Universal behavior of the osmotically compressed cell and its analogy to the colloidal glass transition

Crowding within the cytoplasm seems to have been sufficiently important in the course of evolution that multiple regulatory mechanisms arose to detect and respond to even small variations of intermolecular spacing (18). In addition to the primary volume response dictated by osmotic equilibrium as governed by van 't Hoff's law, secondary mechanisms of cell volume regulation play out over longer scales of time (10). To perturb cell volume acutely, we subjected the isolated human airway smooth muscle (HASM) cell in culture to sudden hypertonic shock via the addition of 400-Dalton polyethylene glycol (PEG) (Methods). Cell volume was measured by atomic force microscopy (AFM) in the adherent cell, and, in separate experiments, by direct microscopic observation of the suspended cell (Fig. 1B) (Methods). Over the wide range of osmotic stresses studied, these 2 methods yielded closely similar volume responses (Fig. 1C). In response to osmotic shock, cell volume decreased and then equilibrated within a few tens of seconds (Fig. 1A) to volumes similar to that of a perfect osmometer (Fig. 1C). Similar observations were made for a different osmotic agent (sucrose), after actin depolymerization with latrunculin A, and with 2 different cell types (lung fibroblasts and SY5Y neuroblastoma cells), suggesting that acute changes in cell volume during these extreme challenges was dominated by the primary osmotic response (Fig. 1C). Overall, the relationship conformed to the well-known Ponder's relationship (19, 20), v = RΠ_iso/Π + (1 − R), with a slope of R = 0.7 (black line, Fig. 1C), where Π is the applied osmotic pressure, Π_iso is the isotonic osmotic pressure, and relative cell volume v is the current cell volume normalized by cell volume under isotonic conditions. Its minimum, v_min, occurs at the intercept corresponding to infinite osmotic stress. As such, v − v_min is the water volume that is osmotically active, v_min is the volume of the cell that is not osmotically active, and the “solid” volume fraction, φ = v_min/v, approaches 1 as v approaches v_min (gray line in Fig. 1C).

To assess the ability of the cell to resist changes of shape, we used optical magnetic twisting cytometry (OMTC) to probe the force-deformation relationship at 0.75 Hz, from which the complex shear modulus, G* = G′ + iG″, can be calculated with the aid of a finite element model (3, 21) (SI Results and Discussion). Here, G′ and G″ are storage and loss moduli, respectively, and we mainly focus on the magnitude G = |G*|. Cell shear stiffness at isotonic condition, G_iso, was stable and close to 2000 Pa. With application of hyperosmotic stress (PEG), cell stiffness increased in a manner that depended on osmolality, consistent with prior reports (22, 23) (Fig. 1D). Like the volume response, the stiffening response reached a plateau within a few tens of seconds. As bath osmolality increased and the volume fraction φ approached 1 the level of the stiffness plateau dramatically increased (Figs. 1D and 2A). Extreme osmotic stress caused shear stiffness of the HASM cell to increase by more than an order of magnitude and, as shown below, in other cell types to increase by more than two orders of magnitude. The magnitude of this response can be appreciated by noting that the most extreme challenge with a potent contractile agonist causes the stiffness of these contractile cells, at most, to double (24). Although shear stiffness increased greatly, images of the actin cytoskeleton revealed no remarkable changes (Fig. S1). We further probed the frequency dependence of the cell stiffness. In isotonic steady-state conditions the storage and loss moduli, G′ and G″, followed a weak power law over 4 frequency decades (Fig. S2). With progressively increasing osmotic stress, both G′ and G″ increased systematically but the power law exponent remained largely constant. Bulk changes of cell volume are described by the osmotic bulk modulus, φ (dΠ/dφ), where Π is the osmotic stress, and it too increased strongly as φ approached 1 (SI Results and Discussion and Fig. S3).

