Biophysical Journal Volume 105 August 2013 609–619
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Cellular Pressure and Volume Regulation and Implications for Cell Mechanics Hongyuan Jiang† and Sean X. Sun†‡* † Department of Mechanical Engineering and ‡Department of Biomedical Engineering and Johns Hopkins Physical Oncology Center, Johns Hopkins University, Baltimore, Maryland
ABSTRACT In eukaryotic cells, small changes in cell volume can serve as important signals for cell proliferation, death, and migration. Volume and shape regulation also directly impacts the mechanics of cells and tissues. Here, we develop a mathematical model of cellular volume and pressure regulation, incorporating essential elements such as water permeation, mechanosensitive channels, active ion pumps, and active stresses in the cortex. The model can fully explain recent experimental data, and it predicts cellular volume and pressure for several models of cell cortical mechanics. Moreover, we show that when cells are subjected to an externally applied load, such as in an atomic force microscopy indentation experiment, active regulation of volume and pressure leads to a complex cellular response. Instead of the passive mechanics of the cortex, the observed cell stiffness depends on several factors working together. This provides a mathematical explanation of rate-dependent response of cells under force.
INTRODUCTION Unlike plant and bacterial cells, animal cells lack a stiff cell wall that resists large changes in cell volume. Instead, the interplay of membrane tension, active contractility, water, and ion flows control the cell shape and volume (1). Conversely, when external forces are applied to the cell, the observed shape and pressure responses are also the combined results of these influences. A quantitative understanding of this important system is still lacking. Most mathematical models of cell-shape dynamics treat the cell as a constant volume of cytoplasm surrounded by a layer of membrane or cortex (see more detailed review in Clark and Paluch (2) and Sabreux et al. (3)). Such models have been successfully used to quantitatively describe problems such as red blood cell mechanics (4) and shape instability of dividing cells (5). However, the transport of water and ions and the subsequent cell-volume change controlled by passive or active ion channels are generally neglected in these models. In studies of cell mechanics with mechanical perturbation measurements (6–9), cells are usually modeled as an elastic or viscoelastic body, without consideration of the possible volume change induced by external forces. Lately, however, cell-volume dynamics is being recognized as an important element in cell mechanics. A recent experiment (10) showed that cell rheological properties can
Submitted April 15, 2013, and accepted for publication June 12, 2013. *Correspondence:
[email protected] Hongyuan Jiang’s present address is Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PRC This is an Open Access article distributed under the terms of the Creative Commons-Attribution Noncommercial License (http://creativecommons. org/licenses/by-nc/2.0/), which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Editor: Charles Wolgemuth. Ó 2013 The Authors 0006-3495/13/08/0609/11 $2.00
change due to the effects of the interstitial fluid and related volume change. Another recent experiment (11) measured volume and pressure changes in metaphase cells after introducing osmotic shocks. The quantitative results of that experiment showed that the cell volume during metaphase can adapt to large external osmotic shocks, and that the adaptation timescale is over tens of minutes. There are large numbers of studies on ion channels in cell osmotic regulation, and cell mechanics. However, there seem to be a limited number of studies so far linking these two fundamental aspects. Here we develop a mathematical model of cell volume and pressure response, combining the influence of cortical tension, water permeation, and ion dynamics. The model is able to compute cell-shape changes during osmotic shock and predict the response of the cell to externally applied mechanical forces. We show that the mechanical response of cells during slow deformations (~mm/min) is dominated by water permeation. Cytoskeletal mechanics is only one of several variables that strongly influence the apparent cellular mechanical response. Cytoplasmic fluid flow has been examined as an important element that controls the overall mechanics of eukaryotic animal cells (12), and a recent experiment showed that the cytoplasm of living cells behaves as a poroelastic material (10) where fluid pressure gradients equilibrate on the scale of tens of seconds. A closely related phenomenon is cellular blebs, where local build up of hydrostatic pressure leads to detachment of the membrane from the actomyosin cortex underneath the membrane (13). The developed bleb bulges out from the cell, and in some cases, the cortex can reform. Blebbing has been observed during cytokinesis (5) and has been implicated in amoeba cell motility (13). Less attention has been paid to how cytoplasmic fluid might flow across the cell membrane. Indeed, the cell membrane is permeable to a variety of ions, small solutes, and water. http://dx.doi.org/10.1016/j.bpj.2013.06.021
