INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL CONSTRUCTION RESULTS DISCUSSION AND CONCLUSION APPENDIX REFERENCES

To maintain their functional integrity, cells must face a number of constraints, including osmotic ones. In particular, animal cells avoid osmotic lysis only because they can equalize osmotic pressure between the two sides of their membrane by actively pumping out Na⁺ ions with an ion-motive pump, the Na/K-ATPase (Hoffman and Simonsen, 1989). Because of this pump-leak mechanism, regulation of cell volume is tightly coupled with that of membrane polarization and of ion concentrations. A further complication in this coupling arises from the electric double layers that are formed at the plasma membrane surface. The surface potential was shown thus to control activity of the Na/K-ATPase by changing the local concentration of its cationic substrates sodium and potassium (Ahrens, 1981, 1983) and its anionic ligands ATP, ADP, and Pi (Nørby and Essmann, 1997). In addition to their effects on specific molecular targets, numerous compounds, which are either endogenous or exogenous, modify charge distribution on the cell membrane surface and should thus affect activity of enzymes like the Na/K-ATPase. It is therefore likely that the surface potential represents an important factor in the control of ion processes and cell metabolism (Wojtczak and Naleçz, 1985). In this context it might be wondered whether electrostatic control of the Na/K-ATPase could bring about a functional advantage to living cells. In other words, in view of the fact that the function of the Na/K-ATPase is to regulate ion concentrations, could electrostatic interactions enhance the stability of these concentrations faced with naturally occurring perturbations?

Most existing models of cell electro-osmotic regulation cannot be used for a general study of this question because they were designed to investigate the properties of specific cell types. However, models like the ones proposed by Kabakov (1994) and Jakobsson (1980) reduce description of electro-osmotic regulation to the basic mechanism of ion pump-leak quoted above, and therefore keep an essentially general character. In principle, the influence of electrostatic interactions on membrane transport can be introduced into these models by substituting superficial ion concentrations for bulk ones into ion transport equations. However, this requires having at one's disposal a model of the surface potential coherent with the full set of properties to be described. Thus, the most obvious approach would be to use the Gouy-Chapman equation or its generalized formulation for electrolytes with mixed valences, the Grahame equation (Hille, 1992; McLaughlin, 1989; Grahame, 1947). Indeed, this theory provides a quantitative description of unspecific (i.e., electrostatic) effects of ionic strength on the activation of numerous ion channels (Hille, 1992) and on activation of the Na/K-ATPase (Ahrens, 1981, 1983). However, this theory describes the solvent as a uniform dielectric continuum and consequently predicts the osmolarity at the charged surface of a membrane should be in excess of a quantity 1/2² with respect to the bulk osmolarity, with ² denoting the inverse of the Debye length,  the dielectric constant of the solution, and _S the surface potential (Winterhalter and Helfrich, 1988). Mechanical equilibrium of the electric double layer actually implicates establishment of an excess pressure at the charged interface (Sanfeld, 1968). One must therefore carefully evaluate to what degree this pressure is challenging theoretical results derived from the application of the Gouy-Chapman-Grahame theory to biological membranes.

The paper is organized as follows. First, we cast equations describing the time evolution of ion concentrations, membrane potential, and volume of a model cell endowed with a superficially charged plasma membrane; the model is derived from Kabakov's (1994) and Jakobsson's (1980) studies and equations make use of the ion distribution predicted by Gouy-Chapman-Grahame theory. Next, we use irreversible thermodynamics (see Durand-Vidal et al., 2000; Sanfeld, 1968) to give a more general description of electrostatic potentials at the cell membrane surface by including pressure in ion potential energy. This formalism is used in the first Results section to show that Gouy-Chapman-Grahame theory provides a very good approximation of the electrostatic potential in an electric double layer where osmotic equilibrium is achieved. The reader most interested in the biophysics of the cell model should turn immediately to subsequent sections, where it is shown that the model admits a unique resting state, the stability of which depends on the presence of charges at the membrane surface. We show that negative charge densities, with values comparable to those found on biological membranes, greatly accelerate recovery of resting concentrations after various perturbations. A local stability analysis, shown in the Appendix, is used to show that this heightened stability stems from an increase in the Na/K-ATPase flux sensitivity to internal Na⁺ ions induced by electrostatic interactions.

