time scale and other invariants of integrative mechanical ......time scale and other invariants of...

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PHYSICAL REVIEW E 68, 041914 ~2003!

Time scale and other invariants of integrative mechanical behavior in living cells

Ben Fabry,1,* Geoffrey N. Maksym,2 James P. Butler,1 Michael Glogauer,3 Daniel Navajas,4 Nathan A. Taback,5

Emil J. Millet,1 and Jeffrey J. Fredberg1

1Physiology Program, Harvard School of Public Health, 665 Huntington Avenue, Boston, Massachusetts 02115, USA2School of Biomedical Engineering, Dalhousie University, 5981 University Avenue, Halifax, Nova Scotia, Canada B3H 3J5

3Faculty of Dentistry, University of Toronto, 150 College Street, Toronto, Ontario, Canada M5S 3E24Unitat Biofısica i Bioenginyeria, Universitat de Barcelona–IDIBAPS, Casanova 143, 08036 Barcelona, Spain

5Department of Biostatistics, Harvard School of Public Health, 665 Huntington Avenue, Boston, Massachusetts 02115, US~Received 7 May 2003; published 27 October 2003!

In dealing with systems as complex as the cytoskeleton, we need organizing principles or, short of that, anempirical framework into which these systems fit. We report here unexpected invariants of cytoskeletal behav-ior that comprise such an empirical framework. We measured elastic and frictional moduli of a variety of celltypes over a wide range of time scales and using a variety of biological interventions. In all instances elasticstresses dominated at frequencies below 300 Hz, increased only weakly with frequency, and followed a powerlaw; no characteristic time scale was evident. Frictional stresses paralleled the elastic behavior at frequenciesbelow 10 Hz but approached a Newtonian viscous behavior at higher frequencies. Surprisingly, all data couldbe collapsed onto master curves, the existence of which implies that elastic and frictional stresses share acommon underlying mechanism. Taken together, these findings define an unanticipated integrative frameworkfor studying protein interactions within the complex microenvironment of the cell body, and appear to set limitson what can be predicted about integrated mechanical behavior of the matrix based solely on cytoskeletalconstituents considered in isolation. Moreover, these observations are consistent with the hypothesis that thecytoskeleton of the living cell behaves as a soft glassy material, wherein cytoskeletal proteins modulate cellmechanical properties mainly by changing an effective temperature of the cytoskeletal matrix. If so, then theeffective temperature becomes an easily quantified determinant of the ability of the cytoskeleton to deform,flow, and reorganize.

DOI: 10.1103/PhysRevE.68.041914 PACS number~s!: 87.16.Ka, 83.85.Vb, 64.70.Pf, 87.19.Rr

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INTRODUCTION

In response to the application of a mechanical stress,cytoskeleton has the ability to deform, flow, and remodel.the living cell, these processes involve the coordinatedsembly and disassembly of cytoskeletal polymers as wethe coupling of those structures to motor proteins. Deformtion, flow, and remodeling are fundamental to a varietyhigher cell functions including cell contraction, adhesiospreading, crawling, invasion, wound healing, and divisiand have been implicated as well in mechanotransductregulation of protein and DNA synthesis, and programmcell death@1#. Our understanding of the empirical constittive laws that govern these mechanical processes remquite fragmentary, however, as does our understandingunderlying mechanism. The manner in which the multituof cytoskeletal and associated proteins interact with oneother within the dynamic microenvironment of the cell boso as to produce integrated mechanical effects remains ajor open question.

Although mechanical behavior of the cell is importantits own right, it can also be seen as being only one instawithin the broader rubric of integrative phenotypic phenoena. We demonstrate in this paper that microrheology ofliving cell may provide a unique window on protein dynamics in complex functional systems, by which we mean tho

*Email address: [email protected]

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dominated by multiple weak effects operating within a nomicroenvironment and far from thermodynamic equilibriumWe address three questions. First, what might comprisfavorable model system for the study of molecular evethat underlie integrative system functions? Second, mithere be generic phenotypic properties that are characterof such systems? Finally, are there fundamental principlebiological design that govern integrative system functions

Within the living cell, the universe of protein interactionis populated unevenly, with a very few highly connecthubs and a great many less connected nodes@2#; indeed, theprobability of any given protein havingk interactions de-creases as a power law withk. This power-law dependencshows that the highly interconnected network of biologicinteractions possesses no internal scale that can typifynumber of interactions per protein and, accordingly, in tregard biological networks are said to be scale-free@2#. Inturn, individual proteins comprising the network have beshown to be characterized by a very special structure;universe of protein structures is populated unevenly, witvery few clusters, folds, and superfold domains thatrather common, but a great many fold domains that are mless common@3#. Interestingly, as protein size increases, tprobability of any given number of fold domains varies aspower law with protein size. This power-law dependenshows that protein structure possesses no internal scalecan typify the number of folds per protein. Like netwotopology, protein structure is also scale-free@3#. The spatialdistribution of structure density within the cell has been m

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FABRY et al. PHYSICAL REVIEW E 68, 041914 ~2003!

sured, and the limited data that are available suggest thtoo shows a power-law distribution and, therefore, appearbe scale-free@4#.

The topology of the protein interaction network, taktogether with the structures of the individual proteins theselves and their distribution in space, comprises the supon which protein dynamics must be played out. We hadout to identify distinct internal time scales, relaxation timeor time constants that might typify mechanical responsethe integrated cytoskeletal matrix; in rheology, mechanirelaxation processes are taken as being the shadow of ulying molecular dynamics@5#. We report here that overspectrum spanning five decades of frequency and in all offive cell types that were investigated, relaxation times wfound to be distributed as a power law, with a great marelaxation processes contributing when the time scale ofimposed forcing was long, but very few as the time scalethe forcing was reduced and slower processes becamegressively frozen-out of the response. As regards proteinteractions within the complex microenvironment of the liing intact cytoskeleton, therefore, there was no internal tscale that could typify their dynamics. All time scales wepresent simultaneously but distributed very unevenly; thetegrated mechanical behavior was scale-free. Scale-freehavior thus pertains at the levels of the topology of tprotein-protein interaction network, the structure and disbution of the individual proteins, and the dynamics of tmatrix that they comprise. As described below, this findstands in contrast with current theories of cytoskeletal behior, all of which posit behavior characterized by one or,most, a few distinct internal time scales.

We found in addition that regardless of the frequencythe imposed oscillation, the cell type that was studied,signal transduction pathway that was activated, or the typmolecule that was manipulated, all data could be collaponto master curves that depended on only a single param@6#. These master curves demonstrate that when the meccal properties of the cell change, they must do so alonspecial trajectory.

These findings, taken together, establish unanticipatedvariants of cytoskeletal mechanical behavior and, in doso, would appear to set limits on what can be predicted abthe integrated mechanical behavior of the cell from conserations of cytoskeletal molecules studied in isolation.particular, they lead to the notion that the integrated mchanical behavior of the cytoskeleton may not dependmuch on the molecular details of particular cytoskeletal ements but instead may depend more on the generic metbility of their microstructural arrangements, as describedgreater detail below. For any such arrangement, an indethis metastability can be given as the level of mechanagitation~noise! present in the microenvironment relativethe depth of the energy well that defines that arrangemThis index is easily measured and can be expressed aeffective temperature of the matrix. As explained below,havior of this kind is consistent with the hypothesis thatcytoskeletal matrix of the living adherent cell behaves asoft glassy material@7#; here and elsewhere in this paper wuse the term ‘‘cytoskeleton’’ in the broadest sense so a

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subsume scaffolding proteins, the contractile apparatus,any attached structures or molecules that contribute apprebly to the integrated mechanical properties of the cell. Tpaper expands on our previous short communication@6# byproviding greater methodological detail, expanded data sand new statistical analysis.

METHODS

To study the rheology of cytoskeletal polymers requireprobe whose operative frequency range spans, in so fapossible, the internal molecular time scales of the rate pcesses in question@8,9#. Here we used magnetic twistincytometry ~MTC! with optical detection of bead motionWith this method, we were able to apply probing frequencranging from 0.01 Hz to 1 kHz@10–14#.

Microbead preparation. As a measurement probe we usferrimagnetic microbeads (4.560.3mm diam) that werebound to specific cell surface receptors. We producedbeads by aerosolizing a water-based, surfactant stabilFe3O4 ferrofluid ~Ferrotec, Nashua, NH! with a spinning topaerosol generator~BGI inc., Waltham, MA!. The aerosolizedparticles were dried in a copper funnel, neutralized withradioactive beta source~americium!, and then sintered forseveral seconds at 800 °C while suspended in an atmospcontaining 2% H2 and 98% N2 ~see @15# for details!. Wecoated the beads overnight or longer at 4 °C with a synthpeptide that contained the sequence RGD~Arg-Gly-Asp!@16# ~Peptite 2000, Integra Life Sciences, San Diego, kinprovided by Dr. Ju¨rg Tschopp! at 50 mg peptide per mgbeads in 1 ml carbonate buffer (pH 9.4!. These beads wereused for all cell types except neutrophils. In experimewith neutrophils we used polystyrene beads~Spherotech,Libertyville, IL! that were coated with chromium dioxidparticles and precoated with goat-anti-mouse immunoglolin G as a primary antibody. We coated those beads witsecondary mouse-anti-human CD-45 antibody~BD PharM-ingen, San Diego, CA! overnight at 4 °C at a concentratioof 200 mg antibodies per 108 beads in 1 ml phosphatebuffered saline.

Bead twisting. After the beads were bound to the ce~described below!, the beads were magnetized horizontaby a brief ~0.1 ms! magnetic field pulse~.0.1 T! and thentwisted vertically by an external homogeneous magnetic fi~,0.01 T! that was varying sinusoidally in time@Fig. 1~c!#.The twisting field induces a mechanical torque that tendsalign the magnetic orientation of the bead with that of tmagnetic twisting field.

The specific torque~T! is defined as the mechanicatorque per unit bead volume, and has dimensions of st~Pa!. T is given by

T5cH cosu, ~1!

whereH is the magnetic twisting field~in gauss!, u is theangle of the bead’s magnetic moment relative to the origimagnetization direction, andc is the bead constant, expressed as torque per unit bead volume per gauss.c is deter-mined by placing beads in a fluid of known viscosity a

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TIME SCALE AND OTHER INVARIANTS OF . . . PHYSICAL REVIEW E 68, 041914 ~2003!

FIG. 1. ~a! Scanning EM of a bead bound to the surface of a HASM cell.~b! Ferrimagnetic beads coated with RGD-containing peptbind avidly to the actin cytoskeleton~stained with fluorescently labeled phalloidin! of HASM cells via cell adhesion molecules~integrins!.~c! A magnetic twisting field introduces a torque which causes the bead to rotate and to displace. Large arrows indicate the directbead’s magnetic moment before~black! and after~gray! twisting. If the twisting field is varied sinusoidally in time, then the microbewobbles to and fro, resulting in a lateral displacement,d, that can be measured.

