Membrane Biology: Electrogenic Properties of The Sodium Pump in A Dynamic Model of Membrane Transport
Membrane Biology: Electrogenic Properties of The Sodium Pump in A Dynamic Model of Membrane Transport
Membrane Biology: Electrogenic Properties of The Sodium Pump in A Dynamic Model of Membrane Transport
Membrane Transport
J.A. Hernandez
1
, S. Chifflet
2
1
Seccion Biof sica, Facultad de Ciencias, Univ. de la Republica, Igua esq. Mataojo, 11400 Montevideo, Uruguay
2
Depto. de Bioqu mica, Facultad de Medicina, Univ. de la Republica, Gral. Flores 2125, 11800 Montevideo, Uruguay
Received: 8 September 1999/Revised: 21 March 2000
Abstract. The general purpose of this theoretical work is
to contribute to understand the physiological role of the
electrogenic properties of the sodium pump, by studying
a dynamic model that integrates diverse processes of
ionic and water transport across the plasma membrane.
For this purpose, we employ a mathematical model that
describes the rate of change of the intracellular concen-
trations of Na
+
, K
+
and Cl
and water
channels, and systems mediating K:Cl and Na:Cl co-
transport processes. The sodium pump is incorporated
via an explicit kinetic diagram [a modification of the one
analyzed by Chapman et al. (1983)]; the uncoupled ionic
fluxes of Na
+
, K
+
and Cl
and
water channels, and K:Cl and Na:Cl cotransporters in its
plasma membrane (see Fig. 1A; see also Introduction).
The changes in the cell volume (V
c
) are determined by
the net water movement between the extracellular and
intracellular compartments, as a response to the intracel-
lular osmolarity changes. The total solute concentration
of the extracellular compartment remains constant, at an
isosmotic value. The cell contains a fixed amount of an
impermeant anion (X
i
) which, for simplicity, we consider
to be monovalent. We assume that the total intracellular
osmolarity is given by the sum of the concentrations of
Na
+
, K
+
, Cl
and of the Na
+
:K
+
:2Cl
symports
(Baumgarten & Feher, 1995)]. In our previous study
(Hernandez & Cristina, 1998) these transport processes
were considered to remain inactive under basal condi-
tions and were only dramatically triggered to large levels
of activation as a result of sudden cell volume changes
determined by anisosmotic shocks. As a major differ-
ence with that work, in the present study the K:Cl and/or
Na:Cl cotransport systems may remain active only at
basal values. Also, for this study, the model does not
include processes (e.g., the K:Cl and Na:Cl cotransport-
ers, or other mechanisms) mediating short-term cell vol-
ume regulation (Baumgarten & Feher, 1995). Finally,
we assumed that the total membrane area available for
solute and water transport (A
c
) remains constant, and
independent of the cell volume changes.
Under these assumptions, the following mathemati-
cal model governs the rate of change of the cell volume
(V
c
), of the intracellular amounts (e.g., in moles) of Na
+
,
K
+
and Cl
(n
Na
, n
K
and n
Cl
), and of the electrical po-
tential difference across the plasma membrane (V
m
, de-
fined as V
m
V
intracellular
V
extracellular
):
dn
Na
dt = A
c
3J
p
+ J
Na
+ J
NaCl
dn
K
dt = A
c
2J
p
+ J
K
+ J
KCl
(1a)
dn
Cl
dt = A
c
J
Cl
+ J
NaCl
+ J
KCl
dV
c
dt = A
c
V
w
P
w
X
i
+ n
Na
+ n
K
+ n
Cl
V
c
e
and N
are
given by the modified Goldman expression (Goldman,
1943; Hodgkin & Katz, 1949):
J
Na
= P
Na
m
Na
+
e
exp u2 n
Na
V
c
exp u2
J
K
= P
K
m
K
+
e
exp u2 n
K
V
c
exp u2 (2)
J
Cl
= P
Cl
m
Cl
e
exp u2 n
Cl
V
c
exp u2,
where u F V
m
/(R T) and
m
u/[exp (u/2) exp
(u/2)], and where F is Faradays constant, R the gas
constant, T the absolute temperature (310 kelvin), and P
i
the ionic permeability coefficient (i Na
+
, K
+
, Cl
).
We assumed that the symport-mediated fluxes of
KCl and NaCl are given by
J
NaCl
= Q
NaCl
Na
+
e
Cl
e
n
Na
n
Cl
V
c
2
(3)
J
KCl
= Q
KCl
K
+
e
Cl
e
n
K
n
Cl
V
c
2
,
where Q
NaCl
and Q
KCl
are characteristic parameters
(Table 1).
In agreement with its main specific objective (see
Introduction), this theoretical study is not intended to be
applied to any specific cell type. In the following section
we perform some numerical studies of the model, in-
tended to illustrate some basic stationary and dynamic
properties of cells following the general design described
above (see Fig. 1A).
Results and Discussion
NUMERICAL METHODS
The methods employed in this work are similar to those
in our previous work (Hernandez & Cristina, 1998), and
are summarized here. To perform the simulations, Eqs.
(1) were integrated numerically employing the Runge-
Kutta fourth order method, except for the determination of
V
m
. After each integration time step, we determined V
m
assuming Eq. (1b), as a solution of the transcendental equa-
tion (Hernandez et al., 1989; Hernandez & Cristina, 1998):
V
m
= (RTF) ln [( + )( + )] (4)
In this equation, the functions (V
m
),
(V
m
), (V
m
) and (V
m
) are given by
= P
Na
Na
+
e
+ P
K
K
+
e
+ P
Cl
n
Cl
V
c
m
= N a
12
a
23
a
34
a
45
a
56
a
61
= P
Na
n
Na
V
c
+ P
K
n
K
V
c
+ P
Cl
Cl
m
= N a
21
a
32
a
43
a
54
a
65
a
16
with
m
defined by Eqs. (2) and with and the a
ij
s
defined in the Appendix.