	MODEL CONSTRUCTION

TOP ABSTRACT INTRODUCTION MODEL CONSTRUCTION RESULTS DISCUSSION AND CONCLUSION APPENDIX REFERENCES

Basic assumptions

Let us consider the cellular model schematized in Fig. 1 into which a cytoplasmic compartment is separated from the external milieu by a membrane charged on its two sides. The membrane is permeant to Na⁺, K⁺, and Cl ions, while the active fluxes of the Na/K-ATPase equilibrate the diffusional flux of each of the two cations. The cell also contains a fixed quantity, n_a, of impermeant monovalent anions and a fixed number, n_d, of divalent cations (we can readily identify the latter to Mg²⁺, the major divalent cation type in animal cells). We first define the following dimensionless potentials:

(1a)

(1b)

(1c)

where _o, _i, and _t denote electric potential differences depicted in Fig. 1. The macroscopic membrane potential difference _m, measurable with microelectrodes, is given by:

(2)

We assume that membrane charges affect ion transport only through the following two processes:

1.	Concentration of ion species j with valency zj effectively "seen" by membrane structures of passive and active transport does not stand for the bulk concentration, but for concentration in the immediate vicinity of the membrane surface. If we put forward the hypothesis that quasi-equilibrium is achieved inside internal and external compartments at each point in time, these concentrations are related by the Maxwell-Boltzmann statistics: (3a) (3b)
	where [j]o, [j]os, [j]i, and [j]is, respectively, denote the concentration of species j in the bulk of external medium, at the external surface of the membrane, in the bulk of the cytoplasm, and at the internal side of the membrane.
2.	Measured binding constants of the sodium pump with its ionized substrates are apparent constants (Nørby and Essmann, 1997; Ahrens, 1983). If we denote these by K and thermodynamic constants by K, we have: (4a) (4b)
	if the binding site is either on the external side (Eq. 4a) or on the internal side of the membrane (Eq. 4b).

View larger version (32K): [in this window] [in a new window]   FIGURE 1   Structure of the model of electro-osmotic regulation within an animal cell. Regulations of cell volume and membrane polarization are coupled by the pump-leak mechanism, which renders the membrane functionally impermeable to Na+ ions. Superficial charges of the membrane modify the ion distribution near the membrane sides. Concentrations appearing in ion transport equations are therefore superficial concentrations, as predicted by the Gouy-Chapman-Grahame theory of the electric double layer. Electrostatic potential o at the membrane outer surface (minus the potential at infinite distance from the cell), the potential i at the inner membrane side (minus the potential at the center of the cell), and the transmembrane potential (difference of potentials between the internal and external sides of the membrane) appear in equations as dimensionless potentials u, v, and w, respectively. It is assumed that the cell membrane remains in osmotic equilibrium and that both inner and outer compartments are electrically neutral at each moment.

Constitutive equations of the model cell

If the small charge imbalance from which the membrane potential difference arises was omitted (see Roux, 1997), the principle of macroscopic electroneutrality would imply that the total charge in the cytoplasmic compartment was zero. Therefore, ions in the solution must equilibrate the intrinsic charges on the inner membrane side. Rapid estimate with typical cell radii, however, shows that this charge corresponds to a negligible number of ions compared to the total number of ions contained in cells. One can therefore approximate that the ion bulk concentration [j]_i of an ion species j is related to the mole number n_j in the cell by:

(5)

where V denotes the cell volume (m³). Assuming that ion diffusion across the membrane obeys the constant field equation of Goldman (1943), the time evolution of the Na⁺ and K⁺ ion mole numbers in the cytoplasm follows the ordinary differential equations:

(6)

(7)

where J denotes the sodium pump active flux of Na⁺ ions and r_p the coupling ratio of Na⁺ on K⁺ active fluxes, while P_Na and P_K denote the integrated permeability of the membrane (m³ s¹) for Na⁺ and K⁺ ions, respectively. Flux J is given by:

