Active volume regulation in adhered cells

Significance

Living cells regulate their volume using a diverse set of mechanisms to maintain their structural and functional integrity. The most widely used mechanism to control cell volume is active ion transport. Recent experiments on adhered cells reveal that their volume is significantly reduced as their basal area is increased. Our physical theory includes electrostatics and cell activity to obtain the osmotic pressure within the cell. We predict a generic relation for how adhered cells regulate their volume in response to changes in their basal area. The theory is in agreement with the recent experiments on different cell types.

Abstract

Recent experiments reveal that the volume of adhered cells is reduced as their basal area is increased. During spreading, the cell volume decreases by several thousand cubic micrometers, corresponding to large pressure changes of the order of megapascals. We show theoretically that the volume regulation of adhered cells is determined by two concurrent conditions: mechanical equilibrium with the extracellular environment and a generalization of Donnan (electrostatic) equilibrium that accounts for active ion transport. Spreading affects the structure and hence activity of ion channels and pumps, and indirectly changes the ionic content in the cell. We predict that more ions are released from the cell with increasing basal area, resulting in the observed volume–area dependence. Our theory is based on a minimal model and describes the experimental findings in terms of measurable, mesoscale quantities. We demonstrate that two independent experiments on adhered cells of different types fall on the same master volume–area curve. Our theory also captures the measured osmotic pressure of adhered cells, which is shown to depend on the number of proteins confined to the cell, their charge, and their volume, as well as the ionic content. This result can be used to predict the osmotic pressure of cells in suspension.

Living cells regulate their volume using a diverse set of mechanisms to maintain their functional integrity (1–3). Cell size is typically uniform across tissues, and volume regulation is performed at the single-cell level (1). The most widely used mechanism to control cell volume is ion release and ion uptake through ion channels and active pumps. Such transport is adaptive and allows regulatory volume increase/decrease in response to transient changes in the cell environment (2, 3). As part of in vitro experiments, on the other hand, it is possible to alter the physical conditions of the cells over long periods of time and manipulate their volumes.

Recent measurements by Guo et al. (4) and Xie et al. (5) quantify how the volume of adhered cells decreases as their basal area is increased. The two independent experiments were carried out on different cell types. They both demonstrate that the volume–area dependence of the cells is independent of how the basal area is changed (e.g., by varying the substrate stiffness or by patterning of substrate adhesion molecules). Both works emphasize the importance of cell activity in volume regulation and show that the cell volume increases when activity is suppressed (e.g., via ATP depletion).

Guo et al. (4) examined also the volume dependence of three mechanical cell moduli: the bulk modulus, cortical shear modulus, and cytoplasmic shear modulus. All three moduli were shown to follow a similar functional form $\sim k_{\mathrm{B}}TV/{\left(V-V_{min}\right)}^2$, where $k_{\mathrm{B}}T$ is the thermal energy, $V$ the cell volume, and $V_{min}$ a minimal volume that accounts for molecular crowding. The measured moduli each differ by several orders of magnitude. For large cell volumes, the bulk modulus was of the order of megapascals, while the cortical one was of kilopascals and the cytoplasmic one of pascals. These results indicate that volume regulation is determined mainly by the osmotic pressure (that is related to the bulk modulus), rather than by cortical tension.

Here we present a minimal model to predict the volume of adhered cells as a function of their basal area. The cell volume is determined by pressure equality with the buffer. The cell pressure originates from proteins that are confined to the cell and ions that can exchange with the buffer. The ionic concentration is tuned by cell area-dependent, active ion transport, related to ion channels and active pumps. This yields a generalization of the Donnan (electrostatic) equilibrium that accounts for cell activity. Our predictions of the osmotic pressure vs. cell volume provide a good fit to the results of Guo et al. (4) and can also be applied for cells in suspension. We show that the separate volume vs. area measurements of Guo et al. (4) and Xie et al. (5) fall on the same master curve, when scaled by the appropriate measurable volume scales. Our theory also lies on this master curve with only two fit parameters. We predict that as the basal area is increased, more ions are pumped actively out of the cell. This serves to maintain the osmotic pressure within the cell.