To assess the generality of these results, we then examined a variety of other cases. When we treated HASM cells with deoxyglucose and sodium azide, which act to deplete adenosine triphosphate (ATP), G_iso decreased by over 3-fold (4, 25), but with increasing volume fraction the shear stiffness increased dramatically nonetheless, reaching a value similar to that of the untreated cells (Figs. 1E and 2A). When we treated these cell with latrunculin A, which acts to depolymerize filamentous actin, G_iso decreased by nearly 20-fold but the shear stiffness increased sharply as φ approached 1, again reaching a value similar to that of the untreated cells (Figs. 1F and 2A). When we used a different impermeant solute (sucrose) or a different bead coating (polyl-Lysine, which couples to the cell nonspecifically, as opposed to RGD, which binds to integrins), we found similar results (see Fig. 2 E and F). Assuming that the same relationship existed between volume fraction and bath osmolality as in the HASM cell, we then examined 3 other cell types: Madin–Darby canine kidney (MDCK) epithelial cells, SY5Y neuroblastoma cells and human lung fibroblasts. Despite significantly different baseline stiffnesses, in every case the stiffness of these different cell types increased dramatically with increasing volume fraction and appeared to converge at the highest volume fraction (Fig. 2C). The maximum value of the stiffness was nearly invariant with G_iso and fell in the range of 10⁴ to 10⁵ Pa (Fig. 2E). Accordingly, with changes of cytoskeletal integrity, ATP availability, cell type, solute, or bead coating, the dependence of cell stiffness on volume fraction φ was robust and qualitatively similar. Thus, the mechanical properties of the cell under compression are dominated by the contents of the crowded intercellular space, becoming progressively stiffer with increasing osmotic compression.

To gain insights into the intracellular relaxation dynamics, we monitored spontaneous nanoscale motions of a microbead that was tightly coupled to the cytoskeleton (4, 26). When mean square bead displacement (MSD) was resolved as a function of time lag Δt, it exhibited subdiffusive behavior for small Δt and superdiffusive behavior for large Δt (Fig. 3A), confirming many previous reports (4, 26, 27). With increasing volume fraction, there was a dramatic reduction in spontaneous bead motions, but even at the maximum volume fraction these motions remained superdiffusive at long time scales indicating persistent nonequilibrium dynamics in the crowded intracellular space (Fig. S4). All these relationships have approximately the same shape and could be scaled onto a single master curve (Fig. 3A Inset); as such, this behavior could be fully characterized by the dependence of remodeling dynamics on only one characteristic time. As a measure of this characteristic time, τ, required to achieve a given extent of structural rearrangement, we set an arbitrary threshold value of the MSD (100 nm²) and then recorded the corresponding time lag needed to achieve that threshold value. As volume fraction increased, τ at first precipitously increased but then reached an approximate plateau (Fig. 3B). When ATP was depleted, τ was far larger at baseline, suggesting that baseline rearrangements are actively driven (4), but as volume fraction progressively increased, τ rose to approximately the same plateau value (Fig. 3B). F-actin depolymerization by latrunculin A accelerated baseline remodeling by an order of magnitude, but with progressive increase in volume fraction, τ again rose to approximately the same plateau (Fig. 3B). Despite such a plateau at high volume fraction, these dynamics depend sensitively on volume fraction near isotonic conditions.

In isotonic conditions the major determinant of the cellular shear stiffness is known to be the cytoskeletal tensile prestress (6–8, 28), and it is conceivable therefore that increases of G caused by osmotic compression were secondary to underlying increases in this tensile prestress. Much evidence argues against this interpretation, however. Depletion of ATP would be expected to blunt any active contractile response, and, indeed, isotonic stiffness in that case was much smaller but osmotically induced increases in cell stiffness were dramatically enhanced (Fig. 1D). Similarly, depolymerization of actin filaments largely ablates cytoskeletal tensile prestress (6), but osmotically induced increases in cell stiffness were seen to be even greater still (Fig. 1E). Finally, we measured changes in prestress by studying the isolated airway smooth muscle (ASM) strip (SI Methods); for technical reasons, traction force microscopy (6, 7) cannot be used in the context of extreme osmotic stress. When challenged with hyperosmotic stress, the longitudinal stiffness of the ASM strip increased in a manner comparable to the ASM cell in culture, as described above, whereas the prestress did not increase and even tended to decrease (Fig. S5). Taken together, these findings rule out the cytoskeletal network and the tensile prestress that it carries as being responsible for the dramatic rise in cell stiffness as cell volume decreases.

