geometricconﬁnement inﬂuences cellular mechanical ... · we investigated the “tensegrity”,...

Copyright c© 2007 Tech Science Press MCB, vol.4, no.2, pp.87-104, 2007

Geometric Confinement Influences Cellular Mechanical Properties I –Adhesion Area Dependence

Judith Su∗, Xingyu Jiang†, Roy Welsch‡, George M. Whitesides§, Peter T. C. So¶

Abstract: Interactions between the cell and theextracellular matrix regulate a variety of cellularproperties and functions, including cellular rhe-ology. In the present study of cellular adhesion,area was controlled by confining NIH 3T3 fibrob-last cells to circular micropatterned islands of de-fined size. The shear moduli of cells adheringto islands of well defined geometry, as measuredby magnetic microrheometry, was found to havea significantly lower variance than those of cellsallowed to spread on unpatterned surfaces. Weobserve that the area of cellular adhesion influ-ences shear modulus. Rheological measurementsfurther indicate that cellular shear modulus is abiphasic function of cellular adhesion area withstiffness decreasing to a minimum value for inter-mediate areas of adhesion, and then increasing forcells on larger patterns. We propose a simple hy-pothesis: that the area of adhesion affects cellularrheological properties by regulating the structureof the actin cytoskeleton. To test this hypoth-esis, we quantified the volume fraction of poly-merized actin in the cytosol by staining with fluo-rescent phalloidin and imaging using quantitative3D microscopy. The polymerized actin volumefraction exhibited a similar biphasic dependenceon adhesion area. Within the limits of our simpli-fying hypothesis, our experimental results permitan evaluation of the ability of established, micro-

∗ Department of Mechanical Engineering, MIT. Present ad-dress: Biochemistry and Molecular Biophysics Option,Caltech.

† Department of Chemistry and Chemical Biology, Har-vard University. Present address: National Center forNanoScience and Technology, China.

‡ Statistics and Management Science, MIT.§ Department of Chemistry and Chemical Biology, Harvard

University¶ Department of Biological Engineering and Department of

Mechanical Engineering, MIT.

mechanical models to predict the cellular shearmodulus based on polymerized actin volume frac-tion. We investigated the “tensegrity”, “cellular-solids”, and “biopolymer physics” models thathave, respectively, a linear, quadratic, and 5/2 de-pendence on polymerized actin volume fraction.All three models predict that a biphasic trend inpolymerized actin volume fraction as a functionof adhesion area will result in a biphasic behaviorin shear modulus. Our data favors a higher-orderdependence on polymerized actin volume frac-tion. Increasingly better experimental agreementis observed for the tensegrity, the cellular solids,and the biopolymer models respectively. Alter-natively if we postulate the existence of a criticalactin volume fraction below which the shear mod-ulus vanishes, the experimental data can be equiv-alently described by a model with an almost lineardependence on polymerized actin volume frac-tion; this observation supports a tensegrity modelwith a critical actin volume fraction.

Keyword: magnetic trap; microcontact print-ing; cytoskeleton; mechanotransduction

1 Introduction

The cytoskeleton – the primary structural compo-nent of the cell – determines the mechanical re-sponse of the cell to an externally applied load(1). The cytoskeleton plays a major role in cellu-lar processes such as migration, mitosis, and ad-hesion (1). Changes in cellular compliance caninfluence the expression of genes and the produc-tion of proteins, and can combine with other fac-tors such as adhesion strength to effect cellularmovement and differentiation (2). The study ofcellular rheology and its molecular basis is an ac-tive area of research in biomechanics, with sev-

88 Copyright c© 2007 Tech Science Press MCB, vol.4, no.2, pp.87-104, 2007

eral fundamentally different models proposed tolink macroscopic properties of the cell with mi-crostructural detail of the cytoskeleton (1). Thesimplest representation of cytoskeletal mechanicsapproximates the cell as a continuum, and usessprings and dashpots to represent its viscoelas-tic behavior (3). Other structurally based mod-els, such as the tensegrity model proposed by In-gber and co-workers, take into account how indi-vidual cytoskeletal components contribute to theelasticity of the cytoskeleton (4). A recent studyby Fabry et al presents results that suggest thatthe cytoskeleton behaves as a soft glassy material,and imply that the cytoskeleton may deform andflow like a liquid (5).

One key accomplishment in the study of cy-toskeletal mechanics has been the developmentof experimental methods to probe the cytoskele-tal mechanics of cells. These methods includemeasurements of indentation performed by “cellsquashing” and “poking” (6), micropipette as-piration (7), magnetic twisting cytometry (5),laser tracking microrheology (8), atomic force mi-croscopy (9), and laser (10) and magnetic tweezer(11, 12, 13, 14) based microrheometers. Thesestudies, however, do not quantitatively link cel-lular rheology to systematic changes in molecu-lar structure. Further, conventional measurementsof cellular rheology do not take into considerationthe fact that cellular properties may be affected bythe interactions between cells and their substrate.Recent developments in micropatterning make itpossible to examine the effects of cellular adhe-sion area’s size and shape (15, 16). Micropattern-ing further allows partial control of the biochem-istry and density of cellular adhesion molecules.Previous work by Whitesides and coworkers hasdemonstrated that the size and shape of cells in-fluence their passage through the cell cycle andtheir probability of initiating apoptosis (17). Con-trol of cellular shape through micropatterning canalso regulate aspects of cellular physiology suchas motility, differentiation, and the assembly ofcells into tissues (18, 19, 20, 21). We seek to elu-cidate the effects of cellular adhesion area on cel-lular rheology, and to correlate these mechanicalchanges with molecular level remodeling of the

cytoskeleton.

