1Physiology Program, Harvard School of Public Health, Boston, Massachusetts 02115; 2Physics Department, Erlangen University, 91054 Erlangen, Germany; and 3Unitat de Biofísica i Bioenginyeria, Facultat de Medicina, Universitat de Barcelona-IDIBAPS, Barcelona 08036, Spain
Submitted 5 February 2004 ; accepted in final form 5 May 2004
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ABSTRACT |
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actin; cytoskeleton; magnetic twisting cytometry; scale free; viscoelasticity
In those studies, it was noted that elastic moduli measured after the different manipulations defined a family of curves, all of which, when extrapolated, appeared to cross in the vicinity of a common intersection, or fixed point, at high frequency. Moreover, Fabry et al. (19, 20) showed that data for all frequencies, all cell types, and all interventions that they studied, when suitably normalized, could be collapsed onto two master relationships, one describing elasticity and the other friction, in which the power law exponent x was the central controlling parameter.
Although the collapse of diverse data is remarkable and has broad implications, in every instance Fabry's measurements probed elastic and loss moduli of the cell by using microbeads coated with the very same ligand: a peptide containing the sequence RGD (Arg-Gly-Asp). RGD-coated beads bind to the cell surface via integrins (69, 77), mostly 5
1, which cluster in localized attachment domains and assemble into adhesion complexes (56, 74, 84, 90). Consequently, this process inevitably alters cellular mechanical responses (8, 12, 43, 44, 61, 84). It is possible, therefore, that the power law behavior reported by Fabry et al. (19, 20), as well as the collapse of all data onto master relationships, might reflect nothing more than the particular ligand-receptor complex through which the cell was probed. If so, then Fabry's results would not be generalizable. In the present study, we used a wide panel of bead coatings to examine that possibility.
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MATERIALS AND METHODS |
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For the ligand group, beads were coated with one of the following: a peptide containing the sequence RGD, vitronectin (VN), urokinase plasminogen activator (uPA), or acetylated low-density lipoprotein (AcLDL). Because each of these ligands binds to the cell in a different way (31), the coupling of the bead to the cytoskeleton (CSK) depends on the bead coating (Fig. 1). RGD peptide binds primarily to 1-integrin. VN is also an RGD site-dependent adhesive glycoprotein and binds mainly to
3-integrin (83); both VN and RGD bind to the underlying CSK with the assembly of focal contacts. uPA binds to the uPA receptor, which in turn binds to integrins through caveolin and mediates mechanical force transmission to the CSK by indirect attachment (9, 65, 66, 87). AcLDL binds to LDL receptor, a nonadhesion scavenger receptor that does not bind to the CSK (36, 69, 70, 84). Each ligand was bound onto the bead surface (50 µg ligand/mg beads) by overnight incubation at 4°C in carbonate buffer (pH 9.4).
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Cell culture. Human tracheas were obtained from lung transplant donors, in accordance with procedures approved by the University of Pennsylvania Committee on Studies Involving Human Beings. Tracheal smooth muscle cells were harvested as previously described (50, 56, 67). The cells were plated in plastic flasks and maintained in nutrient mixture (Ham's) F-12 medium supplemented with 10% FCS, 100 U/ml penicillin, 100 µg/ml streptomycin, 200 µg/ml amphotericin B, 12 mM NaOH, 1.7 µM CaCl2, 2 mM L-glutamine, and 25 mM HEPES. The confluent cells were serum-deprived and supplemented with 5.7 µg/ml insulin and 5 µg/ml human transferrin for 48 h before the experiment. The cells were used in passages 47.
Cell-bead preparation. Cells were trypsinized (0.25% trypsin and 1 mM EDTA), plated at confluence overnight (2 x 104 cells/well) on collagen-coated (500 ng/well) plastic wells (6.4-mm, 96-well Removawells, Immulon II; Dynatech, Chantilly, VA). In the case of beads coated with uPA, however, we used 1 x 104 cells/well grown to subconfluence; this was necessary because beads coated with uPA do not bind to confluent cells (68).