(8)

where coefficient Q scales the maximum flux of the pump and where  and  are two functions with the range [0, 1], given by:

(9a)

(9b)

with:

(9c)

where K_s, K_i, and K_c are ion binding constants (M), U a scaling factor, and _Na the fraction of charge that raises the membrane electric field during deocclusion of Na⁺ ions on the external side of the membrane; parameters m, n, and q are Hill coefficients. Equation 8 accounts for the independence of the effects of cationic substrates of the pump on the two sides of the membrane (Garay and Garrahan, 1973; Apell, 1989; Laüger, 1991). Equations 9, b and c are given by the model of Vasilets et al. (1993).

Jakobsson (1980) noticed that other processes involved in cell volume regulation are much faster than the time evolution of ionic concentrations. We can therefore assume that these processes remain in a pseudo-equilibrium state at each moment (see Appendix), and replace differential equations governing the evolution of fast variables with algebraic equations. Thus, we can assume macroscopic electroneutrality of the cytoplasm at each point in time which, from Eq. 5, gives:

(10)

from which we can deduce two more relations. First, we can write down a flux equation for Cl ions with the help of the Goldman equation. By introducing this equation in Eq. 10 and after differentiation with respect to time, we find that the w potential satisfies:

(11)

Second, we assume that electrostatic potentials on the two membrane sides can be described with the Grahame (1947) equation. When considering a spherical cell, these equations write, with the help of Eq. 10:

(12)

for the internal compartment, and:

(13)

for the outer solution; Q_i and Q_o denote the total charge on the internal and the external side of the membrane, respectively. Surface charges partly recombine with cations (McLaughlin et al., 1971), so that Q_o and Q_i are given by (Genet and Cohen, 1996):

(14a)

(14b)

where Q_ot and Q_it now denote the intrinsic amount of charges on the two sides of the membrane (i.e., without cations) and where the K_j values are thermodynamic binding constants (see Eq. 4).

As the Gouy-Chapman-Grahame theory neglects the excess mechanical pressure induced by ion accumulation at a charged interface, we can write that the osmolarity in the bulk of the cytoplasm remains identical to that in the bulk of the external medium in order for the membrane to be in osmotic equilibrium:

(15)

where [S]_o = _ionsj [j]_o is the bulk osmolarity in the external compartment.

Extension of the Gouy-Chapman-Grahame theory

The above equations make the basic assumption contained in the Gouy-Chapman-Grahame theory, which involves replacing the potential of mean force acting on ions with the mean electrostatic potential (see Durand-Vidal et al., 2000). To examine whether this assumption is physically consistent with the mechanical equilibrium of the membrane, we now give an alternate description of membrane electrostatic potentials, which introduces the mechanical excess pressure induced by electric fields at the charged membrane sides.

The Gibbs-Duhem equation for a polarized system (De Groot and Mazur, 1962, p. 397, Eq. 102) can be written with dimensions that allow one to have the chemical potentials, µ, of the system components (i = 1, ... , r) expressed as molar quantities (J mol¹):

(16)

where T denotes the absolute temperature, p the pressure (N m²), S_v the entropy density (JK¹ m³), C_i the molar concentration of the component i (mol m³), E the electric field vector (V m¹), and P the polarization density vector (C m²). P is related to the molar dipolar moment vector P (C m mol¹) of the system components by:

(17)

If we denote the vector sum of the external force density by F(N m³), neglecting forces arising from the gravitational field, we have:

(18)

By subtracting Eq. 18 from Eq. 16 we obtain:

(19)

into which one recognizes the gradient of electrochemical potential :

(20)

At thermal equilibrium (T = 0), the condition to have mechanical equilibrium of a volume element is that:

(21)

and thereby, that electrochemical potential gradients simultaneously vanish for all species comprising the system:

(22)

Given the hypothesis of local equilibrium, µ writes (De Groot and Mazur, 1962):