Previous studies of rheological properties as afunction of the characteristics describing the ad-hesion of the cell to the substrate involve deter-mining the effect of fibronectin density in hep-atocytes on cellular shear modulus using atomicforce microscopy (22); endothelial cells have alsobeen examined using magnetic twisting cytome-try (23). These studies, however, have been un-able to separate the effects of total cell area fromthe effect of the density of the adhesion moleculein the adhesion area. Micropatterning in combi-nation with magnetic microrheometry allows usto quantify – precisely and systematically – cellu-lar rheological properties and cytoskeletal struc-ture as a function of adhesion area. This tech-nique has allowed us to establish that the cellularshear modulus is a biphasic function of cellularadhesion area. As a rationalization of this bipha-sic behavior, we assume that cellular rheology isdominated by the mechanical properties of thepolymerized actin network. With this simplifyingassumption, standard cytoskeletal microstructuralmodels (tensegrity, cellular solids, and biopoly-mer physics) predict a power law dependence ofcellular shear modulus on the volume fraction ofpolymerized actin (defined as the ratio of the cel-lular volume occupied by the polymerized actin tothe total cell volume). We can therefore advancethe hypothesis that changes in cellular shear mod-ulus are a direct function of the polymerized vol-ume fraction of actin; this fraction is regulated,in turn, by the adhesion area of the cell. To testthis hypothesis, we measured the volume frac-tion of polymerized actin as a function of area ofadhesion of the cell to the substrate using phal-loidin labeling and spinning disk confocal mi-croscopy. The cellular shear modulus can be cal-culated based on polymerized actin volume frac-tion using the microstructural mechanical mod-els. The accuracies of the shear moduli derivedfrom the tensegrity, cellular-solids, and biopoly-mer physics models can then be compared withexperimental magnetic microrheometry measure-ments.

Geometric Confinement Influences Cellular Mechanical Properties I 89

2 Materials and Methods

2.1 Substrate preparation

Micropatterned substrates were created followingthe method of Whitesides and coworkers (24).A 130 Å-thick layer of gold was deposited onthe coverslip bottoms of Bioptechs culture dishes(Bioptechs, Butler, PA) using electron-beam va-por deposition. To promote the adhesion of thegold to glass, 30 Å of titanium was first evapo-rated onto the dish. To create the polydimethyl-siloxane (PDMS, Sylgard 184, Dow Corning,Midland, Michigan) stamp used for microcon-tact printing, negative photoresist (SU8-2015, Mi-crochem, Newton, MA) was spin-coated for 35seconds at 1500 rpm onto a 4” silicon wafer.The photoresist was then baked on a hotplate at65oC for 1 minute and 95oC for 2 minutes. Thephotoresist was exposed to UV light for 12 sec-onds through a transparency photomask (OutputCity, Poway, CA) to create the photoresist master.Following exposure, the wafer was post-bakedagain at 65oC for 1 minute and 95oC for 2 min-utes. The unpolymerized photoresist was washedaway with propylene glycol monomethyl ether ac-etate (PGMEA, Microchem, Newton, MA) andthe wafers were silanized for 15 minutes to pre-vent the PDMS from sticking. PDMS was pouredon the bas relief photoresist structure and bakedin an oven for 2 hours at 65oC. The PDMS stampwas then peeled off and coated with 2 mM hex-adecanethiol (Sigma Aldrich, St. Louis, MO) inethanol. Excess solvent was removed by evapora-tion in a stream of nitrogen, and the stamp waspressed onto the gold coverslip for 30 seconds.The stamp was then gently peeled off and the bot-tom of the bioptechs dish covered with 2mM of apolyethyleneglycol (PEG)-terminated alkanethiol(Prochimia, Poland) for 30 minutes. Afterwardsthe dishes were rinsed once with ethanol and oncewith phosphate buffered saline (PBS) solution,(Invitrogen Life Technologies, Carlsbad, CA).

2.2 Cell culturing and plating

Cells were cultured according to instructionsprovided by the American Type Culture Col-lection (ATCC). NIH 3T3 fibroblasts (ATCC,

Manassas, VA) were grown using high glucoseDubecco’s Modified Eagle Medium (DMEM)supplemented with 10% v/v bovine calf serum(Invitrogen Life Technologies, Carlsbad, CA) and1% penicillin/streptomycin in 10 cm2 tissue cul-tures plates in an incubator at 37oC until 70% con-fluent. Cells were split in a laminar flow hoodwhere the media was vacuum aspirated off witha sterile pipette and 2.5 mL of trypsin-EDTA wasadded. The cells were placed back into the in-cubator for 5-7 minutes until they disassociatedfrom the bottom of the plate. An equivalentamount of serum-containing media was added toinactivate trypsin. The solution of media and cellswas gently mixed using a 5-mL pipette for severalminutes until a homogeneous suspension of cellswas obtained. The desired quantity of cells wasadded to a new tissue culture plate for continuedpropagation and placed back into the incubator.

To plate the cells, 0.5 mL of 0.25 μg/mL ofhuman plasma fibronectin (Sigma Aldrich, St.Louis, MO) in PBS was placed in each stampedBioptechs dish for four hours at 37oC. The fi-bronectin was aspirated out, while the desirednumber of cells was simultaneously pipetted. Ap-proximately 30,000 cells were added per dish.

