Modeling of transcellular Ca transport in rat
duodenum points to coexistence of two mechanisms of apical
entry
Boris M.
Slepchenko1 and
Felix
Bronner2
Departments of 1 Physiology and 2 BioStructure and
Function, University of Connecticut Health Center, Farmington,
Connecticut 06030
 |
ABSTRACT |
Employing
realistic parameters, we have demonstrated that a relatively simple
mathematical model can reproduce key features of steady-state
Ca2+ transport with the assumption of two mechanisms of
Ca2+ entry: a channel-like flux and a carrier-mediated
transport. At low luminal [Ca2+] (1-5 mM),
facilitated entry dominates and saturates with
Km = 0.4 mM. At luminal
[Ca2+] of tens of millimolar, apical permeability is
dominated by the channel flux that in turn is regulated by cytosolic
Ca2+. The model reproduces the linear relationship between
maximum Ca2+ transport rate and intestinal calbindin
D9K (CaBP) content. At luminal [Ca2+] > 50 mM, local sensitivity analysis shows transcellular transport to be most
sensitive to variations in CaBP. At low luminal [Ca2+],
transport becomes sensitive to apical entry regulation. The simulations
have been run within the Virtual Cell modeling environment, yielding
the time course of external Ca2+ and spatiotemporal
distributions of both intracellular Ca2+ and CaBP.
Coexistence of two apical entry mechanisms accords with the properties
of the duodenal Ca2+ transport protein CaT1 and the
epithelial Ca2+ channel ECaC.
intestine; calcium entry; brush border; calbindin D9K; intracellular calcium diffusion
 |
INTRODUCTION |
TRANSEPITHELIAL CALCIUM
TRANSPORT in the small intestine follows two pathways: a
transcellular, vitamin D-dependent route, localized largely to the
duodenum but also present to a minor degree in the upper jejunum, and a
paracellular, concentration-dependent movement that takes place all
along the small intestine (2). The paracellular process is
not subject to regulation beyond that affecting tight junction function
generally. The transcellular pathway is saturable, requires metabolic
energy, and is upregulated in pregnancy and calcium deficiency and
downregulated with increasing calcium, with regulation mediated by the
hormone-like actions of vitamin D (2, 5). Thus
transcellular calcium movement plays a significant role under
conditions of low calcium intake, transporting a substantial proportion
of absorbed calcium (2, 5). When calcium intake goes up,
an increasing proportion is absorbed via the paracellular route,
predominantly in jejunum and ileum.
It is now widely recognized that transcellular calcium transport is a
three-step process consisting of passive calcium entry at the brush
border, facilitated diffusion through the cell, and active extrusion at
the basolateral membrane (2, 7, 13) (see diagram in Fig.
1). Calcium extrusion is mediated by the Ca2+-ATPase (8) and, to a minor degree, by a
Na+/Ca2+ exchanger (19). It has
been shown experimentally (5) that the extrusion capacity
is more than adequate, and therefore this step does not appear to be
rate limiting in the overall transport. A major molecular expression of
vitamin D regulation is calbindin D9K (CaBP), a cytosolic
protein discovered by Wasserman and colleagues (28),
which, by binding calcium, has been shown (5, 15, 29) to
act like an intracellular ferry, thereby increasing the rate of
intracellular calcium diffusion. The role of CaBP in facilitating intracellular calcium diffusion has been demonstrated by mathematical modeling (7). On the other hand, the molecular nature of
the apical entry mechanism has remained obscure until recently
identified structures, the epithelial Ca2+ channel (ECaC)
in rabbit duodenum, jejunum, and kidney (11, 13) and the
duodenal calcium transport protein CaT1 in rat duodenum and proximal
jejunum (24), were shown to mediate calcium entry. These
proteins have 75% structural similarity and exhibit similar calcium
transport features. However, unlike ECaC, CaT1 is not responsive to
1,25-dihydroxyvitamin D3
[1,25(OH)2D3] administration or to calcium
deficiency (10, 24). Controversy still exists as to how
these structures transport calcium ions. Both molecules have pore
regions that would suggest a channel-type mechanism (11,
24). However, their macroscopic kinetic properties, as well as
the fact that no single-channel calcium currents have been detected,
favor a facilitated transporter mechanism that requires calcium binding
to a transporter. Thus each of these two structures, CaT1 or ECaC, can
act as a calcium channel and transporter.

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Fig. 1.
Diagram of the 3-step mechanism of steady-state
transcellular calcium transport. CaBP9K, calbindin
D9K; D, vitamin D; ECaC, epithelial calcium channel; CaT1,
duodenal Ca2+ transport protein. [Adapted from Hoenderop
et al. (12).]
