opposing views on tensegrity as a structural framework for understanding cell mechanics by ingber

8/9/2019 Opposing Views on Tensegrity as a Structural Framework for Understanding Cell Mechanics by Ingber

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89:1663-1678, 2000.J Appl PhysiolBuxbaumDonald E. Ingber, Steven R. Heidemann, Phillip Lamoureux and Robert E.

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controversies in physiology

Opposing views on tensegrity as a structural frameworkfor understanding cell mechanics

DONALD E. INGBERDepartments of Pathology and Surgery, Childrens Hospital

and Harvard Medical School, Boston, Massachusetts 02115

STEVEN 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 are

probably 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, 2024, 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 and

largely 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 and Associate 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, 2224). 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. Fullers more formal definition in his treatise, Synergeticsis Tensegrity describes a structural-relationship principle in which structural shape is guaranteed by thefinitely closed, comprehensively continuous, tensionabehaviors of the system and not by the discontinuousand exclusively local compressional member behaviors (16). Note that there is no mention of rigid strutselastic strings, tensile filaments, internal vs. externamembers, or specific molecular constituents in thisdefinition. In fact, Fuller describes a balloon with noncompressible gas molecules pushing out against a

tensed rubber membrane as analogous to one of hisgeodesic domes when viewed at the microstructuralevel (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 hierarchicatensegrity structures in which individual struts or tensile 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 system of tensile structuring at all size scales and that itprovides a way to mechanically integrate part and

whole (16), a view I recently explored in greater depth(22).In 1948, Fullers student, Kenneth Snelson, con

structed the first stick-and-string tensegrity sculpture, which thrilled Fuller because it visibly communicated 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 Ref5). Snelsons 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 Physio89: 16631678, 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 natures 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 Fullersclose 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 structure before application of an external load. The distinguishing microstructural feature accounting for thisbehavior is that, when placed under load, the discrete

structural elements move, changing orientation andspacing relative to one another, until a new equilibrium configuration is attained. For this reason, a locastress 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 itstiffness 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 multimodular and hierarchical: if I accidentally sever my Achilles 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 oelastin in the extracellular matrix (ECM) relax, buckled bundles of cross-linked (stiffened) collagen filaments straighten, basement membranes tighten, andthe adherent cells and cytoskeletal filaments feel thepull; however, nothing breaks and the deformation isreversible. Tensegrity provides a structural basis to

explain all these phenomena.In the cellular tensegrity model, the stabilizing prestress is generated actively by the cells contractileapparatus and passively by distension through extracellular adhesions, by osmotic forces acting on the cellssurface 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 cytoskeleton, primarily actin microfilaments and intermediatefilaments, and that the cell is both a hierarchical andmultimodular structure (5, 2023) (Figs. 1 and 2). Thisprestress is balanced by interconnected structural elements that resist being compressed at different size

scales, including the cells external adhesions to therelatively inflexible ECM and internal cytoskeletal filaments, specifically microtubules that stretch acrosslarge regions of the cytoplasm and cross-linked bundlesof cytoskeletal filaments that stabilize specialized microdomains of the cell surface (e.g., actin microfilaments in filopodia; microtubules in cilia). In this modelthe internal cytoskeleton is surrounded by an elasticsubmembranous cytoskeleton (e.g., actin-ankyrinspectrin network) and its associated lipid bilayerwhich 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 elastic

nucleus in its center. In engineering terms, this is acontinuum 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 (1720, 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 different size scales. Then why the continued criticisms?Lets 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 be

transmitted into the cell equally at all points on the celsurface. 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 networkas recently demonstrated in the purest form of thisstructure, the red blood cell membrane (11). Dependingon the molecular composition of the attachment substrate (e.g., ECM, surface of another cell) to which acell anchors, this highly elastic cortex may or may notmechanically couple to the internal microfilament-microtubule-intermediate filament lattice, which, in turndistributes 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 moleculabridge between the ECM and the internal cytoskeletonprovide a preferred path for transmembrane mechanical signal transfer and, hence, play a central role incellular mechanotransduction. On the basis of subsequent 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 Wang

et al., unpublished observations).

