Modeling cell volume regulation in nonexcitable cells: the
roles of the Na+ pump and of
cotransport systems
Julio A.
Hernández and
Ernesto
Cristina
Sección Biofísica, Facultad de Ciencias, Universidad
de la República, 11400 Montevideo, Uruguay
 |
ABSTRACT |
The purpose of this study is to contribute to understanding the
role of
Na+-K+-ATPase
and of ionic cotransporters in the regulation of cell volume, by
employing a model that describes the rates of change of the
intracellular concentrations of
Na+,
K+, and
Cl
, of the cell volume, and
of the membrane potential. In most previous models of dynamic cellular
phenomena,
Na+-K+-ATPase
is incorporated via phenomenological formulations; the enzyme is
incorporated here via an explicit kinetic scheme. Another feature of
the present model is the capability to perform short-term cell volume
regulation mediated by cotransporters of KCl and NaCl. The model is
employed to perform numerical simulations for a "typical" nonpolarized animal cell. Basically, the results are consistent with
the view that the Na+ pump mainly
plays a long-term role in the maintenance of the electrochemical
gradients of Na+ and
K+ and that short-term cell volume
regulation is achieved via passive transport, exemplified in this case
by the cotransport of KCl and NaCl.
mathematical models; cell dynamics; electrochemical gradients
 |
INTRODUCTION |
MANY DYNAMIC PROPERTIES OF cells are determined by the
integrated functioning of cell membrane transport processes. As a
classic example, the generation of the electrical potential difference across the plasma membrane of animal cells
(Vm) is the
result of the interaction of several passive and active processes of ionic transport (8, 29, 49). Under steady-state conditions, the resting
Vm can be
approximated by rather straightforward physicochemical formulations of
the diffusive and electrogenic components (7, 17, 22, 24, 28, 41, 42,
53). However, a description of the temporal behavior is more complex
and usually consists of systems of several nonlinear differential
equations governing the rates of change of the membrane potential, of
the cell volume, and of some intracellular ionic concentrations (29,
33, 34, 50, 55). These systems are generally difficult to
analyze, and their study is mainly restricted to numerical simulations of transient and/or periodic changes in the membrane potential of excitable and nonexcitable cells. The complexity of these models is
a consequence of the fact that, in living cells, the transport of ions
across the cell membrane simultaneously affects the cell volume (via
effects on the intracellular osmolarity) and the
Vm (via the
generation of diffusive and/or electrogenic potentials) (29,
56). As one possible simplification, several authors have assumed that
the cell volume remains constant during the course of
electrical membrane phenomena (33, 34, 50). Still, this simplification
does not permit the mathematical difficulties to be completely
overcome, especially those emerging from the basic, nonlinear character
of the ionic flux equations, in which the voltage dependence is
determined by both linear and exponential terms (17, 24, 29, 49).
The
Na+-K+-ATPase
of the plasma membrane plays key roles in most dynamic responses of
cells. In excitable cells, the onset of action potentials and the
subsequent electrical recovery of the cell membrane occur by abrupt
changes in the electrical currents of
Na+ and
K+, in turn determined by steep
conductance changes and by the preexistence of electrochemical
gradients for these ions. For the case of nerve cells, it has been
classically recognized that the
Na+ pump mainly plays a long-term
role in the maintenance of the electrochemical gradients of
Na+ and
K+; the enzyme can indeed be
inhibited with no substantial effect on the development of action
potentials for a considerable period (e.g., see Ref. 8). On the
contrary, in cardiac cells, significant modification of the intra- and
extracellular concentrations of Na+ and
K+ may take place in the course of
action potentials (34). Hence, the
Na+ pump plays a crucial role in
the membrane repolarization that follows, to promptly restore ionic
concentrations to the normal physiological levels that afterward
guarantee the generation of a new action potential (33, 34). In both
excitable and nonexcitable cells, the
Na+ pump plays diverse other roles
associated with membrane and other cellular phenomena (6, 16, 47, 48),
among them the maintenance of cell volume (5, 8).
The mechanisms involved in the regulation of cell volume fall into two
main categories, long-term and short-term cell volume regulation (5,
20). Under normal isotonic conditions, all cells exhibit mechanisms to
counter the natural tendency to increase intracellular tonicity. This
tendency results from passive processes that can be basically
interpreted in terms of a Gibbs-Donnan equilibrium (5, 8). In animal
cells, characterized by distensible membranes, the passive tendency to
gain diffusible cations, and hence to increase the cell volume by
osmotically coupled water entry, is mainly counterbalanced by the
Na+ pump (5, 8). Under the
standard mode of operation characterized by a 3 Na+:2
K+:1 ATP stoichiometric ratio, the
enzyme extrudes one diffusible cation per cycle, thus contributing to
reducing the amount of osmotically active intracellular solutes. The
experimental evidence suggests that, under isotonic conditions, this
protective device provided by the
Na+ pump mainly represents a
"long-term" mechanism of regulating the cell volume (37). A
rather explicit model capable of describing the long-term dynamic
events that follow the "shutting off" of the
Na+ pump was advanced by Jakobsson
(29).