Cells were washed twice with serum-free medium supplemented with 1% BSA, coated beads were then added for 1520 min at 37°C to allow for the binding to the receptors on the cell surface, and wells were washed twice with serum-free medium supplemented with 1% BSA to remove any unbound beads The final concentration was approximately 1 bead/cell. Finally, measurements were performed as described below.
Experiments for each bead type and each bead coating were done on five separate experimental days and measured in 1530 cell wells for each condition. These wells were further distributed among three conditions: cells not treated (baseline); cells treated with cytochalasin D (2 x 106 M), an actin network disruptor, for 15 min; and cells treated with histamine (104 M), a contractile agonist, for 15 min.
In a separate series of experiments, we established the specificity of bead binding. Cells were pretreated for 30 min with the corresponding soluble coating material in the media as a competitor (0, 5, and 50 mg/ml), and the number of beads that subsequently bound was measured with and without the competitor. All coating showed specificity. We also measured cell stiffness as a function of the concentration of bead coating material in which the beads were incubated (as described above). For each bead coating we found that measured stiffness increased with concentration but eventually saturated at high concentration. Similarly, we measured the number of beads bound to each cell as a function of the concentration of bead coating material in which the beads were incubated, and the number of attached beads increased with concentration but eventually saturated at high concentrations. All data reported were obtained with the use of saturating concentrations.
Magnetic twisting cytometry with optical detection. The experimental setup is described elsewhere (19, 20). Briefly, wells containing cells with beads attached were placed on an inverted microscope. Beads were magnetized horizontally and then subjected to a vertical oscillatory field. This oscillatory field causes a mechanical torque that twists the bead toward alignment with the direction of the imposed field. This torque is transmitted from the bead through the ligand-receptor complex to the cell body, which, because of its elasticity and friction, impedes the bead motion. Measurements were performed at oscillatory frequencies between 101 and 103 Hz, and the amplitude of the oscillatory magnetic field was first adjusted to keep the mean bead displacement within the linear range for each of the three conditions with each different coating and was then kept at the same magnetic amplitude across frequencies; data from beads with amplitude >500 nm were discarded. Oscillatory lateral displacements of each bead were recorded using a charge-coupled device camera with an exposure time of 0.1 ms. Bead position was determined using an intensity-weighted center-of-mass algorithm yielding accuracy in the bead position better than 5 nm.
The complex elastic modulus was defined at each radian frequency, , as g*(
) = T*/d*, where T* and d* are the Fourier coefficients of the oscillatory specific torque and displacement,
= 2
f, and the specific torque is the torque per unit bead volume. This is equivalent to computing the components of bead displacement both in phase and out of phase with the applied specific torque. As defined here, this modulus has units of pascals per nanometer. Such measurements can be transformed into traditional elastic and loss moduli (in units of Pa) through multiplication by a geometric scale factor,
, which depends on the shape and thickness of the cell and the degree of the bead embedding. Finite element analysis of cell deformation sets
at roughly 6.8 µm (60), assuming homogeneous and isotropic elastic properties with 10% of the bead diameter embedded in a cell 5 µm high.
Statistical evaluation of power law behavior.
For a material whose stress relaxation function is At 1x, the complex modulus is given by the power law,
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In the limit that x approaches unity, the real part of g* reduces to a perfect elastic Hookean stiffness, and the imaginary part is the simple additive viscous term (20). In the limit that x approaches 2, the real part of g* vanishes, and the imaginary part corresponds to a perfect Newtonian viscous fluid of viscosity A + µ. As such, x describes the transition from solidlike elastic states to fluidlike viscous states.
The variability in g* across the bead population increases with its magnitude and is approximately log normal (19). We therefore modeled the variability in g* as being proportionate; this is equivalent to having an approximately uniform (complex) variability that is additive in ln(g*). Consequently, we used the logarithm of the measured complex modulus for the individual fits to the logarithm of Eq. 1 and for subsequent statistical analysis. In that connection, we used least-squares minimization to evaluate the parameters, and all sums of squared residuals were thus performed using logarithms of data and of prediction (20).
Existence of a fixed point.
