Active Biochemical Regulation of Cell Volume and a Simple Model of Cell Tension Response

Abstract

Active contractile forces exerted by eukaryotic cells play significant roles during embryonic development, tissue formation, and cell motility. At the molecular level, small GTPases in signaling pathways can regulate active cell contraction. Here, starting with mechanical force balance at the cell cortex, and the recent discovery that tension-sensitive membrane channels can catalyze the conversion of the inactive form of Rho to the active form, we show mathematically that this active regulation of cellular contractility together with osmotic regulation can robustly control the cell size and membrane tension against external mechanical or osmotic shocks. We find that the magnitude of active contraction depends on the rate of mechanical pulling, but the cell tension can recover. The model also predicts that the cell exerts stronger contractile forces against a stiffer external environment, and therefore exhibits features of mechanosensation. These results suggest that a simple system for maintaining homeostatic values of cell volume and membrane tension could explain cell tension response and mechanosensation in different environments.

Introduction

Eukaryotic cells can actively exert mechanical forces on their extracellular environment. These forces have been measured in two- (2D) and three-dimensional (3D) cell cultures (1, 2, 3, 4), and have been shown to be important not only during cell migration, tissue and organ formation, and development, but also during cell-volume control in response to osmotic changes (5). Many experiments have shown that cells on stiffer substrates apply stronger contractile forces (1, 6, 7). Biochemical signaling pathways have been implicated in this active force generation. Notably, GTPases such as the Rho family of proteins, are part of the signaling pathway that controls myosin II assembly and force generation. The active form of Rho phosphorylates ROCK, which then activates the myosin light chain (MLC) (8, 9, 10, 11, 12). This leads to the assembly of myosin minifilaments and an increase in contractile forces. Remarkably, Rho itself also responds to externally applied mechanical forces. When cells are mechanically pulled by attached magnetic beads, the active form of Rho also increases and then diminishes in time, presumably correlated with changes in contractile force (12). Related phenomena are seen when cells are subjected to pipet suction. Here, an increase in myosin accumulation is observed at the location of suction force (13, 14), although Rho activation was not directly measured in those experiments. Finally, when cells are subjected to osmotic shock, which changes the mechanical tension across the cell-membrane cortex, myosin contraction has been implicated in restoring the cell volume to preshock values (5, 15). More recent studies have shown that mechanosensitive (MS) membrane channels can regulate the activity of Rho and catalyze the conversion from the inactive form to the active form (8, 16, 17, 18, 19). These experiments are beginning to reveal the feedback loop between active cell force generation and mechanical tension.

In this article, we mathematically examine such a feedback mechanism that controls cell active contraction using a simple mechanical model coupled to a biochemical network. To keep the cell geometry simple and remove complexities from cell adhesions, we consider suspended or mitotic cells where they are spherical, and cylindrical cells between flat cantilevers with fixed adhesion area. The latter situation has been elegantly examined recently in experiments (20, 21). We first describe the balance of forces at the cell boundary, which is made of cell membrane and an actomyosin cortex. By modeling the cortex as an active gel with rapid actin turnover (22, 23, 24), we find that the hydrostatic pressure difference across the cell membrane is balanced by active cortical contraction, passive stress from cortical flow, and membrane tension. Indeed, cell osmotic pressure is partially controlled by MS ion channels and ion transporters in the membrane (15, 25). Recent studies have shown that the MS channel TRPIV is involved in activating Rho in response to osmotic pressure changes (16, 17). Related experiments in Drosophila cells indicate that the transmembrane protein Toll can activate Rho and contraction (8, 18, 19). Further evidence also suggests that membrane tension is a global signal that controls cell polarization (26). Labeling of the active form of Rho in live cells showed that Rho is preferentially activated near the cell leading edge, where membrane tension is likely high (27, 28). Here we demonstrate how to model this system mathematically, and compute the cell response to external changes in osmolarity as well as externally applied forces.

The model appears to unify a number of related phenomena in cell mechanics. First, the proposed system is able to maintain a relatively constant cell volume in response to osmotic changes. Osmotic shocks lead to changes in the hydrostatic pressure difference and membrane tension and cause water flow across the cell membrane. In our model, this leads to ion flows across the membrane and changes in active contraction. The result is a robust adjustment of cell volume and membrane tension back to the preshock values, in accordance with single-cell experiments (5). We also show that neither ion flow nor active contraction alone leads to robust adaptation to osmotic shocks. Both systems are needed to obtain robust volume control. Second, when a cell is stretched between two cantilevers, external mechanical pulling also leads to water flow and membrane-tension changes. This increases active contractile forces that again try to restore cell volume and membrane tension. Cell active contraction therefore changes over time. The contraction dynamics of the cell depends on the rate of pulling, and the final cell tension depends on the total amount of deformation. Third, when a cell is subjected to a jump in externally applied mechanical force, Rho becomes activated and there is a membrane tension jump. However, Rho activation recedes over time because of ion flows, and the membrane tension is restored to prejump values, in accordance with dynamics observed by Zhao et al. (12). Fourth, our model is able to predict that the cell will exert larger steady contractile force against a stiffer substrate. The steady-state cell volume also varies depending on cantilever stiffness. This result indicates that our model can explain some features of cell mechanosensation where stiffness of the cell substrate influences cell contractility. It also indicates that active control of cell contraction can explain cell volume dynamics as well as cellular response to externally applied forces.