Short-term cell volume regulation induced by anisosmotic shocks
represents another characteristic example of dynamic phenomena in
cells. The acute regulatory responses of cells to ambient perturbations of the osmotic pressure are widely distributed throughout the entire
biological world (20, 31a, 62). Despite the large spectrum of ionic and
nonionic solutes involved, these responses exhibit common dynamic
features (5, 20). Basically, the cells respond to the cellular volume
increase produced by acute hyposmotic shocks by augmenting the efflux
of a highly concentrated intracellular species. Osmotic equilibrium is
thus achieved by intracellular water loss and concomitant cell volume
recovery [regulatory volume decrease (RVD)]. Conversely, a
hyperosmotic medium results in a compensatory entry of a diffusive
species, followed by osmotically driven water influx [regulatory
volume increase (RVI)]. Although the elucidation of the basic
role of short-term cell volume regulation under physiological
conditions and of the primary mechanisms determining it remain matters
of active investigation and discussion (3, 11, 20, 21,
45), it is nevertheless clear that this cellular response
constitutes a ubiquitous and highly conserved way of reacting against a
changing environment (9, 26, 31). The literature on the mathematical
modeling of short-term cell volume regulation is not abundant (see, for
instance, Ref. 58) and mainly consists of the incorporation of the
phenomenon into the description of the overall properties of transport
exhibited by specific cellular types (35, 55, 60, 61). A mathematical analysis of the dynamic aspects of the short-term regulatory response per se was performed by Weinstein (58) for the case of an epithelial cell transporting two electroneutral solutes and water. The author employed thermodynamic formalism to study the steady-state and time-dependent behavior of the system and to perform numerical calculations for a model of a urinary epithelium.
The purpose of this study is to contribute to understanding the role of
Na+-K+-ATPase
and of cotransport systems in the dynamic behavior of nonexcitable
symmetrical cells, particularly cell volume regulation. Although this
enzyme has been involved, directly or indirectly, in diverse cellular
processes (see above), we focus here on its direct effects on the ionic
concentrations, the cell volume, and the membrane potential. For this,
we introduce a mathematical model describing the rates of change of the
intracellular concentrations of
Na+,
K+, and
Cl
([Na+]i,
[K+]i,
and
[Cl
]i),
of the cell volume (Vc), and of
the Vm. Taken
together, these five physicochemical variables might constitute the
minimum set of variables necessary for an appropriate description of
the basic dynamic properties of symmetrical animal cells (29). In
contrast to most previous models of dynamic cellular phenomena (29, 35, 50, 55, 58, 60, 61), in which
Na+-K+-ATPase
is incorporated via phenomenological formulations, in the present work
the enzyme is incorporated via an explicit kinetic scheme (10). This
kinetic model has been employed, for instance, as a plausible
alternative to describe the electrogenic contribution of the
Na+ pump (22) and to analyze
repolarization in cardiac cells (34). Another feature of the cell model
studied here is the capability of performing the characteristic
responses of short-term cell volume regulation, RVD and RVI. In this
model, these responses are mediated by a cell swelling-induced
electroneutral flux of KCl (for RVD) and by a cell shrinkage-induced
electroneutral flux of NaCl (for RVI). In view of its complexity (see
above), the model is employed to illustrate, by means of numerical
simulations, the behavior of a "typical" nonpolarized animal
cell. Some particular issues considered here are
1) the dependence of the reference state (steady state) on the activity of the
Na+ pump,
2) the role of the enzyme and of the
cotransporters of KCl and NaCl in long-term and short-term cell volume
regulation, and 3) the enzyme
contribution to cell recovery after anisosmotic shocks. In general, the
results are consistent with the view that the enzyme mainly plays a
long-term role in the maintenance of the electrochemical gradients of
Na+ and
K+, whereas short-term cell volume
regulation is mainly handled by the passive transport systems. Also,
some results suggest that the electrogenic properties of the enzyme may
underlie signaling events in cells.
Glossary
Variables
Vc |
Cell volume
|
mNa,
mK,
mCl |
Intracellular masses of Na+,
K+, and
Cl
|
Vm |
Electrical potential difference across cell membrane
|
t |
Time
|
Parameters
Ac |
Effective permeant area of cell surface
|
PNa,
PK,
PCl |
Permeability coefficients of Na+,
K+, and
Cl
|
Pw |
Osmotic permeability
|
v ,
v+ |
"Threshold" values of cell volume
|
,
|
Kinetic parameters of cell volume-induced fluxes
|
[Na+]e, [K+]e, [Cl ]e, [X]e |
Extracellular concentrations of
Na+,
K+,
Cl , and impermeant solute
under isotonic conditions
|
|
Time delay of cell volume regulatory responses
|
e |
Total extracellular solute concentration under isotonic conditions
|
r |
Osmolarity ratio
|
Xi |
Total amount of intracellular impermeant solute
|
N |
Total
Na+-K+-ATPase
membrane density
|
[ATP]i,
[ADP]i, [Pi]i |
Intracellular concentrations of ATP, ADP, and
Pi
|
k12, · · · , k61 |
Rate constants of transitions
12, · · · , 61
|
k16, · · · , k21 |
Rate constants of transitions
16, · · · , 21
|
Keq |
Equilibrium (dissociation) constant of the reaction ATP ADP + Pi
|
 |
MATHEMATICAL MODEL |
The cell model employed here to derive the mathematical model has the
following general characteristics (Fig. 1;
see also introduction).

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Fig. 1.
Cell scheme representing diverse fluxes taking place across plasma
membrane (i, intracellular; e, extracellular). Scheme includes
Na+-K+-ATPase
with 3 Na+:2
K+ stoichiometric ratio, diffusive
paths for Na+,
K+,
Cl , and water (solid
arrows), and cell volume-induced NaCl and KCl fluxes (dashed arrows).
|
|
Assumption 1.
The cell is nonpolarized. In fact, several nonpolar cell types have
been demonstrated to exhibit short-term cell volume regulation (see,
for instance, Refs. 11, 19, 31).
Assumption 2.