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In the present study, a relatively simple mathematical model is used to
analyze the typical features of calcium transport in rat duodenum
(21-23). On the basis of the volume of the small intestine of rats and their calcium intake, one can estimate that, after a meal, calcium concentrations in the lumen can vary up to >50
mM. Experimentally, solutions of widely ranging calcium concentrations
were instilled into intestinal loops, with calcium transport then
measured in animals that had been on different calcium intakes and,
therefore, had varying levels of CaBP (5). In experiments
done at tens of millimolar of soluble luminal calcium, with the
cellular content of CaBP near maximum, the saturable component of
transepithelial calcium transport was well approximated by a
Michaelis-Menten relationship, with Vm = 21 µmol · h
1 · g
1 and
Km = 51 mM, where Vm
is the maximum rate of luminal calcium efflux and
Km is the calcium concentration at
Vm/2. Moreover, Vm is a positive, linear function of the
cellular calbindin D9K content (5, 21). In the
1-5 mM range, however, as evident from intestinal sac experiments
(5), transcellular calcium saturates with
Km = 0.35 mM and
Vm = 2.2 µmol · h
1 · g
1, findings
that translate into much higher brush-border permeability. Interestingly, a value of Km = 0.35 mM is
in agreement with the experimental data obtained recently in studies of
CaT1 and ECaC (11, 24) done at low luminal calcium concentrations.
Employing realistic parameters, we have demonstrated here that the
model reproduces the key experimental features of steady-state calcium
transport if the coexistence of two mechanisms of calcium entry, a
channel-like flux and a carrier-mediated transport, is assumed. The
mechanism of facilitated entry dominates at low luminal calcium
concentrations and saturates with Km = 0.4 mM. At luminal calcium concentration in the range of tens of
millimolar, channel-like flux regulated by cytosolic calcium at the
inner brush border largely determines brush-border permeability. Given
the similarity in the transport properties of ECaC and CaT1, both
structures can, as mentioned above, function at low and high luminal
calcium concentrations. In other words, each can function as a
transporter at low luminal calcium concentrations and as a channel at
high luminal calcium concentrations (see also DISCUSSION).
The model has been used to simulate the time dependence of luminal
calcium in loop experiments (5) under various initial conditions. The simulations have been run within the Virtual Cell environment (25), which allows one to obtain
spatiotemporal distributions of intracellular calcium and CaBP as well
as the time course of external calcium. In addition, we have used the model to analyze the effect of vitamin D on calcium entry.
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MODEL DESCRIPTION |
To analyze transcellular calcium transport, we have used a
one-dimensional model of intracellular calcium dynamics in the presence
of a mobile buffer. We let c denote the free calcium concentration and b represent the concentration of calcium
bound to CaBP. The model is then described by the equations
|
(1)
|
|
(2)
|
where Dc and Db are
diffusion coefficients of calcium and CaBP, respectively, and
R, the reaction rate of calcium binding to CaBP, is
determined by simple mass action kinetics
|
(3)
|
with kon and koff
denoting kinetic constants and Bt representing
the total concentration of CaBP. The ellipsis in Eq. 1
represents the rates of all intracellular processes in which calcium
can participate, such as binding to fixed binding sites or being pumped into internal calcium stores.
Equation 1 is subject to the boundary conditions that
prescribe fluxes of apical calcium entry (Ja)
and of calcium extrusion at the basolateral membrane
(Jb)
|
(4.1)
|
while the boundary conditions for Eq. 2 reflect zero
flux of CaBP at the cell membrane
|
(4.2)
|
In Eqs. 4.1 and 4.2, L denotes
the length of the cell.
Assuming steady calcium flux and near-instantaneous buffering, the
model reduces to a system of nonlinear algebraic equations for the
total calcium flux density
and the concentrations of intracellular calcium at the boundaries,
c
= (c)x=0 and
c+ = (c)x=L (see
APPENDIX A) such that
|
(5.1)
|
for apical calcium entry (1st stage)
|
(5.2)
|
for facilitated diffusion (2nd stage) and
|
(5.3)
|
for calcium extrusion (3rd stage). In Eq. 5.2,
K is the dissociation constant for calcium binding to CaBP,
K = koff/kon. The experimentally determined value of K is 0.43 µM
(7).
As mentioned above, calcium is extruded from the intestinal cell at the
basolateral membrane, predominantly via Ca2+-ATPase pumps.
The rate of Ca2+-ATPase pumping is commonly described by a
Hill-type equation (17). It is known that in some cell
types the plasma membrane pumps exhibit a threshold behavior as a
function of cytosolic calcium, with the rate increasing in a
near-linear fashion beyond the threshold (9). Taking this
into account, we use the following expression for
Jb
|
(6)
|
The threshold calcium concentration, c0, in
Eq. 6 corresponds to an average concentration of free
intracellular calcium at rest, estimated as 0.07 µM (see also Table
1 for other parameter values). The
maximum flux density of calcium extrusion, Vp,
has been determined as 150 µmol · h
1 · g
1, and the
experimental value for the dissociation constant
Kp of calcium binding to a pump is 0.25 µM
(5). The second term in the upper line of Eq. 6
represents the leak of calcium into a cell across the basolateral
membrane, a flux balanced by pumping under rest conditions.
In the following, the model for apical calcium entry is presented in
detail. We show that the model of a channel regulated by intracellular
calcium is consistent with the experimental data obtained at high
luminal calcium. The model fails, however, in the physiological range
of relatively low luminal calcium concentrations, where brush-border
permeability appears to be much higher, whereas flux saturates at a
much lower level. This can be explained by a transporter-mediated
mechanism of calcium apical entry. The coexistence of both mechanisms
then explains experimental findings over the full range of luminal
calcium concentration.
 |
RESULTS |
Transcellular calcium flux at high luminal calcium concentrations.
Because the Vp is more than seven times greater
than the Vm, extrusion is not a limiting factor,
and therefore all calcium transported to the basolateral membrane can
be readily pumped out of the cell (see below for sensitivity analysis).