We begin by considering mechanics of the cell cortex and membrane subjected to excess osmotic pressure in the cell. We then describe the regulation of osmotic pressure by membrane channels and ion pumps, and a simple model of Rho regulation of myosin contraction. Model predictions for cells subjected to osmotic shocks or mechanical shocks, such as a sudden application of pulling force, are analyzed. Detailed predictions of cell responses to mechanical forces and the mechanism of strain-rate-dependent force response are discussed. We also demonstrate that the model predicts increasing levels of myosin activation when the cell contracts against substrates of increasing stiffness. In comparison with our previous work on cell volume control (25), which considered an elastic constitutive relation for the cell cortex, this work focuses on dynamics for a liquid-like cortex and active control of myosin contraction. The liquid-like cortex is likely the correct description for most tissue cells under normal circumstances. Therefore, our model suggests that the cell volume exists as a stable steady state of a dynamically controlled biochemical system.

Materials and Methods

Force balance at the cell surface

A typical cell is bounded by a surface consisting of an outer plasma membrane and an actomyosin cortex. The plasma membrane is adhered to the actomyosin cortical layer through transmembrane proteins (Fig. 1). The membrane is typically 5 nm in thickness, but the cortical layer thickness, h, is 200–500 nm (29). Within the cell cortex, actin filaments rapidly turn over and small myosin motor assemblies exert active contractile forces (23, 30). There are also dynamic actin cross-linking proteins that can potentially alter the mechanics of the cortex (30). Although the molecular details of the cortex are complex, it can be generically modeled as an active viscoelastic gel-like fluid. Being a viscoelastic fluid, it does not have a reference geometry and will flow under mechanical perturbations. Therefore, force balance and cortex geometry alone cannot determine the global cell shape and volume. To simplify matters in this article, we will ignore actin dynamics associated with focal adhesions and focus on cells with fixed or no contact with substrates.

Figure 1.

(A) Illustration of the Rho signaling pathway that activates myosin assembly and active contraction in the cell cortex. At mechanical equilibrium, the membrane tension must balance both osmotic pressure in the cell and active contraction in the cortex (Eq. 1). (B) In our model, we consider membrane-tension changes that activate MS channels, which then activate Rho and myosin contraction. The contractile force negatively feeds back to membrane tension. The probability of Rho activation, $\text{Λ}\left(T\right)$, starts to increase at a critical tension, $T_{\text{c}}$, and saturates at $T_{\text{s}}$. To see this figure in color, go online.

The cytoplasm is crowded with proteins, RNA, and ions; the osmotic pressure inside the cell is therefore generally higher than that of the extracellular milieu. Therefore, there is an osmotic pressure difference, $\text{Δ}\text{Π}={\text{Π}}_{\text{in}}-{\text{Π}}_{\text{out}}$, between the inside and outside of the cell. ${\text{Π}}_{\text{in}}$ is related to the total osmolytes, n, in the cell, or ${\text{Π}}_{\text{in}}\sim RT{c}_{\text{in}}\propto n/V$, where V is the cell volume (Fig. 2 A), R is the gas constant, and $c_{\text{in}}$ is the osmolyte concentration inside. Water will flow in response to this osmotic pressure difference, but at static equilibrium, the osmotic pressure difference equals the hydrostatic pressure difference: $\text{Δ}\text{Π}=\text{Δ}P$. The hydrostatic pressure difference is balanced by tension in the membrane and mechanical stress in the cortex. Using a simple active gel model, force balance can be solved for a cell surface element (see the Supporting Material for details). It can be shown that the passive viscous shear stress in the cortex is small at steady state, and the active stress is the dominant contribution. The resulting force balance is simple for a spherical cell with radius R:

$$\frac{\text{Δ}PR}{2}=T+{\sigma }^{\text{a}}h,$$ (1)

where ${\sigma }^{\text{a}}$ is the nonzero diagonal part of the active stress tensor: ${\sigma }_{\text{active}}=-{\sigma }^a\left({\mathbf{e}}_{\varphi }\otimes {\mathbf{e}}_{\varphi }+{\mathbf{e}}_{\mathbf{\theta }}\otimes {\mathbf{e}}_{\mathbf{\theta }}\right)$, and T is the tension in the membrane. Here, we assume that myosin contracts in directions tangential to the cell surface. In general, for an arbitrary cell shape with local mean curvature C, where $C=\nabla ·\mathbf{n}$, the hydrostatic pressure is balanced by

$$\frac{\text{Δ}P}{2C}=T+{\sigma }^{\text{a}}h,$$ (2)

and $\mathbf{n}$ is the local cell surface normal vector. Given that actin polymerization and turnover is relatively fast, the above force-balance condition is reached within tens of seconds. In deriving the force balance condition, we have assumed that the cortical mass and density are constant, and the cortex behaves as a Newtonian fluid. Actomyosin networks, however, have complex mechanical properties, and the physics of cortical flow has been extensively discussed (23, 31). The simple approximations followed here allowed us to obtain the analytical estimates above.

