Water Permeability.

The cell has high water permeability and is always at osmotic equilibrium (equilibration is instantaneous on the time scale of the calculations).

Osmotic Equilibrium.

The requirement for osmotic equilibrium (and stable volume) is that osmolarity_out = osmolarity_in. This equality must be so at all times, because the fragile plasma membrane of the cell cannot support a pressure difference.

Electroneutrality (Approximate).

For a cell of normal size, if the membrane voltage has a reasonable (millivolt scale) value, the cationic charge must be almost equal to anionic charge, on both sides of the membrane. The exact equations for a cell with a membrane voltage V_m (voltage inside − voltage outside) are 1 and 2 where C_m (C × V⁻¹) is membrane capacitance. V_m × C_m is the small excess of charge that makes the membrane voltage, an excess of negative charge on one side and positive on the other. It is negligibly small compared with the ∑ cationic charge or ∑ anionic charge, so, at the beginning of a calculation it is justified to assume perfect electroneutrality; i.e., ∑ cationic charge is set equal to ∑ anionic charge both inside and outside. The excess of charge, V_m × C_m, is not used in the computations.

Ion Flux Through Channels.

Following the reasoning given in Appendix, the currents (C⋅cm⁻²⋅s⁻¹) of the permeant ion species through channels in the membrane are given by the equations 3a where P_X (cm⋅s⁻¹) is the number of pores/cm² times the permeability of a single pore ρ_K (cm³⋅pore⁻¹⋅s⁻¹). F (C⋅mol⁻¹) is Faraday's constant, z_X (dimensionless) is the magnitude of the valence (1 for all ions considered here), R is the gas constant, and T is the absolute temperature. ψ (dimensionless) is the ratio of potential (z_x × V_m × F) to random thermal energy (R × T). The current voltage (I–V_m) relations for Na⁺ and K⁺ are shown in Fig. 2. Eqs. 3 are interpretable, for Na⁺ and K⁺, as a voltage-independent influx, and an efflux that varies exponentially with voltage, whereas for Cl⁻, the opposite is true. Alterations of the model that use other single-channel I–V relations are described below.

• Download figure
• Open in new tab
• Download powerpoint

Figure 2

Theoretical current–voltage relations for open Na and K channels. (A) The solid curves are from Eqs. 3a (the Na curve is the almost horizontal line, just negative to zero), and the dashed curves from Eqs. 3b. For both sets of curves, I_Na = −I_K at −80 mV; i.e., the resting potential is −80 mV. (B) Approximately linear current–voltage relations, generated from Eqs. 8. These relations are most similar to the experimentally observed curves, but the model works well when any of the three pairs of current–voltage relations are used, as described in the text.

Ion Flux Through the Na/K Pump.

The equations for Na⁺ and K⁺ flux (mol⋅cm⁻²⋅s⁻¹) through the pump are 4 where [Na⁺]_c is the concentration for half-maximal occupation of a Na⁺-binding site on the pump. For very low concentration the pump rate thus depends on the third power of the [Na⁺]_i. The factor 2.17, the pump rate coefficient, was determined empirically. For simplicity, pump rate is a linear function of ATP concentration and is independent of [K⁺]_o. Neither of these simplifications matters for present purposes. The minus sign in the second equation shows that Na and K flux are in opposite directions. The Na/K pump ratio is normally 3/2 (9). Pump current (C⋅cm⁻²⋅s⁻¹) is 5

Membrane Voltage.

To obtain an easily calculable model, it is necessary to have an equation for membrane voltage that takes into account the contribution of the Na/K pump. Further, the equation must be compatible with the equations for the current of each ion species. The following derivation uses the current–voltage relations shown above (Eqs. 3a) to get an equation for V_m that includes a term for pump activity. As is the case with the GHK equation and the Mullins–Noda equation, the equation derived here is valid only when V_m changes slowly enough that capacitive current is negligible. The derivation assumes the usual relation between net membrane current and the rate of change of V_m: Expanding the currents with Eqs. 3a and assuming that V_m is not changing gives This equation attributes all of the voltage dependence of the fluxes to the efflux (Na⁺ and K⁺), or to the influx (Cl⁻), as described in Appendix. It rearranges to Rearranging and taking the log of both sides gives 6a

Internally Sequestered Impermeant Anions.