Similarly, the steady-state values of the variables
were determined by the iterative procedure employed
previously (Hernandez & Cristina, 1998).
Within the context of this work we considered that
the electrogenic contribution of the sodium pump is
given by the difference between the actual membrane
potential V
m
[Eq. (4)] and the corresponding diffusion
potential V
m
diff
:
V
m
diff
(RT/F) ln (/) (5)
Equation (5), obtained from Eq. (4) by setting N
0, is actually the Goldman-Hodgkin-Katz (Goldman,
1943; Hodgkin & Katz, 1949) explicit equation for the
diffusion potential.
In every run, as a control test, the simulation pro-
gram checked the simultaneous satisfaction of the con-
ditions of osmotic equilibrium and of macroscopic elec-
troneutrality:
(X
i
+ n
Na
+ n
K
+ n
Cl
)/V
c
e
and X
i
+ n
Cl
= n
Na
+ n
K
. (6)
To perform the dynamic studies, transient responses
of the model were produced by perturbing some of the
Table 1. Glossary of symbols
(I) Variables
V
c
: cell volume
n
Na
, n
K
, n
Cl
: intracellular amounts of Na
+
, K
+
and Cl
V
m
: electrical potential difference across the cell membrane
t: time
(II) Parameters
A
c
: effective permeant area of the cell surface
P
Na
, P
K
, P
Cl
: permeability coefficients of Na
+
, K
+
and Cl
P
W
: osmotic permeability
Q
NaCl
, Q
KCl
: kinetic parameters of the symport-mediated
transports of NaCl and KCl
: time constant of parameter modifications
(Na
+
)
e
, (K
+
)
e
, (Cl
)
e
, (X)
e
: extracellular concentations of Na
+
,
K
+
, Cl
(X)
e
: extracellular concentration of impermeant solute
e
: total extracellular solute concentration
X
i
: total amount of intracellular impermeant solute
N: total NaK ATPase membrane density
(ATP), (ADP), (P
i
): cellular concentrations of ATP, ADP and
inorganic phosphate
k
12
, . . . , k
61
: rate constants of transitions 12, . . . , 61
k
16
, . . . , k
21
: rate constants of transitions 16, . . . , 21
K
eq
: equilibrium (dissociation) constant of the reaction
ATP ADP + P
i
44 J. Hernandez and S. Chifflet: Electrogenic Properties of the Sodium Pump
characteristic parameters (Table 1). These parameter
modifications may or may not have an actual experimen-
tal counterpart; as commented above, the purpose is not
to mimic particular behaviors but to understand basic
properties. We assumed that any modification (activa-
tion or inhibition) of the ionic fluxes mediated by the
sodium pump and by the cotransporters followed a time
course given by
Y(t) Y
[1 exp (t/)] + Y
0
exp (t/), (7)
where Y
0
and Y
n
K
/V
c
, and (Cl
)
i
n
Cl
/V
c
.
REFERENCE STATE
We determined reference states for four different cases
(Cells 0, 1, 2 and 3, Table 2). In Cells 0 and 3 the
systems mediating the K:Cl and Na:Cl cotransport pro-
cesses remain inactive (that is, Q
NaCl
Q
KCl
0), in
Cells 1 and 2 they exhibit their basal activities. Cells 0
and 1 exhibit larger ionic permeabilities than Cells 2 and
3. Unless specified, the numerical values of the param-
eters employed for the simulations were the ones shown
in Table 2. They mostly correspond to the ones em-
ployed in our previous work (Hernandez & Cristina,
1998), except for Q
NaCl
, Q
KCl
, the ionic permeabilities
(for the cases of Cells 2 and 3), and for X
i
, that was
estimated here for the case of a smaller cell [e.g., for a
cell with an average volume of 10
9
cm
3
(Jakobsson,
1980; Hernandez & Cristina, 1998)]. The membrane
area available for transport (A
c
) has also been modified
accordingly. The majority of the values listed in Table 2
have an experimental basis (Hernandez & Cristina,
1998), others (e.g., Q
NaCl
, Q
KCl
and ) were heuristically
determined in order to obtain a plausible behavior of the
model. The values for the rate constants of the enzymat-
ic reaction were taken from Chapman et al. (1983) and
from Wuddel & Apell (1995) (see Appendix).