We therefore considered the simplest possible alternative, namely, the notion that the shear stiffness of the cell comprises 2 independent and additive contributions—one attributable to the cytoskeleton network, which is thought to dominate in isotonic conditions (6–8), and the other attributable to the colloidal phase, which becomes increasingly important as water leaves the cell and the volume fraction φ increases. If these contributions are additive and independent, then the colloidal contribution would be isolated by subtracting from the measured stiffness (G) of the osmotically compressed cell the isotonic stiffness of that cell (G_iso). Upon doing this subtraction, the colloidal contribution is seen to depend on volume fraction exponentially (Figs. 2B and 2D),where the parameter F quantifies the sensitivity of the colloidal stiffness to changes in volume fraction φ, and G₀ represents this stiffness at infinite dilution (φ = 0). The parameter F varied inversely with G_iso (Fig. 2F) and G₀ varied positively with G_iso (Fig. S6). This relationship therefore reveals a simple but universal phenomenological rule by which volumetric compression enhances shear stiffness of the eukaryotic cell.

Under compression, cytoskeletal filaments experience smaller tension, perhaps even buckle, and for either reason will tend to soften. Thus, the relative increase in cell shear stiffness that is observed under osmotic compression cannot be attributed to cytoskeletal network tension and seems to arise mainly from the crowded colloidal cytoplasm. If true, we would expect to see general behavior of cells that remains applicable independent of the state of the cytoskeleton. Indeed, such a behavior is found in Eq. 1, regardless of the integrity of the cytoskeleton. This provides strong support for a dominant colloidal contribution to the cell's shear stiffness under compressive stress. The approach to the glass transition is exponential in φ and data converge at high φ in every case, but the sensitivity of stiffness to changes in φ differs depending on circumstances (Fig. 2 B, D, and F). For example, the behavior would appear to depend on the relative amounts of polymerized actin, suggesting a potential role of protein shape, and the presence of ATP, indicating that nonequilibrium factors come into play.

This mechanical behavior of the osmotically compressed cell has interesting physical implications. Living cells are under continuous remodeling and relaxation, such as revealed by the spontaneous motions of adherent beads (Fig. 3A Insets). These spontaneous motions are consistent with the idea that the cell under isotonic conditions is far from being a simple elastic solid (4). Our data further reveal that even under maximum osmotic compression, relaxation processes persist but become dramatically slowed. Indeed, dramatic slowing of relaxation processes is a hallmark of the glass transition (29). For molecular liquids, decreases in temperature lead to increases in relaxation time until the system becomes frozen into an amorphous solid (29), whereas for colloidal suspensions it is decreases in system volume, and corresponding increases in volume fraction, that lead to such a transition (17). Our MSD data therefore suggest that the behavior of cells under compression is reminiscent of the colloidal glass transition in qualitative terms, but how can we understand osmotically compressed cells in the context of the colloidal glass transition in quantitative terms? Spontaneous bead motions are suggestive, but these motions cannot be used to extract the true relaxation time because they are driven by the motor-generated forces, which are hard to measure and likely to vary with compression.

The relaxation time under a constant external force is thus desirable, and can be quantified via the material viscosity and active rheology. To estimate the cytoplasmic viscosity, η, from the oscillatory shear modulus we used the Cox–Merz rule (30). If γ̇ is the shear rate in a constant shear experiment, and if ω is the frequency in an oscillatory experiment, then the Cox–Merz rule requires that η|_{γ̇ = ω} = G(ω)/ω. It is well established that cell rheology exhibits a weak power law over a wide frequency range (3–5, 8, 25, 31, 32), and that this behavior is explained phenomenologically by the soft glassy rheology (SGR) model (33). This model predicts that for the typical power-law slope (≈0.2) observed in cells (Fig. S2), the Cox–Merz rule will underestimate the viscosity by 40–60% (33). Because this establishes the shear viscosity to well within one order of magnitude, and because the power-law slope is little affected by compression (Fig. S2), this error does not appreciably affect the comparison of viscosities for different compressive stresses. Over a range spanning 4 decades of viscosity, rheological measurements on osmotically compressed cells therefore suggest that the cytoplasmic viscosity increases exponentially with volume fraction (Fig. 4A).