The dominant internal anion is assumed to be an S⁻ molecule that bears a charge of −1e. This assumption is made for simplicity, but many other choices are possible. For example, S could be composed of two species present at equal concentrations, one bearing a charge of 0 and the other a charge of −2e. Jakobsson (7) has studied the effect of altering the valence of internally sequestered substances.

Volume.

Volume is normalized relative to the steady-state volume in standard ionic conditions, with 3 mM internal ATP.

Sequence of Computations.

When the model is set running, (i) V_m is calculated from the permeabilities and the initially assigned values for ion concentrations and pump flux. (ii) The passive flux of each ion species is calculated by using Eqs. 3a with the present values of internal and external concentration and the just calculated value of V_m. (iii) The pump fluxes of Na⁺ and K⁺ are calculated from Eqs. 4 and 5. (iv) Net fluxes are calculated by summing passive flux and pump flux (which is zero for Cl⁻). (v) By conceptually delaying water flux for the moment (handled in steps vi and vii), the would-be osmolarity is calculated from the initial concentrations, adjusted by the net fluxes just calculated. Water movement is handled implicitly, by (vi) adjusting the volume as required to make internal osmolarity equal to external, and (vii) by calculating the new values of ion concentrations using this volume. (i) Using the new concentrations, V_m is calculated, and the cycle repeats.

Alternative I–V Relations for the Channels.

Eqs. 3a above predict the current (I) as a function of voltage (V_m) through conducting channels as given by the solid curves in Fig. 2A for K⁺ and Na⁺ (I_Na is the almost horizontal line). The curves have been scaled according to the resting permeabilities, so that at the resting potential inward I_Na is equal in magnitude to the resting potential outward I_K. These curves are not much like the I–V curves of known channels, and two different paths were taken to assess the importance of the disparity. The first path taken was to derive an alternative form of the I–V relations that, for cations, attributes all voltage dependence to the influx (rather than to the efflux as in Eqs. 3a), and the opposite for Cl⁻. 3b Using Eqs. 3b, the equation derived for V_m is 6b The I–V relations for Na⁺ and K⁺ using Eqs. 3b are shown as dashed curves in Fig. 2A. At the resting potential, where I_Na is equal but opposite to I_K, both currents are much larger than with Eqs 3a. To achieve stability, it is necessary to increase the pump rate coefficient by a factor of 13. Once this is done, calculations with Eqs. 3b and 6b give results that are very similar to the ones with Eqs. 3a and 6a.

The second path for exploring the importance of I–V shape is to make the P_Na and P_K terms in Eqs. 3a voltage dependent in such a way as to approximately linearize the I–V relations. 7 where P0_K is normally 1 and P0_Na is normally 0.02. The term e^−0.71ψ approximately linearizes P_K and (70 − V_m) approximately linearizes P_Na. The factors 0.2 in the P_K equation and 106.4 in the P_Na equation were chosen to keep the resting potential unchanged at −80 mV. Combining Eqs. 3a and Eq. 7 gives 8 These I–V curves are shown in Fig. 2B. Thanks to the voltage-dependent permeabilities, both I_K and I_Na are approximately linear over the voltage range in which the model is used. When Eqs. 8 for currents are used collectively with Eq. 6a for V_m, the results are qualitatively similar to those with Eqs. 3a and 6a. Thus, all three sets of I–V_m relations give similar results, leading to the conclusion that the shape of the I–V relation does not affect the behavior of the model in essential ways.

Validation of the Model.

