J Appl Physiol controversies in physiology...scales in the hierarchy of life gain their shape stability and their ability to exhibit integrated mechanical be-havior through use of

controversies in physiologyOpposing views on tensegrity as a structural frameworkfor understanding cell mechanics

DONALD E. INGBERDepartments of Pathology and Surgery, Children’s Hospitaland Harvard Medical School, Boston, Massachusetts 02115STEVEN R. HEIDEMANN, PHILLIP LAMOUREUX, AND ROBERT E. BUXBAUMDepartment of Physiology, Michigan State University, East Lansing, Michigan 48824-1101

Donald E. Ingber: Important new theories in scienceoften ignite heated debates. If they do not, they areprobably of little significance. Thus a strong argumentin support of the importance of the tensegrity model ofcell and tissue architecture first proposed almost 20years ago (23, 24) is the large number of public andprivate criticisms that have been mounted against thistheory. Demonstration of the ability of the tensegritymodel to explain complex mechanical behaviors in vi-ruses, nuclei, cells, tissues, and organs in animals aswell as in insects and plants (reviewed in Refs. 4, 5, 7,10, 17, 20–24, 26, 30, 32, 42) has led to a drasticreduction in the number of these confrontations. Nev-ertheless, some intransigent critics remain. However,their remaining objections are limited in scope andlargely result, I believe, from an overly concrete defi-nition of what tensegrity is and how it can be applied.My purpose here, at the request of the Editor andAssociate Editors of this journal, is to present theargument in support of the tensegrity model and torespond to some of these remaining concerns.

The tensegrity model states that cells, tissues, andother biological structures at smaller and larger sizescales in the hierarchy of life gain their shape stabilityand their ability to exhibit integrated mechanical be-havior through use of the structural principles oftensegrity architecture (5, 20, 22–24). The term,“tensegrity” (contraction of “tensional integrity”) wasfirst created by the architect R. Buckminster Fuller,who first explored use of this form of structural stabi-lization as early as 1927 in his plan for the WichitaDymaxion house, which minimized weight by separat-ing compression members from tension members (31).To create this cylindrical building, Fuller proposed toset a central mast in the earth as a vertical compres-sion strut and to suspend from it multiple circularfloors (horizontal wheels) using tension cables. Tensileguy wires that linked the mast to surrounding anchorsin the ground provided the balancing tension necessaryto stabilize the entire structure. “Fuller called thisspecial discontinuous-compression, continuous-tension

system, the Tensegrity” (31) to emphasize how it dif-fers from conventional architectural systems (e.g.,brick-on-brick type of construction), which depend oncontinuous compression for their shape stability. Full-er’s more formal definition in his treatise, Synergetics,is “Tensegrity describes a structural-relationship prin-ciple in which structural shape is guaranteed by thefinitely closed, comprehensively continuous, tensionalbehaviors of the system and not by the discontinuousand exclusively local compressional member behav-iors” (16). Note that there is no mention of rigid struts,elastic strings, tensile filaments, internal vs. externalmembers, or specific molecular constituents in thisdefinition. In fact, Fuller describes a balloon with non-compressible gas molecules pushing out against atensed rubber membrane as analogous to one of hisgeodesic domes when viewed at the microstructurallevel (i.e., the balloon is a porous, tensed molecularnetwork on the microscale) and explains that bothstructures are classic examples of shape stabilitythrough tensegrity. Fuller also described hierarchicaltensegrity structures in which individual struts or ten-sile elements are themselves tensegrity structures on asmaller scale; key to this concept is that smallertensegrity units require external anchors to othertensegrity units to maintain higher order stability. Infact, he argued that nature utilizes this universal sys-tem of tensile structuring at all size scales and that itprovides a way to mechanically integrate part andwhole (16), a view I recently explored in greater depth(22).

In 1948, Fuller’s student, Kenneth Snelson, con-structed the first “stick-and-string” tensegrity sculp-ture, which thrilled Fuller because it visibly communi-cated the essence of this novel form of shape stability tothose who could not “see” it in more complex structures(e.g., geodesic domes with rigid struts; see Fig. 5 in Ref.5). Snelson’s sculptures contain isolated compressionmembers that are suspended in midair by interconnec-tions with a continuous tensile network. Some of thesestructures require anchorage to the ground to remain

J Appl Physiol89: 1663–1678, 2000.

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stable (e.g., large cantilevered structures); however,most are entirely self-stabilizing. Similar stick-and-string tensegrity models have been used to visualizetensegrity in cells and other biological structures forthose who cannot easily visualize them (Figs. 1 and 2).The appearance of geodesic patterns in biological struc-tures, including viruses, clathrin-coated vesicles, andactin geodomes in the cytoskeleton of mammalian cells,provides additional visual evidence of nature’s use ofthis form of architecture (20, 22).

My own view of tensegrity has been refined over theyears as a result of extensive reading, personal corre-spondence with Fuller, conversations with Fuller’sclose associates (including Snelson), collaboration withexpert mechanical engineers, and many hours of think-ing about how to best respond (experimentally) to somevery intelligent critics. In simplest terms, tensegritystructures maintain shape stability within a tensed

network of structural members by incorporating othersupport elements that resist compression. The stiffnessof the stick-and-string tensegrity structures, and hencetheir ability to resist shape distortion, depends on thelevel of preexisting tension or “prestress” in the struc-ture before application of an external load. The distin-guishing microstructural feature accounting for thisbehavior is that, when placed under load, the discretestructural elements move, changing orientation andspacing relative to one another, until a new equilib-rium configuration is attained. For this reason, a localstress can result in global structural rearrangementsand “action at a distance.”

To visualize tensegrity at work, think of the humanbody: it stabilizes its shape by interconnecting multiplecompression-resistant bones with a continuous seriesof tensile muscles, tendons, and ligaments, and itsstiffness can vary depending on the tone (prestress) inits muscles. If I want to fully extend my hand upwardto touch the ceiling, I have to tense muscles down to mytoes, thus producing global structural rearrangementsthroughout my body and, eventually, upward exten-sion of my fingers. However, the body is also multimo-dular and hierarchical: if I accidentally sever my Achil-les tendon, I lose form control in my ankle module, butI still maintain structural stability in the rest of mybody. Furthermore, every time I breath in, causing themuscles of my neck and chest to pull out on my latticeof ribs, my lung expands, alveoli open, taught bands ofelastin in the extracellular matrix (ECM) relax, buck-led bundles of cross-linked (stiffened) collagen fila-ments straighten, basement membranes tighten, andthe adherent cells and cytoskeletal filaments feel thepull; however, nothing breaks and the deformation isreversible. Tensegrity provides a structural basis toexplain all these phenomena.

In the cellular tensegrity model, the stabilizing pre-stress is generated actively by the cell’s contractileapparatus and passively by distension through extra-cellular adhesions, by osmotic forces acting on the cell’ssurface membrane, and, on a smaller scale, by forcesexerted by molecular filaments extending throughchemical polymerization. The model assumes that theprestress is carried by tensile elements in the cytoskel-eton, primarily actin microfilaments and intermediatefilaments, and that the cell is both a hierarchical andmultimodular structure (5, 20–23) (Figs. 1 and 2). Thisprestress is balanced by interconnected structural ele-ments that resist being compressed at different sizescales, including the cell’s external adhesions to therelatively inflexible ECM and internal cytoskeletal fil-aments, specifically microtubules that stretch acrosslarge regions of the cytoplasm and cross-linked bundlesof cytoskeletal filaments that stabilize specialized mi-crodomains of the cell surface (e.g., actin microfila-ments in filopodia; microtubules in cilia). In this model,the internal cytoskeleton is surrounded by an elasticsubmembranous cytoskeleton (e.g., actin-ankyrin-spectrin network) and its associated lipid bilayer,which may or may not mechanically couple to theinternal, tensed microfilament-microtubule-intermedi-

Fig. 1. A hierarchical tensegrity model of a nucleated cell composedof sticks and elastic string when unanchored and round (top) vs.attached and spread on a rigid adhesive substrate (bottom). Theindependent nuclear tensegrity sphere is mechanically connected tothe surface of the larger tensegrity unit by black elastic filamentsthat are not visible against the black background. This model pre-dicts that rapid pulling on surface receptors that mechanically cou-ple to linking filaments in the cytoplasm may promote immediatechanges in nuclear structure, as confirmed experimentally (Ref. 30and Wang et al., unpublished observations; also see Fig. 3 below).