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then measuring the angular velocity while twisting@10#. Inthis study, we used beads withc51.86 Pa/G. Becauseu wassmall @13#, we could ignore the cosu term in Eq.~1! whencomputing the specific torque from the twisting field~seebelow!. This resulted in errors of less than 5% for small berotations (u,18°) or small lateral bead displacemen~,700 nm in the case of a pivoting motion of the bearound a contact point at the bead perimeter!.

Measurement of bead motion. In response to a twistingfield that varied sinusoidally in time, beads oscillated latally to and fro, mainly along the magnetization direction. Weliminated beads from subsequent data analysis whenmovement in the magnetization direction was smaller ththeir movement orthogonal to the magnetization directithis was the case in about 10% of all beads. Lateral bdisplacement,d, in response to the imposed oscillatotorque was observed with an inverted microscope~Diaphot,Nikon, Kanagawa, Japan! with a 103 objective ~NA 0.2!.The microscope was placed on a vibration isolation ta~Newport, Irvine, CA!. A progressive scan, triggerable blacand-white CCD camera with pixel-clock synchronizati~JAI CV-M10, Glostrup, Denmark! was attached to the camera side-port of the microscope via a camera adapter with3magnification. Image acquisition~exposure time 100ms! wasphase-locked to the twisting field so that 16 images wacquired during a twisting cycle. Heterodyning~a strobo-scopic technique! was used at frequencies.1 Hz. The phaseseparation between consecutive images wasp/8 when het-erodyning was not used, or 2pn1p/8 when heterodyningwas used, withn51 at frequencies of 4.5 Hz or 10 Hz,n53 at 30 Hz,n510 at 100 Hz,n530 at 300 Hz, andn5100 at 1000 Hz. The camera trigger signal and the sigfor the twisting current driver were generated with a 16-microcontroller ~C167, Infineon, Munich, Germany!. Thetwisting current was generated with a current source capof driving up to 2.5 A. The images were analyzed on-liusing an intensity-weighted center-of-mass algorithm fortermining the bead position with an accuracy of 5 nm~rms!~see@14# for details!.

Measurement of elastic moduli. As the bead moves, it deforms the cell to which it is bound@17#. The cell resists thisbead motion by developing internal stresses that depenthe cell’s mechanical properties. Thus, cell mechanical prerties could be calculated from the imposed mechantorque and the measurement of the resulting bead motio

The ratio of the complex specific torqueT to the resulting

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complex bead displacementd defines a complex elastimodulus of the cellg( f )5T( f )/d( f ) as a function of fre-quencyf, and has dimensions of Pa/nm. For each beadcomputed the elastic modulus, or stiffness,g8 ~the real partof g), the loss modulus, or friction,g9 ~the imaginary part ofg), and the loss tangent, or hysteresivity,h ~the ratiog9/g8).These measurements could be transformed into traditioelastic shear (G8) and loss (G9) moduli by a geometric fac-tor a that depends on the shape and thickness of the cell,the degree of bead embedding, whereG5ag, G5G81 iG9, andi 2521. Finite element analysis of cell deformation for a representative bead-cell geometry~assuming ho-mogeneous and isotropic elastic properties with 10% ofbead diameter embedded in a cell 5mm high! setsa to 6.8mm @17#. This geometric factor need serve only as a rouapproximation, however, because it cancels out in the scaprocedure described below, which is model-independent

To test the entire measurement system, we studiedfrequency response of collagen-coated beads that werevalently crosslinked to the surface of a polyacrylamide~7% acrylamide with 0.25% bis-acrylamide as a crosslink!with a low-frequency stiffness of 7 kPa@18#. Our measure-ments indicated a predominantly elastic response of thewith a g8 that was flat between 0.01 Hz and 1000 Hz. Thbehavior closely resembled the behavior of an identicaprepared polyacrylamide gel that we measured wdiffusing-wave spectroscopy using the method of MasonWeitz @19#.

Data analysis. To describe our data, we used the structudamping law~sometimes called hysteretic damping! togetherwith a small additive Newtonian viscosity term@20–25#. Themechanical properties of such a material can be mathemcally expressed either in the time domain or in the frequedomain. In the time domain, the mechanical stress respoto a unit step change in strain imposed att50 is an instan-taneous component attributable to a pure viscous resptogether with a component that rises instantaneouslythen decays over time as a power law,

g~ t !5md~ t !1g0~ t/t0!12x. ~2!

g0 is the ratio of stress to the unit strain measured atarbitrarily chosen timet0 , m is a Newtonian viscous termandd is the Dirac delta. The stress response to unit amplitsinusoidal deformations can be obtained by taking the F

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rier transform of the step response@Eq. ~2!# and multiplyingby iv, which gives the complex modulusg(v) as

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whereh5tan(x21)p/2 andv is the radian frequency 2p f .g0 and F0 are scale factors for stiffness and frequency,spectively, andG denotes the gamma function.g0 and mdepend on bead-cell geometry.h has been called the structural damping coefficient@25#. The elastic modulusg8 cor-responds to the real part of Eq.~3!, which increases for allvaccording to the power-law exponent,x21. The loss modu-lus g9 corresponds to the imaginary part of Eq.~3! and in-cludes a component which also increases as a powerwith the same exponent. Therefore, this component ifrequency-independent fraction (h) of the elastic modulussuch a direct coupling of the loss modulus to the elamodulus is the characteristic feature of structural dampbehavior@25#. The loss modulus also includes a Newtoniviscous term,ivm, which turns out to be small except avery high frequencies. At low frequencies, the loss tangenhapproximatesh. In the limit that x approaches unity, thepower-law slope approaches zero,g8 approachesg0 , andhapproaches zero. In the limit thatx approaches 2, the powelaw slope approaches unity,g9 approachesm, and h ap-proaches infinity. Thus, Eq.~3! describes a relationship between changes of the exponent of the power law andtransition from solidlike (x51,h50) to fluidlike (x52,h5`) behavior.

In the case of human airway smooth muscle~HASM!cells we converted the modulig8 andg9 ~in units of Pa/nm!and viscositym ~in Pa s/nm! into absolute moduliG8 andG9~in Pa! and viscosityM ~in Pa s! using a geometric conversion factor,a, as described above@17,26#. Thus, Eq.~3! canbe reexpressed in the usual units of elastic moduli andcosity with the transformationsG05ag0 , andM5am. Thisconversion does not impact the findings presented below,has only been introduced so that our measurements cacompared with those of others. For other cell types wenot determineda and thus present our unconverted data (g8,g9, andm! only.

Cells. We measured the rheology of five different cetypes: HASM cells, human bronchial epithelial~HBE! cells,mouse embryonic carcinoma cells~F9!, mouse pulmonarymacrophages~J774A.1!, and human neutrophils. HASMcells were harvested from human trachea@27# and culturedas described in@28#. Confluent cells in passage 4–7 weserum-deprived for 24 h. Cells were harvested with trypand plated at 37 °C for 3 to 6 h in plastic wells~6.4 mm,96-well Removawells, Immunlon, IL! at a density of 20 000cells/well. The wells were coated with collagen I~Cohesion,Palo Alto, CA! at a density of 500 ng/cm2. Approximately10 000 beads were added to an individual cell well 30 mprior to measurements of cell mechanics. Within one or tminutes, most beads sunk to the bottom of the well and win contact with the apical surface of the cells. Before adddrugs or—if no drugs were added—before placing the w

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into the microscope stage, the wells were washed twice wmedium to remove unbound beads. Except where notederwise, the experimental protocol for all cell types was idetical to that of HASM cells. HBE cells~Clonetics, San Di-ego, CA! isolated from the trachea and central airways wexpanded to passage 4 in bronchial epithelial growth me~CC-3170, Clonetics!. F9 cells~kindly supplied by WolfgangGoldmann, Harvard Medical School, Boston, MA! were cul-tured as described in@29#. J774A.1 macrophages originatefrom a BALB/c/NIH mouse@30# and were obtained from theGerman Collection of Animal Cell Cultures~Tumorbank,DKFZ Heidelberg, Germany!. J774A.1 cells were plated fo24 h in culture medium, followed by 24 h of bead incubatioNeutrophils were isolated from human blood obtained froconsenting healthy adult volunteers by venipuncture intovol of sodium citrate anticoagulant~Sigma!, using Neutro-phil Isolation Media~NIM; Cardinal Associates, Santa FeNM!. The resulting cell preparation contained.95% neutro-phils as assessed by hematoxylin and eosin staining. Neuphils were suspended in Hanks Balanced Salt Solutionallowed to adhere on a glass surface for 10 min, followed10 min of bead incubation. We performed all proceduwith endotoxin-free solutions and completed all experimewithin 3 hours of blood collection.

Protocol. The protocol for the twisting field applicationwas as follows: We started with one cycle at 0.3 Hz, flowed by three cycles at 0.01 Hz, five cycles each at 0.0.1, 0.3, and 0.75 Hz, 80 cycles each at 4.2 and 10 Hz,cycles at 30 Hz, 800 cycles at 100 Hz, 4800 cycles at 3Hz, and 28 800 cycles at 1000 Hz. We then decreasedfrequency, mirroring the protocol around 1000 Hz. The land the first 0.3 Hz cycle were not included in the data anasis. HBE cells, neutrophils, and macrophages were proonly at frequencies above 0.03 Hz. Drugs were addedminutes prior to measurements of cell mechanics. Measments were not repeated on the same wells under diffedrug treatment conditions.

Reagents. Tissue culture reagents and drugs used in tstudy were obtained from Sigma~St. Louis, MO, USA!, withthe exception of Trypsin-ETDA solution, which was puchased from Gibco ~Grand Island, NY!. Histamine,N-formyl-methionyl-leucylphenylalanine ~FMLP!, andN6,28-O-dibutyryladenosine 38,58-cyclic monophosphate~DBcAMP! were dissolved in distilled water at 0.1M . Cy-tochalasinD was dissolved in DMSO at 0.1M . All reagentswere frozen in aliquots, thawed on the day of use, andluted in media.

Statistics. Equation~3! was fit to the data. In doing so, whad three objectives. The first was to estimate the parameg0 , F0 , x, and m. The second was to assess the extenwhich changes in these parameters with defined interventwere statistically significant. The third objective was to etablish the extent to which the number of free parametcould be reduced. In particular, stiffness data for differeexperimental conditions from cells of the same type seemto indicate a value ofg0 and F0 that could be regarded ainvariant with experimental interventions. Similarly, the frition data seemed to indicate that the viscosity parametemwas nearly invariant with experimental interventions. As w

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We used two complementary statistical analyses. Theand simplest used as a starting point pooled data in whiceach frequency we considered only the median value ogacross all cells for a given cell type and treatment conditiIn the second approach, we used a mixed-effects anathat deals explicitly with the measured frequency and dresponse across all cells within a given cell type.

