Mathematical modeling of intracellular osmolarity and cell volume stabilization: The Donnan effect and ion transport

Cells cannot function without a significant inventory of metabolites and macromolecules, typically around 100 mOsm (Milo and Phillips, 2015). However, this large total concentration of molecules poses a challenge to cellular stability because of the high water permeability of lipid bilayers (Fettiplace and Haydon, 1980). The presence of impermeant molecules within a cell establishes a Donnan (or Gibbs–Donnan) effect that, if left unchecked, will lead to the cell volume increasing because of the influx of water driven by osmosis, until the cell lyses. The Donnan effect can be counteracted by establishing a cell wall that allows the development of a turgor pressure, which is what plants and bacteria do. Animals evolved a strategy, the pump-leak mechanism (PLM) (Tosteson and Hoffman, 1960), whereby Na⁺ is actively pumped out of cells while K⁺ is pumped in. This effectively stabilizes the cell by equalizing the osmolarity across the membrane but requires continuous energy expenditure to preserve this dynamic steady state. The operation of the PLM leads to the marked asymmetry in Na⁺ and K⁺ concentrations inside and outside cells, which together with a higher permeability to K⁺ than Na⁺ leads to the development of a negative membrane potential (Weiss, 1996). The effect of the PLM is to effectively hold the Donnan effect at bay, forestalling a Donnan equilibrium.

The ion transport properties of a cell are typically modeled by measuring or abstracting from the literature the characteristics of the ion channels and transporters, incorporating them into a model, and determining if the model recapitulates the physiology of the cell. In contrast, we are attempting to assess the global characteristics of a set of ion channels and transporters, not just single instances. In our case Na⁺, K⁺, and Cl⁻ ion channels, the Na⁺/K⁺-ATPase (NKA), and three kinds of SLC12 transporters are considered. It is an attempt to determine whether qualitatively different behaviors exist in the model’s parameter space. Conventional simulations can recapitulate behavior; our approach aims to provide a global perspective on such systems.

The cation–chloride co-transporters (CCCs) are a family (SLC12) of the Solute Carrier transporter superfamily, which transport Na⁺ and/or K⁺ and Cl⁻ across the cell membrane (Arroyo et al., 2013). There are three types of electroneutral CCCs: the Na⁺–Cl⁻ co-transporters, NCC; the Na⁺–K⁺–Cl⁻ co-transporters, NKCC; and the K⁺–Cl⁻ co-transporters, KCC. The CCCs are secondary transporters that depend on the nonequilibrium ion gradients established by the NKA to drive Cl⁻ out of equilibrium and perturb the steady state of Na⁺ and K⁺. Hence, in the absence of an active NKA, the CCCs cannot operate.

The SLC12 family plays a significant role in numerous aspects of physiology, and the literature on this topic is extensive. Therefore, we will not attempt to summarize it here. However, we encourage readers to refer to recent summaries for comprehensive information (Delpire and Gagnon, 2018; Chew et al., 2019; Pressey et al., 2022). Our objective is to determine the effect of the CCCs on ions and volume regulation from the mathematical relationships that encapsulate the CCCs behavior.

An important part of modeling the transport characteristics of cells is incorporating water permeability. Doing this requires that one include the mechanics of the cell since differences in osmolarity across the plasma membrane will allow water flux and hence volume changes. Moreover, once one introduces water flux, it opens the possibility that the cell might not be stable since it can now shrink or lyse (Jakobsson, 1980). Most ion transport models, with some notable exceptions (Mackey, 1975; Jakobsson, 1980; Hernández and Cristina, 1998; Armstrong, 2003; Fraser and Huang, 2004; Yurinskaya et al., 2005; Kager et al., 2007; Østby et al., 2009; Ullah et al., 2015; Dijkstra et al., 2016), do not include water flux, which would mask any instabilities that might arise. Part of the objective of this study is to call attention to the importance of including the osmotic flux of water in any models that attempt to model ionic homeostasis in cells.

The pump-leak equations (PLEs) are a set of four differential equations and one algebraic equation that typically model the PLM. These equations describe the dynamics on Na⁺, K⁺, and Cl⁻, the cell volume, and the membrane potential (see Results for more details). The PLEs were first proposed by Tosteson and Hoffman in 1960 and have been employed and extended by others to understand cell volume control in single cells (Jakobsson, 1980; Hernández, 2007; Vereninov et al., 2014) and epithelial (Lew et al., 1979; Sharp et al., 2015). Keener and Sneyd (2009) derived a very useful analytic solution to the PLEs. Mori (2012) established analytical results on the existence and stability of steady states for a general class of PLEs. Here, we have extended the PLEs to include impermeant extracellular molecules and CCCs. We show that the former can help establish an equilibrium that requires no energy to sustain. From this passive case, we then move to the active case with an NKA.

There is a range of models for the NKA from a simple constant model to complex nonlinear models (Garay and Garrahan, 1973; Smith and Crampin, 2004; Keener and Sneyd, 2009; Yurinskaya et al., 2020). We will show that the mathematical form of these models does not affect the steady-state of a cell, and therefore, considering a constant rate model for the NKA pump suffices. Operation of the NKA requires ATP, and it is estimated that it expends ∼20–30% of a cell’s energy (Rolfe and Brown, 1997). Using our equations, we show how cellular conductances and pump stoichiometry influence energy expenditure to preserve a constant volume. In the last third of the paper, we examine the effect of the CCCs on ion distributions and cellular stability. Interestingly, we show that the NCC has the capability of destabilizing cells, which might explain its restricted distribution.

As with any theory, there are implications within the model that are not self-evident. One of our objectives is to delve into the model and make more of its predictions explicit. Expanding the range of predictions of the PLM gives experimentalists more targets for testing the model.