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ate filament lattice depending on the type of adhesioncomplex that forms. The entire cytoskeleton is perme-ated by the viscous cytosol. Most importantly, thismicromechanical model leads to specific predictionsrelating to the mechanical role of distinct cellular andmolecular elements in cell shape control.

In contrast, a conventional model of cell structure(12), which is espoused by my esteemed counterpartsin this article (18), depicts the cell as an elastic cortexthat surrounds a viscous cytoplasm with an elasticnucleus in its center. In engineering terms, this is a“continuum” model, and, by definition, it assumes thatthe load-bearing elements are infinitesimally smallrelative to the size of the cell. It is essentially theballoon model considered by Fuller, but in this case allmicrostructure is ignored. Because they ignore micro-structural features, continuum models cannot providespecific predictions that relate to the functional contri-bution of distinct cytoskeletal filaments to cell mechan-ics. Furthermore, although these models can provideempirical fits to measured mechanical properties incells under specific experimental conditions, they can-not predict how these properties alter under new chal-lenges to the cell.

Future advancement of our understanding of therelation between cell mechanics, molecular structure,and biological function requires a more unified cellmodel. This model must build on our existing knowl-edge of cell microstructure and take into account ex-perimental observations that reveal that the cytoskel-eton is organized as a porous molecular networkcomposed of discrete structural elements that physi-cally interconnect with external support networks inthe ECM and in neighboring cells (14). I would arguethat tensegrity provides this model. In fact, we andothers (including my counterparts in this article) have

shown that both buildable tensegrity structures (17,20, 23, 26, 42) and a theoretical tensegrity model de-veloped from first principles (9, 38, 39, 46) are robust interms of their ability to predict complex cell behaviorsin various experimental systems and across many dif-ferent size scales. Then why the continued criticisms?Let’s explore this in greater detail.

One of the most important features of the tensegritymodel, as opposed to the viscous cytosol model, is thatit predicts that applied mechanical forces will not betransmitted into the cell equally at all points on the cellsurface. In the tensegrity model, the submembranouscytoskeleton (cortical actin-ankyrin-spectrin lattice) isviewed as an independent tensegrity structure, whichis itself stabilized by the presence of a prestress withinits discrete porous (and geodesic) molecular network,as recently demonstrated in the purest form of thisstructure, the red blood cell membrane (11). Dependingon the molecular composition of the attachment sub-strate (e.g., ECM, surface of another cell) to which acell anchors, this highly elastic cortex may or may notmechanically couple to the internal microfilament-mi-crotubule-intermediate filament lattice, which, in turn,distributes loads throughout the cell and to the nu-cleus. A simple example of how the tensegrity modelhas contributed to the advancement of science is that ithas led to the proposal that adhesion receptors, such asintegrins, which form a transmembrane molecularbridge between the ECM and the internal cytoskeleton,provide a preferred path for transmembrane mechan-ical signal transfer and, hence, play a central role incellular mechanotransduction. On the basis of subse-quent experimental confirmation (8, 30, 35, 42), thisrole for integrins is now well established (7, 21).

The point here is that, if cells use tensegrity, thenlong-distance force transfer should be observed in liv-

Fig. 2. A multimodular tensegrity model of a portion of the internal cytoskeleton containing long microtubules(yellow) that interconnect and stabilize multiple smaller polygonal networks comprised of contractile microfila-ments (blue). Microfilament contraction induces compressive buckling in the semiflexible microfilament struts(right vs. left). This model is consistent with the finding that drugs that stimulate cell contraction increasemicrotubule curvature, whereas compounds that suppress this response promote straightening (Ref. 45 and Wanget al., unpublished observations).

1665CONTROVERSIES IN PHYSIOLOGY

ing cells. However, this action at a distance will only beobserved if the correct series of molecular couplings areformed between the surface receptor and the internalcytoskeletal lattice; externally applied stresses woulddissipate at the cell surface under other conditions. Incontrast, the elastic cortex-viscous cytosol model (12,18) would predict that living cells will never exhibitdirected action at a distance inside the cell. Impor-tantly, when we applied mechanical stresses directly totransmembrane integrin receptors using surface-bound micropipettes that were precoated with theECM molecule fibronectin, we observed immediate re-positioning of cytoskeletal filaments and elongation ofnuclei along the applied tension field lines as well asmolecular realignment within nucleoli deep in the cen-ter of the nucleus within living cells (30) (Fig. 3). Incontrast, no changes in intracellular structure wereobserved when tension was applied to other transmem-brane receptors on the cell surface that only couple tothe submembranous actin cytoskeleton. More recently,similar studies were carried out using pipettes to pullon ECM-coated microbeads bound to cell surface inte-grin receptors on cells that were transfected with en-

hanced yellow fluorescent protein (EYFP)-cytochromec to make mitochondria fluorescent throughout theentire cell. Real-time fluorescence microscopic analysesof these living cells revealed coordinated movement ofmitochondria during the entire course of the pull andrealignment of these natural fiducial markers; this wasobserved as far as 20 mm into the depth of the cell (N.Wang, K. Naruse, D. Stamenovic, J. J. Fredberg, S. M.Mijailovich, G. Maksym, T. Polte, and D. E. Ingber,unpublished observation). Again, pulling on othertransmembrane receptors that do not couple to theinternal cytoskeletal lattice (but do couple to the corti-cal actin cytoskeleton) did not result in long-distanceforce transfer as predicted by the tensegrity model.Because mitochondria directly associate with microtu-bules, these results indicate that forces transmitted tomicrofilaments via integrins can result in displace-ment of microtubules at distant sites and that thesedifferent filament networks are mechanically con-nected inside living cells.

The main reason for Dr. Heidemann’s change ofheart regarding tensegrity (he was one of the firstproponents of this model) is described in his recentpublication (18) in which the action at a distance heexpected to see was not observed when he pulled on cellsurface receptors using ECM-coated micropipettes.However, the ECM protein laminin, which was used inthat study, binds to classes of integrin receptors differ-ent from fibronectin and focal adhesion formation wasnot demonstrated in that study. In fact, his results arenot new or surprising: we and others have experimen-tally observed similar local responses and high cellmembrane deformability when cells were probed withbeads coated with antibodies to certain integrin sub-types (44) and even with fibronectin when analyzedduring the first few seconds after binding (i.e., beforefocal adhesion formation) (35) or when dragged overshort distances in the plane of the membrane (i.e.,when the submembranous cytoskeleton is the primaryload-bearing element) (1). Thus, consistent with thetensegrity model, the cell may appear to behave like anelastic cortex surrounding a viscous cytosol, if the sub-membranous cytoskeletal network is probed indepen-dently of the internal cytoskeleton (microfilament-mi-crotubule-intermediate filament lattice). In contrast,action at a distance can be observed when other recep-tors that provide deeper linkages (e.g., integrin a5b1)are ligated, although the specific molecular speciesinvolved will vary depending on cell type.

The cellular tensegrity model also differs from othermodels of cell mechanics in that it predicts that cy-toskeletal prestress is a critical determinant of cellshape stability. This has been demonstrated directly instudies in which cytoskeletal prestress was altered bymodulating actomyosin-based contractility using drugs(19), transfecting cells with constitutively active myo-sin light chain (MLC) kinase (3), varying transmem-brane osmotic forces (3) or quickly distending the flex-ible ECM substrate on which the cell is adherent (34),resulting in immediate changes in the cellular shearmodulus (a quantitative measure of stiffness or shape

Fig. 3. Phase-contrast (left) and polarization optic (right) views of anadherent endothelial cell immediately before (top) and after (bottom)a fibronectin-coated micropipette (visible in bottom) was bound tointegrin receptors on its surface and pulled laterally (downward inthis view) using a micromanipulator. Arrow in bottom left indicatesdownward extension of the nuclear border along the applied tensionfield lines. Arrowheads in bottom right point to white birefringentspots, which indicate induction of molecular realignment withinnucleoli in the center of the nucleus by applying mechanical stress tointegrins micrometers away on the cell surface. These results di-rectly demonstrate that action at a distance can occur in living cellsif external forces are applied via the correct set of transmembranemolecular linkages (e.g., integrins that form intact focal adhesioncomplexes), as predicted by the nucleated tensegrity model shown inFig. 1 (see Ref. 30 for more details).


stability). One may argue (and some have) that it maybe prestress in the cortical cytoskeleton (the elasticcortex in the continuum models, which view the cell asan inflated balloon or rubber ball) that is responsiblefor these effects. However, when cell mechanics wasmeasured by twisting on two differently sized magneticbeads bound to the same type of cell surface integrinreceptor using cell magnetometry, cell stiffness scaleddirectly with bead size for a given applied stress (cellsappeared to be less stiff using the smaller beads) (43);this result is the opposite of what would be predicted bya prestressed membrane cortex model. Furthermore,when cell mechanics was measured through cell sur-face integrins that connect to the internal cytoskeletallattice, cell stiffness was found to be increased inspread vs. round cells (43) and in cells expressingconstitutively active MLC kinase (3), whereas no sig-nificant difference in stiffness was measured when thesame cells were probed through transmembrane recep-tors that only connect to the cortical cytoskeleton inthose studies. Thus differences in shape stability dueto altered prestress in these cells cannot be explainedsolely by changes in the cell cortex.