Pooled data analysis. We compared four models, eachwhich was based on Eq.~3!. The four models differed only inregard to which parameters were held to be common amthe conditions, and which were allowed to differ. In the fimodel all parameters~x, g0 , F0 , and m! were free in thatthere were no constraints across treatments. The semodel constrainedg0 and F0 to be constant across treaments, whilex and m were treatment-dependent. The thimodel further constrained the viscous coefficientsm to beequal among treatments and allowed onlyx to vary. Thefourth model further constrainedF0 to be equal among alcell types, withF052.143107 rad/s.

We observed that the variability ing between beads atgiven frequency and treatment was approximately proptional to the magnitude ofg. This suggests using the logarithm of the complex stiffness measurements for data ansis and modeling. Furthermore, this approach was justibecause the variability among cells has been shown todistributed in a log normal fashion@14#. We used least-squares minimization to evaluate the parameters in theExplicitly, for each model we minimized the squared diffeences between the logarithm of the complex data andlogarithm of the complex model, summed over all frequecies and all treatment conditions; thus we minimized wregard to the sum of the logarithmic magnitude ofg @the realpart of log(g)] and the phase@the imaginary part of log(g)].We compared these models against one another breduction-in-variance F-test. Accordingly, the fractionalduction in the sum of the squared residuals~S!, suitably nor-malized by the appropriate degrees of freedom (df), is anF-statistic, which we used to determine whether there wastatistically significant difference in the fits between the fomodels:

F II2I5~SII2SI!/~df

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SI/dfI , ~4!

where the superscript I refers to the model with the greanumber of parameters, and superscript II refers to the mowith the reduced set of parameters. If the observed F-statexceeded a value corresponding top,0.05, we concludedthat the model with the greater number of parameterssignificantly better than the other.

Mixed-effects model. To estimate confidence intervals othe parametersx, g0 , F0 , andm, we performed a nonlineamixed-effects analysis using the statistical analysis progR ~version 1.6.0!. Equation~3! was fit to the logarithm of thecomplex modulus of the entire data set for a given cell tythis data set consisted of oneg8 and oneg9 value for each

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RESULTS

Bead movement

Lateral bead movement lagged slightly behind the iposed torque@Fig. 2~a!#. When bead displacement was ploted versus torque, elliptical loops resulted@Fig. 2~b!#. Beaddisplacement amplitude in response to sinusoidal twistvaried widely between cells according to a log-normal dtribution that has been analyzed in detail elsewhere@14#, andresulted in a correspondingly wide distribution ofg8 andg9between cells as described in more detail below. The phangle between torque and bead displacement~or equivalentlythe hysteresivityh, which is the tangent of the phase angle!,however, had a comparatively narrow distribution~see be-low!.

When we increased twisting frequency while keepitorque amplitude constant, bead displacement amplitudecreased appreciably, whereas the phase angle changedtively little @Fig. 2~b!#.

Linearity

The relationship between torque amplitude and beadplacement amplitude was linear. When we varied the spec

FIG. 2. ~a! Specific torqueT ~solid line! and lateral displacemend ~filled circles connected by a solid line! vs time in a representativebead. Bead displacement followed the sinusoidal torque witsmall phase lag. The filled circles indicate when the image andacquisition was triggered, which was 16 times per twisting cyc~b! Loops of lateral bead displacement vs specific torque of a rresentative bead at different frequencies. With increasing frequedisplacement amplitude decreased.

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torque amplitude between 1.8 and 130 Pa~at a twisting fre-quency of 0.75 Hz!, bead displacement amplitude changedproportion, with the median amplitude increasing from 3259 nm; the phase lag between torque and bead displaceremained constant. Thus, over a wide range of spectorque amplitudes and bead displacement amplitudes,g8 andg9 were constant, implying linear mechanical behavior ofcells in this range~Fig. 3!. We observed similar behavior og8 and g9 at frequencies of 0.1 Hz and 10 Hz~data notshown!. However, for bead displacement amplitudes in ecess of 500 nm, harmonics became evident, and bothg8 andg9 tended to increase, indicating the emergence of nonlinbehavior. We eliminated beads from subsequent analwhen the amplitude of higher harmonics exceeded 18%the fundamental amplitude. Depending on the cell type,was the case in about 2% to 20% of the beads. This approis in keeping with that of Schmidtet al. and Choquetet al.,who reported that fibronectin-coated microbeads couldsorted into two distinct populations: 80% of all beads apeared to be tightly bound to cytoskeletal structures, whe20% of the beads were loosely bound and could be eamoved in a laser trap@32,33#.

HASM cells: Control conditions

Figure 4 shows the relationship ofG8 andG9 versus fre-quency for HASM cells under control conditions. Each dapoint represents the median value of 256 cells; the variabbetween cells is addressed below. Throughout the frequerange studiedG8 increased with increasing frequency,f, ac-cording to a power law,f x21. Because the axes in Fig. 4 alogarithmic, a power-law dependency appears as a straline with slopex21. We argue below that if the CSK behaves as a soft glassy material, thenx would correspond toan effective noise temperature of the matrix.

The power-law exponent ofG8 was 0.20 (x51.20), indi-cating only a weak dependency ofG8 on frequency.G9 wassmaller thanG8 at all frequencies except at 1 kHz. LikeG8,G9 also followed a weak power law with nearly the samexponent at low frequencies. At frequencies larger thanHz, however,G9 exhibited a progressively stronger frequency dependence, approaching but never quite attaini

FIG. 3. g8 andg9 vs specific torqueT. g8 andg9 were measuredin 537 HASM cells atf 50.75 Hz. Specific torque amplitudesTvaried from 1.8 to 130 Pa.g8 andg9 were nearly constant, implyinglinear mechanical behavior of the cells in this range. Error bindicate6 one standard error.

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power-law exponent of 1, which would be characteristic oNewtonian viscosity.

HASM cells: Challenge

When cells were activated with the contractile agonist htamine ~100 mM!, G8 increased but still exhibited a weapower-law dependence on frequency;x fell slightly to 1.17~Fig. 5!. When cells were relaxed with DBcAMP~1 mM!,G8 decreased, andx increased to 1.28. When the actin ctoskeleton of the cells was disrupted with cytochalasinD ~2

s

FIG. 4. G8 andG9 ~median of 256 HASM cells! under controlconditions measured at frequencies between 0.01 Hz and 1000The solid lines were obtained by fitting Eq.~3! to the data.G8 andG9 ~in units of Pa! were computed from the measured values ofg8and g9 ~in units of Pa/nm! times a geometric factora of 6.8 mm~see Methods!. G8 increased with increasing frequency,f, accordingto a power law,f x21, with x51.20.G9 was smaller thanG8 at allfrequencies except at 1 kHz. At frequencies below 10 Hz,G9 alsofollowed a weak power law with nearly the same exponent asG8; above 10 Hz the power-law exponent increased andproached unity.

FIG. 5. G8 vs f in HASM cells under control conditions~j, n5256), and after 10 min treatment with the contractile agonist htamine (1024 M) ~L, n5195), the relaxing agonist DBcAMP(1023 M) ~l, n5239) and the actin-disrupting drug cytochalasD (231026 M) ~h, n5171). At all frequencies, treatment withistamine causedG8 to increase, while treatment with DBcAMPand cytoD causedG8 to decrease. Under all treatment conditionG8 increased with increasing frequency,f, according to a powerlaw, f x21. x varied between 1.17~histamine! and 1.33~cytoD!. AdecreasingG8 was accompanied by an increasingx, and vice versa.Solid lines are the fit of Eq.~3! to the data. Surprisingly, these lineappeared to cross at a coordinate close to@G0 ,F0/2p#, well abovethe experimental frequency range. According to Eq.~3!, an approxi-mate crossover implies that in the HASM cell the values ofG0 andF0 were invariant with differing treatment conditions.

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mM!, G8 decreased further, andx increased to 1.33. Remarkably, theG8 data defined a family of curves that, when etrapolated, appeared to intersect at a single value (G0) at avery high frequency (F0) ~Fig. 5!.

With all drug treatments,G8 andG9 tended to change inconcert. The relationship betweenG9 and frequency re-mained a weak power law at lower frequencies, andpower-law exponent ofG9 changed in concert with that oG8. At the highest frequencies, the curves ofG9 versus fre-quency for all treatments appeared to merge onto a siline with a power-law exponent approaching unity~Fig. 6!.

Structural damping equation: Reduction of variables

When we fitted the structural damping equation tomedian of the complex moduli of HASM cells, we obtainefor each of the four treatment conditions a parameter setx, G0 , F0 , and M. The goodness of the fit was excelle(r 250.997, Table I!. Nonetheless, two observations indcated that the number of free parameters could be reduwithout compromising the goodness of fit. First, we noticthat viscosity values~M! were similar for all treatments~Table II, model 1!, although cells after treatment with cy

FIG. 6. G9 vs f in HASM cells under control conditions~j, n5256), and after 10 min treatment with histamine (1024 M) ~L,n5195), DBcAMP (1023 M) ~l, n5239), and cytochalasinD(231026 M) ~h, n5171). At all frequencies, treatment with histamine causedG9 to increase, while treatment with DBcAMP ancytoD causedG9 to decrease. Under all treatment conditions,G9increased at frequencies below 10 Hz according to a powerf x21, with exponents that were similar to that of the correspondG8 data~Fig. 5!. Above 10 Hz the power-law exponents increasand approached unity for all treatments; theG9 curves merged ontoa single relationship.

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tochalasinD expressed a somewhat smaller viscosity. Sond, the apparent common intersection of theG8 versusfcurves at high frequencies~Fig. 5! suggested a single valuG0 ,F0 for all treatments. As explained below, this observtion of a common intersection, or fixed point, turns outhave important consequences.

In order to test whetherG0 , F0 , and M were indeedinvariant with drug treatment, we analyzed the residual vaances of the fit of Eq.~3! to the data. Specifically, we evaluated how well the structural damping equation@Eq. ~3!# ac-counted for the pooled~median! data when either allparameters~x, G0 , F0 , andM! were free across drug treament~model 1!, whenG0 andF0 were constrained to be thsame for all drug treatments~model 2!, whenG0 , F0 , andM were all constrained to be the same for all drug treatme~model 3!, or when, in addition to the constraints of modelF0 was constrained to be the same for all cell typ~model 4!.