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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 hit

end-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 of

microtubules 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 their

ECM 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 increase when tubulin monomers are released in cells inwhich cytoskeletal tension is decreased using relaxantdrugs before microtubule disruption (Polte and Ingberunpublished observations). In other words, the increase in MLC phosphorylation observed after microtubule disruption (28) does not result from release o

tubulin monomers; rather, it appears to be a compensatory mechanism that is activated in response totransfer of mechanical stress from microtubules to theECM and the remaining cytoskeleton in these cellsThis is yet another example of a complex behavior thatcan be explained by tensegrity and not by the other celmodels.

Some of those who accept that microtubules bearcompression locally within an otherwise tensed cytoskeleton, a clear example of cellular tensegrity, thenargue whether this contributes significantly to celmechanics. To explore this idea in greater detail, studies were recently carried out in pulmonary airwaysmooth muscle cells cultured on flexible polyacryl

amide gel substrates containing small fluorescent microbeads as fiducial markers, which permit quantitation of cell tractional forces and prestress withinindividual cells (by quantitating bead displacemenrelative 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 constrictor histamine, again to ensure optimal MLC phosphorylation. These studies revealed that microtubulescounterbalanced approximately one-third of the totacellular prestress within an individual histamine-stimulated cell (Wang et al., unpublished observations)

Thus these data confirm that the ability of microtu-bules to bear compression locally contributes significantly to cellular prestress and that prestress, in turnis critical for maintenance of cell shape stability. However, because of complementary (tensegrity-basedforce 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. Thusalthough the demonstration that microtubules do carrycompression in living cells is a strong support fortensegrity, a negative result in a particular cell would

not necessarily rule out this model.Importantly, many biologists fail to recognize theimportant difference between engineering models thatcan describe (curve-fit) a complex cell behavior vsone, 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 droplet, or a spring and dashpot can mimic mechanicabehaviors (e.g., strain-hardening behavior) observed inliving cells and tissues, as can tensegrity. This is trueIn 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).



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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 observations) demonstrate that these a priori predictionsare supported by experimental measurements of staticand dynamic mechanical behaviors in living cells and

thus 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 othe larger cell and tissue (4, 5, 7, 17, 20, 22, 30, 33, 3442). Because the tensegrity model is a mechanicaparadigm, 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 to

explain how mechanical forces regulate cellular biochemistry and influence gene expression (7, 17, 21, 33)The other cell models that still dominate the literaturecannot.

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20. Ingber DE. Cellular tensegrity: defining new rules of biologicaldesign that govern the cytoskeleton. J Cell Sci 104: 613627,1993.

21. Ingber DE. Tensegrity: the architectural basis of cellular mech-anotransduction. Annu Rev Physiol 59: 575599, 1997.

22. Ingber D. The architecture of life. Sci Am 278: 4857, 1998.23. Ingber DE and Jamieson JD. Cells as tensegrity structures:

architectural regulation of histodifferentiation by physical forcestransduced over basement membrane. In: Gene Expression Dur-ing Normal and Malignant Differentiation, edited by AnderssonLC, Gahmberg CG, and Ekblom P. Orlando, FL: Academic, 1985,p. 1332.

24. Ingber DE, Madri JA, and Jamieson JD. Role of basal lam-ina in the neoplastic disorganization of tissue architecture. ProcNatl Acad Sci USA 78: 39013905, 1981.

25. Ingber DE, Prusty D, Sun Z, Betensky H, and Wang N. Cellshape, cytoskeletal mechanics and cell cycle control in angiogen-esis. J Biomech 28: 14711484, 1995.

26. Joshi HC, Chu D, Buxbaum RE, and Heidemann SR. Ten-sion and compression in the cytoskeleton of PC 12 neurites.J Cell Biol 101: 697705, 1985.

27. Kaech S, Ludin B, and Matus A. Cytoskeletal plasticity incells expressing neuronal microtubule-associated proteins. Neu-ron 17: 11891199, 1996.