Model for Cellular Osmotic Pressure

Our model of the experiments considers a cell of volume $V$, whose basal area $A$ is adhered to a substrate and whose apical area is in contact with a buffer. The cell is assumed to be in steady state, such that its volume and basal area remain constant (after initial transients) during the experimental time of minutes. We rely on a minimal description of the cell (Fig. 1). As the experiments of ref. 4 show, the nucleus occupies a fixed fraction $\alpha$ of the cell volume, with $0<\alpha <1$. The remaining volume, $\left(1-\alpha \right)V$, contains the intracellular fluid that consists of water, molecules that are confined to the cell (referred to hereafter simply as “proteins”), and ions that exchange with the buffer. The shape of the adhered cell is usually similar to that presented in Fig. 1. However, as is justified further below, the shape of the adhered cell does not directly enter our theory, and it is sufficient to consider the cell as flat.

Fig. 1.

Sketch of an adhered cell. The nucleus occupies a fixed volume fraction $\alpha$. The cell contains proteins with localized ions (circled with dashed lines) as well as free cations and anions. While the proteins are confined to the cells, free ions exchange with the buffer through ion channels and pumps (not drawn). For simplicity, the proteins are assumed to have an average, effective charge of $-e/2\hspace{0.17em}\left(z=1/2\right)$. Electroneutrality is satisfied in the nucleus, in the cell, and in the buffer.

The buffer is an aqueous solution of ions, some of which are permeable. In some of the experiments, the buffer included added polyethylene glycol (PEG) which cannot enter the cell. For simplicity, we describe it as a monovalent electrolyte, with a single cationic and anionic species, both of concentration $n_b$. These ions can move in and out of the cell through ion channels and pumps.

Proteins are confined to the cell and have a fixed number $N$. To account for excluded volume, we define an effective protein concentration in their available volume, $n_p=N/V_{\mathrm{e}\mathrm{ff}}$. The effective volume is smaller than the total one, $V_{\mathrm{e}\mathrm{ff}}<V$, due to the volumes of the nucleus and of the proteins themselves (the volume occupied by ions is negligible, compared to these volumes). The protein concentration is given by

$$n_p=\frac{N}{\left(1-\alpha \right)V-Nv}=\frac{N_p}{V-V_m},$$ [1]

where $v$ is a typical protein volume, $N_p=N/\left(1-\alpha \right)$ is the effective number of proteins, and $V_m=N_{p}v$ is the minimal possible cell volume. The experiments of ref. 4 show that the nucleus occupies a fixed volume fraction of the cell, which justifies the constant $\alpha$ in our model. Eq. 1 demonstrates that although the volume of the nucleus changes, the excluded volume $V_m$ can be considered as fixed. The nucleus merely defines the effective protein number, $N_p$, via $\alpha$.

The proteins are typically negatively charged and tend to localize ions in a rather involved manner. The association and dissociation of charge groups on the proteins are regulated by the ionic environment (6–8). Furthermore, as proteins are amino acid chains that contain solvent, ions can enter some of their volume, similar to the situation in polyelectrolytes (9, 10), as is illustrated in Fig. 1. In addition, counterions and coions surround the proteins and screen the electric field around them (11, 12), giving rise to an effective charge, analogous to that of charged colloidal systems (13). This localization of the ions near the proteins reduces the osmotic pressure (13), compared with that of an ideal solution of ions in the absence of proteins.

We are interested in the osmotic pressure exerted by the ions, rather than in a microscopic description of their concentration profiles. Therefore, we regard the system as homogeneous and describe the electrostatics in a simpler manner. The bare proteins and their localized ions (whether within the protein volume or the surrounding screening cloud) are treated as a single species (Fig. 1) with a concentration $n_p$ and an effective, average charge $-ze$ that is smaller in absolute value than the bare protein charge.

