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ing capsule has been termed the chondron (Poole, 1992; Poole, 1997). While the
function of the PCM is not known, there has been considerable speculation that the
chondron plays a biomechanical role in articular cartilage (Szirmai, 1974). For ex-
ample, it has been hypothesized that the chondron provides a protective effect for the
chondrocyte during loading (Poole et al., 1987), and others have suggested that the
chondron serves as a mechanical transducer (Greco et al., 1992; Guilak and Mow,
2000).
To determine mechanical properties of the PCM, the solution of a layered elastic
contact problem that models micropipette aspiration of an isolated chondron was de-
veloped. This theoretical solution was applied to measure an elastic Young™s modulus
for the PCM in human chondrons isolated from normal and osteoarthritic sample
groups (Alexopoulos et al., 2003). The mean PCM Young™s modulus of chondrons
isolated from the normal group (66.5 ± 23.3 kPa) was found to be a few orders of
magnitude larger than the chondrocyte modulus (∼1 kPa) and was found to drop sig-
ni¬cantly in the osteoarthritic group (41.3 ± 21.1 kPa, p < 0.001). These ¬ndings
support the hypothesis that the PCM serves a protective mechanical role that may be
signi¬cantly altered in the presence of disease. In a multiscale ¬nite element analy-
sis (Guilak and Mow, 2000), the macroscopic solution for transient deformation of
a cartilage layer under a step load was computed and used to solve a separate mi-
croscale problem to detemine the mechanical environment of a single chondrocyte. In
this study, the inclusion of a PCM layer in the microscale model signi¬cantly altered
the mechanical environment of a single cell. A mathematical model for purely radial
deformation in a chondron was developed and analyzed under dynamic loading in
the range 0“3 Hz (Haider, 2004). This study found that the presence of a thin, highly
stiff PCM that is less permeable than the chondrocyte enhances the transmission of
compressive strain mechanical signals to the cell while, simultaneously, protecting it
from excessive solid stress.
Using the micropipette aspiration test coupled with a linear biphasic ¬nite element
model, recent studies have reported the biphasic material properties of the PCM of
articular chondrocytes (Alexopoulos et al., 2005) (Fig. 5-5). Chondrons were me-
chanically extracted from nondegenerate and osteoarthritic (OA) human cartilage.
Micropipette aspiration was used to examine the creep behavior of the pericellular
matrix, which was matched using optimization to a biphasic ¬nite element model
(Fig. 5-6). The transient mechanical behavior of the PCM was well-described by a
biphasic model, suggesting that the viscoelastic response of the PCM is attributable to
¬‚ow-dependent effects, similar to that of the ECM. With osteoarthritis, the mean
94 F. Guilak et al.




Fig. 5-5. Micropipette aspiration of the pericellular matrix (PCM) of chondrocytes. This region
surrounds cells in articular cartilage, similar to a glycocalyx, but contains signi¬cant amounts of
extracellular matrix collagens, proteoglycans, and other macromolecules. The mechanical properties
of this region appear to have a signi¬cant in¬‚uence on the stress-strain and ¬‚uid-¬‚ow environment
of the cell. From Alexopoulos et al., 2005.


Young™s modulus of the PCM was signi¬cantly decreased (38.7 ± 16.2 kPa vs.
23.5 ± 12.9 kPa, p < 0.001), and the permeability was signi¬cantly elevated (4.19 ±
3.78 — 10’17 m4 /N·s vs. 10.2 ± 9.38 — 10’17 m4 /N·s, p < 0.001). The Poisson ra-
tio was similar for both nondegenerate and osteoarthritic PCM (0.044 ± 0.063 vs.
0.030 ± 0.068, p > 0.6). These ¬ndings suggest that the PCM may undergo enzy-
matic and mechanical degradation with osteoarthritis, similar to that occurring in the
ECM. In combination with previous theoretical models of cell“matrix interactions in
cartilage, these ¬ndings suggest that changes in the properties of the PCM may have
an important in¬‚uence on the biomechanical environment of the cell.
Together, these studies support the utility of in vitro mechanical analyses of iso-
lated functional cell“matrix units. Because cartilage, in particular, is avascular and
aneural, characterization of PCM mechanical and chemical properties is a key step
toward characterizing the in vivo state of the cell and its metabolic response to alter-
ations in the local cellular environment. The triphasic model provides a framework
for developing extended models of the PCM that can delineate effects of the dis-
tinct mechanochemical composition of the PCM, relative to the ECM, on the local
environment of the cell.

Indentation studies of cell multiphasic properties
In addition to micropipette aspiration, various techniques for cellular indentation have
been used to measure the modulus of adherent cells, including cell indentation (Daily
et al., 1984; Duszyk et al., 1989; Zahalak et al., 1990), scanning probe microscopy
(Radmacher et al., 1992; Shroff et al., 1995), or cytoindentation (Shin and Athana-
siou, 1999; Koay et al., 2003). (See also the discussion of experimental approaches in
Chapter 2.) The conceptual basis of these techniques is generally similar, in that a rigid
probe is used to indent the cell and the ensuing creep or stress-relaxation behavior
Multiphasic models of cell mechanics 95

(a) CELL AND PERICELLULAR MATRIX MODEL




(b)


R


R




Fig. 5-6. (a) Biphasic ¬nite element mesh of the micropipette aspiration experiment. The cell and
pericellular matrix were modeled using an axisymmetric mesh with bilinear quadrilateral elements
(342 nodes, 314 elements). (b) Transient response of normal and osteoarthritic chondrons (cell
with the pericellular matrix), and the associated biphasic prediction of their mechanical behavior.
The transient mechanical behavior of the PCM was well-described by a biphasic model, suggesting
that the viscoelastic response of the pericellular matrix is attributable to ¬‚ow-dependent effects,
similar to that of the extracellular matrix. From Alexopoulos et al., 2005 with permission.


is recorded. These techniques have generally used elastic or viscoelastic models to
calculate the equilibrium or dynamic moduli of cells over a range of frequencies. In
one set of cell indentation experiments, MG63 osteoblast-like cells were modeled with
either a linear elasticity solution of half-space indentation or the linear biphasic theory
under the assumption that the viscoelastic behavior of each cell was due to the interac-
tion between the solid cytoskeletal matrix and the cytoplasmic ¬‚uid (Shin and Athana-
siou, 1999). The intrinsic biphasic material properties (aggregate modulus, Poisson™s
ratio, and permeability) were determined by curve-¬tting the experimental surface re-
action force and deformation with a linear biphasic ¬nite element code in conjunction
with optimization routines. These cells exhibited a compressive aggregate modulus
96 F. Guilak et al.