28. Kolodney MS and Elson EL. Contraction due to microtubuledisruption is associated with increasing phosphorylation of my-

osin regulatory light chain. Proc Natl Acad Sci USA 92: 1025210256, 1995.

29. Kolodney MS and Wysolmerski RB. Isometric contraction byfibroblasts and endothelial cells in tissue culture: a quantitativestudy. J Cell Biol 117: 7382, 1992.

30. 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: 849854, 1997.

31. Marks R and Fuller RB. The Dymaxion World of BuckminsterFuller. Garden City, NY: Anchor/Doubleday, 1973, p. 5760.

32. Pickett-Heaps JD, Forer A, and Spurck T. Traction fibre:toward a tensegral model of the spindle. Cell Motil Cytoskele-ton 37: 16, 1997.

33. Pienta KJ and Coffey DS. Cellular harmonic informationtransfer through a tissue tensegrity-matrix system. Med Hy-

potheses 34: 8895, 1991.34. Pourati J, Maniotis A, Spiegel D, Schaffer JL, Butler JP,

Fredberg JJ, Ingber DE, Stamenovic D, and Wang N. Iscytoskeletal tension a major determinant of cell deformability inadherent endothelial cells? Am J Physiol Cell Physiol 274:C1283C1289, 1998.

35. Schmidt CE, Horwitz AF, Lauffenburger DA, and SheetzMP. Integrin-cytoskeletal interactions in migrating fibroblastsare dynamic, asymmetric, and regulated. J Cell Biol 123: 977991, 1993.

36. Sheetz MP, Wayne DB, and Pearlman AL. Extension offilopodia by motor-dependent actin assembly. Cell Motil Cy-toskeleton 22: 160169, 1992.

37. Sims J, Karp S, and Ingber DE.Altering the cellular mechan-ical force balance results in integrated changes in cell, cytoskel-etal, and nuclear shape. J Cell Sci 103: 12151222, 1992.

38. Stamenovic D and Coughlin MF. The role of prestress andarchitecture of the cytoskeleton and deformability of cytoskeletalfilaments of adherent cells: a quantitative approach. J TheorBiol 201: 6374, 1999.

39. Stamenovic D, Fredberg JJ, Wang N, Butler JP, and Ing-ber DE. A microstructural approach to cytoskeletal mechanicsbased on tensegrity. J Theor Biol 181: 125136, 1996.

40. Thoumine O, Ziegler T, Girard PR, and Nerem RM. Elon-gation of confluent endothelial cells in culture: the importance offields of force in the associated alterations of their cytoskeletalstructure. Exp Cell Res 219: 427441, 1995.

41. Vasiliev JM. Actin cortex and microtubular system in morpho-genesis: cooperation and competition. J Cell Sci Suppl 8: 118,1987.

42. Wang N, Butler JP, and Ingber DE. Mechanotransductionacross the cell surface and through the cytoskeleton. Science 26011241127, 1993.

43. Wang N and Ingber DE. Control of cytoskeletal mechanics byextracellular matrix, cell shape, and mechanical tension. Biophys J66: 21812189, 1994.

44. Wang N and Ingber DE. Probing transmembrane mechanicacoupling and cytomechanics using magnetic twisting cytometry Biochem Cell Biol 73: 327335, 1995.

45. Waterman-Storer CM and Salmon ED. Acto-myosin based

retrograde flow of microtubules in the lamella of migratingepithelial cells influences microtubule dynamic instability andturnover and is associated with microtubule breakage and treadmilling. J Cell Biol 139: 417434, 1997.

46. Wendling S, Oddou C, and Isabey D. Stiffening response of acellular tensegrity model. J Theor Biol 196: 309325, 1999.

