Many cells have a more difficult problem to solve, that of maintaining their cell volume in widely varying conditions, while transporting large quantities of ions through the cell. Here we present a simplified model for transport and volume regulation in a Na+-transporting epithelial cell.

As are virtually all models of transporting epithelia, the model is based on that of Koefoed-Johnsen and Ussing (1958), the so-called KJU model. In the KJU model, an epithelial cell is modeled as a single cell layer separating a mucosal solution from the serosal solution (Fig. 2.16). (The mucosal side of an epithelial cell is that side on which mucus is secreted and from which various chemicals are withdrawn, for example, from the stomach. The serosal side is the side of the epithelial cell facing the interstitium, wherein lie capillaries, etc.) Na+ transport is achieved by separating the Na+ pumping machinery from the channels that allow Na+ entry into the cell. Thus, the mucosal membrane contains Na+ channels that allow Na+ to diffuse down its concentration gradient into the cell, while the serosal membrane contains the Na+-

K+ ATPases, which remove Na+ from the cell. The overall result is the transport of Na+ from the mucosal side of the cell to the serosal side. The important question is whether the cell can maintain a steady volume under widely varying concentrations of Na+ on the mucosal side.

We begin by letting N, K, and C denote Na+, K+, and Cl— concentrations respectively, and letting subscripts m, i, and s denote mucosal, intracellular and serosal concentrations. Thus, for example, Ni is the intracellular Na+ concentration, and Nm is the mucosal Na+ concentration. We now write down the conservation equations for Na+, K+, and Cl— at steady state. The conservation equations are the same as those of the pump-leak model with some minor exceptions. First, instead of the linear I-V curve used in the pump-leak model, we use the GHK formulation to represent the ionic currents. This makes little qualitative change to the results and is more convenient because it simplifies the analysis that follows. Second, we assume that the rate of the Na+-K+ pump is proportional to the intracellular Na+ concentration, Ni, rather than N?, as was assumed in the generalized version of the pump-leak model. Thus,

Ci Csev

Note that the voltage, v, has been scaled by F/(RT) and that the rate of the Na+-K+ pump is pNi. Also note that the inward Na+ current is assumed to enter from the mucosal side, and thus Nm appears in the GHK current expression, but that no other ions enter from the mucosa. Here the membrane potential is assumed to be the same across the lumenal membrane and across the basal membrane. This is not quite correct, as the potential across the lumenal membrane is typically —67 mV while across the basal membrane it is about —70 mV.

There are two further equations to describe the electroneutrality of the intracellular space and the osmotic balance. In steady state, these are, respectively, w(Ni + Ki — Ci) + zxX = 0, (2.112)

w where X is the number of moles of protein, each with a charge of zx < — 1, that are trapped inside the cell, and w is the cell volume. Finally, the serosal solution is assumed to be electrically neutral, and so in specifying Ns,Ks, and Cs we must ensure that

Since the mucosal and serosal concentrations are assumed to be known, we now have a system of 5 equations to solve for the 5 unknowns, Ni,Ki,Ci,v, and \x = w/X. First, notice that (2.109), (2.110), and (2.111) can be solved for Ni, Ki, and Ci, respectively, to et wv) = + ;Nm(;v v), (2.115)

Next, eliminating N, + K between (2.112) and (2.113), we find that

We now use (2.117) to find that

and thus, using (2.119) to eliminate / from (2.112), we get

Ni(v) + Ki(v) = [-2zx + ev(1 + zx)] = $(v). (2.120)

Since zx - 1 < 0, it must be (from (2.119)) that v < 0, and as v ^ 0, the cell volume becomes infinite. Thus, we wish to find a negative solution of (2.120), with Ni(v) and Ki(v) specified by (2.115) and (2.116).

It is instructive to consider when solutions for v (with v < 0) exist. First, notice that 0(0) = Cs. Further, since zx <-1, $ is a decreasing function of v, bounded above, with decreasing slope (i.e., concave down), as sketched in Fig. 2.17. Next, from (2.115) and (2.116) we determine that Ni(v) + K(v) is a decreasing function of v that approaches to as v ^ -c» and approaches zero as v ^to. It follows that a negative solution for v exists if Ni(0) + Ki(0) < Cs, i.e., if