of 2.05 ± 0.89 kPa with a Poisson ratio of 0.37 ± 0.03. These properties are on the
same order of magnitude as the elastic properties determined using other techniques
(Trickey et al., in press), although the permeability of 1.18 ± 0.65 — 10’10 m4 /N·s
is several orders of magnitude higher than that estimated for chondrocytes using
micropipette aspiration (Trickey et al., 2000).
Analysis of cell“matrix interactions using multiphasic models
Previous studies suggest that cells have the ability to respond to the local stress-strain
state within the extracellular matrix, thus suggesting that cellular response re¬‚ects
the history of the time-dependent and spatially varying changes in the mechanical
environment of the cells. The use of multiphasic models for cells has been of particular
value in theoretical models of cell“matrix interactions that seek to model the stress-
strain and ¬‚uid-¬‚ow environment of single cells within a tissue matrix. However, the
relationship between the stress-strain and ¬‚uid-¬‚ow ¬elds at the macroscopic “tissue”
level and at the microscopic “cellular” level are not fully understood. To directly test
such hypotheses, it would be important to have accurate knowledge of the local stress
and deformation environment of the cell. In this respect, theoretical models of cells
and tissues are particularly valuable in that they may be used to provide information
on biophysical parameters that cannot be measured experimentally in situ at the
cellular level, for example, the stress-strain, physicochemical, and electrical states in
the immediate vicinity of the cell.
Based on existing experimental data on the deformation behavior and biomechan-
ical properties of articular cartilage and chondrocytes, a multiscale biphasic ¬nite
element model was developed of the chondrocyte as a spheroidal inclusion embed-
ded within the extracellular matrix of a cartilage explant (Fig. 5-7). In these studies,
the cell membrane was neglected, and it was assumed that the cell was freely perme-
able to water to allow for changes in volume via transport of interstitial water in an out
of the cell. Finite element analysis of the stress, strain, ¬‚uid ¬‚ow, and hydraulic ¬‚uid
pressure were made of a con¬guration simulating a cylindrical cartilage specimen
(5 mm — 1 mm) subjected to a step load in an uncon¬ned compression experiment.
A parametric analysis was performed by varying the mechanical properties of the
cell over 5“7 orders of magnitude relative to the properties of the ECM. Using a
range of chondrocyte biphasic properties reported in the literature (E ∼ 0.5 ’ 1 kPa,
k ∼ 10’10 ’ 10’15 m4 /N·s, ν ∼ 0.1 ’ 0.4) (Shin and Athanasiou, 1999; Trickey
et al., 2000; Trickey et al., 2006), the distribution of stress at the cellular level was
found to be time varying and inhomogeneous, and it differed signi¬cantly from that in
the bulk extracellular matrix. At early time points (<100 s) following application of the
load, the chondrocytes were exposed primarily to shear stress and strain and hydraulic
¬‚uid pressure, with little volume change. At longer time periods, changes in cell shape
and volume were predicted coincident with exudation of the interstitial ¬‚uid (Fig. 5-7).
The large difference (∼3 orders of magnitude) in the elastic properties of the chondro-
cyte and of the extracellular matrix results in the presence of stress concentrations at
the cell“matrix border and a nearly two-fold increase in strain and dilatation (volume
change) at the cellular level, as compared to that at the macro-level. The presence of
a narrow “pericellular matrix” with different properties than that of the chondrocyte
or extracellular matrix signi¬cantly altered the principal stress and strain magnitudes
Multiphasic models of cell mechanics 97


(a) Macro-scale model Micro-scale model

Macro-scale
Element




0.5 mm




2.5 mm

100 µm
Extracellular
Matrix



Chondrocyte


Micro-scale
Pericellular Element
Matrix

50 µ
µm
10 µm




(b)
t*=0 t*=0.1 t*=0.2
Normalized
Principal
Stress
1.8
1.5

1.2
t*=0.5 t*=1 t*=2
0.9
0.6

0.3
0.0
1 0 µm

Fig. 5-7. (a) A biphasic multiscale ¬nite element method was used to model the mechanical envi-
ronment of a single cell within the cartilage extracellular matrix. The “macro-scale” response of
a cartilage explant in a state of uncon¬ned compression was the ¬rst model. From this solution, a
linear interpolation of the time-history of the kinematic boundary conditions within a 50 — 100 µm
region were then applied to a “micro-scale” ¬nite element mesh that incorporated a chondrocyte
(10 µm diameter) and its pericellular matrix (2.5 µm thick) embedded within and attached to the ex-
tracellular matrix. Using this technique, it is assumed that due to their low volume fraction (<10%),
the cells do not contribute mechanically to the macroscopic properties and behavior of the extra-
cellular matrix. (b) Predictions of the compressive stress in the cell and extracellular matrix versus
time. A gray-scale image of one-quarter of the cell is shown within the matrix. Stress is normalized
to the far-¬eld extracellular matrix stress at equilibrium, and time is normalized to the biphasic gel
time (t — = t/„gel ). At early times following loading, low magnitudes of solid stress were observed, as
the total stress in the tissue was borne primarily by pressurization of the interstitial ¬‚uid. With time,
stresses were transferred to the solid phase and increased stress concentrations are observed at the
cell“matrix boundary. From Guilak and Mow, 2000.


within the chondrocyte, suggesting a functional biomechanical role for this tissue re-
gion. These ¬ndings suggest that even under simple compressive loading conditions,
chondrocytes are subjected to a complex local mechanical environment consisting of
tension, compression, shear, and hydraulic pressure. Knowledge of the magnitudes
98 F. Guilak et al.

and distribution of local stress/strain and ¬‚uid-¬‚ow ¬elds in the extracellular ma-
trix around the chondrocytes is an important step in the interpretation of studies of
mechanical signal transduction in cartilage explant culture models.
In other tissues, anisotropic behavior may play an important role in de¬ning the
micromechanical environment of the cell. For example, cellular response to me-
chanical loading varies between the anatomic zones of the intervertebral disc, and
this difference may be related to differences in the structure and mechanics of both
cells and extracellular matrix, which are expected to cause differences in the phys-
ical stimuli (such as pressure, stress, and strain) in the cellular micromechanical
environment (Guilak et al., 1999; Baer et al., 2003). In other studies, ¬nite element
analyses have been used to model ¬‚ow-dependent viscoelasticity using the biphasic
theory for soft tissues; ¬nite deformation effects using a hyperelastic constitutive
law for the solid phase; and material anisotropy by including a ¬ber-reinforced con-
tinuum law in the hyperelastic strain energy function. The model predicted that the
cellular micromechanical environment varies dramatically depending on the local
tissue stiffness and anisotropy. Furthermore, the model predicted that stress-strain
and ¬‚uid-¬‚ow environment is strongly in¬‚uenced by cell shape, suggesting that the
geometry of cells in situ may be an adaptation to reduce cellular strains during tissue
loading.
With similar multiscaling techniques, other studies have used triphasic consti-
tutive models to predict the physicochemical environment of cells within charged,
hydrated tissues (Likhitpanichkul et al., 2003). These studies also show that in ad-
dition to nonhomogeneous stress-strain and ¬‚uid-¬‚ow ¬elds within the extracellular
matrix, cells may also be exposed to time- and spatially varying osmotic pressure
and electric ¬elds due to the coupling between electrical, chemical, and mechan-
ical events in the cell and in the surrounding tissues. Such methods may provide
new insight into the physical regulatory mechanisms that in¬‚uence cell behavior
in situ.