As a result, the intracellular calcium concentration at the basolateral
membrane will be close to c0 at any time, and we
can set c+
c0 in Eq. 5.2. We also take into account that
DbBt/Dcc
1 when there is an adequate amount of CaBP. This means that only a
small fraction of calcium flux is in the form of free calcium in the presence of CaBP. Equation 5.2 can therefore be reduced to
|
(7.1)
|
where
c is by definition
c
c0 and
km is by definition
c0 + K = 0.5 µM.
Equation 7.1 helps us understand why the experimental data
obtained at high luminal calcium concentrations
([Ca2+]lumen) can be well approximated by a
Michaelis-Menten function
|
(7.2)
|
A comparison of Eqs. 7.1 and 7.2 immediately
yields km =
Km
with
= 10
5 and
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(8.1)
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(8.2)
|
Equation 8.2 states, in agreement with experimental
findings (5), that the maximum calcium transport is
linearly dependent on the total amount of CaBP. The latter is therefore
a limiting factor at high luminal calcium concentrations. We confirm
this conclusion below by means of sensitivity analysis.
Interestingly, the slope of this dependence is determined by the CaBP
diffusion coefficient. To estimate Db from
Eq. 8.2, we must evaluate the CaBP concentration,
Bt, and convert experimental flux values,
expressed as
µmol · h
1 · g
1, into the
units of flux density, µM · µm · s
1.
The conversion of units yields 3.6 µmol · h
1 · g
1 = (1012/N
0)
µM · µm3 · s
1, where
N is the number of cells per gram of tissue and
0 is the cross-sectional area of a cell.
Assuming N = 108 and
0 = 102 µm2
(5, 7), we arrive at the conversion relationship 1 µmol · h
1 · g
1 = (102/3.6) µM · µm · s
1,
the value used below. The CaBP content can be estimated from the fact
that rats on a low-calcium diet (0.06% Ca, 0.2% P) will have up to
100 nmol of Ca bound to CaBP per gram of mucosa (6), with
two calcium binding sites per one CaBP (27). (However, the
transport equations account for only one Ca2+ per CaBP,
because the other is very tightly bound.) With the assumption, as
above, of 108 cells per gram of mucosa, each with a
volume of ~2,000 µm3, Bt
equals 250 µM. With K = 0.43 µM,
c0 = 0.07 µM, L = 20 µm, and Bt =250 µM, we then obtain for
Db the value of 56.8 µm2/s.
Interestingly, the estimate of Db from the
comparison with the self-diffusion coefficient of calcium in cytoplasm
yields a similar value. Indeed, Dc is estimated
as 300 µm2/s (1). The diffusion coefficient
of CaBP can then be roughly evaluated as Db = Dc(mCa/mCaBP)1/3
[1/6]Dc = 50 µm2/s, where
m refers to molecular mass.
We next turn to Eq. 5.1 to show that the hypothesis of
calcium apical entry via channels, regulated by intracellular calcium, is consistent with the experimental data on calcium transport at high
luminal calcium concentrations. At the brush border, the calcium flow
through a pore would take place at a favorable electrochemical gradient
and can be approximated by the Goldman-Hodgkin-Katz equation (Ref.
14, p. 53). This equation, because the
membrane potential is normally negative and
[Ca2+]lumen is greater than
c
by several orders of magnitude, can be
simplified to
|
(9)
|
with P = g(
/
0)[1
exp(
/
0)]
1, where
is the membrane
potential,
0 = RT/2F
(R is the gas constant, T is the absolute temperature, F is the Faraday number), and the channel
permeability g can be a function of the membrane potential
and can be regulated by binding of intracellular calcium and/or some
other molecules to a channel (channel gating). Although P is
nominally a function of the membrane potential,
does not explicitly
enter the equations below because we are concerned only with changes in
the chemical potential.
There is increasing experimental evidence that the apical entry is
regulated by intracellular calcium (7, 12, 24). Mathematically, this means that P should be a function of
c
. To determine this function, we use
Eq. 8.1 to rewrite Eq. 7.2 as follows
|
(10)
|
where Ki = km
c0 = K. From Eqs. 5.1 and 9, we then
conclude that P
(c
+ Ki)
1. This can be interpreted as
channel inhibition by the binding of intracellular calcium to the
channel-inhibiting binding sites with a dissociation constant
Ki. In this case, channel permeability is
proportional to the fraction of uninhibited channels
|
(11)
|
where P0 is the maximum permeability. It is
interesting to note that the functional form of Eq. 11 with
Ki = 0.5 µM was postulated earlier
(7). By combining Eqs. 5.1, 9,
10, and 11, we find
We thus conclude that brush-border permeability determines the
ratio of the Michaelis-Menten parameters. The value of
P0 obtained for Vm = 21 µmol · h
1 · g
1 is 0.01 µm/s.
The numerical solution of Eqs. 5, 6,
9, and 11 supports the conclusions of the preliminary
analysis above. Parameter values used in computations are given in
Table 1. We first fixed the concentration of CaBP at 250 µM and
computed transcellular calcium flux J (along with
c
and c+) as a function
of [Ca2+]lumen. These results are presented
in Table 2 and Fig.
