Mapping the cytoskeletal prestress

Chan Young Park,1 Dhananjay Tambe,1 Adriano M. Alencar,1,2 Xavier Trepat,1,3 En Hua Zhou,1

Emil Millet,1 James P. Butler,1,4 and Jeffrey J. Fredberg1

1Molecular and Integrative Physiological Sciences, Department of Environmental Health, Harvard School of Public Health,Boston, Massachusetts; 2Department of Pathology, Medical School, University of Sao Paulo, Sao Paulo, Brazil; 3Institute forBioengineering of Catalonia, Universitat de Barcelona, and Ciber Enfermedades Respiratorias, Barcelona, Spain; and4Department of Medicine, Harvard Medical School, Boston, Massachusetts

Submitted 16 September 2009; accepted in final form 9 February 2010

Park CY, Tambe D, Alencar AM, Trepat X, Zhou EH, Millet E,Butler JP, Fredberg JJ. Mapping the cytoskeletal prestress. Am JPhysiol Cell Physiol 298: C1245–C1252, 2010. First published Feb-ruary 17, 2010; doi:10.1152/ajpcell.00417.2009.—Cell mechanicalproperties on a whole cell basis have been widely studied, whereaslocal intracellular variations have been less well characterized and arepoorly understood. To fill this gap, here we provide detailed intracel-lular maps of regional cytoskeleton (CSK) stiffness, loss tangent, andrate of structural rearrangements, as well as their relationships to theunderlying regional F-actin density and the local cytoskeletal pre-stress. In the human airway smooth muscle cell, we used micropat-terning to minimize geometric variation. We measured the local cellstiffness and loss tangent with optical magnetic twisting cytometryand the local rate of CSK remodeling with spontaneous displacementsof a CSK-bound bead. We also measured traction distributions withtraction microscopy and cell geometry with atomic force microscopy.On the basis of these experimental observations, we used finiteelement methods to map for the first time the regional distribution ofintracellular prestress. Compared with the cell center or edges, cellcorners were systematically stiffer and more fluidlike and supportedhigher traction forces, and at the same time had slower remodelingdynamics. Local remodeling dynamics had a close inverse relation-ship with local cell stiffness. The principal finding, however, is thatsystematic regional variations of CSK stiffness correlated only poorlywith regional F-actin density but strongly and linearly with theregional prestress. Taken together, these findings in the intact cellcomprise the most comprehensive characterization to date of regionalvariations of cytoskeletal mechanical properties and their determi-nants.

cell mechanics; stiffness; remodeling; heterogeneity

EVOLUTIONARY SUCCESS of the eukaryotic cytoskeleton (CSK) isbased on its abilities to perform tasks that demand contrastingmaterial properties (31). For example, cell spreading (11),crawling (34), and invasion (72) require traction forces and me-chanical strength on the one hand but adaptability and remodeling(8) on the other. During pattern formation (27) and gene expres-sion (21) the CSK mediates transduction of mechanical cues intochemical signals. For most of these activities, the cell needs totune CSK material properties not only globally at the whole celllevel but also locally within the single cell (26). It is now knownthat corners of the micropatterned cell correspond to the region ofgreatest focal adhesion density (6), highest traction forces (3), andmost active lamellar extension (45), but detailed mapping of CSKmaterial properties within a single cell has yet to be defined.

Since Petersen et al.’s (48) measurements of intracellularstiffness, regional differences in cellular mechanical propertieshave been heavily emphasized (23, 28, 53, 68, 73). For exam-ple, Su et al. (61) have shown that in the migrating cell theleading edge is about two times stiffer than the trailing edge.Similarly, Shroff et al. (55) have shown that the local stiffnessvaries up to fivefold within a single cell and that the peripheralregions are the stiffest, while the cell center is the softest.Similar heterogeneities have been found in other cell types andwith other methodologies (2, 29, 30, 39). To account forregional differences in cell stiffness it has been suggested thatstress fibers are locally thicker or denser at stiffer locations (2,29, 30, 40, 61, 62), and common to all these studies is thefurther assertion—based upon only limited data—that localvariations in cell mechanical properties must be closely linkedto variations in the local density of the F-actin network.

Nonetheless, both the tensegrity model and experimentsfrom prestressed actin gels and stretched cells would suggestthat the prestress should be the central factor determiningcellular stiffness (22, 25, 49, 51, 52, 59, 60, 70, 71). However,as a potential contributor to the local variations of CSKstiffness, the local variation of the CSK prestress has not yetbeen evaluated experimentally in the living cell. To addressthis issue, we provide here comprehensive maps of local staticand dynamic mechanical properties of the human airwaysmooth muscle (HASM) cell. We plated each cell on a mi-cropatterned substrate with square extracellular matrix (ECM)islands. Micropatterning provided several experimental advan-tages. Across all cells that were studied, this approach createdwell-defined cell corners with no appreciable variation in cellspreading or cell shape. For that reason, responses could beaveraged across an ensemble of cells with virtually identicalgeometries, and statistical power could be amplified further bytaking advantage of the eightfold symmetry condition impliedby square cell geometry and the associated rectilinear coordi-nate system onto which all data could be unambiguouslymapped. We studied 462 positions on a total of 193 cells.

We measured three static properties locally: cell height,using atomic force microscopy (AFM) (54), F-actin density,using immunofluorescent imaging (56), and traction forces thatthe cell exerts on its elastic substrate, using Fourier-transformtraction microscopy (FTTM) (10). In addition, we measuredthe rate of CSK structural rearrangements, using spontaneous(unforced) nanoscale displacements of a CSK-bound bead asan index (8, 9), and the complex elastic modulus, using forcedbead displacements and optical magnetic twisting cytometry(OMTC) (16–18, 67). Finally, from the measured tractiondistribution and geometry we were able to compute for the firsttime the internal distribution of the local cytoskeletal prestress.

Address for reprint requests and other correspondence: J. J. Fredberg,Molecular and Integrative Physiological Sciences, Dept. of EnvironmentalHealth, Harvard School of Public Health, Boston, MA 02115 (e-mail:jeffrey_fredberg@harvard.edu).