For the parameter values listed in Table 2, the ref-
erence cellular states are determined by the correspond-
ing steady-state values of the variables:
Cell 0: V
c
(0): 1.00 10
9
cm
3
; V
m
(0): 5.45 10
2
V; V
m
diff
(0): 5.17 10
2
V; (Na
+
)
i
(0), (K
+
)
i
(0), (Cl
)
i
(0): 2.2 10
5
, 1.27 10
4
, 1.8
10
5
mol cm
3
Cell 1: V
c
(0): 1.04 10
9
cm
3
; V
m
(0): 5.17 10
2
V; V
m
diff
(0): 4.71 10
2
V; (Na
+
)
i
(0), (K
+
)
i
(0), (Cl
)
i
(0): 3.1 10
5
, 1.18 10
4
, 2.5
10
5
mol cm
3
Cell 2: V
c
(0): 1.09 10
9
cm
3
; V
m
(0): 5.40 10
2
V; V
m
diff
(0): 1.70 10
2
V; (Na
+
)
i
(0), (K
+
)
i
(0), (Cl
)
i
(0): 2.4 10
5
, 1.25 10
4
, 3.1
10
5
mol cm
3
Cell 3: V
c
(0): 1.62 10
9
cm
3
; V
m
(0): 1.83 10
2
V; V
m
diff
(0): 1.24 10
2
V; (Na
+
)
i
(0), (K
+
)
i
(0), (Cl
)
i
(0): 1.2 10
5
, 1.37 10
4
, 7.0
10
5
mol cm
3
As can be seen, Cells 0, 1 and 2 exhibit a basically
similar reference state, characterized by comparable val-
ues of the membrane potential, cell volume and intracel-
lular ionic concentrations. The major difference between
these cells concerns the mechanisms underlying the gen-
eration of the membrane potential. As revealed by the
particular values of V
m
diff
(0), while in Cells 0 and 1 the
membrane potential is basically determined by electro-
diffusion, Cell 2 maintains a similar V
m
, albeit the value
for V
m
diff
(0) is significantly lower. Since the ionic per-
meabilities are substantially smaller, the membrane po-
tential of Cell 2 is mainly determined by the electrogenic
activity of the sodium pump. Since Cl
is only subject to
passive transport and since Na
+
already exhibits a neg-
ligible permeability in Cell 1, the main difference be-
tween Cell 1 and Cell 2, from the point of view of the
electrical properties of the plasma membrane, is related
to the potassium permeability. For low ionic permeabili-
ties and in the absence of both the K:Cl and Na:Cl co-
transport processes (the case of Cell 3), the sodium pump
proves to be ineffective in maintaining a membrane po-
tential similar to the ones of the three other referential
cells.
Table 2. Numerical values of the parameters
A
c
: 5 10
6
cm
2
P
Na
, P
K
, P
Cl
: 7 10
8
, 7 10
7
, 10
6
cm sec
1
(Cell 0 and Cell 1)
2 10
8
, 3 10
8
, 2 10
8
cm sec
1
(Cell 2 and
Cell 3)
P
w
: 1.5 10
2
cm sec
1
Q
NaCl
, Q
KCl
: 0, 0 (Cell 0 and Cell 3)
7 10
4
, 5 10
3
cm
4
mol
1
sec
1
(Cell 1 and Cell 2)
: 14 sec
(Na
+
)
e
, (K
+
)
e
, (Cl
)
e
, (X)
e
: 1.4 10
4
, 10
5
, 1.4 10
4
, 10
5
mol cm
3
e
: 3 10
4
mol cm
3
X
i
: 1.3 10
13
mol
N: 1.25 10
13
mol cm
2
(ATP), (ADP), (P
i
): 5 10
6
, 6 10
8
, 4.95 10
6
mol cm
3
k
12
: 2.5 10
11
mol
3
lt
3
sec
1
; k
21
: 424563 sec
1
(*)
k
23
: 10
4
sec
1
; k
32
: 10
5
mol
1
lt sec
1
k
34
: 360 sec
1
; k
43
: 8.5 10
3
mol
3
lt
3
sec
1
k
45
: 1.5 10
7
mol
2
lt
2
sec
1
; k
54
: 2 10
5
mol
1
lt sec
1
k
56
: 2 10
6
mol
1
lt sec
1
; k
65
: 30 sec
1
k
61
: 1.15 10
4
sec
1
; k
16
: 6 10
8
mol
2
lt
2
sec
1
K
eq
: 239000 mol lt
1
(*) Determined from the detailed balance condition [Eq. (A4)].
45 J. Hernandez and S. Chifflet: Electrogenic Properties of the Sodium Pump
The finding of comparable reference states for Cells
0, 1 and 2 reveals the relevant role of electroneutral ionic
exchange (for the model analyzed here, net Na
+
:K
+
ex-
change) for the determination of a physiological V
m
in a
predominantly electrogenic fashion (as in Cell 2). In es-
sence, this is a consequence of the particular electrical
and biochemical properties of the sodium pump. The
consequences of these noteworthy properties of the so-
dium pump on the determination of the membrane po-
tential can be interpreted with the aid of Eq. (4). Under
conditions of a predominantly electrodiffusional mecha-
nism of generation of the V
m
(e.g., for a large potassium
permeability), V
m
becomes more electronegative with
(K
+
)
i
[Eq. (4)]. In this case (Cells 0 and 1) the enzyme
contributes to the physiological V
m
in a biochemical
fashion, by maintaining a large (K
+
)
i
. However, when
the enzymatic terms predominate (e.g., for low ionic per-
meabilities or relatively large pump activity, the case of
Cell 2) V
m
becomes more electropositive with (K
+
)
i
and,
correspondingly, with decreasing Na
+
)
i
[see Eq. (4)].
In electrical terms, this results from the fact that the
electrogenic current mediated by the sodium pump con-
sists of an outward flow of positive charges. Hence, in
this latter case, the effect of an increase in the intracel-
lular concentration of (K
+
)
i
and of a concomitant de-
crease in (Na
+
)
i
(a consequence of the predominance of
the pump activity) is a more depolarizing membrane po-
tential. The presence of a net electroneutral Na
+
:K
+
ex-
change (a consequence of simultaneous KCl and NaCl
symport-mediated transports, the case of Cell 2) prevents
large modifications in the intracellular concentrations of
these ions under conditions of predominance of the ac-
tive ionic fluxes, and thus permits the maintenance of V
m
at physiological values. For the case of low ionic per-
meabilities and absence of both the K:Cl and Na:Cl co-
transport processes (the case of Cell 3) the effects pro-
duced by the predominant sodium pump-mediated ionic
fluxes cannot be counterbalanced. As a consequence, in
Cell 3 both (Na
+
)
i
and (K
+
)
i
modify to larger extents than
in the case of Cell 2 (see the reference values). Since the
enzymatic terms nevertheless are dominant, a lower
(Na
+
)
i
and a larger (K
+
)
i
simultaneously contribute to a
more depolarizing (unphysiological) value of V
m
[Eq.