(23)

where N_i, _i, and _i respectively denote the mole fraction, the partial molar volume (m³ mol¹), and the partial molar entropy (JK¹ mol¹) of component i, which are defined with the usual thermodynamic relations:

(24a)

(24b)

(24c)

(24d)

The sum in Eq. 23 represents the contribution to µ of the ith component and of its interactions with the other components. From statistical mechanics, the former can be shown to obey:

(25)

Finally, the electric field E must satisfy the Poisson equation from electrostatics:

(26)

where _o (F m¹) is the permittivity of vacuum, and _r the dielectric constant. Recall that:

(27)

Following the Gouy-Chapman theory assumptions, we neglect intermolecular interactions in Eq. 23, so that, at thermal equilibrium, Eq. 22 becomes:

(28)

We also assume 1) that _r has a uniform value, so that Eq. 26 now reads:

(29)

and 2) that the solvent (referred to by subscript 1) is the only component having a dipolar moment given by:

(30)

if one neglects dielectric saturation (Eyges, 1980);  = _r  1 denotes the electric susceptibility. Given that the electric double-layer thickness is negligible compared to the curvature radius of any cell, we apply these equations to a plane surface of infinite lateral extension normally oriented to the x-axis and in contact with a solution of a 1:1 electrolyte. Subscript 2 refers to the cation and subscript 3 to the anion, and the origin of the x-axis is put on the surface; the surface bears uniformly smeared charges with density  (C m²). By noting that the sum of mole fractions is 1, that E is related to the electric potential  by E = , and that the problem is monodimensional, we can write Eq. 28 explicitly for the three species composing the solution and Eq. 29 in terms of  to obtain the following system of equations:

(31a)

(31b)

(31c)

(31d)

which was recast into nondimensional form by introducing the reduced variables:

(32a)

and

(32b)

with

(32c)

In addition to equation system 31, we have the following boundary conditions:

(33a)

(33b)

(33c)

(33d)

whereas Gauss theorem together with the condition of electroneutrality of the charged plan-solution set allows us to establish that:

(33e)

We recall finally that the Gouy-Chapman equation writes:

(34)

where X' = x with

(35)

with boundary conditions:

(36a)

(36b)

Parameters

Equations 12 and 13 are written in terms of Q_ot and Q_it rather than charge densities, because the latter change with cell volume. However, physiological perturbations of cell ionic composition occur on much shorter time scales than the turnover of membrane components. The intrinsic charge of the membrane can therefore be considered constant for the study of such perturbations. It was expressed as a multiple of a reference value Q_ref = 1.0 × 10⁹ e by defining the ratio  = Q_ot/Q_ref. Similarly, we introduce the ratio  = Q_it/Q_ot to study the effects of the relative proportion of charges between the two membrane sides. This model describes transport of Na⁺, K⁺, and Cl ions only. Thus, any discrimination between other ion species excepted on the basis of their electric valency or binding constant with surface charges is irrelevant. Nevertheless, we will refer to divalent cations as to calcium ions, which are the major divalent species in the extracellular space. The reference composition for the external solution was (in mM): [Na]_o = 120, [K]_o = 5, [Ca]_o = 2.5 with osmolarity adjusted to 310 mM with the chloride salt of a monovalent cation inert with respect to membrane ion transport. Other parameters are listed in Table 1.

Numerical methods

Most computations with the algebro-differential system 6, 7, 10, 11, 12, 13, 15 were performed with MAPLEV, using the ODE/DAE solver BESIRK for its integration.