We assessed Eq. 1 using three statistical models. In model I, all parameters (xn, An, and µn) for any given coating were free to vary among the n experimental conditions (n = 3: baseline, histamine, cytochalasin D). In model II, An was given by , where only xn was free to vary, implying a convergence of the curves for all three conditions in the vicinity of the coordinates (g0,
0). In model III, An and µn were free to vary, but the exponent x was constrained to be common over the experimental conditions, implying a parallel shift (log g* vs. log f) with treatment. Model I vs. II and model I vs. III were then compared by reduction-of-variance F-test. If the observed P value of the pairwise comparison was <0.05, we concluded that there were significant differences between models, implying that the fit of the model with fewer parameters (e.g., model II or III) was not as good as the fit using model I. Models II and III were not compared, because neither model is a subset of the other. Finally, for model II, we evaluated 95% confidence regions of g0 and
0. Software used for these procedures was the R statistical program (version 1.6.2).
Chemicals.
Ham's F-12 culture medium, FCS, insulin, human transferrin, penicillin-streptomycin solution, L-glutamine, NaOH, CaCl2, histamine, cytochalasin D, and BSA were obtained from Sigma Chemicals (St. Louis, MO). Fungizone and 0.02% trypsin were purchased from GIBCO (Gaithersburg, MD). Phosphate-buffered saline was obtained from BioWhittaker (Walkersville, MD). Type I rat tail collagen was purchased from Upstate Biotechnology (Lake Placid, NY). Synthetic RGD-containing peptide (Arg-Gly-Asp; Peptide 2000) was purchased from Telios Pharmaceuticals (San Diego, CA). AcLDL was purchased from Biomedical Technologies (Stoughton, MA). Human urokinase was obtained from American Diagnostica (Greenwich, CT). Human monoclonal antibodies against 1-integrin (clones MAB1987 and MAB 2000) and against
v
3-integrin were purchased from Chemicon (Temecula, CA). Monoclonal antibody against
1-integrin (clone K20) was purchased from Beckman-Coulter (France). Polystyrene beads (4.5-µm diameter) with a CrO2 surface layer, precoated with goat anti-mouse IgG for the antibody studies or precoated with reactive carboxyl groups for a separate RGD control, were purchased from Spherotech (Libertyville, IL).
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RESULTS |
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When beads were coated with the peptide sequence containing RGD, the stiffness g' of cells in baseline conditions increased as a power law with frequency throughout the measurement range (Fig. 2A, circles). However, these power law relationships were quite weak. The loss modulus g" also approximated a power law with the same exponent at low frequencies but showed stronger frequency dependence at higher frequencies. When the actin network was disrupted by cytochalasin D (2 x 106 M), both g' and g" fell dramatically, and their dependence on frequency increased (Fig. 2A, squares). When the cells were activated by the contractile agonist histamine (104 M), g' and g" increased and their dependence on frequency decreased (Fig. 2A, triangles). All data were well described by Eq. 1 (fits denoted by solid lines, Fig. 2A) and were consistent with data reported previously (16, 19, 20). Bead-by-bead variability is quantified in Table 1.
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Existence and locus of the fixed point.
For each bead coating we expressed the behavior over the bead population as the geometric mean of the scale factor A (Eq. 1), and the bead-by-bead variability as the geometric standard error (Table 1). We then assessed the existence of a common intersection, or fixed point. In the case of most bead coatings, constraint of the model (Eq. 1) to have a fixed point, implying common values of g0, 0 (model II) was not significantly different from allowing all parameters to be independent (model I) (see results in Table 3). Therefore, we adopted model II for these cases.
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Because power law responses were observed in every case, each of the fits (g' vs. f) depicted in Figs. 2 and 3 can be defined by one point and one slope. The point was arbitrarily chosen as the value of g' measured at 0.75 Hz, and the slope is x 1. All relationships implied by the stiffness data in Figs. 2 and 3 could be represented compactly, therefore, by a graph of g' (0.75 Hz) vs. x (Fig. 5).