2. J, c
, and c+, as well as
the fraction of flux carried by free calcium, were obtained by solving
the system of nonlinear algebraic Eqs. 5.1-5.3 by means
of Newton iterations. The results are accurate to 10
5
(relative error). An average free calcium concentration was determined for selected values of [Ca2+]lumen by running
full spatial simulations based on Eqs. 1-4 for a
sufficiently long time. The results are accurate within 1% of relative
error. Figure 2 shows that, as expected, the model gives a virtually
perfect Michaelis-Menten curve, with Vm = 23.3 µmol · h
1 · g
1 and
Km = 48.7 mM. The numerical results in
Table 2 confirm the assumptions made in the preliminary analysis. The
concentration of intracellular calcium at the basolateral membrane
undergoes only slight changes (Table 2, c+) and
CaBP carries >90% of the transported calcium (Table 2, Free calcium
flux as a fraction of total).

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Fig. 2.
Simulation results for steady-state transcellular calcium
transport in the duodenal loop preparation (5) at high
luminal calcium. [Ca2+]lumen, luminal calcium
concentration; J, transcellular calcium flux. Data points
( ) represent simulation results superimposed on a
Michaelis-Menten relationship, with Vm = 23.3 µmol · h 1 · g 1 and
Km = 48.7 mM, where
Vm is the maximum rate of luminal calcium efflux
and Km is the luminal calcium concentration at
Vm/2.
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To better understand the key elements of the three-step mechanism of
steady-state calcium transport, we performed a sensitivity analysis
(the equations for sensitivities are derived in APPENDIX B). The logarithmic sensitivity
log J/
log
, where
stands for any model parameter, represents a
relative change in calcium flux, J, divided by a small
relative change in the parameter. The results of the sensitivity
analysis for the parameter set of Table 1 are given in Fig.
3. These results indicate that at high
[Ca2+]lumen, calcium transport is most
sensitive to the CaBP content, while at low
[Ca2+]lumen it is mainly controlled by the
brush-border permeability. Thus, in the range of high luminal calcium
concentrations, in agreement with the previous analysis, the
Michaelis-Menten parameter Vm is determined by
the properties of CaBP, while the ratio
Vm/Km depends largely on
the brush-border permeability. Also, as expected, the sensitivity of
calcium transport to the parameters of calcium extrusion is low over
the whole range of luminal calcium concentrations. Interestingly, there
can be a change in sign of the sensitivity of the calcium transport
with respect to the affinity K of CaBP to calcium. This is
consistent with the fact that J is a non-monotone function
of K (see Eq. 5.2).

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Fig. 3.
Results of sensitivity analysis for the indicated parameters, where
represents any parameter. P0, maximum
channel permeability; Ki, dissociation constant
for channel inhibiting binding site; Kp,
dissociation constant of calcium binding to Ca2+-ATPase;
Vp, maximum extrusion flux; K,
dissociation constant for calcium binding to CaBP;
Bt, total CaBP concentration.
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Calcium transport at low luminal calcium concentrations.
Experiments with everted duodenal sacs, carried out at relatively
low luminal calcium concentrations, indicated flux saturation with
Km = 0.35 mM and
Vm = 2.2 µmol · h
1 · g
1
(5). Experiments on ECaC and CaT1 also indicated a value
of Km in the range of 0.2-0.4 mM (11,
24). These data yield a brush-border permeability of the order
of Vm/Km
0.2 µm/s, a value that is 20 times greater than the estimate of
P0 obtained above. This difference indicates the
presence of an additional entry mechanism with substantially higher
brush-border permeability but saturating at relatively low luminal
calcium. In principle, this could be a different channel, with much
higher permeability and an inhibiting binding site for intracellular
calcium with Ki of the order of 0.01 µM; this
is unlikely. In light of the reported macroscopic kinetic properties of
CaT1 and ECaC, the mechanism is more likely to be a transporter, as
previously suggested (30). With this hypothesis we show
below that the higher brush-border permeability and the low-level
saturation can both be accounted for, consistent with the results of
the previous section. Figure 4 displays a
diagram describing a transporter. In the two states on the left-hand
side of Fig. 4, T0 and
T1, the calcium binding site of a transporter is
exposed to external calcium, and in states T2
and T3 on the right-hand side, it is exposed to
internal calcium. To derive the expression for the transporter-mediated
calcium flux, Jtr, we consider steady flux
conditions. In addition, we assume that calcium binding on the luminal
side of the membrane is near instantaneous. Jtr
can then be found from the following equations
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(12)
|
In Eqs. 12, k
and
k
are the rates of transition between
transporter states as shown in Fig. 4;
k2 is the rate of calcium dissociation from the
transporter in the cytosol; and K
and
Ko are the inside and outside transporter
dissociation constants, respectively, for binding calcium. Solving the
system of Eqs. 12 for Jtr (see
APPENDIX C for details) yields
|
(13)
|
where the parameter Vtr is determined by
the total density of transporter binding sites and a certain average of
the kinetic constants k
,
k
, and k2, whereas
the parameter Ktr is determined by the
dissociation constant Ko, modified by a ratio of
kinetic constants as specified in APPENDIX C.

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Fig. 4.
Transporter mechanism for the calcium flux component that
saturates at low luminal calcium concentration. In transporter states
T0 and T1, the calcium
binding site of a transporter is exposed to external calcium (cell
outside), and in states T2 and
T3, it is exposed to internal calcium (cell
inside). k and k ,
rates of transition between transporter states;
k2, rate of calcium dissociation from the
transporter in the cytosol.