Figure 2.

Model calculations for a spherical cell during osmotic shock. (A) Cartoon of the cell showing components important in our model. For the active ion pumps, we have used ${\text{Δ}\text{Π}}_{\text{c}}=1.1{\text{Π}}_{\text{out}}$. (B) The cell is subjected to a hypotonic shock and then a hypertonic shock. The shock magnitudes are 0.5 ${\text{Π}}_{\text{out}}$ and then 0.75 ${\text{Π}}_{\text{out}}$. The cell volume can recover to close to the preshock value. Indeed, volume, membrane tension, Rho-MLC activation level, and pressure difference all can recover, meaning that the cell can adapt to the new osmotic environment. There is a slight overshoot after recovery, because ${\text{Δ}\text{Π}}_{\text{c}}$ is proportional to ${\text{Π}}_{\text{out}}$. (C) The steady-state cell volume depends on ${\text{Π}}_{\text{out}}$. The model predicts that the steady volume after recovery is smaller after a hypotonic shock. Here, ${\text{Δ}\text{Π}}_{\text{c}}=1.1{\text{Π}}_{\text{out}}$. To see this figure in color, go online.

Biochemical regulation of myosin contraction

The force-balance relation in Eq. 1 alone does not determine global cell shape and size. This is achieved by dynamic myosin contraction, i.e., ${\sigma }^{\text{a}}$ is not a constant, but changes in response to mechanical forces applied to the cell. From the force-balance relation in Eq. 1, we see that osmotic pressure, hydrostatic pressure, or any mechanical forces on the cell will change both membrane tension and tension in the cortex. Therefore, we propose that membrane tension could be the upstream signal that catalyzes the activation of Rho. Other possibilities also exist, e.g., transient cortical stress could change myosin binding to actin and power-stroke kinetics (13, 32). This additional complexity is discussed later in the Discussion and Supporting Material. All these models suggest a feedback mechanism where increases in membrane tension and mechanical stress in the cortex lead to increasing myosin contraction and ${\sigma }^{\text{a}}$. This then restores the membrane tension and passive mechanical stress. Indeed, our model predicts that the cell can maintain essentially a constant membrane tension with this mechanism.

The chemical signaling network we propose to examine is shown in Fig. 1. Rho activation is triggered by membrane tension, T. Rho activates the MLC, which increases the fraction of myosin minifilaments, M. The concentration of myosin minifilaments is directly proportional to the active stress,

$${\sigma }^{\text{a}}=K_{\text{max}}M,$$ (3)

where $K_{\text{max}}$ is a maximum contractile stress parameter and M is the fraction of activated myosin. $K_{\text{max}}$ is the maximum stress the cell can exert if all of the myosins are activated. The biochemical equations are

$$\begin{array}{c}\frac{\partial \rho }{\partial t}=a_1\text{Λ}\left(T\right)\left(1-\rho \right)-d_1\rho \\ \frac{\partial M}{\partial t}=a_2\left(1-M\right)\rho -d_{2}M,\end{array}$$ (4)

where $a_1\text{and}{d}_1$ are the activation and deactivation rates, respectively, of Rho; ρ and M are percentages of activated Rho and myosin, respectively; and $a_2$ and $d_2$ are the myosin assembly and disassembly rates, respectively. Here, for simplicity, we do not explicitly include ROCK in the pathway. Instead, ROCK dynamics is included in the equation for Rho by using effective rate constants $a_1$ and $d_1$. $\text{Λ}$ is an activation function of ρ, which depends nonlinearly on membrane tension, T. We propose that $\text{Λ}=\left(T-T_{\text{c}}/T_{\text{s}}-T_{\text{c}}\right)$ when $T_{\text{c}}<T<T_{\text{s}}$. $\text{Λ}=1$ when $T>T_{\text{s}}$ and $\text{Λ}=0$ when $T<T_{\text{c}}$. $T_{\text{c}}$ is the critical membrane tension at which Rho activation starts and $T_{\text{s}}$ is a saturating tension (Fig. 1). The functional form of $\text{Λ}$ is essentially the same as a Michaelis-Menten type of enzymatic kinetics.