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For all bead coatings in which a fixed point could be determined (i.e., all but uPA and anti-v
3-integrins; Table 4), we used g0 as an intrinsic stiffness scale and then defined a normalized stiffness G as the ratio of g' measured at 0.75 Hz to g0. When stiffness data in Fig. 5 were normalized in this way, they collapsed onto a single master relationship (Fig. 6A). We scaled the loss modulus g" (0.75 Hz) by expressing it as a fraction of g' at the same frequency; the ratio is
= g"/g', known as the loss tangent or the hysteresivity. Similarly, despite enormous differences in g" between experimental conditions, coating ligands, and bead type, all data collapsed onto a single relationship without exception (Fig. 6B). Some systematic discrepancies were noted, however, at larger values of x, where
varied with x in a way that was stronger than predicted.
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DISCUSSION |
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There are severe limitations inherent in the magnetic microbead approach. However, these limitations are counterbalanced in many applications by its unique capabilities. We begin by addressing those limitations and then move on to address power law behavior, the marked differences in responses that are revealed when various molecular bead coatings are used, and finally, the commonality of those responses that are revealed when data are appropriately normalized.
Methodological limitations.
Whether by ligation and/or mechanotransduction, the interaction of the magnetic microbead with the cell induces local remodeling events that alter the structure that is being probed (11, 14, 18, 22, 33, 59, 75). This remodeling is the principal weakness of the microbead approach but, at the same time, is one of its greatest strengths. While the bead has caused the cell to remodel locally (see Figs. 1 and 7), these remodeled structures are no different in kind from those that form at anchorage sites to the extracellular matrix (ECM) upon which the cell is adherent; indeed, from the point of view of the adherent cell, the bead is merely another piece of ECM upon which adhesion sites might be generated (11, 62, 74, 84). Because force transmission across the cell membrane occurs preferentially through adhesion molecules, and the adhesion complex in particular (4, 29, 42, 43, 62, 74, 84, 90), from the point of view of physiological relevance these would seem to be the most appropriate molecular pathways through which to probe cell mechanics. Probing the cell through a ligand-coated or antibody-coated bead offers the further advantage, therefore, of a defined molecular coupling rather than a nonspecific coupling. For these reasons, bead binding and mechanical loading have become standard tools for probing cell mechanics in the arsenal of approaches that are used for the investigation of adhesion site formation, mechanotransduction, and reinforcement (10, 12, 32, 33, 46, 86).
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A principal strength of the approach is that the bead can apply a mechanical load to the cell body in the physiological range of stress, from below 1 Pa to >100 Pa (20, 56). These loads are transmitted to the cell body via specific receptor-ligand systems, and the resulting bead displacements can be resolved on the molecular scale [as small as 5 nm (19, 20)]. Dynamic responses can be measured at frequencies as high as 1 kHz (19). Even at the highest frequencies studied, inertia does not come into play, and the effects of viscous loads associated with the medium are smaller than the loads associated with the cell by several orders of magnitude (20). Also, artifacts associated with the local heating caused by laser traps are not an issue.
Mechanical properties of cells in culture are innately variable from cell to cell. Detecting differences of 25% using an unpaired, one-tailed t-test requires roughly 50 cells per group, and detecting differences of 25% in baseline stiffness requires roughly 75 cells per group (21). The requirement of relatively large sample sizes is problematic for techniques that study one cell at a time, such as the use of atomic force microscopy or laser tweezers. Many beads can be tracked simultaneously by using magnetic twisting cytometry (MTC) with optical detection, by contrast, and data from hundreds or even thousands of cells can be collected within a relatively short time (20, 21).