The reality is that transmission of tension acrossmolecular connections within the cytoskeletal networkinfluences shape stability throughout the entire cell.For example, the shape and stiffness of the cell, inter-nal cytoskeleton, and nucleus can be altered by usingdrugs (30, 42) or genetic techniques (e.g., vimentinknock-out mice; Ref. 13) to disrupt the intermediatefilament lattice, which is known to extend throughoutthe depth of the cytoplasm. Coordinated retraction androunding of the entire cell, cytoskeleton, and nucleusalso were observed in membrane-permeabilized cellswhen ATP was added under conditions that supportedmicrofilament contraction but not when a syntheticpeptide that specifically blocks actomyosin filamentsliding was present (37). Quantitation of changes incell stiffness in these permeabilized cells confirmedthat tension within the internal cytoskeleton directlydetermined cell and nuclear shape stability, indepen-dently of transmembrane osmotic forces (43), clearlydemonstrating the inappropriateness of the “water bal-loon” or “inflated rubber ball”-type models of the cell.Finally, Dr. Heidemann’s own elegant studies on neu-rites show that the elastic cortex-viscous cytosol modelalone is not sufficient to explain how nerve cells pro-duce highly extended processes such as neurites (17,26). These cells also must be able to shift mechanicalforces between tensile microfilaments in the cortex,central microtubule compression struts, and externalECM tethers to extend these specialized projections. Inshort, continuous transmission of tension through thedepth of the cytoskeleton and between the cytoskeletonand ECM tethers is critical for cell shape stability.

Probably the most common concern raised over theyears has been, Where are the compression elements?The answer depends on the size scale and hierarchicallevel that one examines. If we ask how the whole cellcontrols its shape in living tissues (the ultimate ques-tion), then we have to take into account the contribu-

tion of the cell’s adhesions to ECM and to other cells aswell as internal support elements. The reality is thatmost cells cannot stabilize their shape in the absence ofthese adhesions: cells with highly specialized formsretract and round when detached from their anchoringsubstrate in vivo as well as in vitro. The reason that anadhesive substrate must be stiff (relative to the cell) topromote cell spreading is that isolated regions of thesubstrate located between the two integrin-containingfocal adhesions that form at the opposite ends of eachcontractile stress fiber must resist local compressionproduced by the contraction and shortening of eachfiber. The finding that cells can spread over multiplefocal adhesion-sized ECM dots that are separated bynonadhesive regions many micrometers in length (6)clearly demonstrates this point.

However, if the ECM were the only compressionelement, then all cells would be flat and smooth as afried egg. The reality is that cells also use many differ-ent types of internal compression struts to furtherrefine their shape, both in microdomains and at thewhole cell level. Internal microtubule struts are used tostabilize local regions of the cytoplasm (25, 41), tostiffen the mitotic spindle (32), and, when orientedvertically, to maintain a cylindrical cell form (2). Bun-dles of cross-linked (and, hence, further stiffened) mi-crotubules help to create specialized membrane exten-sions, such as cilia, and long cell processes, as inneurites (26). Stiffened bundles of cross-linked actinfilaments similarly stabilize the shape of exploratoryprojections (filopodia) that extend from the cell surfaceat the leading edge of migratory cells (36). These locallyrigidified structural elements are interconnected by acontinuous cytoskeletal lattice that is otherwise undertension; severing the cell in any location results inspontaneous cell retraction (34). Again, we see localcompression balanced by continuous tension, the defin-ing features of Fuller’s tensegrity systems.

What is the evidence that these structures actuallybear compression in living cells? Cilia and filopodia,which are rigid enough to resist distortion when probedby micropipettes (36), clearly must act locally to resistthe inwardly directed compression caused by thetensed cortical membrane to maintain shape stability,regardless of the theoretical model one favors. Micro-tubules have also been directly shown to resist com-pression in the mitotic spindles of living cells: when anultraviolet microbeam was used to sever one microtu-bule, the remaining microtubules buckled as expectedif the same total compressive load was now distributedamong a decreased number of semi-flexible compres-sion struts (32). This is an example of tensegrity at alower hierarchical level. Importantly, studies withgreen fluorescent protein (GFP)-labeled microtubulesalso revealed local buckling in the cytoplasm whenpolymerizing microtubules impinge end-on onto sur-rounding cellular structures and thus become com-pressed (Ref. 27 and Wang et al., unpublished obser-vations) (Fig. 4). My counterparts in this editorial haveargued that this form of microtubule buckling involvesvery small compressive loads; hence, it could result


from fluid flow in the surrounding cytosol (18). How-ever, analysis of time-lapse video recordings of cellsexpressing GFP-microtubules reveals no evidence offlow; rather, individual buckled microtubules can beseen to immediately straighten when they slip by anobstacle and then only buckle again when they hitend-on on a second obstacle (Ref. 27 and Wang et al.,unpublished observations). Furthermore, when cellscontaining EYFP-mitochondria or GFP-microtubuleswere repeatedly extended and compressed, with theextension sometimes held for more than 2 min beforerelease, no evidence of intracellular cytoskeletal flowcould be observed (Wang et al., unpublished observa-tions). In addition, the curvature of GFP-microtubules(a visual read-out of compressive buckling) decreaseswhen drugs are used to inhibit tension generation inthe surrounding actin cytoskeleton, whereas bucklingincreases when constrictors are added (Ref. 45 andWang et al., unpublished observations). Disruption ofmicrotubules also significantly reduces the shear mod-ulus (stiffness) of the cell and induces retraction of longprocesses in various cell types (26, 41, 42), thus con-firming the structural importance of their compres-sion-bearing role.

If microtubules are compression elements that main-tain cell shape stability by supporting a substantialpart of the tensile prestress, then their disruptionshould cause the prestress (or a significant portion ofit) to be transferred to the ECM, thereby increasing thetraction at the cell-ECM interface. In contrast, if mi-crotubules were tension elements, then their disrup-tion would inhibit transfer of traction to the ECM. Infact, many cell types increase tractional forces on theirECM substrate when treated with microtubule depoly-merizing agents (10, 20, 29), whereas disruption oftensile microfilaments dissipates stress (29). However,part of the effect of microtubule disruption has beenattributed by some to increases in MLC phosphoryla-tion in response to release of free tubulin monomersafter microtubule depolymerization rather than to atensegrity-based force balance (28). Importantly, simi-lar transfer of prestress from microtubules to the ECMwas recently demonstrated in cells that were pre-treated with chemical constrictors to optimally stimu-late MLC phosphorylation before microtubule disrup-

tion (Wang et al., unpublished observations) and wehave found that MLC phosphorylation does not in-crease when tubulin monomers are released in cells inwhich cytoskeletal tension is decreased using relaxantdrugs before microtubule disruption (Polte and Ingber,unpublished observations). In other words, the in-crease in MLC phosphorylation observed after micro-tubule disruption (28) does not result from release oftubulin monomers; rather, it appears to be a compen-satory mechanism that is activated in response totransfer of mechanical stress from microtubules to theECM and the remaining cytoskeleton in these cells.This is yet another example of a complex behavior thatcan be explained by tensegrity and not by the other cellmodels.