Steven R. Heidemann, Phillip Lamoureaux, andRobert E. Buxbaum: We have been thinking abouttensegrity architecture for cells since a scientific meeting, 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) o

the classic anti-cytoskeletal drug experiments o Yamada et al. (32) on neurons. He and others hashown 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 measurements on axons before and during treatment withanti-microtubule and anti-actin drugs seemed to confirm this mechanical hypothesis. Tension in axons in-creased when microtubules were depolymerized, andtension decreased when axons were treated with actindisrupting drugs. Furthermore, increased tension inaxons caused microtubule depolymerization (9). Combined with the well-known spatial arrangement o

these 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 compressive 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 tensegrity model more critically and compared it to oldermodels of cell architecture. In our view, the tensegrity

model of cells has at least two necessary features, bothfundamental according to Fullers own account otensegrity (12). One, implied by the name derived fromtensional integrity, is that continuous tension in theactin cortex fully integrates overall shape and structure. Thus global integration of the cellular structure iskey; local mechanical inputs should produce distributed cytoskeletal responses (action at a distancebecause 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



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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 soliddish (cells round up when trypsinized from the dish),it would appear that, at best, cells can only approxi-mate true tensegrity structures as envisioned by

Fuller. 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 seen

distributed, 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 experimentsglass needles were treated with an adhesion proteinlaminin, 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 spot

of actin to move with the needle along the surfacewithout disturbing the underlying actin or microtubules. 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 predicted 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 arrangement of microtubules directly adjacent the extension. In addition, it was clear that the attachment hadan effective functional connection to the actin cortex in

that 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 particularly surprised by the lack of evidence for any signifi-cant cytoskeletal interconnections in our recent experiments. 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 contradicted the global integration of the cytoskeletonrequired for the tensegrity model of the cell but generally changed our view of the extent of interconnectionamong cytoskeletal elements.

Dr. Ingber and colleagues (17, 18, 26) have definedtensegrity as continuous tension and local compression. However, we find this definition too broad. Onthis basis, tensegrity would include pup tents (i.eFullers 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 that

long predate Fullers conception of tensegrity. Wewould define tensegrity structures as those with ten-sion-induced structural integrity resulting from a continuous structure of tension-bearing elements and discontinuous compressive elements integrally connectedto but dispersed within the structure by the tensionelements. In other words, in our view, cellular tensegrity requires architectural features that are quite similar in mechanical and shape properties to those of theclassic string-and-strut tensegrity sculptures of Snelson. These sculptures, it should be noted, have beenused repeatedly as the models, illustrations, and the



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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 fairly

call 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 hardening

has 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-N 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 tensegrity

model 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 cytoskeletal integration in the form of action at a distance 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: streshardening of a liquid drop; analysis of values from Table VI in themodel of Yoneda et al. (33) of the mechanical properties of a liquid

drop. As height of the drop (z) is decreased by external compressionsurface 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 heigh(ds/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 (stiffness 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 empiricameasurements were taken on an inflated rubber ball. Weights weresequentially added to compress a rubber ball whose relative heighwas 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 deformationsplotted here as a function of the compressing force on the rubber ball



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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 the

other 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 fundamentally tensile and share mechanical propertieswith tensegrity architecture, such as prestress and

stress 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 functiongrowth and development of plants and animals, andcellular differentiation and morphogenesis. Howeverunlike chemical signaling, for which textbooks andprofessional articles are adorned with elaborate, RubeGoldberg-like sequences of molecular cause and effect

Fig. 6. Cytoskeletal action at a distance in culturedchick forebrain neurons. Experimental tension was applied to a chick forebrain neuron with an attachedcalibrated glass needle. As previously reported for thiand other neuronal types, this causes the axon to elongate. 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 cytoplasm

returned to its original location by 5 h. Particularlyintriguing was that, throughout the 5-h observationdendrites 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. Specificallythis 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.



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(epinephrine activates the -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.

REFERENCES

1. Buxbaum RE and Heidemann SR. A thermodynamic modelfor force integration and microtubule assembly during axonalelongation. J Theor Biol 134: 379390, 1988.

2. Buxbaum RE and Heidemann SR. An absolute rate theorymodel for tension control of axonal elongation. J Theor Biol 155:409426, 1992.