(4)]. Also for the case of Cell 3, a significant accumu-
lation of intracellular K
+
and Cl
)
i
, an increase in V
c
, and membrane de-
polarization. This agrees with classic experimental evi-
dence (Byrne & Schultz, 1988) and is similar to previous
model simulations of the behavior of animal cells em-
bodied with a large potassium permeability (that is, simi-
lar to Cell 1) (Jakobsson, 1980; Hernandez & Cristina,
1998). In particular, for the case of a complete inhibition
of the enzyme, the membrane depolarization consists of
two phases: a rapid initial one corresponding to the in-
hibition of the electrogenic component and a second
slow phase associated with the gradual changes of the
intracellular ionic concentrations (Hernandez & Cristina,
1998). The value reached at the depolarizing peak cor-
responds to the electrodiffusive membrane potential, and
is approximately determined by the initial ionic concen-
trations [Eq. (5)]. Inhibition to a lower degree (Fig. 2A)
is compatible with a steady state approximately similar
to the referential state of Cell 2. The transient only af-
fects the membrane potential and, similar to the case of
a large inhibition, also consists of two phases: a rapid
depolarization corresponding to the initial electrogenic
inhibition followed by a slower relaxation to the final
steady state. The rapid initial depolarization is a conse-
quence of the sudden perturbation of the membrane po-
tential provoked by the abrupt modification in the en-
46 J. Hernandez and S. Chifflet: Electrogenic Properties of the Sodium Pump
zyme density. The intracellular ionic concentrations
have not undergone any changes yet, therefore they cor-
respond to the initial values. Once the peak is reached,
the process undergoes a slower relaxation towards its
final steady state. Different from the case of the com-
plete inhibition of the enzyme (see above), the slower
relaxation exhibited by the partial inhibition consists in a
recovery of the resting membrane potential. For the case
shown in Fig. 2A, the partial enzyme inhibition deter-
mines a small increase in (Na
+
)
i
and a small decrease in
(K
+
)
i
, as a consequence the final membrane potential is
more electronegative than the depolarizing peak [Eq. (4)]
but not significantly different from the initial V
m
. The
effects of a partial inhibition of the sodium pump, just
described for Cell 2, are similar to those encountered
from the study of models of cells of the type of Cell 1
(Hernandez & Cristina, 1998). This dynamic response to
a partial inhibition of the sodium pump, therefore exhib-
ited both by Cells 1 and 2, may provide an alternative
explanation of actual experimental findings of spontane-
ous recovery of the membrane potential after the admin-
istration of sodium pump inhibitors (De Weer & Gedul-
dig, 1978; Brismar & Collins, 1993).
Figure 2B shows the effects produced on Cell 2 by a
fourfold activation of the sodium pump. The activation
of the enzyme produces the expected consequences on
the intracellular ionic concentrations: an increase in
(K
+
)
i
, a decrease in (Na
+
)
i
and almost no modifications in
(Cl
)
i
. V
c
also experiences an insignificant decrease, as a
consequence of a larger extrusion of osmotic particles by
the sodium pump. Again, the intermediate transient
characterizing the time course of V
m
can be interpreted in
similar terms as previously: the abrupt hyperpolarizing
peak reveals the activation of the electrogenic compo-
nent under initial conditions, while the subsequent slow
depolarizing relaxation is a consequence of the slower
gradual changes produced in (K
+
)
i
and (Na
+
)
i
by the
enzyme activation [Eq. (4)]. It must be noticed that the
rapid initial electrical response is basically a conse-
quence of the relatively small value assumed for the time
constant () of the enzyme modifications [Table 2, Eq.
(7)], the effect of larger values of on the dynamic
response has not been explored here. Figure 2B there-
fore reveals that the final steady-state value achieved for
V
m
is not significantly different from the initial one. This
is a consequence of the modifications produced on the
ionic concentrations. Under the electrogenic mode of
generation of V
m
, the increase in (K
+
)
i
and the decrease
in (Na
+
)
i
produced by the activation of the sodium pump
compensate the initial hyperpolarizing tendency [Eq.
(4)]. Analogous to the case of a partial inhibition (see
above), the effects produced on Cell 2 by activation of
the sodium pump within physiological values are also
similar to those exhibited by cell models of the type of
Cell 1 (Hernandez & Cristina, 1998).
The results shown in Fig. 2 therefore permit us to
conclude that the individual modifications of the sodium
pump determine changes on Cell 2 that are basically
similar to those produced on cells of the type of Cell 1.
A partial inhibition and an activation of the enzyme may
be compatible with the maintenance of nearly physi-
ological cellular steady states. In particular, no signifi-
cant modifications on the final steady-state membrane
potential were produced by modifications of the enzyme
Fig. 2. Dynamic responses of Cell 2 to a partial
inhibition (A) and activation (B) of the sodium
pump. At t 40 sec, the enzyme density
undergoes a change from its initial value N(0) (as
for N in Table 2) toward a final value N
[Eq.