Some of those who accept that microtubules bearcompression locally within an otherwise tensed cy-toskeleton, a clear example of cellular tensegrity, thenargue whether this contributes significantly to cellmechanics. To explore this idea in greater detail, stud-ies were recently carried out in pulmonary airwaysmooth muscle cells cultured on flexible polyacryl-amide gel substrates containing small fluorescent mi-crobeads as fiducial markers, which permit quantita-tion of cell tractional forces and prestress withinindividual cells (by quantitating bead displacementrelative to the traction-free state of the gel after thecells are released using trypsin). Colchicine was usedto disrupt microtubules in adherent cells that wereactivated with a saturating dose of the chemical con-strictor histamine, again to ensure optimal MLC phos-phorylation. These studies revealed that microtubulescounterbalanced approximately one-third of the totalcellular prestress within an individual histamine-stim-ulated cell (Wang et al., unpublished observations).Thus these data confirm that the ability of microtu-bules to bear compression locally contributes signifi-cantly to cellular prestress and that prestress, in turn,is critical for maintenance of cell shape stability. How-ever, because of complementary (tensegrity-based)force interactions between microtubules, contractilemicrofilaments, and ECM, microtubules may bear lesscompression in cells when high levels of stress areborne by a rigid ECM substrate, just as tent poles maybear less compressive load if the tent is partially se-cured by tethers to an overlying tree branch. Thus,although the demonstration that microtubules do carrycompression in living cells is a strong support fortensegrity, a negative result in a particular cell wouldnot necessarily rule out this model.

Importantly, many biologists fail to recognize theimportant difference between engineering models thatcan describe (“curve-fit”) a complex cell behavior vs.one, such as tensegrity, that can explain and predictmultiple behaviors at many different size scales frommechanistic principles. For example, one can arguethat a tensed (prestressed) rubber ball, a liquid drop-let, or a spring and dashpot can mimic mechanicalbehaviors (e.g., strain-hardening behavior) observed inliving cells and tissues, as can tensegrity. This is true.In fact, living cell aggregates can be modeled with

Fig. 4. Two sequential time-lapse immunofluorescence views of thesame endothelial cell expressing GFP-tubulin showing a straightmicrotubule that extends through a large region of the cytoplasm(left), which then buckles locally due to compression (indicated byarrowhead) when it elongates through polymerization and impingesend-on on the stiffened cell cortex (right).


quantitative accuracy as liquids (15). However, weknow that these biological structures are not con-structed in this manner, and, indeed, the viscous cy-tosol-elastic cortex model (12,18) does not mesh withthe microarchitectural complexity that is observedwithin the cytoplasm of living cells (14). Essentially,these are all ad hoc models, and, as such, they do notprovide a means to explain these behaviors in mecha-nistic or molecular terms and do not lead to specificpredictions that are independent of the experimentalsystem. In contrast, Stamenovic and colleagues haveformulated a theoretical description of the tensegritymodel of the cytoskeleton starting from first principlesof mechanics (9, 38, 39). This micromechanical modelprovides multiple a priori predictions of which thestrain-hardening behavior of living cells is only one.For example, another key quantitative prediction aris-ing from the tensegrity model is that the static shearmodulus of the cell should change approximately lin-early with the prestress, that is, with the internaltensile stress that preexists in the cytoskeleton beforestress application (this is distinct from strain-harden-ing behavior). This model also suggests that cell me-chanical impedance can be decomposed into the prod-uct of a prestress-dependent component and afrequency-dependent component. Specifically, tenseg-rity predicts that, at a given frequency, both the stor-age and loss moduli should increase with increasingprestress, whereas the hysteresivity coefficient (thefraction of the frictional energy loss relative to theelastic energy storage) should be independent of pre-stress. Recent studies (Wang et al., unpublished obser-vations) demonstrate that these a priori predictionsare supported by experimental measurements of staticand dynamic mechanical behaviors in living cells andthus clearly demonstrate the validity and relativevalue of the tensegrity model. In short, the tensegritymodel provides mechanistic, theoretical, and quantita-tive bases to begin to define the molecular basis of cellmechanics as well as mechanotransduction; the rubberball model leaves us with, well, a rubber ball.

In summary, I hope that I have convinced you that,although the elastic membrane-viscous cytosol modelembraced by my counterparts in this discussion may beable to describe certain behaviors of cells, it cannotexplain others. This continuum model also does notprovide insight into the molecular basis of cell mechan-ics or the hierarchical basis of cell organization. Incontrast, tensegrity represents a unified model.Tensegrity can explain and predict from mechanisticprinciples how complex cellular behaviors observed atdifferent size scales and under different experimentalconditions emerge from collective interactions amongspecific molecular components. The cellular tensegritytheory also takes into account the molecular intricacyof living cells and can incorporate increasing levels ofcomplexity, including multimodularity and the exis-tence of structural hierarchies (5, 20, 22). These fea-tures may help to explain how molecular structures inspecialized regions of the cell are independently stabi-lized on progressively smaller size scales, although also

displaying integrated mechanical behavior as part ofthe larger cell and tissue (4, 5, 7, 17, 20, 22, 30, 33, 34,42). Because the tensegrity model is a mechanicalparadigm, it does not per se explain chemical behaviorin living cells. However, as many investigators (includ-ing Dr. Heidemann) have shown, tensegrity provides aframework to distribute and focus mechanical forces onspecific molecular components; hence, it may help toexplain how mechanical forces regulate cellular bio-chemistry and influence gene expression (7, 17, 21, 33).The other cell models that still dominate the literaturecannot.

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Steven R. Heidemann, Phillip Lamoureaux, andRobert E. Buxbaum: We have been thinking abouttensegrity architecture for cells since a scientific meet-ing, 15 years ago, at which Dr. Ingber pointed out to usthat our evidence on the mechanical roles of actin andmicrotubules in neurons fit a tensegrity structure. Wehad just conducted a mechanical reinvestigation (19) ofthe classic anti-cytoskeletal drug experiments ofYamada et al. (32) on neurons. He and others hadshown that depolymerizing microtubules caused axonsto retract suddenly, suggesting to us that the axon maybe under tension, which was normally balanced bycompression of the microtubules. Direct force measure-ments on axons before and during treatment withanti-microtubule and anti-actin drugs seemed to con-firm this mechanical hypothesis. Tension in axons in-creased when microtubules were depolymerized, andtension decreased when axons were treated with actin-disrupting drugs. Furthermore, increased tension inaxons caused microtubule depolymerization (9). Com-bined with the well-known spatial arrangement ofthese filaments in axons, the simplest interpretationwas that the outer actin network of axons is under asustained tension that is normally supported in part bythe inner bundle of microtubules. This complementaryforce balance between separate tensile and compres-sive elements is a basic feature of tensegrity. On thisbasis, we also proposed an idea related to, but separatefrom, tensegrity per se that shifts in this force balanceregulate microtubule assembly during axonal growth(1, 2).

The problems began when we assessed the tenseg-rity model more critically and compared it to oldermodels of cell architecture. In our view, the tensegritymodel of cells has at least two necessary features, bothfundamental according to Fuller’s own account oftensegrity (12). One, implied by the name derived from“tensional integrity,” is that continuous tension in theactin cortex fully integrates overall shape and struc-ture. Thus global integration of the cellular structure iskey; local mechanical inputs should produce distrib-uted cytoskeletal responses (“action at a distance”)because cytoskeletal elements are interconnectedthroughout the cell (4). Indeed, pull on one side of aclassic stick-and-wire tensegrity sculpture and thestructure as a whole shifts slightly toward the side


with relative motion even on the other side of thestructure. In support of this aspect of cells as tenseg-rity structures, Maniotis et al. (21) showed that aneedle attached to one side of a cell and pulled causedthe relatively distant nucleus to change shape. Thesecond necessary feature that cells should manifest iftensegrity is to be a useful model is that a significantportion of the compression balancing the surface ten-sion must be borne by discontinuous internal elements,e.g., cytoplasmic microtubules, not by attachment tothe dish or general compression of a fluid interior, as ina balloon. This requirement for internal compressivesupport is a key difference between tensegrity struc-tures and other tensile structures. That is, Fuller (12)makes a clear distinction between tensegrity and othertensile structures based on anchoring to an externalcompressive support:

“I also saw that man had long known of tensionalstructures and had experienced and developed thosetensional capabilities but apparently only as a second-ary accessory of primary compressional structuring.For instance, he inserted a solid mast into a hole in‘solid’ earth and rammed it in as a solid continuity ofthe unitary solid earth. However, to keep it from blow-ing over and breaking off when hurricane raged, headded a set of tension stays triangulated from the topof the mast-head to the ground, thus taking hold of theextreme end of the potential mast-lever at the point ofhighest advantage against motion. But these tensionswere secondary structuring actions.”

Insofar as cultured cell shape is clearly dependent onattachment to and compression of the “unitary solid”dish (cells round up when “trypsinized” from the dish),it would appear that, at best, cells can only approxi-mate true tensegrity structures as envisioned byFuller. Nevertheless, we would find tensegrity useful ifaxial compression along the microtubules would beseen to hold up part of the tensile forces known to existin the actin cytoskeleton and this structuring wasinterconnected throughout much of the cell.