Summary
Multiphasic approaches have important advantages and disadvantages relative to more
classic single-phase models. The disadvantages are based primarily on the added
complexity required for computational models. In most cases, analytical solutions
are intractable and numerical methods such as ¬nite element modeling are required.
Furthermore, additional experimental tests are necessary to determine the intrinsic
mechanical contributions of the different phases to the overall behavior of the cell.
However, multiphasic models may provide a more realistic representation of the phys-
ical events that govern cell mechanical behavior. Furthermore, as most current multi-
phasic models are based on a continuum approach, the constitutive models describing
each phase can be selected independently to best describe the empirically observed
behavior of the cell. A multiphasic approach may also be combined with other struc-
turally based models (such as, tensegrity models), and thus may provide a versatile
modeling approach for examining the interactions of the different constitutive phases
governing cell mechanical behavior.
Multiphasic models of cell mechanics 99


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6 Models of cytoskeletal mechanics based
on tensegrity

Dimitrije Stamenovic
´




Cell shape is an important determinant of cell function and it provides a regu-
ABSTRACT:
latory mechanism to the cell. The idea that cell contractile stress may determine cell shape
stability came with the model that depicts the cell as tensed membrane that surrounds viscous
cytoplasm. Ingber has further advanced this idea of the stabilizing role of the contractile stress.
However, he has argued that tensed intracellular cytoskeletal lattice, rather than the cortical
membrane, con¬rms shape stability to adherent cells. Ingber introduced a special class of
tensed reticulated structures, known as tensegrity architecture, as a model of the cytoskeleton.
Tensegrity architecture belongs to a class of stress-supported structures, all of which require
preexisting tensile stress (“prestress”) in their cable-like structural members, even before ap-
plication of external loading, in order to maintain their structural integrity. Ordinary elastic
materials such as rubber, polymers, or metals, by contrast, require no such prestress. A hallmark
property that stems from this feature is that structural rigidity (stiffness) of the matrix is nearly
proportional to the level of the prestress that it supports. As distinct from other stress-supported
structures falling within the class, in tensegrity architecture the prestress in the cable network
is balanced by compression of internal elements that are called struts. According to Ingber™s
cellular tensegrity model, cytoskeletal prestress in generated by the cell contractile machinery
and by mechanical distension of the cell. This prestress is carried mainly by the cytoskeletal
actin network, and is balanced partly by compression of microtubules and partly by traction at
the extracellular adhesions.
The idea that the cytoskeleton maintains its structural stability through the agency of con-
tractile stress rests on the premise that the cytoskeleton is a static network. In reality, the
cytoskeleton is a dynamic network, which is exposed to dynamic loads and in which the
dynamics of various biopolymers contribute to its rheological properties. Thus, the static model
of the cytoskeleton provides only a limited insight into its mechanical properties (for example,
near-steady-state conditions). However, our recent measurements have shown that cell rheolog-
ical (dynamic) behavior may also be affected by the contractile prestress, suggesting thereby
that the tensegrity idea may also account for some features of cell rheology.
This chapter describes the basic idea of the cellular tensegrity hypothesis, how it applies to
problems in cellular mechanics, and what its limitations are.


Introduction
A new model of cell structure to explain how the internal cytoskeleton of adher-
ent cells mediates alterations in cell functions caused by changes in cell shape was
103
104 D. Stamenovic
´

proposed in the early 1980s by Donald Ingber and colleagues (Ingber et al., 1981;
Ingber and Jamieson, 1985). This model is based on a building system known as
tensegrity architecture (Fuller, 1961). The essential premise of what is known as the
cellular tensegrity model is that the cytoskeletal lattice carries preexisting tensile
stress, termed prestress, whose role is to confer shape stability to the cell. A second
premise is that this cytoskeletal prestress is partly balanced by forces that arise at cell
adhesions to the extracellular matrix and partly by internal, compression-supporting
cytoskeletal structures (for example, microtubules). The cytoskeletal prestress is gen-
erated actively, by the cytoskeletal contractile apparatus. Additional prestress is gen-
erated passively by cell mechanical distension through adhesions to the substrate, by
cytoplasmic swelling pressure (turgor), and by forces generated by ¬lament polymer-
ization. The prestress is primarily carried by the cytoskeletal actin network and to a
lesser extent by the intermediate ¬lament network (Ingber, 1993; 2003a).
There is a growing body of experimental data that is consistent with the cellular
tensegrity model. The strongest piece of evidence in support of the tensegrity model
is the observed proportional relationship between cell stiffness and the cytoskeletal
contractile stress (Wang et al., 2001; 2002). Experimental data also show that micro-
tubules carry compression that, in turn, balances a substantial portion of the prestress,
which is another key feature of tensegrity architecture (Wang et al., 2001; Stamenovi´ c
et al., 2002a). Together, these two ¬ndings have provided so far the most convincing
evidence in support of the cellular tensegrity model.
In successive sections, this chapter describes basic concepts, de¬nitions, and un-
derlying mechanisms of the cellular tensegrity model; describes experimental data
that are consistent and those that are not consistent with the tensegrity model; and
describes results from mathematical modeling of typical tensegrity-based models of
cell mechanics and compares predictions from those models to experimental data
from living cells. Then the chapter brie¬‚y discusses the usefulness of the tensegrity
idea in studying the dynamic behavior of cells and ends with a summary.


The cellular tensegrity model
It is well established that cell shape is critical for the control of many cell behaviors,
including growth, motility, differentiation, and apoptosis and that the effects of cell
shape are mediated through changes in the intracellular cytoskeleton (see Ingber,
2003a and 2003b). To explain how cells generate mechanical stresses in response
to alterations in their shape and how those stresses affect cellular function, various
models of cellular mechanics have been advanced, as other chapters here extensively
discuss. All these models can be divided into two distinct classes: continuum models,
and discrete models.
Continuum models (Theret et al., 1988; Evans and Yeung, 1989; Fung and Liu,
1993; Schmid-Sch¨ nbein et al., 1995; Bausch et al., 1998; Fabry et al., 2001a) assume
o
that the stress-bearing elements within the cell are small compared to the length scales
of interest and that they uniformly ¬ll the space within the cell body. The microscale
behavior of these elements is given by equations that describe local deformation and
mass, and momentum and energy balance (see Chapter 3 of this book). This leads to
descriptions of stress and strain patterns that are continuous in space within the cell.
Models of cytoskeletal mechanics based on tensegrity 105