The rest of the paper is organized as follows: In Results, we review the PLEs for a single cell. In the following sections, we examine the effect of three factors on cell stabilization. In the section Passive cells: The effect of…, we consider a passive cell (i.e., one where the NKA is not active) and show how extracellular impermeant molecules stabilize a cell. In this section, we extend the work of Fraser and Huang (2007) to include Na⁺ permeability. In the section Active cells: The effect of…, we first show that the mathematical form of the NKA pump does not affect the steady states, and only its stoichiometry matters. Then, we consider a constant pump, derive an explicit steady state, and show how various model parameters affect the existence and stability of such steady states. In the section Active cells: Interaction of…, we incorporate three co-transporters into the model of an active cell and explore their effect on cell volume regulation. Finally, we conclude at the end, and provide more details in Materials and methods and the Appendix.

All computations were performed using MATLAB, release R2020b (The MathWorks, Inc.). All graphs were produced with OriginPro (OriginLab).

A small capacitance (C) allows rapid changes in voltage. In this case, the PLEs become “stiff,” and special numerical solvers are required to solve this system. In this work, for the case of the non-linear NKA models, the system of equations was solved using the MATLAB stiff differential equation solver ode15s or ode23tb (the results are similar, only their speeds are different for different parameters.)

In this paper, we examine the conditions that will stabilize the volume of a cell or work against it using the PLM. As shown in Fig. 1, we consider a simple cell with a pliant membrane that is permeable to Na⁺, K⁺, Cl⁻, and water, immersed in an extracellular medium that has a fixed concentration. In this work, for the sake of simplicity, we disregard pH and therefore H⁺ and HCO₃⁻; however, we leave them for future explorations and reference the work of Li et al. (2021). We further assume that concentrations and voltages are uniform, showing no spatial gradients. For most single cells, this is a realistic assumption since diffusion is rapid on the micron scale. For two adjacent cells coupled through the cleft gap, spatial variability becomes essential (Weinstein and Stephenson, 1979; Dvoriashyna et al., 2018).

To begin, we review the construction of the PLEs and show their firm grounding in physical principles. There are two major physico-chemical forces that are at play across the membrane and exert a significant role in determining cell volume. First, water moves across the plasma membrane if there is a gradient of osmolarity across the membrane. Second, the interaction between charges in the solution, governed by Maxwell’s laws, ensures that no significant imbalances in charge arise (Dickinson et al., 2011). We write down a set of equations that determine how ions and water move in and out of the cell.

The PLEs are a set of parametric algebraic-differential equations that can be used to predict how the intracellular ion concentrations, voltage, and volume will change over time. The parameters are the ion conductances, extracellular concentrations, and transporters’ rates. For all non-zero and finite steady-state solutions of the PLEs, the osmolarity inside the cell is equal to that outside and the intracellular solution is approximately electroneutral. There are two essential questions concerning the PLEs: (1) which parameter sets result in a unique and stable steady state, and (2) how do these steady states vary when parameters are altered?

Mori addressed the first question in 2012 by considering a general PLE with constant pump/co-transporters and identified a single relationship between the parameters that ensure the existence of a unique globally stable steady state. This guarantees that the system returns to its steady state after any perturbations and precludes stable limit cycles or chaotic behavior. Mori showed that PLEs obey a free energy relationship with a single minimum, similar to the Gibbs Free Energy that governs the equilibrium of chemical reactions. He constructed a Lyapunov function (Strogatz, 1994) based on the free energy and illustrated that it decreases along the trajectories of the PLE and converges to a steady state that continuously dissipates energy. Free energy relationships are used in chemical thermodynamics to predict the equilibria of systems. However, free energy relationships can be extended to cover nonequilibrium systems that are not too far from equilibrium (De Groot and Mazur, 1962).

Our work aims to address the second question. We first derive the steady state solutions analytically for a constant NKA, without co-transporters, where all time derivatives are zero, and which gives the concentrations of all the ions, the voltage, and the cell volume. We demonstrate that these steady states are globally asymptotically stable according to Mori’s theory and then examine their dependence on different parameter values.

In the section Active cells: Interaction of…, we introduce CCCs and derive the steady states. We cannot use Mori’s result to establish the stability of these steady states, so we demonstrate their stability numerically and investigate how they depend on the model parameters. We leave the analytical analysis of the stability of such steady states for future research.

It is sometimes claimed that animal cell membranes are not permeable to water in the absence of aquaporin; however, there is much evidence that lipid bilayers are very permeable to water (Fettiplace and Haydon, 1980). The average osmotic permeability coefficient of artificial lipid bilayers is 3 × 10⁻⁴ cm s⁻¹, while that of red blood cells, which contain aquaporins, is 2 × 10⁻² cm s⁻¹ (Finkelstein, 1987).

We will call Eqs. 2, 4, and 6, a–c the PLEs (Keener and Sneyd, 2009). In Mori (2012) and Kay (2017), the authors studied the PLEs and employed the NKA pump as a mechanism to stabilize the volume of a single cell. Here, we generalized the PLEs by incorporating the CCCs and impermeant molecules in extracellular fluid. Our goal is to understand the effect of each of these on controlling the stability of a single cell.

All the parameter values are given in Table 1 unless otherwise stated.

The primary source of the osmotic instability that requires the continuous action of the NKA is the impermeant molecules within a cell. That this is so can be demonstrated by showing that if a cell does not have impermeant molecules and if all ionic species are permeable, a stable passive equilibrium can be achieved. However, if impermeant molecules are introduced, the cell becomes unstable and expands continuously. This has been shown for the case of uncharged molecules by Post and Jolly (1957), and we will show that is so for charged molecules. We will refer to the destabilizing effect of impermeant molecules as the Donnan effect.

• (1)

  The extracellular space is infinite, and hence the extracellular concentrations are fixed, and the net charge of the solution is zero.

• (2)

  The cell is assumed to be spherical, with no concentration or voltage gradients. For most single cells, this is a realistic assumption since diffusion is rapid on the micron scale and the cytoplasmic conductivity is large.