All other ions are treated as free ions. The concentration of free cations in the cell is $n_{+}$ and that of free anions is $n_{-}$. Electroneutrality in the cell dictates $n_{+}-n_{-}=z{n}_p$. We treat the electrostatics within the Donnan approximation (14–16), which neglects the local electric field and assumes a fixed electrostatic potential within the cell, $\psi <0$, compared with a zero potential in the buffer. Within our model, the effects of local electric fields are encapsulated in the localized ions and effective protein valency $z$.

The ionic concentrations in the cell, $n_{+}$ and $n_{-}$, are determined by the exchange of ions between the cell and the buffer. In thermal equilibrium, this reduces to the equality of electrochemical potentials. However, due to active ion pumps, the cells are out of thermodynamic equilibrium with a chemical-potential difference between the cell and the buffer, $\mathrm{\Delta }{\mu }_{\pm }$ for the cations and anions. Explicitly, the differences $\mathrm{\Delta }{\mu }_{\pm }$ satisfy $k_{\mathrm{B}}T\ln n_{\pm }{a}^3\pm e\psi =k_{\mathrm{B}}T\ln n_b{a}^3+\mathrm{\Delta }{\mu }_{\pm }$, where $a$ is a microscopic length. This relation, together with the electroneutrality condition, leads to

$$n_{+}+n_{-}=\sqrt{{\left(z{n}_p\right)}^2+{\left(2n_b{\mathrm{e}}^{\delta }\right)}^2},$$ [2]

where we have defined the (dimensionless) activity, $\delta =\left(\mathrm{\Delta }{\mu }_{+}+\mathrm{\Delta }{\mu }_{-}\right)/\left(2k_{\mathrm{B}}T\right)$, as the average ionic chemical-potential difference in units of $k_{\mathrm{B}}T$. The electrochemical-potential difference can be related to the zero-net transport rate of ions through channels and active pumps, as is further justified below.

The osmotic pressure within the cell, $P_{\mathrm{i}\mathrm{n}}$ is obtained by adding the concentrations of proteins and free ions (Eqs. 1 and 2, respectively), $P_{\mathrm{i}\mathrm{n}}=k_{\mathrm{B}}T\left(n_p+n_{+}+n_{-}\right)$. We find that (see SI Appendix for further details)

$$\frac{P_{\mathrm{i}\mathrm{n}}}{k_{\mathrm{B}}T}=\frac{N_p}{V-V_m}+\sqrt{{\left(\frac{z{N}_p}{V-V_m}\right)}^2+{\left(2n_b{\mathrm{e}}^{\delta }\right)}^2}.$$ [3]

This expression accounts for steric interactions of the proteins via their excluded volume $V_m$ and for the effect of the localized ions via the effective protein charge $-ze$. Here we have assumed that all of the proteins are free to explore their available volume and thus contribute to the pressure. This microscopic quantity can be replaced by a mesoscopic and measurable volume scale, as is shown below.

As the cell volume decreases and approaches the minimal volume, $V_m$, Eq. 3 obtains the simple form, $P_{\mathrm{i}\mathrm{n}}=\left(1+z\right)N_p{k}_{\mathrm{B}}T/\left(V-V_m\right).$ This is the same functional form that was reported in ref. 4, where volume vs. pressure data for different substrates collapse onto a single curve as $V$ approaches $V_m$ (figure S2 of ref. 4). The translational entropy of ions in the cell becomes very small in this limit and they are all released to the buffer, except for $z{N}_p$ neutralizing counterions that remain confined to the cell. Note that cell activity, described within our formulation by the parameter $\delta$, is negligible in this limit.