Continuum models can run from simple to very complex and multicompartmental,
from elastic to viscoelastic (Chapter 4) or even poroelastic (Chapter 5).
Discrete models (Porter, 1984; Ingber and Jamieson, 1985; Forgacs, 1995; Satcher
and Dewey, 1996; Stamenovi´ et al., 1996; Boey et al., 1998) consider discrete stress-
c
bearing elements of the cell that are ¬nite in size, sometimes spanning distances that
are comparable to the cell size (for example, microtubules). The cell is depicted as
being composed of a large number of these discrete elements that do not ¬ll the space.
The behavior of each discrete element is subject to conditions of mechanical equi-
librium and geometrical compatibility at every node. At this point, a coarse-graining
average can be applied and local stresses and strains can be obtained as continuous
¬eld variables. Within the class of discrete models there is a special subclass, known as
stress-supported (or prestressed) structures. While ordinary elastic materials such as
rubber, polymers, or metals, by contrast, require no such prestress, all stress-supported
structures require tensile prestress in their structural members, even before the appli-
cation of external loading, in order to maintain their structural integrity. A hallmark
property that stems from this feature is that structural rigidity (stiffness) of the matrix
is proportional to the level of the prestress that it supports (Volokh and Vilnay, 1997;
Stamenovi´ and Ingber, 2002). Tensegrity architecture falls within this class. As in the
c
case of continuum models, discrete models, of which the tensegrity architecture is one,
can range from very simple to very complex, multimodular, and multicompartmental.
It has long been known that many cell types exist under tension (prestress) (Harris
et al., 1980; Albrecht-Buehler, 1987; Heidemann and Buxbaum, 1990; Kolodney and
Wysolmerski, 1992; Evans et al., 1993). Theoretical models that depict the cell as a
tensed (that is, prestressed) membrane that surrounds viscous cytoplasm have been
proposed in the past (Evans and Yeung, 1989; Fung and Liu, 1993; Schmid-Sch¨ nbein o
et al., 1995). However, none of those studies show that this cell prestress may play a
key role in regulating cell deformability. In the early 1980s, Donald Ingber (Ingber
et al., 1981; Ingber and Jamieson, 1985) introduced a novel model of cytoskeletal
mechanics based on architecture that secures structural stability through the agency
of prestress. This model has become known as the cellular tensegrity model. Basic
features and mechanisms of this model and how they apply to mechanics of cells are
described in the coming sections.


De¬nitions, basic mechanisms, and properties of tensegrity structures
Tensegrity architecture is a building principle introduced by R. Buckminster Fuller
(Fuller, 1961). He de¬ned tensegrity as a system through which structures are stabi-
lized by continuous tension carried by the structural members (like a camp tent or a
spider web) rather than continuous compression (like a stone arch). Fuller referred to
this architecture as “tensional integrity,” or “tensegrity” (Fig. 6-1).
The central mechanism by which tensegrity and other prestressed structures develop
restoring stress in the presence of external loading is by geometrical rearrangement
(that is, by change in spacing and orientation and to a lesser degree by change in length)
of their pre-tensed members. The greater the pre-tension carried by these members,
the less geometrical rearrangement they undergo under an applied load, and thus the
less deformable (more rigid) the structure will be. In the absence of prestress, these
106 D. Stamenovic
´




Fig. 6-1. A cable-and-strut tensegrity dome (“Dome Image c 1999 Bob Burkhardt”). In this struc-
ture, tension in the cables (white lines) is partly balanced by the compression of the struts (thick
black lines) and partly by the attachments to the substrate. At each free node one strut meets several
cables. Adapted with permission from Burkhardt, 2004.


structures become unstable and collapse. This explains why the structural stiffness
increases in proportion with the level of the prestress.
An interesting (although not an intrinsic) property of tensegrity structures is a long-
distance transfer of mechanical disturbances. Ingber referred to this phenomenon as
the “action at a distance” effect (Ingber, 1993; 2003a). Because tensegrity structures
resist externally applied loads by geometrical rearrangements of their structural mem-
bers, any local disturbance should result in a global rearrangement of the structural
lattice and should be manifested at points distal from the point of an applied load.
This is quite different from continuum models where local disturbances produce only
local responses, which dissipate inversely with the distance from the point of load ap-
plication. In complex and multimodular tensegrity structures, this action at a distance
may not be easily observable because the effect of an applied mechanical disturbance
may be dissipated through the multi-connectedness of structural members and fade
away at points distal from the point of load application.


The cellular tensegrity model
In the cellular tensegrity model, actin ¬laments and intermediate ¬laments of the
cytoskeleton are envisioned as tensile elements (cables) that carry the prestress. Mi-
crotubules and thick cross-linked actin bundles, on the other hand, are viewed as
Models of cytoskeletal mechanics based on tensegrity 107

compression elements (struts) that partly balance the prestress. The rest of the pre-
stress is balanced by the extracellular matrix, which is physically connected to the
cytoskeleton through the focal adhesion complex. In highly spread cells, however,
intracellular compression-supporting elements may become redundant and the ex-
tracellular matrix may balance the entire prestress. In other words, the cytoskeleton
and the extracellular matrix are viewed as a single, synergetic, mechanically sta-
bilized system, or the “extended cytoskeleton” (Ingber, 1993). Thus, although the
cellular tensegrity model allows for the presence of internal compression-supporting
elements, they are neither necessary nor suf¬cient for the overall stress balance in the
cell“extracellular matrix system.


Do living cells behave as predicted by the tensegrity model?
This section presents a survey of experimental data that are consistent with the cellular
tensegrity model, as well as those that are not.


Circumstantial evidence
Data obtained from in vitro biophysical measurements on isolated actin ¬laments
(Yanagida et al., 1984; Gittes et al., 1993; MacKintosh et al., 1995) and microtubules
(Gittes et al., 1993; Kurachi et al., 1995) indicate that actin ¬laments are semi¬‚exible,
curved, of high tensile modulus (order of 1 GPa), and of the persistence length (a
measure of stiffness of a polymer molecule that can be described as a mean radius
of curvature of the molecule at some temperature due to thermal ¬‚uctuations) on the
order of 10 µm. On the other hand, microtubules appeared straight, as rigid tubes, of
nearly the same modulus as actin ¬laments but of much greater persistent length,
order of 103 µm. Based on these persistence lengths, actin ¬laments should appear
curved and microtubules should appear straight on the whole cell level if they were
not mechanically loaded. However, immuno¬‚uorescent images of the cytoskeleton
lattice of living cells (Fig. 6-2) show that actin ¬laments appear straight, whereas
microtubules appeared curved (Kaech et al., 1996; Eckes et al., 1998, 2003a). It
follows, therefore, that some type of mechanical force must act on these molecular
¬laments in living cells: conceivably, the tension in actin ¬laments straighten them
while compression in microtubules result in their bending (caused by buckling).
On the other hand, Satcher et al. (1997) found that in endothelial cells the average
pore size of the actin cytoskeleton ranges from 50“100 nm, which is much smaller
than the persistence length of actin ¬laments. This, in turn, suggests that the straight
appearance of actin cytoskeletal ¬laments is the result of their very short length
relative to their persistence length.
It is well established that the prestress borne by the cytoskeleton is transmitted to
the substrate through transmembrane integrin receptors. Harris et al. (1980) showed
that in response to the contraction of ¬broblasts cultured on a ¬‚exible silicon rub-
ber substrate, the substrate wrinkles. Similarly, contracting ¬broblasts that adhere
to a polyacrylamide gel substrate cause the substrate to deform (Pelham and Wang,
1997). Severing focal adhesion attachments of endothelial cells to the substrate by
trypsin results in a quick retraction of these cells (Sims et al., 1992), suggesting that
108 D. Stamenovic
´




Fig. 6-2. Local buckling of a green ¬‚uorescent protein-
labeled microtubule (arrowhead) in living endothelial cells
following cell contraction induced by thrombin. The micro-
tubule appears fairly straight prior to cell contraction (a) and
assumes a typical sinusoidal buckled shape following con-
traction (b). The white lines are drawn to enhance the shape of
microtubule; the scale bar is 2 µm. Adapted with permission
from Wang et al., 2001.