We also note that within our model, it is possible to consider nonlinear behavior in the mechanics of the actin cortex and strain-hardening behavior in actin networks. Multiple experiments show that there is a change in the number of actin cross-linkers when cells are subjected to rapid changes in force (13, 20). This complex behavior would influence myosin assembly and contraction. These results suggest that myosin assembly and disassembly rates, $a_2\text{and}{d}_2$, depend on transient passive stress in the cortex. One possible way to incorporate this effect is to write

$$a_2=a_{20}\left(1+f\left(T_{\text{shear}}\right)\right),$$ (5)

where $a_{20}$ is a constant and $T_{\text{shear}}$ is a passive transient force per unit length in the cortex (see the Supporting Material for details). $f\left(T_{\text{shear}}\right)$ is an activation function that depends on $T_{\text{shear}}$. For Newtonian fluids, $T_{\text{shear}}$ is proportional to the flow rate in the cortex. This phenomenological model is consistent with the idea that myosin assembly and force production depend on the shear stress (rate of deformation) in the cortex. This model is also related to the observed strain-rate-dependent force change (see Fig. 4). However, the details of the model will have to depend on a better understanding of the relationship between cortex mechanics and myosin assembly and contraction. Other mechanisms that regulate myosin contraction are also possible; for example, calcium influx from tension change can also regulate myosin contraction (33, 34). It is likely that multiple mechanisms are at play to different degrees in different kinds of cells.

Figure 4.

Cylindrical cell response to vertical displacements. (A) The vertical dimension of the cell, H, is changed at different velocities. The cortical thickness is h and the contact radius with the cantilevers is R. The model is used to compute the necessary pulling force, F. (B) As H is changed, the cell force response goes through several phases. Here, red and blue curves represent two different velocities of vertical displacement. The velocity of vertical displacement affects the transient force developed by the cell. If vertical displacement is fast, there is a large jump in force, because osmotic pressure and active stress cannot adapt quickly. The final steady-state force also depends on the displacement velocity. The final steady-state F depends on the rate of mechanical pulling, or strain rate. During pulling, the model also can compute the changes in cell osmotic pressure, $\text{Δ}P$, and the level of myosin active stress. (C) Comparisons of the model results with experimental results from Webster et al. (20). The final steady-state force agrees well with experimental results. The model can also fit a short-term transient force jump. (D) Model prediction of steady-state force as a function of vertical strain for a slow pulling case ($\sim 1\mu$ m/min). Curves at faster pulling rates are shown in Fig. 5. Note that the force is not zero at zero strain. Zero force is reached when the strain is $\sim -10\%$. To see this figure in color, go online.

Active regulation of cell volume and membrane tension

In addition to active regulation of myosin contraction, the cell can also adjust its internal osmotic pressure, ${\text{Π}}_{\text{in}}$, leading to cell-volume adaptation to osmotic shocks (5, 15). Equations for water and ion fluxes were discussed in a recent article (25). They are

$$\begin{array}{c}\frac{\partial V}{\partial t}=-\alpha A\left(\text{Δ}P-\text{Δ}\text{Π}\right)\\ \frac{\partial n}{\partial t}=A\left(J_1+J_2\right),\end{array}$$ (6)

in which V and A are cell volume and surface area, respectively. $\text{Δ}P$ and $\text{Δ}\text{Π}$ are hydrostatic and osmotic pressure differences across the membrane. $\partial V/\partial t$ is the rate of cell-volume change due to water flow. n is the total number of osmolytes in the cell. $J_1$ is the ion flux out of the cell through passive membrane channels. These passive ion channels could be MS and open in response to changes in membrane tension. Thus, $J_1=-\beta {\text{Λ}}^{\prime }\left(T\right)\text{Δ}\text{Π}$. The negative sign indicates the outflow of ions. ${\text{Λ}}^{\prime }\left(T\right)$ is a function similar to $\text{Λ}\left(T\right)$ in Fig. 1 B, and it depends on parameters $T_1\text{and}{T}_2$, which are the critical and saturation tensions of MS channels.

$J_2$ describes flux through active ion pumps, which pump against concentration gradients. $J_2$ can be computed from a simple channel model. Assuming that the pump uses energy input, $\text{Δ}{G}_{\text{a}}$, to generate a pump force of $\text{Δ}{G}_{\text{a}}\text{/}\delta$, where δ is the membrane thickness, the steady-state flux through a single channel is approximately $j_2=-D\partial c\text{/}\partial x-D\text{/}{k}_{\text{B}}T\left(\text{Δ}{G}_{\text{a}}\text{/}\delta \right)c$, where c is the ion concentration profile within the channel and D is an effective diffusion constant. c satisfies boundary conditions $c\left(0\right)=c_{\text{in}}$ and $c\left(\delta \right)=c_{\text{out}}$. This gives the single pump flux as

$$j_2=\frac{D\text{Δ}{G}_{\text{a}}}{k_{\text{B}}T\delta }\left(\frac{\text{Δ}\text{Π}{e}^{-\text{Δ}{G}_{\text{a}}/k_{\text{B}}T}}{RT\left(1-e^{-\text{Δ}{G}_{\text{a}}/k_{\text{B}}T}\right)}-c_{\text{out}}\right).$$ (7)

From this, we can obtain ${\text{Δ}\text{Π}}_{\text{c}}$, the critical concentration at which the flux is zero:

$${\text{Δ}\text{Π}}_{\text{c}}={\text{Π}}_{\text{out}}\left(e^{\text{Δ}{G}_{\text{a}}/k_{\text{B}}T}-1\right).$$ (8)

The total pump flux therefore can be written as

$$J_2=n_{\text{a}}{j}_2=-\gamma \left(\text{Δ}\text{Π}-{\text{Δ}\text{Π}}_{\text{c}}\right),$$ (9)

where γ is another effective permeation constant that contains the number density of the ion pumps, $n_{\text{a}}$. In general, depending on the molecular mechanism of the ion pump, the flux expression, $J_2$, can be quite complicated. If multiple species of ions are considered, the different flux would depend on individual ion concentrations. Equation 17 is the simplest model. More complex models, such as those in Gao et al. (35) and Armstrong (36), can be explored as well.

