Note that this parameter set has been used in all figures unless otherwise specified.

The work presented here derives from a long line of work in cellular physiology. I thought it useful to present it in part as a tutorial, since some of the information is collected in rather specialized publications, often directed at excitable cells. I wanted to present it as one concise narrative rather than as a tedious opera. I hope to make clear that there is a direct connection between the number of impermeant ions in a cell and its size. The elementary physics that is required to understand electrical current flow in biological systems is succinctly covered in the appendices of the following textbooks (Nicholls et al., 2011; Blaustein et al., 2012).

In what follows I will set up a model that incorporates all the forces and constraints that act on ions and water, both within and outside the cell, to determine how the ions and water will distribute as time moves on. A crucial feature of the model is that the membrane is assumed to be freely distensible, so that as water moves in it will expand correspondingly and shrink if water moves out.

In addition, I will assume that the composition of the solution in the extracellular space is fixed, which is reasonable since the volume of the extracellular space if far larger than that of a single cell. I will also set aside spatial effects and assume that the cell, for the purposes of this paper, can be considered a single iso-potential sphere (i.e., all points within the cell are at the same potential) and the voltage in the extracellular space is zero.

To improve the flow of the equations I have adopted the following conventions: capitalized solutes represent concentrations, and lower cases represent amounts in moles. Extracellular ions are denoted by a subscript “o” for “outside,” intracellular ions have no subscript.

Animal cell membranes, being somewhat fragile, cannot sustain much transmembrane pressure nor expand much (see however Sachs and Sivaselvan, 2015). Therefore, because membranes are permeable to water, the osmolarity of the extracellular fluid should match closely that of the intracellular fluid. Over 70 years ago Boyle and Conway (1941) realized that the impermeant molecules (metabolites, proteins, nucleic acids etc.) which cells must contain, impose an osmotic load on cells.

To see how this arises I will consider the effect of impermeant ions on the osmotic balance of cells. Since the osmotic strength of a solution is a colligative property (Atkins and de Paula, 2014), all the impermeant molecules can be lumped together and considered as a single chemical species, without any loss of precision, which I will term x (where X is its osmolarity and x the number of osmoles) with an average charge z. It should be noted that some molecules might have osmotic coefficients less than one.

It can be shown (see below) that a cell which has a pliant membrane, permeant to monovalent ions and containing impermeant molecules, immersed in a saline solution cannot reach a stable steady state without the input of energy and will burst (Weiss, 1996). Whereas in the absence of impermeant molecules the “cell” is stable if the solution within the cell is identical to that outside.

The influence of the impermeant molecules is called the Donnan (or Gibbs-Donnan) effect, which has been and continues to be a concept that has been misapplied and misunderstood even in contemporary textbooks. Although a Donnan equilibrium is not possible in a live animal cell, x does exert an effect on the cell and cannot be ignored. Tosteson and Hoffman (1960) showed that the operation of an energy consuming NKA stabilizes cell volume in the presence of x and water permeability. It is this PLM that I will discuss in the rest of this article.

To see how x and z, together with the distributions of Na⁺, K⁺, and Cl⁻ determine cell volume I consider two physico-chemical constraints on cells:

Osmotic balance. Since the membrane of animal cells cannot support much of a transmembrane pressure, the osmotic strength of the intracellular solution should be equal to that of the extracellular solution (Π_o): (1)Na + K + Cl + X = Πo

Where w is the cell volume. This also holds true for the extracellular solution.

From these two conditions I can derive an expression for the cell volume:

Where V is the transmembrane potential. The charge imbalance that gives rise to the transmembrane potential is so small as to be chemically undetectable (Burton, 1975). The only assumption for the second part of this equation is that the chloride distribution is at equilibrium.

From this equation it is clear that the volume has a simple dependence on x and its charge, z. This equation was derived by Boyle and Conway (1941), and I follow Fraser and Huang (2004) in calling it the “constraint equation.”

The constraint equation shows for a given z, which values of X and V are consistent with a stable volume (Figure 1, see gray lines). However, it only determines the sum of Na⁺ and K⁺. To determine the individual concentrations, I have to consider the forces acting on all the ions.

Two important implications of the constraint equation are that to accommodate x, the voltage needs to be negative, and with a voltage of zero the volume goes to infinity i.e., the cell is unstable. A useful way of thinking about the PLM is that it creates a negative potential and Cl⁻ follows its equilibrium making space to accommodate x.