the cytoskeleton carries prestress and that this prestress is transmitted to and balanced
by traction forces that act at the cell-anchoring points to the substrate.
Experimental observations support the existence of a mechanical coupling between
tension carried by the actin network and compression of microtubules, analogous to
the tension-compression synergy in the cable-and-strut tensegrity model. For example,
as migrating cultured epithelial cells contract, their microtubules in the lamellipodia
region buckle as they resist the contractile force exerted on them by the actin network
(Waterman-Storer and Salmon, 1997). Extension of a neurite, which is ¬lled with
microtubules, is opposed by pulling forces of the actin micro¬laments that surround
those microtubules (Heidemann and Buxbaum, 1990). Microtubules of endothelial
cells, which appear straight in relaxed cells, appear buckled immediately following
contraction of the actin network (Fig. 6-2) (Wang et al., 2001). In their mechanical
measurements on ¬broblasts, Heidemann and co-workers also observed the curved
shape of microtubules. However, they associated these con¬gurations with ¬‚uid-
like behavior of microtubules because they observed slow recovery of microtubules
following mechanical disturbances applied to the cell surface (Heidemann et al., 1999;
Ingber et al., 2000). Contrary to these observations, Wang et al. (2001) observed
relatively quick recovery of microtubules in endothelial cells following mechanical
disturbances.
Cells of various types probed with different techniques exhibit a stiffening effect,
such that cell stiffness increases progressively with increasingly applied mechanical
load (Petersen et al., 1982; Sato et al., 1990; Alcaraz et al., 2003). This, in turn,
implies that stress-strain behavior of cells is nonlinear, such that stress increases
Models of cytoskeletal mechanics based on tensegrity 109

faster than strain. This stiffening is also referred to as a strain or stress hardening. In
discrete structures, this nonlinearity is primarily a result of geometrical rearrangement
and recruitment of structural members in the direction of applied load, and less due
to nonlinearity of individual structural members (Stamenovi´ et al., 1996). In their
c
early works, Ingber and colleagues considered this stiffening to be a key piece of
evidence in support of the cellular tensegrity idea, as various physical (Wang et al.,
1993) and mathematical (Stamenovi´ et al., 1996; Coughlin and Stamenovi´ , 1998)
c c
tensegrity models exhibit this behavior under certain types of loading. It turns out
that this is an inconclusive piece of evidence that neither supports nor refutes the
tensegrity model for the following reasons. First, the stress/strain hardening behavior
characterizes various types of solid materials, many of which are not at all related to
tensegrity. Second, the stress/strain hardening behavior is not an intrinsic property of
tensegrity structures because they can also exhibit softening “ that is, under a given
loading their stiffness may decrease with increasingly applied load (Coughlin and
Stamenovi´ , 1998; Volokh et al., 2000) “ or they may, under certain conditions, have
c
constant stiffness, independent of the applied load (Stamenovi´ et al., 1996). Third,
c
recent mechanical measurements in living airway smooth muscle cells showed that
their stress-strain behavior is linear over a wide range of applied stress, and thus they
exhibit neither stiffening nor softening (Fabry et al., 1999; 2001).
Based on the above circumstantial evidence and differing interpretations of the
evidence, it is clear that rigorous experimental validation of the cellular tenseg-
rity model was needed to demonstrate a close association between cell stiffness and
cytoskeletal prestress, and to show that cells exhibit the action-at-a-distance behav-
ior. Also essential is quantitative assessment of the contribution of the substrate vs.
compression of microtubules in balancing the prestress, and also understanding the
role of intermediate ¬laments in the context of the tensegrity model. New advances
in cytometry techniques made it possible to provide direct, quantitative data for these
behaviors. These data are described below.


Prestress-induced stiffening
An a priori prediction of all prestressed structures is that their stiffness increases
in nearly direct proportion with prestress (Volokh and Vilnay, 1997). A number of
experiments in various cell types have shown evidence of prestress-induced stiffening.
For example, it has been shown that mechanical (Wang and Ingber, 1994; Pourati et al.,
1998; Cai et al., 1998), pharmacological (Hubmayr et al., 1996; Fabry et al., 2001),
and genetic (Cai et al., 1998) modulations of cytoskeletal prestress are paralleled
by changes in cell stiffness. Advances in the traction cytometry technique (Fig. 6-3)
made it possible to quantitatively measure various indices of cytoskeletal prestress
(Pelham and Wang, 1997; Butler et al., 2002; Wang et al., 2002). These data are
then correlated with data obtained from measurements of cell stiffness. It was found
(see Fig. 6-4) that in cultured human airway smooth muscle cells whose contractility
was altered by graded doses of contractile and relaxant agonists, cell stiffness (G)
increases in direct proportion with the contractile stress (P); G ≈ 1.04P (Wang et al.,
2001; 2002). Although this association between cell stiffness and contractile stress
does not preclude other interpretations, it is the hallmark of structures that secure
110 D. Stamenovic
´

Fig. 6-3. (a) A human airway smooth muscle cell cultured
a
on a ¬‚exible polyacrylamide gel substrate. As the cell con-
tracts (histamine 10 µM), the substrate deforms, causing ¬‚u-
orescent microbead markers embedded in the gel to move
(arrows). From measured displacement ¬eld of the markers
and known elastic properties of the gel, one can calculate trac-
tion („ ) that arises at the cell-gel interface (Butler et al., 2002).
Because the cytoskeletal prestress (P) is balanced partly by „ ,
one can asses P(Wang et al., 2002). (b) A free body diagram
of a cell section depicting a three-way force balance between
the cytoskeleton (P), substrate (PS ), and microtubules (PQ ):
PS = P ’ PQ where PS indicates the part of P that is bal-
anced by the substrate and PQ indicates the part of P that is
balanced by compression-supporting microtubules. At equi-
librium, the force balance requires that „ A = PS A where
A and A are interfacial and cross-sectional areas of the cell
section, respectively. Because „, A and A can be directly
measured, one can obtain PS . A and A were measured for
b many optical cross-sections of the cell. For each section, PS
was calculated and the average value was obtained (Wang et
al., 2002). Note that in the absence of internal compression
structures (for example, upon disruption of microtubules),
PQ = 0 and the entire prestress P is balanced by „ (i.e.,
PS = P).


shape stability through the agency of the prestress. Other possible interpretations of
this ¬nding are discussed below.
In addition to generating contractile force, it has been shown that pharmacological
agonists also induce polymerization of the actin network (Mehta and Gunst, 1999;
Tang et al., 1999). Thus, the observed stiffening in response to contractile agonists
could be nothing more than the result of actin polymerization. However, An et al.
(2002) have shown that agonist-induced actin polymerization in smooth muscle cells
accounts only for a portion of the observed stiffening, whereas the remaining portion
of the stiffening is associated with contractile force generation. Another potential
mechanism that could explain the data in Fig. 6-4 is the effect of cross-bridge recruit-
ment. It is known from studies of isolated smooth muscle strips in uniaxial extension
that both muscle stiffness and muscle force are directly proportional to the number of
attached cross-bridges (Fredberg et al., 1996). Thus, the proportionality between the
cell stiffness and the prestress could re¬‚ect nothing more than the effect of changes
in the number of attached cross-bridges in response to pharmacological stimulation.
A result that goes against this possibility is obtained from a theoretical model of the
myosin cross-bridge kinetics (Mijailovich et al., 2000). This model predicts a qualita-
tively different oscillatory response from the one measured in airway smooth muscle
cells (Fabry et al., 2001). Thus, the kinetics of cross-bridges cannot explain all aspects
of cytoskeletal mechanics.