Activity and Its Area Dependence

We now examine in more detail the physical origin of the activity, $\delta =\left(\mathrm{\Delta }{\mu }_{+}+\mathrm{\Delta }{\mu }_{-}\right)/2k_{\mathrm{B}}T$, introduced in Eq. 2. Ions move in and out of the cell through ion channels and pumps. The transport through channels is passive, down the electrochemical potential gradient, while that through pumps is active, up the gradient (17, 18). Ions can enter (or exit) the cell through either channels or pumps, depending on their electrochemical potentials. The most common ion pump in the cell is the sodium–potassium pump, which exports three sodium ions and imports two potassium ions for each ATP molecule that is hydrolyzed. The ionic currents through pumps and channels are equal in steady state. This condition relates the electrochemical potential difference, $\mathrm{\Delta }\mu$, with microscopic properties of ion channels and pumps.

Consider a single ion passing through a channel. The transport of an ion through a channel is driven by the electrochemical potential gradient across the two sides of the channel, and it vanishes in equilibrium. Close to equilibrium (linear response), the ion velocity, $u$, is given by $u=-\left(D/k_{\mathrm{B}}T\right)\mathrm{\Delta }\mu /h$, where $D$ is the diffusion coefficient and $h$ the channel length. Here we have approximated the electrochemical potential gradient by $\mathrm{\Delta }\mu /h$. Consequently, the time it takes an ion to traverse the length $h$ of the channel is ${\tau }_D/\left|\delta \right|$, where ${\tau }_D=h^2/D$. is the diffusion time to cross the channel in the absence of any electrochemical potential differences (as in equilibrium). The transport rate of ions through all of the channels in the cell is thus $r_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}=N_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}\left|\delta \right|/{\tau }_D$, where $N_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}$ is the total number of ions in channels at a given time.

Pumps, on the other hand, can be considered as machines with a characteristic cycle time, $1/r$. The transport rate of ions through all of the pumps is thus $r_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}=N_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}r$, where $N_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}$ is the number of ions in pumps at a given time. In steady state, the transport rates through channels and pumps must be equal, leading to

$$\delta =-\frac{N_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}}{N_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}}r{\tau }_D.$$ [4]

While the equilibrium diffusion time, ${\tau }_D$, is expected to remain relatively constant for different cell volumes and basal areas, the ratio $N_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}/N_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}$ is sensitive to many microscopic details, such as the number of pumps and channels, as well as the microscopic conformation and configuration of their proteins (which may also affect the pumping rate, $r$). Furthermore, ion channels are known to be mechanosensitive (17) in many cases. Therefore, the activity $\delta$ is expected to depend on the tension exerted on channel proteins by the cell membrane, which is coupled by proteins to the cortex (19).

Since cell adhesion results in membrane traction forces that might also originate in the coupling to cortical tension (19), the activity is a function of the basal area, $A$. Furthermore, the assembly and hence the number of channels and pumps may also vary with $A$. In light of the many mechanisms that take part in this dependence, we do not attempt a microscopic derivation of the area dependence $\delta \left(A\right)$, but rather assume a simple linear form in terms of the strain,

$$\delta ={\delta }_0+{\delta }_1\frac{A-A_s}{A_s}\equiv {\delta }_0+{\delta }_1\left(\stackrel{\sim }{A}-1\right).$$ [5]

Here ${\delta }_0$ is the activity for the reference area $A=A_s$, related to the cell volume in suspension by $A_s=V_s^{2/3}$. The dimensionless basal (adhered) area is given by $\stackrel{\sim }{A}=A/A_s$. In Eq. 5, ${\delta }_1$ is a phenomenological model parameter that describes the change in $\delta$ with $A$. Note that the difference ${\delta }_0-{\delta }_1$ is the activity in suspension, for which there is no contact with any substrate and $A=0$.