To complete the model of the cell, equations are required that describe the forces acting on the three mobile ions, Na⁺, K⁺, and Cl⁻. These forces are chemical potential (diffusion), electric, and active ion transport. This gives three equations for the rate of change of the intracellular ion concentrations:

Where A is the membrane area (assumed to be constant) and F Faraday's constant. To break the expressions down into their components, the term gNa[V-RTFln(NaoNa)] is Ohm's law, which combines the chemical and electrical potentials, and RTTln(NaoNa) is often referred to as the “Nernst potential.” While, the NKA is represented by a constant pump rate p with n and q being the number of Na⁺ and K⁺ ions, respectively transported per cycle. As I show in the Appendix (A2, Figure A1) the simplified form of the NKA, where it is represented by constant terms in Equations (4) and (5), has little impact on my overall conclusions. The last term in the equations reflects the influence of volume changes on ion concentrations. This term is very small and can be dropped without incurring much error.

The assumption of constant area is adopted to reflect the fact that over a short time scale the number of channels and transporters is unlikely to change much.

The flow of water can be modeled by the following equation (Fettiplace and Haydon, 1980):

Where P_f is the osmotic water permeability coefficient of the membrane and ν_w is the partial molar volume of water (18 cm³ mol⁻¹). Water permeability even in the absence of aquaporins is higher than the ionic permeability (Verkman, 1992) and I will assume that water instantaneously equilibrates across the membrane, unless otherwise noted.

From the definition of capacitance, the voltage is given by the following equation (Varghese and Sell, 1997):

I shall refer to the system of Equations (1), (2), (4), (5), (6), and (8) as the “pump-leak equations” (PLE). In 1998, Keener and Sneyd found an analytical solution for the steady state of the PLEs, but this important work has as yet not made its way into the biological literature. The advantage of an analytical solution is that it makes evident the influence of the various factors at play in regulating cell volume. Moreover, it offers a convenient check of numerical methods used for solving the PLEs. Keener and Sneyd's solutions (KSS) to the PLEs are given in Appendix A1.

No closed-form solutions are available for the approach of the system to steady-state, hence the system has to be solved by numerical integration.

To solve numerically the PLEs I integrate Equations (4–7) and use the algebraic Equation (8) to calculate V. Fraser and Huang (2007), termed this the “charge-difference” (CD) method, which explicitly takes into account the impermeant ions, x and their charge, z. I have found that the CD approach converges to the KSS, with reasonable initial conditions (data not shown).

If I begin the simulation with some arbitrary values of ion concentrations and turn on the NKA the system approaches the steady state given by the KSS. If after reaching steady state the NKA is turned off, the ions equilibrate across the membrane and the volume increases continuously, showing that the system is unstable in the absence of NKA activity (Figure 2). An important feature of the PLM is its robustness; a stable volume can be achieved with a very wide array of combinations of channel conductances and extracellular ion concentrations, so long as nNaogNa>qKogK and the p not too high (Keener and Sneyd, 2009; Mori, 2012). The PLM also stabilizes cells against sudden changes in extracellular osmolarity, which is followed by the cell changing to a new stable steady state volume (Figure 3).

It is worth noting that if a cell has no impermeant ions, which is clearly impossible, the system is stable in the absence of an NKA. However, with no x, the operation of an NKA has the paradoxical effect of rendering the system unstable (see Appendix A3).

The impermeant molecules x exert an osmotic effect on the system and constrain the size of the cell. The mean charge on these molecules, z, also has a powerful influence on the ion distributions, voltage and volume of the cell. This can be seen by plotting these variables as a function of z with a fixed number of moles of x (Figure 4).

The action of the NKA can be made evident by plotting ion concentrations, V and volume as a function of the pump rate, p What I will term, Cp curves, provide a way of visualizing the forces at play as p is ramped up and the effect of p on the apportioning of ions. When one plots the concentration of intracellular ions as a function of pump rate the ion concentrations form a braid (symmetrical for the case of z = −1 because of the osmotic and charge constraints (Figure 5, 2nd panel from the top and see Appendix A4).

If, as in Figure 5 (top panel), the cations and anions in the Cp plots are grouped separately, the preservation of charge and osmotic balance become obvious; the sum of all intracellular species is equal to Π_o, while the sum of the positive charges is equal to that of the negative charges. This also holds for z ≠ −1 but relationships between the ion concentrations and p become somewhat more complicated (see Appendix A5, Figure A2).

As p is increased, although Na⁺ and K⁺ are being pumped, it is Cl⁻ and X that first respond to this applied force. Why is this so? Because the net action of the NKA during a cycle is to move one positive charge out of the cell, it makes the inside of the cell more negative (i.e., hyperpolarizes), which drives Cl⁻ out of the cell passively requiring little energy, while setting up the asymmetric Na⁺/K⁺ distribution, which is far out of equilibrium, requires a greater energetic input.

An alternate way of viewing the PLM is shown in Figure 1 where X vs. V derived from the KSEs is plotted for a given p (blue lines). The intersections between the CE equation and the X vs. V serve as attractor points, so that if the system starts at a nonequilibrium point it will move to the intersection when the pump is turned on. If the pump is turned off the system moves to V = 0 and the volume goes to infinity.

