1Physiology Program, Department of Environmental Health, Harvard School of Public Health, Boston, Massachusetts; and 2Physics Department, Erlangen University, Erlangen, Germany
Submitted 11 February 2005 ; accepted in final form 9 July 2005
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ABSTRACT |
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structural damping; scale-free; glass
The hypothesis that the CSK might behave similarly to a glassy material was first suggested by Fabry et al. (14, 15), who measured stiffness and friction in a variety of cell types in culture, including the human airway smooth muscle (ASM) cell. Their report contained two surprising results. They had set out to identify distinct internal time scales (i.e., molecular relaxation times or time constants) that might reflect molecular dynamics of proteins integrated within the cytoskeletal lattice, and in particular they had expected to find relaxation time scales corresponding to myosin-actin cycling rates. Surprisingly, across a spectrum spanning five decades of frequency, f, and in all five of the cell types that were investigated, they found no characteristic time scales; rather, stiffness increased as fx 1, implying that relaxation times were distributed as a power law, with a great many relaxation processes contributing when the frequency of the imposed deformation was small but fewer contributing as the frequency was increased and slower processes became progressively frozen out of the response. Thus no distinct internal time scale could typify protein-protein interactions; all time scales were present simultaneously but were distributed broadly. Because no distinct time scales characterize the response, behavior of this kind is referred to as being scale-free. Using other techniques and differing cell types, other researchers have confirmed that cytoskeletal dynamics are scale-free (1, 3, 8, 25, 32).
The second surprise was an unexpected relationship between changes in cell mechanics that arise when cytoskeleton constituents are modulated. In the living cell, the rheology of the CSK matrix is influenced by many structural proteins and their interactions. Despite this complexity, if stiffness or friction data were appropriately scaled and then plotted against x (which is readily determined from the power law exponent of stiffness vs. frequency), all data collapsed onto the same relationship regardless of the cell type studied, the signal transduction pathway activated, or the particular molecule manipulated (13, 14).
Purely as a matter of phenomenology, the observations described above firmly establish that the parameter x determines where the CSK sits along a continuous spectrum of solidlike vs. fluidlike states (15). In the limit in which x approaches 1, the behavior approaches that of a Hookean elastic solid and, in the limit in which x approaches 2, the behavior approaches that of a Newtonian viscous fluid. Values of x for adherent living cells fall in the range of 1.11.3, placing them closer to the solidlike than to the fluidlike state. Moreover, the parameter x was subsequently found to bring together into one phenomenological picture not only cytoskeletal stiffness and friction but also remodeling (5, 36).
With regard to mechanism of action, classical theories of viscoelasticity and equilibrium systems fail to account for these observations (5), and the physical significance of the parameter x remains elusive. Our working hypothesis holds that the CSK behaves as a glassy system and that the parameter x, which is easily measured, corresponds to an effective temperature of the CSK lattice (5, 13, 14, 36). In that case, the system is regarded as being frozen when x is close to 1 and melted when x is close to 2. Although the underlying physics remain poorly understood, strong experimental support for x as an effective temperature of the cytoskeletal lattice has recently been reported (5, 36).
Master relationships reported to date were obtained using only a small panel of experimental interventions, however, and used only one agent targeted specifically at actin (cytochalasin D). In the present study, we targeted actin using a range of interventions that was substantially wider than any studied previously. We show that these data confirm and substantially extend previous reports. We conclude by speculating about mechanisms that might account for the ability of an effective temperature to unify these observations.
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MATERIALS AND METHODS |
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Materials. DMEM-Hams F-12 medium, Fungizone, and 0.02% trypsin were purchased from GIBCO (Gaithersburg, MD). PBS was obtained from BioWhittaker (Walkersville, MD). Other reagents for tissue culture were ordered from Sigma (St. Louis, MO). Synthetic arginine-glycine-aspartic acid (RGD)-containing peptide (Peptite 2000) was obtained from Integra Life Sciences (San Diego, CA). Collagen I was procured from Cohesion (Palo Alto, CA). Drugs were acquired from Sigma, Calbiochem (La Jolla, CA), or Tocris Cookson (Ellisville, MO). They were reconstituted in sterile water [serotonin, dibutyryl adenosine 3',5'-cyclic monophosphate (DBcAMP)], DMSO [cytochalasin D, latrunculin A, 1-(5-iodonaphthalene-1-sulfonyl)-1H-hexahydro-1,4-diazepine (ML-7), genistein, phalloidin oleate, and jasplakinolide], or methanol (phallacidin). Controls for the solvents were shown previously to have no effect on cellular mechanics (2).
