodiffusion.
Calculations were conducted in a manner similar to that in Fig. 3.
A: transepithelial water flux. B: transepithelial
sodium flux. C: transepithelial potassium flux.
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Basic State
The model could simulate total water and solute reabsorptions as well
as luminal potential of the distal tubule in the basic state (Table
9). 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. 6A) 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 6B) 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. 6B). 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.
6C), 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).
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Fig. 5.
Luminal variables in presence of thiazide (solid lines) or in the basic
state (broken lines). A: luminal electrical potential.
B: tubular flow rate. C: luminal sodium concentration.
D: luminal potassium concentration.
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Fig. 6.
Transepithelial water and solute fluxes in presence of thiazide (solid
lines) or in the basic state (broken lines). A: transepithelial
water flux. B: transepithelial sodium flux. C:
transepithelial potassium flux. Positive values indicate net
reabsorption.
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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.
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Fig. 7.
Luminal variables in presence of amiloride (solid lines) or in the
basic state (broken lines). A: luminal electrical potential.
B: tubular flow rate. C: luminal sodium concentration.
D: luminal potassium concentration.
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Fig. 8.
Transepithelial water and solute fluxes in presence of amiloride (solid
lines) or in the basic state (broken lines). A: transepithelial
water flux. B: transepithelial sodium flux. C:
transepithelial potassium flux. Positive values indicate net
reabsorption.
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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.
10A). 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 (Table
10). 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. Figure
11 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.
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Fig. 9.
Luminal variables with high sodium load (solid lines) or in the basic
state (broken lines). A: luminal electrical potential.
B: tubular flow rate. C: luminal sodium concentration.
D: luminal potassium concentration.
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Fig. 10.
Transepithelial water and solute fluxes with high sodium load (solid
lines) or in the basic state (broken lines). A: transepithelial
water flux. B: transepithelial sodium flux. C:
transepithelial potassium flux. Positive values indicate net
reabsorption.
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Fig. 11.
Relationship between sodium reabsorption and sodium load. Effect of
sodium load on total sodium reabsorption was examined by solving the
model equations with various amounts of sodium load. Sodium load was
manipulated by altering initial flow rate and tubular sodium
concentration by means of linear interpolation of the basic state and
the sodium-loaded state used in the definition of the penalty function
(Table 6). Specifically, Fv and
Cm,0Na were changed using the following
equations with x varying from 0.15 to 1.0: Fv = 7.99 + (35.02 7.99) x (in nl/min);
Cm,0Na = 51.06 + (61.74 51.06) x (in mM); where x is a dummy
parameter with x = 0 and x = 1 corresponding to the
basic state and the sodium-loaded state (as used in the penalty
function), respectively. Solid circles indicate experimental results
from Refs. 13 and 37.
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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.
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Fig. 12.
Relationship between potassium secretion and tubular flow rate. Effect
of tubular flow rate on total potassium secretion was examined by
solving the model equations with various amounts of initial flow rate.
Initial flow rate as well as initial tubular potassium concentration
were altered by means of interpolation of the basic state and the high
flow state used in the definition of the penalty function (Table 6).
Specifically, Fv and Cm,0K were
changed using the following equations with x varying from
0.15 to 1.0: Fv = 7.99 + (38.06 7.99) x
(in nl/min); Cm,0K = 2.30 + (1.86 2.30) x (in mM); where x is a dummy
parameter with x = 0 and x = 1 corresponding to the
basic state and the high flow state (as used in the penalty function),
respectively. Solid circles indicate experimental results from Refs. 38
and 51.
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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 12C)
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 12S). 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 J mcNaCl,max 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
(PmcNa) 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
PmcNa 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 viv
TOP
ABSTRACT
INTRODUCTION