(7)]. (A) Plots of (Na
+
)
i
, (K
+
)
i
, (Cl
)
i
, V
c
and V
m
as functions of time, for N
0.4 N(0). B
Similar to Fig. 2A, but for N
4 N(0).
47 J. Hernandez and S. Chifflet: Electrogenic Properties of the Sodium Pump
activity within a physiological range. However, this type
of modification determined a characteristic transient re-
sponse, in the form of a depolarizing (for the case of
partial inhibition) or a hyperpolarizing (for pump activa-
tion) electrical signal, both for a cell with the character-
istics of Cell 1 (Hernandez & Cristina, 1998) and for Cell
2 (this work). From these results we may conclude that,
even for a cell generating its membrane potential in an
electrogenic fashion, the electrogenic properties of the
sodium pump may mostly be revealed by the character-
istic short-duration electrical signals induced by modifi-
cations in the enzyme activity within physiological val-
ues.
Figure 3A shows the effects of a partial inhibition of
the K:Cl cotransporter (to one half of the basal value of
Q
KCl
) on Cell 2. The immediate consequence of the in-
hibition is an increase in the intracellular mass of Cl
.
The increase in (Cl
)
i
results in membrane depolarization
[Eq. (4)]. V
c
increases, thus compensating for the in-
crease in (Cl
)
i
by diluting the impermeant intracellular
anion. Figure 3B shows the effects of a twofold activa-
tion of the K:Cl cotransporter. The effects produced are
the opposite than in the case of inhibition: the activation
determines a net decrease in (Cl
)
i
, membrane hyperpo-
larization and a compensatory decrease in the cell vol-
ume. A partial inhibition of the Na:Cl cotransporter (to
one half of the basal value of Q
NaCl
, Fig. 4A) produces a
decrease in the intracellular mass of Na
+
and Cl
.
Electroneutrality imposes a reduction in V
c
, in order to
compensate for the decrease in (Cl
)
i
by concentrating
the impermeant intracellular anion. The consequent in-
crease in (K
+
)
i
and the simultaneous decrease in (Na
+
)
i
result, under the electrogenic mode, in a more depolar-
ized V
m
[Eq. (4)]. Figure 4B shows the effects produced
on Cell 2 by a twofold activation of the Na:Cl cotrans-
porter. This activation results in opposite effects to the
case of inhibition: an increment in V
c
, in order to com-
pensate for the increase in (Cl
)
i
, plus a decrease in (K
+
)
i
and a simultaneous increase in (Na
+
)
i
that determine,
under the electrogenic mode, a more hyperpolarized V
m
[Eq. (4)]. It is interesting to note that, due to the alter-
ations produced on (K
+
)
i
and (Na
+
)
i
, the modifications on
the Na:Cl cotransport process determine larger changes
on V
m
than similar relative modifications on the K:Cl
cotransporter. It is also noteworthy that modifications on
each one of the ionic cotransport processes determine
significantly larger changes on the cell volume than rela-
tively large modifications on the sodium pump activity
(compare Figs. 3 and 4 with Fig. 2).
The results obtained so far permit us to conclude that
modifications induced on some of the individual trans-
port systems of Cell 2 mostly produce either nonsignif-
icant effects on the final V
m
(e.g., partial inhibition or
activation of the sodium pump) or relatively important
changes on the membrane potential accompanied by sig-
nificant modifications on other variables, like the cell
volume (e.g., inhibition or activation of the individual
Na:Cl and K:Cl cotransport processes). We now illus-
trate the effects of modifications in the rate of net elec-
troneutral Na
+
:K
+
exchange. As mentioned above, this
exchange is the result of the simultaneous activity of
both the Na:Cl and K:Cl cotransport processes.
Figure 5 shows the effects produced on Cell 2 by a
simultaneous partial inhibition (to one half of the basal
value of Q
KCl
and of Q
NaCl
, respectively, Fig. 5A) and by
a simultaneous twofold activation (Fig. 5B) of both the
Fig. 3. Dynamic response of Cell 2 to a
modification in the K:Cl cotransporter activity. At
t 40 sec, Q
KCl
undergoes a change from its
initial value Q
KCl
(0) (as for Q
KCl
in Table 2)
toward a final value Q
KCl
[Eq. (7)]. (A) Plots of
(Na
+
)
i
, (K
+
)
i
, (Cl
)
i
, V
c
and V
m
as functions of
time, for the case of a 50% inhibition [Q
KCl
0.5 Q
KCl
(0)]. (B) Similar to Fig. 3A, but for a
twofold activation [Q
KCl
2 Q
KCl
(0)].
48 J. Hernandez and S. Chifflet: Electrogenic Properties of the Sodium Pump
K:Cl and Na:Cl cotransport processes. The inhibition of
the ionic cotransport processes (Fig. 5A) determines the
expected increase in (K
+
)
i
and simultaneous decrease in
(Na
+
)
i
(upper panel). These modifications result in a
more depolarizing membrane potential (Fig. 5A, lower
panel), as a consequence of the predominance of the
enzymatic terms in the determination of V
m
[Eq. (4)].
For the case of Cell 1, characterized by a predominantly
electrodiffusional mechanism of generation of the mem-
brane potential, an analogous perturbation in the cotrans-
port processes does not determine changes in V
m
(not
shown). It is noteworthy that the simultaneous inhibition
of the cotransporters does not determine significant
modifications in the cell volume, either for the case of
Cell 1 (not shown) or Cell 2 (Fig. 5A, lower panel).
Both in Cell 1 (not shown) and in Cell 2 (Fig. 5B) the
Fig. 4. Dynamic response of Cell 2 to a
modification in the Na:Cl cotransporter activity.