2.3 Actin staining

Cells were fixed with 3.7% formaldehyde (Z-fix, Anatech LTD, Battlecreek, MI) and their F-actin stained with Alexa Fluor 488-conjugatedphalloidin (Molecular Probes, Eugene, Oregon)according to directions provided by MolecularProbes. The media was first aspirated and thecells were washed twice with PBS. Z-fix wasadded for 10 minutes and then removed by aspi-ration. The Bioptechs dishes were washed againtwice with PBS and a solution of 0.1% Triton X-100 was added for a total of 5 minutes to allowfor entry of the dye. Triton X-100 was removedand the bioptechs dishes were washed twice withPBS. The dishes were soaked then in a 1% bovineserum albumin (BSA) solution for 30 minutes toreduce non-specific binding. The staining solu-tion consisting of 200 μL of PBS and 15 μL ofmethanolic dye solution was then added to eachdish for 20 minutes after which the dishes were


washed again twice with PBS, once with distilledwater, and gently blown dried with nitrogen.

2.4 Whole-cell stain

For total cell volume measurement, cells wereincubated using a 1-μM solution of cell-trackergreen (CMFDA) in media for 15 minutes at 37oC.The dye-containing solution was then replacedwith media for 30 minutes, after which cells wereready for imaging.

2.5 Gel and actin filament preparation

Actin (1 mg, Cytoskeleton Inc., Denver, CO) wasdissolved in 100 μL of ddH2O. General actinbuffer (GAB), (Cytoskeleton Inc., Denver, CO)was added to the actin vial to create a concentra-tion of 0.5 mg/mL. This mixture was placed on icefor one hour. Actin was polymerized by mixing100 μL of the actin and GAB mixture with 10 μLof actin polymerizing buffer (ABP), (Cytoskele-ton Inc., Denver, CO). Alexa Fluor-488 phalloidinwas added at the desired concentration and thismixture was placed on ice in the dark for one hour.The solution was centrifuged for 30 minutes at10,000 rpm to pellet the actin and wash out the ex-cess dye. Equal volumes of filament solution weremixed with ProLong (Molecular Probes, Eugene,Oregon) on a glass slide, and the mixture left tosolidify.

2.6 Magnetic bead preparation

Paramagnetic polystyrene beads (4.5 μm diame-ter, Dynal, Oslo, Norway) with a tosyl-activatedcoating were covalently conjugated to fibronectinaccording to instructions provided by the manu-facturer. The solution containing the magneticbeads was placed on top of a magnet to drawthe beads to the bottom. The storage solutionwas then removed by aspiration and the beadswashed once in a borate buffer solution with a pHof 9.4. Fibronectin was added (5 μg/107 beads)and the solution was gently agitated for 10 min-utes at 37oC. BSA was added until its concentra-tion was 0.1%, and the entire mixture was agitatedovernight at 37oC. The beads were washed threetimes with 0.1% BSA in PBS, and once with Trisbuffer with 0.1% BSA (pH 8.5). Prior to use, the

beads were mixed in 1% BSA in PBS for 5 min-utes, and resuspended in media. This mixture wasadded to the cells 24 hours before experimentswere performed.

2.7 Magnetic trap setup

The magnetic trap (Figure 1) was constructed fol-lowing the design of Huang, et. al (25). Aferromagnetic CMI-C rod (Cold Metal ProductsInc) was machined and heat treated to improve itsmagnetic properties (25). The trap was wrappedapproximately 550 times with 21 gauge copperwire which was held in place by epoxy.

Figure 1: Magnetic trap schematic

A computer-controlled current sent through thecoil generated a magnetic field that exerts force onthe magnetic beads. The force increases exponen-tially as distance from the tip decreases (26). Thedisplacement of the bead over time in responseto a step force was recorded to determine the ef-fective stiffness of a cell. To calibrate the trap,a force was applied to a magnetic bead in a so-lution of known viscosity (polydimethylsiloxane)(Sigma Aldrich, St. Louis, MO) (26). Imageswere recorded at 30 fps using a CMOS camera(Silicon Imaging, Costa Mesa, CA). The steadystate velocity of the bead was found by measur-ing the displacement over time using a customparticle-tracking program (27) written in Matlab(Mathworks, Natick, MA).

For experiments in the cell, the tip of the mag-


Figure 2: (top) In order from left to right: an array of 30 micron cells at 10x magnification stained withAlexaFluor488-phalloidin and Sytox Orange, two 30 micron cells at 25x magnification, and a 30 microncell at 100x magnification. (bottom) 3D reconstruction of cell on a 50 micron pattern with an endocytosedmagnetic bead. The view on the right is a section view (vertical cut) and the view on the bottom is a sectionview (horizontal cut). Cross hairs are located on the bead. Cells on 50 micron patterns have the lowestaverage height. Beads are completely internalized after 24 hours.

netic trap was wiped with 70% ethanol to preventcontamination and placed in the same plane asthe bead. Cells were kept at 37 oC through theuse of a temperature-controlled stage (Delta TC3,Bioptechs, Butler, PA), and an objective heater(Bioptechs, Butler, PA). For each experiment, a5-second step-forcing function of 20 ± 1 nN offorce was applied. All experiments were done atthe same distance between the magnetic bead andthe pole piece.