3. Carton RW, Dainauskas J, and Clark JW. Elastic propertiesof single elastic fibers. J Appl Physiol 17: 547551, 1962.

4. Chicurel ME, Chen CS, and Ingber DE. Cellular control liesin the balance of forces. Curr Opin Cell Biol 10: 232239, 1998.

5. Choquet D, Felsenfeld DP, and Sheetz MP. Extracellularmatrix rigidity causes strengthening of integrin-cytoskeletallinkages. Cell 88: 3948, 1997.

6. Cole KS. Surface forces of the Arbacia egg. J Cell Comp Physiol1: 19, 1932.

7. Coughlin MF and Stamenovic D. A tensegrity structure withbuckling compression elements: application to cell mechanics.J Appl Mech 64: 480486, 1997.

8. Daily B, Elson EL, and Zahalek GI. Cell poking: determina-tion of elastic area compressibility modulus of the erythrocytemembrane. Biophys J 45: 671682, 1984.

9. Dennerll TE, Joshi HC, Steel VL, Buxbaum RE, and Hei-demann SR. Tension and compression in the cytoskeleton of PC12 neurites. II. Quantitative measurements. J Cell Biol 107:665674, 1988.

10. Dong C, Skalak R, and Sung K-L P. Cytoplasmic rheology ofpassive neutrophils. Biorheology 28: 557567, 1991.

11. Evans E and Yeung A. Apparent viscosity and cortical tensionof blood granulocytes determined by micropipet aspiration. Bio-phys J56: 151160, 1989.

12. Fuller RB. Tensegrity. Portfolio Artnews Annu 4: 112127,1961.

13. Fung YCB. Elasticity of soft tissues in simple elongation. Am JPhysiol 213: 15321544, 1967.

14. Heidemann SR, Kaech S, Buxbaum RE, and Matus A.Direct observations of the mechanical behaviors of the cytoskel-eton in living fibroblasts. J Cell Biol 145: 109122, 1999.

15. Hiramoto Y. Mechanical properties of sea urchin eggs. I. Sur-face force and elastic modulus of the cell membrane. Exp Cell Res56: 201208, 1963.

16. Hoh JH and Schoenenberger C-A. Surface morphology andmechanical properties of MDCK monolayers by atomic forcemicroscopy. J Cell Sci 107: 11051114, 1994.

17. Ingber DE. Cellular tensegrity: defining new rules of biologicaldesign that govern the cytoskeleton. J Cell Sci 104: 613624,1993.

18. Ingber DE. Tensegrity: the architectural basis of cellular mech-anotransduction. Ann Rev Physiol 59: 575599, 1997.

19. Joshi HC, Chu D, Buxbaum RE, and Heidemann SR. Ten-sion and compression in the cytoskeleton of PC 12 neurites.J Cell Biol 101: 697705, 1985.

20. Ludin B and Matus A. GFP illuminates the cytoskeleton.Trends Cell Biol 8:7277, 1998.

21. Maniotis AJ, Chen CS, and Ingber DE. Demonstration ofmechanical connection between integrins, cytoskeletal fila-ments, and nucleoplasm that stabilize nuclear structure. ProcNatl Acad Sci USA 94: 849854, 1997.

22. Mitchison JM and Swann MM. The mechanical properties ofthe cell surface. III. The sea urchin egg from fertilization tocleavage. J Exp Biol 32: 734750, 1955.

23. Needham D and Hochmuth RM. A sensitive measure ofsurface stress in the resting neutrophil. Biophys J 61: 16641670, 1992.

24. Oakes BW and Bialkower B. Biomechanical and ultrastructural studies of the wing tendon from the domestic fowl. J Ana123: 369387, 1977.

25. Peterson NO, McConnaughey WB, and Elson EL. Dependence of locally measured cellular deformability on position onthe cell, temperature and cytochalasin B. Proc Natl Acad ScUSA 79: 53275331, 1982.