At t 40 sec, Q
NaCl
undergoes a change from its
initial value Q
NaCl
(0) (as for Q
NaCl
in Table 2)
toward a final value Q
NaCl
[Eq. (7)]. (A) Plots of
(Na
+
)
i
, (K
+
)
i
, (Cl
)
i
, V
c
and V
m
as functions of
time, for the case of a 50% inhibition [Q
NaCl
0.5 Q
NaCl
(0)]. (B) Similar to Fig. 4A, but for a
twofold activation [Q
NaCl
= 2 Q
NaCl
(0)].
Fig. 5. Dynamic responses of Cell 2 to a
simultaneous partial inhibition (A) and
simultaneous activation (B) of the K:Cl and Na:Cl
cotransport processes. At t 40 sec Q
KCl
and
Q
NaCl
undergo changes from their initial values
Q
KCl
(0) and Q
NaCl
(0) (as for Q
KCl
and Q
NaCl
in
Table 2) toward final values Q
KCl
= 0.5 Q
KCl
(0)
and Q
NaCl
= 0.5 Q
NaCl
(0) (A) and final values
Q
KCl
= 2 Q
KCl
(0) and Q
NaCl
2 Q
NaCl
(0) (B) [Eq. (7)]. The figure shows plots of (Na
+
)
i
,
(K
+
)
i
, (Cl
)
i
, V
c
and V
m
as functions of time.
49 J. Hernandez and S. Chifflet: Electrogenic Properties of the Sodium Pump
simultaneous activation of the ionic cotransporters deter-
mines a decrease in (K
+
)
i
and a simultaneous increase in
(Na
+
)
i
. Also, both cells experience a very small increase
in the cell volume associated with a slight increase in
(Cl
)
i
, in turn a consequence of a somewhat larger rate of
Na:Cl cotransport. In the case of Cell 1 the membrane
potential only experiences a negligible depolarization
(not shown). Under the electrogenic mode of generation
of the membrane potential, the changes in the intracel-
lular concentrations of K
+
and Na
+
experienced by Cell
2 (Fig. 5B) determine a significant hyperpolarization
[Eq. (4)]. This result is consistent with those obtained by
Jacob et al. (1984) from the analysis of a steady-state
model generalizing the classic Mullins-Noda phenom-
enological approach (Mullins & Noda, 1963).
From the above we may conclude that the simulta-
neous modifications in both ionic cotransporters produce
larger changes on the membrane potential of Cell 2 than
similar independent modifications in each one of the co-
transport processes (compare Fig. 5 with Figs. 3 and 4).
Also, the conservation of the osmotic balance implicit in
the net K
+
:Na
+
exchange determines that the changes
produced on the cell volume are significantly smaller.
Therefore, the modification in the rate of net K
+
:Na
+
exchange represents a good candidate for the modulation
of the membrane potential under the electrogenic mode
of generation. However, for the case of a cell embodied
with a large potassium permeability (e.g., as Cell 1), the
modifications in the rate of K
+
:Na
+
exchange do not
produce significant changes in the membrane potential
(see discussion above). As shown above, for the case of
Cell 2, the modifications of the sodium pump activity
within the physiological range do not determine signifi-
cant final changes in the membrane potential (see Fig. 2).
Figure 6 summarizes these ideas, by comparing the final
steady-state values achieved by Cell 2 as a function of
the rate of K
+
:Na
+
exchange (Fig. 6A) and of the sodium
pump activity (Fig. 6B). As can be seen, the main dif-
ference refers to the membrane potential: while V
m
hy-
perpolarizes linearly to large extents with the rate of
ionic exchange, it experiences almost no modification
with the pump density within physiological val-
ues. Taken together (see also Reference States), these
results are consistent with the idea suggested by some
authors (Bashford & Pasternak, 1986) that electroneutral
ionic exchange may play a relevant role in the determi-
nation of the membrane potential in an electrogenic fash-
ion (see Introduction).
Conclusions
In summary, the results of this study permit us to con-
clude that, for the cell model considered, the mainte-
nance of the plasma membrane potential at nearly physi-
ological values by means of the electrogenic contribution
of the sodium pump requires the simultaneous existence
of a relatively large rate of electroneutral Na
+
:K
+
ex-
change and of a low potassium permeability. As men-
tioned, these results are consistent with experimental
findings about this type of cell (Bashford & Pasternak,
1986; Ishida & Chused, 1993). From the analysis per-
Fig. 6. Plots of the steady-state values of (Na
+
)
i
,
(K
+
)
i
, (Cl
)
i
, V
c
, V
m
, and V
m
diff
as functions of the
cotransporter activity ratio (A) and the sodium
pump density (B), for the case of Cell 2. The
cotransporter activity ratio (r) is defined as: r
Q
KCl
/Q
KCl
(0) Q
NaCl
/Q
NaCl
(0), where
Q
KCl
and Q
NaCl
are the modified values and
where Q
KCl
(0) and Q
NaCl
(0) are the initial values
(as in Table 2) of Q
KCl
and Q
NaCl
, respectively.
50 J. Hernandez and S. Chifflet: Electrogenic Properties of the Sodium Pump
formed here we may conclude that the main reason for
these requirements is that the Na
+
:K
+
exchange prevents
large modifications in the intracellular concentrations of
these ions when the enzyme-mediated ionic fluxes be-
come prevailing. In this way, a cell can maintain similar
macroscopic properties (membrane potential, cell vol-
ume, intracellular ionic concentrations) in the presence
of different mechanisms of generation of the membrane
potential (predominantly electrogenic or predominantly
electrodiffusional). The numerical studies of the model
also show that, under the electrogenic mode of genera-
tion of the membrane potential, modifications in the so-
dium pump activity within physiological values do not
determine significant changes in the steady-state mem-
brane potential. The electrogenic properties of the en-
zyme are specially revealed in the characteristic short-
duration electrical responses in the transient behavior.