Am J Physiol Cell Physiol 298: C1245–C1252, 2010.First published February 17, 2010; doi:10.1152/ajpcell.00417.2009.

0363-6143/10 $8.00 Copyright © 2010 the American Physiological Societyhttp://www.ajpcell.org C1245

on May 25, 2010

ajpcell.physiology.orgD

ownloaded from

Our principal finding is that systematic regional variations ofCSK stiffness correlate quite poorly with regional F-actindensity but correlate strongly and linearly with the regionalprestress.

METHODS

Cell culture. HASM cells were isolated from tracheal smoothmuscle of human lung transplant donors (approved by the Universityof Pennsylvania Committee on Studies Involving Human Subjects) asdescribed previously (44). Cells at passages 4–7 were used in allexperiments. After cells reached confluence in plastic dishes, theywere serum deprived for 24 h before being trypsinized and seeded onmicropatterned substrates.

Acrylamide gel. We adapted the technique of polyacrylamidesubstrate preparation described previously (47). The mixing ratio ofacrylamide solution was adjusted to obtain a Young’s modulus of 4kPa (74). It is well established that the stiffness of the cell substrateinfluences both cell traction forces and cell stiffness (46, 58). How-ever, in this report we restrict attention to the case of regionalvariations with a single value of substrate stiffness falling within thephysiological range.

Micropatterning. We used the MEMPAT (membrane-based pat-terning) technique to make square cells (15, 43). In brief, polydim-ethylsiloxane (PDMS) membranes with 50 � 50-�m square holeswere fabricated with conventional micromachining procedures andput onto the center of a glass-bottomed well or on top of theacrylamide gel surface. We then coated the open surfaces with bovinecollagen type I (Cohesion). Collagen solutions (5 �g/ml and 100�g/ml,) respectively) were used to coat glass and gel surfaces,respectively (1, 17). After incubation at 4°C overnight, we lifted offthe membrane and blocked the uncoated area with 1% bovine serumalbumin (Sigma).

The size of the 50-�m square was chosen based on the average areaof isolated HASM cells (66). We plated serum-deprived cells (�1,500cells/well) sparsely, washed them two times with fresh medium after1 h, and incubated them overnight before starting experiments. Singlecells conforming to the micropatterned square shape were used formeasurements.

Atomic force microscopy. We measured the height profile of themicropatterned cell with AFM (Asylum Research); three-dimensionalimages were taken in contact mode with a pyramid tip [springconstant (k) �0.03 N/m, Novascan].

Optical magnetic twisting cytometry. Detailed descriptions andvalidations of this technique have been given elsewhere (9, 17, 18, 50,67). Briefly, to probe the rheology of the CSK, ferrimagnetic beads(4.1-�m diameter) were coated with a synthetic peptide containing theArg-Gly-Asp (RGD) sequence (Integra Lifesciences) and allowed toadhere randomly to the apical cell surface. RGD binds mostly with�1-integrin, which has been shown to be distributed relatively homo-geneously (41). The effect of RGD’s binding with integrin in OMTCmeasurement has been discussed in depth elsewhere (50, 67). In brief,integrins are activated by binding with RGD, and their activationinduces local remodeling, but these remodeled structures are nodifferent from those that all adherent cells form at anchorage sites thatconnect cells to the ECM. Also, we showed previously (9) thatcellular stiffness measured with beads coated with ligands that acti-vate integrins is similar to the stiffness measured with poly-L-lysine-coated beads, which do not form focal adhesion. We then measuredthe complex modulus as a function of frequency by applying oscilla-tory magnetic fields and measuring the resultant bead motions. At agiven frequency of oscillatory magnetic fields (f), the complex mod-ulus is g̃(f) � T̃/D̃, where T is specific torque, D is the displacement,and the tilde denotes Fourier transform. The real part of the complexmodulus is the storage modulus (g=), and the imaginary part is the lossmodulus (g�). The ratio g�/g= is the loss tangent.

We first recorded spontaneous motions of beads as a measure of therate of structural rearrangements (described below). Beads were thenmagnetized with a short (�1 ms) magnetic field (�1,000 G) andtwisted with a small magnetic field (10 G) in the OMTC protocol. Toavoid time-dependent changes, we used a low specific torque (�20Pa) and finished measurements within 30 min (12).

Within the linear range, we established previously (16, 18) thatCSK rheology corresponds to the power law response g= fx1 overa wide range of frequencies. In such systems, the power law exponentx describes a range of CSK states ranging from Hookean elasticsolidlike behavior (x � 1) to Newtonian fluidlike behavior (x � 2)(16, 57).

For each bead, the structural damping model, Eq. 1, was fitted tothe complex modulus (g̃) in the frequency domain with respect to thethree parameters g, x, and �,

g� � g · �i��x�1���2 � x�� � i�� (1)

where g is a prefactor that we used as the elastic modulus or stiffness, �is angular velocity (2�f), [] is the gamma function, � is a parameter to

account for the effect of Newtonian viscosity, and i is ��1 (16). Theunits of g are pascals per nanometer.

Structural rearrangements. Bead positions were recorded at afrequency of 12 frames/s for maximum time (tmax) �160 s aspreviously described (8, 9). We defined mean square displacement(MSD) of an individual bead as

MSD��t� � ��r�t � �t� � r�t��2�

where r(t) is the bead position at time t, �t is the time lag, and anglebrackets indicate an average over t (8). MSD of most beads increasedwith time according to a power law relationship, MSD(�t) � D*(�t/�t1)�, where �t1 � 1 s, thus nondimensionalizing the time depen-dence. The coefficient D* has dimensions in square nanometers, andthe exponent � of each bead was calculated from a least-square fit ofa power law to the MSD data for �t between 4 s and tmax/4 (9). Allthe beads used for MSD data were included in OMTC data.

Fourier-transform traction microscopy. Bright field and fluores-cent images of the micropatterned cell on a gel were recorded. Afterthe cell was trypsinized, another fluorescent image was taken tocalculate the displacement field generated by the cell. Using theseimages, we determined the displacement field of the gel (65). Theconstrained traction field within a 50-�m square was calculated fromthe displacement field implementing the solution described by Butleret al. (10).