26. Pourati J, Maniotis A, Spiegel D, Schaffer JL, Butler JPFredberg JJ, Ingber DE, Stamenovic D, and Wang N. Icytoskeletal tension a major determinant of cell deformability in

adherent endothelial cells? Am J Physiol Cell Physiol 274C1283C1289, 1998.

27. Putman CAJ, van der Werf KO, de Grooth BG, van HulsJF, and Greve J. Viscoelasticity of living cells allows highresolution imaging by tapping mode atomic force microscopyBiophys J 67: 17491753, 1994.

28. Stamenovic D, Fredberg JJ, Wang N, Butler JP, and Ingber DE. A microstructural approach to cytoskeletal mechanicsbased on tensegrity. J Theor Biol 181: 125136, 1996.

29. Thoumine O, Cardoso O, and Meister J-J. Changes in themechanical properties of fibroblasts during spreading: a micromanipulation study. Eur J Biophys 28: 222224, 1999.

30. Thoumine O and Ott A. Time-scale dependent viscoelastic andcontractile regimes in fibroblasts probed by microplate manipulation. J Cell Sci 110: 21092116, 1997.

31. Wang N, Butler JP, and Ingber DE. Mechanotransductionacross the cell surface and through the cytoskeleton. Science 26011241127, 1993.

32. Yamada KM, Spooner BS, and Wessels NK. Axon growthrole of microfilaments and MTs. Proc Natl Acad Sci USA 6612061212, 1970.

33. Yoneda M. Tension at the surface of sea urchin eggs on the basiof liquid drop concept. Adv Biophys 4: 153190, 1973.

REBUTTALS

Donald E. Ingber: I believe that most of the concerns

raised by Dr. Heidemann and co-workers were addressed in my original editorial; however, there are afew points that deserve further clarification and emphasis. The major issue is the definition of tensegrityI used Fullers own formal definition, which can befound in the definitive text of his lifes work (Synergetics). A more thorough definition of tensegrity thatclosely matches my own can be found in laymansterms in A Fuller Explanation by Amy Edmondson, aclose associate of Fullers (3). The fact that tents, spiderwebs (e.g., stabilized by attachment to compressionresistant tree branches), and ships riggings (whichFuller often described in terms of tensegrity) existedfor years before Fullers birth is of no import. Fuller did

not invent this architectural method; he discovered theuniversality of its use and inspired its application byothers (e.g., Snelson). To arbitrarily narrow and makeconcrete Fullers definition and then to cite a randomFuller quote out of context, which was written for anartists journal, seems unreasonable to me; however, leave 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 thatsome compressive support for neurite tension isclearly provided by the dish attachment (see above).



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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 through

the 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 nicely

with 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-wiredtensegrity 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 previously demonstrated action at a distance when wpulled 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 a

recent study using GFP-labeled intermediate filaments(8). Furthermore, buckling of microtubules was actually demonstrated in the study by Dr. Heidemann et alin 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 increased numbers of focal adhesions along the cell baseAs described above, only microdomains of the multimodular cytoskeletal lattice between the fixed focal adhesions would be expected to rearrange or significantly

deform in response to a local mechanical manipulationunder these conditions. In fact, this is exactly what DrHeidemanns 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 celbody before application of a similar incision. Interestingly, the nerve cells that Dr. Heidemann studies form

relatively few substrate adhesions beneath the celbody; this may be why he can more easily visualizeaction at a distance at the whole cell level in thasystem.

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 tensegrity model is consistent with all of these findingsFurthermore, only tensegrity can also predict many othese results a priori. For example, reconstituted gelsof intermediate filaments can also exhibit strain hardening (12); however, living cells still exhibit strainhardening after intermediate filaments are chemically

disrupted or knocked out genetically (20). It is also truethat others structures, including rubber balls, liquiddrops, tensed cable networks, and tensed cortical membrane/viscous cytosol models, may exhibit strain-hardening behavior and approximately linear dependencesof stiffness on prestress. However, these other tensedmodels are not consistent with many other experimental results (as I described in my original discussionabove) or with the microarchitecture that we observe inliving cells (dense cytoskeletal networks throughouthe cytoplasm, straight microfilaments, curved microtubules). More importantly, Dr. Heidemanns pre



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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 to

predict 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 cellHowever, in the end, it will be in the forging of a newunderstanding of the relation between mechanics, molecular structure, and biochemical function that theimportance of higher order architecture and prestresswill become most clear and in which tensegrity wilprovide its greatest value.