The changes in the Vc are
determined by the net water movement between the extracellular and
intracellular compartments, as a response to the generation of osmotic
gradients across the cell membrane. The total solute concentration of
the extracellular compartment remains constant. The cell contains a
fixed intracellular amount of an anionic impermeant species
(Xi), which, for simplicity, we
assume to be monovalent. As an approximation, we assume that the total
intracellular osmolarity is given by the sum of the concentrations of
Na+,
K+,
Cl
, and the impermeant
species. We also assume ideal osmotic behavior for all the species. In
contrast to previous models in which the assumption of instant osmotic
equilibrium is employed to determine cell volume changes (29, 35, 55,
58), the present work considers the water flow explicitly.
Assumption 3.
The plasma membrane contains diffusive paths for
Na+,
K+, and
Cl
(ionic channels) and a
Na+ pump with a fixed 3 Na+:2
K+ stoichiometric ratio. In
APPENDIX A, we derive
expressions for the steady-state fluxes of
Na+ and
K+ mediated by the enzyme, from
the analysis of an explicit kinetic model (10, 22, 34) (Fig.
2). The employment of these expressions in
the overall dynamic model implicitly assumes that the enzyme reaction
achieves the steady-state condition on a different time scale than the
Vc or the ionic intracellular
concentrations. This has been a usual procedure in previous models of
macroscopic dynamic phenomena incorporating explicit kinetic schemes of
transport systems (33, 34, 59).

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Fig. 2.
State diagram of process of transport of
Na+ and
K+ mediated by
Na+-K+-ATPase
(after Ref. 10).
N1, · · · , N6 are
intermediate states of enzyme. Na+i,
K+i, Na+e
and K+e are intracellular and
extracellular Na+ and
K+, respectively.
|
|
Assumption 4.
Cell volume regulation is mediated by coupled fluxes of
K+ and
Cl
(for RVD) and of
Na+ and
Cl
(for RVI). In many
animal cells, RVD is mediated by the loss of
K+ and
Cl
, whereas RVI is achieved
by an interaction of transport systems leading to a net gain of
Na+,
K+, and
Cl
(for references, see
Refs. 5, 20). Although, in most cases, the individual ionic fluxes
appear to be uncoupled (2, 12, 25), we assume here that they are
coupled via electroneutral cotransport systems. Under a hyposmotic
shock, Vc increases. We assume
that, when Vc reaches a threshold
value v+
(v+ > reference
Vc), a sudden activation of
systems leading to a net efflux of KCl takes place. Conversely, we
assume that a shrinkage-induced influx of NaCl is triggered when
Vc reaches the threshold value v
(v
< reference
Vc). In previously published
models incorporating short-term cell volume regulation, the acute
responses are mediated by volume-induced stepwise modifications of the
basal ionic permeabilities (55, 58). We explore here a formulation in
which the regulatory paths are independent of the basal ionic channels
and remain inactive under reference conditions. These transport systems
are turned on by cell volume changes and, as mentioned, determine
electroneutral fluxes of KCl (for RVD) and NaCl (for RVI), in the form
of cotransport systems. Once activated, these systems are modulated by
the cell volume (see below). Indeed, there is evidence that some of
these cotransport systems are implicated in the regulatory responses of
diverse cell types (for references, see Ref. 5). The molecular mechanisms underlying these abrupt changes in the ionic fluxes and
their dependence on the cell volume are not considered here. A
plausible mechanism for determining modifications of the permeabilities mediating the regulatory responses is that the change in the
concentration of the intracellular macromolecules constitutes the
primary signal (38).
Assumption 5.
The total area of the cell surface available for solute and water
transport (Ac)
remains constant and independent of the changes in cell volume.
This approximation can be justified by assuming that all solute and
water transport processes are mediated by integral membrane proteins,
present in amounts that do not change in the course of cell volume
modifications.
Under assumptions
1-5,
the following mathematical model governs the rates of change of
Vc, of
mNa,
mK, and
mCl, and of
Vm (defined as
intracellular potential
extracellular
potential)
|
(1a)
|
To obtain the time dependence of
Vm, we employed
a stationary solution of the electroneutral condition
(APPENDIX B)
|
(1b)
|
In Eqs.
1a and 1b, the cycle flux
Jp is defined by
Eq. A1;
JNa,
JK, and
JCl are the
corresponding electrodiffusive fluxes, and
Na and
K are the cell volume-induced
fluxes of NaCl and KCl, respectively.
Vw is the partial
molar volume of water, and the rest of the symbols represent parameters
and variables listed in the Glossary.
From inspection of Eqs. 1a and 1b, one promptly notices that, under
nonregulatory conditions of the cell volume (that is, for
Na =
K = 0), the steady state
requires equivalence of the magnitudes of the active and passive fluxes
of Na+ and
K+, electrochemical equilibrium of
Cl
, and the condition of
osmotic equilibrium. As is the usual practice in physicochemical models
of integrated membrane transport processes, the model does
not include terms describing the development of transmembrane tension.
The basal electrodiffusive fluxes of
Na+,
K+, and
Cl
are given by the Goldman
expression (17), as modified by Hodgkin and Katz (24)
|
(2)
|
with u = FVm/RT
and
m = u/[exp(u/2)
exp(
u/2)] and where
F is Faraday's constant,
R is the gas constant, and
T is the absolute temperature in
kelvin.
We assume that the cell volume-induced fluxes mediating the short-term
regulatory responses are given by
|
(3)
|
For the physiological range of Vc
values, we assume that QNa and
QK depend on
Vc according
to
|
(4)
|
where
and
are characteristic parameters
(Glossary).
In the absence of cell volume regulatory mechanisms,
QK = QNa = 0 for any value of
Vc.