3 Results

3.1 Quantification of cellular rheological pa-rameters as a function of area of adhesion

We used micropatterned planar circular islands of10-, 20-, 30-, 40-, and 50- micron diameters toconfine cells (Figure 2 (top)). The size range ofthe islands was chosen to be as wide as possiblewith the following consideration for the upper andlower limits: if the islands were too small (∼5


microns) the cells would undergo apoptosis (14);if the islands were too large, then one single cellcannot fully spread to occupy the entire area ofthe island. To quantify the effective stiffness ofthe cells, magnetic beads were dropped on a dishof cells cultured on these substrates, allowed to in-ternalize for 24 hours, and pulled on for 5 secondsusing the step-force protocol at 20 nN amplitudeas described previously. Only cells with one mag-netic bead per cell were examined. The 24-hoursinternalization period was designed to eliminatethe effect of bead rolling which occurs in beadson the cell surface because the magnetic force in-duces a torque around the bottom of the bead inaddition to a translational movement (28). Theadded rotational movement of the bead causes asignificant difference between the measured beadcenter displacement and the inferred displacementof the cell membrane thus contributing a signif-icant error to this class of magnetic-force-basedrheological measurements (28). We further foundthat internalized particles provided more consis-tent data than beads on the cell surface (where ad-hesion strength may vary with local receptor den-sity). In confirmation with another published pa-per (29), beads were found to be internalized af-ter 24 hours via confocal imaging (Figure 2 (bot-tom)).

The measured displacement of magnetic particleswas fitted to a Voigt model (3) in series with adashpot (Figure 3) to obtain the values for thethree associated viscoelastic parameters that char-acterize the response curve. A four-parametermodel consisting of a Kelvin model in series witha dashpot was also tried, but was found to be un-necessarily complicated: a three-parameter modelfit the data as well as did the four-parametermodel.

We can interpret the meaning of the viscoelasticparameters as follows: The free dashpot and theVoigt element experience the same force,τ , withthe total deformation ε as the sum of the deforma-tion of the Voigt element,ε1, and the free dashpotdeformation, ε2.

From the definition of a dashpot, Dε2 = τη2

,where Dis the time differential operator,τ is theforce, and η2 represents the free dashpot as

Figure 3: Voigt element in series with a free dash-pot

Figure 4: Response of a Voigt element in se-ries with a dashpot for a unit step function forc-ing. The curve asymptotically approaches the topdashed line which represents the limiting functionfor large times. The lower dashed line indicatesthe contribution from the free dashpot.

shown in Figure 3. Therefore ε2 = τη2D and

ε1 = τμ+η1D Thus, ε = τ

η2D + τμ+η1D and it can

then be shown that the modeling equation is:η2με̇ + η1η2ε̈ = μτ + (η1 +η2) τ̇. The solu-tion of this equation under a step force, F ·H(t)is: ε(t) = F

[t

η2+ 1

μ(1−e(−t/tε ))]H(t), where

F is the force amplitude, H(t), is the unit stepand tε is the relaxation time, which equals η1/μ .This equation is plotted as Figure 4. When theforce is a rectangular function, a superpositionwith a time shift results in: τ (t) = F [H (t − t1)]−


F [H (t − t2)],

ε(t) = F

{t − t1

η2+

1μ

[1−e−(t−t1)/tε

]}H(t− t1)

−F

{t − t2

η2+

1μ

[1−e−(t−t2)/tε

]}H(t − t2)

which is represented in the following plot (Figure5):

Figure 5: (top) Solution of a Voigt model in serieswith a dashpot for a rectangular forcing function.(bottom) Representative data and correspondingcurve fit for a 30-micron diameter cell. Datapoints are shown at 1s intervals so that the curvefit may be seen clearly through the points.

Figure 6 shows measured values for the threeviscoelastic parameters. Error bars in all of thegraphs represent the standard deviation of themean (that is, the standard deviation of the dis-tribution divided by the square root of the numberof samples).

For each cell size, the rheological response fromtwenty cells was analyzed. We found that thevariance in the three viscoelastic parameters wassignificantly less than the variance in the threeviscoelastic parameters obtained from the control(unpatterned cells). We noted that in the graphof η2, due to the large difference in values be-tween the control and patterned cells, that a dif-ferent scaling was used with the break mark indi-cated on the vertical axis. If a uniform scale wasapplied, the error bars for the control would ap-pear significantly larger than the error bars for thepatterned cells. Statistical significance betweenvalues was determined by use of the Student’s t-test and the (*) was used to indicate the existenceof statistical significance between selected pair ofmeasurements. It appears that μ and η1 displaya definite biphasic response as a function of pat-tern diameter with a minimum at 30 microns. Thespring constant, μ is the measure of the elastic-ity and is demonstrated later to be proportional tothe shear modulus, G. The values of μ and η1

for the 20- and 30- micron diameter cells, are lessthan the values of these parameters for cells 10-,40-, and 50-microns in diameter; this differenceis statistically significant. We will demonstratelater that this minimum in shear modulus corre-lates with a minimum in the volume fraction ofpolymerized actin. In the graph of η2 vs. patterndiameter (Figure 6, bottom-left), we notice thatthere is a significant difference between the valueof the control cell and the patterned cells. Thehigh value of η2 of the control cell indicates thatthe control cell has the least residual deformation.The relaxation time,tε = η1/μ , (Figure 6, bottom-right) provides the characteristic time measure forthe cell to reach its full elastic deformation; tε ison the order of 1 second.