Integration of equation system 31 presented a basic difficulty, as the system is associated to boundary conditions on the charged plane and at infinite distance (Eqs. 33, a-e), whereas numerical procedures require a finite interval. Several strategies allow bypassing this problem but they present difficulties (see Stafiej et al., 1996). We therefore followed the approach of Forsten et al., (1994) by simply truncating the infinite interval at length L to rewrite condition 33b on Y at X = L; equations were integrated either with the shooting method capability of the XPP software or a P1 finite element method, the two methods giving closely matching results. Truncating the integration interval prevented the condition of zero electric field to be strictly satisfied at X = L and, due to this approximation, the solution proved highly sensitive to the interval length for small L values. But above L = 30, the surface potential Y(X = 0) adopted an asymptotic behavior, its value changing only on the fourth decimal for small increments in the interval length above L = 50. The following partial volumes of monovalent cations were used for computations: ₁ = 18 × 10⁶ m³ mol¹; ₂ = ₃ = 25 × 10⁶ m³ mol¹. For comparison with the Grahame equation, the partial volume of divalent cations was given values, ranging from one to three times that given to monovalent ions, that induce negligible incidence on the difference between the solution of Eqs. 31 and that of Eq. 34.

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL CONSTRUCTION RESULTS DISCUSSION AND CONCLUSION APPENDIX REFERENCES

Adequacy of the Gouy-Chapman-Grahame theory to describe membrane electrostatic potentials

Fig. 2 compares the numerical solution of the TIP equations (Eq. 31) to the analytical solution of the Gouy-Chapman equation (Eq. 34) as functions of  for N₀ = 0.0027 (giving a concentration C₀  150 mM, which corresponds well to that of the intracellular milieu). Graph A compares values of the reduced surface potential given by the two equations, while graph B gives pressure values at the charged plane as predicted by system 31. In the interval of compiled estimations for mean charge density on biological membranes (delineated by the two vertical bars on graph A), the two equations give the same surface potential within 0.1%. The region of larger charge densities in graph A reveals that the surface potential predicted by system 31 is, in fact, slightly more negative than the one given by the Gouy-Chapman equation, the difference increasing with charge density. This difference is due to the excess pressure, which ensures the mechanical equilibrium of the electric double layer (graph B); this pressure limits accumulation of cations at the charged wall, thereby reducing the screening of surface charges. However, the difference only amounts to 2% with the highest charge density used,  = 0.0765 C m², for which the profiles of the variables of system 31 are illustrated on graphs C, D, and E of Fig. 2. Because biological solutions comprise divalent cations, we also applied the TIP equations to a solution of electrolytes with mixed valences to compare it with the Grahame equation (Grahame, 1947), which relies on the same basic assumptions as the Gouy-Chapman theory. Again, results did not differ by >2% within the millimolar range of divalent cation concentrations encountered in biological media (not illustrated).

View larger version (16K): [in this window] [in a new window]   FIGURE 2   Comparison between TIP equations (Eq. 31) and the Gouy-Chapman theory (Eq. 34) for description of electrostatic potential in the electric double layer. (A) Reduced surface potential Y(X = 0) versus surface charge density  according to Eq. 31 (solid line) and Eq. 34 (dotted line); (B) pressure on the charged plane predicted by TIP equations. Lower graphs illustrate profiles of reduced potential (C), ion mole fractions (D), and pressure (E) versus reduced distance from the charged plane, as given by TIP equations for  = 0.0765 Cm2. All data were computed for a 150 mM solution of a 1-1 electrolyte (N2 and N3 denote mole fractions of the anion and the cation, respectively). The two vertical lines in graph (A) delimit the interval of mean charge densities measured at the surface of biomembranes (0.035, 0.008 Cm2; Wojtczak and Naleçz, 1985).

These results confirm Laüger et al.'s (1967) insight that the influence of pressure on ion potential energy in the double layer is negligible in the range of parameter values corresponding to biological membranes. Within this range, the Gouy-Chapman-Grahame theory therefore provides a reliable approximation of the electric potential profile at a single charged interface, where all components in the solution, including the solvent, are in thermodynamic equilibrium. However, biomembranes achieve two such interfaces. Thus, it follows that, in addition to conditions of the type 33e on the electric field, continuity conditions on the electric displacement also hold at the membrane boundaries. These conditions entail an electrostatic coupling across the membrane, the intensity of which depends both on the membrane specific capacitance and magnitude of the transmembrane electric potential difference (Genet et al., 2000). Here, we suppose that this difference is not too large (absolute value <100 mV). Thus, and given that the capacitance of the double layers is larger than that of the membrane, the electrostatic coupling can be neglected (see Peitzsch et al., 1995). In contrast, a pressure discontinuity will occur across the membrane in the case of unequal charge densities on the two membrane sides. However, because solute activity coefficients are ignored in the Gouy-Chapman-Grahame theory, the solvent chemical potential profile will be uniform across the membrane, thereby guaranteeing membrane osmotic equilibrium, providing the bulk osmolarity is the same in the two solutions that are in contact with the membrane boundaries.