Equations 1, 3, 4, and 6 are six equations that describe cell-volume changes in response to changes in osmotic pressure, mechanical forces, and active motor activity. For a spherical cell, the unknowns are $\left(\rho ,M,T,\text{Δ}P,{\sigma }^{\text{a}},R,\text{and}n\right)$. These equations are not closed, and they require one other relationship, the constitutive law for the cell membrane, which relates the membrane tension to overall area changes of the membrane. A simple linear relationship (37) is

$$T=\kappa \left(\frac{A-A_0}{A_0}\right),$$ (10)

where κ is an effective elastic modulus and $A_0$ is the reference membrane area when it is not under tension. $A_0$ is set by the total number of lipid molecules. $A_0$ also can depend on lipid trafficking, which may be also triggered by membrane tension (38, 39). In addition, $A_0$ includes possible entropic properties of the membrane and reflects the fact that the membrane is typically highly folded in the cell (40). A is the stretched membrane area. Here, κ could arise from entropic elasticity. Lipid trafficking occurs on a timescale of hours, and here we regard $A_0$ as a constant.

Results

Robust control of cell volume and membrane tension

Using this model, we can mathematically describe the dynamics of the cell volume during osmotic shock (Fig. 2). Results show the behavior of a stable dynamical system arising from active control, where a stable volume is determined not by any reference geometry but by cell parameters such as $K_{\text{max}}$ and n. A sudden decrease in ${\text{Π}}_{\text{out}}$ causes water influx into the cell across the membrane, decreasing $\text{Δ}P$, which leads to an increase in membrane tension. Membrane tension changes trigger chemical activation of the Rho-MLC pathway, as well as opening of ion channels at the cell surface. This active contraction and the ion fluxes help membrane tension to recover from the initial changes. Because the cell cortex is a viscoelastic fluid without any reference shape, active contractile stress must adjust to maintain a constant cell volume. If there is no active control, the cell volume cannot adapt properly to osmotic shocks (Fig. 3 A). Under hypertonic shock, ion pumps are essential in regulating cell volume, since myosin contraction does not play a role. Without ion fluxes, the cell is unable to recover after hypertonic shock (Fig. 3). Without myosin active contraction, the cell is still able to recover, but there is a large overshoot in the cell-volume change after recovery.

Figure 3.

Biochemical control of contraction and ion permeation help to maintain cell volume and membrane tension. (A) The model predicts that when Rho signaling control of myosin is turned off and ${\sigma }^{\text{a}}$ is a constant (red line), or ion fluxes are turned off and $J_1=J_2=0$ (black line), the cell does not recover cell volume or membrane tension effectively. When both systems are active (green line), the cell volume and membrane tension can effectively maintain a homeostatic value. Note that the initial cell volume before shock is kept the same in the model, but different initial starting points for n and ${\sigma }^{\text{a}}$ are needed to achieve the same initial volume. (B) Steady-state cell volume, membrane tension, and hydrostatic pressure are determined by maximum possible myosin active stress $\left(K_{\text{max}}\right)$ and ${\text{Π}}_{\text{out}}$. As $K_{\text{max}}$ increases, the cell volume decreases and pressure increases. The membrane tension decreases. Decreasing ${\text{Π}}_{\text{out}}$ decreases steady-state cell volume, although the volume does transiently increase at first. The degree of volume overshoot, $\text{Δ}V\text{/}{V}_{\text{initial}}$, after osmotic shock also depends on $K_{\text{max}}$. To see this figure in color, go online.

Critical parameters in this volume adaptation system are ${\text{Δ}\text{Π}}_{\text{c}}$, permeation constants γ and β, and maximum contractile stress, $K_{\text{max}}$. These parameters all can influence the final steady-state volume of the cell (Figs. 3 B and S2). As we noted in Eq. 8, ${\text{Δ}\text{Π}}_{\text{c}}$ is a function of ${\text{Π}}_{\text{out}}$; therefore, the cell volume after recovery is not the same as before the osmotic shock (Fig. 2 C). Instead, assuming everything else is constant within the cell, the steady-state cell volume decreases as ${\text{Π}}_{\text{out}}$ decreases, even though volume initially increases transiently after ${\text{Π}}_{\text{out}}$ decreases. γ and β are permeation constants of the passive and active ion channels. These parameters are related to overall expression levels of these membrane proteins and their molecular properties. We see that these parameters determine the overall steady-state solute content in the cell, and therefore the steady-state volume (Fig. S2).