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Eqs. 9, 11, and 13, combined, provide
the complete description of calcium apical entry as
|
(14)
|
The expression inside the parentheses in Eq. 14 can be
interpreted as the macroscopic permeability of the apical membrane, modulated by both intracellular and luminal calcium. With the parameters Vtr and Ktr
set so that P0
Vtr/Ktr and
Vtr/[Ca2+]lumen
P0 in the range of tens of millimolar of luminal
calcium, then, at relatively low [Ca2+]lumen
(on the order of Ktr or smaller), the
brush-border permeability will be much greater than
P0, while tending to approach
P0 as [Ca2+]lumen
increases. At high luminal calcium concentrations, therefore, simulations based on the complete model of the apical entry (Eq. 14) will yield results similar to those described in the preceding section, whereas at low luminal calcium concentrations, entry saturates
in accordance with Eq. 13. Figure
5 shows the results obtained with
Vtr = 2.2 µmol · h
1 · g
1,
Ktr = 0.2 mM,
P0 = 0.01 µm/s, and all other parameters
as defined in Table 1. At high luminal calcium, the simulation results
are again well approximated by a Michaelis-Menten equation, with
Vm = 21.3 µmol · h
1 · g
1 and
Km = 50.0 mM. At low luminal calcium, a
plateau is reached (see Fig. 5, inset), characterized by
Vm = 3.1 µmol · h
1 · g
1 and
Km = 0.4 mM.

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Fig. 5.
Simulation of transcellular calcium transport with the
complete model based on Eq. 14. Data points
( ) represent simulation results superimposed on a
Michaelis-Menten relationship, with Vm = 20.7 µmol · h 1 · g 1 and
Km = 50 mM at high luminal calcium
concentrations. Inset: at low luminal calcium
concentrations, Vm = 3.1 µmol · h 1 · g 1 and
Km = 0.37 mM.
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Finally, we used the complete model to determine the steady-state
spatial distributions of free and bound calcium inside a cell (Fig.
6). To do this, we ran full spatial
simulations based on Eqs. 1-4, 6, and
14 long enough to reach a steady state. In these spatial
simulations we also used the near-instantaneous approximation for
calcium binding to CaBP (26).

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Fig. 6.
Distributions of free
(A) and bound (B) calcium inside the duodenal
cell under steady-state transcellular calcium transport. Distance is
measured from the brush border and varies from 0 to L, the
length of the cell. Simulations are based on Eqs. 1-4,
6, and 14, with calcium binding to calbindin
D9K considered near instantaneous.
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Calcium absorption and transport in duodenal loops.
In loop experiments (21-23), the intestine is
exteriorized from an anesthetized, laparotomized animal, the most
proximal 10 cm of the small intestine is tied proximally and distally,
and a calibrated quantity of a calcium-containing buffer is instilled into the previously rinsed intestinal loop. The decrease in calcium concentration as a function of time is established on the basis of
measurements in a series of loops; a single time point is typically the
mean of three to six loop measurements. Figure
7A represents typical results.

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Fig. 7.
Time dependence of luminal calcium transport in duodenal
loops. A: experimental findings. [Reprinted from Bronner et
al. (5).] B: simulation results.
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To simulate these findings, it is necessary to take into account both
paracellular and transcellular calcium movement, with the rate of the
former 0.13-0.16 h
1 (5). Equation 15 describes the time-dependent changes in the luminal calcium
concentration
|
(15)
|
where Vl and Sl are the loop
volume and internal surface area, respectively;
Jpara = k[Ca2+]lumen is the flux
density of the paracellular transport with the appropriate rate
constant k, and Ja is the flux
density of calcium entry at the brush border from Eq. 14. To
solve Eq. 15, we need to estimate the loop surface-to-volume
ratio,
1 = Sl/Vl. Because
Ja is, according to Eq. 14,
significantly regulated internally, Eq. 15 has to be solved
simultaneously with Eqs. 1-4, 6, and
14. To do this rigorously, one has to fill in the ellipsis
of Eq. 1 by introducing the mechanisms of intracellular
calcium uptake. This, however, leads to a myriad of additional
parameters with uncertain values. If these mechanisms are ignored, it
is still possible to obtain reasonable simulation results in the two
limiting cases of high and low initial lumen calcium concentrations.
Because the principal interest is to simulate events that have a time scale of minutes and hours (cf. Fig. 7A), the very fast
component of intracellular calcium uptake, i.e., rapid binding to fixed sites, may be ignored inasmuch as equilibrium is attained very rapidly
and the binding capacity is low.
At low initial luminal calcium concentration, the process of absorption
is brief. Therefore, the relatively slow uptake by intracellular
calcium stores is likely to have little effect on calcium transport.
When the initial luminal calcium concentration is high, the other of
the two limiting situations, intracellular calcium uptake by fixed
sites and via endoplasmic calcium pumps, is bound to be small compared
with total transcellular calcium transport and will have been completed
well before transport is completed. Hence, it again can be ignored. At
intermediate luminal calcium concentrations, however, results of
simulation studies that do not take intracellular calcium uptake into
account may deviate significantly from the experimental findings.