Immunofluorescent imaging and quantification of F-actin amount.We stained F-actin with rhodamine phalloidin (Molecular Probes)(56). The structure of F-actin was scanned with a confocal microscope(Leica Microsystems) with fixed step size (0.2 �m), the images werestacked vertically, and intensities were summed (64). The fluorescentintensities were used as a measure of F-actin amount based on the factthat the scanning conditions and staining conditions were same.

Bead position and symmetry conditions. With bright field images,local positions of beads on a square cell were calculated relative to thelower left corner of the square with the same center of mass algorithmthat we used for bead tracking (17). The symmetry of micropatternedsquare lattice allowed us to map all data onto a single trianglecorresponding to one-eighth of the square, thus increasing the pointdensity by eightfold (7). With the four symmetry lines of a square(horizontal, vertical, and 2 diagonals), all data were mapped onto aright-triangular octant. We also applied the same symmetry transfor-mation on tractions, cell heights, and F-actin fluorescent intensities.

Statistical analysis. Statistical analyses were performed with un-paired t-test assuming unequal variance. Statistical significance wasdetermined at the level of P � 0.05.

Mapping schemes. From pooled data over all cells studied, wecreated regional distribution maps with a grid spacing of 0.5 �m,smoothed over a 6-�m-radius circle. Maps presented here depict

C1246 MAPPING CYTOSKELETAL PRESTRESS

AJP-Cell Physiol • VOL 298 • MAY 2010 • www.ajpcell.org

on May 25, 2010

ajpcell.physiology.orgD

ownloaded from

geometric mean (g and D*) or arithmetic mean (x) values over eachcircle, consistent with the distribution of these variables previouslynoted (9, 17). Arithmetic means of traction data, cell height, andF-actin density were taken over all cells.

Prestress calculation. We define prestress as the maximum princi-pal stress within our cell model, as described here. To calculate theprestress we used finite element methods (FEM). We constructed athree-dimensional finite element model of one-eighth of the cell (seeFig. 1A, inset) based on the AFM measurements. Displacementsnormal to the reflection-symmetry planes and the cell-substrate inter-face are constrained to be zero. Tractions obtained from FTTM werereversed in direction and applied to the bottom plane of the cell. Thisis a precise application of Newton’s third law: the reciprocity of actionand reaction. With these boundary conditions and the assumptions ofan isotropic, linearly elastic material (Young’s modulus � 10 kPa,Poisson ratio � 0.499), we calculated the stress distribution within thecell. It should be noted that as long as the strains are small, therecovered stresses given traction boundary conditions do not dependon the elastic modulus of the medium. We therefore chose a Young’smodulus sufficiently high to ensure remaining in the linear regime.The two-dimensional contour map of prestress was generated byaveraging the stress along the axis normal to the basal plane of thecell.

RESULTS

Random variability. We obtained mechanical data from atotal of 462 beads distributed among 193 different micropat-terned cells. To minimize the effect of bead-bead interaction,all beads studied were apart from any other beads by at least 2bead diameters (17). Data for all beads were mapped into onecell octant (Fig. 1A, inset). The distribution of bead positions

showed no obvious spatial bias and spanned the space ofinterest uniformly. The local elastic modulus or stiffness, g,was coded by both the color and the size of each datum andshowed that local stiffness is highly variable (Fig. 1A). Thehistogram of g conformed to a log-normal distribution, as hasbeen reported previously by ourselves (17, 76) and confirmedwidely by others (4, 14, 36, 69). Even though there wasminimal variation in cell size or cell shape, the variation in gapproached 2 orders of magnitude.

Systematic regional distribution. As a crude first analysis,we segregated beads into three groups: center, edge, and corner(Fig. 1B, inset). The data in Fig. 1B show that the geometricmean of the elastic storage modulus (g=) increased as a weakpower law function of frequency (16) in all three regions. Thecorner was approximately twice as stiff as the center, with theedge being intermediate; this result is compatible with previousreports (5, 55, 61). The loss modulus (g�) was also weaklydependent on frequency and showed regional differences com-parable to the storage modulus. When we compared data ineach region, the stiffness (g) was significantly higher in thecorner than the other regions and the center was the softestregion (Table 1). The Newtonian viscosity term � in the cornerfollowed a trend similar to that of g, but the effect was small.The power law exponent x was higher in the corner than theother regions; this difference was quantitatively small, but withthe large size of the data set did reach statistical significance.There was no difference between the edge and the center. Insummary, the corner region was stiffer, more viscous (theNewtonian �), and more fluidlike than the other regions.

Fig. 1. A: local elastic modulus g values(Pa/nm) measured by optical magnetic twist-ing cytometry (OMTC) were, by symmetry,folded onto an octant of the square cell (darktriangle, inset). B: raw data were segregatedinto 3 groups: center (ce), edge (ed), andcorner (co) (inset). Average storage (g=) andloss (g�) modulus of each region show powerlaw dynamics as a function of frequency.Data presented are as in Table 1; g= and g� ineach region were significantly different fromeach other at all frequencies except for g= at 1kHz.

Table 1. Average values of x, g, �, D*, and � from each region and P values between regions

No. of Beads x g, Pa/nm �, 104 Pa � s � nm1 No. of Beads D*, nm2 �

Center 149 1.176 � 0.003 0.178 � 0.013 2.51 � 0.16 108 43.9 � 3.7 1.58 � 0.02Edge 221 1.180 � 0.003 0.222 � 0.015 3.28 � 0.20 178 31.1 � 2.0 1.55 � 0.02Corner 92 1.201 � 0.007 0.289 � 0.027 5.12 � 0.45 78 29.4 � 2.9 1.62 � 0.03P value ce-ed 0.48 0.026 0.0021 N/A 0.0014 0.26

ce-co 0.0017 �0.0001 �0.0001 N/A 0.0026 0.45ed-co 0.0053 0.025 �0.0001 N/A 0.65 0.059

Data shown are means � SE for power law exponent x, viscosity (�), and exponent � and geometric means � the log SE times the geometric mean for stiffness(g) and local mean square displacement prefactor D*. ce, Center; ed, edge; co, corner; N/A, not applicable.