Sadly, our recent experiments with cell structurefailed to support either of these key properties oftensegrity. We tested cell tensegrity (14) by pushing,pulling, prodding, and cutting the cytoskeleton of fibro-blasts whose actin and microtubule arrays were visu-alized in real time using cytoskeletal proteins labeledwith GFP (20). If actin and microtubules are highlyintegrated by a tensegrity interaction, or indeed at-tached to one another in any way, we should have seendistributed, generalized changes in cell shape and/or inthe filament array when force was applied to variousregions of the cell. Rather than integrated, spatiallybroad responses to forces, we repeatedly observedhighly local responses. The outer actin network didbehave elastically, but the internal microtubule cy-toskeleton behaved primarily like a fluid. Most disap-pointing, the outer elastic network of actin behavedindependently of the underlying cytoplasm with itsmicrotubules and other organelles. The most tellingseries of experiments were those in which glass needlesat the surface were strongly and effectively engaged

with the underlying actin cortex. In these experiments,glass needles were treated with an adhesion protein,laminin, to engage integrin receptors. As predicted bycurrent models of cell adhesion, we observed a rapidrecruitment of GFP-actin on the cytoplasmic side of theneedle tip, which had a robust mechanical attachmentto the cell (Fig. 8 in Ref. 14). Relatively weak tensionexerted by the needle caused the newly recruited spotof actin to move with the needle along the surfacewithout disturbing the underlying actin or microtu-bules. Larger forces exerted by actin-attached needlescaused the cell to change shape, but only a local exten-sion of cytoplasm formed. Rather than the cell and itssubstructure moving toward the pulled side, as pre-dicted by tensegrity, the cell shape changed so thatmost cytoplasm moved away from the needle! Mostdamaging to our view of tensegrity was that quite largeforces exerted by or on these short cellular extensionsproduced little change in the shape, position, or ar-rangement of microtubules directly adjacent the exten-sion. In addition, it was clear that the attachment hadan effective functional connection to the actin cortex inthat the cell was able to exert large contractile forceson the needle (Fig. 10 in Ref. 14). Whether the attachedneedle exerted forces on the cell or the cell exertedforces on needle, we repeatedly observed independenceof actin and microtubule behaviors among themselvesand failed to observe any effect of actin deformation onmicrotubule arrangements. Indeed, we were particu-larly surprised by the lack of evidence for any signifi-cant cytoskeletal interconnections in our recent exper-iments. Our deformations of the cell, with and withoutneedle linkage to the cortical cytoskeletal, producedmovements only among those microtubules or actinfilaments directly contacted by the needle. Even cy-toskeletal fibers quite near to the site of interventionwere unaffected. Thus our observations not only con-tradicted the global integration of the cytoskeletonrequired for the tensegrity model of the cell but gener-ally changed our view of the extent of interconnectionamong cytoskeletal elements.

Dr. Ingber and colleagues (17, 18, 26) have definedtensegrity as continuous tension and local compres-sion. However, we find this definition too broad. Onthis basis, tensegrity would include pup tents (i.e.Fuller’s compressional mast stabilized by a tensilecloth in place of discrete guy wires), suspensionbridges, and rubber membranes stretched out on aboard with multiple pins. These are all structures thatlong predate Fuller’s conception of tensegrity. Wewould define tensegrity structures as those with ten-sion-induced structural integrity resulting from a con-tinuous structure of tension-bearing elements and dis-continuous compressive elements integrally connectedto but dispersed within the structure by the tensionelements. In other words, in our view, cellular tenseg-rity requires architectural features that are quite sim-ilar in mechanical and shape properties to those of theclassic string-and-strut tensegrity sculptures of Snel-son. These sculptures, it should be noted, have beenused repeatedly as the models, illustrations, and the


sources of predictions for cellular tensegrity (7, 17, 28,31).

In addition to the multiplicity of tensile structures,old and new, that fit a broad definition of tensegrity, wepropose our more narrow definition of cellular tenseg-rity because two properties of the tensegrity modelemphasized by Dr. Ingber and colleagues are shared bycell models quite distinct from anything we could fairlycall “tensegrity.” These properties are prestress of theouter actin network and linear stiffening of the cell inresponse to deformation or increased stress (4, 7, 17,18, 26, 28, 31). In addition, these properties have bothbeen well described for years, if not decades, withoutrecourse to tensegrity models. Thus sustained tensionof the cell surface, i.e., prestress in the actin cell cortex,has been analyzed as far back as the 1930s (6), andthere have been well-controlled measurements of resttension at the cell surface and associated modelingthroughout the decades (10, 11, 15, 22, 23, 25, 33).Perhaps the simplest mechanical models for cells havebeen an inflated rubber ball (15) and a liquid drop (33).Although both of these structures would have unmis-takable prestress on the surface (due to elasticity andsurface tension, respectively), it is clear that neitherqualify as tensegrity structures.

As shown in Fig. 5, our analysis of these classic dataon liquid drops (Fig. 5A) and rubber balls (Fig. 5B)indicates that they share with tensegrity structuresthe property of being linearly strain/stress hardening,i.e., of behaving like an increasingly stiff spring withgreater force loads or extensions (15, 33). This is aninteresting property, but it cannot be regarded as di-agnostic for tensegrity. Indeed, cellular stiffening isfascinating to us because it appears at every scale offorce, length, and time. That is, stress/strain hardeninghas been measured with subcellular deformations andforces in the pico- to nanonewton range using atomicforce microscopy and laser optical trapping (5, 16, 27);in whole cells with deformations in micrometers andforces in the 0.1- to 1-mN range by plates, needles, andsuction (8, 10, 14, 29, 30); and in cell layers and tissueswith forces in the dyne to gram range with deforma-tions in the millimeters (3, 13, 24). However, thisstiffening effect can be explained by a wide variety ofmodels in addition to tensegrity, including simple vis-coelasticity (27), a liquid drop surrounded by an elasticcortex (10, 30), and active responses (5, 14).

Thus we are skeptical of the value of data on cellularprestress and/or stiffening for support of the tensegritymodel of cells. In our view, tensegrity requires clearevidence for cell-wide integration of the cytoskeletalstructure as seen by motion integration. We furtherrequire evidence of discontinuous, dispersed compres-sive support for the universally observed tension in theanimal cell cortex. This compressive support could besupplied by microtubules or other discrete cytoplasmicelements in arrangements similar to the classic strut-and-string tensegrity structures. For these reasons, wecontinue to hold out some hope that neurons will beshown to resemble classic tensegrity structures be-cause their axonal microtubules appear to be dispersed

and under compression and because they show cy-toskeletal integration in the form of action at a dis-tance in response to local mechanical disturbances. Forexample, Fig. 6 shows a reproducible shape-changephenomena in which towing of a chick forebrain axonat the distal end causes a significant migration of thecell body cytoplasm, including the nucleus, into theaxon shaft. When tension is relieved, the nucleus andcytoplasm then migrate back to the original positionamong the dendrites. At this time, we have no other

Fig. 5. Stress hardening of liquid drops and of rubber balls. A: stresshardening of a liquid drop; analysis of values from Table VI in themodel of Yoneda et al. (33) of the mechanical properties of a liquiddrop. As height of the drop (z) is decreased by external compression,surface area (s) increases. Because of the surface tension, increasedarea corresponds to increased energy. Thus the change in surfacearea (ds, effectively change in energy) with respect to relative height(2ds/dz) is a force (because energy is force acting through a distance).We calculated values of this force change at 8 drop heights down to80% of its original height and calculated values for stiffness (stiff-ness 5 force change/%height change), which are plotted here as afunction of the 8 corresponding forces. B: stress hardening of arubber ball; analysis of Fig. 8 in Hiramoto (15) in which empiricalmeasurements were taken on an inflated rubber ball. Weights weresequentially added to compress a rubber ball whose relative heightwas measured after application of 5 different forces from 0 to 8 kg. Ateach such step of deformation, the difference in force was divided bythe difference in height to provide a stiffness between deformations,plotted here as a function of the compressing force on the rubber ball.


interpretation of these data except that it is the sort ofgeneralized shape change typical of classic tensegrity:pulling on one end caused significant changes at theother end of the structure. Furthermore, althoughsome compressive support for neurite tension is clearlyprovided by dish attachment (neurites retract whengrowth cones are dislodged from substratum), thisshows the structure is under tension. In addition, ourolder work (outlined above) continues to suggest thatsubstantial compressive support may also be providedinternally by microtubules that are dispersed and in-tegrally connected to the tension structure.