The exponential manner in which viscosity varies with volume fraction sheds light on the properties of constituent particles and the interactions among them. The viscosity of hard sphere colloids approaching a glass transition varies with volume fraction φ as described by Mooney's equation (34), η = η_sexp[νφ/(φ₀ − φ)], whereas the viscosity of a molecular liquid approaching a glass transition varies with temperature T as described by the Vogel–Fulcher–Tammann (VFT) equation (29), η = η₀exp[DT₀/(T − T₀)]. For the former equation, η_s is the solvent viscosity, φ₀ is 0.64 for hard spheres, and ν is the crowding factor (1.1 for hard spheres); for the latter, η₀ is the viscosity at infinite temperature, T₀ is the so-called VFT temperature, and D controls the deviation of this relationship from the Arrhenius law. The similarity between these 2 equations suggests that in these systems φ and 1/T play analogous roles in the approach to the glass transition. We note in particular that in certain limits the Mooney's equation becomes exponential in φ, i.e., Arrhenius-like, and fits very well data for cells under compression (Fig. 4A). As such, the concept of fragility, which has been instrumental in the categorization of strong versus fragile molecular glass formers (29), can be extended to colloidal glassy systems. We define φ_g, the glass transition volume fraction, as the volume fraction at which viscosity reaches an arbitrarily chosen high value, 40,000 Pa·s, and the fragility, m, for the colloidal glass transition as the slope of the relationship between log η versus φ/φ_g as the latter approaches 1. The fragility for hard spheres is seen to be quite high, whereas the fragility of cells is much lower (Fig. 4B). Our data therefore show that the osmotically compressed cell behaves very differently from a suspension of hard spheres; whereas the latter behaves as a fragile glass-former, the eukaryotic cell is reminiscent of a strong glass-former (29).

That a soft cell under compressive stress behaves as a strong glass is a clear finding and represents the major result of this report. The underlying structures and processes that might account for this finding remain a good deal less clear, however, and represent an open question. We found that ATP depletion strongly modulates the glass transition behavior of the cell, consistent with the notion that nonequilibrium processes may modulate glass transition behavior (35). Similarly, we found that F-actin disruption substantially increases the fragility, consistent with theoretical work demonstrating the effects of particle geometry on glass transition behavior (36). In addition, macromolecules, and organelles within the eukaryotic cytoplasm are far from being compact hard spheres. Even at isotonic volume fraction, it is likely that they interact with each other strongly, unlike the situation for hard sphere suspensions at a similar volume fraction, in which case the particles barely interact. Such interactions potentially explain the much higher viscosity of the cytoplasm compared with that of the hard sphere suspension at the same concentration. More importantly, these “particles” are deformable; for example, under the influence of molecular crowding proteins have been shown to deform and change conformation (14, 37). In this respect, then, the cytoplasm resembles a colloidal system of repulsive, deformable particles, exemplified by suspensions of star polymers, block-copolymer micelles, and soft microgel particles. Indeed, these systems often exhibit more gradual increases in viscosity than those found for hard sphere colloids during the approach to glass transition (38, 39). Furthermore, Mattsson et al. demonstrated a direct association between the deformability of individual particles and the fragility characterizing the colloidal glass transition (Mattsson J, Wyss HM, Fernandez-Nieves A, Miyazaki K, Hu Z, Reichman DR, Weitz DA (personal communication) Soft colloids make strong glasses.) Although the mechanistic connection between particle deformability and glass fragility remains unclear (40–42), the strong glassy dynamics of the cytoplasm closely resembles those found in a crowded suspension of repulsive soft colloidal particles (Fig. 4).

In summary, the eukaryotic cell possesses a cytoskeleton that is under tension but also a crowded intracellular space that is under compression. The mechanical behavior of the former is positively determined by tension (6–8) and is reasonably well described by the model for soft glassy rheology (3, 33). By contrast, the behavior of the latter is positively determined by compression and is reminiscent of a suspension of soft, deformable particles capable of undergoing a colloidal glass transition. These findings highlight the rich mechanical behavior of the cell, but also suggest a new mechanism by which the cell could regulate its mechanical behavior.