Volume of Adhered Cells as a Function of Their Basal Area

The combination of Eqs. 3 and 5 provides a relation between the cell volume and basal area in terms of the osmotic pressure within the cell. This pressure can be inferred in steady state from the buffer pressure, $P_{\mathrm{o}\mathrm{u}\mathrm{t}}$, via the Young–Laplace law (11, 20), $P_{\mathrm{i}\mathrm{n}}=P_{\mathrm{o}\mathrm{u}\mathrm{t}}+2\gamma /R$, where $\gamma$ is the surface tension and $R$ the radius of curvature. Compared with the osmotic pressures of order megapascals, the term due to the surface tension is negligible due to the much smaller cortical modulus (4) and the large radius of curvature. Therefore, the volume–area dependence is determined to a very good approximation by the osmotic-pressure equality, as if the cell is flat.

The buffer remains unchanged during the $V-A$ experiments of refs. 4 and 5, indicating that the cell pressure is constant. As Eq. 3 shows, the pressure increases with increasing cell volume. The principal way, in which the pressure can remain fixed (as the cell area increases and volume decreases), is by pumping out more ions, as the basal area is increased. Therefore, ${\delta }_1$ of Eq. 5 must be negative, ${\delta }_1<0$.

The minimal possible cell volume for a given buffer pressure is obtained when the maximal number of ions is pumped out of the cell. This corresponds to the limit $\exp \left(2\delta \right)\ll 1$ in Eq. 3. For a buffer that contains only ions (which is a good approximation of the experimental measurements of volume vs. area in refs. 4 and 5), the buffer pressure is $P_{\mathrm{o}\mathrm{u}\mathrm{t}}=2n_b{k}_{\mathrm{B}}T$, and this minimal value, $V_0$ is given by $V_0=V_m+\left(1+z\right)N_p/\left(2n_b\right)$ (see SI Appendix for further details).

We now clarify the difference between the two volume scales, $V_0$ and $V_m$. $V_0$ is the “desalinated volume,” below which cells contain water, proteins, and counterions, with no extraneous salt. Volumes smaller than $V_0$ are obtained by squeezing out water without any osmolytes. Such volumes can be obtained only for buffers with larger pressures. The smallest possible volume, obtained in the limit of infinite buffer pressure, is $V_m$. This is the “dried” volume, where all free water molecules have been squeezed out. Note that both $V_0$ and $V_m$ can be measured as the minimal volumes in different experiments; $V_0$ is the minimal volume in a spreading experiment with a fixed buffer pressure, while $V_m$ is the minimal volume in an osmotic compression experiment. The combination of the volume scales defines a natural dimensionless volume,

$$\stackrel{\sim }{V}=\frac{V-V_0}{V_0-V_m}.$$ [6]

A summary of the different volume scales used in this paper and their meanings is given in Table 1.

Table 1.

Different measurable volume scales and dimensionless volumes used in this work

$V_m$	Dried volume
$V_0$	Desalinated volume
$V_s$	Cell volume in suspension
$\stackrel{\sim }{V}$	Dimensionless volume, $\stackrel{\sim }{V}=\frac{V-V_0}{V_0-V_m}$
${\stackrel{\sim }{V}}_s$	Dimensionless volume in suspension, ${\stackrel{\sim }{V}}_s=\frac{V_s-V_0}{V_0-V_m}$

We rewrite the cell pressure in terms of dimensionless volumes and equate it to the buffer pressure. This yields the following relation between the cell volume and basal area,

$${\mathrm{e}}^{2{\delta }_1\stackrel{\sim }{A}}=f\left(\stackrel{\sim }{V}\right)/f\left({\stackrel{\sim }{V}}_s\right),$$ [7]

where we have defined the function (see SI Appendix for further details)

$$f\left(y\right)=\frac{\left(1+z\right)y^2+2zy}{{\left(1+y\right)}^2}.$$ [8]

Note that the microscopic parameters $N_p$ and ${\delta }_0$ are expressed through the mesoscale volumes $V_0$ and $V_s$, which enter the dimensionless variables $\stackrel{\sim }{V}$ and ${\stackrel{\sim }{V}}_s$. Eq. 7 is satisfied, in particular, for suspended cells, since $\stackrel{\sim }{V}={\stackrel{\sim }{V}}_s$ for $\stackrel{\sim }{A}=0$.