The simulation results displayed in Fig. 7B have utilized
the parameters of a complete model (Table 1) plus two additional parameters, i.e.,
l = 0.015 µm
1 and k = 0.0022 µm/s, which in
turn yield k ·
l = 0.12 h
1, a value comparable to what has been found
experimentally (5). A typical radius of an intestinal loop
is r
2 mm. This value would yield a
surface-to-volume ratio of 2/r = 0.001 µm
1. However, intestinal villi markedly increase the
available surface area. A 15-fold surface amplification factor is
compatible with the estimate (7) of 108
cells/g duodenum used above.
At initial luminal calcium concentrations of 1 mM or 50-200 mM,
there is good agreement between the experimental findings (Fig.
7A) and the simulation results (Fig. 7B). In the
intermediate range of 10-25 mM, agreement between experiment and
simulation is relatively poor, intracellular calcium uptake having been
ignored. The discrepancy between simulation and experiment in this
range of luminal calcium can, however, provide information on the rate and capacity of intracellular calcium uptake by duodenal cells. For
instance, the discrepancy in calcium absorption at an initial luminal
calcium concentration of 25 mM (Fig. 7) indicates that intracellular
calcium uptake is complete at 45 min. A rough estimate of uptake
capacity yields a value of 25 nmol/106 cells.
Interestingly, these estimates are consistent with the experimental
finding (4) that isolated duodenal cells immersed in 3 mM
calcium have taken up calcium at the rate of 10 nmol (106
cells)
1 · h
1, with uptake not yet
complete at 30 min.
Effect of vitamin D on calcium entry.
As shown previously (5, 15, 29) and earlier in this paper,
calbindin D9K, a
1,25(OH)2D3-dependent protein, significantly facilitates transcellular calcium transport. At the same time, calbindin D9K acts as a mobile calcium buffer that helps
keep intracellular free calcium at a low level, even when there is a
large steady-state calcium flux through the cytosol. In the case of
vitamin D deficiency, therefore, when no calbindin D9K is
expressed in the transporting duodenal cell, a prohibitively high
concentration of free cytosolic calcium would build up unless vitamin D
also upregulated calcium entry. Indeed, our model (Eqs. 5.1-5.3, 6, and 14) predicts that in
the duodenal cells exposed to 50 mM of luminal calcium in the absence
of CaBP, the concentration of free calcium at the inner brush border
will rise to >6 µM. To keep free calcium concentration <1 µM, the
brush-border permeability must be reduced to 7% of its nominal value.
We thus conclude that the vitamin D-independent component of calcium
entry, presumably mediated by CaT1, amounts to <10% of the total
calcium entry that takes place in the presence of vitamin D, with the
vitamin D-dependent ECaC then mediating the increase in calcium entry.
Experiments with brush-border vesicles (18) showed that in
the case of vitamin D deficiency, calcium entry decreased by some
30-50%. This value was probably an underestimate because in
vesicles, unlike intact cells, the calcium influx is substantially reduced as a result of calcium accumulation inside a vesicle. The
vitamin D dependence of calcium entry can also be inferred from in situ
experiments (21-23) in which both the saturable
component of calcium transport and the CaBP content were evaluated. As
shown above, the brush-border permeability can be estimated from the Michaelis-Menten parameters as being equal to
Vm/Km, whereas the CaBP
content is a measure of the degree of vitamin D deficiency or
sufficiency. Our model also permits estimation of the effect of vitamin
D on calcium entry if two additional assumptions are made:
1) the vitamin D-insensitive component constitutes 7% of the total of calcium entry (see the estimate above), and 2)
transcellular calcium transport is a linear function of the CaBP content.
Figure 8 is a comparison of experimental
findings with a modeling function. The experimental data for the
transcellular transport with Km values
relatively close to 50 mM (Table 3) were
obtained under similar conditions in animals of varying ages and
varying levels of vitamin D sufficiency (22, 23).
Permeability was calculated as
Vm/Km. The permeability
values of the modeling function were derived from the modeled linear
relationship between calcium flux and CaBP content. To facilitate
comparison, all values were normalized to their respective maxima. The
reasonable agreement between the experimental data and the modeling
function supports the assumptions made above. In particular, taking
into account the vitamin D dependence of calcium entry from Fig. 8, we
can now compute transcellular calcium transport as a function of the luminal calcium concentration for a wide range of CaBP concentrations. The computation results in Fig.
9A are fitted to
Michaelis-Menten functions. As expected, the maximum calcium transport
rates Vm, derived from the fitted functions,
depend linearly on the CaBP content (Fig. 9B), in full
agreement with the experimental data (5, 21).

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Fig. 8.
Relationship between CaBP content and calcium entry, with
both parameters being vitamin D dependent. Experimental values
( ) were obtained from animals of varying ages (see
Table 3). Permeability was estimated as being equal to
Vm/Km. To facilitate
comparison with modeling results (solid curve), all values were
normalized to their respective maxima.
|
|

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Fig. 9.
Effect of CaBP on calcium transport. A:
simulated calcium flux, J, as a function of luminal calcium
concentration at varying tissue concentrations of CaBP ([CaBP]).