Action at a distance
To investigate whether the cytoskeleton exhibits the action-at-a-distance effect,
Maniotis et al. (1997) performed experiments in which the tip of a glass micropipette
Models of cytoskeletal mechanics based on tensegrity 111




Fig. 6-4. Cell stiffness (G) increases linearly with increasing cytoskeletal contractile stress.
Measurements were done in cultured human airway smooth muscle cells. Cell contractility was
modulated by graded doses of histamine (constrictor) and graded doses of isoproterenol (relaxant).
Stiffness was measured using the magnetic cytometry technique and the prestress was measured by
the traction cytometry technique (Wang et al., 2002). The slope of the regression line is 1.18 (solid
line). The measured prestress represents the portion of the cytoskeletal prestress that is balanced by
the substrate (that is, PS from Fig. 6-3). Because in those cells microtubules balance on average ∼14%
of PS (Stamenovi´ et al., 2002a), the slope of the stiffness vs. the total prestress (P) relationship
c
should be reduced by 14% and thus equals 1.04 (dashed line). The stiffness vs. prestress relationship
displays a nonzero intercept. This is due to a bias in the method used to calculate the prestress (in other
words, the cell cross-sectional area A from Fig. 6-3 is an overestimate) (Wang et al., 2002). In the
absence of this artifact, the stiffness vs. prestress relationship would display close-to-zero intercept,
that is, G ≈ 1.04P (Wang et al., 2002). (Redrawn from Wang et al. (2002) and Stamenovi´ et al. c
(2002b); G is rescaled to take into account the effect of bead internalization. From Mijailovich et al.,
2002.



coated with ¬bronectin and bound to integrin receptors of living endothelial cells was
pulled laterally. Because integrins are physically linked to the cytoskeleton, then if
the cytoskeleton is organized as a discrete tensegrity structure, pulling on integrins
should produce an observable deformation distal from the point of load application.
The authors observed that the nuclear border moved along the line of applied pulling
force, which is a manifestation of the action at a distance. A more convincing piece of
evidence for this phenomenon was provided by Hu et al. (2003). These investigators
designed the intracellular tomography technique that enabled them to observe dis-
placement distribution within the cytoskeleton region in response to locally applied
shear disturbance. Lumps of displacement concentrations were found at distances
greater than 20 µm from the point of application of the shear loading, which is indica-
tive of the action-at-a-distance effect (Fig. 6-5). Interestingly, when the actin lattice
was disrupted (cytochalasin D), the action-at-a-distance effect disappeared (data not
shown), suggesting that connectivity of the actin network is essential for transmission
of mechanical signals throughout the cytoskeleton. The action-at-a-distance effect has
also been observed in neurons (Ingber et al., 2000) and in endothelial cells (Helmke
et al., 2003). On the other hand, Heidemann et al. (1999) failed to observe this phe-
nomenon in living ¬broblasts when they applied various mechanical disturbances by
a glass micropipette to the cell surface through integrin receptors. They found that
112 D. Stamenovic
´




Fig. 6-5. Evidence of the action-at-a-distance effect. Displacement map in living human airway
smooth muscle cells obtained using the intracellular tomography technique (Hu et al., 2003). Load
is applied to the cell by twisting a ferromagnetic bead bound to integrin receptors on the cell apical
surface. The bead position is shown on the phase-contrast image of the cell (inset), the black dot
on the image is the bead. The white arrows indicate the direction of the displacement ¬eld and
the gray-scale map represents its magnitude. Displacements do not decay quickly away from
the bead center. Appreciable “lumps” of displacement concentration could be seen at distances
more than 20 µm from the bead, consistent with the action-at-a-distance effect. The inner el-
liptical contour indicates the position of the nucleus. Adapted with permission from Hu et al.,
2003.



such disturbances produced only local deformations. However, the authors did not
con¬rm formation of focal adhesions at points of application of external loading,
which is essential for load transfer between cell surface and the interior cytoskeleton
(Ingber et al., 2000). Thus, their results remain controversial.



Do microtubules carry compression?
Microscopic visualization of green ¬‚uorescent protein-labeled microtubules of living
cells (see Fig. 6-2) shows that microtubules buckle as they oppose contraction of the
actin network (Waterman-Storer and Salmon, 1997; Wang et al., 2001). It was not
known, however, whether the compression that causes this buckling could balance
a substantial fraction of the contractile prestress. To investigate this possibility, an
energetic analysis of buckling of microtubules was carried out (Stamenovi´ et al.,
c
2002a). The assumption was that energy stored in microtubules during compres-
sion was transferred to a ¬‚exible substrate upon disruption of microtubules. Thus,
measurement of an increase in elastic energy of the substrate following disruption
of microtubules should indicate compression energy stored in microtubules prior to
their disruption. Elastic energy stored in the substrate was obtained from traction
microscopy measurements as a work done by traction forces during cell contraction.
It was found in highly stimulated and spread human airway smooth muscle cells that
Models of cytoskeletal mechanics based on tensegrity 113

following disruption of microtubules by colchicine, the work of traction increases
on average by ∼30 percent relative to the state before disruption, and equals 0.13
pJ (Stamenovi´ et al., 2002a). This result was then utilized in the energetic analysis.
c
Based on the model of Brodland and Gordon (1990), the microtubules were assumed
as slender elastic rods laterally supported by intermediate ¬laments. Using the post-
buckling equilibrium theory of Euler struts (Timoshenko and Gere, 1988), the energy
stored during buckling of microtubules was estimated as ∼0.18 pJ, which is close
to the measured value of ∼0.13 pJ (Stamenovi´ et al., 2002a). This is further evi-
c
dence in support of the idea that microtubules are intracellular compression-bearing
elements. Potential concerns are that disruption of microtubules may activate myosin
light-chain phosphorylation (Kolodney and Elson, 1995) or could cause a release
of intracellular calcium (Paul et al., 2000). Thus, the observed increase in traction
and work of traction following disruption of microtubules could be due entirely to
chemical mechanisms rather than through mechanical load transfer. These concerns
are alleviated by observations indicating that microtubule disruption results in an in-
crease of traction even when the level of myosin light-chain phosphorylation and the
level of calcium do not change (Wang et al., 2001; Stamenovi´ et al., 2002a).
c
From the same experimental data used in the energetic analysis, the contribution of
microtubules to balancing the prestress was obtained as follows (Wang et al., 2001;
Stamenovi´ et al., 2002a). An increase in traction following microtubule disruption
c
indicates the part of the prestress balanced by microtubules that is transferred to the
substrate (see Fig. 6-3b). It was found that this increase ranges from ∼5“30 percent,
depending on the cell, and is on average ∼14 percent, suggesting that microtubules
balance only a small fraction of the cytoskeletal prestress and that the substrate bal-
ances the bulk of it (Stamenovi´ et al., 2002a). An increase in traction in the response
c
to disruption of microtubules had been observed previously, in different cell types, by
other investigators, but has not been quanti¬ed (Kolodney and Wysolmersky, 1992;
Kolodney and Elson, 1995). More recently, Hu et al. (2004) showed that the contri-
bution of microtubules to balancing the prestress and to the energy budget of the cell
depends on the extent of cell spreading. Using the traction cytometry technique, these
investigators found that in airway smooth muscle cells, changes in traction and the
substrate energy following disruption of microtubules decrease with increasing cell
spreading. For example, as the cell projected area increases from 500 to 1800 µm2 ,
the percent increase in traction following disruption of microtubules decreases from
80 percent to a very small percent. Because in their natural habitat cells seldom exhibit
highly spread forms, the above results suggest that the contribution of microtubules
in balancing the prestress cannot be overlooked.