In the following section, we perform numerical studies of the model
employing values corresponding to a typical ideal animal cell. These studies are not exhaustive and are intended only to illustrate some basic properties.
 |
NUMERICAL RESULTS AND DISCUSSION |
Numerical methods.
To perform the simulations, Eqs. 1a and 1b were integrated numerically,
employing the Runge-Kutta fourth-order method, except for the
determination of
Vm. As mentioned
above, after every time step,
Vm was calculated
assuming Eq. 1b by employing a
stationary approximation including the electrogenic component (Ref. 22; APPENDIX B). Besides the conditions
given by Eqs. 3 and 4, a time delay (
) was introduced
in the activation of short-term cell volume regulation. If
t is the instant at which
Vc reaches the corresponding
threshold value, the induced response shall be triggered at
t +
. Time delays have been
employed as auxiliary tools for the study of metabolic pathways in the
absence of a detailed knowledge of the kinetic properties of the
intermediate enzymes (4, 39). For the case of short-term cell volume
regulation, the activation of the membrane transporters that determine
the regulatory responses requires a complex and incompletely understood chain of intracellular signaling events, mediated by multienzymatic paths [e.g., the phosphatase and kinase cascades (44)]. A
time lag ranging from a few seconds to some minutes is to be expected between a perturbation in the concentration of the initial substrate of
an enzymatic path and its effect on a final product (14, 15). In
concordance with these ideas, we arbitrarily assumed here a
of 20 s
for the activation of short-term cell volume regulation (Table
1), which is characteristic of a relatively fast response. A detailed knowledge of the intermediate enzymes of the
signaling paths will permit evaluation of the time delay in terms of
the kinetic properties of the enzymes (14, 39).
For the case of osmotic shocks determined by anisosmotic values of
e
([
e]an),
the extracellular ionic concentrations were corrected according to
([ion]e)an = r[ion]e, where r = (
e)an/
e. The osmotic shocks were thus determined not solely by the modification of the ambient concentration of a nontransported ("inert")
solute, but via modifications of the concentrations of all the
extracellular ionic species.
For the particular conditions, the steady-state values of the variables
were initially obtained from the corresponding time integrations and
confirmed by an iterative procedure (APPENDIX C). The dependent variables were plotted as the
absolute values of
Vm,
Vc, and the corresponding ionic
concentrations
([Na+]i = mNa/Vc,
[K+]i = mK/Vc,
and
[Cl
]i = mCl/Vc).
In every run, the following control tests were performed to guarantee
the physicochemical consistency of the simulation. 1) Once a steady state is achieved,
the conditions of osmotic equilibrium and of macroscopic
electroneutrality must be simultaneously satisfied
|
(5a)
|
2) In the absence of
volume-induced regulatory fluxes (that is, for
Na =
K = 0), the steady-state values
of mCl,
Vc, and Vm must satisfy
the Nernst equilibrium for
Cl
|
(5b)
|
Reference state.
We determined reference states for two different cases
(cell 1 and cell
2, Table 1). Cell 1 contains a smaller amount of the fixed anion (and, as a consequence, a
larger amount of intracellular Cl
in the reference state;
see below) than cell 2. Unless
otherwise specified, the numerical values of the parameters employed
for the simulations were those shown in Table 1. Among these values, the osmotic permeability
(Pw)
corresponds to that determined for several animal cells (for
references, see Ref. 57). The basal ionic permeabilities
(PNa,
PK, and
PCl) are also
within the corresponding experimental ranges (e.g., see Ref. 54). As
can be seen, both cell 1 and
cell 2 are characterized by relatively
large PK and PCl and by a
lesser PNa. The
total amount of impermeant solute (Xi) is a reasonable estimate
for a cell with an average volume of
10
8
cm3 (29). The extracellular ionic
concentrations
([Na+]e,
[K+]e,
[Cl
]e,
and [X]e) were
arbitrarily fixed and approximately correspond to the usual isotonic
extracellular values (54) under physiological conditions. Because
Cl
is the only diffusible
anion considered in the model, its extracellular concentration was set
slightly higher than the usual physiological value, to
compensate for other diffusible anions. The density of
Na+-K+-ATPase
(N) falls within the physiological
range (10, 18, 22), as also do the
[ATP]i,
[ADP]i, and
[Pi]i
(32). The equilibrium constant
(Keq) is
characteristic of the ATP hydrolysis (32). The rate constants of the
enzymatic reaction were taken from Chapman et al. (10). The values
chosen for the threshold cell volumes are arbitrary and characteristic
of a highly sensitive cell; actual control values could correspond to
an ~1-5% change in the reference volume of the cell (26, 46).
In this respect, studies performed with threshold values corresponding
to a less sensitive cell reveal behaviors basically similar to the ones
shown here, except that the volume recovery after anisosmotic shocks
occur (with close approximation) not to the initial reference values
but to the threshold ones (e.g., see Fig. 10). The other parameters of
the model were heuristically determined by a trial-and-error method, to
obtain a plausible behavior of the model. For the parameter values
listed in Table 1, the reference values of the variables satisfy the
steady state under the isotonic condition
(
e = 3.0 × 10
4
mol/cm3).
The reference values of variables for cell
1 were Vc(0), 1 × 10
8
cm3;
Vm(0),
4.1 × 10
2 V;
[Na+]i(0),
1.84 × 10
5
mol/cm3;
[K+]i(0),
1.32 × 10
4
mol/cm3; and
[Cl
]i(0),
3.02 × 10
5
mol/cm3.
For cell 2, they were
Vc(0), 1 × 10
8
cm3;
Vm(0),
5.2 × 10
2 V;
[Na+]i(0),
1.65 × 10
5
mol/cm3;
[K+]i(0),
1.33 × 10
4
mol/cm3; and
[Cl
]i(0),
2.01 × 10
5
mol/cm3.