In summary, our view of tensegrity is that it shoulddenote a quite specific type of architecture and must becarefully distinguished from other tensile structures.

As we have discussed here, most cell models are fun-damentally tensile and share mechanical propertieswith tensegrity architecture, such as prestress andstress hardening. Therefore, it will be important todevelop clear predictions that distinguish tensegritystructures from other tensile structures. Interest inmechanotransduction is increasing rapidly; the formathere is too brief to allow us to cite the reviews on therole of forces and mechanical properties in sensorytransduction, ventilation and other organ function,growth and development of plants and animals, andcellular differentiation and morphogenesis. However,unlike chemical signaling, for which textbooks andprofessional articles are adorned with elaborate, Rube-Goldberg-like sequences of molecular cause and effect

Fig. 6. Cytoskeletal action at a distance in culturedchick forebrain neurons. Experimental tension was ap-plied to a chick forebrain neuron with an attachedcalibrated glass needle. As previously reported for thisand other neuronal types, this causes the axon to elon-gate. Of particular relevance to tensegrity, at relativelyhigh tensions, the cytoplasm of the soma, including thenucleus, migrates into the axon during towing to leavea somatic “ghost” at the original site. When the needlewas pulled free, after ;2.5 h, the somatic cytoplasmreturned to its original location by 5 h. Particularlyintriguing was that, throughout the 5-h observation,“dendrites” from the cell body remained motile whetheror not the soma contained the nucleus and phase-densecytoplasm. The movement of cytoplasmic mass shownhere to change cell shape at one end of the neuron inresponse to tension applied at the other end is highlyreproducible, if technically demanding. Specifically,this cytoplasmic migration requires placing the neuriteunder tension just below that at which it would detachfrom the needle, i.e., pulling hard but not too hard.


(epinephrine activates the b-receptor that activates Gs,activating adenylate cyclase, and so forth), there arefew detailed models for mechanotransduction. Like thediagrams for chemical signaling, architectural modelssuch as tensegrity will help in visualizing and compre-hending mechanotransduction, but only if they areapproached critically and skeptically.

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REBUTTALS

Donald E. Ingber: I believe that most of the concernsraised by Dr. Heidemann and co-workers were ad-dressed in my original editorial; however, there are afew points that deserve further clarification and em-phasis. The major issue is the definition of tensegrity.I used Fuller’s own formal definition, which can befound in the definitive text of his life’s work (Synerget-ics). A more thorough definition of tensegrity thatclosely matches my own can be found in layman’sterms in A Fuller Explanation by Amy Edmondson, aclose associate of Fuller’s (3). The fact that tents, spiderwebs (e.g., stabilized by attachment to compression-resistant tree branches), and ship’s riggings (whichFuller often described in terms of tensegrity) existedfor years before Fuller’s birth is of no import. Fuller didnot invent this architectural method; he discovered theuniversality of its use and inspired its application byothers (e.g., Snelson). To arbitrarily narrow and makeconcrete Fuller’s definition and then to cite a randomFuller quote out of context, which was written for anartist’s journal, seems unreasonable to me; however, Ileave that to the reader. Perhaps most befuddling isthat Dr. Heidemann and colleagues then ignore theirown new narrowed definition when they “see” tenseg-rity in their own experimental system: they admit that“some compressive support for neurite tension isclearly provided by the dish attachment” (see above).


Another point of confusion is the concept that localmechanical inputs must result in action at a distancein a tensegrity structure. The point is that action atdistance (global structural rearrangements) can occurin tensegrity structures; however, this is not alwaysthe case and especially so in multimodular and hierar-chical arrays, as observed in living cells, tissues, andorganisms. For example, stress transmitted throughthe network will dissipate locally if it is passed to asupport element that is highly flexible; in essence, thisis why forces dissipate at the highly compliant lipidbilayer/submembranous cytoskeleton, whereas theypass deep into the cell across stiffer integrin connec-tions. In fact, local variations in the compliance ofdifferent cytoskeletal support elements may be howstresses are selectively transmitted to and focused onparticular transducing molecules (e.g., stress-sensitiveion channels) during the process of cellular mechano-transducton. Local accommodation and dissipation offorce may also be observed if the multimodular cy-toskeleton is tethered at many points to a fixed ECM;responses may only extend between neighboring adhe-sions and progress no farther. Multimodularity and theexistence of multiple tethers to extracellular scaffoldsalso may permit the cell to remove or dynamicallyrearrange a local support element without loss of me-chanical integrity in the larger structure. This form ofstructural memory could play an important role inmaintenance of cell form as well as tissue regeneration.

Yet another misconception by some is that tensegrityis only relevant for describing static behaviors. Alltensegrity structures exhibit characteristic dynamic(frequency dependent) behaviors; in fact, we have re-cently shown that a priori predictions from the tenseg-rity model relating to dynamic responses match nicelywith experimental results (N. Wang, K. Naruse, D.Stamenovic, J. J. Fredberg, S. M. Mijailovich, G.Maksym, T. Polte, and D. E. Ingber, unpublished ob-servations). Furthermore, in the cell, it is the three-dimensional arrangement of support elements withinthe tensegrity-stabilized array that channels and fo-cuses mechanical energy on the cytoskeleton-boundmolecules that mediate its remodeling. Thus tensegrityis also critical for slower time-dependent responsesbecause it guides how one instantaneous “hard-wired”tensegrity configuration will be transmuted into thenext; it its absence, pattern integrity would be lost overtime. Finally, it is well known that the different cy-toskeletal filament systems exhibit their own time-dependent (viscoelastic) responses (12); however, theseproperties are not sufficient to explain complex cellbehaviors, unless architecture and prestress (andhence, tensegrity) are also taken into account (Refs. 15and 19 and Wang et al., unpublished observations).

My colleagues’ major claim that negative resultsobtained in one study (7) using a single cell type and apoorly characterized method (pulling on cell mem-branes with laminin-coated micropipettes) are suffi-cient to disprove tensegrity and to discount the resultsfrom the various publications I cited in my editorial isabsurd to say the least. I also do not understand why

these authors did not consider that there might bepossible caveats in their work, given that we had pre-viously demonstrated action at a distance when wepulled on fibronectin receptors that we knew formedintact focal adhesions and not when we pulled on otherreceptors that only connected to the submembranousactin cytoskeleton (15). Action at a distance also hasbeen observed by other groups (16), including in arecent study using GFP-labeled intermediate filaments(8). Furthermore, buckling of microtubules was actu-ally demonstrated in the study by Dr. Heidemann et al.in which the cells’ adhesions were quickly detached(Fig. 6 in Ref. 7). However, so far, they have ignoredthis point. Another caveat not considered that wasraised in a prior publication (15) is that the flexibilityof the cytoskeleton becomes greatly reduced when cellsare cultured on rigid dishes coated with high densitiesof matrix molecules that promote formation of in-creased numbers of focal adhesions along the cell base.As described above, only microdomains of the multimo-dular cytoskeletal lattice between the fixed focal adhe-sions would be expected to rearrange or significantlydeform in response to a local mechanical manipulationunder these conditions. In fact, this is exactly what Dr.Heidemann’s team showed in well-spread fibroblasts: alocal incision in the cell resulted in a local retractionresponse (7). This was used as additional evidence toclaim the absence of “action at a distance” in cells andthus to invalidate the tensegrity model. We observedsimilar local responses when we cut highly adhesivecells (17); however, we found that we could obtain moreglobal responses by first using a micropipette like aspatula to loosen the basal adhesions beneath the cellbody before application of a similar incision. Interest-ingly, the nerve cells that Dr. Heidemann studies formrelatively few substrate adhesions beneath the cellbody; this may be why he can more easily visualizeaction at a distance at the whole cell level in thatsystem.

The most important point of this discussion is thatmy critics are correct in that there are alternativeexplanations and models that can explain the resultsfrom any single experiment. However, only the tenseg-rity model is consistent with all of these findings.Furthermore, only tensegrity can also predict many ofthese results a priori. For example, reconstituted gelsof intermediate filaments can also exhibit strain hard-ening (12); however, living cells still exhibit strainhardening after intermediate filaments are chemicallydisrupted or knocked out genetically (20). It is also truethat others structures, including rubber balls, liquiddrops, tensed cable networks, and tensed cortical mem-brane/viscous cytosol models, may exhibit strain-hard-ening behavior and approximately linear dependencesof stiffness on prestress. However, these other tensedmodels are not consistent with many other experimen-tal results (as I described in my original discussionabove) or with the microarchitecture that we observe inliving cells (dense cytoskeletal networks throughoutthe cytoplasm, straight microfilaments, curved micro-tubules). More importantly, Dr. Heidemann’s pre-


ferred elastic membrane-viscous cytosol-elastic nu-cleus model clearly does not fit with the structuralcomplexity that cell biologists know exist at the molec-ular level in the cytoplasm of living cells (4,9). It also isnot consistent with experimental results from manylaboratories including my own, which showed that in-termediate filaments connect nuclei to surface recep-tors, that microtubules and other cytoskeletal fila-ments play a key role in cell and nuclear shape control,and that there are filamentous load-bearing elementswithin the depth of the nucleus (2, 5, 14, 15, 19). Theother established continuum models of cell mechanics,although useful at the whole cell level in particularsituations, similarly offer no handle on the mechanicalrole of specific molecular structures or any mechanisticbasis for complex mechanical behavior in cells. Onlytensegrity satisfies all of these requirements.