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Therefore, both hydrostatic and osmotic pressures control fluid flow across the membrane. Since water is essentially incompressible, the flow of fluid is directly related to cell volume change. This phenomenon is typically seen when cells are subjected to large osmotic shocks. Equally important, ion and osmolyte flow across membrane channels and pumps can also change osmotic pressure. Flows of small molecules and ions across the membrane are principally controlled by two types of membrane proteins. Mechanosensitive channels are passive membrane proteins that can open in response to changes in membrane tension. Subsequent ion flows across the channel are driven by concentration differences. In contrast, active ion pumps are energyconsuming membrane proteins that pump solutes and ions against concentration gradients. Finally, the actomyosin cortex underneath the membrane and the active stress generated by molecular motors can also influence membrane tension and cellular hydrostatic pressure, indirectly influence solute and ion flow, and thus influence the cell volume. The membrane tension will also influence the opening and closing of mechanosensitive channels. One experiment (11) revealed that when the cortex is disrupted with blebbistatin to inhibit myosin II contraction and latrunculin A to depolymerize actin filaments, the volume of mitotic cells slightly increases. Therefore, all of these elements must be considered together to understand cell shape and pressure changes. Understanding the combined effects of water, ions, cytoskeletal mechanics, and contractility is important for understanding the overall mechanical response of cells, tissues, and developmental mechanics in general. There are now experiments utilizing atomic force microscopy (AFM) or microrheology to measure the mechanical response of the cell (6–9). When forces are applied to the cell, hydrostatic pressure and cortical tension can change, leading to changes in water and ion flow. We use our model to compute the shape and force response of the cell during a constant-velocity indentation. The results reveal the underlying molecular mechanism that controls the rate-dependent viscoelastic response of the cell. More generally, it has been proposed that by placing the membrane channels and pumps nonuniformly, the cell can introduce influx and efflux of water at different locations (14,15). This mechanism may be crucial for cell motility in neutrophils, and may be important in cell-shape changes in general. A MINIMAL MODEL OF CELLULAR VOLUME AND PRESSURE REGULATION Let us consider a minimal model of volume and pressure regulation for a spherical cell, e.g., a mitotic cell or a cell in suspension. The cellular cytoplasm is enclosed by a cortical layer, which is a network of actin filaments and contractile myosin motors (Fig. 1 a). Enclosing and adhering to the cortex is one layer of cell membrane, in which passive mechanosensitive (MS) ion channels and active ion transBiophysical Journal 105(3) 609–619
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FIGURE 1 (a) A minimal model of cell volume and pressure regulation. We consider a spherical cell enclosed by an actomyosin cortex and the cell membrane. Embedded in the membrane are several families of passive MS ion channels and active ion pumps. The MS channels and active ion pumps can change the internal ion concentration, cin , and the intneral osmotic pressure, Pin , leading to changes in water flux across the membrane. Net flow of water leads to cell volume changes. (b) Opening probability versus cortical stress for an MS channel. The red curve shows the simplified piecewise linear function used in Eq. 2. (c) The flux of ions transported by ion transporters as a function of osmotic pressure difference, DP. This flux is modeled by Eq. 3. (d) The steady-state phase diagram of the cell as a function of model parameters DPc and sc. The model predicts two regimes. In the static regime, there are no ion fluxes at steady-state cell size. All of the channels and pumps are inactive. In the pump-and-leak regime, the influx and efflux of ions balance, and the cell maintains a steady-state size.