B: relationship between CaBP content and maximum calcium
transport (Vm).
|
|
 |
DISCUSSION |
The mathematical model of transcellular calcium transport in rat
duodenum developed here indicates the coexistence of two regulatory
mechanisms that operate at the brush border: a channel calcium flux
regulated by intracellular calcium and a facilitated transporter
mechanism with calcium binding to the transporter. This conclusion
accords with the properties of the recently discovered epithelial
calcium channel, ECaC (12), and the calcium transport protein, CaT1 (24). We believe that either of these
molecular structures can accommodate the two mechanisms of calcium entry.
As shown above, at a macroscopic level both mechanisms result in a
Michaelis-Menten-type equation for calcium transport, but with very
different values of Km. As a consequence, when
luminal calcium is low (1-5 mM), the facilitated entry mechanism
dominates calcium transport but saturates at a relatively low flux
level. In the range of tens of millimolar of luminal calcium, apical transport is largely determined by the channel flow of calcium. Indeed,
as follows from Table 2, the channel permeability of 0.015 µm/s,
which fits the experimental data at high luminal calcium, yields only
0.5 µmol · h
1 · g
1 of
calcium flux at [Ca2+]lumen = 1 mM. With a reasonable channel density of 103 per
cell and the fraction of open channels of the order of 0.01 (20), this calcium flux translates into a single-channel
current amplitude of 0.03 pA, well below the limits of detectability. This can explain why single-channel events associated with calcium influx have not been detected (24). The model, however,
predicts that in the range of tens of millimolar of luminal calcium,
these events should become detectable.
The channel mechanism assumes importance under conditions of calcium
deficiency when an animal is faced with the opportunity of increased
calcium intake. In the absence of a channel mechanism, the relatively
low level of saturation of the transporter would limit calcium entry
and, therefore, the amount absorbed. Because of the existence of the
channel mechanism, calcium entry is no longer limited when calcium
intake goes up, inasmuch as the channel permits calcium to enter in
proportion to the luminal calcium concentration.
The model also allows estimation of the density of transporter binding
sites, T (see APPENDIX C). With the typical average transporter rate of 103 s
1 and the
maximum transporter flux Vtr = 2.2 µmol · h
1 · g
1, the value
for T is ~40 µm
2, or 4,000 transporter
binding sites per cell, a not unreasonable number.
It has been shown that intracellular calcium can inhibit both ECaC
(12) and CaT1 (24). The model predicts that
the channel mechanism is inhibited by binding intracellular calcium
with a dissociation constant of ~0.5 µM (see Eq. 11).
Note that to describe the detailed kinetics on a fraction-of-a-second
time scale, this equation should be replaced with P = P0h, where the fraction of uninhibited channels, h, should be described by an
additional equation, ht = k+[Ki
(Ki + c
)h], with
k+ being an on-rate constant (16).
The transporter mechanism also slows down at high intracellular
calcium, with Vtr being generally a decreasing
function of c
(see APPENDIX C).
Both contributions to the apical calcium flux are voltage dependent.
The channel permeability is naturally sensitive to membrane potential
(see Eq. 9 and related text). The rates of facilitated transport, k
, are also affected by the
intramembrane electric field. This is in agreement with the observed
ECaC activation by hyperpolarization (12) and the voltage dependency of CaT1-mediated currents (24). The detailed
quantitative description of the response of each of the mechanisms to
the change in the electrochemical potential requires further
experiments and additional model development.
Although similar in mediating calcium transport, ECaC and CaT1 differ
in that only the former is vitamin D dependent (10, 24).
Transcellular, i.e., active, calcium transport assumes physiological
importance under conditions of low calcium intake, which lead to an
increase in vitamin D-dependent processes in the cell, including
neosynthesis of CaBP. This upregulates transcellular movement. Inasmuch
as ECaC is vitamin D dependent, its contribution to calcium entry
increases in proportion to the increase in
1,25(OH)2D3 in the circulation.
At high luminal calcium, the ability of the epithelial cell to restrict
calcium entry is increasingly challenged, and downregulation of
facilitated diffusion is likely accompanied by a shutting down of the
ECaC entry system. The epithelial cell thus increasingly depends on the
CaT1 entry system, a logical response to the need of the cell to
maintain a low intracellular free calcium concentration. In view of the
fact that even when the luminal calcium concentration is 1 mM the
intracellular free calcium concentration is being maintained below 1 µM, severe entry restriction is needed at all times to
prevent cellular flooding with calcium.
When suitably modified, the model developed here can be applied to
calbindin-mediated transcellular calcium transport in other tissues,
e.g., proximal jejunum (23), kidney (10),
cecum (3), and colon (3). For example, in the
proximal jejunum (23), with the calbindin D9K
content found to be 45% of that in the duodenum, transcellular
transport was well approximated by a Michaelis-Menten function with
Vm = 8 µmol · h
1 · g
1 and
Km = 20 mM. In reproducing these findings,
our model yields a permeability value for the jejunum equal to 65% of
that in the duodenum.
In summary, a simple model of transcellular calcium transport in rat
duodenal cells has been formulated. On the assumption of the
coexistence of two mechanisms of calcium apical entry, the model
faithfully reproduces the relationship between luminal calcium and
calcium transport. At high luminal calcium, transcellular calcium
transport is largely a function of the intracellular concentration of
calbindin D9K, a vitamin-D-dependent mobile calcium binding protein that acts like a calcium ferry. The model reproduces the positive, linear dependence of Vm, the maximum
flux rate derived from a Michaelis-Menten relationship, on the
calbindin D9K content, with the slope of this relationship
being a function of the rate of calbindin D9K diffusion
through the cell.