Importantly, cells probed with this technology display mechanical responsiveness that is consistent with physiological responses measured at the tissue and organ levels. Panettieri et al. (67) showed that nontransformed HASM cells that are serum deprived and grown to confluence retain smooth muscle-specific contractile protein expression (-actin and desmin), though not as much as freshly dissociated cells. These cells retain physiological responsiveness, including cytosolic Ca2+ release and cAMP production, in response to histamine, leukotrienes, bradykinin, platelet-activating factor, substance P, and thromboxane analogs (21, 41, 49, 67, 78). Using MTC, Hubmayr et al. (41) showed that cultured HASM cells stiffen when challenged with a panel of contractile agonists reported to increase intracellular Ca2+ concentration or inositol 1,4,5-trisphosphate formation and that the extent of cell stiffening rank is in order with the relative potency of these same agonists in mediating bronchoconstriction at the level of isolated muscle strips (2426, 35). Conversely, cell stiffness decreases progressively with increasing doses of bronchodilating agonists that are known to increase intracellular cAMP and cGMP levels (41, 78). Shore and colleagues (41, 64, 78) showed that these cells retain functional coupling to
-adrenergic receptors over many population doublings. An et al. (3) found that serotonin (5-HT) increases cell stiffness in a dose-dependent fashion and also elicits rapid formation of F-actin, whereas a calmodulin antagonist, a myosin light chain kinase inhibitor, and a myosin ATPase inhibitor each ablate the stiffening response but not the F-actin polymerization induced by 5-HT. However, agents that inhibit the formation of F-actin attenuate both baseline stiffness and the extent of cell stiffening in response to 5-HT (3).
As with many of the available methods for measurement of cell mechanics (1, 6, 7, 12, 17, 38, 39, 72, 93), a length scale , described in MATERIALS AND METHODS, must be invoked to convert raw data into a proper elastic modulus. Such a length scale can be computed using a finite element model of cell deformation (60) or estimated from simple dimensional arguments. In many instances, however, dynamic data either can be expressed as relative changes or can be nondimensionalized in such a way that this length scale cancels out (Fig. 6) and, therefore, does not come into play. Below we deal in more detail with the microbead approach, implications of differences in bead coating, and the way that these data can be scaled.
Power law behavior.
Power law behavior according to Eq. 1 pertained over a wide range of circumstances without exception (Figs. 2 and 3). Moreover, power law behavior persisted on a cell-by-cell and bead-by-bead basis, and the power law exponent showed only modest variability within each experimental condition (Table 1; Refs. 19, 20). These observations rule out artifacts in which a multiplicity of relaxation time scales might be attributable to population averages that pool together data sampled from many different individual cells and from very different regions on each cell. In addition, power law responses are also observed when the cell is probed using atomic force microscopy (1), parallel-plate extension of single cells (79), optical tweezers (5, 15), or laser tracking microrheology (91). Taken together, these observations strongly support the conclusion that power law behavior is an intensive property of the cellular material that is being probed.
It might be argued in this connection that, rather than a power law, these data could be fit equally well by using a viscoelastic model comprising roughly two relaxation time scales per frequency decade, or eight in all for the frequency range reported in the present study. While this is true, such an interpretation requires assignment of an ad hoc distribution of time constants and represents nothing more than a different parameterization of the data, albeit one requiring eight free parameters instead of one.
The great value of using a mechanical assay to probe protein-protein dynamics is that the elastic modulus gives a direct indication of the number of molecular interactions, and the loss modulus (expressed as the hysteresivity, or loss tangent, ) gives a direct indication of their rate of turnover (19, 26, 37). Using that strategy, Fabry et al. (20) had set out initially to identify a small number of distinct internal time scales, molecular relaxation times, or time constants that might typify mechanical responses of the integrated cytoskeletal matrix. However, the power law responses that they found precluded that possibility, suggesting instead the existence of a great many relaxation processes contributing when the frequency of the imposed forcing is small, but very few as the frequency of the forcing is increased and slower processes become progressively frozen out of the response. Therefore, with regard to protein-protein interactions within the complex microenvironment of the intact living cytoskeleton, there is no internal time scale that can typify the dynamics. Instead, all time scales are present simultaneously but distributed very unevenly; the dynamics are scale free (89). Scale-free behavior is thus found to pertain not only at the level of topology of the protein-protein signaling network (45), the level of protein structure (48), and the level of the spatial distribution of individual proteins (23) but also at the level of dynamics of the protein-protein interactions (20).