This condition is sufficient for the existence of a solution, but not necessary. That is, if this condition is satisfied, we are assured that a solution exists, but if this condition fails to hold, it is not certain that a solution fails to exist. The problem, of course, is that negative solutions are not necessarily unique, nor is it guaranteed that increasing Nm through Ns j+p causes a negative solution to disappear. It is apparent from (2.115) and (2.116) that N/(v) and Ki(v) are monotone increasing functions of the parameter Nm, so that no negative solutions exist for Nm sufficiently large. However, for Nm = Ns to be the value at which the cell bursts by increasing Nm, it must also be true that

Figure 2.17 Sketch (not to scale) of the function ), defined as the right-hand side of (2.120), and of N,(v) + K,(v), where N, and K, are defined by (2.115) and (2.116). 0(v) is sketched for zx < — 1.

or that

For the remainder of this discussion we assume that this condition holds, so that the failure of (2.122) also implies that the cell bursts.

According to (2.122), a transporting epithelial cell can maintain its cell volume, provided that the ratio of mucosal to serosal concentrations is not too large. When Nm/Ns becomes too large, \x becomes unbounded, and the cell bursts. Typical solutions for the cell volume and membrane potential, as functions of the mucosal Na+ concentration, are shown in Fig. 2.18.

Obviously, this state of affairs is unsatisfactory. In fact, some epithelial cells, such as those in the loop of Henle in the nephron (Chapter 20), must work in environments with extremely high mucosal sodium concentrations. To do so, these Na+-transporting epithelial cells have mechanisms to allow operation over a much wider range of mucosal Na+ concentrations than suggested by this simple model.

From (2.122) we can suggest some mechanisms by which a cell might avoid bursting at high mucosal concentrations. For example, the possibility of bursting is decreased if pn is increased or if pk is decreased. The reasons for this are apparent from (2.115) and (2.116), since Ni(v) + Ki(v) is a decreasing function of pn and an increasing function of pk. From a physical perspective, increasing Nm causes an increase in Ni, which increases the osmotic pressure, inducing swelling. Decreasing the conductance of sodium ions from the mucosal side helps to control this swelling. Similarly, increasing the conductance of potassium ions allows more potassium ions to flow out of the cell, thereby decreasing the osmotic pressure from potassium ions and counteracting the tendency to swell.

It has been conjectured for some time that epithelial cells use both of these mechanisms to control their volume (Schultz, 1981; Dawson and Richards, 1990; Beck et al., 1994). There is evidence that as Ni increases, epithelial cells decrease the Na+ conduc-

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Figure 2.18 Numerical solutions of the model for epithelial cell volume regulation and Na+ transport. The membrane potential I/, the scaled cell volume and the intracellular Na+ concentration [Na+],- are plotted as functions of the mucosal Na+ concentration. The solid lines are the solutions of the simpler version of the model, where PNa and PK are assumed to be constant. The dashed lines are the solutions of the model when PNa is assumed to be a decreasing function of N/, and PK is assumed to be an increasing function of w, as described in the text. Parameter values are Ks = 2.5, Ns = 120, Cs = 122.5, P = 2, y = 0.3, zx = -2. All concentrations are in mM.

tance on the mucosal side of the cell, thus restricting Na+ entry. There is also evidence that as the cell swells, the K+ conductance is increased, possibly by means of stretch-activated K+ channels (Ussing, 1982. This assumption was used in the modeling work of Strieter et al., 1990).

To investigate the effects of these mechanisms in our simple model, we replace PNa by PNa20/M (20 is a scale factor, so that when Ni = 20 mM, PNa has the same value as in the original version of the model) and replace PK by PKw/w0, where w0 is the volume of the cell when Nm = 100 mM. As before, we can solve for v and ¡x as functions of Nm, and the results are shown in Fig. 2.18. Clearly the incorporation of these mechanisms decreases the variation of cell volume and allows the cell to survive over a much wider range of mucosal Na+ concentrations.

The model for control of ion conductance used here is extremely simplistic, as for example, there is no parametric control of sensitivity, and the model is heuristic, not mechanistic. More realistic and mechanistic models have been constructed and analyzed in detail (Lew et al., 1979; Civan and Bookman, 1982; Strieter et al., 1990; Weinstein, 1992, 1994, 1996; Tang and Stephenson, 1996).

2.9 Exercises_

1. Find the maximal enhancement for diffusion of carbon dioxide via binding with myoglobin using Ds = 9 x 10-4 cm2/s, k+ = 2 x 108 cm3/M • s, k- = 1.7 x 10-2a. Compare the amount of facilitation of carbon dioxide transport with that of oxygen at similar concentration levels. (Hint: For oxygen p = 14, whereas for carbon dioxide, p = 2.)