The residual variances of the fit of model 1 and modeto the pooled data were not significantly different from eaother (p50.086, Table I!. Model 2 is thus to be preferredover model 1 because it can account for the data with feparameters. This finding supports the notion thatG0 andF0were invariant with drug treatment. The residual varianc(r ss) of the fit of model 3 and 4 to the data were slightly bsignificantly (p,0.001) larger than those of models 1 orhowever~Table I!. This indicates that the Newtonian viscoity ~M! term changed in response to drug treatment. Thchanges were small compared to drug-induced changeG8 and G9 at low frequencies, however. Moreover, as eplained below, drug-induced changes ofm in cell types otherthan HASM did not reach statistical significance and, in tcase of treatment with cytochalasinD, were inconsistent between cell types~Table III!. Thus, for all practical purposesm could be regarded as being approximately invariant wdrug treatment. Consequently, we were able to fit the strtural damping equation to the entire data set depicted in F6 and 7~solid lines! using only a single value ofG0 , a singlevalue ofF0 , and a single value ofM, as shown in Table II.Despite this profound reduction in the number of parametthe goodness of fit remained excellent (r 250.993). Thus, allchanges inG8 and G9 that occurred with changes of frequency and in response to drug treatment were accounteby changes ofx alone. A comparison between models 1–using a mixed-effect analysis of the bead-by-bead~non-pooled! data gave similar results~Table I, right!. The good-ness of fit (r 2) of Eq. ~3! to the data remained high for a

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TABLE I. Statistical evaluation of the fit of the structural damping equation@Eq. ~3!# to pooled~median!data and data from individual beads~mixed effects! in HASM cells. Models 1–4 denote different constrainof Eq. ~3! ~see text!.

median mixed effects

model 1 model 2 model 3 model 4 model 1 model 2 model 3 mode

r2 0.997 0.997 0.993 0.993 0.948 0.948 0.947 0.947rss @Pa2/nm2# 0.216 0.231 0.511 0.527 1028 1028 1047 1047

df 76 78 81 82 18928 18930 18933 18934pa 0.082 ,0.0001 ,0.0001 0.001 ,0.0001 ,0.0001

aCompared to model 1.

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models (r 2.50.947), although model 1 gave a slightly beter fit which, because of the very large number of data po(n518 942), reached statistical significance (p,0.001).

We estimated the confidence intervals of the fitted paraeters from Eq.~3! using a nonlinear mixed-effect analys~Table III!. The 95% confidence interval forG0 ranged from4.91 Pa/nm to 7.39 Pa/nm, and that ofF0 ranged from9.383108 rad/s to 4.893107 rad/s~Fig. 7!. The 95% confi-dence interval forx was 0.2% around thex estimates, and the95% confidence interval forM was 6% around theM esti-mates. The parameter estimates obtained from the nonlimixed-effect analysis were in all cases close to the valobtained from the fit to the pooled data.

Other cell types

We measured the dependence ofg8 andg9 on frequencyin four other cell types: F9 mouse embryonic carcinocells, human bronchial epithelial cells, J774A.1 mouse m

TABLE II. Parameter estimates~mixed effects analysis! of thefit of the structural damping equation@Eq. ~3!# to the data fromindividual beads on HASM cells. Models 1–4 denote different costraints of Eq.~3! ~see text!.

model 1 model 2 model 3 model 4

G0 ~kPa! 40.96 51.40 41.47F0 ~rad/s! 2.143107 5.243107 2.143107

x control 1.205 1.204 1.206 1.20x hist 1.165 1.166 1.171 1.17x DBcAMP 1.279 1.277 1.281 1.282x cytoD 1.327 1.329 1.319 1.32M control ~Pa s! 0.68 0.68M hist ~Pa s! 0.92 0.91M DBcAMP ~Pa s! 0.73 0.74M cytoD ~Pa s! 0.41 0.40M common~Pa s! 0.60 0.60

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rophages, and human neutrophils~Fig. 8!. With the few ex-ceptions noted below, these cell types behaved qualitativmuch as did the HASM cells, although the numerical valuof the fitted parameters differed~Table III!. In all cell types,and regardless of intervention,g8 increased according toweak power law with exponentx21, althoughx differedamong cell types.

Similarly, in all cell typesg9 also followed a weak powelaw at low frequencies. As in the HASM cells, at the highefrequenciesg9 approached a Newtonian-type viscous behior, and the curves ofg9 versus frequency from differentreatments merged at the highest frequencies. Further, incell types and under all interventions,g8 was larger thang9for frequencies below 300 Hz.

In all cell types, an increase ing8 induced by drug treat-ment was accompanied by an increase ofg9 and a decreasein x. Conversely, a decrease ing8 was accompanied by adecrease ofg9 and an increase ofx. For example, whenneutrophils were stimulated with FMLP~10 nM!, g8 andg9increased, andx fell slightly. All cell types responded simi-larly to treatment with the actin-disrupting drug cytochalasD ~2mM!: g8 andg9 fell considerably, andx increased.

Generally, the power-law exponent ofg9 at low frequen-cies was similar to that ofg8, but we observed some systematic departures from that behavior. In three cell types~humanbronchial epithelial cells, macrophages, and neutrophils!, wefound that the behavior ofg9 under baseline conditions afrequencies below 10 Hz departed systematically frompower-law behavior implied by Eq.~3!; in those cell typesg9became nearly frequency-independent.

In all cell types, theg8 versusf curves for different drugtreatments intersected at a very high frequency (F0). Theestimated values forF0 differed between cell types, but th95% confidence intervals ofF0 were wide and overlapped~Fig. 7!. These wide confidence intervals arose becauserange of power-law slopes was small. Accordingly, the strtural damping equation@Eq. ~3!# was able to fit the data in al

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TABLE III. Parameter estimates and 95% confidence intervals from a mixed effects analysis of thethe structural damping equation~Eq. 3, constraint model 2! to the data from individual beads on different cetypes.

F9 HASM HBE Macro Neutro

g0 ~Pa/nm! 0.520.261.04 6.024.91

7.39 2.862.083.93 2.611.30

5.25 2.501.145.48

f0 ~rad/s! 3.2031061.80* 1045.70* 108

2.1431079.38* 1064.89* 107

1.0931063.00* 1053.92* 105

6.8631045.47* 1038.60* 105

1.9031075.14* 1057.05* 108

x control 1.1951.1901.200 1.2041.202

1.206 1.1731.1691.178 1.2001.191

1.209 1.1861.1781.194

x FMLP 1.1571.1481.166

x histamine 1.1661.1631.189

x DBcAMP 1.2771.2751.280

x cytoD 1.3021.2941.310 1.3291.326

1.332 1.3191.3151.323 1.3381.327

1.348 1.2521.2461.259

m controla 0.470.430.51 1.000.94

1.07 0.640.590.69 1.951.71

2.22 0.630.560.71

m FMLPa 0.820.730.93

m histaminea 1.341.251.44

m DBcAMPa 1.091.041.14

m cytoDa 0.650.600.70 0.590.56

0.62 0.550.510.59 3.022.71

3.35 0.860.920.81

aIn 1024 Pa s/nm.

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cell types using only a single value forF0 (2.143107 rad/s) without appreciably decreasing the goodnesfit ~Table I, model 4!. The stiffnessg0 differed between celltypes, as did the high-frequency viscositym ~Table III!.Among all cell types, macrophages turned out to havehighestm value, whereas mouse embryonic carcinoma c~F9! had the lowestg0 value andm value. As in HASM cells,m changed only little with drug treatment~Table III!. In allcell types, therefore, the relationships ofg8 and g9 versusfrequency and their changes with drug treatments werefit by Eq. ~3! ~model 4! using only one free parameter,x ~Fig.8 solid lines!.

Master curves: Scaling the data

In order to compare responses between drug interventand between differing cell types, we normalized the datafollows. We defined a normalized cell stiffnessGn as g8measured at 0.75 Hz~an arbitrary choice! divided by g0 ,whereg0 served as an internal stiffness scale for eachtype; this normalization causes the geometric factora to can-cel out. To normalize the cell’s frictional properties, we usthe ratiog9/g8 ~the hysteresivityh! at 0.75 Hz, which alsocauses the geometric factora to cancel out. We estimatedxfrom the fit of Eq.~3!, model 1~no constraints!, to the pooled~median! data. We then plottedGn versusx and h versusx~Fig. 9, black symbols!.

FIG. 7. 95% confidence region ellipses forF0 ~in Hz! andg0 infive different cell types as estimated with a mixed-effects analyThe confidence intervals forF0 were very wide and overlappebetween different cell types.

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In HASM cells, drugs that increasedx caused the normalized stiffnessGn to decrease@Fig. 9~a!, black symbols# andthe hysteresivityh to increase@Fig. 9~b!, black symbols#.The normalization that leads to these relationships is moindependent. Nonetheless, these relationships fell close topredictions from the structural damping equation~solid blacklines!.

Surprisingly, the normalized data for the other cell typcollapsed onto the very same relationships that we foundHASM cells ~Fig. 9!. In all cases, drugs that increasedxcaused the normalized stiffnessGn to decrease and hysteesivity h to increase. These relationships thus representversal master curves in that a single parameter,x, defined theconstitutive elastic and frictional behaviors for a varietycytoskeletal manipulations, for five frequency decades,for diverse cell types.

Variability across individual cells

The distribution ofg8 among individual cells varied ovea range spanning more than two orders of magnitude~geo-metric SD52.1, Fig. 10!; because of the large number ocells measured, the geometric standard error was withthan 0.05 logs smaller than the symbol size in Figs. 4–8.g9varied over a similarly wide range, butg9 always remainedtightly coupled tog8 such that their ratioh varied little be-tween cells (SD,0.1) ~Fig. 11!. Thus, insofar as geometricafactors contributed to differences ofg9 andg9 across cells,these geometrical factors canceled out when calculatingteresivity, which in any given cell is the ratiog9/g8.

Within individual cells,g8 andg9 increased with twistingfrequency much like the behavior seen in the populationerage. Single cell behavior was generally well matchedthe structural damping equation@Eq. ~3!#, with an averagecorrelation coefficient (r 2) of 0.95 in the case of HASMcells ~Table I, mixed effects!. In agreement with this observation, we found that not only in population averages balso in individual cellsh varied withx according to the re-lationship expressed in Eq.~3! ~Fig. 11!.

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FIG. 8. g8 and g9 vs frequency in five different cell types under control conditions~j!, and after 10 min treatment with histamin(1024 M) ~L!, DBcAMP (1023 M) ~l!, cytochalasinD (231026 M) ~h!, and FMLP (1028 M) ~n!. The solid lines are the fit ofx, m,andg0 @according to Eq.~3!, constraint model 4# to the data. In all cells,F0 was set to 2.143107 rad/s, andm was set to be constant withina given cell type~i.e., independent of drug treatment!. In all cell types, the relationships ofg8 andg9 vs frequency and their changes witdrug treatments were well fit by Eq.~3! using only one free parameter,x. However, small but systematic discrepancies between theg9 dataand the fit are noted in HBE cells, J741.A cells, and neutrophils~see text!.