In a network in which the formation of new bonds and the breaking of old ones are thermally driven, relaxation processes are characterized by exponential decay. The cytoskeletal matrix of the living cell, by contrast, tends to relax with functions that decay much more slowly, as do other nonequilibrium condensed systems (13, 20, 92). If scale-free responses prevail, then interpretations based on processes characterized by one or even several molecular relaxation time scales must be ruled out (6, 7, 47, 91, 93). Instead, Fabry and colleagues (1820, 34) associated the exponent x with an effective temperature of the cytoskeletal matrix as described in the theory of soft glassy rheology of Sollich (80, 81).
Probe specificity.
What then might account for the differences in measured values of g' and g" associated with the various probes that were employed? As noted above, the binding of a bead to the cell via any particular ligand-receptor complex inevitably leads to a cascade of signaling events and structural remodeling in the vicinity of the bead. RGD-coated beads, for example, are known to bind to integrins (mostly but not exclusively to v
1), which cluster and recruit other proteins into adhesion complexes (12, 33, 52, 62, 63, 70, 77, 84). These complexes are tightly coupled to deeper cytoskeletal structures such as the stress fibers and the contractile machinery of the cell (3, 19, 33, 40, 41, 50, 58, 76).
Among the bead coatings studied, stiffness values measured through beads coated with RGD showed the biggest changes with CSK manipulations, contractile activation, and deactivation (Figs. 2 and 3; Refs. 21, 41, 50, 78, 88) and showed stiffness values 10-fold greater than those seen through beads coated with AcLDL. These observations and others (2, 51, 69) suggest that measurements using RGD-coated beads tend to emphasize mechanical properties of the deep cytoskeletal structures such as stress fibers and the cell's contractile machinery (Fig. 7). Particularly relevant in that regard are recent data showing cytoskeletal remodeling induced by the presence of a RGD-coated bead (14). Supporting this interpretation further, Hu et al. (40) showed in the living adherent cell that stresses applied at the apical surface are transmitted via stress fibers over very long distances.
Beads coated with AcLDL bind to scavenger receptors, do not induce the formation of adhesion complexes, and are mechanically connected to stress fibers only indirectly if at all (11, 36, 70, 84, 85). As such, we speculate that they tend to emphasize the mechanics of the structures to which they are connected, which, in this case, would be cortical structures. If so, then the stiffness is likely to depend on lipid bilayer in-plane tension, membrane-CSK adhesions, and the strength of the link between the bead and those structures (73). This interpretation would help to explain the relatively smaller stiffness observed through AcLDL-coated beads as well as the attenuated effect of the CSK manipulations (Figs. 26). This result is also consistent with studies on the plasma membrane by Raucher and colleagues (29, 73).
In contrast with the behavior of beads (solid Fe3O4) coated with RGD, those coated with AcLDL reflect a value of x that is substantially smaller (1.118 ± 0.002 vs. 1.181 ± 0.002, at baseline). Similarly, beads coated with nonactivating and blocking anti-1-integrin antibodies, which do not promote the adhesion complexes formation and, hence, are only loosely connected to the deep CSK, also show relatively smaller values of x. Recalling that x is an index along the spectrum from solidlike (x = 1) to fluidlike (x = 2) states (Eq. 1), these observations suggest the interpretation that cortical structures are floppy but solidlike in character nonetheless, like a flimsy but stable elastic membrane, whereas stress fibers are closer to a fluidlike state.
For reasons that are unclear, RGD coatings on the two different bead types yielded very different mechanical responses (Fig. 5), with median stiffness values being different by more than one decade. Beads coated with activating anti-1-integrin were intermediate in that regard. Plausible explanations of these discrepancies might be differences in the amount of binding of peptide or antibody to the bead surface, differences in the availability of the molecular binding site to the receptor, or differences in geometrical arrangement of the binding site on the bead. These factors, in turn, would influence the extent to which the bead becomes embedded in the cell and the subcellular microenvironment that bead is probing (47, 57).
Scaled data.