Because the study is mainly illustrative, these values do not
correspond to any specific cell type. For a roughly spherical cell, a
Vc of
10
8
cm3 would approximately correspond
to a cell diameter of 25 µm. Together with the rest of the values
corresponding to the reference states and parameters (Table 1), these
values could approximately characterize, for instance, some cells of
the hematopoietic lines, like large lymphocytes or granulocytes (27,
51, 52).
Effect of
Na+-K+-ATPase
on the reference state.
Figure 3,
A and
B, shows the dependence of the cell
steady state on N, for a physiological
range (18) and for the case of cell 2.
To obtain the final steady-state values, the model was run employing
the reference state as the initial state and subject to the
perturbation determined by the particular value chosen for
N. As mentioned, the values obtained
were then employed as the inputs of an iterative procedure to determine
the roots (APPENDIX C). In every
case, this procedure confirmed that the system had indeed reached the
steady state after the time run. Typically, the system achieved the
steady state at ~2,000 s. For lower values of
N, the behavior of the system was not
significantly different from the one corresponding to the complete
inhibition of the enzyme (see below).

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Fig. 3.
Plots of steady-state values of intracellular concentrations of
Na+,
K+, and
Cl
([Na+]i,
[K+]i,
and
[Cl ]i)
and of cell volume (Vc) and
electrical potential difference across the cell membrane
(Vm) vs. total
Na+-K+-ATPase
membrane density (N) under
nonregulatory (A) and regulatory
(B) conditions, for case of
cell 2.
|
|
For the interval considered, and in the absence of short-term cell
volume regulation (Fig. 3A),
[K+]i
increases and
[Na+]i
decreases with N. There is a small
variation in Vm,
since its steady-state value is basically given by the electrodiffusive component. For this reason and also due to the fact that in the absence
of cell volume regulatory mechanisms
[Cl
]i
is determined by electrochemical equilibrium, there is a negligible variation of
[Cl
]i
with N. As expected,
Vc increases with decreasing
N, since fewer diffusible cations are
being extruded from the cell. Although the global final effects are
basically similar, the variation of N
in the presence of short-term cell volume regulation (Fig. 3B) determines some distinctive
features. Because Vc increases with decreasing N, the mechanisms for
RVD are activated. The additional net efflux of KCl induced by this
activation produces a larger decrease of
[K+]i
than in the nonregulatory case and compensates for the increase in
[Na+]i.
For the interval of values of N
considered, the regulatory mechanisms succeed in maintaining
Vc at the reference value, despite a significant diminution of the enzyme density.
Effect of
Na+-K+-ATPase
inhibition under isotonic conditions.
The effect of the complete inhibition of the
Na+ pump on the time-dependent
behavior is shown in Fig. 4,
A and
B, for the case of
cell 2. In the absence of short-term
cell volume regulation (Fig. 4A),
the pump inhibition immediately produces the expected consequences: a
decrease in
[K+]i,
an increase in
[Na+]i
and
[Cl
]i,
and an increase in Vc and membrane
depolarization. These results are consistent with classic experimental
evidence (8) and with previous model simulations employing
phenomenological formulations of the enzyme (29). In the presence of
active mechanisms of short-term cell volume regulation (Fig.
4B), the cell is capable of
maintaining Vc at the reference
value until
[K+]i
becomes sufficiently low. At this instant, the regulatory flux of KCl
turns from efflux into influx,
[Cl
]i
starts to increase, and the fall of
[K+]i
becomes less steep. These results could provide a means to interpret
controversial evidence of cells in which the pump inhibition does not
produce an immediate increase in cell volume (3, 30, 36). As for the
nonregulatory case, however, the cell finally evolves toward a
Gibbs-Donnan equilibrium (see above). In both plots, the initial abrupt
depolarization step exhibited by the Vm curves (see
also below) corresponds to the sudden inhibition of the electrogenic
component of the enzyme.

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Fig. 4.
A: dynamic response of
cell 2 to a complete inhibition of
Na+ pump
(N = 0), for nonregulatory case.
[Na+]i,
[K+]i,
[Cl ]i,
Vc, and
Vm are plotted as
functions of time. Initial state: reference state.
B: analogous to
A, but for regulatory condition.
|
|
Restitution of the pump to its normal physiological density produces a
complete recovery of the cell to its reference steady state. In Fig.
5, we show the recovery of
cell 2 in the presence of short-term
cell volume regulation. In particular,
Vm gradually becomes more electronegative. The initial abrupt hyperpolarization exhibited by the
Vm curve
corresponds to the sudden turning on of the electrogenic component.

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Fig. 5.
Recovery of cell 2 after inhibition of
Na+ pump, for regulatory case.
Initial state:
[Na+]i,
[K+]i,
[Cl ]i,
Vc, and
Vm after 3,800 s
of complete pump inhibition (see Fig. 4).
N corresponded to physiological value
(Table 1).
[Na+]i,
[K+]i,
[Cl ]i
(top), Vc, and
Vm
(bottom) are plotted as functions of time.
|
|
Responses to anisosmotic shocks.
Figures 6 and 7
show the time course of the changes induced by anisosmotic shocks, as
obtained from the simulation of cell 1 subject to hypotonic (Fig. 6) and hypertonic (Fig. 7) perturbations of
the extracellular medium. In the presence of volume regulatory mechanisms, cell 1 exhibits the
typical dynamic responses corresponding to RVD (Fig. 6) and RVI (Fig.
7) (5, 20). When RVD is triggered (Fig.