• (3)

  All components of the solutions are characterized exactly by their concentrations. However, the relationship between the concentration of all chemical species and their osmolarity is likely to be nonlinear. This is particularly marked in the case of macromolecules (Weiss, 1996). However, this assumption has no impact on the qualitative behavior of the PLM since each concentration term is simply multiplied by an ion-specific constant factor.

• (4)

  The membrane is assumed to be pliant and expand freely without generating tension, as in the original PLM (Tosteson and Hoffman, 1960), and there is no transmembrane pressure. Membrane tension can be linked to pressure through Laplace’s law and can be introduced into Eq. 2 (Düsterwald et al., 2018; Venkova et al., 2022). We have chosen to omit the membrane tension to bring the ion transport into sharper relief.

• (5)

  The reflection coefficients for all ions are assumed to be unity (Blaustein et al., 2019). This is approximately true since the water permeability is much greater than the passive permeabilities for the ions used in this paper. The reflection coefficient (σ_i) for a particular ion enters Eq. 2, by being multiplied with its corresponding concentration terms, inside and outside. For σ_i ≠ 0 this has no impact on the final steady state osmolarity, only the rate at which the steady state is approached.

In this section, we consider a passive cell, Eqs. 2, 4, and 6, a–c, without any pumps or co-transporters, and analytically study the effect of extracellular impermeant molecules on cellular stability. To this end, we compute the equilibrium values and explore the effect of [Y]_e and z_Y on them.

We will show that for a passive cell, namely one that does not expend energy, a stable thermodynamic equilibrium can be attained simply by the provision of impermeant extracellular molecules ([Y]_e > 0), which stabilizes the volume of the cell.

This contradicts the fact that [Y]_e > 0. Thus, we can conclude that [Y]_e ≠ 0 is not only a sufficient condition but also a necessary condition for a cell containing impermeant molecules (X_i ≠ 0) to maintain a finite volume.

Note that a small amount of [Y]_e (e.g., [Y]_e = 1 mM, 1/3 of extracellular potassium concentration) reduces the cell volume. See Fig. 2. Therefore, to obtain a finite equilibrium w^eq, we assume that [Y]_e ≠ 0 and use it as a mechanism to stabilize a cell in the absence of the NKA.

Substituting $[Cl−]e,[Na+]e$ (from Eqs. 7a and 7b) and $[X]ieq$ (from Eqs. 13 or 14a and 14b) in Eqs. 12, a–e, we obtain new expressions for the equilibrium values of the ion concentrations and voltage in terms of z_Y,[Y]_e, and fixed parameters $[K+]e$ and $Oe$⁠. We do not show the new formulae here but plot them in Fig. 2. In this figure, we plot the equilibrium values of ion concentrations, impermeant molecule, volume, and voltage as [Y]_e changes. Each equilibrium is plotted for five different values of z_Y:−2, −1, 0, 1, 2. In this figure, z_X = −1. Qualitatively, the behavior of the cell is similar for z_X = 1 or z_X ≠ ±1 (the simulations are not shown here).

Based on Eq. 13 and Fig. 2, the chloride concentration and volume always decrease as Y_e increases. For z_Y < −1, sodium and voltage are increasing functions of Y_e while potassium is a decreasing function of Y_e. For z_Y = −1, sodium, potassium, and voltage are constant functions of Y_e. For z_Y > −1, sodium and voltage become decreasing functions of Y_e while potassium becomes an increasing function of Y_e.

For the rest of the paper, we fix [Y]_e = 1 mM and z_Y = −1.

Note that the equilibria do not depend on g_Cl explicitly. However, to derive these equations, we assumed that g_Cl is non-zero (see Eqs. 8 a–c). The chloride conductance is essential for the operation of the PLM since without it there is no exchangeable anion to compensate for the change in [X]_i that results from changes in cell volume (see Fig. 7).

In Fig. 3, we compare the ATP consumption rate of the constant and nonlinear pumps, namely $JATPp1=p1/F$ and $JATPp2=p2/F$⁠, as a function of the pump rate p. Notice that for the latter, the ATP consumption rate comes close to an asymptotic value of about 0.75 nmol⁻¹dm⁻² when p ≈ 100 μA dm⁻² but carries on increasing very slowly as the pump rate increases.

In Fig. 4, we plot the steady state of the system as a function of the ATP consumption rate for the constant and the nonlinear models of NKA. As the figure shows, the steady-state values coincide exactly. We observed that this also occurs for less complicated nonlinear dynamics such as $pNKA[Na+]i$ (Yurinskaya et al., 2020; Keener and Sneyd, 2009), $pNKA([Na+]i/[Na+]e)3$ (Keener and Sneyd, 2009), and $pNKA([K+]e/[K+]i)2([Na+]i/[Na+]e)3$ (Manicka and Levin, 2019), $pNKA([Na+]i/[Na+]e)3pNKA([K+]e/[K+]i)2([Na+]i/[Na+]e)3$⁠). Therefore, the precise form of the NKA model is not important in determining the steady state of the PLEs.

In what follows, we compute the steady-state values of the PLM analytically and explore the effect of various model parameters on these steady states.

Keener and Sneyd (2009) derived equations for the steady state for the case of z_X ≤ −1. Later, Mori (2012) relaxed this condition and proved the existence of a steady state for any z_X. Here, we extend their work by including impermeant extracellular molecules, arbitrary z_X, and arbitrary NKA stoichiometry. We show that the steady states exist for a specific range of p_NKA; see Eq. 19 below.