$K_{\text{max}}$ is a parameter that describes the maximum active contractile stress the cell can generate when all available myosin is fully activated. Myosin active stress also depends on an intact cell cortex, so depolymerizing actin would impact $K_{\text{max}}$. We see that the steady-state cell volume and membrane tension decline as $K_{\text{max}}$ increases, suggesting that the cell volume will increase if active stress is reduced (Fig. 3). Larger $K_{\text{max}}$ also increases the hydrostatic pressure difference at steady state. However, larger $K_{\text{max}}$ provides stronger control of cell volume after osmotic shocks (Fig. 3 B), where the volume change after osmotic shock is minimized with respect to before the shock.

The rate of cell-volume adaptation depends on the water permeation parameter, α, and the chemical rate constants $a_1$, $a_2$, $d_1$, and $d_2$ that govern the speed of Rho and myosin activation. In this work, we have set these rate constants such that adaptation occurs within 10 min. This is consistent with results seen in Stewart et al. (5) and Zhao et al. (12), where activation of Rho occurred $\sim$ 5 min after cells were pulled by magnetic beads. This is also the same timescale as for myosin activation after pipet aspiration, which changes the hydrostatic pressure difference across the membrane (13).

As mentioned earlier, we have assumed that the lipid reference area, $A_0$, is a constant and that cell volume changes from fluxes through membrane channels such as aquaporins. Cells can also change volume through endocytosis, whose rates should also depend on membrane tension. At longer timescales of hours, lipid trafficking from the Golgi to the cell surface can occur. This would potentially change $A_0$, and it suggests a third mechanism of cell membrane-tension control. Previous work suggests that membrane tension can be controlled if the rates of lipid addition and subtraction are functions of membrane tension (41). This phenomenon is likely if the membrane tension has changed persistently for long periods, and myosin contraction and ion transport are unable to restore membrane tension. Lipid trafficking can provide the final mechanism of restoring cell integrity.

A simple model of cell tension homeostasis

The proposed model not only can describe how cells respond to osmotic changes, but can also predict cell response to external applied mechanical forces. Many experiments using different techniques have examined how cells respond to mechanical forces. Here, we focus on a simple geometry where the cell is between two plates. One of the plates is actuated vertically at velocity v, leading to an overall change in cell height, H (Fig. 4). Such an experiment was performed recently, where the cell adhesion areas at the two plates are fixed (20, 42). This implies that there is negligible change in cell adhesion during pulling. We model the mechanics of the cell within this experiment. The goal is to explain changes in mechanical tension on the cantilever.

For a cylindrical cell, the cell volume and surface area are determined by the overall height, H, and cell radius, R. During fast pulling or large strain, strictly speaking, the cell no longer maintains a cylindrical shape, and the cell radius will vary along the z direction (Fig. 4). The full geometry is mechanically complicated to analyze. For simplicity, we limit our discussion to the regime where the pulling rate is slower than the water permeation rate and approximate the cell as a cylinder throughout. The cell adhesion areas at the two plates are fixed, $z=0$ and $z=H$. Therefore, we assume that R remains constant through the pulling/compressing process $\left(\partial R\text{/}\partial t\sim 0\right)$. In this case, the volume change is simply related to a change in H, i.e., $\partial V\text{/}\partial t=\pi R^2\partial H\text{/}\partial t$.

Under these assumptions and an active contractile stress of the form ${\sigma }_{\text{active}}=-{\sigma }^a\left({\mathbf{e}}_{\mathbf{z}}\otimes {\mathbf{e}}_{\mathbf{z}}+{\mathbf{e}}_{\theta }\otimes {\mathbf{e}}_{\theta }\right)$, the membrane tension for a cylindrical cell under a constant force F at $z=H$ can be computed (see the Supporting Material for details) as

$$\begin{array}{c}T=\left(\text{Δ}P+\frac{F}{\pi R^2-\pi {\left(R-h\right)}^2}\right)R\\ -{\sigma }^{a}h\left(1+\frac{h}{R}\right)\left(1-\frac{h}{2R}\right),\end{array}$$ (11)

where h is the cortical thickness. This result relies on the fact that $h\text{/}H$ is a small parameter and the pulling velocity is slow, and therefore, the rate of cortical volume change is slow. We see that the membrane tension is again a combination of hydrostatic pressure, pulling force, cell dimensions, and active contractile stress.

Together with the constitutive relation in Eq. 10, water and ion permeation in Eq. 6, and biochemical regulation in Eq. 4, we again have a close set of equations. From these, we can predict how cells respond to external dimensional change and mechanical force, as well as osmotic changes.