We can further approximate a frequency-dependent complex shear modulus, G∗(ω), witha real part, G′(ω), and an imaginary part G′′(ω),by modeling our experiments as a spherical beadembedded in a linear, infinite, uniform, andisotropic viscoelastic medium (30). Note thatthese validity conditions are only approximatelysatisfied for endocytosed particles in cells. Inparticular, the cell’s membrane adjusts to accom-


Figure 6: Viscoelastic parameters extracted from curve fit. (top-left) spring constant (top-right) dashpot inVoigt element (bottom-left) free dashpot (bottom-right) relaxation time. Twenty cells were measured foreach cell size. The * above the brackets indicates statistical significance (p < 0.05) between values. Errorbars in all graphs represent the standard deviation of the mean.

modate the height of the bead. We would like tonote that despite the close presence of the cellboundary Figure 2 (bottom), these assumptionsare widely assumed in all bead based rheologicalcalculations as there exists at present no analyticexpression which takes into account the bound-ary conditions for the calculation of the shearmodulus. A versatile, precise, and efficient finiteelement method to account for such complicatedboundary conditions has yet to be developed.As current rheological methods are unable toseparate out the contributions of the membranefrom the cytoskeleton, we view the shear modulustaken from this calculation as a lumped moduluswhich takes into account all contributions. Takingthese considerations into account leads to the

following established analytical expression forthe complex shear modulus:

G∗(ω) =f (ω)

6πRx(ω)

where f (ω) is the force exerted as a function offrequency, x(ω) is the resulting complex bead dis-placement, and R is the radius of the bead. We de-fine a stiffness G to be the value of the real part ofG∗(ω), G′(ω), measured at 0.05 Hz. This valuewas chosen to be much greater than the recipro-cal of the relaxation time, tε = η1/μ . The zerofrequency value, although representing the staticcomponent, was not chosen because the validityof the model breaks down due to the presence ofthe free dashpot which would predict infinite de-


Figure 7: Representative plot of G′(ω) and G′′(ω)for 30-micron diameter cell. These curves are de-termined analytically from curve fits of the data.

formation at zero frequency under any force. Fig-ure 7 shows a representative graph of G′(ω) andG′′(ω) (imaginary component of G∗(ω)) Figure8 shows G as a function of pattern diameter.

Figure 8 shows a plot of G as a function of patterndiameter with error bars representing the stan-dard deviation of the mean value. The value ofG that we obtain for the control cells is on theorder of 103 Pa; this value is within an order ofmagnitude of the values reported in the litera-ture for fibroblast cells by magnetic trapping (104

Pa, (11)) and similar to the values reported byatomic force microscopy (103 Pa, (9)). It maybe possible that the shear modulus for the con-trol value is larger than any other size on the pat-terns due to the fact that the cell achieves max-imum size if allowed to spread freely on an ad-hesive substrate. It is also possible that effectssuch as cell shape may play some role. As dis-cussed previously, our model for G∗(ω) neglectsboundary effects. One can not rule out the possi-bility that a change in average cell height on thedifferent patterns can affect the accuracy of shearmodulus measurement. Nonetheless, because theaverage cell height (total cell volume/base area)monotonically decreases while the shear modu-lus shows a biphasic behavior that decreases andthen increases, we can conclude that while heightmay contribute to the uncertainty in shear modu-lus measurement, it is at most a secondary factorin our measurement.

Figure 8: Shear modulus as a function of patterndiameter. Error bars represent the standard devia-tion of the mean (standard deviation of the distri-bution / square root of the number of samples).

The distribution of G for patterned and unpat-terned cells is shown in Figure 9. We note thatthe shear modulus histogram for patterned cellshas a much tighter distribution than the histogramfor unpatterned cells. We considered several mea-sures of variance including median absolute devi-ation and interquartile range (IQR). Specifically,IQR is a robust measure of spread that is designedto not be overly influenced by outliers. In bothcalculations, there was an increased variance ofthe shear modulus on the unpatterned substrates.On average, the unpatterned cells correlate witha larger area which lends credence to the notionthat size rather than shape creates variability. Oneinterpretation is that the cellular rheological prop-erties are more homogeneous in smaller areas.When the cell gets larger, there may be regionaldifference accounting for the variance.

We measured the actin cytoskeleton structural pa-rameters as a function area of adhesion (Fig-ure 10). Polymerized actin was visualized basedon phalloidin labeling. To obtain the volumeof polymerized actin present in each cell, three-dimensional images of AlexaFluor488-phalloidinlabeled cells confined to 10-, 20-, 30-, 40-, and50-micron diameter circles were taken using aspinning disk confocal microscope (Perkin ElmerUltraview, Wellesley, MA). A 100x Plan Apo


Figure 9: Distribution statistics of the shear modulus, G (value of G′(ω), measured at 0.05 Hz) for patternedand unpatterned cells.


Nikon objective with a numerical aperture of 1.45was used for imaging. Two-dimensional sliceswere taken every 100 nm for a total of 1000 slices.Each three-dimensional stack was deconvolved toremove the out-of-focus light using Huygens Es-sential (Scientific Volume Imaging, The Nether-lands) and reconstructed in the image visualiza-tion software Imaris (Bitplane AG, St. Paul, MN)where background was subtracted from the im-ages and a consistent threshold was used to de-termine the presence of actin.

Figure 10: Corrected actin volume of the cell asa function of pattern diameter. Inset demonstrateslinearity between actin concentration and total in-tensity. Error bars represent the standard devia-tion of the mean.