In summary, we have shown that for p < p₀ = p_max/A₀, there exists a unique steady state given in Eqs. 16, 17, and 18 a–e. Without computing the steady states, Mori (2012) (Propositions 3.3, 3.4, and Theorem 3.5) proved that PLEs with a constant pump that satisfies some conditions (given by Mori’s Eq. 3.5) admit a unique and globally asymptotically stable steady state. This means that starting at any “acceptable” values of the cell ion concentrations, volume, and membrane potential, the system will evolve to that steady state after a transition time. Here, by “acceptable” values, we mean positive values for the concentrations and the volume, but the membrane potential may be negative too. Although the condition given by Mori’s Eq. 3.5 looks complicated, one can show that it is exactly equivalent to Eq. 19. In addition, Mori proved that if his Eq. 3.5, or equivalently our Eq. 19, does not hold, then the volume converges to infinity, which is consistent with what we discussed above when p ≥ p₀. Hence, Mori’s result, Theorem 3.5, ensures that the steady state given in Eqs. 16, 17, and 18, a–e is globally asymptotically stable.

Besides the extracellular ion concentrations and osmolarity, which we assume are constant (see Table 1), there are four groups of parameters that affect the steady states: (1) the NKA rate, (2) the valence z_X, and (3) the sodium (g_Na) and potassium (g_K) conductances. In what follows, we discuss the effect of each of these groups of parameters. Any of these parameters can be used as a control parameter to maintain the cell at a desired steady-state level. For instance, to maintain the cell volume at its minimum with a minimum amount of energy, we need to keep the ratio g_Na/g_K small (see Fig. 6) below.

First, we focus on how the steady state depends upon the pump rate. To do this, we have calculated how the steady state varies as the pump rate is increased. We have previously referred to these as Cp plots (Kay, 2017). In addition, we derive conditions that allow for a stable steady state. We then move on to consider the influence of the pump rate and how a cell could establish an energetically efficient pump rate.

We fix all the parameters at the values given in Table 1 and vary the NKA rate p_NKA = pA₀. We vary p, which is the density of NKA pumps in the membrane, multiplied by the ATP hydrolysis rate of a single pump.

In Fig. 5, we plot all the steady-state values as functions of p where p changes from 0 to p_max/A₀. In these calculations, [Y]_e = 1 mM, ensuring that in the absence of NKA activity the steady-state volume is finite. As is evident from the figure, as the pump rate increases, the steady-state volume initially decreases and then increases. This “switchback” was shown in Keener and Sneyd (2009) and,Mori (2012, Theorem 3.7) but does not seem to have been remarked on by others. Since it is also observed in the case of nonlinear pump mechanisms, it appears to be an inherent characteristic of the PLM. We will discuss the switchback further in the section The relation between NKA pump… below when we give a relationship between p_NKA and free energy.

It seems likely that cells expend only as much energy as necessary to stabilize cell volume. As the pump rate increases, driving $[K+]i$ up and $[Na+]i$ down, there comes a point where further increases in pump rate do not change the volume further. In this regime, fluctuations in p do not lead to large changes in volume, whereas for lower values of p they do. For the case where κ > 0, there is a minimum inflection point in the Cp curves for the volume and voltage. We will call this the p_min, which serves as a useful proxy for the optimal pump rate.

Note that for κ = 0, p_min does not exist. Indeed, when κ = 0, the steady-state volume is a monotonically decreasing function of p (see Eq. 18d). Moreover, when $κ=0,[K+]i$ and $[Cl−]i$ exhibit a Donnan equilibrium for K⁺ and Cl⁻.

The average intracellular valence of impermeant ions, z_X, is an important part of the PLM and exerts a strong effect on how ions are distributed across the membrane in the steady state. z_X can be estimated by measuring the concentrations of the predominant permeable cations and anions in the cell; a nontrivial task. Estimates of z_X in several organisms put it in the range of −1.5 to −0.5 (Burton, 1983). For the rest of the paper, we will choose z_X = −1. z_X is as important as any other physiological measure, like osmolarity; however, it has not attracted much attention (Model et al., 2023) and it is unclear whether cells actively attempt to regulate its magnitude.

In Fig. 5, we have plotted the Cp curves of the PLEs at five different values of z_X (−3,−1, 0, 1, 1.5). Straightforward calculations show that $[X]iss$ is differentiable with respect to $zX2$ and its derivative is strictly negative. Therefore, as $zX2$ increases, $[X]iss$ decreases. Note that the sign of z_X does not affect $[X]iss$⁠. Similarly, the cell volume also depends on $zX2$⁠, but unlike $[X]iss$⁠, the cell volume is a monotonically increasing function of $zX2$⁠.

From Eqs. 18, a–e, we can conclude that the steady-state values of $[Cl−]i$ and voltage are decreasing functions of z_X while the steady-state values of $[Na+]i$ and $[K+]i$ are increasing functions of z_X.

The leak conductance of the cell, in particular to the monovalent cations ions (g_Na and g_K), exerts a strong effect on the energy consumption of the NKA. Fig. 6 shows p_min as a function of g_Na and g_K when $Na+e=147 mM$⁠, $K+e=3 mM,ν=3$⁠, and κ = 2. Also, we assume that g_K > g_Na since this is known to be so in most cells, as most cells have a resting potential less than −20 mV (Weiss, 1996). The resting potential is determined by the conductances and their associated Nernst potentials. As g_Na becomes larger, it will shift the resting potential toward its reversal potential that is greater than −20 mV. To minimize p_min, g_Na should be kept low. Note that when g_Na is small, p_min remains small for any value of g_K. This means that when g_Na is small, p_min is less sensitive to any perturbations to g_K. It is worth pointing out that our results are robust over a very wide range of leak conductances, which includes the low conductances of red blood cells (Lew and Bookchin, 1986).

It may seem as if the passive conductances are incidental to the operation of the PLM, however they are required for its normal functioning. We can demonstrate this by systematically setting each one of the conductances in turn to zero, either while the pump is on or while it is off. In what follows we will assume that there are no impermeant ions in the extracellular space. Blocking g_K = 0 will destabilize a cell with an active NKA, so the cell swells without ceasing. If the conductance is turned off in a passive cell, it does not change the instability of the cell (data not shown).