Resting point of the model

Three distinct stationary points were found for 5 <  < 1 and 0.1 <  < 1.3 with the reference value of other parameters. Existence of multiple stationary points is due to Eqs. 12 and 13, which involve the square of Q_ot and of Q_it. One therefore finds solutions corresponding to the sign given to these parameters and nonphysical solutions corresponding to the opposite sign. Two of the points found correspond to such nonphysical states, whereas the third one appears consistent with the sign chosen for Q_ot and Q_it. Fig. 3 illustrates dynamics of the model following three distinct perturbations of its stationary state in the case where  = 1.5 and  = 1; that is, for a symmetrical distribution of negative charges on the two membrane sides. These perturbations consisted of changing the value of a single parameter and then finding coordinates of the corresponding resting state. These were used as initial conditions for the integration of equations with the reference value of the parameters. Illustrated perturbations are a partial neutralization of charges on the inner side of the membrane, an increased P_Na and an increased [K]_o. In all cases, the model consistently relaxed toward the same resting point.

View larger version (15K): [in this window] [in a new window]   FIGURE 3   Trajectories of the model following three different perturbations of its resting state. Curves show the time evolution of Na+ and K+ ions' mole numbers in the cell with parameters set at their reference value (Table 1), and membrane charge distribution set by  = 1.5 and  = 1 (i.e., with negative charges symmetrically distributed between the two membrane sides). Initial conditions for each curve corresponded to the resting state coordinates of the model with one of its parameters changed from its reference value: (a) PNa × 1.2 (increased sodium permeability); (b) [K]o = 7 mM (hyperkalemia); (c) Qit × 0.8 (partial neutralization of charges on the inner membrane side). The system also converged toward the same steady state with all types of perturbations tested, and qualitatively identical results were obtained with different charge distributions on the membrane.

Furthermore, we found the model to admit a resting state with physical significance in an extended range of ionic compositions of the external medium (hypo/hyperkalemia (2 < [K]_o < 8 mM), hypo/hypernatremia (100 < [Na]_o < 150 mM), and hypo/hypercalcemia (2 < [Ca]_o < 20 mM), as well as for significant departures from the reference value of all other parameters. In every case, the model converged to the resting point following finite perturbations of the kind illustrated in Fig. 3.

As there was no sign suggesting existence of a limit cycle, we can assume that the model has a unique resting point because nonphysical points were never reached as long as consistent initial conditions were chosen for integrating equations. Moreover, the above results cover a priori the range of parameter values corresponding to pathophysiological situations. This suggests strongly that the stationary state is globally stable even if formal demonstration of this property would require finding a Lyapunov function for the model under investigation.

Negative charges on the membrane increase the resting state stability

We examined the influence of membrane charges on the model dynamics by integrating the equation system with various assumptions on charge distribution for several sets of initial conditions departing from the resting state. Fig. 4 displays an example where the initial state presented an increased n_Na and a decreased n_K by 5 × 10¹⁵ mol, corresponding to a 0.4 mM deviation from resting value in each of the two cation concentrations. This perturbation simulates ion composition changes that neurons withstand following a brief, sustained discharge of action potentials. Rather than plotting absolute values of state variables, we have represented their deviations from resting value to highlight the influence of membrane charges on the rate of relaxation.