The role of intermediate ¬laments
Cytoskeletal-based intermediate ¬laments also carry prestress and link the nucleus
to the cell surface and the cytoskeleton (Ingber, 1993; 2003a). In support of this
view, vimentin-de¬cient ¬broblasts were found to exhibit reduced contractility and
reduced traction on the substrate in comparison to the wild-type cells (Eckes et al.,
1998). Also, it was observed that the intermediate ¬lament network alone is suf¬cient
114 D. Stamenovic
´

to transfer mechanical load from cell surface to the nucleus in cells in which the
actin and microtubule networks are chemically disrupted (Maniotis et al., 1997).
Taken together, these observations suggest that intermediate ¬laments play a role in
transferring the contractile prestress to the substrate and in long-distance load transfer
within the cytoskeleton. Both are key features of the cellular tensegrity model. In
addition, inhibition of intermediate ¬laments causes a decrease in cell stiffness (Wang
et al., 1993; Eckes et al., 1998; Wang and Stamenovi´ , 2000), as well as cytoplasmic
c
tearing in response to high applied strains (Maniotis et al., 1997; Eckes et al., 1998).
In fact, it appears that the intermediate ¬laments™ contribution to a cell™s resistance to
shape distortion is substantial only at relatively large strains (Wang and Stamenovi´ , c
2000). Another role of intermediate ¬laments is suggested by Brodland and Gordon
(1990). According to these authors, intermediate ¬laments provide a lateral stabilizing
support to microtubules as they buckle while opposing contractile forces transmitted
by the cytoskeletal actin lattice. This description is consistent with experimental data
(Stamenovi´ et al., 2002a).
c

Summary
Results from experimental measurements on living adherent cells indicate that their
behavior is consistent with the cellular tensegrity model. It was found that cell stiffness
increases directly proportionally with increasing contractile stress. It was also found
that microtubules carry compression that, in turn, balances a substantial portion of
the cytoskeletal prestress. This contribution of microtubules is much smaller in highly
spread cells, roughly a few percent, whereas in poorly spread cells it can be as high
as ∼50 percent. The majority of data from measurements of the action-at-a-distance
phenomenon indicate that cells exhibit this type of behavior when the force is applied
through integrin receptors at the cell surface and focal adhesions were formed at the
site of force application. Intermediate ¬laments appear to be important contributors
to cell contractility and thus to supporting the prestress. They serve as molecular
“guy wires” that facilitate transfer of mechanical loads between the cell surface and
the nucleus. Finally, intermediate ¬laments appear to stabilize microtubules as the
latter balance the cytoskeletal prestress. Taken together, these observations provide
strong evidence in support of the cellular tensegrity model. Although they can have
alternative interpretations, there is no single model other than tensegrity that can
explain all these data together.

Examples of mathematical models of the cytoskeleton based
on tensegrity
Despite its geometric complexity, its dynamic nature, and its inelastic properties, the
cytoskeleton is often modeled as a static, elastic, isotropic, and homogeneous network
of idealized geometry. The idea is that if the mechanisms by which such an ideal-
ized model develops mechanical stress are indeed embodied within the cytoskeleton,
then, despite all simpli¬cations, the model should be able to capture key features that
characterize mechanical behavior of cells under the steady-state. With the tensegrity
model, however, each element is individually taken into account for a discrete formu-
lation of the model. This section describes three types of prestressed structures that
have been commonly used as models of cellular mechanics: the cortical membrane
Models of cytoskeletal mechanics based on tensegrity 115




Fig. 6-6. (a) Ferromagnetic beads bound to the apical surface of cultured human airway smooth
muscle cells (unpublished data kindly supplied by Dr. B. Fabry). (b) A free-body diagram of a
magnetic bead of diameter D half embedded into an elastic membrane of thickness h. The bead
is rotated in a vertical plane by speci¬c torque M through angle θ. The rotation is resisted by the
membrane tension (prestress) (Pm ).

model; the tensed cable net model; and the cable-and-strut model. All three models
are stabilized by the prestress. They differ from each other in their topological and
structural organization, and in the manner by which they balance the prestress. Results
obtained from the models are compared with data from living cells.

The cortical membrane model
This model assumes that the main force-bearing elements of the cytoskeleton are
con¬ned either within a thin (∼100 nm) cortical layer (Zhelev et al., 1994) or several
distinct layers (Heidemann et al., 1999). The cortical layer is under sustained tension
(that is, prestress) that is either entirely balanced by the pressurized cytoplasm in
suspended cells, or balanced partly by the cytoplasmic pressure and partly by traction
at the extracellular adhesions in adherent cells. This model has been successful in
describing mechanical features of various suspended cells (Evans and Yeung, 1989;
Zhelev et al., 1994; Discher et al., 1998). However, in the case of adherent cells,
this model has enjoyed limited success (Fung and Liu, 1993; Schmid-Sch¨ nbein o
et al., 1995; Coughlin and Stamenovi´ , 2003). To illustrate the usefulness of this
c
model, a simulation of a magnetic twisting cytometry measurement is described
below (Stamenovi´ and Ingber, 2002).
c
In the magnetic twisting cytometry technique, small ferromagnetic beads (4.5-µm
diameter) bound to integrin receptors on the apical surface of an adherent cell are
twisted by a magnetic ¬eld, as shown in Fig. 6-6a. Because integrins are physically
linked to the cytoskeleton, twisting of the bead is resisted by restoring forces of the
cytoskeleton. Using the cortical membrane model, magnetic twisting measurements
are simulated as follows.
A rigid spherical bead of diameter D is half-embedded in an initially tensed (pre-
stressed) membrane of thickness h. A twisting torque (M) is applied to the bead in
the vertical plane (Fig. 6-6b). Rotation of the bead is impeded by the prestress (Pm ) in
the membrane. By considering mechanical balance between M and Pm it was found
(Stamenovi´ and Ingber, 2002) that
c