If g_Na is set to zero while the NKA is operating the system will settle into a new stable steady state, with higher $[K+]i$ and lower $[Na+]i$⁠, as shown in Fig. 7 A.

If g_Na is set to zero after the NKA is turned off, since Na⁺ and K⁺ cannot exchange, both of their concentrations will be fixed close to their values when the conductance is turned off, as shown in Fig. 7 B. There is some change in concentrations, volume, and voltage that occurs as the system settles into equilibrium. Whatever Na⁺ is in the cell, it is trapped there and cannot get out, but the volume can change since water can cross the membrane, and this gives rise to changes in $[Na+]i$⁠. After turning off the sodium conductance, $[Cl−]i$ and [X]_i remain close to their values just prior to the change. It is notable that after turning off g_Na, the volume stabilizes and the system is at a true thermodynamic equilibrium. Blocking g_Na traps the cell in a stable equilibrium. This is somewhat like the double Donnan equilibrium that was proposed by Leaf (1959).

It is worthwhile asking why Cl^−- does not equilibrate with the extracellular compartment in the above example? If Cl⁻ were to diffuse in, it would need to be accompanied by K⁺ to prevent charge accumulation. However, the countervailing K⁺ gradient prevents this from occurring. This can be seen from the fact that if a Na⁺ conductance is opened, the ion gradients will dissipate and the cell will swell.

We can prove formally that blocking the g_Na will stabilize a cell. If both p_NKA and g_Na are zero, $e−νFRTgNapNKA$ stays below 1, and hence condition specified by Eq. 19 still holds. The reason is as follows: Let $y=e−νFRTgNapNKA$⁠. Then, $gNaln(y)=−νFRTpNKA$⁠. Therefore, when p_NKA = 0 and $gNa≠0,ln(y)$ must be zero which implies y = 1. However, if g_Na = 0, there is no reason for In (y) to be equal to zero.

In contrast to the case of sodium conductance, if the chloride conductance is blocked while the NKA is active, the $[Cl−]i,[X]i$ and the volume “freeze” and the cell is stable, Fig. 7 C. After the conductance is turned off, $[Na+]i$ and $[K+]i$ relax to a new stable steady state. This behavior occurs because Cl⁻ is the only exchangeable anion, so that when its conductance is blocked, the total number of anions becomes fixed; Cl⁻ is now effectively impermeant. Hence the total number of cations is fixed to match the charge of the anions. However, because Na⁺ and K⁺ can exchange, they will relax to a new steady state.

In the case where pump is off, blocking the g_Cl, still freezes the intracellular anion concentrations and volume. However, in this case because the pump is not active, the intracellular cation concentrations relax to their extracellular concentrations. Fig. 7 D.

There is a direct relationship between the pump rate and ATP consumption; for each cycle of the pump a single ATP is hydrolyzed and hence the energy utilized. Since energy is at a premium, there is strong evolutionary pressure on organisms to minimize energy consumption and the PLEs can provide some insight into the factors that influence energy expenditure. In Eq. 20, we saw that p_min depends on the stoichiometry of the pump, i.e., the number of Na⁺(ν) and K⁺(κ) transported per cycle. Not all stoichiometries are consistent with a stable cell. For example, if ν = 0 and κ > 0, from Eq. 20, the cell is unstable, as has been pointed out by Jakobsson (1980). Indeed, for a constant κ, increasing ν decreases p_min. On the other hand, for a fixed ν, increasing κ decreases p_min. Putting these two observations together, we conclude that to achieve a minimum p_min, both κ and ν should be maximized. However, there is an energetic limit on how large κ can be, which we take up in the next paragraph.

Following Clarke et al. (2013), we use the value of ∆G_ATP = −55 kJ mol⁻¹. The energy required to drive a single cycle of the pump can be calculated from thermodynamics using the chemical potentials of all the species involved (Läuger, 1991).

The derivation is given in Appendix 4. For the default parameters in Table 1 and for stoichiometry 3 Na⁺:2 K⁺, Eq. 27 holds if p < 6.24 × 10⁻⁵, which is less than p_max/A₀ ≈ 17.5 × 10⁻⁵Adm⁻² computed from Eq. 19. In this case, since p_min > p_ATP the switchback in Fig. 5 does not occur, but for larger values of −ΔG_ATP this inequality may be reversed.

The thermodynamic limit of the system is p_ATP; in reality, it is unlikely to be achieved because the second law of thermodynamics implies that a fraction of the energy will be dissipated as heat.

In this section, we introduce three cation-coupled co-transporters into the pump-leak scenario (see Fig. 1 for an illustration).

• (1)

  K⁺–Cl⁻ cotransporter (KCC), where one K⁺ and one Cl⁻ are transported out of the cell, with rate c_KCC,

• (2)

  Na⁺–Cl⁻ cotransporter (NCC), where one Na⁺ and one Cl⁻ are transported into the cell, with rate c_NCC, and

• (3)

  Na⁺–K⁺–Cl⁻ cotransporter (NKCC), where one Na⁺, one K⁺, and two Cl⁻are transported into the cell with rate c_NKCC.

If a CCC is introduced into this passive cell, it cannot drive ions since there is no energy input into the systems. We show this for the case of the KCC, the other co-transporters are similar. From the above equation, $[K+]e[K+]ieq=[C−]ieq[Cl−]e$ implies $[K+]e[Cl−]e=[K+]ieq[Cl−]ieq$⁠, where Eqs. 29a and 29b demonstrates that the ionic flux through the KCC co-transporter is zero. Similarly, all other CCC currents become zero. This implies that adding any of these co-transporters to the NKA-pump leak model with the pump off cannot change the equilibrium. However, they may change the rate at which the cell convergence to the steady state if the NKA is turned off (Yurinskaya and Vereninov, 2022).