porters are embedded. The mechanical tension in the membrane is a complex issue. The overall membrane tension is controlled by the hydrostatic pressure difference across the membrane, membrane-cytoskeleton adhesion, and active stresses in the cell cortex. However, the tension is also known to be related to the total membrane area, and therefore is also regulated by membrane superstructures (e.g., membrane folds and blebs) and membrane trafficking (e.g., exocytosis and endocytosis) (16). In particular, when the hydrostatic pressure in the cell becomes large, the membrane may detach from the cortex, leading to the formation of cellular blebs (17). Therefore, the cortical tension is not necessarily equivalent to the membrane tension. Here, to simplify the problem, we neglect the membrane superstructures and membrane trafficking, and treat the cell membrane and cortex as a single mechanical structure. Kinetics of water Let Pin and Pin be the hydrostatic pressure and the osmotic pressure, respectively, inside the spherical cell. The corresponding values in the extracellular environment are Pout and Pout (Fig. 1 a). Osmotic pressure can be estimated using the Van’t Hoff equation P ¼ cRT, where c is the molar concentration of solutes, R is the gas constant, and T is absolute temperature. For an enclosed volume, this can be written as PV ¼ nRT, where V is the net volume and n is the total
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number of solutes. For high solute concentrations and crowded cellular environments, the osmotic pressure is similar, except that one should include an activity coefficient that measures the deviation away from the ideal solution approximation above. The chemical potential of water on both sides is given by Jin ¼ Pin Pin and Jout ¼ Pout Pout . Therefore, the flux of water can be modeled as Jwater ¼ aDJ (18), where a is a rate constant, DJ ¼ ðDP DPÞ is the chemical potential difference, and DP ¼ Pin Pout and DP ¼ Pin Pout are the hydrostatic and osmotic pressure differences across the membrane. The volume change of a spherical cell with radius r is then d=dtðð4=3Þpr3 Þ ¼ 4pr2 Jwater , which gives dr ¼ Jwater ¼ aðDP DPÞ: dt
(1)
In some cells, membrane water permeability can be enhanced by aquaporins, also known as water channels (19). This additional mechanism for water transport simply increases the constant a in our model. Thus, the permeability constant a is related to both the basal permeability of the cell membrane and water flux through specialized water channels. Kinetics of ions and small molecules In a vesicle enclosed by a semipermeable membrane, the flux of solutes is zero and the flux of water is sufficient to describe the volume change. In living cells, there are many mechanosensitive (MS) channels (20) and active ion and small-molecule transporters (or pumps) (21,22) in the cell membrane, which enable the cell to actively control the influx and efflux of ions and other osmolytes. Therefore, the kinetics of the solute must be included to study the volume and pressure regulation of living cells. To reduce complexity, we include only one species of MS channel and one species of ion transporter in our model. MS channels are membrane proteins that can open a pore in response to mechanical stimuli (20). They act as emergency valves to release solutes in response to hypotonic shocks. These channels can vary in permittivity from nonselective to highly selective (20). The opening probability, Popen , of MS channels follows a Boltzmann function (23) (Fig. 1 b). If there are N MS channels on the cell membrane, and there are NPopen open channels for a given stress. Therefore, the ion flux across MS channels is proportional to NPopen Dc=h0 , where Dc=h0 represents the concentration gradient of ions, and h0 is the membrane thickness. By approximating the Boltzmann function by a piecewise linear function (Fig. 1 b, red curve), the ion flux due to the opening of MS channels can be modeled as 8 if s%sc 90 Pa. For ATP-driven pumps, DPc z 3 104 MPa, as discussed above. This implies that Eq. 6 is always satisfied for a typical cell in metaphase. Therefore, cells in Stewart et al. (11) are in the pump-and-leak regime. It should be noted that the static regime is an idealization. If we use the Boltzmann distribution proposed in Sukharev et al. (23), there should be three regimes: the saturated efflux regime, the quasistatic regime, and the intermediate regime. In the quasistatic regime, the efflux of ions is much smaller than that in the other regimes but is not zero. In this case, the boundary between the two regimes is no longer the sharp interface depicted in Fig. 1 d. Cells can adapt to osmotic shocks After applying a hypotonic shock to a cell that is initially in dynamic equilibrium (Fig. 2 a), the decrease of medium osmotic pressure drives the influx of water. The influx of water leads to an increase of both the cell radius (volume) and hydrostatic pressure difference. The influx of water also leads to an increase of cortical stress, which in turn opens more MS channels and increases the efflux of ions. Therefore, the net ion flux becomes negative (efflux), so that the total number of ions, n, decreases. As n decreases, the cortical stress and the efflux of ions decrease. When DP