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FIG. 9. ~Color! Normalized stiffnessGn ~left! and hysteresivityh ~right! vs x of HASM cells~black,n5256), human bronchial epitheliacells ~blue, n5142), mouse embryonic carcinoma cells~F9! cells ~pink, n550), mouse macrophages~J774A.1! ~red, n546), and humanneutrophils~green,n542) under control conditions~j!, treatment with histamine~h!, FMLP ~l!, DBcAMP ~n!, and cytochalasinD ~m!.x was obtained from the fit of Eq.~3!, model 1~no constraints!, to the pooled~median! data. Drugs that increasedx caused the normalizedstiffness Gn to decrease and hysteresivityh to increase, and vice versa. The normalized data for all types collapsed onto therelationships. The structural damping equation by Eq.~3! is depicted by the black solid curves: lnGn5(x21)ln(v/F0) with F052.143107 rad/s, andh5tan„(x21)p/2…. Error bars indicate6 one standard error. Ifx is taken to be the noise temperature, then these dsuggest that the living cell exists close to a glass transition and modulates its mechanical properties by moving between glassy sta‘‘hot,’’ melted, and liquidlike, and states that are ‘‘cold,’’ frozen, and solidlike. In the limit thatx approaches 1 the system behaves as an idHookean elastic solid, and in the limit thatx approaches 2 the system behaves as an ideal Newtonian fluid@Eq. ~3!#.

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Surprisingly, the value ofx derived from any given beadunder a given treatment condition covaried with the valueg8 ~Fig. 10!. Cells with smallerg8 tended to have lowerxvalues, and cells with largerg8 tended to have higherx val-ues. This behavior was different from the behavior seenpopulation averages, where an increase ofx in response todrug challenge was consistently associated with a decreag8 ~Fig. 10; population averages depicted by large symbo!.

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DISCUSSION

We have measured the elastic (g8) and the frictional (g9)moduli of five different cell types over five frequency dcades and with a variety of interventions. Our principal finings are as follows. Bothg8 andg9 increased with probingfrequency according to a weak power law. This frequendependence was well described by the structural dampequation@Eq. ~3!#. Data from all cells, all frequencies, and aexperimental conditions could be scaled in such a way

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FIG. 10. ~Color! Relationship betweeng8 vs xin HASM cells under control~j! conditions, andafter treatment with histamine~L!, DBcAMP~L!, and cytochalasinD ~h!. g8 of individualbeads were measured at 1 kHz.x was determinedfrom the exponent of a power-law regressiontheg8 data of individual beads measured betwe0.01 Hz and 1 kHz. Each small symbol represethe data from one bead, while the large symborepresent the median of all beads measured unone condition. The thin solid lines are the expnential regression through the data for individubeads. The correlation coefficient (r 2) for the re-gression was 0.48 under control conditions, a0.68 under all other conditions. The correlatiocoefficients decreased when the relationshwere analyzed forg8 values measured at lowefrequencies than 1 kHz, but remained significafor g8 values measured above 0.3 Hz (p,0.05). The thick solid line is the exponentiaregression through the median values.

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they collapsed onto two master relationships, one describstiffness and the other friction; to a reasonable approximtion, changes of stiffness and friction induced by pharmalogical interventions were accounted for solely by changeonly one parameter, namely the power-law exponentx. Therheology of living cells in culture was seen to differ inqualitative fashion from the rheology that has been repofor reconstituted actin gels containing specific accrosslinkers. These latter simplified systems have playedimportant role in developing our understanding of the funtions of specific proteins@34,35# but at the same time, thefail to capture some essential aspects of the biology.

In this section, we critique the methods that were eployed and contrast our findings against those reported inliterature. We then go on to discuss time-scale invariancethe collapse of all data onto master curves, wherein the msured exponentx is shown to play a major unifying roleleading in turn to a major empirical simplification. These aexperimental findings, they are model-independent, andstand independently of interpretation or putative mechani

With regards to mechanism, we conclude by showing tthe mechanical behavior reported here satisfies many oempirical criteria that define the class of soft glassy mater@7#. If the CSK is indeed a soft glass, then the theory of sglassy rheology~SGR! would identify the parameterx asbeing an effective temperature of the matrix, distinct frothe thermodynamic temperature. Moreover, SGR theleads to a novel mechanistic perspective on the natureprotein-protein interactions and their contribution to ingrated mechanical functions of the intact living cell, incluing the ability of the cytoskeleton to deform, flow, and rmodel.

Critique of methods

The data presented here were not entirely general.studies were restricted to only five types of adherent cand to small sinusoidal stresses and deformations. As a pwe used beads of only a single size~4.5 mm! that werecoupled to the cytoskeleton via integrins. The binding b

FIG. 11. Relationship between hysteresivityh vs x in HASMcells under control conditions. Each symbol represents thefrom one bead (n5256). h was determined from the ratiog9/g8 ofindividual beads measured at 0.75 Hz.x was determined from theexponent of a power-law regression to theg8 data of individualbeads measured between 0.01 Hz and 1 kHz. The solid line resents the relationship betweenh andx as predicted by the structuradamping equation@Eq. ~3!#: h5tan„p/2(x21)….

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tween integrin receptors and ligands on the bead may hinduced changes of cell mechanics. Further, these stuwere limited in that we did not discriminate mechanical rsponses based on cell size, shape, or location of the beathe cell, all of which have been shown to influence cell mchanics@11,32,38,37#. It is possible, therefore, that our results might be not so much a reflection of intrinsic propertof the cytoskeleton but instead are peculiar to the probewe used, or are attributable to receptor kinetics or membrmechanics. The evidence described below argues agthose possibilities, however.

Binding of RGD-coated beads with integrin receptorsthe cell surface has been shown to trigger a host of cellresponse that are likely to have time-dependent effectscell mechanics, including but not limited to focal adhesiformation, sodium-proton antiporter activation, increasecyclic adenosine monophophate levels, protein kinasetranslocation, and a multitude of gene activation@11,32,38–40#. These responses can be considered as entirely phologic, however, in that they closely mimic the responsobserved when these cells adhere to and spread on dcoated with integrin ligands@38#. To separate mechanicaresponses that may have been triggered by ligand bindevents from measurements of passive cell mechanics,employed four strategies. First, we investigated the effectdifferent bead-coatings, including extracellular matrix prteins ~vitronectin and collagen!, activating antibodies, andimportantly, nonactivating antibodies to different integrsubunits~data not shown!. Despite the differences in the signaling cascades that may be activated by these diffeligands, in each case power-law behavior prevailed~M. Puigde Morales, personal communication!. Therefore, the exis-tence of power-law behavior did not depend on the detailsthe coupling of the bead to the cell. Second, we carefucontrolled the timing so that measurements were perform30 minutes after bead addition~24 h in the case of macrophages!. Third, synchronous detection of bead movement wused to distinguish periodic bead motions caused bymagnetic field from any nonsynchronous bead movemeguaranteeing that we measured purely mechanical respoto the mechanical perturbations of the cells. Finally, eameasurement on each bead was performed at two time pso that the effects of twist versus the effects of time coulddiscerned.

We found that cell rheology greatly changed after trement with drugs that altered actin polymerization or acmyosin bridge formation. In particular, we found that in rsponse to the contractile agonist histamine smooth mucells increased stiffness by 1.4-fold, in response to DBcAMdecreased stiffness by 3.4-fold, thus confirming that thcells express a considerable degree of contractile tonbaseline, and in response to cytochalasinD decreased stiff-ness by 8.3-fold~at 0.75 Hz!, Other cell types responded tcytochalasinD with a decrease in stiffness of at least threfold. Neutrophils increased stiffness in response to FMLP1.4-fold. In all cases cell stiffness changed in the directone would expect based on the well-characterized effectthese drugs on the cytoskeleton. Moreover, these resultsin good agreement with a large number of studies that re

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the effect of these drugs on cell mechanics but emplodifferent measurement techniques, including indentation wa glass fiber@41#, scanning atomic force microscopy@42#,and measurement of the transit time of cells through calary pores@43#. In a separate set of experiments we also ubeads coated with acetylated low-density lipoprot~acLDL!; those beads bind to scavenger receptors whichnot form focal adhesion complexes@11,40#. We found thatbaseline values ofG8 andG9 measured with acLDL-coatebeads were smaller by fivefold compared with beads boto integrins, and demonstrated an attenuated response totractile agonists~data not shown!.

Taken together, these findings support the conclusioncell stiffness and friction as reported here predominantlyflect the mechanical properties of cytoskeletal structuThese data cannot be explained by receptor-ligand dynaas would be the case if there were repetitive peeling ofbead away from the cell surface and subsequent reattachwith each bead oscillation. Neither can these data be reaexplained by changes in cell membrane mechanics. Rathese data are more consistent with the notion that the bbinds avidly to the cytoskeleton via focal adhesions, andthe cytoskeleton is deformed as the bead rotates~Fig. 1!.This view is in accord with those of Schmidtet al. @32# andPlopper et al. @40#, who showed that beads bound to tapical cell surface viab1 integrins are tightly connected tdeep cytoskeletal structures via a series of protein interdiates. Moreover, several studies have demonstrated thastiffness of smooth muscle cells probed with magnetic bebound tob1 integrins can be modulated over a wide rangeinhibiting or activating myosin motors@13,14,28,44–47#.

Variability. Values for stiffness (g8) and friction (g9) be-tween cells of the same type and under the same treatmvaried over more than two orders of magnitude~geometricSD52.1, Fig. 10!. This variability may have been caused bdifferences in the attachment geometry as reflected byscaling factora, local variations of cell mechanics, proximity of neighboring cells, variations in bead size or magnemoment. Nonetheless, we found that the behavior in invidual cells of every cell type and under all treatment contions conformed well to the structural damping equation@Eq.~3!#, regardless of the magnitude ofg ~Tables I–III!. Thevariability of x between cells for a given drug treatment wsmall (SD,0.05).

The value ofx derived from any given bead under a givetreatment covaried with the magnitude of the measured sness, however~Fig. 10!. Beads reflecting higher stiffnestended to exhibit stronger frequency dependence~corre-sponding to higher values ofx! compared to those reflectinsmaller stiffness. We speculate that greater stiffness maindicative of a deeper internalization of the bead into the cbody and a closer association of the bead with the intecytoskeleton, while smaller stiffness may be indicativelittle internalization of the bead and its association maiwith cortical structures@14,17,26#. If true, then the covari-ance betweenx and g8 may be attributable to differencebetween cortical and internal cytoskeletal structurwhereby the cortical cytoskeleton~as probed by marginallyinternalized beads with smallerg8) exhibits smallerx values,

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and the internal cytoskeleton~as probed by deeply internaized beads with largerg8) exhibits largerx values. Below weargue thatx may reflect the degree of metastability of thcytoskeletal matrix. If true, then these data would suggthat the cortical cytoskeleton is more stable than is the innal cytoskeleton.