The panel of agents and the concentrations used, as well as the presumed mechanisms of action, are summarized in Table 1. In separate experiments, we determined the incubation time for each agent to cause its maximum effect. Latrunculin A and cytochalasin D result in net actin depolymerization by binding to monomeric actin (G-actin) and capping filamentous actin (F-actin), respectively (27). Phalloidin is an actin-stabilizing peptide. Phallacidin and phalloidin oleate are two membrane-permeable phalloidin analogs (33). Jasplakinolide results in net actin polymerization by promoting both linear and branched growth of actin fibers (26). Genistein is a nonspecific blocker of tyrosine kinases (6). These kinases are necessary for the phosphorylation of the complex of actin-binding proteins at the branch sites of actin fibers.
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Cell preparation. When ASM cells (passages 38) were confluent, they were serum deprived for 24 h. To create ITS medium, we added sodium selenite (6.7 mg/ml), transferrin (5.5 mg/ml), and insulin (10 mg/ml) to serum-free DMEM-Hams F-12 medium. Cells were released from the flasks using trypsin and then plated at 30,000 cells/well. Plastic wells (96-well plate; Corning, Corning, NY) were previously coated with collagen at 500 ng/cm2. Cells were allowed to adhere to the matrix overnight before use.
Bead preparation. Ferrimagnetic microbeads (Fe3O4, 4.5-µm diameter) were produced in our laboratory at the Harvard School of Public Health (15, 29). Beads were coated for at least 24 h with RGD peptide (200 µg/mg beads) in carbonate buffer (pH 9.4). On the day of the experiments, beads were added to FBS-free medium with 1% BSA. The presence of BSA permitted the specific binding of the beads to the integrin receptors on the surface of the cells.
Optical magnetic twisting cytometry. Cell mechanical properties were measured using magnetic twisting cytometry with optical detection of bead motion (15, 42). Serum-deprived cells were plated on collagen-coated wells, incubated with ferrimagnetic beads (20 min), and then washed with medium to remove unbound beads. Wells were held on the stage of an inverted microscope. The apparatus contained two sets of coils and a temperature controller. All experiments were performed at 37°C. The beads were magnetized horizontally so that their magnetic moments were aligned in the plane of the plates surface. A second magnetic field, the twisting field, was applied vertically. This second field caused the beads to rotate and translate, tending to orient their magnetic moment with the direction of applied field. As the beads twisted, they exerted a stress on the adhesion receptors, the focal adhesion complex, and other CSK structures that had formed around the beads (7, 15, 42). The stress was transmitted to the cell and its underlying structures, which, because of their own elastic and frictional properties, impeded the movement of the bead. A frequency scan was performed by increasing frequency and then rescanned with decreasing frequency to check for history dependence. The frequencies imposed were 0.01, 0.03, 0.1, 0.3, 0.75, 4.2, 10, 30, 100, 300, and 1,000 Hz.
Experimental protocols. After ferrimagnetic beads were added and wells were washed, cells were ready for investigation. Cells were tested at the start of each experimental day for their sensitivity to the drugs. These wells were used for one run and then discarded. For frequency scans, each well was run twice: once in baseline conditions and once after the addition of drug. In this way, each well served as its own control. For each well, the first scan was performed, and then the well was removed from the microscope and set aside while the chosen drug was added. The well was then placed back onto the microscope stage, remagnetized, and the second frequency scan was performed. Only one drug was tested in each well. For a negative control (sham), ITS medium was added. Each drug was studied in multiple wells on multiple days. Table 2 shows the number of beads from which data were collected for each drug.