REFERENCES

1. Caspar DLD. Movement and self-control in protein assemblies Biophys J32: 103138, 1980.

2. Eckes B, Dogic D, Colucci-Guyon E, Wang N, Maniotis AIngber D, Merckling A, Aumailley M, Koteliansky V, Babinet C, and Krieg T. Impaired mechanical stability, migrationand contractile capacity in vimentin-deficient fibroblasts. J CelSci 111: 18971907, 1998.

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

4. Fey EG, Capco DG, Krochmalnic G, and Penman S. Epithelial structure revealed by chemical dissection and unembedded electron microscopy. J Cell Biol 99: 203S208S, 1984.

5. Goldman RD, Khuon S, Chou YH, Opal P, and Steinert PM

The function of intermediate filaments in cell shape and cytoskeletal integrity. J Cell Biol 134: 971983, 1996.6. Heidemann SR and Buxbaum RE. Tension as a regulator and

integrator of axional growth. Cell Motil Cytoskeleton 17: 6101990.

7. Heidemann SR, Kaech S, Buxbaum RE, and Matus ADirect observations of the mechanical behaviors of the cytoskeleton in living fibroblasts. J Cell Biol 145: 109122, 1999.

8. Helmke BP, Goldman RD, and Davies PF. Rapid displacement of vimentin intermediate filaments in living endotheliacells exposed to flow. Circ Res 86: 745752, 2000.

9. Heuser JE and Kirschner MW. Filament organization re vealed in platinum replicas of freeze-dried cytoskeletons. J CelBiol 86, 212234, 1980.

10. Huang S, Chen CS, and Ingber DE. Control of cyclin D1, p27Kip

and cell cycle progression in human capillary endothelial cells bycell shape and cytoskeletal tension. Mol Biol Cell 9: 31793193

1998.11. Ingber D. The architecture of life. Sci Am 278: 4857, 1998.12. Janmey PA, Eutenauer U, Traub P, and Schliwa M. Vis

coelastic properties of vimentin compared with other filamentous biopolymer networks. J Cell Biol 113, 155160, 1991.

13. Lazarides E. Actin, -actinin, and tropomyosin interactions inthe structural organization of actin filaments in nonmuscle cellsJ Cell Biol 68: 202219, 1976.

14. Maniotis A, Bojanowski K, and Ingber DE. Mechanical continuity and reversible chromosome disassembly within intacgenomes microsurgically removed from living cells. J Cell Biochem 65: 114130, 1997.

15. Maniotis AJ, Chen CS, and Ingber DE. Demonstration omechanical connections between integrins, cytoskeletal filaments, and nucleoplasm that stabilize nuclear structure. ProNatl Acad Sci USA 94: 849854, 1997.

16. Mathur AB, Truskey GA, and Reichert WM. Atomic force

and total internal reflection fluorescence microscopy for thstudy of force transmission in endothelial cells. Biophys J 7817251735, 2000.

17. Pourati J, Maniotis A, Spiegel D, Schaffer JL, Butler JPFredberg JJ, Ingber DE, Stamenovic D, and Wang N. Icytoskeletal tension a major determinant of cell deformability inadherent endothelial cells? Am J Physiol Cell Physiol 274C1283C1289, 1998.

18. Schutt CE, Kreatsoulas C, Page R, and Lindberg U. Plugging into actins architectonic socket. Nat Struct Biol 4: 1691721997.

19. Wang, N, Butler, JP, and Ingber, DE. Mechanotransductionacross the cell surface and through the cytoskeleton. Science 26011241127, 1993.