View larger version (31K): [in this window] [in a new window]   FIGURE 4   Influence of membrane charges on the rate of relaxation of the model following a perturbation of its internal ionic composition. Perturbation consisted in increasing nNa by 5 × 1015 mol and decreasing nK by the same amount from resting values. To highlight membrane charge effects, graphs display the time evolution of deviations of the model variables with respect to their resting values. (A) internal sodium concentration [Na]i. (B) Internal potassium concentration [K]i; the negative of the cation concentration deviation is plotted for comparison with (A). (C) Transmembrane potential t. (D) Cell volume V. For these simulations, membrane charges were equally distributed between the two membrane sides, by setting  = Qit/Qot to 1, while their density was varied by changing  = Qot/Qref (Qref = 1.0 × 109 e); thick curves correspond to the neutral membrane case. Note the prominent increase in the rate of relaxation of Na+ and K+ induced by negative membrane charges.

The top half of Fig. 4 displays relaxation curves of sodium (A) and potassium (B) concentration deviations for various amounts of charges distributed symmetrically on the two membrane sides (i.e.,  = 1). With an uncharged membrane ( = 0, thick curves), cation concentrations have only slightly changed at the end of the illustrated time interval of 2 s (full recovery took ~30 s); but negative charges on the membrane speed up relaxation at such a point that both concentrations have fully resumed their resting value after the same time span for 4 <  < 1. Dashed curves that connect relaxation curves at several discrete times give clear evidence that a maximum rate of relaxation occurs around  = 2. Positive charges exert an opposite influence, but it is barely perceptible on the graphs (a shorter interval of integration would have highlighted this effect, but it would have prevented a better resolution of the curve for  < 0). It is interesting that even though P_Na is 10 times smaller than P_K, membrane charges about equally change the relaxation rate of the two cation concentrations. This evidently points the origin of membrane charge effects on the rate of relaxation to the process of ion active transport by the sodium pump, rather than to passive ion fluxes.

The bottom half of Fig. 4 plots three significant samples of the transmembrane potential (C) and cell volume (D) relaxation curves corresponding to upper ion concentration curves. The membrane potential curve for a neutral membrane ( = 0) exhibits a transient 0.5 mV hyperpolarization, whose rate of decay matches that of cation concentration deviations on graphs A and B (thick curves). Membrane charges profoundly changed the magnitude and temporal pattern of this transient hyperpolarization. Its amplitude was larger (>2 mV) with negative charges ( = 1) on the membrane while it decayed much faster, with about the same kinetics than cation concentrations on top of the figure. With positive charges ( = 1), the hyperpolarization became so small (<0.01 mV) that the curve confounds with baseline potential, but its decay was very much prolonged with respect to the neutral membrane case, like that of cation concentrations on graphs A and B. This transient hyperpolarization unambiguously arose from a stimulation of the electrogenic sodium pump by the perturbation that had increased [Na]_i. Graph C therefore shows that this stimulation was larger with negative charges on the membrane than without, the pump stimulation becoming extremely weak with positive charges. This again suggested that the origin of increased rate of recovery of cation concentrations induced by negative charges should reside in a modulation of the sodium pump activity by electrostatic interactions.

The cell model also underwent a transient shrinkage during relaxation toward steady state (Fig. 4 D). This shrinkage represents no more than a fraction of a percent change in resting volume (V  0.108 ×10⁴ µl), owing to the initial internal osmolarity having the same 310 mM value than in the steady state. This transient shrinkage can be related to the sodium pump activation identified above, as this pump moves three Na⁺ ions from the inside of the cell to the outside for two K⁺ ions in the opposite direction; its sudden activation must therefore result in a small ion flux imbalance that transiently lower internal osmolarity. Consistent with this interpretation, the lower activation of the sodium pump found with positive charges on the membrane resulted in slower rise and decay of the transient shrinkage (curve labeled  = 1 on graph D). However, negative charges would have been expected to increase both amplitude and rate of decay of the cell shrinkage, whereas the curve labeled  = 1 on graph D exhibits just the opposite. This suggested that some factor may have prevented negative charges to accelerate relaxation of the cell volume like that of cation concentrations and membrane potential. The most likely candidate for such a factor was the chloride permeability P_Cl, as temporal changes of [Cl]_i mirrored that of cell volume in all simulations (not illustrated). Thus, negative charges on the membrane resulted in faster cell volume transients changes than without charges when P_Cl was raised to 100 times its reference value (not illustrated). Most interestingly, curves of cation concentration relaxation remained almost identical to those illustrated in Fig. 4, whether P_Cl was given 10 or 100 times its reference value. This result agrees well with Jakobsson's (1980) simulations, which show that the magnitude of P_Cl weakly affects temporal changes in Na⁺ and K⁺ concentrations. Moreover, qualitatively similar data were obtained with much larger (up to several mM) deviations in cation concentrations for the specific perturbation illustrated in Fig. 4, as well as with other kinds of perturbations, such as those illustrated in Fig. 3. Overall, negative charges on the membrane consistently heightened the stability of resting cation concentrations, despite the fact that they had adverse effects on cell volume, depending on the magnitude of P_Cl.