For the particular case of a partial inhibition of the so-
dium pump, the biphasic curve conforming the transient
consists in a rapid depolarization followed by a slower
recovery of the resting membrane potential. This behav-
ior is similar to that occasionally encountered in some
actual experimental systems, where a depolarization in-
duced by an inhibition of the enzyme is followed by a
spontaneous recovery of the membrane potential (De
Weer & Geduldig, 1978; Brismar & Collins, 1993). Fi-
nally, our theoretical study also suggests that, for the
case of nonexcitable cells maintaining a predominantly
electrogenic membrane potential, the most efficient
mechanism for regulating the steady-state membrane po-
tential may consist in modifications in the rate of Na
+
:K
+
exchange. For the model studied here, small modifica-
tions in this rate determine relatively larger changes in
the membrane potential and less significant changes in
the cell volume than similar relative modifications in the
individual transport systems.
This work was supported by grants from the Programa para el Desar-
rollo de las Ciencias Basicas (PEDECIBA) and from the Comision
Sectorial de Investigacion Cient fica (CSIC) de la Universidad de la
Republica, Uruguay.
References
Bashford, C.L., Pasternak, C.A. 1985. Plasma membrane potential of
neutrophils generated by the Na
+
pump. Biochim. Biophys. Acta
817:174180
Bashford, C.L., Pasternak,C.A. 1986. Plasma membrane potential of
some animal cells is generated by ion pumping, not by ion gradi-
ents. Trends Biochem. Sci. 11:113116
Baumgarten, C.M., Feher, J.J. 1995. Osmosis and the regulation of the
cell volume. In: Cell Physiology. Source Book. N. Sperelakis, edi-
tor. pp. 180211. Academic Press, New York
Borst-Pauwels, G.W.F.H. 1993. Mutual interaction of ion uptake and
membrane potential. Biochim. Biophys. Acta 1145:1524
Brismar, T., Collins, V.P. 1993. Effect of external cation concentration
and metabolic inhibitors on membrane potential of human glial
cells. J. Physiol. 460:365383
Byrne, J.H., Schultz, S.G. 1988. An Introduction to Membrane Trans-
port and Bioelectricity. pp. 6692. Raven Press, New York
Chapman, I.B., Johnson, E.A., Kootsey, J.M. 1983. Electrical and bio-
chemical properties of an enzyme model of the sodium pump. J.
Membrane Biol. 74:139153
De Weer, P., Geduldig, D. 1978. Contribution of sodium pump to
resting potential of squid giant axon. Am. J. Physiol. 235:C55C62
Goldman, D.E. 1943. Potential, impedance and rectification in mem-
branes. J. Gen. Physiol. 27:3760
Gradmann, D., Tittor, J., Goldfarb, V. 1982. Electrogenic Cl
pump in
Acetabularia. Phil. Trans. R. Soc. Lond. B 299:447457
Hernandez, J.A., Cristina, E. 1998. Modeling cell volume regulation in
nonexcitable cells: the roles of the Na
+
pump and of cotransport
systems. Am. J. Physiol. 275:C1067C1080
Hernandez, J.A., Fischbarg, J., Liebovitch, L.S. 1989. Kinetic model of
the effects of electrogenic enzymes on the membrane potential. J.
Theor. Biol. 137:113125
Hill, T.L. 1977. Free Energy Transduction in Biology. pp. 132. Aca-
demic Press, New York
Hodgkin, A.L., Katz, B. 1949. The effect of sodium ions on the elec-
trical activity of the giant axon of the squid. J. Physiol. 108:3777
Ishida, Y., Chused, T.M. 1993. Lack of voltage sensitive potassium
channels and generation of membrane potential by sodium potas-
sium ATPase in murine T lymphocytes. J. Immunol. 151:610620
Jacob, R., Piwnica-Worms, D., Horres, C., Lieberman, M. 1984. Theo-
retical effects of transmembrane electroneutral exchange on mem-
brane potential. J. Gen. Physiol. 83:4756
Jacquez, J.A. 1971. A generalization of the Goldman equation, includ-
ing the effects of electrogenic pumps. Math. Biosci. 12:185196
Jakobsson, E. 1980. Interactions of cell volume, membrane potential,
and membrane transport parameters. Am. J. Physiol. 238:C196
C206
Kabakov, A.Y. 1994. The resting potential equations incorporating
ionic pumps and osmotic concentration. J. Theor. Biol. 169:5164
Lauger, P. 1991. Electrogenic Ion Pumps. pp. 314. Sinauer Associ-
ates, Sunderland, MA
Lemieux, D.R., Roberge, F.A., Savard, P. 1990. A model study of the
contribution of active Na-K transport to membrane repolarization in
cardiac cells. J. Theor. Biol. 142:133
Lemieux, D.R., Roberge, F.A., Joly, D. 1992. Modeling the dynamic
features of the electrogenic Na,K pump of cardiac cells. J. Theor.
Biol. 154:335358
Moreton, R.B. 1969. An investigation of electrogenic sodium pump in
snail neurons using the constant field theory. J. Exp. Biol. 51:181
201
Movileanu, L., Flonta, M.L., Mihailescu, D., Frangopol, P.T. 1998.
Characteristics of ionic transport processes in fish intestinal epithe-
lial cells. BioSystems 45:123140
Mullins, N.J., Noda, K. 1963. The influence of sodium-free solutions
on the membrane potential of frog muscle fibers. J. Gen. Physiol.