Cell tension and step change in displacement

Some model results are shown in Fig. 4, where we compute the force response of the cell following the methods used in the experiments of Webster et al. (20). Direct comparisons between our model results and experimental data from Webster et al. (20) are shown in Fig. 4, C and D. We first apply a vertical displacement of 1.0 μm at $z=H$. This displacement is applied at different speeds, as illustrated in Fig. 4. We find that the force response of the cell depends on the rate of the external displacement. A faster displacement drives a stronger development of active contractile force. When a tensile displacement takes place, $\text{Δ}\text{Π}$ decreases due to the volume expansion, which also changes $\text{Δ}P$. Tension also increases from stretching in the membrane. These are the initial passive mechanical events upon a sudden tensile displacement. Subsequently, ions start to flow inside the cell $\left(J_2\right)$ and myosin contraction becomes activated. This leads to contractile stress changes and adaptation of osmotic pressure. As the osmotic pressure difference recovers, the tensile force also decreases.

Our model shows features that are consistent with experiments on cell tension. When cells are pulled, the force jumps significantly but then recovers to a smaller value. The final steady-state force depends on the overall deformation and the strain rate, matching experimental results. As the strain becomes larger, the cell activates a larger portion of myosin in response to pulling and this force eventually saturates. However, the cell pressure is not dependent on strain rate and depends only on ion flux rates. The model also shows that in the negative strain direction, the cell adjusts osmotic pressure but generally resists compression. Here, active contraction does not play a significant role. In previous experimental studies, the cell was also subjected to a step-function-like displacement (with the strain rate approaching infinity), with the result that the final force is smaller than the final force when the pulling rate is smaller: 1 μ/min. Currently, our model does not predict such nonmonotonic strain response, because we assume that the cell maintains cylindrical shape and water permeation is fast.

Cell tension depends on cantilever stiffness

Remarkably, this model predicts that the cell should be sensitive to the stiffness of the cell environment. The following experiment can be modeled by our equations. A flexible cantilever is placed at one end of the cell and the cell is allowed to contract against it (Fig. 5). When the force exerted by the cell is equal to the cantilever force, $F=-K\text{Δ}H$, where K is the cantilever stiffness, the system will reach mechanical equilibrium. Graphically, mechanical equilibrium is equivalent to the intersection of the F-versus- $\text{Δ}H$ curve and $F=-K\text{Δ}H$. The result shows that the cell contractile force increases with increasing cantilever stiffness. Experimental results from traction-force microscopy have shown a similar trend (1, 43). Here, the cell volume and membrane tension are both higher when the cantilever is stiffer, and the cell exerts stronger contractile force to try to reduce volume and membrane tension. Note that adhesion in this problem is kept fixed, so cell adhesion changes do not play a role in our model.

Figure 5.

(A) A cell contracting against a flexible cantilever. The force of the cantilever is $F=-K\text{Δ}H$, where K is the cantilever stiffness. (B) Mechanical equilibrium is reached when the cantilever force is equal to the cell contractile force. This represents the intersection between the F-versus-strain curve and $F=-K\text{Δ}H$ (red line). Our model predictions for F versus strain are taken from Fig. 4D. Different F-versus-strain curves represent results from different strain rates. Results show that the cell increases contractile force as K becomes larger (green curve). Here, for an adhesion radius of 20 μm, $K=10$ nN/μm is equivalent to a Young’s modulus of 500 Pa. (C) The model also predicts that cell volume is larger with a stiffer cantilever (blue curve). The contractile force generated by the cell also increases with increasing cantilever stiffness. This is because the activated form of myosin increases with increasing cantilever stiffness, which results from the increasing activation of Rho. To see this figure in color, go online.

In addition to cell tension, the steady-state cell volume also increases with cantilever stiffness (Fig. 5 B). These results suggest that while trying to maintain cell volume and membrane tension, the cell will exert forces that depend on the stiffness of the environment or the surrounding extracellular matrix. Although our model does not currently describe cells on a flat substrate, it is possible that the cell on flat surfaces may increase adhesions by spreading more broadly, thus increasing overall cell volume and membrane tension. Contractile forces are then increased to compensate for volume and tension changes. In addition, the strain-rate-dependent force response results in a different force-strain curve, as shown in Fig. 5 B. This adds a strain-rate-dependent dimension to the cell’s response to changing cantilever stiffness (Fig. 5 C).

In addition, the model predicts that the steady-state amount of active myosin also depends on overall strain and the stiffness of the cantilever. The amount of myosin activation increases with increasing cantilever stiffness. With higher overall myosin activation and active tension in the cortex, the effective stiffness of the actin network in the cortex should be higher.

Cells during a step change in force

Figs. 4 and 5 show the cell response under a step change in displacement. Alternatively, one can apply a step change in force, and the results are shown in Fig. 6. Here, a jump in applied force of $F=300\text{nN}$ is applied. The model predicts that the force jump leads to a rapid increase in membrane tension and lowering of osmotic pressure. This leads to activation of Rho and contraction. Concomitantly, the cell also adjusts the osmotic content, but this is slower. As the osmotic content rises, the contractile force falls and the membrane tension adjusts to close to its value before the force jump. The increase and decrease in Rho predicted by the model are in agreement with data from Zhao et al. (12), where cells were pulled by magnetic beads while the active form of Rho was measured. Note that our model assumes that water flow is relatively fast, and that the cell volume increases at the same rate as the force application. This is in general not realistic. Cell volume would slowly increase, which means that the initial jump in Rho activation should be slower. The full problem where water permeation rate is slow is complex and requires sophisticated computational approaches. It is beyond the scope of this article. Nevertheless, the model predicts an increase and subsequent decrease in Rho activation from cell osmotic adaptation. Also, notice that the intracellular pressure does not recover completely after application of force. This is because pressure at steady state is determined by osmolyte concentrate, and transient ion fluxes may lead to a lower steady-state osmotic pressure. In the Supporting Material, we discuss the influence of the ion channel and pump properties on pressure recovery.