Because the diameter of a single actin filament (6-8 nm) is significantly less than the microscopypoint spread function of about 300 nm, themeasured total volume of the fluorescent voxelsgreatly overestimates the volume fraction of poly-merized actin in the cell. However, the poly-merized actin volume can be estimated using in-tensities by assuming that the local polymerizedactin density is proportional to the bound phal-loidin density and hence, to the intensity of fluo-rescence. The polymerized actin volume can thenbe determined if the fluorescence intensity emit-ted from a unit volume of polymerized actin canbe measured. This scaling factor was obtained byimaging a gel containing similarly fluorescently

labeled actin filaments at known weight concen-tration and measuring the corresponding inten-sity. Weight concentration was converted to vol-ume concentration using the known density foractin filaments (732 mg/mL, CRC Handbook ofBiochemistry). This calibration experiment pro-vides the needed scaling factor to convert the flu-orescence intensity at each voxel to the volumeof polymerized actin. To further validate this ap-proach, a series of actin-gel experiments were per-formed to establish linearity between actin con-centration and measured fluorescence intensity(Figure 10, inset). This result is corroboratedby the fact that the binding of AlexaFluor488-phalloidin and polymerized actin is stoichiomet-ric (1:1, Molecular Probes Handbook). The pro-portionality of actin concentration and measuredintensity is adequately verified by linear regres-sion obtaining a reduced χ2 value of 1.1 and a y-intercept of 0.1×105±0.78×103. The correctedtotal actin volume of the cell linearly increaseswith pattern diameter and is plotted as Figure 10.

To determine the actin density as a function of pat-tern diameter, a measurement of the volume of thecell is needed. The total volume of each cell (Fig-ure 11) was measured according to the conven-tional protocol of cell volume measurements (31)by fluorescently labeling live cells with the wholecell-permeant marker 5-chloromethylfluoresceindiacetate and using confocal microscopy to imagetwo-dimensional slices, in the present case, every100 nm. A consistent threshold was applied to de-termine the extent of the cell and total volume wascalculated by summing the number of fluorescentvoxels (Figure 11).

The measured total cell volume (Figure 11) indi-cates that cell volume increases with pattern di-ameter and appears to approach a plateau value.The variation of total cell volume with patterndiameter deviates from the expectation that thevolume would be conserved as adhesion area in-creases. Our cell tracker labeling result is inagreement with an independent cell volume mea-surement using actin labeling where a surfaceis numerically superimposed over the actin fila-ment images and the volume underneath calcu-lated (data not shown).


Figure 11: (left) Total cell volume as a function of pattern diameter. Error bars represent the standard devi-ation of the mean. (right) Average cell height as a function of pattern diameter. The height monotonicallydecreases whereas for the stiffness data a minimum occurs at 30 microns demonstrating that cell height isnot the most critical factor in determining cell stiffness.

The ratio of the corrected actin volume to totalcell volume determines the actin density or vol-ume fraction,φ , of actin in each cell. (Figure 12).The value that we obtain of 1-2% is similar tothe values reported for the volume fraction of en-dothelial cells (32).

Figure 12: Volume fraction of actin as a functionof pattern diameter. Error bars represent the stan-dard deviation of the mean.

4 Discussion

The main goal of our work is to understand howcellular structural changes dictate variations inrheology. By independently measuring rheolog-ical and structural parameters as a function of ad-hesion area, we can relate measured rheologicalproperties to measured cytoskeletal properties viacellular micro-mechanical models. This processalso helps to evaluate the validity of these mod-els. More specifically, we can independently pre-dict the relative change in shear modulus obtainedfrom magnetic trapping by substituting the vol-ume fraction of actin present in each cell into cel-lular micro-mechanical models that give a scal-ing relation for the Young’s modulus of the cy-toskeletal network (based on the amount of actinpresent, and on its material properties) (1). Thesepredictions provide a molecular-level explanationfor the change in effective cell stiffness as mea-sured by magnetic trapping.

Three of the best developed cellular micro-mechanical models are the tensegrity (33),cellular-solids (32), and biopolymer physics (34)model. For the tensegrity model, the structuralintegrity of the cytoskeleton is maintained by theinteraction of elements in tension and compres-sion. In the cellular solids theory, the cytoskeletalmeshwork is treated as a compilation of staggered


unit cells connected at their midpoint by struts.In the biopolymer physics model, the cytoskele-tal network is treated as a polymer gel in whichthere are entangled or cross-linked filaments. Weemphasize the obvious: that these microstruc-tural models are highly idealized representationsof cellular structures. Nonetheless, they providea starting point in discussions of the influence ofcellular structure on cellular rheology. Althoughmany molecular structural components, such asactin, microtubules, and intermediate filaments,can contribute to cellular rheology, we choose asimplifying assumption focusing only on the actincomponent of the cytoskeleton. Specifically, wehave tested a simple hypothesis: changes in poly-merized actin volume fraction regulate changesin cellular shear modulus. Controlling the areaof cellular adhesion using microcontact printingprovides a useful way to vary cellular actin vol-ume fraction. Note that this hypothesis neglectsmany factors including structural contributionsfrom other cytoskeletal components such as inter-mediate filaments, possible changes in actin pre-tension, or changes the Young’s modulus of indi-vidual actin filaments. Nonetheless, as a first or-der hypothesis, it is reasonable to first test the roleof actin density in regulating the observed cellularrheological differences.

For various formulations of the tensegritymodel (e.g. two-dimensional two-strut andthree-dimensional six-strut models), the networkmodulus,En, scales linearly with volume fraction(1). We fit our data into the two-strut tensegritymodel, in which the Young’s modulus of the net-work is linearly dependent on the volume frac-tion, φ , and the Young’s modulus of an actinfilament (En ∼ E f φ ) and the six-strut tensegritymodel, in which the Young’s modulus of the net-work scales linearly with the volume fraction ofactin and the pre-stress (σc) and pre-strain (ε) ofthe network. For the theory of cellular solids,the network Young’s modulus scales as E f φ 2 (29)and for the biopolymer physics model the networkYoung’s modulus equals kBT l2

pa−5φ 5/2 where kB

is the Boltzmann’s constant, T is temperature, lp

is the persistence length, and a is the radius of anactin filament (34). Assuming an incompressible

material we can further relate the shear modulusmeasured experimentally with the Young’s mod-ulus, E, in the models (E = 3G).