In the classical PLM, since Cl⁻ is not actively transported in the steady state, E_Cl is equal to the membrane potential, i.e., the transmembrane Cl⁻ gradient is at equilibrium. Hence, g_Cl appears not to play much of a role in establishing the steady state (note it does not appear in the equations that determine the steady state). However, in the absence of a Cl⁻ conductance, the PLM is essentially frozen to preserve electroneutrality and $[Cl−]i$ needs to be able to change. It is also worth noting that in the PLM, the magnitude of g_Cl can influence the rate at which the steady state is approached. When a CCC is introduced into the scenario, since Cl⁻ is now actively transported, the magnitude of g_Cl now becomes important.

In Figs. 8, 9, and 10, the steady-state values of the ions concentrations, intracellular impermeant molecule, volume, and voltage are plotted at different cotransporter rates when only one cotransporter is active. To compute these steady states, we solved a system of five algebraic equations, setting the righthand side of Eq. 6, a–c, and Eq. 2 to zero and combining them with the constraint, Eq. 4. The analytic expressions for these steady states are given in Appendix 5. We note that Düsterwald et al. (2018) previously derived an analytical solution to the PLEs with a KCC, but with a different mathematical form of the NKA.

Figs. 8, 9, and 10 are plotted for p = p_min, the value of p where the cell takes its minimum volume and voltage at the steady state and for an intermediate value of p, p = 0.6 p_min or p = 0.75 p_min. The behavior of the cell is qualitatively similar for other values of p (results are not shown).

Mori (2012) in his Proposition 4.4 and Theorem 4.8 proved that if Eq. 21 holds for “sufficiently small” NKA and CCCs rates, namely p_NKA and c_CCC, the PLEs with both active NKA and CCCs possess asymptotically stable steady states¹. Mori’s result does not guarantee uniqueness. Here, we derived these steady states in Appendix 5, and the derivation guarantees the uniqueness. Since we have the explicit expressions for the steady states, similar to the case of PLEs with an active NKA in the section Active cells: The effect of…, we can find the range of p_NKA and c_CCC in which the steady states exist, i.e., we determine how small these parameters must be to ensure the existence of the steady states. Mori’s results ensure that our unique steady states are asymptotically stable for small p_NKA and c_CCC. For larger p_NKA and c_CCC, we show the stability numerically. A rigorous stability analysis of the steady states with arbitrary large p_NKA and c_CCC will be a topic for future investigations.

Since $[Na+]i$ increases as c_KCC increases, the voltage which is a decreasing function of $[Na+]i$⁠, must decrease as c_KCC increases.

Since $[K+]i$ decreases, and voltage is a decreasing function of $[K+]i$⁠, voltage increases as $[K+]i$ decreases. Equivalently, voltage increases as c_NCC increases, see Fig. 9 (middle panels). In contrast to the KCC, the action of the NCC can destabilize a cell. Above a critical value of c_NCC the volume does not stabilize, although the concentrations and voltage reach a steady state. As far as we know, this aspect of the NCC has not been noted before and only becomes evident when the NCC is modeled in the context where the membrane is water permeable and the membrane is pliant.

As shown in Fig. 10, the steady-state behavior of a cell with an active NKCC is similar to a cell with an active NCC. When NKCC is active, one Na⁺, one K⁺, and two Cl⁻ enter the cell. So, it is expected that the Na⁺ and K⁺ level increase; however, the sum of $[Na+]i$ and $[K+]i$ must remain constant. Hence, one of these ions must decrease. Since g_K is much higher than g_Na, some K⁺ escape through the membrane, and therefore, the K⁺ level decreases. The rest of the characteristics of NKCC is similar to NCC co-transporter, except that it does not destabilize the cell volume.

The primary role of the CCCs appears to be to shift the $[Cl−]i$ away from equilibrium. In the case of KCC below equilibrium, NCC and NKCC shift it above its equilibrium value. In addition, their activation can shift the steady-state volume of the cell. In the case of KCC decreasing the volume, NKCC and NCC increase the volume. It is clear from our work that activation of either KCC or NKCC leads to predictable shifts in ion distributions without perturbing cellular stability. However, excessive activation of an NCC can lead to the loss of cellular stability. We have not investigated how the simultaneous deployment of different CCCs influences ionic distributions, leaving that for another time.

The flux of ions through the co-transporters is determined by what we will term the “driving force,” F_CCCs defined in Eqs. 31 a–c. If the term is negative, the ions are driven out of the cell (KCC); conversely, if the driving force is positive the ions are driven into the cell (NCC and NKCC). Notice that all of the cotransporters are symporters, i.e., all the transported ions always move together in the same direction.

We will use the analytical PLE to determine if it is possible to reverse the direction of the co-transporter (Payne, 1997; Kakazu et al., 2000). The concentrations within the driving force terms are primarily set by the NKA when the transport rate of the CCC is low. As the cotransporter rate is increased, the system is driven toward a steady state where the driving force term tends to zero. The steady state is determined by the competition between the NKA and the CCCs.

It is worth noting that all the CCCs preserve their default direction of ion transport, regardless of the choice of the parameters for the PLE, so long as the parameters generate a stable system. The CCCs could be reversed if the ion concentrations are perturbed away from their steady state values, but the system will return to a steady state with the CCCs transporting ions in their default direction.

Instead of attempting to model a particular kind of cell, we have explored what sorts of steady-state ion distributions can be achieved by combing the classical PLEs with CCCs and extracellular impermeant molecules. We have also identified some mathematical principles that undergird such systems.

There are several analytical tools for making sense of how voltage-gated ion channels interact to generate neuronal excitability (Izhikevich, 2007; Sterrat et al., 2011) but little for the case where one incorporates ion transporters. The analysis becomes a lot more complicated as soon as one allows for the accumulation of ions and the flux of water driven by changes in osmolarity.

Analytical solutions to models have a significant advantage over piecemeal simulation since they can make evident the dependence of the system’s behavior on its parameters. Additionally, in simulations of a system, one is in a sense working blind, while analytical solutions give one an overview of the system.