Figure 6.

A cylindrical cell subjected to a jump in pulling force (300 nN) at $t=0$ and $K_{\text{max}}$ =100 kPa. The jump in applied force increases the activation of ρ and the active stress. The vertical strain therefore decreases slowly after a sudden increase. Water and ion flows also help to adjust the overall tension in the membrane. After a transient jump in pressure, tension, and Rho activation, the cell eventually recovers to the steady-state ρ and pressure. To see this figure in color, go online.

Discussion

Unlike plant and bacteria cells, animal cells lack a stiff outer cell wall to maintain their shape. Thus, active processes must compensate for changes in mechanical tension to maintain the cell shape and volume. Membrane tension and cell volume are both important for many cellular processes. Therefore, a key question is how these variables are maintained. In this article, we relate cell membrane and cortical tension to the osmotic pressure difference. We unify water and ion permeation, cell cortical mechanics, and myosin active contraction in a single model of cell mechanical and osmotic response, and we show that osmoregulation and regulation of myosin contraction can work together to maintain cell volume and membrane tension. Biochemical pathways that can adjust cell contractility have been identified. We study a model where membrane tension directly signals Rho activation and active contraction, and we show that such a model can maintain cell volume and membrane tension for a variety of environmental perturbations. Of course, other feedback mechanisms are possible. In particular, it has been noted that myosin II accumulation and contraction itself is tension-sensitive (13). Transient stress in the actin network may also activate Rho and myosin contraction. However, the actin cortex is highly dynamic and turns over quickly on a timescale of seconds. Sustained signaling from actin or myosin alone would require complex coordination. Nevertheless, it is possible multiple feedback mechanisms are at play, and further experiments are needed. Although details of the kinetics of the biochemical pathway are currently not available, we have used generic forms of activation and deactivation. Therefore, we expect qualitative agreement with experimental observations.

In this model, the cell membrane has a reference size, described by the constant $A_0$. This reference size is the equilibrium area of the membrane when there are no forces acting on it. It is determined by the total amount of lipid in the membrane and any possible thermal fluctuations that can generate folds. In the cell, $A_0$ is further regulated by lipid trafficking from the Golgi (44). If the membrane is under high tension, addition of lipid may become more likely than subtraction (41). Therefore, there is a third regulatory mechanism that controls $A_0$. Lipid trafficking, however, is likely quite slow, occurring on a timescale of hours, although rapid vesicle fusion has been observed in some situations (45). Therefore, in our current model, $A_0$ is assumed to be constant. It is also possible to develop a more comprehensive model incorporating dynamics of $A_0$. The actual quantitative predictions on homeostatic membrane tension depend on parameters such as $T_{\text{c}}$ and $T_{\text{s}}$, and detailed measurements are needed to determine these parameters in MS channel activation.

In our model, we assume that the total protein content in the cell is constant and there is no active production of osmolytes (described by n) in the cytoplasm. In a live cell during the G1 phase of the cell cycle, n of course changes due to transcription and translation, and the overall cell volume also increases. How the total protein content is controlled remains to be studied. There appear to be other signaling networks that control the total protein content, possibly by coupling to the active cell-volume control system. In addition, our model currently does not consider charges and voltage effects. This would require a more detailed model where Na, K, and Cl ions, and their respective channels, pumps, and exchangers are considered. This more detailed model requires other unknown parameters, and is beyond the scope of this article.

Results of our model show that a feedback control algorithm governing active cell contraction and osmotic regulation can maintain cell volume and control cellular force generation. Multiple feedback systems are potentially at play in maintaining cell volume and tension. Our model can be extended to consider cells at the multicellular tissue scale as well. At this larger scale, our model exhibits behaviors that are somewhat similar to those observed in the cellular Potts model used in tissue mechanics, although the details are not completely the same (46, 47, 48). In addition, with changes in parameters, the model can also exhibit oscillatory behavior (49). The missing model elements are signals that govern actin polymerization (through the Rac pathway) and signaling from cell adhesion. These elements are critical for understanding cell mechanics during migration and interaction with extracellular matrices. Cell adhesion and subsequent signaling affect both actin polymerization and myosin contraction. Once again, it is possible that membrane tension also plays a role. Experiments with beads have shown that integrin engagement alone can trigger actin polymerization (50). Modeling of these different pathways will yield new predictions about cell behavior in a wide variety of settings.

Author Contributions

J.X.T. and S.X.S. designed the research, J.X.T. performed the research, and J.X.T. and S.X.S. wrote the article.

Editor: Charles Wolgemuth.