Linearity. We found that loops of bead displacement vsus specific torque appeared as simple ellipses in most@Fig. 2~b!#, and thatg8 andg9 did not change appreciably aspecific torque amplitude was varied between 1.8 and 130~Fig. 3!. Therefore, cell mechanics could be regarded asing linear over the range of stresses and deformationsployed in this study. We did find departures from linear bhavior in some beads with displacement amplitudes in excof 500 nm, however. These beads caused larger cellularformations around their attachment site, and they rotamore than beads with smaller displacement amplitudes@17#.A nonlinear torque-displacement relationship in those becould arise from three sources: geometric nonlinearitiesduced at large deformations, intrinsic nonlinearities of tmaterial, and the dependence of torque on the cosine ofangle between bead magnetization and the direction oftwisting field @Eq. ~1!# @17,26#. In this study we excluded albeads with large displacement amplitude from further anasis. For the remaining beads, which was more than 80%the total, we found linear responses~Fig. 3! and thus can ruleout nonlinearities of the cell or of the measurement tenique.

The finding of linear or nearly linear cell mechanicsconsistent with reports by Bauschet al., who probed mac-rophages over forces ranging from 500 pN to 2 nN usmagnetic tweezers@48#, by Thoumine and Ott, who imposestrains of over 1.5 onto fibroblasts that were held betweforce plates@49#, and by Alcarazet al., who probed endot-helial cells with atomic force microscopy over indentatioforces ranging from 0.1 nN to 0.9 nN@50#.

It is interesting, however, that the finding of linear mchanical behavior would appear to be in conflict with tview that cells express inherently nonlinear mechanicalhavior, including shear thinning and strain hardening@51,52#.Although this view is widely held, evidence in the literatudemonstrating nonlinearities is scant and can be tracetwo primary sources. First, strain hardening is occasionconfused with the increase in stiffness that is seen in mcell types after they are mechanically stimulated. Thecrease in stiffness is the result of active responses of theto these stimuli, such as strengthening of the focal adhesites after beads attached to integrins were immobilizedoptical traps@33#, cytoskeletal remodeling after exposurefluid shear flow@53#, or contractile activation of cells that arcompressed or stretched between microplates@49#. These ac-tive responses of the cell do not become evident until sevseconds to hours after the onset of the mechanical stimuand are distinct from the passive mechanical responses ocytoskeleton we have measured here. Second, reportsour own laboratory are often cited to support strain harding @11,12,54,55#. We have recently shown, however, ththis apparent strain-hardening is an artifact that comes awhen an external magnetometer is used to measure the

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age rotation of the entire bead population within a cell wcontaining several thousands of beads and a like numbecells. That population inevitably includes a small percentaof beads that are poorly bound to the cells@14,26,56# Be-cause they are loosely bound, these beads can rotate esively even at very low twisting fields, and contribute sustantially to the net magnetic signal. But because they canrotate in excess of 90 degrees, their contribution quicsaturates at a higher twisting field. The mistaken impressof strain hardening arises from an increasing numberloosely bound beads that saturate as the twisting field iscreased. Here, by contrast, we have presented evidenceindividual cells, when probed with firmly attached beaddisplay linear material properties over a wide range of scific torque amplitudes between 1.8 and 130 Pa. Whileresult does not rule out nonlinear material behavior at higstresses or strain rates, we conclude that the apparent shardening previously reported with magnetic twisting cytoetry is an artifact arising from population averages thatclude loosely bound beads@13,26#. This artifact is avoidedby using optical detection of bead motion, which allows usexclude beads from further analysis when they are pobound to the cell, or when they are arranged in clusters oclose proximity to neighboring beads.

Compatibility with previous reports. The power-law de-pendence ofg8 andg9 on frequency reported here is constent with data reported for atrial myocytes and fibroblameasured with atomic force microscopy~AFM!, for pelletsof mouse embryonic carcinoma cells measured with a drheometer, for airway smooth muscle cells measured woscillatory magneto-cytometry, and for kidney epithelcells measured by laser tracking of Brownian motion oftracellular granules@13,36,57–59#. Yamadaet al. also mea-sured cell mechanics before and after microfilaments wdisrupted with Latrunculin A, and obtained two curves ofg8versusf that intersected at a frequency comparable tovalue we report here@36#.

Of particular relevance are data recently reported forman bronchial endothelial cells measured over frequenranging from 0.01 Hz to 100 Hz with atomic force microcopy @50#. The AFM probe used in that study was uncoatand, unlike previous reports using AFM, the measuremewere corrected for the viscous forces of the cell culture mdium acting on the AFM cantilever. These data conformedthe structural damping equation@Eq. ~3!# and were entirelyconsistent with the data reported here. Moreover, frictionthose cells approached the behavior of a viscous fluidfrequency range comparable to our findings.

Taken together, data reported previously in different ctypes, measured with techniques that are fundamentallyferent in regard to probe size, coupling to the cell, and aplitude of deformations, strongly suggest that the resultsported here are not peculiar to the measuremmethodology and instead reflect intrinsic cytoskeletal prerties. Compared with other methods that are availableprobe cell mechanics, the method used here has the partiadvantage to be able to measure responses of large numof cells in a short time and over a wide frequency ranHowever, because the position at which the beads settle

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the cell surface cannot be controlled, our method offers oa limited ability to resolve local variations in cell mechanicresponses.

Time-scale invariance

With the few exceptions highlighted below, in all casreported here cells conformed to the structural dampequation@Eq. ~3!#. Wheng8 and g9 were measured overwide range of frequencies, in the resulting data no spefrequency, or molecular relaxation time, or resonant fquency stood out. Instead,g8 and g9 increased with fre-quency in a featureless fashion following a weak power laf x21. Except for a small additive viscous term that emergonly at high frequencies, mechanical responses were notto any specific frequency scale, and in that respect canregarded as being scale-free. Scale-free dynamics stancontrast with the frequency-dependent properties of commpolymer matrices, whose spectra typically display distinplateaus, shoulders and inflections; these features are intive of transitions from one dominant molecular mechanito another@60#. For example, rubbers show a transition frodominance of chain entropy at lower frequencies, whstiffness is in the MPa range, to dominance of van der Wabonds at higher frequencies, where stiffness is in the Grange. Such transitions do not occur in scale-free mediawhich case molecular relaxation processes in the matrixnot tied to any particular internal time scale or any distinmolecular rate process.

In that connection, the inverse Fourier transform of tstructural damping equation predicts that stress relaxaafter a unit step deformation should follow a power lawtime, t12x @Eq. ~2!#, and that displacements in response tounit step torque~i.e., the creep response! follow a similarpower law, tx21. Experiments in which we have measurebead displacements in response to a step change oftorque have confirmed power-law creep responses in HAcells over more than three decades of time~3 ms to 3.2seconds; G. Lenormand, personal communication!. Power-law stress relaxation, power-law creep, and structural daing behavior have been reported in a variety of inert abiological materials@20–22,25,61–64#.

Most studies of single cell rheology report data that sponly relatively small ranges of time or frequency, typicalone or at most two decades. Almost invariably, authors hfit simple viscoelastic models embodying a small numberdistinct elastic and viscous components to these datanecessarily, these models express only a limited rangecharacteristic relaxation times@11,12,37,48,54,55,65–69#.Associated model parameters were often given physicalterpretations such as cytoplasmic viscosity@10,37,49,69–74#,static elasticity of the cytoskeleton@11,12,49,54,55,75#, ortension, surface elasticity and bending modulus of thecortex @71,72,74,76#. Reported values for the model parameters, however, varied greatly between different visco-elamodels, experimental methods and conditions, and cell tystudied@52,67,68,70,73#.

Scale-free rheological behavior as reported here callsquestion all such models and associated interpretati

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When power-law behavior prevails, the fit of any lumplinear visco-elastic model to data obtained over a limirange of frequency~or time! will result in apparent relaxationtime scales that are determined mostly by the time~or fre-quency! range probed in the experiment, as opposed tomaterial properties themselves@77#. That is to say, whatevelimited range might be probed, one or two time constantsthat range will always be found to fit the available daEquivalently, with data spanning only a limited time or frquency range, it is difficult to distinguish an exponential lafrom a power law. Not surprisingly therefore, those reportswhich cell responses have been measured over a large rof frequency or time have systematically abstained froming simple visco-elastic models to fit their data@36,50,57–59#.

It has been suggested by Buxbaumet al. and Tsaiet al.that the cytoplasm does not behave as a Newtonian fluidexhibits power-law shear thinning, and that measurementcell rheology do not so much depend on the frequencytime scale of the imposed forces or deformations but instdepend on the shear rate@52,73#. To test this possibility, weprobed cells over a large range of shear rates using twoferent protocols; first we changed the deformation amplituwhile maintaining a constant frequency, and secondchanged the frequency while maintaining a constant amtude. We found no shear thinning when the deformation aplitude was changed and the probing frequency was kconstant~Fig. 3!. However, cell rheology did vary accordinto a power law with frequency when the deformation amptude was kept constant~Fig. 4!. Thus, our measurements ocell rheology did not depend on shear rate but instead ontime scale or frequency range over which the shear wasplied.

Other invariants of cell mechanical behavior: Collapse ontomaster curves

Our measurements spanned five frequency decades,different cell types and challenges with a variety of druNonetheless, the mechanical behavior of cells was restrito vary only in a very particular way@Eq. ~3!#, and with onlya small set of options to explain differences in mechanproperties: the power-law exponentx, the viscositym, andscale factors for stiffness (g0) and frequency (F0). A majorfinding of this study is that from those options, the paramex alone was to a reasonable extent sufficient to charactethe changes of both the elastic and the frictional behavio

First, we unexpectedly found that for any cell type theg9curves from different treatment conditions merged at hfrequencies. This indicated that the Newtonian viscosity tem was negligible over most of the frequency range studiand was relatively insensitive to drug treatment. Indeed,fit of Eq. ~3! to the data of a given cell type gave similvalues~Table III!. m did change appreciably in responsecytochalasinD in the case of macrophages and HASM ce~Table III!, albeit in different directions. But even in thoscell types, the fit of the structural damping equation todata remained excellent even when we used a single v

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for m that was independent of drug treatment~Fig. 8, TablesI and II!.

Second, for any given cell type we unexpectedly fouwhat appeared to be a common intersection, or fixed poof the g8 versus frequency curves at a very high frequenindicating thatg0 andF0 were invariant with different drugtreatments~Fig. 5!. Statistical analysis of the pooled datacells with multiple treatments~HASM and neutrophils! re-vealed that theg8 curves from multiple treatment conditionindeed could be accounted for with common values forg0andF0 . The value ofF0 differed between cell types, butmixed effect analysis of the unpooled data revealed that95% confidence intervals of theF0 estimates were very wideand overlapped between cell types~Table III, Fig. 7!. Thus,to a reasonable approximation the data for all cell typcould be accounted for with only a single value forF0 .