Eq. 7 is our main result. It describes the dependence of the cell volume on its basal area in terms of measurable volume scales and only two phenomenological parameters. The parameter ${\delta }_1$ expresses the change in activity as the basal area is increased, and $-ze$ is the average, effective protein charge in the presence of ions.

The volume–area dependence has two simple limits. For spread cells with small $\stackrel{\sim }{V}\ll 1$, the volume decays exponentially with the spread area, $\stackrel{\sim }{V}\sim \exp \left(-2\left|{\delta }_1\right|\stackrel{\sim }{A}\right)$. For small basal areas, on the other hand, the volume falls off linearly: $\stackrel{\sim }{V}-{\stackrel{\sim }{V}}_s\sim -\stackrel{\sim }{A}$.

Comparison with Experiments

We compare our predicted volume–area relation of Eq. 7 with the experimental results of refs. 4 and 5. First, the normalization of volumes and areas requires knowing the three volume scales (Table 1): $V_s$, $V_0$, and $V_m$. While these three volumes can generally be measured, not all of them were measured in the experiments.

Guo et al. (4) performed osmotic-compression measurements, by adding PEG molecules to the buffer in different concentrations, and inferred $V_m=\mathrm{2,100}\hspace{0.17em}\mathrm{\mu }{\mathrm{m}}^3$, while $V_s$ and $V_0$ were not determined. Xie et al. (5), on the other hand, performed cell-spreading experiments and measured $V_0\simeq \mathrm{1,500}\hspace{0.17em}\mathrm{\mu }{\mathrm{m}}^3$ and $V_s\simeq \mathrm{3,900}\hspace{0.17em}\mathrm{\mu }{\mathrm{m}}^3$, without finding $V_m$. We estimate the remaining volumes by assuming a typical protein volume of $v\simeq 5.3\hspace{0.17em}{\mathrm{n}\mathrm{m}}^3$, which is considered a reasonable value (21). The value of $v$ relates $V_m$ and $V_0$ via $V_0/V_m=1+\left(1+z\right)/\left(2n_{b}v\right).$ Furthermore, we assume that the ratio $V_m/V_s$, which corresponds to the volume fraction of proteins in the intracellular fluid for cells in suspension, is equal for both cell types.

The abovementioned considerations yield the necessary unknown volumes for a given value of $z$. We determine the values of $z$ and ${\delta }_1$ by fitting Eq. 7 with experimental measurements. As is evident from Fig. 2, the fit works well. Experimental data from the two independent experiments, those of ref. 4 marked by blue circles and those of ref. 5 by red squares, fall on the same $\stackrel{\sim }{V}\hspace{0.17em}\hspace{0.17em}\left(\stackrel{\sim }{A}\right)$ curve, as predicted from the theory. Furthermore, the theory coincides with this curve for values $z=0.26$ and ${\delta }_1=-0.14$. The asymptotic expressions for $\stackrel{\sim }{V}\hspace{0.17em}\hspace{0.17em}\left(\stackrel{\sim }{A}\right)$ listed in the previous section can be used to robustly analyze future, more detailed experiments; this would permit extraction of the model parameters from the small/large volume regimes, without the need for global fits.

Fig. 2.

Measurements of cell volume as a function of their basal area from ref. 4 (blue circles) and ref. 5 (red squares), compared with the theoretical prediction of Eq. 7 for $z=0.26$ and ${\delta }_1=-0.14$. (A) Results in their original units. (B) Results in terms of the dimensionless area, $\stackrel{\sim }{A}$, and volume, $\stackrel{\sim }{V}$. The two experiments fall on the same curve.