Whether the steady state of a model of cellular ion transport is stable or not is a matter of great importance in evaluating models. Weinstein (1997) studied the “local” stability by linearization about a physiologic reference condition. The framework introduced by Mori (2012) of viewing PLEs as being governed by a free energy principle is very useful. It allows one to get a global overview of the system and predict what behaviors may or may not occur without computing the steady states. However, it cannot fully characterize the full range of parameters in which the steady states exist and are stable. In this work, we were able to derive the steady states explicitly and found the full range of the parameters in which the steady states exist and numerically showed that these steady states are stable. Mori’s work ensures “global” stability of the steady states for PLEs with a constant pump; however, it cannot determine their global stability in the presence of co-transporters. In this work, we explicitly derived unique steady states and numerically showed their stability. A rigorous stability analysis will be left for another time. In Weinstein and Sontag (2009), the authors go beyond stability analysis and apply linear optimal control theory to stabilize Na⁺ flux, cell volume, and cell pH.

The operation of the PLM has wide-ranging consequences for physiology. In what follows, we discuss some of these while reviewing our findings. The discussion may seem disjointed, but it reflects the diverse implications of the PLM. However, our discussion is by no means comprehensive, omitting, for example, the role of the PLM in development (Levin, 2021).

The NKA utilizes ∼20–30% of all cells’ energy budget, its continuous action is required to forestall an osmotic catastrophe induced by the presence of impermeant molecules. This is an ongoing process that does not relent. Given that the NKA constitutes so much of a cell’s energy budget, cells have probably evolved mechanisms to constrain energy consumption. If the pump rate is too low small fluctuations in the pump rate will lead to large ones in cell volume. See Fig. 5, lower right panel.

We have shown that if one incorporates the thermodynamics of ion transport into the PLEs, a natural limit to the rate of ATP consumption emerges which prevents excessive ATP consumption (Eq. 28). This to the best of our knowledge has not been demonstrated before and it is a feature that only becomes apparent if the action of the NKA is viewed in the framework of the PLEs in conjunction with the energy of the pump.

A complete model of the NKA would require a characterization of all the steps in its cycle, with their dependence on ion concentration and voltage. However, we have shown in the section The relation between NKA pump… that if one is only interested in the steady state of the PLEs and the precise form of the NKA is of no importance. Where it becomes important is in the system’s transient response, which we have not considered here.

Theoretically, we can contemplate any stoichiometry; however, the constraints of protein chemistry probably limit the number of ions that can be coordinated at one time. So, for example, there are no cases where more than five ions or molecules are transported during one cycle of a transporter.

Our model makes predictions as to what would happen to the distribution of ions if the NKA stoichiometry were changed, but at present this is not something that can be altered experimentally. Recently, Artigas et al. (2023) demonstrated that the NKA of brine shrimp, which can live in supersaturated salt solutions, has a stoichiometry of 2:1. They argue that this stoichiometry has evolved since the ambient concentration gradients makes a 3:2 stoichiometry unable to be driven by the Gibbs free energy of ATP, but can by a 2:1 stoichiometry.

Many cellular transport models omit a crucial aspect (with some exceptions noted in the Introduction), namely water movement. It is certainly possible to simulate the system without this, but it runs the risk of arriving at a model which might be unstable if water transport is included, as it ultimately must since all membranes are water permeable.

The omission of water permeability may veil an instability in the system. One may find that a model produces the correct electrical phenomenology. However, the volume may be unstable when one introduces water permeability and cell pliancy.

The values of g_K and g_Na are primary determinants of ATP utilization. Moreover, although this is not well known, leak conductances can strongly impact the action potential threshold (Kay, 2014). But little is known about how cells set and regulate their resting input conductances. If the input conductance is very low, the opening of ion channels can lead to pronounced changes in voltage which may be damaging.

Lipid bilayers have a unit membrane resistance that varies from 0.2 to 4 MΩcm² (Miyamoto and Thompson, 1967). For a cell with a 5 µm radius cell with a unit membrane resistance of 2 MΩcm², the cell’s input resistance would be ∼600 GΩ. Assuming Ohmic behavior, if 1 pA of current flows into the cell the change in potential would be ∼600 mV. So very small changes in current would lead to very large changes in voltage. Thus, cells need to keep their input resistance relatively low to avoid large swings in membrane potential, which could rupture the membrane. Since ions channels open in a stochastic fashion, varying fluxes are inevitable.

In addition, if the input resistance is too high, the transport of metabolites like electrogenic co-transporters will induce significant changes in the membrane potential (Berndt and Holzhütter, 2013).

There is a large class of channels that are open at rest that determines the baseline permeability of cells and their input resistance. These include K_2P channels (Enyedi and Czirják, 2010), NALCN channels (Ren, 2011), Kir channels (Nichols and Lopatin, 1997), and HCN channels (Wahl-Schott and Biel, 2009).

Since some confusion has been generated by the terms Donnan potential and equilibrium, it is worthwhile clarifying the terminology. If at the cell’s steady state, where all of the permeant ions conform to the Donnan equilibrium, (Eqs. 9, a–c), then the system is at thermodynamic equilibrium. We suggest that this is the only case where one should refer to the potential as being a Donnan potential or system being at a Donnan equilibrium. However, if one permeant ion does not obey Eqs. 9 a–c, an active transport process must be driving one or more of the ions. The membrane potential in this case is determined by the distribution of ions and the conductances of all the permeant ions. In this case, the impermeant molecules are exerting a Donnan effect, which is being cancelled out by the operation of the NKA. An interesting situation arises when the stoichiometry is 3 Na⁺:0 K⁺, K⁺ and Cl⁻ conform to the Donnan equilibrium but Na⁺ does not, but because it is being actively driven, hence the membrane potential is not a Donnan potential (Kay, 2017).