Mechanism of heightened stability induced by negative membrane charges

To understand the origin of membrane charge effects on the cell model dynamics, we examined the local stability of the model in a neighborhood of its resting state. The Appendix shows that stability analysis can be reduced to that of a two-dimensional model with mole numbers n_Na and n_K as variables. Absolute value of the eigenvalues of the Jacobian matrix J of this system, evaluated at the resting state coordinates, are the reciprocal of the two characteristic times of relaxation of the model. The three-dimensional plots in Fig. 5 illustrate how these eigenvalues depended on the density of charges on the membrane (set by ) and on their partition between the two membrane sides (set by ). First, one sees that both eigenvalues were real and negative-valued across the entire domain of the (, ) plane considered; this proves that the resting point is asymptotically stable over a broad range of the parameter values that set membrane charge distribution. Second, one can see from the different vertical scales on the two graphs of Fig. 5 that the model has two broadly different relaxation times. The ₁ eigenvalue corresponds to short times ranging from tenths of seconds to seconds, while ₂ corresponds to time scales ~100 times larger.

View larger version (35K): [in this window] [in a new window]   FIGURE 5   Membrane charge effects on the eigenvalues of the Jacobian matrix of equation system 6, 7 evaluated at the resting state. The absolute values of these eigenvalues are the reciprocal of the two characteristic times governing the rate of relaxation of the model following perturbations of its resting state (see Appendix). Parameter  = Qot/Qref (Qref = 1.0 × 109 e) sets the algebraic value of membrane charges, while parameter  = Qit/Qot sets their distribution between the two membrane sides. The resting state is always stable with all charge distributions (both eigenvalues are strictly negative).

It is clear from Fig. 5 that membrane charges exert adverse influences on the two eigenvalues, and hence on the two relaxation times of the model. On the one hand, positive charges ( > 0) render ₁ less negative than when the membrane bears no charge, thereby lengthening the short relaxation time. Negative charges ( < 0) have opposite and much more marked effects on this relaxation time, which can be reduced from ~3 s, with no membrane charges, down to 150 ms with  = 2 and  = 1.2 (Fig. 5, upper graph).

On the other hand, both negative and positive charges render ₂ less negative and therefore lengthen the long relaxation time of the model. For example, this relaxation time, which is ~25 s with no charge on the membrane, is increased to >100 s by the large density of negative charges (4 <  < 2), whatever their distribution between the two membrane sides.

In principle, the overall degree of stability of the model is determined by the eigenvalue with the largest algebraic value, that is, ₂ (see Bourlès, 1986). The ₂ plot in Fig. 5 would therefore suggest that membrane charges, whatever their sign, should destabilize the resting point, but the above simulations have highlighted an increase in stability induced by negative charges, which is in better agreement with charge effects on ₁. This suggested that relaxation velocity of the model depends mostly on ₁. This was confirmed by further calculations showing that ₂ contributes 100 times less than ₁ to the rate of relaxation of the Na⁺ and K⁺ concentrations. Moreover, this conclusion agrees with the minimum in ₁ for  = 1, which occurs precisely for  = 2 (Fig. 5), like the maximum rate of recovery of ion concentrations found in Fig. 4. This feature of the model stems from the fact that negative charges shorten the relaxation time corresponding to ₁ so much that the model has alre