## Abstract

Recently reported data from mechanical measurements of cultured airway smooth muscle cells show that stiffness of the cytoskeletal matrix is determined by the extent of static contractile stress borne by the cytoskeleton (Wang N, Tolić-Nørrelykke IM, Chen J, Mijailovich SM, Butler JP, Fredberg JJ, and Stamenović D. *Am J Physiol Cell Physiol* 282, C606-C616, 2002). On the other hand, rheological measurements on these cells show that cytoskeletal stiffness changes with frequency of imposed mechanical loading according to a power law (Fabry B, Maksym GN, Butler JP, Glogauer M, Navajas DF, and Fredberg JJ. *Phys Rev Lett* 87: 148102, 2001). In this study, we examine the possibility that these two empirical observations might be interrelated. We combine previously reported data for contractile stress of human airway smooth muscle cells with new data describing rheological properties of these cells and derive quantitative, mathematically tractable, and experimentally verifiable empirical relationships between contractile stress and indexes of cell rheology. These findings reveal an intriguing role of the contractile stress: although it maintains structural stability of the cell under applied mechanical loads, it may also regulate rheological properties of the cytoskeleton, which are essential for other cell functions.

- contractility
- power law
- stiffness
- cytometry
- mechanics

we attempt here to bring together two empirical observations concerning the deformability of the cytoskeleton (CSK) of adherent cells. On the one hand, Wang et al. (25) have shown that the stiffness of the CSK matrix of cultured human airway smooth muscle (HASM) cells depends mainly on the static contractile stress that is borne by the CSK microfilaments and varies with that contractile stress in a linear fashion. The greater the contractile stress, the stiffer the cell is. Although other interpretations are possible, such a finding is consistent with the idea that the CSK behaves mechanically mainly as a tensed cable network (25). In that case, elastic restoring forces arise mainly from changes in the orientation and spacing of tensed filaments as the network is deformed (cf. Refs. 19, 21). The contribution of this mechanism to the matrix stiffness is a consequence of the internal balance of forces. This simple mechanism is not mutually exclusive with other factors that might add to CSK stiffness, but it sets a firm lower bound on that stiffness. Interestingly, in HASM cells, this lower bound seems to account for most of the observed stiffness (25).

On the other hand, Fabry et al. (6, 7) have established that the stiffness (as well as other rheological properties) of the CSK matrix of cultured HASM cells increases with frequency of the imposed deformation as a weak power law, ω^{α}, where ω is loading frequency and α is the power-law exponent (0 ≤ α ≤ 1). This power-law behavior extends over five decades of frequency (10^{-2}-10^{3} Hz) and persists over a wide range of experimental conditions. As explained below, α is an index of transition between solid-like (α = 0) and Newtonian fluid-like (α = 1) behaviors. Fabry et al. (6) recognized that this power-law behavior conforms to what is known in the engineering literature as structural (hysteretic) damping, an empirical law suggesting that the phase lag between the frictional and elastic stresses that develop within a material body is independent of the frequency of the loading (cf. Refs. 8, 13). Although the mechanistic basis of this behavior remains unclear, these data rule out viscoelastic models with one or even several time constants, as well as the structural assumptions and dynamics on which such models rest. For example, data of Fabry et al. (6, 7) rule out the Voigt model, the Maxwell model, and the standard linear solid model. In addition, the studies by Fabry et al. (6, 7) showed a substantial range of linear responses and no yield stress in various types of adherent cells, thereby ruling out, as well, Bingham-type plastic behavior (1) as well as so-called power-law fluids (12). On the other hand, observations of Fabry et al. (6, 7) are consistent with the hypothesis that the ability of the CSK to deform, to flow, and to remodel corresponds to the behavior of soft glassy materials existing close to a glass transition (18).

In this study, we examine the possibility that these two empirical observations (linear dependence of the stiffness on the static contractile stress and power-law dependence of the stiffness on the frequency of the imposed loading) might be interrelated. We address the extent to which these observations are compatible and complementary or contradictory. We combine previously reported data for contractile stress of HASM cells (25) with new data describing rheological properties of HASM cells and derive quantitative, mathematically tractable, and experimentally verifiable empirical relationships between contractile stress and indexes of cell rheology. Tentative mechanistic interpretations of these findings are discussed.

## METHODS

We used our previously reported data for contractile stress measured in cultured HASM cells in response to graded doses of histamine and isoproterenol (25). We measured rheological properties of cultured HASM cells in which contractility was modulated by agonists and doses that matched those that were used in the previous measurements of contractile stress. We then combined those two sets of data and derived empirical relationships between the contractile stress and indexes of cell rheology. A brief description of cell culture and rheological measurements are given below.