The parameterg0 differed between cell types, as did thviscosity parameterm ~Table III!. These differences could ba reflection of a number of factors including cell type, spcific differences in the attachment properties betweenand bead, different rheological properties between cell typdifferences in cell shape, height, cytoskeletal organizatiamount of cytoskeletal or crosslinking proteins, activationmotor-proteins, etc. Whatever the origin of the differencbetween cell types, the contributing factors were canceout by our normalization of theg8 andg9 data. Normalizingg8 ~measured at 0.75 Hz! by g0 , and normalizingg9 by g8~both measured at 0.75 Hz! lead to a collapse of the datbased onx alone ~Fig. 9!: the normalized stiffnessGn de-creased monotonically, and the normalized frictionh in-creased monotonically with increasingx. The very same re-lationships ofGn versusx andh versusx were found in allcell types that we studied. Those relationships closely follthe predictions from the structural damping equation, withxbeing the only free parameter.

The observation of universal scaling based onx leads tothe prospect of a major empirical simplification. Those retionships suggest that key functions of diverse cytoskelproteins, such as their influence on the ability of the cytoeleton to deform and to flow, may be understood maithrough their ability to modulatex. Such behavior is remi-niscent of systems that exhibit only a weak dependenceaggregate mechanical behavior on the details of underlymolecular interactions. Familiar examples of such systeinclude the depression of the freezing point of liquidsaddition of arbitrary solutes, or Van’t Hoff’s law of osmotipressure. In each of these processes, specific molecular iactions are at some level surely important, but to explainmacroscopic behavior, the molecular details are of only sondary importance. Instead, these systems are colligativthat their macroscopic behavior depends primarily onnumber of molecules and their temperature.

The data and analysis presented above establish emcally a series of universal properties that spans diversetypes. We now turn attention to the question of mechanisthat might underly this behavior. Below we provide a ratinale for considering a recently developed theory of sglassy rheology to explain these findings; this theory derithe structural damping equation from considerations of s

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tistical mechanics. The details of this theory have beenscribed elsewhere@7,78,79#.

Soft glassy materials

The class of soft glassy materials~SGM! comprises whatwould at first glance seem to be a remarkably diverse grof substances that includes foams, pastes, colloids, esions, and slurries. Yet, the mechanical behavior of eacthese substances is surprisingly alike. The common empicriteria that define this class of materials are that theyvery soft ~in the range of Pa to kPa!, that bothG8 and G9increase with the same weak power-law dependences onquency, and that the loss tangenth is frequency insensitiveand of the order 0.1@7,78#. The data presented here establthat the cells that we studied fulfilled these empirical criterThe studies described here were not designed to investaging phenomena or rejuvenation caused by imposed swhich would represent even stronger evidence of glassyhavior @80#. Nonetheless, hints of aging, as discussed belwere evident in some of the data.

The physics of glasses in general, and of soft glasseparticular, are much studied but remain poorly understo@81#. In the case of soft glasses, Sollich reasoned thatcause the materials comprising the class are so diversecommon rheological features must be not so much a refltion of specific molecules or molecular mechanisms as tare a reflection of generic system properties that play ousome higher level of structural organization@7#. The genericfeatures that all soft glassy materials share are that eaccomposed of elements that are discrete, numerous and agated with one another via weak interactions. In additithese materials exist far away from thermodynamic equirium and are arrayed in a microstructural geometry thainherently disordered and metastable. All of these featualso apply to the CSK of living cells.

Taken together, the cytoskeleton of the living cell satisfithese mechanical criteria as well as the generic featuresdefine the class of soft glassy materials. Accordingly,propose the working hypothesis that the cytoskeleton ofliving cell is a soft glassy material.

To describe the interaction between the elements withe matrix, Sollich developed the theory of soft glassy rhology ~SGR! using earlier work by Bouchaud as a pointdeparture@82#. SGR theory considers that each individuelement of the matrix exists within an energy landscape ctaining many wells, or traps, of differing depthE. These trapsare formed by interactions of the element with neighborelements. In the case of living cells those traps mightplausibly thought to be formed by binding energies betweneighboring cytoskeletal elements including but not limitto crosslinks between actin filaments, cross-bridges betwactin and myosin, hydrophilic interactions between varioproteins, charge effects or simple steric constraints.

In Bouchaud’s theory of glasses, an element can escapenergy well and fall into another nearby well; such hoppevents are activated by thermally driven random fluctuatio

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As distinct from Bouchaud’s theory, in Sollich’s theory osoft glassy materials each energy well is regarded as beindeep that the elements are unlikely to escape the welthermal fluctuations alone. Instead, elements are imaginebe agitated, or jostled, by their mutual interactions wneighboring elements@7#. A clear notion of the source of thenonthermal agitation remains to be identified, but this agtion can be represented nonetheless by an effective tempture, or noise level,x. Whenx.1, there is sufficient agitationin the matrix that the element can hop randomly betwewells and, as a result, the system as a whole can flowbecome disordered. Whenx approaches 1, however, the elments become trapped in deeper and deeper wells fwhich they are unable to escape—the system exhibits a gtransition and becomes a simple elastic solid with stiffnG0 .

In SGR theory, the effective noise temperaturex is a tem-perature to the extent that the rate at which elements canout of a trap assumes the form exp(2E/x), wherex takes theusual position of a thermal energykT in the familiar Boltz-mann exponential. By analogy,x has been interpreted bSollich as reflecting jostling of elements by an unidentifibut nonthermal origin. This ambiguity has been considerecentral limitation of SGR theory@78#. There is no evidenceto suggest that the ambiguity surroundingx might be re-solved in the case of living cells~as opposed to the inermaterials for which SGR theory was originally devised! byan obvious and ready source of non-thermal energy injectnamely, those proteins that go through cyclic conformatiochanges and thus agitate the matrix by mechanisms thaATP-dependent.

Remarkably, in the limit that the frequency is small~com-pared toF0) and the imposed deformations are small, Soch’s theory leads directly to the structural damping equat@Eq. ~3!#. Except for a small discrepancy in some cell typat low frequencies, which we discuss further below, the dreported here establish firmly that the mechanical behaof cells conform well to Eq.~3! ~Table I!. If Sollich’s quan-titative theory and underlying ideas are assumed to applthe data reported here, then the parameters in Eq.~3! can beidentified as follows.

The parameterx is identified as being the noise temperture of the cytoskeletal matrix. The measured values ofx incells lie between 1.15 and 1.35, indicating that cells exclose to a glass transition.

G0 is identified as being the stiffness of the cytoskeletat the glass transition (x51). In this connection, Satcher anDewey @83# developed a static model of cell stiffness bason consideration of cell actin content and matrix geomeAll dynamic interactions were neglected in their model,would be the case in SGR theory in the limit thatx ap-proaches 1, when all hopping ceases. As such, it mighexpected that their model would predict this limiting valuethe stiffness,G0 , as defined in Eq.~3!. Indeed, we havefound a remarkably good correspondence between theirdiction ~order of 10 kPa! and our estimate forG0 ~41 kPa inHASM cells!.

Finally, F0 is identified in Sollich’s theory as being thmaximum rate at which cytoskeletal elements can esc

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their traps. However, for soft glassy materials in general,the case of living cells in particular, the factors that detmineF0 remain unclear. Statistical analysis of our data sgests thatF0 did not vary with drug treatments and possibnot even across cell type~Fig. 7!. But why F0 is invariant isnot at all clear, and is not explained by SGR theory.

Identification of the living cell in culture as belonging tclass of soft glasses at first surprised us, but on further ansis seemed mechanistically plausible. Our data suggestthe living cell exists close to a glass transition and modulaits mechanical properties by moving between glassy stthat are ‘‘hot,’’ melted and liquidlike, and states that a‘‘cold,’’ frozen and solidlike~Fig. 9!. More than a superficiametaphor, this is a precise mechanistic statement of the ghypothesis and has implications that are quantitativetestable.

The consequences of this hypothesis are interestingopposed to focusing attention on the details of specific mlecular interactions, the glass hypothesis focuses atteninstead on a higher level of structural organization. Thisnot meant to imply that molecular details are unimportantthe contrary, there are many individual proteins whose inbition or dysfunction has catastrophic consequenceswhose molecular pathways are critically important. Nonetless, the existence of master curves~Fig. 9! suggests thamolecular interactions may contribute to cytoskeletal dynaics and remodeling mainly to the extent that they are ablemodulate the noise temperaturex. This point of view is novelin that it suggests a generic quality of the contributionindividual molecular species to the noise temperatureassociated mechanical behaviors of the CSK. As such,glass hypothesis to a certain extent subordinates the roindividual molecular species and puts them instead intdeeply integrative context. On the one hand, it implies inhent limits to reductionism, for to study specific moleculesisolation might be to destroy the very interactions that neto be studied. On the other hand, to elucidate a particmolecular-level event in situ without regard to its impactthe noise temperature might be to miss a major facet ofunction.

It is interesting that, from a mechanistic point of view, tparameterx plays a central role in the theory of soft glasmaterials. At the same time, from a purely empirical pointview, the parameterx is found to play a central organizinrole leading to the collapse of all data onto master cur~Fig. 9!. Whether or not the measured value ofx might ulti-mately be shown to correspond to a noise temperature,empirical analysis would appear to provide a unifying framwork for studying protein interactions within the compleintegrative microenvironment of the cell body.

Departures from structural damping behavior—Hintsaging. In three cell types~human bronchial epithelial cellsmacrophages and human neutrophils!, we found that the be-havior of g9 departed systematically from the power labehavior implied by Eq.~3!; g9 under baseline conditionbecame nearly frequency independent below 10 Hz~Fig. 8!.These discrepancies disappeared when the cells were trwith cytochalasinD, however.

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Closely similar behavior has been reported in soft marials in which the mechanical properties are found to depon the elapsed time since the sample was prepared@80#.Fielding et al. @80# suggested that this behavior is attribuable to aging, wherein elements comprising the matriximagined to populate deeper and deeper energy wells asprogresses. The system never achieves a steady state,fore, and instead behaves as though its longest relaxatime is of the order of its own age. Hence, the longest relation time is not a property of any specific molecule of whithe system is comprised. Rather, it is a property of the ingrated system and its history. SGR theory provides a conctual framework to study such aging phenomena and predaging as an integral part of the low-frequency rheologiresponses near the glass transition@7,80#. The studies re-ported here were not designed to investigate aging phenena. Nonetheless, the particular departures of the dataEq. ~3! at low frequencies conform to aging phenomena.cannot exclude the possibility, however, that the aging-lbehavior that we observed was instead attributable to sunspecified relaxation processes occurring at frequencieslow our measurement range.