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The theory of soft glass rheology predicts material behavior according to the structural damping equation (Eq. 1). In that theory, x isassociated with athermal mechanisms of agitation in the microenvironment and plays a role akin to a temperature and for that reason has been called the effective temperature (5, 15, 38). Moreover, the parameter x and the structure-damping coefficient (loss tangent) are directly related to one another; when
is small compared with unity,
= 1 x, but more generally,
tan[(x 1)
/2].
g' and g" have units of Pa/nm.
These values can be converted into traditional elastic and loss moduli (in Pa units) through multiplication by a geometric scale factor that depends on the shape and thickness of the cell and the degree of the bead embedding. Finite element analysis of cell deformation previously published by Mijailovich et al. (28) fixed
at roughly 6.8 µm. This finding assumed a homogeneous and isotropic elastic area surrounding the site of bead embedding, a 5-µm cell height, and 10% embedding of the bead into the cell.
Statistical analysis. We compared three models, all based on the structural damping equation. These models differed only with regard to which parameters were held to be common among the conditions and which were allowed to differ.
Model I is fully unconstrained. There were separate G, x, and µ values for each drug. Taken together, g' vs. frequency f, all data across separate drug interventions appeared to converge at a common intersection point, which we denote g0,f0. Because of the function in Eq. 1, there is not one common intersection, but all curves do cross in the same vicinity. Regarding Models II and III (see below), the choice of model influenced the position of the apparent common intersection point, (g0,f0). In Model II, all relationships were forced to have a common (g0,f0) parameter pair that was determined by the fit. This finding corresponds to the empirical observation in some sets of data that the stiffness plots (g') all seemed to intersect at one point (g0,f0). In Model III, the further constraint was imposed that all data had the same viscosity (µ), which might reflect the background cytosol through which cytoskeletal components move.
Fits to the data were obtained by minimizing the sum of the squared magnitudes of the difference between the logarithm of the complex data and the logarithm of the complex prediction from Eq. 1. Logarithms were used both because a power law becomes linear in log space, and because the observed variability is approximately log-normal and becomes normal in log space (15, 16). Fits to the data were done using the R statistical analysis package version 1.6.2 (The Comprehensive R Archive Network, http://cran.r-project.org/). This procedure was repeated for all three models.
ANOVA was performed to evaluate the parameters of the fits and to compare these models using a reduction-in-variance F-test. If the observed P value was <0.05, we concluded that there were significant differences between models.
Methodological limitations. Methodological limitations of the bead-twisting approach for the assessment of cell rheology have been described in detail in a recent publication (32).
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RESULTS |
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Friction. The baseline measurement friction data (Fig. 1B) also exhibited weak frequency dependence. g" followed the same weak power law behavior as did g' at lower frequencies, but at larger frequencies it exhibited a stronger frequency dependence approaching a power law exponent of 1, which is characteristic of Newtonian viscosity. As with g', when the actin network was disrupted, g" fell and the exponent of the low-frequency power law increased. Again, as with g', when the cells were activated or actin was polymerized, g" increased and the exponent of the low-frequency power law component decreased.
Structural damping equation. All data were well described by Eq. 1 (fits denoted by solid lines in Fig. 1, Model II). When we fit the structural damping equation to the complex moduli for each of the treatment conditions, we obtained a parameter set for x, g0, f0, and µ. We evaluated how well the structural damping equation accounted for the data when either all parameters were free across drug treatments (Model I), when (g0, f0) was constrained to be the same for all drug treatments (Model II), or when g0, f0, and µ were constrained to be the same for all drug treatments (Model III). The residual variances of the fits of Models I and II were significantly different from each other (P = 0.0016). Thus ANOVA rejected Model II compared with Model I. However, both correlation coefficients were high: R2 values were 0.9933 and 0.9926, respectively. We thus argue that Model II, with fewer parameters, captured the essence of the data.