2. Devise a model to determine the rate of production of product for a spherical enzyme capsule of radius R0 in a bath of substrate at concentration S0. Assume that the enzyme cannot diffuse within the capsule but that the substrate and product can freely diffuse into, within, and out of the capsule. Show that spheres of small radius have a larger rate of production than spheres of large radius.

Hint: Reduce the problem to the nondimensional boundary value problem

and solve numerically as a function of a. How does the radius of the sphere enter the parameter a?

3. Suppose a membrane that contains water-filled pores separates two solutions.

(a) Suppose that the solution on either side of the membrane contains an impermeant solute. Show that the hydrostatic pressure of the water within the pores must be less than the hydrostatic pressure of the solutions on either side of the membrane.

(b) Show that if the solute can permeate the pore freely, no such drop in hydrostatic pressure exists.

(c) Show that if the solution on one side of the membrane contains both permeant and impermeant solutes, while the solution on the other side contains only the perme-ant solute, it is possible for water to flow against its chemical potential gradient (at least temporarily). This problem of wrong-way water flow has been observed experimentally, and it is discussed in detail by Dawson (1992).

4. Red blood cells have a passive exchanger that exchanges a single Cl- ion for a bicarbonate (HCO-) ion. Develop a model for this exchanger and find the flux.

5. Modify (2.52) to take into account the fact that the concentration of ATP affects the rate of reaction (since pumping should stop if there is no ATP).

6. Generalize (2.53) to account for the fact that with each turn of the sodium-potassium pump three sodium ions are exchanged for two potassium ions.

7. Almost immediately upon entering a cell, glucose is phosphorylated in the first reaction step of glycolysis. How does this rapid and nearly unidirectional reaction affect the transmembrane flux of glucose as represented by (2.41)? How is this reaction affected by the concentration of ATP?

8. How does the concentration of ATP affect the rate of the sodium-potassium pump?

9. The process by which calcium is taken up into the sarcoplasmic reticulum (SR) in muscle and cardiac cells is similar to the sodium-potassium ATPase, but simpler. Two intracellular calcium ions bind with a carrier protein with high affinity for calcium. ATP is dephospho-rylated, with the phosphate bound to the carrier. There is a conformational change of the carrier protein that exposes the calcium to the interior of the SR and reduces the affinity of the binding sites, thereby releasing the two ions of calcium. The phosphate is released and the conformation changed so that the calcium binding sites are once again exposed to the intracellular space.

Formalize this reaction and find the rate of calcium uptake by this pump.

10. A 1.5 oz bag of potato chips (a typical single serving) contains about 200 mg of sodium. When eaten and absorbed into the body, how many osmoles does this bag of potato chips represent?

11. Generalize formula (2.86) to take into account that the two fluids have different densities and to allow the columns to have different cross-sectional areas.

12. Two columns with cross-sectional area 1 cm2 are initially filled to a height of one meter with water at T = 300K. Suppose 1 gm of sugar is dissolved in one of the two columns. How high will the sugary column be when equilibrium is reached? Hint: The weight of a sugar molecule is 3 x 10-22 gm, and the force of gravity on 1cm3 of water is 980 dynes.

13. Suppose an otherwise normal cell is placed in a bath of high extracellular potassium. What happens to the cell volume and resting potentials?

14. Based on what you know about glycolysis from Chapter 1, how would you expect anoxia (insufficient oxygen) to affect the volume of the cell? How might you incorporate this into a model of cell volume? Hint: Lactic acid does not diffuse out of a cell as does carbon dioxide.

15. Suppose 90% of the sodium in the bath of a squid axon is replaced by inert choline, preserving electroneutrality. What happens to the equilibrium potentials and membrane potentials?

16. Determine the effect of temperature (through the Nernst equation) on cell volume and membrane potential.

17. Write and analyze the balance equations for a cell in a finite bath. Hint: In a finite bath the total volume is conserved as are the total number of sodium, potassium, and chloride ions.

18. Simulate the time-dependent differential equations governing cell volume and ionic concentrations.

19. Many animal cells swell and burst when treated with the drug ouabain. Why? Hint: Ouabain competes with K+ for external potassium binding sites of the Na+-K+ ATPase. How would you include this effect in a model of cell volume control?

CHAPTER 3

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