Alternative models. Despite a number of open questionSollich’s SGR model is simple, intuitive, and predicts strutural damping behavior@Eq. ~3!#. But soft glassy dynamicsas proposed by SGR theory is not the only mechanismcould lead to scale-free mechanical behavior as expressestructural damping and power-law stress relaxation. Powlaw behavior could be explained by models containinglarge number of viscoelastic compartments with a particudistribution of time constants, but such models lack a mecnistic basis, they require many parameters with no appaphysical meaning, and these parameters would change ecally between different drug treatments of the cell. Powlaw stress relaxation has been reported for a number of crlinked polymers at or near the gelation point@84#. However,potential mechanisms leading to this behavior, such ascolation phenomena in self-similar polymer clusters, are sa matter of intense debate@85#. Interestingly, it has beensuggested that both the glass transition and critical gelamay arise from a common mechanism, namely kinetic ardue to crowding of clusters, which is a manifestation of mogeneral jamming transitions@86#. Finally, we cannot rule outthe possibility that relaxation processes within the proteinsthe cytoskeleton themselves, for instance during folding aunfolding processes and during conformational changmight account for power-law behavior of the entire cell@87#.

CONCLUSIONS

Time-scale invariance, collapse of data onto mascurves, and SGR theory, when taken together, point to mstability of interactions and disorder of the matrix as beigeneric features that govern integrative mechanical behaof the cells studied here. It is conceivable that these genfeatures might transcend cytoskeletal mechanics and cover into the dynamics of protein interactions whose mmanifestation are not at all mechanical, but this possibility

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not presently amenable to experimental attack. When stuing muscle contraction, Hill suggested that the special vaof heat release measurements is that they place firm phylimits on underlying mechanical and molecular processalthough they do not point unequivocally to specific moleclar events@5#. Similarly, we would say that the special valuof the noise temperature is its intimate relation to the mchanical and chemical changes involved, and the sensitand speed of the measurement methods that are now aable. Measurement of the noise temperature admittedly dnot point in a specific way to the individual molecular relaation processes that occur, but it does provide a firm empcal framework to which those processes seem to conform

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ACKNOWLEDGMENTS

We thank Stephanie Shore, Paul Moore, Wolfgang Gomann, and Daniel Tschumperlin for help with cell cultureSrbolub Mijailovich for providing finite element analysis ocell deformation, Winfried Mo¨ller and Maureen Harp forhelp with bead production, Reynold Panettieri for providiHASM cells, Rick Rogers, Rebecca Sterns, and Marina Pde Morales for support with microscopy, Peter Malave aBivash Dasgupta for performing the control experimentspolyacrylamide gels, and C. Forbes Dewey and David Wefor helpful comments and suggestions. This study was sported by National Institute of Health Grant Nos. HL3300HL65960, HL59682, and SAF-2002-03616~Spain!.

m-

@1# P. A. Janmey, Physiol. Rev.78, 763 ~1998!.@2# H. Jeonget al., Nature~London! 407, 651 ~2000!.@3# E. V. Koonin, Y. I. Wolf, and G. P. Karev, Nature~London!

420, 218 ~2002!.@4# Y. L. Kim et al., IEEE J. Sel. Top. Quantum Electron.~to be

published!.@5# A. V. Hill, Trails and Trials in Physiology~E. Arnold, London,

1965!, pp. 14 and 15.@6# B. Fabryet al., Phys. Rev. Lett.87, 148102~2001!.@7# P. Sollich, Phys. Rev. E58, 738 ~1998!.@8# P. G. de Gennes,Scaling Concepts in Polymer Physics~Cor-

nell University Press, Ithaca, 1979!, pp. 134–152.@9# J. D. Ferry,Viscoelastic Properties of Polymers~Wiley, New

York, 1980!.@10# P. A. Valberg and J. P. Butler, Biophys. J.52, 537 ~1987!.@11# N. Wang, J. P. Butler, and D. E. Ingber, Science260, 1124

~1993!.@12# N. Wang and D. E. Ingber, Biophys. J.66, 2181~1994!.@13# G. N. Maksymet al., J. Appl. Physiol.89, 1619~2000!.@14# B. Fabryet al., J. Appl. Physiol.91, 986 ~2001!.@15# W. Moller, W. Stahlhofen, and C. Roth, J. Aerosol Sci.21,

S435~1990!.@16# W. S. Craiget al., Biopolymers37, 157 ~1995!.@17# S. M. Mijailovich et al., J. Appl. Physiol.93, 1429~2002!.@18# R. J. Pelham, Jr. and Y. Wang, Proc. Natl. Acad. Sci. U.S

94, 13661~1997!.@19# T. G. Mason and D. A. Weitz, Phys. Rev. Lett.74, 1250

~1995!.@20# W. Weber, Ann. Phys.~Leipzig! 54, 1 ~1841!.@21# F. Kohlrausch, Ann. Phys.~Leipzig! 128, 1 ~1866!.@22# J. Hildebrandt, Bull. Math. Biophys.31, 651 ~1969!.@23# Y. C. Fung, Am. J. Physiol.213, 1532~1967!.@24# S. H. Crandall, J. Sound Vib.11, 3 ~1970!.@25# J. J. Fredberg and D. Stamenovic, J. Appl. Physiol.67, 2408

~1989!.@26# B. Fabryet al., J. Magn. Magn. Mater.194, 120 ~1999!.@27# R. A. Panettieriet al., Am. J. Physiol.256, C329~1989!.@28# J. D. Laporteet al., Am. J. Physiol.275, L491 ~1998!.@29# J. L. Coll et al., Proc. Natl. Acad. Sci. U.S.A.92, 9161~1995!.@30# P. Ralph and I. Nakoinz, Nature~London! 257, 393 ~1975!.@31# J. C. Pinheiro and D. M. Bates,Mixed-Effects Models in S an

S-Plus~Springer, New York, 2000!.

.

@32# C. E. Schmidtet al., J. Cell Biol.123, 977 ~1993!.@33# D. Choquet, D. P. Felsenfeld, and M. P. Sheetz, Cell88, 39

~1997!.@34# P. A. Janmeyet al., Nature~London! 345, 89 ~1990!.@35# P. A. Janmeyet al., J. Biol. Chem.269, 32503~1994!.@36# S. Yamada, D. Wirtz, and S. C. Kuo, Biophys. J.78, 1736

~2000!.@37# M. Yanai et al., Am. J. Physiol.277, C432~1999!.@38# C. J. Meyeret al., Nat. Cell Biol.2, 666 ~2000!.@39# J. Chen et al., Am. J. Physiol. Cell Physiol.280, C1475

~2001!.@40# G. Plopper and D. E. Ingber, Biochem. Biophys. Res. Co

mun.193, 571 ~1993!.@41# T. Wakatsukiet al., J. Cell. Sci.114, 1025~2001!.@42# C. Rotschet al., Cell Biol. Int. 21, 685 ~1997!.@43# R. S. Frank, Blood76, 2606~1990!.@44# R. D. Hubmayret al., Am. J. Physiol.271, C1660~1996!.@45# S. A. Shoreet al., Am. J. Respir. Cell Mol. Biol.16, 702

~1997!.@46# N. Wang et al., Am. J. Physiol. Cell Physiol.282, C606

~2002!.@47# S. S. Anet al., Am. J. Physiol. Cell Physiol.283, C792~2002!.@48# A. R. Bauschet al., Biophys. J.75, 2038~1998!.@49# O. Thoumine and A. Ott, J. Cell. Sci.110, 2109~1997!.@50# J. Alcarazet al., Biophys. J.84, 2071~2003!.@51# D. E. Ingber, J. Appl. Physiol.89, 1663~2000!.@52# R. E. Buxbaumet al., Science235, 1511~1987!.@53# M. Sato, N. Ohshima, and R. M. Nerem, J. Biomech.29, 461

~1996!.@54# N. Wang and D. E. Ingber, Biochem. Cell Biol.73, 327~1995!.@55# N. Wanget al., Am. J. Physiol.268, C1062~1995!.@56# F. J. Alenghatet al., Biochem. Biophys. Res. Commun.277,

93 ~2000!.@57# S. G. Shroff, D. R. Saner, and R. Lal, Am. J. Physiol.269,

C286 ~1995!.@58# R. E. Mahaffyet al., Phys. Rev. Lett.85, 880 ~2000!.@59# W. H. Goldmann and R. M. Ezzell, Exp. Cell Res.226, 234

~1996!.@60# I. M. Ward, Mechanical Properties of Solid Polymers~Wiley,

Chichester, 1985!.@61# W. Weber, Ann. Phys.~Leipzig! 34, 247 ~1835!.@62# J. J. Fredberget al., J. Appl. Physiol.74, 1387~1993!.

4-17

io

.

si

J.


@63# J. H. Bateset al., Ann. Biomed. Eng.22, 674 ~1994!.@64# Z. Hantoset al., J. Appl. Physiol.72, 168 ~1992!.@65# G. W. Schmid-Schonbeinet al., Biophys. J.36, 243 ~1981!.@66# S. Chien and K. L. Sung, Biophys. J.46, 383 ~1984!.@67# G. I. Zahalak, W. B. McConnaughey, and E. L. Elson, J. B

mech. Eng.112, 283 ~1990!.@68# J. P. Butler and S. M. Kelly, Biorheology35, 193 ~1998!.@69# W. Moller et al., Biophys. J.79, 720 ~2000!.@70# P. A. Valberg and D. F. Albertini, J. Cell Biol.101, 130~1985!.@71# A. Yeung and E. Evans, Biophys. J.56, 139 ~1989!.@72# E. Evans and A. Yeung, Biophys. J.56, 151 ~1989!.@73# M. A. Tsai, R. S. Frank, and R. E. Waugh, Biophys. J.65, 2078

~1993!.@74# O. Thoumine, O. Cardoso, and J. J. Meister, Eur. Biophys

28, 222 ~1999!.@75# M. Sato, M. J. Levesque, and R. M. Nerem, Arteriosclero

~Dallas! 7, 276 ~1987!.

04191

-

J.

s

@76# D. V. Zhelev, D. Needham, and R. M. Hochmuth, Biophys.67, 696 ~1994!.

@77# T. A. Wilson, J. Appl. Physiol.77, 1570~1994!.@78# P. Sollichet al., Phys. Rev. Lett.78, 2020~1997!.@79# B. Fabry and J. J. Fredberg, Respir. Physiol. Neurobiol.~to be

published!.@80# S. M. Fielding, P. Sollich, and M. E. Cates, J. Rheol.44, 323

~2000!.@81# S. Torquato, Nature~London! 405, 521 ~2000!.@82# J. Bouchaud, J. Phys. I2, 1705~1992!.@83# R. L. Satcher, Jr. and C. F. Dewey, Jr., Biophys. J.71, 109

~1996!.@84# H. H. Winter and F. Chambon, J. Rheol.30, 367 ~1986!.@85# K. Broderix et al., Physica A302, 279 ~2001!.@86# P. N. Segreet al., Phys. Rev. Lett.86, 6042~2001!.@87# A. L. Lee and A. J. Wand, Nature~London! 411, 501 ~2001!.

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time scale and other invariants of integrative mechanical ......time scale and other invariants of...

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