C1247MAPPING CYTOSKELETAL PRESTRESS

AJP-Cell Physiol • VOL 298 • MAY 2010 • www.ajpcell.org

on May 25, 2010

ajpcell.physiology.orgD

ownloaded from

Stiffness data were smoothed and mapped back onto theoriginal square cell geometry (Fig. 2). Stiffness decreasedgradually from the corners toward the center. The edge wasrelatively soft, and there was little change in stiffness alonghorizontal and vertical lines of symmetry (Fig. 2A). However,corners had slightly higher values of the power law exponent x(Fig. 2B) and were thus more fluidlike than other cell regions.The cell center, overlying the nucleus, was very slightly morefluidlike than surrounding regions, and a ring area around thenucleus was the most solidlike region. The distribution of xwas comparable to that in a recent report (63).

Structural rearrangements. Among all 364 beads we mea-sured, spontaneous bead motions showed superdiffusive MSDsat lag times �t � 1 s; this result is comparable to those inprevious reports (8, 9). These data indicated, further, that for alltime lags spontaneous motions were larger at the cell centerthan at edges or corners (data not shown). When the MSD wasfitted to a power law at large time lags (�t � 4 s), the prefactorD* was significantly higher in the center than the other regions,but there was no significant difference between the edge andthe corner (Table 1). The exponent � showed no significantdifference between groups (P � 0.05). When we mapped D*onto a square grid, D* was highest in the center and smallest inthe corners (Fig. 3A). To compare directly the local cellremodeling with local stiffness, we divided the maps of D* andstiffness (g) into small squares (5 � 5 �m2) and cross-plottedthe mean values of D* and g within each square (Fig. 3B).Consistent with our previous work in semiconfluent cells that

are not micropatterned (8), there was a strong inverse associ-ation between regional distributions of D* and those of g.

Height profile. To examine the relationship, if any, betweenregional cell height (h) and local cell stiffness, h was measuredwith AFM (Fig. 4A). When we compared local cell height hand local stiffness g, the local stiffness increased as the localcell height decreased. However, for reasons addressed in theDISCUSSION, quantifying the direct influence that cell height hason the stiffness as measured by OMTC is not possible.

F-actin distribution. To further investigate the role of F-actin in regional heterogeneity of cell mechanical properties,we stained F-actin with rhodamine phalloidin and measured theregional distribution of the amount of F-actin by confocalmicroscopy. We projected the fluorescent intensities withineach slice onto the bottom plane, summing the intensities; thisrepresents the amount of F-actin per unit area. F-actin wasconcentrated around the cell corners and cell center (Fig. 4C).There was no clear relationship between local cell stiffness andlocal F-actin amount (Fig. 4D). This failure is not surprisingbecause stiffness would be expected to be related to F-actinconcentration rather than amount, which in turn is proportionalto local cell height. We therefore divided the local F-actinamount by the local cell height to find the local F-actin density(i.e., amount of F-actin per unit volume). F-actin density wasalmost constant in the cell center and perinuclear regions andhigh in the corner and edge regions (Fig. 4E). The relationshipbetween the local stiffness and the local F-actin density wasgenerally monotonic, with stiffness increasing with F-actin

Fig. 2. Distribution of local stiffness g (A) andpower law exponent x (B).

Fig. 3. A: distribution of local mean squaredisplacement (MSD) prefactor D*, evaluatedfrom the MSD at lag time �t � 1 s. B: crossplot of local MSD prefactor D* vs. localstiffness g. Plotted are mean and SD withinsmall squares (5 � 5 �m2) into which wedivided the maps of D* and stiffness g.

C1248 MAPPING CYTOSKELETAL PRESTRESS

AJP-Cell Physiol • VOL 298 • MAY 2010 • www.ajpcell.org

on May 25, 2010

ajpcell.physiology.orgD

ownloaded from

density, but the relationship was not simple (Fig. 4F). Inparticular, at higher densities a roughly linear relationship wasfound, whereas at lower densities the stiffness appeared to varyalmost independently of F-actin content. As we show below,the complexity of this relationship arises in part because thereis a hidden independent variable, the regional prestress.

Traction forces. For a square micropatterned cell on anelastic gel, cell-induced displacements of the elastic substratewere biggest at the cell corners, and, as others have found (3,6, 45, 65), the traction forces were concentrated in thesecorners (Fig. 5A, inset). There was minimal traction in the cellcenter, and the directions of traction vectors were mostlycentripetal (data not shown).

Prestress. We computed the full tensor for intracellularstress as a function of x,y,z, using finite element analysis of a

continuum model for a cell subjected to the tractions obtainedfrom FTTM. At each x,y location, we computed the averageover z of the maximum principal stress and defined this scalaras the local prestress. Similarly, the orientation of the prestresswas taken as the average over z of the orientation of themaximum principal stress. We confirmed that the averageprestress over an imaginary plane transecting the cell wasequivalent, by a balance of forces, to the summation of trac-tions at the cell-gel interface (59). The first finding was that theprestress within a single cell was not homogeneous. The prestresswas largest in the corners and decreased along the diagonal linetoward cell center, where it was smallest but not zero (Fig. 5A).The direction of prestress along the diagonal was centripetal andparallel to the direction of traction. The prestress along the edge ofa square cell, however, was parallel to the cell boundary and

Fig. 4. A: average height of cell measuredwith atomic force microscopy (AFM) (n �5). B: cross plot of local stiffness g vs. localcell height. C: mean fluorescent intensity ofF-actin in a square cell (n � 36). A.U.,arbitrary units. D: cross plot of g vs. localF-actin amount. E: distribution of the localF-actin density calculated by dividing F-ac-tin amount by cell height. F: cross plot of gvs. local F-actin density.

C1249MAPPING CYTOSKELETAL PRESTRESS

AJP-Cell Physiol • VOL 298 • MAY 2010 • www.ajpcell.org

on May 25, 2010

ajpcell.physiology.orgD

ownloaded from

perpendicular to the direction of traction. When we compared theregional stiffness to the regional prestress, a simple and closepositive relationship was revealed (Fig. 5B).