Specifically, the relationship between the Young’smodulus and the actin volume fraction can be ex-pressed for all three models as follows:

E = Aφ where A ∝ σc3

1+4ε1+12ε for the six-strut

tensegrity model

E = Aφ 2 where A ∝ 38E f for the cellular solids

model

E = Aφ 5/2 where A ∝ kBT l2pa−5 for the biopoly-

mer physics model

where A is the proportionality constant encom-passing factors such as pre-stress and pre-strainin the tensegrity model, the Young’s modulus ofactin in the cellular solids model, and tempera-ture, persistence length, and radius of an actin fil-ament in the biopolymer physics model. A non-linear regression can be performed to fit our mea-sured E as a function of φ by minimizing theχ2 merit function (Experimental Data Analyst,Mathematica) (Figure 13), where xi and σxi rep-resent the value and the standard deviation of themean for φ respectively, and σEi is the standarddeviation of the mean of the Young’s modulus.

χ2(k) =N

∑i=1

(Ei −Axi)2

σ2Ei

+A2σ2xi

This curvefitting approach allows us to extract theproportionality constant, A, from our data. Com-paring the fits of the three microstructural mod-els, the biopolymer physics model and the cellu-lar solids model appear to provide better agree-ment than the tensegrity model (Figures 13 and14). We can analyze this in a more quantitativefashion by using a two sample independent t-testthat compares the mean of all 100 observationson log E (modulus) with the mean of all 100 ob-servations on log φ multiplied by the exponent ofinterest (1, 2, 2.5) plus the log of A. Since thereare 20 observations for each pattern diameter, weweight each batch of 20 data points by our esti-mate of their standard error before computing themean of all 100 points. Our probability model isthat the two means are normally distributed andthe difference in the means is hypothesized to be


Figure 13: (top) Tensegrity model: non-linear fitof Young’s modulus as a function of volume frac-tion, (middle) Cellular-solids model: non-linearfit of Young’s modulus as a function of volumefraction squared, (bottom) Biopolymer physicsmodel: non-linear fit of Young’s modulus as afunction of volume fraction to the 5/2 power.There are five data points, one for each patterndiameter.

Figure 14: Young’s modulus predicted by tenseg-rity, cellular solids, and biophysical polymermodel compared to measured values. Error barsrepresent the standard deviation of the mean. Thepoints are slightly separated laterally for clarity.

zero. The t-test gives a difference in the meansof 2.25 for the tensegrity model, 1.91 for the cel-lular solids model, and 1.81 for the biopolymerphysics model. At the 95% significance level thecritical difference is 2 which supports our quali-tative observation that the biopolymer and cellu-lar solids models appear to provide better agree-ment and argues that the exponent is not correctfor the tensegrity model and that it may be re-jected. The error would need to be further reducedto be able to definitively distinguish between thecellular solids and biopolymer physics model.

Using the constant of proportionality, A, obtainedfrom the linear regression, we can further calcu-late the Young’s modulus predicted by each modelusing measured values of volume fraction, andcompare these values to the measured Young’smodulus (Figure 14).

The pre-factor, A, is an experimentally obtainedscale factor associated with each model. Becausethese models do not predict these pre-factors apriori, the goodness of fit of the exponent of φ ,is the only way to compare the various models.In the case of the six-strut tensegrity model, weobtain an A value of 9.16×104 which representsthe pre-stress and pre-strain factor. This value isthe same as the value calculated by Stamenovic


and Coughlin (35) for a six-strut tensegrity model.This agreement is not surprising, as the A valuesfrom both measurements were obtained by fittingthe tensegrity model to experimentally measuredcellular rheological data.

For the cellular solids model, the scaling pre-factor was obtained through rheological measure-ments of actin gels and various fibrous materials.As φ approaches 1, the proportionality constantrepresents the Young’s modulus of the network.For the cellular solids model, we obtain an E f

value of 5 × 106. This value, according to themodel, represents the Young’s modulus of an in-dividual strut. This value is sometimes assumedto be the Young’s modulus of an individual actinfilament, E f (109 Pa, (36), (37) (38)). In actuality,however, E f of an individual strut is unknown be-cause the struts are most likely composed of fiberbundles which slide relative to each other and sothe strut would exhibit a significantly lower mod-ulus than that of a single filament. Further, thereare many other factors such as the irregularitiesof the actin network which also influence the truevalue of the coefficient. Our value for E f is a fac-tor of 100 lower than the value used by Satcherand Dewey measurement (32). This differencein actin Young’s modulus between our value andthe Satcher and Dewey value comes from the factthat our volume fraction (which is close to 2%)is slightly higher than the volume fraction valueused by Satcher and Dewey (close to 1%).

Similarly, the scaling pre-factor in the biopoly-mer model was also obtained through rheologi-cal measurements of actin gels. The pre-factor forbiopolymer physics model that we obtain (3.77)is two orders of magnitude larger than the num-ber (2 × 105) obtained from using rheologicalmeasurements of actin gels (34). The differencebetween these values represents differences be-tween the bending stiffness, kB, and the persis-tence length, lp, of the filaments in the actin gelversus the cell where factors such as the degree offilament cross-linking may be different.