Despite the marked differences that were apparent in Figs. 2, 3 and 5, these diverse responses collapsed onto the same master curves (Fig. 6). As mentioned before, the values of the internal stiffness scale g0 varied substantially among coatings, a result that was not unexpected. Confidence intervals for 0 were extremely wide as a result of the combination of appreciable variability together with the small exponents of the power laws (Fig. 4). Nonetheless,
0 could be set roughly in the neighborhood of 1010 Hz and, because of the wide confidence interval, could be taken as the same for all bead coatings.
The collapse of data imply that differences between RGD and AcLDL represent the extremes along a continuous spectrum of possibilities; the different bead types and various bead coatings that were studied always fell within the same class of responses (power law behavior) but with different power law exponents. In particular, the choice of the bead and its coating set the stiffness scale g0, but once that stiffness scale was set, responses merely sampled different regions of the very same master curve (Fig. 6). Accordingly, except for the scale factor of normalization (g0), the dynamics of the system, as characterized by the dependencies of g' and g" on frequency, were set by x alone (Fig. 6; Eq. 1).
Special cases were beads coated with uPA and anti-v
3. These coatings conformed to the universal scaling with regard to friction (Fig. 6B) but not stiffness. The departure of these cases from the others stems from the fact that the power law responses of g' with f were nearly parallel, thereby making determination of a fixed point (and the internal stiffness scale g0) not meaningful. We do not understand why these two instances are different in that regard. More generally, the finding of a fixed point stands as a serendipitous observation with no known explanation. As such, the physical meanings of the fixed point and the associated frequency scale
0 remain unclear.
Surprising generality of these results.
Elastic and frictional moduli reported in this study in cells measured under different circumstances and probed through different molecular windows were closely similar in character to measurements in a totally different kind of system, namely, colloid suspensions close to a jamming transition (71). Prasad et al. (71) interpreted their result with a very simple physical picture. They suggested that colloidal particles aggregate to form a solid network interspersed with background fluid. They reasoned that both the solid network and the viscous fluid contribute independently to the measured moduli. Because the background fluid is purely viscous, they argued that it contributes only to the loss modulus g" and yields an additive term that increases linearly with frequency; Fabry et al. (20) came to much to the same conclusion. The scaling suggested by Prasad et al. (71), however, was flawed by an unrecognized but strong covariance of their scaling axes (for a weakly frequency-dependent complex stiffness plus a viscous term, their scaling is essentially a plot of g' vs. g'). Nevertheless, we can find no reason to suggest that the physical picture they describe might not apply to cytoskeletal dynamics. Moreover, they argued that if this picture is correct, then the viscosity should be independent of the factors that determine the configuration of the network, and, consistent with that argument, they found that the measured viscous term remained roughly independent of the network. Like theirs, our viscosity data (Table 1) are roughly consistent with that argument.
In summary, the findings reported here extend and generalize the results of Fabry et al. (19, 20), showing that power law responses and the collapse of data onto unifying master relationships (Fig. 6) are not peculiar to the RGD probe. Equation 1 describes cell rheology without exception, and accordingly, we suspect that power law behavior must be a signature of some generic feature of underlying protein-protein interactions (20). In most cases, data collapsed onto unifying master relationships, implying that x, an index from solidlike (x = 1) to fluidlike (x = 2) behavior, is the central parameter that sets cytoskeletal dynamics (Fig. 6). These master relationships demonstrated that when a cell modulates its mechanical properties, it does so along a special trajectory.
Moreover, frequency responses of elastic and frictional moduli measured in different circumstances, and probed through a variety of different molecular windows, are similar to those found in colloid systems close to a jamming transition. We have speculated elsewhere that the cytoskeleton, like certain colloid suspensions and other soft media, may belong to the class of soft glassy systems (19, 20, 34, 80, 81). Recently, such soft materials have been related to the jamming concept (54, 71, 82). Dynamics of soft glassy materials are entirely determined by the parameter x, which is called an effective matrix temperature (19, 20, 80, 81). All of these systems have the disordered molecular state of fluid and, at the same time, the rigidity of a solid. If these ideas apply to cytoskeletal behavior, they would point to metastability of interactions and disorder of the matrix as being key features of underlying protein-protein dynamics.
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GRANTS |
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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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