*Cell culture.* Cells were isolated from tracheal muscle of human lung transplant donors (approved by the University of Pennsylvania Committee on Studies Involving Human Subjects) as described previously (15). Cells at *passages 3-6* were used in all experiments. These cells maintain smooth muscle morphology and physiological responsiveness to agonists until at least *passage 8*. After cells reached confluence in plastic dishes, they were serum deprived for 48 h before being trypsinized. The cells were then plated in a serum-free medium on collagen I-coated (0.2 mg/ml) wells (96-well plate, Immunon II, Dynetec) at a density of 20,000 cells/well.

*Magnetic oscillatory cytometry.* Detailed description and validation of this technique is given elsewhere (6, 7). Briefly, small (4.5-μm diameter) ferrimagnetic beads were coated with RGD (Arg-Gly-Asp) peptide, which binds specifically to integrin receptors on the cell apical surface. Beads were added for 20 min to cells plated for 4-6 h on the collagen I-coated wells. Unbound beads were removed by washing with a serum-free medium, and then oscillatory measurements were performed as follows. The beads (1 bead/cell on average) were first magnetized by horizontal magnetic field and then twisted by a sinusoidally varying (∼10^{-1}-10^{3} Hz) vertical magnetic field. The horizontal displacements of each bead were traced by a charge-coupled device camera (JAI CV-M10) mounted on an inverted microscope.

*Protocol.* We started by measuring the frequency response of HASM cells between ∼10^{-1} and ∼10^{3} Hz twice, first in an ascending order and then in a descending order. We then recorded the cell response continuously at 0.75 Hz while adding an agonist to the well (0.1 and 10 μM isoproterenol, and 10 μM histamine or sham). This was done to ensure that the cells responded to the agonists. Our laboratory showed previously (25) that 10 μM is a saturating dose for both agonists, and thus they achieve a wide range of contractility in HASM cells. Fifteen minutes after adding the agonists, we again measured the cell frequency response between ∼10^{-1} and ∼10^{3} Hz twice. All measurements were repeated in five cell wells. We analyzed the displacements of ∼120-150 beads from each well. Beads that displaced by >500 nm or moved at an angle of >45° relative to the magnetization direction were discarded. For each bead, we computed an apparent complex modulus as the ratio between specific torque (amplitude order of 10^{1} Pa) and bead displacement (order of 10^{2} nm). This value had dimensions of (force/area)/length and was then transformed into a traditional dynamic (complex) modulus *G*^{*} (dimensions force/area) by multiplication with a geometric factor that depends on the shape and thickness of the cell and on the degree of bead internalization in the cell (14). The real part of *G*^{*} is the elastic (storage) modulus (*G*′), and the imaginary part is the frictional (loss) modulus (*G*″). For each tested frequency, median values of *G*′ and *G*″ were found and averaged over four (2 ascending and 2 descending) runs. This procedure was repeated for each of the five wells, and the mean values of *G*′ and *G*″ were obtained for each frequency and each cell treatment.

The structural damping equation (*Eq. 1*) was fit to these data 1

where *G*_{0} and Φ_{0} are scale factors for stiffness and frequency, respectively, η = tan(απ/2) has been called the structural damping coefficient (cf. 13), μ is the viscous damping coefficient, and indicates the out-of-phase behavior. *G*′, which corresponds to the real part of *Eq. 1*, increases for all frequencies ω according to a power law with exponent α. *G*″, which corresponds to the imaginary part of *Eq. 1*, includes a component that also increases as a power law with the same exponent. Therefore, this component is a frequency-independent fraction (η) of *G*′; such a direct coupling of *G*′ and *G*″ is the characteristic feature of structural damping behavior (8). *G*″ also includes a Newtonian viscous term, μω, which turns out to be small except at very high frequencies. In the limit of α → 0, *G*′ → *G*_{0} and η → 0; i.e., elastic solid behavior. In the limit of α → 1, *G*″ becomes proportional to ω and η → ∞, i.e., Newtonian fluid behavior. Thus *Eq. 1* describes a relationship between changes of the exponent of the power law and the transition from solid-like (α = 0, η = 0) to Newtonian fluid-like (α = 1, η = ∞) behavior. The Newtonian viscous term μω in *Eq. 1* should not be confused with the Newtonian fluid behavior in the limit when α = 1. This term indicates the prevailing behavior of *G*″ at very high frequencies regardless of what value α has.