A number of tests were performed to ensure that the model gives reasonable and internally consistent results, including the following. The first was a test of the stability of the model cell in normal conditions. In a steady state, concentrations, volume, and V_m should not change. When the model is run for the equivalent of 100,000 min, none of these variables changed detectably. This result shows that the equations are compatible with each other and that the model cell is completely stable. A second test was performed to observe the effects of turning off the pump. Because the pump is electrogenic, turning it off should cause a small positive change in V_m. A quick +2-mV change was observed, as expected, and slower changes followed as described below. The pump rate is faster if the internal [Na⁺] is high, making the quick jump in V_m larger. Changes of ion concentrations, addition of external nonelectrolytes, and permeability changes all gave results that closely resembled the behavior of living cells as described in the literature (6, 7, 13).

The Mutual Interaction of Cl⁻ and Membrane Potential.

The experiments of this section help to understand the influence of Cl⁻ on V_m, and, on the other side of the coin, the influence of V_m on Cl⁻ distribution. In Fig. 3A, V_m is changed by increasing [K⁺]_o from 5 to 20 mM at time 0, with a compensatory decrease of [Na⁺]_o from 145 to 130 mM. V_m quickly changes from −80 to −64 mV, followed by a slow climb to −50 mV. The quick phase is predicted by Eq. 6a, assuming that all internal concentrations are the same before and just after the change. This phase is only about half the alteration of E_K calculated from the Nernst equation, because current through Cl channels pulls V_m toward the as-yet-unchanged Cl⁻ equilibrium potential (E_Cl), −80 mV. Cl⁻, however, is no longer in equilibrium: the less-negative interior allows Cl⁻ to move inward. As [Cl⁻]_i increases, both the E_Cl and V_m drift in the positive direction, each influenced by the other. Finally, a new equilibrium for Cl⁻ is achieved at −50 mV where E_Cl and V_m are equal. Cl⁻ is not pumped in any way into the model cell and it must come to equilibrium (unless its permeability is zero) at the voltage dictated by the concentrations and permeabilities of the pump-regulated ions, Na⁺ and K⁺. There is a small increase in volume, accounted for by an increase in the cell's content of K⁺ and Cl⁻.

• Download figure
• Open in new tab
• Download powerpoint

Figure 3

Cl permeability and membrane voltage. Traces were generated by the model in the text to simulate a classical experiment of Hodgkin and Horowicz (13). (A) To the left of 0 on the time axis, the cell is in normal conditions. At 0 min, [K⁺]_o was increased from 5 to 20 mM, causing a quick change of V_m. A slower change follows as Cl⁻, no longer at equilibrium, drifts into the cell, causing a small volume increase. During the interval when Cl⁻ is not in equilibrium (0 min to ≈9 min), there is a Cl⁻ current that keeps V_m negative to its steady-state level, −50 mV. The current weakens with time as equilibrium ([Cl⁻]_i = 23 mM, V_m = −50 mV) is approached. (B) The experiment is repeated after setting P_Cl to zero at the time indicated. When [K⁺]_o is increased to 20 mM at time 0, V_m rises within milliseconds to −50 mV. The time course is quicker than in A because there is no Cl⁻ current. Notice that [Cl⁻]_i, the very bottom trace, and volume do not change.

If the experiment is repeated with P_Cl = 0 the results are quite different. In Fig. 3B the trace begins in standard conditions, and P_Cl is lowered at the time indicated. As predicted by Eq. 6a, there is no change in V_m, because Cl⁻ is at equilibrium and the net flux through an open Cl channel is zero. When [K⁺]_o is increased ([Na⁺]_o decreased in compensation), V_m jumps immediately to −50 mV, with no delay because of Cl⁻ readjustments. Note that there is a small volume increase with P_Cl = 2, but none when P_Cl = 0. These experiments are based on the classic work of Hodgkin and Horowicz (13), and show clearly that [Cl⁻]_i in the model cell is set in the steady state by V_m, which is, in turn, set (in the steady state) by the gradients and permeabilities of Na⁺ and K⁺. These results also show that when (and only when) it is not in equilibrium, Cl⁻ has an influence on V_m by means of the current flowing through open Cl channels. When Cl⁻ is at equilibrium (E_Cl = V_m) the current is zero (Eqs. 3).