A constrained (common) viscous term over treatments (Model III) resulted in a fit with R2 = 0.9886, which was statistically poorer than Model I or Model II (P = 1.7 x 1014, Model II vs. Model III). This finding would indicate that the Newtonian viscosity term changed in response to drug treatment. On the other hand, the changes in viscosity were small compared with drug-induced changes of g' and g" at lower frequencies. Thus, for our purposes µ could also be regarded as being approximately invariant with drug treatment. Values for g', µ (both from Model II), and x (from Model I) are listed in Table 2.
Scaling data onto master curves.
Because power law responses are straight lines on log-log graphs, each of the relationships of g' vs. f could be defined by one point (which we chose arbitrarily as the value measured at 0.75 Hz) and one slope (x 1). To compare responses between drug interventions, we normalized the data using the approach described previously by Fabry et al. (15). We used the common intersection stiffness g0 as a stiffness scale and then defined normalized cell stiffness, Gnorm, as g'0.75 Hz/g0. To scale the cells frictional properties, we used the ratio g"/g' (hysteresivity, or loss tangent, ) at 0.75 Hz. We used x from the fit of the structural damping equation using Model I and then plotted Gnorm vs. x and
vs. x (Fig. 2), for which Model I was used because it is the least constrained. When the stiffness and friction data were normalized in this way, they collapsed onto two master relationships. Drugs that increased x caused the normalized stiffness to decrease and the hysteresivity to increase. These relationships fell close to the predictions from the structural damping equation (Fig. 2).
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DISCUSSION |
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We addressed methodological limitations in MATERIALS AND METHODS and note that the panel of interventions used in this study was wide and the results obtained were mutually consistent, and as such, the chance of misinterpretation on the basis of the nonspecific effects of any one agent was small. Each intervention served as a positive control for the others, and stiffness and frictional moduli showed large changes in the expected directions.
Glassy dynamics. Fabry and colleagues (14, 15) noted that the power law behavior observed in cells is strongly analogous to the behavior observed in soft glassy materials such as foams, pastes, and colloids, and they speculated that the parameter x defined in Eq. 1 might be associated with the effective temperature that had been postulated by Sollich and colleagues (38, 39) to control the dynamics of such systems. In a recent publication, this analogy was supported by a broader range of biophysical observations (5, 36).
This analogy suggests the following physical picture. Cytoskeletal dynamics in the ASM are proposed to fit within the framework of the trapping of a CSK structural protein or protein complex in a deep energy well, i.e., a well so deep that thermal collisions are insufficient to drive the protein or protein complex out of the well (5, 1315, 21, 36). If thermal forces are insufficient, then hopping out of the well is imagined to be driven by nonthermal energies that are much larger than kBT (where kB is Boltzmanns constant) but can be expressed nonetheless as an effective temperature of the CSK matrix.
Formally, the term "temperature" means molecular motion and carries with it the connotation of molecular collisions and resulting agitation caused by ongoing molecular bombardments of thermal origin. Herein the term "effective temperature" is meant to carry with it a similar connotation of molecular agitation but caused by processes that may be of nonthermal origin. A protein conformational change fueled by hydrolysis of ATP, for example, releases energy that is 25-fold greater than thermal energy (22) and does so at the rate of >1 x 104 events·s1·µm3 of cytoplasm (21). Accordingly, such events would have the potential to jostle a neighboring structure rather substantially and dislodge it from a relatively deep energy well.
Physical interactions that lead to trapping might include molecular crowding, hydrogen bonding, or weak cross linking (1012, 34, 41). With regard to molecular crowding, the volume fraction of macromolecules within the cell approaches the maximum packing fraction that is possible, with the mean distance between macromolecules being only 2 nm (31). The molecular space within a cell is so crowded, in fact, that crowding can change binding affinities between specific proteins by orders of magnitude (9, 10, 12, 20). If trapping is due to nonspecific mechanisms such as molecular crowding or hydrogen bonding, then the specific properties of cross-linking molecules might be of only secondary importance. One rubric illustrating this possibility is shown in Fig. 4, in which the effective temperature x is shown as the central factor controlling CSK dynamics. If trapping has its origins in the cross linking of CSK filaments, however, then the density and the specific type of cross links and CSK filaments, as well as the dynamics of their specific interactions, might be crucial (19). Many of the cytoskeletal modulations that we have used to date alter both macromolecular packing and the dynamics of cross-linked filament networks (27), and thus they cannot determine which of the two interactions might play a dominant role. Regardless of the particular interaction, trapping is hypothesized to arise as a result of the insufficiency of thermal energy to drive CSK structural rearrangements (41). The collapse of all data onto master curves reported herein suggests that specific molecular interactions influence CSK stiffness and friction, but only to the extent that these interactions can modify the effective temperature.