We can further analyze the data in a less model-dependent fashion, by fitting the data to the powerlaw: E = Aφ b. This regression (Figure 15) returnsthe exponent of b = 3.48±0.02 and using the t-

test, a difference in the means of log E and logφ plus log A of 0.09 which suggests a differentmodel with a steeper polymerized actin fractiondependence. A higher order φ dependence, withb = 7/2, may be obtained, for example, by modi-fying the biopolymer physics model. In the origi-nal biopolymer physics model, the length betweenentanglements,Le, and the mesh size, ξ , are as-sumed to both scale with the concentration of thechains (34); however, as noted by Mackintosh, etal., the equal scaling of these quantities with con-centration need not hold when Le ≥ ξ (39) andmay result in a greater than 5/2 dependence ofmodulus on φ .

Figure 15: Fitting E = Aφ b. For a pure powerlaw model, the curve is forced to pass through theorigin. There are five data points, one for eachpattern diameter.

All three existing models stipulate that the curveof Young’s modulus versus polymerized actin vol-ume fraction should go through the origin; alter-natively our data is consistent with the Young’smodulus vanishing as the polymerized actin vol-ume fraction goes to 0.013 ±0.0028. This ob-servation suggests the possibility of alternativemodels with a critical volume fraction,φcritical, ofactin below which the cell will collapse on itselfand have zero stiffness. Qualitatively, this modelis plausible since a solid actin network requirespolymerized actin fragments to reach a percola-tion threshold.

We consider an alternative model,E = A(φ −


Figure 16: Modified tensegrity model fit. Thereare five data points, one for each pattern diameter.

φcritical)b, which incorporates the notion of a criti-cal actin volume fraction (Figure 16). With thismodel, an exponent b = 1.0 ±0.1 is obtained.Given the linear dependence of the shear modu-lus on polymerized actin volume fraction, this re-sult suggests a modified tensegrity model whichincorporates a critical actin volume fraction.

We can only speculate as to how cellular adhe-sion area affects actin density on the molecularlevel. A recent study by McBeath et. al., indicatesthat the proteins RhoA and ROCK are more ac-tive in spread than unspread cells, and that cellu-lar shape affects cellular differentiation via RhoAand ROCK. These proteins have been establishedto regulate cytoskeletal tension (20) and this ob-servation may explain the link between cellularadhesion area and cytoskeletal mechanics.

5 Conclusions

Our study has resulted in two major findings. Onefinding is that controlling the area of cellular ad-hesion reduces the variation in mechanical prop-erties compared with unpatterned cells by a fac-tor of approximately 4 (Figure 9). This reductionprovides not only a means of engineering a cellin a specific mechanical state, but also a methodof achieving higher reliability in cell mechanicsmeasurements than has previously been possible.The large difference between the variances in theYoung’s modulus for the un-patterned cells and

the patterned cells indicates that cell shape playsa role in determining cellular stiffness. This topicis an issue for further study. A second major find-ing is that adhesion area dependent changes in theshear moduli of cells that correlate with area ofadhesion may be largely mediated by changes inthe polymerized actin volume fraction.

In addition, although micro-mechanical mod-els, such as the tensegrity, cellular-solids, andbiopolymer physics model, have been proposedfor over a decade, there is no study where modelparameters are systematically varied and the pre-dicted rheological parameters compared with ex-perimental measurements. Soft lithography pro-vides a powerful method with which to controlcellular shear modulus and cytoskeletal structureby controlling adhesion area (and probably shape,although we have not explored this issue). Thisstudy provides the experimental data needed forcritical evaluation of these three models. By relat-ing the cellular shear modulus to the polymerizedactin volume fraction, we infer that models witha higher order φ dependence such as the biopoly-mer physics model and cellular solids model, pro-vide better agreement than the original tensegritymodel. Alternatively, a modified tensegrity modelwith a critical actin volume fraction can provideequivalent agreement to our data.

Given that the real significance of a complex sys-tem is always less than the statistical significancesince so many sources of error are omitted, ne-glected and underestimated, our hypothesis mustbe evaluated in future experiments. Specifically,measuring cellular rheology over a broader rangeof actin volume fractions, and examining corre-lations of cellular rheology with other cytoskele-tal structural components, should provide valu-able insight. Further, significant uncertainty inour analysis lies in the actin volume fraction mea-surement. This uncertainty in volume fractionpartially originates from the fact that the diffu-sion of dye in the actin filament gel may be differ-ent than the diffusion of dye within the cell. Wewill account for this effect in future experimentsby using stably transfected GFP-actin cells andmeasuring the F-actin/G-actin content using flu-orescence recovery after photobleaching (FRAP),


(40). This error in the volume fraction measure-ment may be furthered reduced by labeling GFP-actin cells with a whole cell stain and measuringactin volume and total cell volume on the samecell.

This paper only addresses the issue of the ef-fect of circular adhesion area on cellular mechani-cal properties. The investigation of axisymmetricshapes is of great interest in cell differentiationand migration. This is addressed in the compan-ion paper.

Acknowledgements: We would like to thankJorge Ferrer (MIT, Biological Engineering) andAlec Robertson (MIT, Mechanical Engineering)for the actin filament gels, John Wu (Harvard,Biology) for imaging Figure 2 (top), and mem-bers of the So Lab for helpful discussions. Weare also grateful to Prof. Forbes Dewey (MIT,Mechanical Engineering, Biological Engineering)and Prof. Roger Kamm (MIT, Mechanical En-gineering, Biological Engineering) for their com-ments and critical reading of the manuscript. Thiswork was supported by NIH PO1HL64858 anda NSF graduate research fellowship (JS). Thesalary of Xingyu Jiang was provided by NIHGM065364.

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