We recall that $z$ is defined as the average, effective valency of proteins, screened by their localized ions. This low value justifies the Donnan framework that neglects the electric field within the cell. The value ${\delta }_1=-0.14$ implies activity changes of $-0.7\hspace{0.17em}{k}_{\mathrm{B}}T$ during cell spreading from $\stackrel{\sim }{A}\simeq 1$ to $\stackrel{\sim }{A}\simeq 6$ (Fig. 2). This is the same order of magnitude of other thermal effects in our system, such as the counterion entropy. Such a difference in the activity can, therefore, be maintained by the cell, making the value of ${\delta }_1$ sensible.

Using the same values of $z$ and ${\delta }_1$, we compare our result for the osmotic pressure (Eq. 3) with $V\left(P\right)$ curves from compression experiments in ref. 4. This fit also works rather well. For this fit we have used a constant value of $A=\mathrm{1,500}\hspace{0.17em}\mathrm{\mu }{\mathrm{m}}^2$, in accordance with the experiments. Note that the uncertainties in volume measurements, both in Fig. 2 and in Fig. 3, are of order $\mathrm{1,000}\hspace{0.17em}\mathrm{\mu }{\mathrm{m}}^3$ and are larger for larger volumes.

Fig. 3.

Measurements from ref. 4 of cell volume as a function of the external buffer pressure, compared with the theoretical prediction of Eq. 3. The curve corresponds to the same values of Fig. 2, $z=0.26$ and ${\delta }_1=-0.14$, used to fit the volume area measurements in Fig. 2, and a constant area of $A=\mathrm{1,500}\hspace{0.17em}\mathrm{\mu }{\mathrm{m}}^2$, in accordance with the experiment.

Discussion

We have shown how adhered cells regulate their volume, by water efflux and active ion transport. Two conditions determine the cell volume: 1) mechanical equilibrium with the extracellular medium (buffer) and 2) a generalization of Donnan equilibrium that accounts for cell activity, based on the zero net transport rate of ions through channels and pumps. The basal area of adhered cells affects the function of ion channels and pumps, and as a result of condition 2, it tunes indirectly the electrochemical potential difference between the cell and the buffer. This is the origin of the $V\left(A\right)$ dependence observed in the experiments of refs. 4 and 5.

Our theory demonstrates that the cell volume can be determined solely by osmotic pressure equality, regardless of cell shape. This is because the number of proteins in the cell is fixed. Therefore, the osmotic pressure that depends on the protein concentration dictates the cell volume. In principle, there are surface-tension corrections to this effect, due to the Young–Laplace law. However, the surface-tension terms are negligible in the experimental systems that we examine, as already noted by ref. 4, due to the large radius of curvature of the adhered cells and the relatively small cortical modulus that they measured.

An estimation of the magnitude of the surface-tension term can be obtained by considering cells with a cortical thickness $d=1\hspace{0.17em}\mathrm{\mu }\mathrm{m}$ and a radius of curvature $R=10\hspace{0.17em}\mathrm{\mu }\mathrm{m}$. We multiply the measured (4) cortical modulus of order 10 kPa (for large cell volumes) by $d/R$ and obtain a surface-tension term of the order of kilopascals. Even for such a thick cortex and small curvature, the surface tension is three orders of magnitude smaller than the measured bulk modulus, which is of the order of megapascals for large cell volumes. This clarifies why we carefully examine the volume- and area-dependent effects in the osmotic pressure equalities, as required for mechanical equilibrium.

Cortical tension is essential, however, in determining the cell shape (22, 23). We note that surface tension and possibly also the cell shape play an indirect, but important role in volume regulation. They are encapsulated within our theory in the parameter ${\delta }_1$ that relates the basal area with the electrochemical potential difference between the cell and buffer.

The activity, $\delta$, depends on a wide range of parameters of the cell and the cell–substrate interface. For example, it is affected by the substrate stiffness, the functionality of ion-channel proteins, and the activity of ion pumps. We expect that most of the different experimental conditions that were tested in refs. 4 and 5, including varying adhesion energy density, ATP depletion, and ion-channel and myosin inhibition, can be accounted for within our theory by their combined effects on the single parameter ${\delta }_1$.