The PLEs can be further simplified by assuming that the NKA can actively transport only one of the monovalent cations.

If Na⁺ transport is eliminated but all other aspects of the PLM are left intact, no stable solutions are possible (see Appendix A6). This happens because the pump generates an inward current that depolarizes the cell and draws Cl⁻ in. Na⁺ cannot follow the voltage because of its high extracellular concentration and it remains out of register with the V throughout the Cp curve. This mechanism is incapable of generating a stable volume and generates a strong depolarization.

If K⁺ transport is eliminated but all other aspects of the PLM are left in place, there is very little difference in the operation of the mechanism as seen in a Cp plot (Figure 5). K⁺ and Cl⁻ can move passively and attempt to follow the voltage. As mentioned above the Na⁺ pump hyperpolarizes the cell. This drives Cl⁻ out and K⁺ in. For this case, the K⁺ and Cl⁻ follow the Donnan relationship (viz. K Cl = K_oCl_o) exactly and are both in equilibrium with the voltage (see Appendix A6). Because the system requires ATP to sustain a steady-state, the system as whole is not at equilibrium.

The actual NKA requires both Na⁺ and K⁺ to operate, however my analysis shows that Na⁺ transport is the only necessary component if one thinks of its role in volume regulation. Dropping active K⁺ transport simplifies the PLEs so that it becomes possible to arrive at analytical solutions to the intersection points on Cp plots and to see how they depend on the systems parameters (see Appendix A4).

The effect of the Cl⁻ conductance on the PLM is entirely passive but it is an essential one, since if the conductance is blocked Cl⁻ cannot equilibrate and the mechanism is blocked. In many cells Cl⁻ is not passively distributed but is pumped and this will have an effect on the volume regulatory mechanism but is beyond the scope of this paper (Kaila et al., 2014; Vereninov et al., 2016).

The pump rate (p) is a direct measure of the energy needed to sustain the volume and voltage of the cell, since it is directly coupled to ATP hydrolysis. In the PLM, as the pump rate is increased, the volume declines and reaches a minimum at what I will call p_min (Figure 5). If p is increased above this, the volume then increases very slowly. For the case where z = −1, p_min can be found by differentiating Equation (A12) with respect to p, setting this derivative equal to zero and solving for p:

Equation (9) encapsulates how the energy required to maintain cell volume depends on the systems parameters. p_min can be minimized by keeping g_Na small but is less sensitive to the magnitude of g_K (Figure 6). It is worth noting for the case where q = 0 the volume declines to a minimum but does not increase as p is increased.

A standard method for predicting the electrical properties of cells is to build up an equivalent circuit model using capacitors, batteries and conductances; where the latter are often voltage gated (Kay, 2014). Because of ion accumulation and the variable volume, there is no simple way of devising such a circuit to represent the PLM. However a partial circuit representation can be developed, if I use the KSEs to estimate the ion distribution and steady state voltage at a given pump rate. A current-voltage (IV) relationship of the Na⁺ and K⁺ currents can then be used to show how current flow is apportioned. This is shown in Figure 7 for two different values of the pump current. The Nernst potentials of the K⁺ (i.e., RTFln(KoK)) and Na⁺ (i.e., RTFln(NaoNa)) conductances can be calculated from the their respective intra- and extracellular concentrations. Notice that at the resting potential both the passive Na⁺ and K⁺ currents are exactly balanced by an equal and opposite pump current, in the ratio 3:2. The system is stable because deviations from the resting potential will induce currents that restore it to rest.

The PLM is not exclusively dependent upon the operation of a NKA; there are other pumps that can substitute. All that is required is a mechanism for pumping Na⁺ out of the cell, together with passive Na⁺, K⁺, and Cl⁻ conductances, and water permeability. Mycoplasma, which are bacteria that do not have cell walls, provide a nice example for looking at alternative PLMs. Like other bacteria and archea they do not have a NKA, but they do have a proton pump (Krulwich et al., 2011) and a Na⁺/H⁺ transporter (Padan and Landau, 2016), where the proton gradient drives the outward flux of Na⁺. Figure 8 shows that the operation of a proton pump in conjunction with a Na⁺/H⁺ exchanger can stabilize a cell. Blocking the proton pump induces mycoplasma to lyse (Linker and Wilson, 1985).

The Donnan effect is frequently introduced in physiological textbooks, however the reader is often left hanging as to its precise implications for cell function. As I have shown, the effect of trapped molecules within the cell has a significant impact on the behavior of the cell; but it should be emphasized that a true (i.e., with no energy input) Donnan equilibrium cannot be attained, unless the membrane can sustain a very high transmembrane pressure. In the hypothetical case where Na⁺ is pumped actively while K⁺ and Cl⁻ distribute passively, these ions distribute in accord with the Donnan relationship (see above), the system is in a dynamic steady-state. Misunderstanding the Donnan effect, as has been pointed out by others (Voipio et al., 2014), can lead to erroneous ideas about the influence of impermeant ions on ion distributions.