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20. Wang N, and Stamenovic D. Contribution of intermediatefilaments to cell stiffness, stiffening and growth. Am J PhysiolCell Physiol 279: C188C194, 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 seen

the 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. Ingbers 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 whats 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 oour esteemed counterparts initial position piece is anideal example of the importance of such viscoelasticphenomena in this debate. This figure shows an uncontested example of elastic cellular response and evidence of connectedness between the cell surface andthe underlying cytoskeleton. If the time scale of thisobservation was 10 min, we would agree that it represents 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 scale

of this observation is 10 s or less, we would bet ourmoney on viscoelasticity. Contrary to Dr. Ingbers as-sertions concerning viscous models, which apply toNewtonian fluids, viscoelastic fluids manifest both substructure of filaments within the fluid (e.g., Ref. 3) andinterconnectedness due to passive entanglements opolymer filaments (4). On the basis of Maniotis et a(8), which is also the basis of Dr. Ingbers 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 crawlGiven the images and the time scale, our honest assessment of Fig. 3 is that it is most likely due to

viscoelastic behavior, like threads embedded in SillyPutty becoming aligned by pulling on the puttyWhether or not cytosolic viscosity is a better modelthan tensegrity structures for what is happening inFig. 3, we think viscoelastic structure, the possibility ofluid interconnectedness, and general time dependenceof mechanical behaviors have all been completely overlooked 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 celmechanics. Our recent analogy of cytoskeletal and celsurface connections to that of an automobile transmission with clutches that engage and disengage an acto

myosin motor was intended to highlight this engagement-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 itnow you dont aspect of cytomechanics. Tensegritystructures behave purely elastically, all the time, andthe stability of the interconnections fundamentally underlies tensegritys 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



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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.Ingbers 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 isnt a better description for the nonfluid behaviorsof the cell. As argued above by us, the solid state is thesimplest representation of durable interconnections

among 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. Ingbers 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 analysis

of 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 if

we only mean transmission of tension across molecu-lar connections within the cytoskeletal network andlong-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 usefu

model 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 distinguish from elastic, and even viscoelastic, models. Wethink it is unlikely that any cell will correspond with astrict definition of tensegrity, but some aspects otensegrity may well be applicable to some cells, at sometimes, under some conditions. Too broad a definition o

tensegrity 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 behaviors in cells.

REFERENCES

1. Brookes M. Hard cell, soft cell. New Scientist 164: 4246, 19992. Buxbaum RE, Dennerll T, and Heidemann SR. F actin and

microtubule suspensions as indeterminate fluids. Science 23515111514, 1987.

3. Collings PJ. Liquid Crystals. Princeton, NJ: Princeton UnivPress, 1990, p. 323.

4. Graessley WW. Viscosity of entangling polydisperse polymersJ Chem Phys 47: 19421953, 1967.

5. Guterman L. A biologists maverick theory likens cell form toFullers geodesic domes. Chron Higher Educ February 4: A19A22, 2000.

6. Heidemann SR and Buxbaum RE. Cell crawling: first themotor, now the transmission. J Cell Biol 141: 14, 1998.

7. Kerst A, Chmielewski C, Livesay C, Buxbaum RE, andHeidemann SR. Liquid crystal domains and thixotropy of Factin suspensions. Proc Natl Acad Sci USA 87: 42414245, 1990

8. Maniotis AJ, Chen CS, and Ingber DE. Demonstration omechanical connection between integrins, cytoskeletal filamentsand nucleoplasm that stabilize nuclear structure. Proc Natl AcadSci USA 94: 849854, 1997.

9. Mark R.Experiments in Gothic Structure. Cambridge, MA: MITPress, 1982, p. 5057

10. Seifriz W. A Symposium on the Structure of Protoplasm. Ames

IA: Iowa St. Univ. Press, 1942, p. 1284.11. Suter DM, Errante LD, Belosertkovsky V, and Forscher PThe Ig superfamily cell adhesion molecule, ApCAM, mediategrowth cone steering by substrate-cytoskeletal coupling. J CelBiol 141: 227240, 1998.


opposing views on tensegrity as a structural framework for understanding cell mechanics by ingber

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