47:117132
Rakowski, R.F., Bezanilla, P., De Weer, P., Gadsby, D.C., Holmgren,
M., Wagg, J. 1997a. Charge translocation by the Na/K pump. Ann.
N.Y. Acad. Sci. 834:231243
Rakowski, R.F., Gadsby, D.C., De Weer, P. 1997b. Voltage depen-
dence of the Na/K pump. J. Membrane Biol. 155:105112
Schultz, S.G. 1980. Basic Principles of Membrane Transport. Cam-
bridge University Press, Cambridge, UK
Scriven, D.R. 1981. Modeling repetitive firing and bursting in a small
unmyelinated nerve fiber. Biophys. J. 35:715730
Segel, G.B., Simon, W., Lichtman, M.A. 1979. Regulation of sodium
51 J. Hernandez and S. Chifflet: Electrogenic Properties of the Sodium Pump
and potassium transport in phytohemagglutinin-stimulated human
blood lymphocytes. J. Clin. Invest. 64:834841
Severini, A., Prasad, K.V.S., Almeida, A.F., Kaplan, J.G. 1987. Regu-
lation of the number of K
+
, Na
+
-pump sites after mitogenic acti-
vation of lymphocytes. Biochem. Cell Biol. 65:95104
Sjodin, R.A. 1984. Contribution of electrogenic pumps to resting mem-
brane potentials: the theory of electrogenic potentials. In: Electro-
genic Transport. Fundamental Principles and Physiological Impli-
cations. M.P. Blaustein and M. Lieberman, editors. pp. 105127.
Raven Press, New York
Slayman, C.L. 1987. The plasma membrane ATPase of Neurospora: A
proton-pumping electroenzyme. J. Bioenerg. Biomembr. 19:120
Thomas, R.C. 1972. Electrogenic sodium pump in nerve and muscle
cells. Physiol. Rev. 52:563594
Tosteson, D.C., Hoffman, J.F. 1960. Regulation of cell volume by
active cation transport in high and low potassium sheep red cells. J.
Gen. Physiol. 44:169194
Wuddel, I., Apell, H.-J. 1995. Electrogenicity of the sodium transport
pathway in the Na,K-ATPase probed by charge-pulse experiments.
Biophys. J. 69:909921
Appendix
STEADY-STATE ANALYSIS OF THE NaK ATPASE
KINETIC MODEL
The transport of Na
+
and K
+
mediated by the NaK ATPase is described
by the diagram shown in Fig. 1B. For this diagram, k
12
, . . . , k
61
(k
16
,
. . . , k
21
) are the rate constants governing the corresponding transitions
in the clockwise (counterclockwise) direction. The analysis of this
diagram has been performed previously (Chapman, Johnson & Koot-
sey, 1983; Hernandez et al., 1989; Lemieux, Roberge & Savard, 1990),
we summarize the main results here. In the steady state the cycle flux
J
p
(considered positive in the clockwise direction) can be expressed,
employing the diagram method (Hill, 1977), as
J
p
= 1 (A1)
In this equation and are defined by
= N a
12
a
23
a
34
a
45
a
56
a
61
and = N a
21
a
32
a
43
a
54
a
65
a
16
(A2a)
with
a
12
= k
12
n
Na
V
c
3
a
23
= k
23
a
34
= k
34
a
45
= k
45
K
+
e
2
a
56
= k
56
ATP
a
61
= k
61
a
21
= k
21
a
32
= k
32
ADP
a
43
= k
43
Na
+
e
3
a
54
= k
54
P
i
a
65
= k
65
a
16
= k
16
n
K
V
c
2
(A2b)
is the sum of all the directional diagrams of the model, and is
therefore a function of all the rate constants and ligand concentrations
[see Hernandez et al. (1989) for explicit expression].
In their original work, Chapman et al. (1983) performed the nu-
merical studies assuming that step N
5
N
6
was voltage-dependent.
More recent evidence suggests that the main electrogenic steps corre-
spond to the successive bindings of the extracellular sodium ions
(Wuddel & Apell, 1995; Rakowski et al., 1997b). Since the extracel-
lular sodium concentration is a parameter of the model studied here
(Table 1), these successive steps (Fig. 1C) can be reduced to a single
step (e.g., to transition N
3
N
4
), governed by pseudo-first order rate
constants (Hill, 1977). Thus, if states N
and N
e
2
+ g
1f
g
3b
Na
+
e
+ g
2f
g
1f
.
(A2c)
The rate constants k
34
and k
43
are assumed here to depend on V
m
according to
k
34
= k
34
exp FV
m
2RT; k
43
= k
43
exp FV
m
2RT (A3)
where k
34
and k
43
are independent of V
m
.
For zero voltage, the parameter values employed by Wuddell &
Apell (1995) were
g
3f
: 1400 sec
1
; g
3b
: 14000 mol
1
lt sec
1
g
2f
: 700 sec
1
; g
2b
: 467 mol
1
lt sec
1
g
1f
: 4000 sec
1
; g
1b
: 14000 mol
1
lt sec
1
Substitution of these values in Eqs. (A2c) yields the values for
k
34
amd k
43
shown in Table 2.
The detailed balance condition imposes the following restriction:
K
eq
= k
12
k
23
k
34
k
45
k
56
k
61
k
65
k
54
k
43
k
32
k
21
k
16
(A4)
where K
eq
is the dissociation constant of the reaction ATP + H
2
O
ADP + P
i
.
The active fluxes of Na
+
and K
+
(positive in the inward direction)
are respectively equal to 3J
p
and 2J
p
.
52 J. Hernandez and S. Chifflet: Electrogenic Properties of the Sodium Pump