DISCUSSION

Here we report three major findings. First, the adherentHASM cell has regionally heterogeneous distributions of CSKmechanical properties; corners are more fluidlike and stifferand have slower remodeling dynamics. Second, the regionalremodeling rate is inversely correlated with the regional stiff-ness. Third, regional mechanical properties correlate onlypoorly with regional actin density but correlate strongly andlinearly with the regional prestress. In this discussion we firstaddress the experimental limitations and cytoskeletal dynam-ics, and then go on to address the relationship of our findingsto data already in the literature. We conclude with new inter-pretations of these findings.

Limitations surrounding OMTC have been described indetail elsewhere (9, 18, 67). Of particular importance to ourstudy of systematic regional variations of cytoskeletal mechan-ical properties, regional variations in cell height complicate thepicture because bead rotation or displacement depends on boththe degree of bead embedding into the cell and the proximityof the cell-glass interface (37, 42). To infer cytoskeletal elasticmoduli from raw mechanical data, all experimental probes ofcell mechanics require some model of cytoskeletal deforma-tion. In general, regardless of the experimental probe, varia-tions of model validity across different cell regions, and thebiases they might engender, are unknown. In particular, as thecell height decreases (such as in the corners) any model’srelationship between the elastic modulus and the bead displace-ment diverges, and, in consequence, it is in precisely theseregions that any predictions from the model are exquisitelysensitive to small changes in cell height and bead embedding.Both of these are difficult to measure, especially the latter, andfor these reasons we feel it most appropriate to focus on theinterpretation of our primary measurements rather than using amodel analysis of questionable utility. Nevertheless, it is truethat these factors do contribute to the regional stiffness heter-ogeneity that we report here; their specific quantitative roleremains an open question.

In contrast to prior investigations of the regional distribu-tions of cell material properties, this report includes character-ization of dissipation (friction) by two different measurements.

First, from the complex elastic modulus measured with OMTC,we computed the loss tangent, given by the ratio of lossmodulus to storage modulus [this is traditionally quantified byx � 1 � (2/�)tan1 (loss tangent) (18)]. Second, we measuredthe rate of CSK remodeling through the MSD of spontaneousmotion of the beads. In previous studies, we have found thatthe loss modulus is proportional to the storage modulus,implying near constancy of the loss tangent or hysteresivity,over a wide frequency range in lung parenchyma (19), smoothmuscle strips (20), and even whole cells (16). Here we foundthat within a single cell the loss tangent is nearly constant,leading to a small, albeit systematic, variation in x (Fig. 2B). Itshould be noted that this variation is quite small compared withthe stiffness variations (Fig. 2A). Hence, the local loss modulusis mostly determined by the local storage modulus, and wherestiffness is higher, such as in the cell corners, the loss modulusis correspondingly higher. Although the fluctuation-dissipationtheorem and the generalized Stokes-Einstein relationship breakdown in the living CSK (8, 13, 33, 67, 75), in regions where theloss modulus is the greatest there is nonetheless a generalslowing down of remodeling dynamics (Fig. 3B).

Stress fibers are concentrated in the corner regions where thestiffness is highest, but also a large amount of fibers is distrib-uted in the cell center where the stiffness is lowest (Fig. 4C).In consequence, the F-actin amount was found to have at besta poor correlation with the local stiffness (Fig. 4D). Thenucleus itself is known to be quite stiff (32) and is wellconnected with the cytoskeletal network (35), but its role indetermining local stiffness seems to be limited. Also, cytosolicactin filaments may exist in the center of the cell that are notpart of adhesive structures but serve other functions and there-fore do not contribute to cell stiffness. As noted in RESULTS, themore appropriate comparison would be between the localF-actin density (amount divided by height) and the localstiffness (Fig. 4F). In particular, the general trend is mono-tonic, and it even suggests the possibility of two distinctregimes: 1) in the center and perinuclear regions, the localstiffness varies significantly without appreciable change inF-actin density, and 2) in the corners and edges, the localstiffness changes roughly linearly with F-actin density (Fig. 4,E and F). These two different regimes confirm a role forF-actin density in setting the local cell stiffness that is complexand less than straightforward.

Fig. 5. A: distribution of prestress within asquare cell calculated with finite elementmethods (FEM). Direction and length of linesare the direction and magnitude of the pre-stress. These were determined from the cellheight and tractions at the cell-substrate in-terface, the latter shown in inset (n � 13),with the magnitude of the tractions colorcoded. B: cross plot of local stiffness g vs.local prestress. Dotted line is a linear fit.

C1250 MAPPING CYTOSKELETAL PRESTRESS

AJP-Cell Physiol • VOL 298 • MAY 2010 • www.ajpcell.org

on May 25, 2010

ajpcell.physiology.orgD

ownloaded from

By contrast, for explaining cell-to-cell variations of stiffnessit is well established that the single most important factor is thecytoskeletal prestress; this relationship is found to be one ofsimple direct proportionality (70) and holds true over a widevariety of pharmacological interventions (71) as well as instressed reconstituted actin gels (22). We reasoned, therefore,that a similar relationship might govern interregional variationsof the local prestress and the local stiffness, but interregionalvariations of the prestress had never before been measured.Here we were able to accomplish this by quantifying both localintracellular prestress, with cell height and traction data, andlocal stiffness. Our findings confirmed that the local stiffness isstrongly and linearly related to the local prestress as quantifiedby maximum principal stress (Fig. 5B). Although regional cellstiffness increased with regional F-actin density, the regionalstiffness was related more closely and more simply to theregional prestress. In that connection, myosins are distributedpreferentially along cell periphery (38), but it remains unclearwhether the distribution of regional prestress might be ex-plained by local acto-myosin activity because cytoskeletaltension is transmitted over great distances (24) and is balancedthrough the interconnected F-actin network. While a closerelationship between average cell stiffness and average cellprestress has been established previously across a population ofcells (71), this result is the first to show that this relationshipextends to intracellular variations.

In conclusion, our results support three major findings. First,cell corners are stiffer than the cell center. The corner of amicropatterned cell mimics the periphery of an unmicropat-terned adherent cell where it builds focal adhesions, connectedin turn to stress fibers bearing tensile stress and contributing toprestress. Second, remodeling dynamics are slower in thecorner regions. This is consistent with the loss modulus in-creasing proportionately to the increasing stiffness, resulting inslower remodeling dynamics. Finally, systematic regional vari-ations of CSK stiffness correlated only poorly with regionalF-actin density but strongly and linearly with regional pre-stress. These findings quantify both the random and the sys-tematic variations of intracellular mechanical properties thatunderlie localized cellular functions.