Finally, I agree with Dr. Heidemann and his coau-thors when they state that “cellular stiffening is fasci-nating to us because it appears at every scale of force,length, and time.” However, again, only tensegrity canexplain why this behavior is observed at these differentsize scales. Clearly, our bodies and tissues are notconstructed like a liquid droplet, a rubber ball, or evena worm-like polymer; rather, they are prestressed hi-erarchical networks composed of contractile cells andextracellular matrices that can bear tension or com-pression locally. The finding that both the musculo-skeleton and mitotic spindle gain their stabilitythrough use of tensegrity and that regions of the actincytoskeleton (geodomes), organelles (clathrin-coatedvesicles), enzyme complexes, viruses, and protein fila-ments all exhibit tensegrity-based geodesic architec-ture (1, 11, 13, 18) provides perhaps the strongestargument that this model is the most robust theoreti-cal formulation of biological structure available at thepresent time.

FINAL STATEMENT

The critics of any new paradigm in science will alwaysbring up new problems and continue to “raise the bar.”However, a new theory will succeed if it is found to beuseful and if it provides new mechanistic insights tothe wider community. Although the current embodi-ment of the tensegrity model may not incorporate all ofthe features one might assume to be critical, experi-mental results confirm that it apparently does incor-porate the subset of features that are sufficient topredict many complex cell mechanical behaviors, infact, many more and diverse responses than any otherexisting model. More importantly, the introduction ofthe tensegrity model has also changed the way we viewcell regulation and has led to the recognition of thecritical importance of cytoskeletal prestress for controlof cell shape stability (Ref. 17 and Wang et al., unpub-lished observations) as well as for regulation of bio-chemical functions, including dynamic force-dependentremodeling of the cytoskeleton (6) and shape-depen-dent control of cell cycle progression (10). Like anytheoretical model, tensegrity is a work in progress that

will need to be continually refined as we gain moreinformation about the complex system we call the cell.However, in the end, it will be in the forging of a newunderstanding of the relation between mechanics, mo-lecular structure, and biochemical function that theimportance of higher order architecture and prestresswill become most clear and in which tensegrity willprovide its greatest value.

REFERENCES

1. Caspar DLD. Movement and self-control in protein assemblies.Biophys J 32: 103–138, 1980.

2. Eckes B, Dogic D, Colucci-Guyon E, Wang N, Maniotis A,Ingber D, Merckling A, Aumailley M, Koteliansky V, Babi-net C, and Krieg T. Impaired mechanical stability, migration,and contractile capacity in vimentin-deficient fibroblasts. J CellSci 111: 1897–1907, 1998.

3. Edmondson AC. A Fuller Explanation: The Synergetic Geome-try of R. Buckminster Fuller. Boston, MA: Birkhauser, 1987.

4. Fey EG, Capco DG, Krochmalnic G, and Penman S. Epi-thelial structure revealed by chemical dissection and unembed-ded electron microscopy. J Cell Biol 99: 203S–208S, 1984.

5. Goldman RD, Khuon S, Chou YH, Opal P, and Steinert PM.The function of intermediate filaments in cell shape and cy-toskeletal integrity. J Cell Biol 134: 971–983, 1996.

6. Heidemann SR and Buxbaum RE. Tension as a regulator andintegrator of axional growth. Cell Motil Cytoskeleton 17: 6–10,1990.


8. Helmke BP, Goldman RD, and Davies PF. Rapid displace-ment of vimentin intermediate filaments in living endothelialcells exposed to flow. Circ Res 86: 745–752, 2000.

9. Heuser JE and Kirschner MW. Filament organization re-vealed in platinum replicas of freeze-dried cytoskeletons. J CellBiol 86, 212–234, 1980.

10. Huang S, Chen CS, and Ingber DE. Control of cyclin D1, p27Kip1

and cell cycle progression in human capillary endothelial cells bycell shape and cytoskeletal tension. Mol Biol Cell 9: 3179–3193,1998.

11. Ingber D. The architecture of life. Sci Am 278: 48–57, 1998.12. Janmey PA, Eutenauer U, Traub P, and Schliwa M. Vis-

coelastic properties of vimentin compared with other filamen-tous biopolymer networks. J Cell Biol 113, 155–160, 1991.

13. Lazarides E. Actin, a-actinin, and tropomyosin interactions inthe structural organization of actin filaments in nonmuscle cells.J Cell Biol 68: 202–219, 1976.

14. Maniotis A, Bojanowski K, and Ingber DE. Mechanical con-tinuity and reversible chromosome disassembly within intactgenomes microsurgically removed from living cells. J Cell Bio-chem 65: 114–130, 1997.

15. Maniotis AJ, Chen CS, and Ingber DE. Demonstration ofmechanical connections between integrins, cytoskeletal fila-ments, and nucleoplasm that stabilize nuclear structure. ProcNatl Acad Sci USA 94: 849–854, 1997.

16. Mathur AB, Truskey GA, and Reichert WM. Atomic forceand total internal reflection fluorescence microscopy for thestudy of force transmission in endothelial cells. Biophys J 78:1725–1735, 2000.


18. Schutt CE, Kreatsoulas C, Page R, and Lindberg U. Plug-ging into actin’s architectonic socket. Nat Struct Biol 4: 169–172,1997.

19. Wang, N, Butler, JP, and Ingber, DE. Mechanotransductionacross the cell surface and through the cytoskeleton. Science 260:1124–1127, 1993.


20. Wang N, and Stamenovic D. Contribution of intermediatefilaments to cell stiffness, stiffening and growth. Am J PhysiolCell Physiol 279: C188–C194, 2000.

Steven R. Heidemann, Phillip Lamoureux, andRobert E. Buxbaum: Our distinguished counterpartin this debate is, of course, an eloquent and forcefuladvocate for tensegrity and cytomechanics generally.In addition, there is a great deal in his “pro” discussionwith which we wholeheartedly agree. Indeed, we havethe feeling that our behavioral model for the mechanicsof the cell is not much different. Even so, we continueto have strong doubts about tensegrity, both as a gen-erally useful model for these behaviors and as a clearlydefined word. We seem to agree that fluid vs. solidbehaviors by the cell is the key to this debate. Does thecell respond mechanically with elastic, sustained equi-libria between force and amount of deformation? Ordoes the cell respond with viscous dissipation of simpledeformations and force transmission by flowing? Dr.Ingber writes of viscous cytoplasm penetrating thecytoskeleton and he notes that his laboratory has seenthe same type of fluid behaviors that we recently pub-lished. Thus we all seem to agree that cells show bothbasic kinds of behaviors, although that is hardly anoriginal insight by any of us (10). More crucially, weagree that “transmission of tension across molecularconnections within the cytoskeletal network influencesshape stability throughout the entire cell.” We recentlypointed to the importance of such connections, demon-strated in experiments on growth cone crawling inAplysia (6). These experiments (11) quite directlyshowed transmission of actomyosin-generated tensionfrom the cell surface to the microtubule-rich, centralcytoplasm. Dr. Ingber’s view also implicitly acceptsthat these behaviors change with time. Cells showtensegrity behaviors only “if the correct series of mo-lecular couplings are formed” and “the elastic cortex-viscous cytosol model . . . will never exhibit directedaction at a distance” (see above). Notwithstanding ourdisagreement that the viscous model suggests any suchthing (see discussion of viscoelasticity below), there is atemporal dimension contained within “if” and “never.”Thus we probably agree that sometimes the cell re-sponds fluidly and at other times elastically. Theseagreements paradoxically lead to one of our two prin-cipal disagreements. From the standpoint of physicalbehaviors and analysis, the lack of a temporal dimen-sion to the ideas and mental images surrounding ar-chitectural tensegrity seriously compromises its valuefor biology. Whether one points to the Dymaxion house,geodesic domes, or string-and-strut models as illustra-tive tensegrity structures, their mechanical propertiesdo not change with time. The lack of a temporal dimen-sion to tensegrity has important implications both forits utility in cell modeling and in the physical evidenceused to distinguish between it and other models.