*Data analysis.* The fitting of *Eq. 1* to the oscillatory data was done by a least-squares method by minimizing the squared differences between log*G*^{*} of the data and of the model summed over all frequencies and all drug treatment conditions, with the constraint that *G*_{0} and Φ_{0} remain constant for all cell treatments (5). The values of α and μ were free to vary with cell treatment. The reason for keeping *G*_{0} and Φ_{0} fixed was that they represent coordinates of the approximate point of intersection of the *G*′ vs. ω curves (see results). The minimization was performed using Microsoft Excel solver. We analyzed the sensitivity of the fit of *Eq. 1* to variability (noise) in the data by determining the 95% confidence interval of the parameter estimates using a previously described method (5).

## RESULTS

The elastic modulus *G*′ increased with increasing frequency ω according to a weak power law over the observed frequency range (10^{-1}-10^{3} Hz) (Fig. 1*A*). On a log-log scale, the power law dependence has a linear appearance with the slope being equivalent to the power-law exponent α. The frictional modulus *G*″ also followed the weak power law initially (10^{-1}-10 Hz), but above 10 Hz the frequency dependence became progressively stronger (Fig. 1*B*). The power-law behavior persisted after agonist treatment, although the exponent α changed. In histamine-treated cells, both *G*′ and *G*″ increased while their dependences on ω decreased relative to the control conditions. In isoproterenol-treated cells, both *G*′ and *G*″ decreased while their dependences on ω increased relative to the control conditions in a dose-dependent fashion (Fig. 1). When extrapolated to high frequencies, the *G*′ vs. ω relationships intersected at an approximately common fixed point (Fig. 1*A*, *inset*). All these observations are consistent with the previously reported data for HASM cells (6).

The structural damping equation (*Eq. 1*) provided an excellent fit to the oscillatory data (Fig. 1). We obtained a unique set of values for *G*_{0} and Φ_{0}, whereas the values for α and μ depended on cell treatment (Table 1). The exponent α exhibited a systematic dependence on the cell treatment. In stimulated cells, α decreased, whereas in relaxed cells α gradually increased relative to the controls with increasing doses of isoproterenol. This, in turn, suggests that rheological behavior of stimulated cells is closer to the behavior of an elastic solid than that of relaxed cells. The μ changed with cell treatment, but these changes were not systematic and were considerably smaller than those of *G*′ or *G*″ measured at low frequencies, suggesting that the state of cell contractility has little effect on μ.

To establish whether there exists a fixed empirical relationship between *G*′ and *G*″ on the one hand and the active stress (*P*) on the other, we combined the oscillatory data with previously published data for *P* as a function of doses of histamine and isoproterenol (25). *P* was obtained from the Fourier transform traction microscopy technique (3, 25). This technique measures traction that arises at the interface of a cell and a flexible substrate in response to cell contraction (Fig. 2*A*). Because traction balances contractile stress (20, 25), the latter can be computed from traction measurements (Fig. 2*B*). It was found that *P* increases or decreases in a dose-dependent fashion relative to the control conditions after histamine or isoproterenol were added (Fig. 2*C*). By cross-plotting these data and the oscillatory data (Fig. 1), we obtained that at each frequency both *G*′ and *G*″ increased with increasing *P* (Fig. 3). We chose to examine the behavior at low frequencies (∼0.1 Hz). The reason for selecting the lowest frequency from the measured range of frequencies is the mathematical simplicity of the *G*′ vs. *P* relationship at low frequencies. For reasons explained in discussion, this selection had no effect on the results we obtained. We found that, at 0.1 Hz, *G*′ conforms to a simple, proportional dependence on *P* (Fig. 3*A*, *inset*), which was consistent with our previous result in HASM cells (25) 2

where *c* = 2.96 is a factor of proportionality whose value was determined from the slope of the linear regression line (Fig. 3*A*, *inset*).

*Compatibility between frequency- and stress-dependent behavior.* For *Eqs. 1* and *2* to be mathematically consistent at 0.1 Hz′ the real part of *Eq. 1* has to be equal to the right-hand side of *Eq. 2*, i.e., 3

Since Φ_{0} = 4.07 × 10^{7} Hz (Table 1), it follows from *Eq. 3* that *P* decreases as α increases from 0 to 1. The α vs. *P* relationship was calculated from *Eq. 3* by taking values of *G*_{0} = 6.09 × 10^{4} Pa, Φ_{0} = 4.07 × 10^{7} Hz, and *c* = 2.96. The relationship of α vs. *P* obtained in this way was consistent with experimental data (Fig. 4), where *P* was obtained from the traction measurements (Fig. 2*C*) and the corresponding values for α from the log-log slope of the *G*′ vs. ω relationships (Fig. 1*A* and Table 1). It is important to note that, although the α vs. *P* relationship (*Eq. 3*) was derived at 0.1 Hz, it is not limited to this particular frequency because neither α nor *P* are frequency-dependent variables. Thus α depends only on *P* and nothing else. The contractile stress alone determined cell transition between solid-like (α = 0) and fluid-like (α = 1) behavior.