ACKNOWLEDGMENTS

We thank Drs. G. Whitesides and B. Mayers of Harvard University for thehelp with micropatterning and the Center for Nanoscale Systems of HarvardUniversity for the support of micromachining and AFM measurement. We alsoappreciate valuable comments and help from J. J. Fredberg’s lab members.

DISCLOSURES

No conflicts of interest are declared by the author(s).

REFERENCES

1. An SS, Fabry B, Trepat X, Wang N, Fredberg JJ. Do biophysicalproperties of the airway smooth muscle in culture predict airway hyper-responsiveness? Am J Respir Cell Mol Biol 35: 55–64, 2006.

2. Azeloglu EU, Bhattacharya J, Costa KD. Atomic force microscopeelastography reveals phenotypic differences in alveolar cell stiffness. JAppl Physiol 105: 652–661, 2008.

3. Balaban NQ, Schwarz US, Riveline D, Goichberg P, Tzur G, SabanayI, Mahalu D, Safran S, Bershadsky A, Addadi L, Geiger B. Force andfocal adhesion assembly: a close relationship studied using elastic mi-cropatterned substrates. Nat Cell Biol 3: 466–472, 2001.

4. Balland M, Desprat N, Icard D, Fereol S, Asnacios A, Browaeys J,Henon S, Gallet F. Power laws in microrheology experiments on living

cells: comparative analysis and modeling. Phys Rev E Stat Nonlin SoftMatter Phys 74: 021911, 2006.

5. Bausch AR, Moller W, Sackmann E. Measurement of local viscoelas-ticity and forces in living cells by magnetic tweezers. Biophys J 76:573–579, 1999.

6. Bershadsky AD, Balaban NQ, Geiger B. Adhesion-dependent cellmechanosensitivity. Annu Rev Cell Dev Biol 19: 677–695, 2003.

7. Brock A, Chang E, Ho CC, LeDuc P, Jiang X, Whitesides GM, IngberDE. Geometric determinants of directional cell motility revealed usingmicrocontact printing. Langmuir 19: 1611–1617, 2003.

8. Bursac P, Lenormand G, Fabry B, Oliver M, Weitz DA, Viasnoff V,Butler JP, Fredberg JJ. Cytoskeletal remodelling and slow dynamics inthe living cell. Nat Mater 4: 557–561, 2005.

9. Bursac P, Fabry B, Trepat X, Lenormand G, Butler JP, Wang N,Fredberg JJ, An SS. Cytoskeleton dynamics: fluctuations within thenetwork. Biochem Biophys Res Commun 355: 324–330, 2007.

10. Butler JP, Tolic-Norrelykke IM, Fabry B, Fredberg JJ. Traction fields,moments, and strain energy that cells exert on their surroundings. Am JPhysiol Cell Physiol 282: C595–C605, 2002.

11. Chen CS, Mrksich M, Huang S, Whitesides GM, Ingber DE. Geomet-ric control of cell life and death. Science 276: 1425–1428, 1997.

12. Deng L, Fairbank NJ, Fabry B, Smith PG, Maksym GN. Localizedmechanical stress induces time-dependent actin cytoskeletal remodelingand stiffening in cultured airway smooth muscle cells. Am J Physiol CellPhysiol 287: C440–C448, 2004.

13. Deng L, Trepat X, Butler JP, Millet E, Morgan KG, Weitz DA,Fredberg JJ. Fast and slow dynamics of the cytoskeleton. Nat Mater 5:636–640, 2006.

14. Desprat N, Richert A, Simeon J, Asnacios A. Creep function of a singleliving cell. Biophys J 88: 2224–2233, 2005.

15. Duffy DC, Jackman RJ, Vaeth KM, Jensen F, Whitesides GM.Patterning electroluminescent materials with feature sizes as small as 5 umusing elastomeric membranes as masks for dry lift-off. Adv Mater 11:546–552, 1999.

16. Fabry B, Maksym GN, Butler JP, Glogauer M, Navajas D, FredbergJJ. Scaling the microrheology of living cells. Phys Rev Lett 87: 148102,2001.

17. Fabry B, Maksym GN, Shore SA, Moore PE, Panettieri RA Jr, ButlerJP, Fredberg JJ. Selected Contribution: time course and heterogeneity ofcontractile responses in cultured human airway smooth muscle cells. JAppl Physiol 91: 986–994, 2001.

18. Fabry B, Maksym GN, Butler JP, Glogauer M, Navajas D, TabackNA, Millet EJ, Fredberg JJ. Time scale and other invariants of integra-tive mechanical behavior in living cells. Phys Rev E Stat Nonlin SoftMatter Phys 68: 041914, 2003.

19. Fredberg JJ, Stamenovic D. On the imperfect elasticity of lung tissue. JAppl Physiol 67: 2408–2419, 1989.

20. Fredberg JJ, Jones KA, Nathan M, Raboudi S, Prakash YS, Shore SA,Butler JP, Sieck GC. Friction in airway smooth muscle: mechanism,latch, and implications in asthma. J Appl Physiol 81: 2703–2712, 1996.

21. Fredberg JJ, Kamm RD. Stress transmission in the lung: pathways fromorgan to molecule. Annu Rev Physiol 68: 507–541, 2006.

22. Gardel ML, Nakamura F, Hartwig JH, Crocker JC, Stossel TP, WeitzDA. Prestressed F-actin networks cross-linked by hinged filamins replicatemechanical properties of cells. Proc Natl Acad Sci USA 103: 1762–1767,2006.

23. Heidemann SR, Wirtz D. Towards a regional approach to cell mechanics.Trends Cell Biol 14: 160–166, 2004.

24. Hu S, Chen J, Fabry B, Numaguchi Y, Gouldstone A, Ingber DE,Fredberg JJ, Butler JP, Wang N. Intracellular stress tomography revealsstress focusing and structural anisotropy in cytoskeleton of living cells. AmJ Physiol Cell Physiol 285: C1082–C1090, 2003.