From the standpoint of physical evidence used tosupport or falsify tensegrity (or any other cell model),we think the time scale of this evidence is crucial to aclearer understanding of “what’s going on.” This arises

from another likely area of agreement: that cytoplasmin its fluid responsiveness is a viscoelastic fluid, notNewtonian like water. As anyone who has played withSilly Putty knows, whether it behaves like a solid or afluid depends on the rapidity or abruptness of theinput. Silly Putty can both bounce like a rubber balland flow slowly over the edge of a desk: pull it abruptlyand it breaks; pull it steadily and it flows. Similarly,both actin and microtubule suspensions, though liquidat long time scales and high stresses, behave as a solidat short time scales or small stresses (2, 7). Figure 3 ofour esteemed counterpart’s initial position piece is anideal example of the importance of such viscoelasticphenomena in this debate. This figure shows an uncon-tested example of elastic cellular response and evi-dence of connectedness between the cell surface andthe underlying cytoskeleton. If the time scale of thisobservation was 10 min, we would agree that it repre-sents the sort of solid interconnections envisaged bytensegrity and provides support for it. (We would arguethat our Fig. 6 in our original discussion above is justsuch an example.) On the other hand, if the time scaleof this observation is 10 s or less, we would bet ourmoney on viscoelasticity. Contrary to Dr. Ingber’s as-sertions concerning viscous models, which apply toNewtonian fluids, viscoelastic fluids manifest both sub-structure of filaments within the fluid (e.g., Ref. 3) andinterconnectedness due to passive entanglements ofpolymer filaments (4). On the basis of Maniotis et al(8), which is also the basis of Dr. Ingber’s Fig. 3 (asnoted), it would seem that this response occurred in 2 sor less, much shorter than the 5- to 10-min time scaleover which fibroblasts maintain cell shape and crawl.Given the images and the time scale, our honest as-sessment of Fig. 3 is that it is most likely due toviscoelastic behavior, like threads embedded in SillyPutty becoming aligned by pulling on the putty.Whether or not cytosolic viscosity is a better modelthan tensegrity structures for what is happening inFig. 3, we think viscoelastic structure, the possibility offluid interconnectedness, and general time dependenceof mechanical behaviors have all been completely over-looked by the tensegrity model.

If we agree that solidlike, cytoskeletal connectionscome and go, then we would argue that such temporalaspects may be the most important for modeling of cellmechanics. Our recent analogy of cytoskeletal and cellsurface connections to that of an automobile transmis-sion with clutches that engage and disengage an acto-myosin motor was intended to highlight this engage-ment-disengagement aspect of cell mechanics (6). Inour view, the ideas and images surrounding tensegritydo not help at all in thinking about this “now you see it,now you don’t” aspect of cytomechanics. Tensegritystructures behave purely elastically, all the time, andthe stability of the interconnections fundamentally un-derlies tensegrity’s beauty as a physical model and asaesthetic objects. At the very least, if the come-and-goconnectedness of a solidlike transmission is closer toreality than the temporally stable connections andbehaviors of tensegrity structures, then cellular


tensegrity must incorporate force, frequency, or time-dependent transmission criteria. Do cells respond elas-tically 10 or 90% of the time, in response to 10 or 90%of experimental deformations? Do 10 or 90% of me-chanical adhesions at the surface produce long-term(.1 min) connections to the underlying cytoskeleton? Ifnot, one is left in the untenable position that when thecell behaves in an interconnected fashion it is tenseg-rity, but the rest of the time and the remainder of thecellular responses are meaningless.

In addition to the absence of a temporal dimensioninherent in tensegrity, our other major disagreement isindeed with the meaning of the word tensegrity andour “overly concrete definition.” We continue to find Dr.Ingber’s account of tensegrity far too broad to be of realuse. That is, a “structural framework” that includesballoons, the human body, geodesic domes, and Snel-son sculptures is an accommodating framework in-deed. We wonder aloud whether the simple word “sol-id” isn’t a better description for the nonfluid behaviorsof the cell. As argued above by us, the solid state is thesimplest representation of durable interconnectionsamong elements. We pointed out in our initial discus-sion above that a wide variety of different structuresmanifest prestress, local compression and are stress/strain hardening, important features of tensegritystructures. In Dr. Ingber’s discussion above, he states,“One of the most important features of the tensegritymodel, as opposed to the viscous cytosol model, is thatit predicts that applied mechanical forces will not betransmitted into the cell equally at all point on the cellsurface.” In fact, most complex solid structures do nottransmit forces equally to/at all points on the surface.A particularly dramatic example, as long as we aretalking about architecture, is the photoelastic analysisof stresses in Gothic cathedrals (9). These show forcesat one location (the vault weight and clerestory windloads) transmitted by the flying buttresses to the pierbuttresses at a distance. Surely, Notre Dame de Parisis not a tensegrity structure!

With time, we have found tensegrity to be a less andless useful model or mental image to generate originalpredictions about cytomechanics. We repeatedly foundthat the qualitative mechanical predictions of tenseg-rity (such as surface prestress, strain hardening, localcompression, and nonuniform stress transmission) arewidespread among solid objects and that time-depen-dent behaviors are not modeled at all by man-madetensegrity structures. Is tensegrity a good metaphor ifwe only mean “transmission of tension across molecu-lar connections within the cytoskeletal network” and“long-distance force transfer” when this is common-place in solids? Tensegrity is attractive largely because

of its rare and innovative structural features (1, 5).(How frequently have you seen geodesic domes, and didyou mistake them for typical architecture?) If cellulartensegrity does not share at least some unique featuresof man-made tensegrity objects, but only common, solidproperties, then we ask whether tensegrity, among themyriad of man-made structures to potentially describecytomechanics, is the best or even a generally usefulmodel and metaphor.

FINAL STATEMENT

As we noted in our original argument above, we arekeeping an open mind about tensegrity as a model forcellular mechanics. However, for us at the moment, itdoes not seem to be general enough for cellular re-sponses or specific enough to experimentally distin-guish from elastic, and even viscoelastic, models. Wethink it is unlikely that any cell will correspond with astrict definition of tensegrity, but some aspects oftensegrity may well be applicable to some cells, at sometimes, under some conditions. Too broad a definition oftensegrity eliminates its usefulness and originality asa model. An important advance would be to identifyand agree on mechanical properties of man-madetensegrity structures that are not widespread amongelastic structures so that we can look for these behav-iors in cells.

REFERENCES

1. Brookes M. Hard cell, soft cell. New Scientist 164: 42–46, 1999.2. Buxbaum RE, Dennerll T, and Heidemann SR. F actin and

microtubule suspensions as indeterminate fluids. Science 235:1511–1514, 1987.

3. Collings PJ. Liquid Crystals. Princeton, NJ: Princeton Univ.Press, 1990, p. 3–23.

4. Graessley WW. Viscosity of entangling polydisperse polymers.J Chem Phys 47: 1942–1953, 1967.

5. Guterman L. A biologist’s maverick theory likens cell form toFuller’s geodesic domes. Chron Higher Educ February 4: A19–A22, 2000.

6. Heidemann SR and Buxbaum RE. Cell crawling: first themotor, now the transmission. J Cell Biol 141: 1–4, 1998.

7. Kerst A, Chmielewski C, Livesay C, Buxbaum RE, andHeidemann SR. Liquid crystal domains and thixotropy of F-actin suspensions. Proc Natl Acad Sci USA 87: 4241–4245, 1990.

8. Maniotis AJ, Chen CS, and Ingber DE. Demonstration ofmechanical connection between integrins, cytoskeletal filaments,and nucleoplasm that stabilize nuclear structure. Proc Natl AcadSci USA 94: 849–854, 1997.

9. Mark R. Experiments in Gothic Structure. Cambridge, MA: MITPress, 1982, p. 50–57

10. Seifriz W. A Symposium on the Structure of Protoplasm. Ames,IA: Iowa St. Univ. Press, 1942, p. 1–284.

11. Suter DM, Errante LD, Belosertkovsky V, and Forscher P.The Ig superfamily cell adhesion molecule, ApCAM, mediatesgrowth cone steering by substrate-cytoskeletal coupling. J CellBiol 141: 227–240, 1998.


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