6A), both
[K+]i
and
[Cl
]i
decrease; however, while
[K+]i
is restored by the pump activity,
[Cl
]i
remains at lower values.
Vm becomes more
electronegative, as a consequence of the fall in
[Cl
]i
and of the rather large Cl
permeability. After the transient phase of the regulatory response, Vc recovers to its reference
value. Throughout the response,
[Na+]i
remains approximately constant. When RVI is triggered (Fig. 7A), both
[Na+]i
and
[Cl
]i
increase. Vm
becomes more electropositive, as a consequence of the increase in
[Cl
]i.
Correspondingly,
[K+]i
experiences a slight fall. In the case shown,
Vc recovers to its reference
value. In Figs. 6B and
7B, one will notice that the acute
phase of short-term cell volume regulation is not affected by the
inhibition of the Na+ pump.
However, once the acute phase is over, the long-term effects described
above for the case of complete pump inhibition become noticeable (e.g.,
Vc slowly increases; other effects
not shown). Also, a concomitant activation (within physiological
values) of the Na+ pump did not
produce any effect on the short-term regulatory response (not shown),
either for RVD or for RVI. Figures 6B
and 7B also include the osmometer
behavior of cells in the absence of regulatory mechanisms, when an
increased (decreased) steady-state Vc results from exposure to a
hypotonic (hypertonic) medium.

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Fig. 6.
Response of cell 1 to a hyposmotic
shock ( e = 2.6 × 10 4
mol/cm3). Initial state:
reference state. A: plots of
[Na+]i,
[K+]i,
[Cl ]i
(top traces), Vc, and
Vm (bottom
traces) as functions of time, during regulatory response.
B: plots of
Vc as a function of time for 3 cases: 1) under nonregulatory
conditions (osmometric curve), 2)
under regulatory conditions and physiological
N [regulatory volume decrease
(RVD) curve], and 3) under
regulatory conditions and complete pump inhibition
(N = 0; superposed RVD curve).
|
|

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Fig. 7.
Similar to Fig. 6, but for a hyperosmotic shock
( e = 3.4 × 10 4
mol/cm3).
A: plots of
[Na+]i,
[K+]i,
[Cl ]i
(top traces), Vc, and
Vm (bottom
traces) as functions of time, during regulatory response.
B: plots of
Vc as a function of time for 3 cases: 1) under nonregulatory
conditions (osmometric curve), 2)
under regulatory conditions and physiological
N [regulatory volume increase
(RVI) curve], and 3) under
regulatory conditions and complete pump inhibition
(N = 0; superposed RVI curve).
|
|
Figures 8 and 9
show the temporal responses of cell 1 (Figs. 8A and
9A) and cell
2 (Figs. 8B and
9B) to anisosmotic shocks, for
different ambient osmolarities. For the case of hyposmotic shocks, both
cell 1 (Fig.
8A) and cell
2 (Fig. 8B) are
incapable of eliciting an appropriate RVD response at sufficiently low
osmolarities. Because, in this case, cell
1 nevertheless exhibits a better capability than
cell 2, we can relate these partial
responses to an insufficient availability of intracellular
Cl
. Similar conclusions
have been obtained from studies of previous models of cell volume
regulation mediated by Cl
permeability changes (55). Because both
Na+ and
Cl
are present at large
extracellular concentrations, neither cell 1 (Fig. 9A) nor
cell 2 (Fig.
9B) exhibits limits to the
capability to perform RVI. From these results, cell volume regulatory
responses by which the initial reference value is not completely
restored might occur by two different mechanisms:
1) with threshold values for the
regulatory response that are significantly different from the reference
Vc (for both RVD and RVI; see also
Reference state and Fig.
10) or
2) with partial intracellular
Cl
availability (for RVD,
in cells in which this response is mediated by volume-induced KCl
efflux). It has been suggested that incomplete cell volume recovery
from short-term regulation could play a role in cellular adaptation
(21).

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Fig. 8.
Responses of cell 1 (A) and cell
2 (B) to hyposmotic
shocks of increasing hypotonicity (from
bottom to
top in both
A and
B:
e = 2.75 × 10 4, 2.5 × 10 4, and 2.0 × 10 4
mol/cm3).
|
|

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|
Fig. 9.
Responses of cell 1 (A) and cell
2 (B) to
hyperosmotic shocks of increasing hypertonicity (from
top to
bottom in both
A and
B:
e = 3.25 × 10 4, 3.5 × 10 4, and 4.0 × 10 4
mol/cm3).
|
|

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|
Fig. 10.
Responses of cell 1 to hyposmotic
shocks (A and
C;
e = 2.6 × 10 4
mol/cm3) and hyperosmotic shocks
(B and
D;
e = 3.4 × 10 4
mol/cm3) for different threshold
values of cell volume (v ,
v+) and kinetic parameters of
cell volume-induced fluxes ( and
). Initial state: reference state.
From top to
bottom:
= 0.1, 1, and 10 cm4 · mol 2 · s 1
(A);
= 0.1, 0.01, and 0.001 cm4 · mol 2 · s 1
(B);
v and
v+ = 9 × 10 9 and 1.1 × 10 8, 9.9 × 10 9 and 1.01 × 10 8, and 9.99 × 10 9 and 1.001 × 10 8
cm3
(C); and
v and
v+ = 9.99 × 10 9 and 1.001 × 10 8, 9.9 × 10 9 and 1.01 × 10 8, and 9 × 10 9 and 1.1 × 10 8
cm3
(D).
|
|
Figure 10 shows the temporal responses of cell
1 to anisosmotic shocks for different values of
v
and
v+ and of the kinetic parameters
of the cell volume-induced fluxes (
and
). As expected, the regulatory
responses depend critically on the magnitude of the kinetic parameters
of the cotransport systems (Fig. 10, A
and B). As can also be seen, the
regulatory response drives the Vc
toward the corresponding threshold value (Fig. 10,
C and
D).