25. Ingber DE. Cellular tensegrity: defining new rules of biological designthat govern the cytoskeleton. J Cell Sci 104: 613–627, 1993.

26. Ingber DE. Mechanosensation through integrins: cells act locally butthink globally. Proc Natl Acad Sci USA 100: 1472–1474, 2003.

27. Ingber DE. Mechanical control of tissue morphogenesis during embryo-logical development. Int J Dev Biol 50: 255–266, 2006.

28. Ji L, Loerke D, Gardel M, Danuser G. Probing intracellular forcedistributions by high-resolution live cell imaging and inverse dynamics.Methods Cell Biol 83: 199–235, 2007.

29. Kidoaki S, Matsuda T, Yoshikawa K. Relationship between apicalmembrane elasticity and stress fiber organization in fibroblasts analyzed

C1251MAPPING CYTOSKELETAL PRESTRESS

AJP-Cell Physiol • VOL 298 • MAY 2010 • www.ajpcell.org

on May 25, 2010

ajpcell.physiology.orgD

ownloaded from

by fluorescence and atomic force microscopy. Biomech Model Mechano-biol 5: 263–272, 2006.

30. Kidoaki S, Matsuda T. Shape-engineered vascular endothelial cells:nitric oxide production, cell elasticity, and actin cytoskeletal features. JBiomed Mater Res A 81: 728–735, 2007.

31. Krishnan R, Park CY, Lin Y, Mead J, Jaspers RT, Trepat X,Lenormand G, Tambe D, Smolensky AV, Knoll AH, Butler JP,Fredberg JJ. Reinforcement versus fluidization in cytoskeletal mechano-responsiveness. PLoS One 4: e5486, 2009.

32. Kuo KH, Seow CY. Contractile filament architecture and force transmis-sion in swine airway smooth muscle. J Cell Sci 117: 1503–1511, 2004.

33. Lenormand G, Bursac P, Butler JP, Fredberg JJ. Out-of-equilibriumdynamics in the cytoskeleton of the living cell. Phys Rev E Stat Nonlin SoftMatter Phys 76: 041901, 2007.

34. Li S, Guan JL, Chien S. Biochemistry and biomechanics of cell motility.Annu Rev Biomed Eng 7: 105–150, 2005.

35. Maniotis AJ, Chen CS, Ingber DE. Demonstration of mechanical con-nections between integrins, cytoskeletal filaments, and nucleoplasm thatstabilize nuclear structure. Proc Natl Acad Sci USA 94: 849–854, 1997.

36. Massiera G, Van Citters KM, Biancaniello PL, Crocker JC. Mechanicsof single cells: rheology, time dependence, and fluctuations. Biophys J 93:3703–3713, 2007.

37. Mijailovich SM, Kojic M, Zivkovic M, Fabry B, Fredberg JJ. A finiteelement model of cell deformation during magnetic bead twisting. J ApplPhysiol 93: 1429–1436, 2002.

38. Murata K, Hirano K, Villa-Moruzzi E, Hartshorne DJ, Brautigan DL.Differential localization of myosin and myosin phosphatase subunits insmooth muscle cells and migrating fibroblasts. Mol Biol Cell 8: 663–673,1997.

39. Nagayama M, Haga H, Kawabata K. Drastic change of local stiffnessdistribution correlating to cell migration in living fibroblasts. Cell MotilCytoskeleton 50: 173–179, 2001.

40. Nagayama M, Haga H, Takahashi M, Saitoh T, Kawabata K. Contri-bution of cellular contractility to spatial and temporal variations in cellularstiffness. Exp Cell Res 300: 396–405, 2004.

41. Nishizaka T, Shi Q, Sheetz MP. Position-dependent linkages of fibronec-tin-integrin-cytoskeleton. Proc Natl Acad Sci USA 97: 692–697, 2000.

42. Ohayon J, Tracqui P, Fodil R, Fereol S, Laurent VM, Planus E,Isabey D. Analysis of nonlinear responses of adherent epithelial cellsprobed by magnetic bead twisting: a finite element model based on ahomogenization approach. J Biomech Eng 126: 685–698, 2004.

43. Ostuni E, Kane R, Chen CS, Ingber DE, Whitesides GM. Patterningmammalian cells using elastomeric membranes. Langmuir 16: 7811–7819,2000.

44. Panettieri RA, Murray RK, DePalo LR, Yadvish PA, Kotlikoff MI. Ahuman airway smooth muscle cell line that retains physiological respon-siveness. Am J Physiol Cell Physiol 256: C329–C335, 1989.

45. Parker KK, Brock AL, Brangwynne C, Mannix RJ, Wang N, OstuniE, Geisse NA, Adams JC, Whitesides GM, Ingber DE. Directionalcontrol of lamellipodia extension by constraining cell shape and orientingcell tractional forces. FASEB J 16: 1195–1204, 2002.

46. Paszek MJ, Zahir N, Johnson KR, Lakins JN, Rozenberg GI, Gefen A,Reinhart-King CA, Margulies SS, Dembo M, Boettiger D, HammerDA, Weaver VM. Tensional homeostasis and the malignant phenotype.Cancer Cell 8: 241–254, 2005.

47. Pelham RJ Jr, Wang Y. Cell locomotion and focal adhesions areregulated by substrate flexibility. Proc Natl Acad Sci USA 94: 13661–13665, 1997.

48. Petersen NO, McConnaughey WB, Elson EL. Dependence of locallymeasured cellular deformability on position on the cell, temperature, andcytochalasin B. Proc Natl Acad Sci USA 79: 5327–5331, 1982.

49. Pourati J, Maniotis A, Spiegel D, Schaffer JL, Butler JP, Fredberg JJ,Ingber DE, Stamenovic D, Wang N. Is cytoskeletal tension a majordeterminant of cell deformability in adherent endothelial cells? Am JPhysiol Cell Physiol 274: C1283–C1289, 1998.

50. Puig-de-Morales M, Millet E, Fabry B, Navajas D, Wang N, Butler JP,Fredberg JJ. Cytoskeletal mechanics in adherent human airway smoothmuscle cells: probe specificity and scaling of protein-protein dynamics.Am J Physiol Cell Physiol 287: C643–C654, 2004.