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The mechanism that accounts for the intersection remains a matter of speculation. Protein-protein interactions of various kinds exhibit molecular relaxation time scales that span 15 orders of magnitude (4), and this profound heterogeneity might plausibly account for the wide range of relaxation times implied by Fig. 1. Heterogeneity by itself cannot explain why the distribution of relaxation times follows the special form of a weak power law, however, and neither can it explain the special relationship among the power law exponents implied by the approximate intersection at f0 and the collapse of all data as demonstrated in Fig. 2.
That such a simple phenomenological framework can describe such a complex system is counterintuitive. Factors that are common to every case, however, are the presence of water and molecular crowding. Surrounding hydrating shells strongly influence protein-protein dynamics, and in turn the presence of a nearby protein surfaces strongly influences water dynamics. Water becomes ordered in the vicinity of some macromolecules, and within the cell the average distance between macromolecules is only 2 nm, or roughly 10 water molecules across (31). Suzuki and co-workers (18, 23, 40) have suggested that water molecules within the cell interact with protein surfaces in broad categories. Bulk water forms a hydrogen-bonding network but remains relatively mobile. Water molecules that interact with protein surfaces are typically less mobile than water molecules in bulk water because the protein reinforces the hydrogen-bonding network of neighboring water molecules. Water that is strongly bound in deep clefts is the least mobile. Interestingly, water in the vicinity of F-actin is hypermobile, perhaps because actin induces breaking in the water network. Suzuki et al. (40) pointed out that the volume fraction of hypermobile water is as great as 80% of the molecular volume of G-actin.
The rotational mobility of water has been measured using microwave dielectric spectroscopy, which shows that the relaxation frequency of hypermobile water is 40 GHz, that of bulk water is 17 GHz, and that of hydrating water near globular proteins is 48 GHz or less (18, 23, 40). This range of relaxation frequencies coincides with the range of estimates for the intersection frequency f0 shown in Fig. 3. We speculate that if the cell were perturbed mechanically at a frequency corresponding to the rotational relaxation frequency of water, the water network might dominate the mechanical response of the system and differences in stiffness with various CSK interventions might vanish. Nonetheless, it remains unclear whether molecular crowding and resulting water-protein interactions might help to account for the simplicity and nonspecificity of the findings reported herein or the existence of the intersection.
In summary, we have discovered general physical laws that appear to govern the mechanical behavior of the cytoskeleton (5, 14, 15, 25, 32, 36), and in the present study, we have extended the generality of those behaviors. These laws bring together into one physical picture cytoskeletal elasticity, friction, and remodeling and show that that the cell is a strange intermediate form of matter that is neither solid nor fluid but retains features of both; classical theories of viscoelasticity, equilibrium systems, and the Stokes-Einstein relationship fail to account for these observations (5). Moreover, these observations establish that the cell possesses a temperature-like property that determines where the CSK sits along a continuous spectrum of solidlike vs. fluidlike states (Fig. 2).
Major unsolved problems in science were recently highlighted (24), and among those identified were the structure of water, protein-protein interactions, the dynamics of nonequilibrium systems, and the nature of the glassy state. The mechanics of the CSK would appear to find itself at a convergence of these unsolved problems, and this convergence represents both a challenge and an opportunity. Although the mechanism of action remains uncertain, the unifying phenomenological framework reported herein is striking. Even in the absence of a mechanism, it is possible that this framework may provide a simplified way to think about bronchospasm, vasospasm, wound repair, embryonic development, cell invasion, and any other cellular processes that have important mechanical components.
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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