In Fig. 11, we show the recovery of
cell 1 to anisosmotic shocks in the
presence of short-term cell volume regulation. As can be seen,
restitution of the ambient osmolarity to the normal physiological value
results in a complete recovery of the cell to its reference steady
state. In the case of a complete inhibition of the
Na+ pump (not shown), the cell
cannot recover from an anisosmotic shock, and evolves in a fashion
similar to the isotonic case (see above). Notice that the RVD response
under recovery conditions (Fig.
11A) is steeper than the one
elicited by a hyposmotic shock (Fig. 6). This is basically due to the
fact that, for the recovery process, the initial
[Cl
]i
is significantly larger than in the case of the hyposmotic shock. This
more intense response determines a transit across the threshold cell
volume values (manifested by the "undershoot" in Fig.
11A,
bottom) and the consequent
activation of the RVI response after the corresponding time delay.

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Fig. 11.
A: recovery of cell
1 from a hyperosmotic shock. Initial state:
[Na+]i,
[K+]i,
[Cl ]i,
Vc, and
Vm after 2,000 s
of run after a hyperosmotic shock
( e = 3.4 × 10 4
mol/cm3; see Fig. 7). For run,
e was set at 3 × 10 4
mol/cm3 (isotonic).
B: similar to
A, but depicting recovery from a
hyposmotic shock (Fig. 6).
|
|
Activation of
Na+-K+-ATPase
under isotonic conditions.
In the long term, modification of N
within the physiological range does not determine significant changes
in Vm (Fig. 3). For those values of N, the overall
steady-state contribution of the enzyme to the membrane depolarization
is therefore small, mainly as a consequence of the dominant diffusive
component. The cellular response to the activation of the enzyme can be
examined more closely in the time-dependent behavior,
during the initial phases after the perturbations. In Fig.
12, we show the effects of pump
activation under nonregulatory (Fig.
12A) and regulatory (Fig.
12B) conditions of cell volume. For
both the nonregulatory and regulatory cases, after the initial abrupt
change toward hyperpolarization, Vm undergoes
relaxation to the reference state. Throughout the simulation,
Vc and the ionic concentrations
experience slight modifications.
[Na+]i
decreases and
[K+]i
increases slowly in response to enzyme activation;
[Cl
]i
remains basically unchanged. In particular, Fig.
12B reveals the sudden (although
small) change in Vc and
Vm determined by the activation of the mechanisms of cell volume regulation.

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Fig. 12.
Na+-K+-ATPase
activation (N = 2.5 × 10 13
mol/cm2) for
cell 2 under nonregulatory
(A) and regulatory
(B) conditions. Initial state:
reference state.
|
|
The inactivation of the Na+ pump
abruptly drives
Vm toward the
diffusive membrane potential (see initial phase in Fig. 4,
A and
B).
Vm afterwards
experiences the slow depolarization associated with the long-term ionic
concentration changes determined by the pump inhibition. Similar
results can be obtained by dramatic decreases in
[ATP]i (not shown).
The abrupt initial change in
Vm in response to
enzyme inhibition (or, equivalently, to ATP depletion) is consistent with experimental evidence about the electrogenic contribution of the
Na+ pump to the membrane potential
(13). This response can be interpreted in electrical terms as the
sudden shutting off of an electric current source (1). Analogously, the
abrupt hyperpolarizing response to enzyme activation (Fig. 12) can be
interpreted as the sudden turning on of the current source associated
with the electrogenic component. As can be seen in Fig. 12, this
electrogenic ionic current is sufficient to determine macroscopic
electrical effects of short duration (in the form of the transient
modification of
Vm) but does
not contribute to significant changes in the ionic concentrations or in
the Vc. For the physiological
values of N, the physicochemical conditions imposed by the overall dynamic model (e.g., tendency to
simultaneously achieve macroscopic electroneutrality and isotonic equilibrium, high Cl
and
K+ permeabilities, and so forth)
afterwards forces the relaxation of
Vm to values
similar to that corresponding to the reference state.
Taken together, the changes in N and
[ATP]i determine both
short- and long-term effects. The short-term effects (Figs. 4 and 12)
depend on modifications of the electrogenic component of the Na+ pump and consist of electrical
signals of short duration. Within certain intervals of
N and
[ATP]i values, no
significant long-term modifications (e.g., of the ionic concentrations
and consequently of the Vc and
membrane potential) take place (Figs. 3, 4, and 12). However, the same
changes in enzyme activity would determine a significant short-duration
signal. If they took place in actual cells, the short-term electrical
signals triggered in response to a slight activation of the enzyme
(Fig. 12) might play a role in intra- or intercellular signaling (43).
Thus the activation of the electrogenic component might, for instance,
mediate fast cellular responses to effectors that would act via the
Na+ pump, like certain hormones
(for references, see Ref. 21).
In conclusion, the model consistently describes properties associated
with the long-term role of
Na+-K+-ATPase.