The Pump, Cl⁻ Ion, and Cell Volume.

The influence of the pump on cell volume is made clear by the experiment in Fig. 4A. The model cell is in standard conditions until the arrow, when the pump is turned off. At this point, [Na⁺]_i begins to rise immediately and [K⁺]_i begins to fall. V_m quickly goes positive by ≈2 mV and then rises more slowly toward 0 mV. Volume and [Cl⁻]_i rise after a lag, i.e., with a sigmoid time course. Cl⁻ movements are controlled by V_m and the rapid phase of Cl⁻ entry does not occur until V_m has fallen to a low value, in response to falling [K⁺]_i. The internal Cl⁻ concentration slowly approaches 150 mM, the concentration in the extracellular solution, while the volume continues to climb toward infinity. The rate of volume increase slows as the surface-to-volume ratio of the cell decreases, but the increase is inexorable: with the pump stopped, there is no equilibrium until the volume is infinite. In these conditions there is no osmotic room for the vital S⁻ substances in the cell, but they are trapped inside nonetheless. The result is that swelling continues until S⁻ is infinitely dilute.

• Download figure
• Open in new tab
• Download powerpoint

Figure 4

The pump, Cl⁻, and cell volume. (A) Turning off the pump of an unperturbed cell leads to an exchange of Na⁺ for K⁺ internally. As V_m rises, Cl⁻ is no longer in equilibrium and enters the cell. The volume increases as the result of increased cell content of Na⁺ and Cl⁻ (Na⁺ enters faster than K⁺ is lost). Swelling continues, in theory, until volume is infinite. (B) P_Cl is decreased to zero at the arrow. When the pump is turned off, the changes are similar to those in A except that Cl⁻ cannot enter and volume remains constant. On restoring P_Cl to 2, V_m jumps negative, pulled by Cl⁻ current, then moves toward zero as Cl⁻ equilibrates. The negative jump of V_m causes a transient increase in [K⁺]_i as internal negativity pulls K⁺ inward. The quick volume increase causes a transient dilution of Na⁺. Volume then increases slowly toward infinity, as Na⁺ and Cl⁻ enter the cell. The total cell content of S⁻ remains fixed, but [S⁻] falls as volume increases.

The crucial importance of low internal [Cl⁻] to osmotic stability is made evident when this experiment is repeated with P_Cl made zero by adding an imaginary toxin. In Fig. 4B the toxin is added to a cell in standard conditions at the time indicated (P_Cl⁻ > 0), producing no detectable effect because Cl⁻ is in equilibrium. At the second arrow the pump was turned off, causing (i) an immediate efflux of K⁺; (ii) an immediate influx of Na⁺; and (iii) a small quick jump of V_m, followed by a slower increase toward 0 mV that is mainly the result of falling [K⁺]_i. Thanks to the absence of Cl⁻ permeability, [Cl⁻]_i does not change. Most importantly for the present argument, there is no volume change. It is important to remember that [Cl⁻]_i was initially low in the cell, making osmotic room for S⁻. Volume remains normal (1.0) because Cl⁻ cannot enter.

After the model reaches a steady state, the imaginary toxin is removed (P_Cl = 2). V_m jumps to −68 mV thanks to I_Cl (E_Cl is −80 mV) and then rises toward zero as Cl⁻ enters the cell. The most important point is that volume rises concurrent with the entry of Cl⁻ and continues to rise until S⁻ is infinitely dilute. There is a transient increase in [K⁺]_i as internal negativity draws K⁺ into the cell. The transient decrease of [Na⁺]_i observed is simply because of dilution: volume rises faster than Na⁺, with its limited permeability, can follow.