51. Rosenblatt N, Hu S, Chen J, Wang N, Stamenovic D. Distending stressof the cytoskeleton is a key determinant of cell rheological behavior.Biochem Biophys Res Commun 321: 617–622, 2004.

52. Rosenblatt N, Hu S, Suki B, Wang N, Stamenovic D. Contributions ofthe active and passive components of the cytoskeletal prestress to stiffen-ing of airway smooth muscle cells. Ann Biomed Eng 35: 224–234, 2007.

53. Rotsch C, Jacobson K, Radmacher M. Dimensional and mechanicaldynamics of active and stable edges in motile fibroblasts investigated by usingatomic force microscopy. Proc Natl Acad Sci USA 96: 921–926, 1999.

54. Sato M, Suzuki K, Ueki Y, Ohashi T. Microelastic mapping of livingendothelial cells exposed to shear stress in relation to three-dimensionaldistribution of actin filaments. Acta Biomater 3: 311–319, 2007.

55. Shroff SG, Saner DR, Lal R. Dynamic micromechanical properties ofcultured rat atrial myocytes measured by atomic force microscopy. Am JPhysiol Cell Physiol 269: C286–C292, 1995.

56. Small J, Rottner K, Hahne P, Anderson KI. Visualising the actincytoskeleton. Microsc Res Tech 47: 3–17, 1999.

57. Sollich P, Lequeux F, Hebraud P, Cates ME. Rheology of soft glassymaterials. Phys Rev Lett 78: 2020–2023, 1997.

58. Solon J, Levental I, Sengupta K, Georges PC, Janmey PA. Fibroblastadaptation and stiffness matching to soft elastic substrates. Biophys J 93:4453–4461, 2007.

59. Stamenovic D, Coughlin MF. The role of prestress and architecture of thecytoskeleton and deformability of cytoskeletal filaments in mechanics ofadherent cells: a quantitative analysis. J Theor Biol 201: 63–74, 1999.

60. Stamenovic D, Suki B, Fabry B, Wang N, Fredberg JJ. Rheology ofairway smooth muscle cells is associated with cytoskeletal contractilestress. J Appl Physiol 96: 1600–1605, 2004.

61. Su J, Brau RR, Jiang X, Whitesides GM, Lange MJ, So PT. Geometricconfinement influences cellular mechanical properties II—intracellularvariances in polarized cells. Mol Cell Biomech 4: 105–118, 2007.

62. Su J, Jiang X, Welsch R, Whitesides GM, So PT. Geometric confine-ment influences cellular mechanical properties I—adhesion area depen-dence. Mol Cell Biomech 4: 87–104, 2007.

63. Sunyer R, Trepat X, Fredberg JJ, Farre R, Navajas D. The tempera-ture dependence of cell mechanics measured by atomic force microscopy.Phys Biol 6: 25009, 2009.

64. Thery M, Pepin A, Dressaire E, Chen Y, Bornens M. Cell distributionof stress fibres in response to the geometry of the adhesive environment.Cell Motil Cytoskeleton 63: 341–355, 2006.

65. Tolic-Norrelykke IM, Butler JP, Chen J, Wang N. Spatial and temporaltraction response in human airway smooth muscle cells. Am J Physiol CellPhysiol 283: C1254–C1266, 2002.

66. Tolic-Norrelykke IM, Wang N. Traction in smooth muscle cells varieswith cell spreading. J Biomech 38: 1405–1412, 2005.

67. Trepat X, Deng L, An SS, Navajas D, Tschumperlin DJ, GerthofferWT, Butler JP, Fredberg JJ. Universal physical responses to stretch inthe living cell. Nature 447: 592–595, 2007.

68. Tseng Y, Kole TP, Wirtz D. Micromechanical mapping of live cells bymultiple-particle-tracking microrheology. Biophys J 83: 3162–3176, 2002.

69. Van Citters KM, Hoffman BD, Massiera G, Crocker JC. The role ofF-actin and myosin in epithelial cell rheology. Biophys J 91: 3946–3956,2006.

70. Wang N, Naruse K, Stamenovic D, Fredberg JJ, Mijailovich SM,Tolic-Norrelykke IM, Polte T, Mannix R, Ingber DE. Mechanicalbehavior in living cells consistent with the tensegrity model. Proc NatlAcad Sci USA 98: 7765–7770, 2001.

71. Wang N, Tolic-Norrelykke IM, Chen J, Mijailovich SM, Butler JP,Fredberg JJ, Stamenovic D. Cell prestress. I. Stiffness and prestress areclosely associated in adherent contractile cells. Am J Physiol Cell Physiol282: C606–C616, 2002.

72. Yamazaki D, Kurisu S, Takenawa T. Regulation of cancer cell motilitythrough actin reorganization. Cancer Sci 96: 379–386, 2005.

73. Yanai M, Butler JP, Suzuki T, Sasaki H, Higuchi H. Regional rheo-logical differences in locomoting neutrophils. Am J Physiol Cell Physiol287: C603–C611, 2004.

74. Yeung T, Georges PC, Flanagan LA, Marg B, Ortiz M, Funaki M,Zahir N, Ming W, Weaver V, Janmey PA. Effects of substrate stiffnesson cell morphology, cytoskeletal structure, and adhesion. Cell MotilCytoskeleton 60: 24–34, 2005.

75. Zhou EH, Trepat X, Park CY, Lenormand G, Oliver MN, MijailovichSM, Hardin C, Weitz DA, Butler JP, Fredberg JJ. Universal behaviorof the osmotically compressed cell and its analogy to the colloidal glasstransition. Proc Natl Acad Sci USA 106: 10632–10637, 2009.

76. Zhou EH, Lim CT, Quek ST. Power-law rheology analysis of cellsundergoing micropipette aspiration. Biomech Model Mechanobiol (Febru-ary 2010). doi:10.1007/s10237-010-0197-7.

C1252 MAPPING CYTOSKELETAL PRESTRESS

AJP-Cell Physiol • VOL 298 • MAY 2010 • www.ajpcell.org

on May 25, 2010

ajpcell.physiology.orgD

ownloaded from