While the linear dependence of ${\delta }_1$ on the adhered area (Eq. 5) is phenomenological, it is sensible. In a power-series expansion of ${\delta }_1$ around a reference area, the first nonvanishing term is indeed expected to be linear. Otherwise, if it were quadratic, the cell would respond equally to both positive and negative area changes. Furthermore, for relatively flat cells with comparable basal and apical areas, Eq. 5 relates the activity to the apical strain.

Our theory is written in terms of the cell volume in suspension, $V_s$, and two additional and measurable minimal volumes. The minimal desalinated cell volume, $V_0$, is obtained for large basal areas, where all of the extraneous salt is released from the cell, leaving only the counterions that neutralize the proteins within the cell. Any additional volume reduction is obtained by water efflux and is limited by the buffer pressure. Larger pressures squeeze more water out of the cell. The minimal possible volume is the dried cell volume, $V_m$, which contains only proteins, counterions, and bound water molecules. Assuming that most cells have comparable effective protein valencies, $z$, and protein volumes $v$, then $V_0$ and $V_m$ are directly related to one another, and it suffices to measure only one of them.

While we focus on adhered cells, our result for the osmotic pressure within cells, Eq. 3, can be used also for cells in suspension. This requires a redefinition of $A$ as the surface area and, consequently, an adjustment of the parameter ${\delta }_1$. This parameter would then refer to activity changes in response to cell swelling and shrinkage. Note that the $V\left(A\right)$ behavior of adhered cells is in accord with the natural adaptivity of cells in suspension. Suspended cells with increased surface areas have increased volumes. The natural cell response would be to decrease its volume via active ion transport.

Eq. 3 can be tested experimentally in several ways. For example, the volumes of suspended cells can be measured as the buffer pressure is varied. We refer here to a slow variation of the pressure, which allows cells to relax to steady state, compared to osmotic-shock experiments that test the cell response to transient extracellular changes. The area dependence of the activity might also be determined by experiments on reconstituted membranes containing both pumps and channels, where the area is varied mechanically. In addition, it is possible to change the cellular protein content and measure the resulting volume changes.

Ref. 5 also includes a model for the volume regulation of adhered cells (based partially on ref. 24), which is used to explain the dependence of cell volume on substrate properties, but not the volume–area dependence. The theory is rather elaborate, involving 17 coupled differential equations and many parameters. Our theory provides simple intuition into the role of both electrostatics and pump activity in regulating cell volume as a function of area and includes only two model parameters; this allows us to analytically capture the volume–area dependence within a single equation (Eq. 7).

We highlight two key differences between our minimal model and the equivalent ingredients that appear in ref. 5. The cell components of ref. 5 are a single neutral osmolyte species that can be released passively through channels or pumped actively into the cell. Our model, on the other hand, distinguishes between cations and anions that can exchange with the buffer and charged proteins that are confined to the cell. These proteins are important for charge neutrality and their confinement has a major contribution to the osmotic pressure. Furthermore, this distinction clarifies that, as a whole, ions are pumped out of the cell rather than into it.

Because our theory is based on a minimal model, we necessarily coarse grain several details that affect the inner cell pressure and the ionic transport in and out of the cell; no distinction is made between different ionic species of the same charge [and their respective transport through channels/pumps (25)] or between different protein species and their various possible configurations. In addition, the cortex and other macromolecular assemblies (that do not contribute to protein osmotic pressure) are not addressed explicitly, although their properties may affect the activity, $\delta$. We assume that the cell content is fixed and neglect, for example, metabolic processes. Furthermore, as we are concerned with steady state, we neglect cell dynamics, including water efflux and its concurrent hydrostatic pressure gradients.

To conclude, our theory accounts in an intuitive and quantitative manner for the observed volume regulation in adhered cells, using a minimal model. Its predictions for the osmotic pressure in cells can be useful in a wide range of works and can be explored further by experimental studies in the future.

Data Availability.

All data discussed in this paper are available in the main text and SI Appendix.