At low luminal calcium, the regulation of apical entry is, according to
the model, a major limiting factor for transcellular calcium transport.
This regulation can be described in terms of facilitated entry
saturated at relatively low luminal calcium. The coexistence of both
mechanisms is required for a consistent description of transcellular
calcium transport in the full range of luminal calcium concentration.
The model qualitatively describes the calcium transient in duodenal
loops. The simulation results are reasonable in the limiting cases of
the high and low initial concentrations of luminal calcium and allow
for approximate estimates of the characteristics of calcium uptake.
In addition, when applied to an analysis of available
experimental data, the model shows that the vitamin D-insensitive
component of calcium entry, presumably CaT1, accounts for <10% of
calcium flux, with the vitamin D-dependent ECaC then mediating the remainder.
 |
APPENDIX A |
We derived Eqs. 5.1-5.3 from Eqs. 1-4
in the text by applying steady-state conditions and assuming
near-instantaneous calcium binding to calbindin D9K (CaBP).
Under steady-state conditions, all intracellular processes of calcium
uptake are completed, and the corresponding terms in Eq. 1
in the text (denoted by the ellipsis) cancel out. Because the time
derivatives on the left-hand side of Eqs. 1 and 2
disappear at steady state, adding them up yields
|
(A1)
|
The expression in the parentheses in Eq. A1 is the
total flux of free and bound calcium, J, which, as follows
from Eq. A1, is constant in time and independent of
x
|
(A2)
|
Applying Eq. A2 to x = 0 and x
= L and using boundary conditions Eqs. 3 and 4 in the text, we obtain Eqs. 5.1 and 5.3. Integration of Eq. A2 over the segment [0,
L] yields
|
(A3)
|
We now use the near-instantaneous approximation for calcium
binding to CaBP by assuming R = 0 in Eq. 3. This
yields
|
(A4)
|
where K = koff/kon. Applying
Eq. A4 to the boundary points, we get
|
(A5)
|
Substituting Eqs. A5 into Eq. A3, we
obtain Eq. 5.2.
 |
APPENDIX B |
Here we derive equations for the steady-state sensitivity
analysis of the model described by Eqs. 5.1-5.3 in the
text. In the general case of a system of n algebraic
equations with n variables xi (i = 1, ..., n) and m parameters
aj (j = 1, ...,
m)
|
(B1)
|
we introduce the sensitivity matrix
ij =
xi/
aj. The
sensitivities
ij are to be found by solving
the linear system of equations, which is derived by differentiating Eqs. B1 with respect to parameters
|
(B2)
|
where Jij
Fi/
xj is the
Jacobian matrix of the system in Eqs. B1 and Rij =
Fi/
aj. In our case,
we deal with the system of Eqs. 5.1-5.3
(n = 3) with the variables
x0
J,
x1
c+, and x2
c
|
(B3)
|
The Jacobian matrix obtained from Eqs. B3 is
We now enumerate parameters as
a0 = P0,
a1 = Ki,
a2 = Vp,
a3 = Kp,
a4 = K, and
a5 = Bt. The matrix
R = {Rij} then is
The sensitivities
ij can then be
transformed into the logarithmic sensitivities
x
aj
ij. The results for
x
aj
0j
are presented in Fig. 3.
 |
APPENDIX C |
Solving the system of Eqs. 12, accompanied by a
thermodynamic constraint,
k
k
K
= k
k
Ko, yields
|
(C1)
|
with
|
(C2)
|
where T is the density of transporter binding sites,
T = [T0] + [T1] + [T2] + [T3], and
|
(C3)
|
Note that, generally, both Vtr and
Ktr are functions of the intracellular calcium
concentration, c
. In particular, Vtr is a decreasing function of
c
, which means that a transporter can in
effect be inhibited by elevation of intracellular calcium. However,
with the simplifying assumption that once calcium gets inside the cell,
it can be readily released from the transporter, i.e.,
c
/K
1, a
characteristic of any effective transporter under normal physiological
conditions, the parameters Vtr and
Ktr can be regarded as constants determined by
kinetic rates k
,
k
, and k2:
Vtr = Tkav, with the
average rate kav = k
k
k2[k2(k
+ k
) + k
(k
+ k
)]
1 and
Ktr = Ko(k
+ k
)(k2 + k
)[k2(k
+ k
) + k
(k
+ k
)]
1. Finally, taking into
account [Ca2+]lumen
c
, Eq. C3.1 reduces to Eq. 13
of the text.
 |
NOTE ADDED IN PROOF |
According to recent evidence (E. M. Brown, personal
communication), duodenal ECaC is now thought to be essentially the same molecule as CaT1. Our model then predicts that calcium entry mediated by this molecule should be at least partly vitamin D responsive.
 |
ACKNOWLEDGEMENTS |
B. Slepchenko thanks I. Moraru for helpful discussions.
 |
FOOTNOTES |
The authors thank the National Institutes of Health for the support of
National Center for Research Resources Grant RR-13186.
Address for reprint requests and other correspondence: F. Bronner, Dept. of BioStructure and Function, Univ. of Connecticut Health Center, Farmington, CT 06030-3705 (E-mail:
bronner{at}sun.uchc.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 30 October 2000; accepted in final form 16 February 2001.
 |
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