It is worthwhile making a distinction between the PLM and the so-called Double-Donnan mechanism, because these two are sometimes conflated. The Double-Donnan mechanism (Leaf, 1959) proposes that the asymmetric ion distribution is achieved by the operation of a NKA together with a membrane that is permeable to K⁺, Cl⁻ and water but not to Na⁺. It is probably best to avoid this term since all membranes have a finite permeability to Na⁺. Moreover, if the sodium conductance is zero, the system of equations is unstable, with the Na⁺ concentration becoming negative, which is clearly impossible.

x is a heterogeneous collection of molecules, yet both it and z can be calculated. To do this one needs to have good estimates of the extracellular osmolarity and the intracellular concentrations of the predominant permeable anions and cations. One can then use Equation (1) to estimate x and Equation (2) z. The estimates are of course only as good as the intracellular concentration estimates, which as yet are difficult to obtain, absent sensitive and accurate monovalent ion sensors. Estimates of z have been made for muscle and some cell lines and ranges between −2 and −0.7 (Burton, 1983a; Model, 2014). Both x and z are unlikely to be fixed and will vary as impermeant molecules are metabolized or transported and as the pH changes. A precise accounting for the contributions to z has yet to be done. The contributions of nucleic acids and proteins will depend on how tightly counter ions bind (Raspaud et al., 2000). Some texts refer to x as the “anion gap,” which is sometimes incorrectly attributed to intracellular proteins.

If one is unaware of the role of the NKA in volume regulation, the presence of Na⁺, K⁺, and Cl⁻ leaks seems counterproductive as they impose an energetic load on the cell. It is only if one takes into account the central role of the NKA in volume regulation, that the function of the leak conductances become apparent as key components of the PLM. There are a large variety of leak channels (Ren, 2011; Feliciangeli et al., 2015) and it seems likely that they may have evolved to help stabilize cell volume. Moreover, g_Na has a profound effect on the energy utilization of a cell, with energy consumption in direct proportion to g_Na (Figure 6). Driving the NKA is one of the most energetically costly processes for animals, with an estimated 20% of a cell's energy consumption needed to keep it running (Rolfe and Brown, 1997; Milo and Phillips, 2015). This large energy investment is a clear indication of the importance of the PLM.

As I have shown, there is a minimum pump rate to achieve a stable volume. Operating above this rate does not change the volume much. As the pump rate declines below p_min, the volume increases monotonically; however, even for very low pump rates, the volume is stable. This constitutes something of a problem for evaluating the PLM experimentally; even with very low pump rates the volume may be stable. In addition it is possible that other pumps may take over and sustain the volume. Moreover, as my simulations show (Figure 2), it may take a long time for ion gradients to dissipate after the application of a pump blocker like ouabain.

It is worth noting that the leak current is also an essential part of the mechanism that makes cells excitable. If one were to remove all leak currents, which is clearly not possible, from a cell with voltage gated Na⁺ and K⁺ channels, there would be no stable resting potential (Kay, 2014). Furthermore the PLM provides a simple explanation for the small changes in axon dimensions induced by action potential propagation (Cohen, 1973; Lee and Kim, 2010).

Equivalent circuit models are often used to simulate neurons, but most models do not, with some exceptions (Kager et al., 2002; Ostby et al., 2009; Ullah et al., 2015), incorporate water and ion fluxes, impermeant molecules and distensible membranes. Cellular swelling occurs after cerebral trauma, stroke, spreading depression and epilepsy (Rungta et al., 2015; Stokum et al., 2016). There is a pressing need for pharmacological interventions that limit swelling and prevent the damage resulting from cells being damaged while expanding into the closed volume of the cranium. Developing such agents requires a detailed understanding of all the factors that play a role in cell volume regulation and the theory that undergirds this mechanism. The operation of the PLM in excitable cells like neurons, with more vigorous passive ion fluxes than in other cells, imposes a more severe energetic demand on neurons and muscle. This by itself may explain why neurons are so sensitive to oxygen deprivation.

The maintenance and restoration of Na⁺ and K⁺ gradients is the most conspicuous function of the NKA, yet its role in volume regulation is seldom recognized. Stein (1995) has argued that the evolution of the NKA and the PLM is a defining characteristic of animal cells. It endows them with an osmoregulatory mechanism that does not require rigid cells and it gives them a negative resting potential, which turned out to be very convenient for evolving excitable cells (Jakobsson, 1980; Armstrong, 2015). The PLM also liberates cells from the straitjacket of cell walls allowing the evolution of contractile mechanisms.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.