For the physiological range of
Na+-K+-ATPase
densities, the reference state of the cell depends on whether the
volume regulatory mechanisms are present or not. In particular, in
the absence of short-term cell volume regulation, the cell volume
increases when the enzyme density decreases, since a gradually
declining number of diffusible cations is extruded from the cell. In
the presence of an active mechanism of short-term cell volume
regulation, and for the interval considered, the cell is capable of
maintaining its volume despite a decrease in the enzyme density. Some
authors suggest that, to assure proper macromolecular activity, it is
critical that the value of the cell volume be kept within a narrow
interval. If this is the case, our results suggest that
the permanent triggering of acute volume regulatory mechanisms might
also play a role in chronic cell homeostasis, by contributing to the
maintenance of the cell volume in the event of a partial inactivation
of the Na+ pump. In the
time-dependent behavior, the complete inhibition of the enzyme
reproduces the classic experimental results (decrease in
[K+]i
and in Vc, increase in
[Na+]i
and
[Cl
]i,
and membrane depolarization). In the presence of regulatory mechanisms,
the cell is capable of maintaining its volume at the reference value
until
[Cl
]i
becomes sufficiently low.
The model introduced here represents a plausible description of the
short-term responses of a schematic cell to anisosmotic shocks. In the
absence of cell volume regulation, the model describes the osmometer
behavior. In the presence of active mechanisms of volume regulation,
the model consistently describes both RVD and RVI. In the present work,
these responses are achieved via the activation of ionic cotransport
systems that remain inactive under reference conditions and are
activated by minute cell volume changes. These short-term responses
exhibit a negligible dependence on the enzyme parameters. The variation
of N and
[ATP]i both have undetectable effects on RVD and RVI, although they affect the cell
recovery. From a biological point of view, this would imply that the
acute regulatory responses are not significantly affected by the pump
condition. This is consistent with the view that an enzyme of the
Na+-K+-ATPase
type mainly plays a long-term role in the generation of electrochemical
gradients. However, the preexistence of these gradients is crucial
for the short-term responses. The simulations performed
here also suggest that partial or incomplete RVD responses might be
related to insufficient intracellular
Cl
availability.
The method introduced here to calculate
Vm underlies the
characteristic time-dependent behaviors of the model. If the method turns out to be an appropriate approximation, the model predicts short-term electrical responses to the
Na+ pump activation, which are a
consequence of the electrogenic properties of the enzyme. Because the
enzyme is a target of diverse effectors of cellular responses, we
suggest that these short-duration electrical signals might be involved
in signaling mechanisms.
 |
APPENDIX A |
Steady-State Analysis of
Na+-K+-ATPase
Kinetic Model
The active transport of Na+ and
K+, mediated by
Na+-K+-ATPase,
is described by the state diagram shown in Fig. 2. In this diagram, N1, · · · , N6 are the
intermediate states of the enzyme, and k12, · · · , k61 and
k16, · · · , k21 are the rate
constants governing the corresponding transitions in the clockwise and
counterclockwise directions, respectively. The meanings of the rest of
the symbols are specified in the
Glossary. The steady-state analysis of
this diagram has already been performed (10, 22, 34); we show here the
main expressions. In the steady state, the cycle flux Jp (considered to
be positive in the clockwise direction) can be expressed, employing the
diagram method (23), as
|
(A1)
|
In
this equation,
and
are functions, defined as
|
(A2)
|
with
and
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 Ref. 22 for explicit expression).
The rate constants
k56 and
k65 are assumed
to depend on Vm
according to
|
(A3)
|
where
k56° and
k65° are
independent of
Vm.
The detailed balance condition imposes the following restriction on the
rate constants
|
(A4)
|
where
Keq is the
equilibrium constant of the reaction ATP + H2O
ADP + Pi.
Because there is only one cycle, the active fluxes of
Na+ and
K+ (considered to be positive in
the inward direction) are respectively equal to
3Jp and
2Jp.
 |
APPENDIX B |
Stationary Solution for Vm
After each integration time step, we determined
Vm, assuming
Eq. 1b, as a solution of the
transcendental equation (22)
|
(B1)
|
where
the functions
=
(Vm),
=
(Vm), µ = µ(Vm), and
=
(Vm)
are given by
with
m defined by
Eqs. 2,
and the
aij terms
defined in APPENDIX A, and
a56° and
a65° given by
It must be noted that
is also a function of
Vm (see
APPENDIX A).
Vm was determined
from Eq. B1 by an iteration of the
type
(Vm)n+1 = f(Vm)n,
which yielded convergent solutions for all the simulations. In the
absence of electrogenic transport (e.g., for
N = 0), Eq. B1 becomes the Goldman-Hodgkin-Katz (17, 24) explicit
equation for the diffusion potential
|
(B2)
|
 |
APPENDIX C |
Determination of Steady-State Values
In the steady state, Eqs. 1a satisfy
|
(C1)
|
From Eqs. 1b and C1, we can express the steady-state
values of the variables by the following system of equations
|
(C2)
|
where
the last equation is actually Eq. B1
(see above for the meaning of the symbols).
Similar to the procedure described in APPENDIX
B, the roots of Eqs.
C2 were determined by an iteration of the type (y)n+1 = fi(mNa,
mK,
mCl,
Vc,
Vm)n
(y = mNa,
mK,
mCl,
Vc,
Vm;
i = 1, 2, · · · , 5),
which yielded convergent solutions for all the simulations.
 |
ACKNOWLEDGEMENTS |
This work was supported by grants from the Programa para el
Desarrollo de las Ciencias Básicas and from the Comisión
Intersectorial de Investigación Científica de la
Universidad de la República, Uruguay.
 |
FOOTNOTES |
A preliminary version of this work was presented at the III Congreso
Iberoamericano de Biofísica, held at Buenos Aires, Argentina, in September 1997.
Address for reprint requests: J. A. Hernández, Sección
Biofísica, Facultad de Ciencias, Iguá s/n, esq. Mataojo,
11400 Montevideo, Uruguay.
Received 3 November 1997; accepted in final form 11 June 1998.
 |
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