To test the consistency of this α vs. *P* relationship with other experimental data, we derived from *Eqs. 1* and *3 G*′ vs. *P* and *G*″ vs. *P* relationships for different frequencies (Fig. 5) and compared them with the corresponding experimental data (Fig. 3). Because μ varied little with the cell treatments (Table 1), we took the average μ = 1.83 Pa·s of the values from Table 1 when calculating the *G*″ vs. *P* relationships. Relationships of *G*′ vs. *P* and *G*″ vs. *P* computed in this way were consistent with experimental data (Fig. 5).

## DISCUSSION

Cellular functions demand a trade-off between deformability on the one hand and mechanical robustness on the other (4). Under certain conditions (e.g., crawling, spreading, division, invasion), cells need to be highly deformable, almost fluid like, whereas to maintain their structural integrity cells have to behave as an elastic solid. Our results establish a novel empirical relationship between an index of transition from solid-like to fluid-like cell behavior on the one hand and the contractile mechanical stress borne by the CSK on the other (*Eq. 3*; Fig. 4). This relationship has not been tied to any theoretical model of cell rheological behavior, nor does it assume any particular biophysical or biochemical mechanism; it represents pure empiricisms, the physical basis of which remains to be explained. Before discussing potential mechanisms, we critically evaluate our findings.

According to *Eq. 1*, the complex modulus *G*^{*} tends toward zero as the frequency ω tends toward zero, which is a characteristic of fluids. That is, at long time scales, there is no “spring constant”; all elastic behavior is lost, and force is dissipated by flow. However, this is true only when α is greater than zero. When α = 0, by contrast, *G*^{*} = *G*′ = *G*_{0} = constant for all ω, which, in turn, is indicative of solid-like behavior and corresponds to a static elastic limit. Moreover, according to *Eq. 3*, for α = 0, *G*_{0} = *cP*, i.e., the elastic modulus is determined by the level of the CSK contractile stress. This feature is consistent with the behavior of tensed cable networks, including tensegrity, which we used in our previous studies as a model of CSK elasticity (cf. Refs. 19, 21). However, taking into account that *G*_{0} ≈ 61 kPa (Table 1) and *c* ≈ 3, it follows that *P*, corresponding to the elastic limit of α = 0, would be an order of magnitude greater than the highest value of *P* measured in HASM cells treated with the saturating dose of histamine (Fig. 2*C*). Thus, in the physiological range of *P*, the HASM cell most likely does not attain the elastic limit. At the same time, it comes close to that limit in the sense that the observed range of α is 0.1-0.2. Therefore, in the spectrum of possible behaviors, with 0 corresponding to the limit of a Hookean elastic solid and 1 corresponding to the limit of a Newtonian fluid, these cells are closer to the former than the latter.

A concern could be raised that the direct proportionality between *G*′ and *P* at 0.1 Hz (*Eq. 2*) was established on the basis of only four data points (Fig. 3*A*, *inset*). However, in a previous study (25), our laboratory obtained a very similar relationship with twice as many data points. Thus, at low frequencies, the proportionality between *G*′ and *P* seems to be realistic. At higher frequencies, however, this proportionality ceases and the *G*′ vs. *P* relationship becomes nonlinear (Fig. 5*A*). Nevertheless, *G*′ collapses to zero when *P* = 0 at all frequencies (Fig. 5*A*). This, in turn, suggests that the contractile stress is a key determinant of the CSK shape stability not only under quasi-static conditions but under dynamic conditions as well.

Another concern is that the oscillatory data (Fig. 1) may not be a reflection of mechanics of the CSK but instead are attributable to the probe we used, receptor kinetics, or membrane mechanics. These issues are critically evaluated in the recent publication of Fabry et al. (7). The evidence they provided argues against those possibilities.