In this paper, we have introduced a simple extension to the PLEs that allows the development of a stable potential and volume by introducing impermeant extracellular molecules. If the NKA is inactive, the system represents a stable Donnan equilibrium at thermodynamic equilibrium. This stratagem of using extracellular molecules to stabilize the cell has only been mentioned by (Fraser and Huang, 2007). Although impermeant extracellular molecules could stabilize a cell, it does not account for the asymmetric Na⁺ and K⁺ distribution encountered in actual cells. Moreover, it would require a very high concentration of Y (50–100 mM) to sustain a comparable concentration of X, which is needed for normal cell function.

Many years ago, Donnan (1942), among others, thought that cells might be in a passive Donnan equilibrium. However, he did not consider the possible influence of impermeant extracellular molecules. We will consider one case where we believe that a high concentration of impermeant extracellular molecules could serve to establish a stable Donnan equilibrium. The larvae of the African Chironimid midge, Polypedilium vanderplanki can survive the loss of up to 94% of its water (ahydrobiosis). During the dehydration process, the trehalose content steadily increases to about 20% of the total wet weight of the larva (Watanabe et al., 2004). It seems likely that the trehalose serves as an impermeant extracellular molecule that stabilizes cells against the Donnan effect as the hemolymph dehydrates. The trehalose also stabilizes proteins during dehydration preventing their denaturation (Olgenblum et al., 2024). Without trehalose if the NKA stopped operating and the extracellular solution increased in osmolarity through evaporation, the intracellular solution will also increase its osmolarity but the cells will be unstable and will grow in volume until they lyse. However, the trehalose concentration increases in the extracellular space and it will stabilize cell volume. What we anticipate happening is that as the extracellular trehalose concentration is increased by synthesis and export and the osmolarity is further increased by dehydration, the intracellular solution will concentrate by passive water movement.

It has been claimed that impermeant molecules can influence the equilibrium potential of chloride (Glykys et al., 2014), but this has been contested (Voipio et al., 2014; Kay, 2017) and refuted by others (Düsterwald et al., 2018).

The addition of the CCCs cannot change the osmotic and electroneutrality constraints. Moreover, the CCCs do not change the qualitative dynamics of the system. We showed that if a CCC is added to stable PLEs, it cannot induce qualitatively different dynamics, i.e., the PLEs possess unique globally stable steady states. However, it is possible that the addition of a CCC could change the rate at which the cell converges to a new steady state.

A novel finding that emerged from our study is that NCC can induce cellular instability at high pump rates. The overall effects of the NKCC and NCC are the same, increasing $[Na+]i,[Cl−]i$⁠, and volume. Therefore, it is something of a puzzle why there are two mechanisms. It is interesting to note that NCC has a very restricted expression, only being found in the kidney and bone (Subramanya, 2020). Although NKCC2 is only found in the kidney, NKCC1 has widespread expression (Delpire and Gagnon, 2018). NKCC can be considered to have what might be thought of as a safety mechanism that limits the level of $[Cl−]i$ that can be achieved and the cell volume expanding uncontrollably. This case of the NCC illustrates the power of theory, which allows one to go beyond intuitions to make explicit and unexpected predictions.

It is relevant to mention that NKCC or NCC and KCC can be coexpressed in non-epithelial and epithelial cells. In the case of non-epithelial, they are both expressed in erythrocytes and activated under different conditions (Kregenow, 1981). In the case of epithelial cells, the NKCC may be expressed on apical surface and the KCC on the basolateral surface, as they are in the thick ascending loop of Henle (Haas and Forbush, 2000). Investigating the behavior of a cell and regulating its volume when multiple co-transporters are present is a topic for future exploration.

It has been claimed by some that the operation of some transporters (viz. NKCC, KCC) leads to the active transport of water against an osmolarity gradient (Zeuthen, 1995). The evidence for this has been contested by others (Lapointe et al., 2002; Boron and Boulpaep, 2016), and whether active water transport indeed occurs, remains a possibility. It is straightforward to include active water transport into the PLM. Active water transport will manifest as a virtual pressure in Starling’s law (Eq. 2), which allows a steady state to be established with a persistent gradient in osmolarity across the membrane but no transmembrane hydrostatic pressure. If reliable, accurate in situ osmosensors were available that could provide the basis for detecting active water transport, which would be signaled by a sustained gradient in osmolarity but no net transmembrane hydrostatic pressure and membrane tension.

It is by no means obvious that the incorporation of CCCs into the PLM would not lead to the development of novel dynamics. In this paper, we have shown that they cannot do so, and that the parameter space of this system does not harbor regions with exotic dynamics. An analytical approach makes evident features of the system that would not be apparent in the absence of explicit mathematical expressions. This allowed us to show that the CCCs are unlikely to reverse and reveal the precise coupling between the NKA and ATP utilization, as well as the number of new features of the PLM.

We believe that our work will help interpret experiments. In considering experiments where the ionic composition or osmolarity of the bathing saline is changed, the results have to be considered in the light of a theory. To imagine that experiments can be interpreted without presuming some theory is to court an illusion. If one does not posit a specific theory, one implicitly assumes that the cell behaves like a pliant bag that is permeable to water, and therefore one makes a theoretical assumption.

Our work also points to significant gaps in the experimental arsenal (viz., techniques for measuring the intracellular osmolarity and transmembrane pressure) that will need to be filled if we are to be able to address significant questions about cell volume regulation.

Accurate models have the additional benefit of predicting the probable effects of pharmacological interventions. In view of the urgent need to reduce neuronal and glial swelling that typically follows a stroke, there is a critical need for effective agents (Stokum et al., 2016). Our analysis indicates that blocking chloride and/or sodium leak conductances could represent a promising strategy (Rungta et al., 2015), but that blocking aquaporin channels is unlikely to prove effective.

Solving the above equation for [X]_i, we obtain Eq. 11.