*Mechanistic interpretation.* It is often assumed that macroscopic deformation of the CSK lattice involves microscale motion of molecules over potential energy barriers. These barriers may be intermolecular (e.g., resistance to filament sliding) or intramolecular (e.g., changes in the conformation of the molecular chain). It has been shown (cf. Ref. 9) that the relationship between the macroscopic flow () and the applied stress (σ) is given as , where Δ*H* indicates the potential energy barrier, *k* is the Boltzman's constant, and θ is the absolute temperature. The physics of molecular motion over energy barriers is also known as the Eyring energy activation process (cf. Ref. 12). In the limit of Δ*H* → ∞, i.e.,

very high energy barrier, (i.e., no flow) implies the solid-like behavior. In the limit of Δ*H* → 0, i.e., very low energy barrier, (i.e., proportionality between stress and flow) which in turn, implies Newtonian fluid-like behavior. Thus, analogous to α in the structural damping model, Δ*H* in the energy activation model is an index of transition between solid-like and fluid-like behavior. Because our findings show that α changes with *P* (Fig. 5*A*), we speculate that Δ*H* may also change with *P*. An increase in *P* within the CSK lattice (mainly due to actomyosin cross-bridge formation) may increase the potential energy level of the CSK lattice. One reason could be that, under contractile stress, long chain molecules of the actin CSK network become taut like guitar strings, which, in turn, reduces their effective flexibility. A decrease in flexibility of these molecules increases the effective energy barrier associated with their conformational changes. Thus modulations of the contractile stress might change the level of potential energy barriers within the CSK and thereby alter the cell transition between fluid-like and solid-like cell behavior. Furthermore, it has also been shown that molecular motion over energy barriers may lead to rheological behavior, which is consistent with power-law behavior (cf. Ref. 9). Thus the physics of molecular motion over energy barriers may serve as a good basis for explaining the empirical link between the CSK contractile stress and the cell rheological behavior given by *Eq. 3*.

There are other energy activation theories that could explain the transition between solid and liquid states. One of them is the soft glass rheology proposed by Sollich (18). In this theory, however, the activation energy is thought to be greater than the thermal energy associated with the Brownian motion. This theory has been successful in explaining the oscillatory behavior of various cell types (6, 7). However, the physical origin of the energy activation processes described in this theory has not yet been fully identified in the CSK.

The transition between solid-like and fluid-like cell behavior is often framed in the context of a gel-sol transition (11, 22, 23). Gels near a critical gelation point exhibit power-law phenomena (12, 16, 26, 27) similar to that observed in soft glasses near a glass transition (18) and the data reported here (Fig. 1). In that connection, it is important to note that both the glass transition and critical gelation are thought to arise from a common mechanism, namely, kinetic arrest due to crowding of clusters, which is a manifestation of more general jamming transitions (17). Recent findings indicate that agonist-induced actin polymerization may not be sufficient to explain the observed stiffening response of smooth muscle cells, however, and that myosin activation is also needed to account for the stiffening (2).

Finally, it is important to point out that phenomena, which have been observed in reconstituted actin networks with myosin II dispersed in them, fail to explain results obtained in living cells. In such reconstituted systems, activation of dispersed myosin by ATP causes fluidization of entangled actin polymer gels and a dramatic reduction of matrix stiffness (10). This result is opposite in sign from the results obtained in living HASM cells, which exhibit stiffening when molecular motors are activated in response to stimulation by histamine (Fig. 3). A possible explanation for this discrepancy is that, within the polymerized actin gel, myosin activation does not induce contractile stress because there are no sites for mechanical anchorage that could counterbalance this stress. Instead, the force generated by the molecular motors pushes the actin filaments through the network, thereby enhancing their longitudinal motion and thus causing an apparent increase in gel fluidity (10). On the other hand, in living cells, the contractile force generated by the molecular motors is transmitted by the actin network to stress fibers, microtubules, and, ultimately, via focal contacts to the extracellular matrix, which counterbalance these forces and thus stabilize the CSK (20, 25). Thus contractile stress enhances stability rather than induces fluidity of the CSK of intact living cells.

In conclusion, we have shown in this study that mechanical stability and rheology of the CSK matrix are linked through the agency of CSK contractile stress. Although the physical basis for this finding remains to be explained, it suggests the possibility that CSK contractile stress may regulate transition between solid-like and fluid-like cell behavior. Activated processes, which have been used to explain rheology of polymer networks, may serve as a basis for explaining the role of the contractile stress in cell transition between solid-like and fluid-like behavior. Results of this study may provide a new perspective on a number of cellular functions that are critically dependent on cell contractility and cell rheology and may set the stage for further investigations in these areas.

## GRANTS

This work was supported by National Heart, Lung, and Blood Institute Grants HL-33009 and HL-59215.

## Acknowledgments

Present address of Dr. B. Fabry: Fridrich-Alexander-University of Erlangen, Turnstrasse 5, D-91054 Erlangen, Germany.

## Footnotes

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- Copyright © 2004 the American Physiological Society