M = D 2 Pm h sin θ, (6.1)

where θ is the angle of bead rotation. In magnetic twisting measurements, a scale for
the applied shear stress (T ) is de¬ned as the ratio of M and 6 times bead volume,
116 D. Stamenovic
´

where 6 is the shape factor, and shear stiffness (G) as the ratio of T and θ (Wang
et al., 1993; Wang and Ingber, 1994). Thus it follows from Eq. 6.1 that
h sin θ
1
G= .
Pm (6.2)
π Dθ
In the limit of θ ’ 0, G ’ (1/π )Pm (h/D) and represents the shear modulus of
Hookean elasticity. It follows from Eq. 6.2 that G increases in direct proportion with
Pm , a feature consistent with the behavior observed in living cells during magnetic
twisting measurements (see Fig. 6-4). Taking into account experimentally based val-
ues for h = 0.1 µm, D = 4.5 µm, and Pm = O(104 ’105 ) Pa it follows from Eq. 6.2
that G = O(102 ’103 ) Pa, which is consistent with experimentally obtained values
for G (see Fig. 6-4). [Pm was estimated as follows. It scales with the cytoskeletal
prestress P as the ratio of cell radius R to membrane thickness h. Experimental data
show that P = O(102 ’103 ) Pa (Fig. 6-4), R = O(101 ) µm and h = O(10’1 ) µm,
thus Pm = O(104 ’105 ) Pa.]
Despite this agreement, several aspects of this model are not consistent with ex-
perimental results. First, Eq. 6.2 predicts that G decreases with increasing angular
strain θ, in other words, softening behavior, whereas magnetic twisting measurements
show stress hardening (Wang et al., 1993) or constant stiffness (Fabry et al., 2001).
Second, Eq. 6.3 predicts that G decreases with increasing bead diameter D, whereas
experiments on cultured endothelial cells show the opposite trend (Wang and Ingber,
1994). One reason for these discrepancies could be the assumption that the cortical
layer is a membrane that carries only tensile force. In reality, the cortical layer can
support bending, for example in red-blood cells (Evans, 1983; Fung, 1993), and hence
a more appropriate model may be a shell-like rather than a membrane-like structure.
Regardless, the assumption that the cytoskeleton is con¬ned within a thin cortical
layer that surrounds liquid cytoplasm contradicts observations in adherent cells that
mechanical perturbations applied to the cell surfaces are transmitted deep into the
cytoplasmic domain (Maniotis et al., 1997; Wang et al., 2001; Hu et al., 2003). These
observations suggest that mechanical force transmission through the cell is facilitated
through the molecular connectivity of the intracellular solid-state cytoskeletal lattice.
Taken together, these inconsistencies lower our enthusiasm for the cortical-membrane
model as an adequate depiction of the mechanics of adherent cells. However, it re-
mains a good mechanical model for suspended cells where the cytoskeleton appears
to be organized within a thin cortical membrane (Bray et al., 1986).


Tensed cable nets
These are reticulated networks comprised entirely of tensile cable elements (Volokh
and Vilnay, 1997). Because cables do not support compression, they need to carry
initial tension to prevent their buckling and subsequent collapse in the presence of
externally applied load. This initial tension de¬nes prestress that is balanced externally
(for example, by attachment to the extracellular matrix), and/or internally (such as,
by cytoplasmic swelling). A simple illustration of key features of tensed cable nets
can be obtained by using the af¬ne network model. A key premise of such a model is
that local strains follow the macroscopic (continuum) strain ¬eld. (This assumption is
Models of cytoskeletal mechanics based on tensegrity 117

known as the af¬ne approximation.) Using this approach and assuming that initially
all cable orientations in the network are equally probable, one can obtain that the shear
modulus (G) (Stamenovi´ , 2005) is
c
G = (0.8 + 0.2B) P (6.3)
where P is the prestress, B ≡ (dF/dL)/(F/L) is nondimensional cable stiffness, and
the F vs. L dependence represents the cable tension-length characteristsic (Budiansky
and Kimmel, 1987). In general, B may depend on the level of cable tension (that is,
on P). In that case, according to Eq. 6.3, the G vs. P relationship is nonlinear. If,
however, B is constant, then G is directly proportional to P. The ¬rst term on the
right-hand side of Eq. 6.3 represents the sum of the contributions of changes in spacing
and orientation of the cables (0.5P + 0.3P) to G, whereas the second term (0.2BP)
is the contribution of the lengthening of the cables to G.
To test whether the prediction of Eq. 6.3 is quantitatively consistent with exper-
imental data from living cells (see Fig. 6-4), we estimate B from measurements of
force-extension properties of isolated acto-myosin interactions (Ishijima et al., 1996).
Based on these measurements, (dF/dL)/F = 0.024 nm’1 for a wide range of F. Thus,
for a 100-nm long actin ¬lament B = 2.4. The choice of ¬lament length of L = 100
nm is based on the observation of the average pore-size of the actin cytoskeletal net-
work of endothelial cells (Satcher et al., 1997). By substituting this value into Eq.
6.3, it follows that G = 1.28P. This is a modest overestimate of the experimentally
obtained result G = 1.04P (Fig. 6-4).
The most favorable aspect of this model is that it provides a mathematically trans-
parent insight into mechanisms that may determine cytoskeletal deformability; G
is primarily determined by P through change in spacing and orientation of the ca-
ble elements, and to a lesser extent by their stiffness. The model can also provide
a reasonably good quantitative correspondence to experimental G vs. P data. The
latter is obtained under the crude assumptions of the af¬ne strain approximation and
of equally probable distribution of cable orientations. These assumptions are known
to lead toward an overestimate of G (Stamenovi´ , 1990). The model also assumes
c
a homogeneous distribution of the prestress throughout the cytoskeleton, although
measurements show that the prestress is greatest near the cell edges and decreases
toward the nuclear region (Toli´ -Nørrelykke et al., 2002). However, in the experi-
c
mental data for the G vs. P relationship (Fig. 6-4), P represents the mean value of
the prestress distribution throughout the cell, and thus the model assumption of uni-
form prestress is reasonable. The model focuses only on the contribution of the actin
network and ignores potential contributions of other components of the cytoskeleton.
These contributions will be considered shortly. Nevertheless, the model provides a
reasonably good prediction of the G vs. P relationship suggesting that the tensed actin
network plays a major role in determining cell mechanical properties. The model also
describes the cytoskeleton as a static, elastic network, whereas the cytoskeleton is a
dynamic and inelastic structure. This issue is discussed in the section on tensegrity
and cellular dynamics.
It is noteworthy that two-dimensional cable nets also have been used to model the
cortical membrane. In those models the cortical membrane has been depicted as a
two-dimensional network of triangles (Boey et al., 1998) and hexagons (Coughlin
118 D. Stamenovic
´

and Stamenovi´ , 2003). In the case of suspended cells, this model provides very good
c
correspondence to experimental data. For example, the model of the spectrin lattice
successfully describes the behavior of red blood cells during micropipette aspiration
measurements (Discher et al., 1998). However, in the case of adherent cells, the model
of the actin cortical lattice has enjoyed only moderate success. While it provides a
reasonably good correspondence to data from cell poking measurements, it exhibits
only some qualitative features of the cell response to twisting and pulling of magnetic
beads bound to integrin receptors (Coughlin and Stamenovi´ , 2003). Taken together,
c
the above results show that the two-dimensional cable net model is incomplete to
describe mechanical behavior of adherent cells; however the results also show that
prestress is a key determinant of the model response.


Cable-and-strut model
This is a cable net model in which the prestress in the cables is balanced by internal
compression-supporting struts rather than by in¬‚ating pressure. At each free node, one
strut meets several cables (see Fig. 6-1). Cables carry initial tension that is balanced
by compression of the strut. Together, cables and struts form a self-equilibrated and
stable form in the space. This structure may also be attached to the substrate (Fig. 6-1).
In this case, the anchoring forces of the substrate also contribute to the balance of
tension in the cables. The main difference between these structures and the cable nets
is that in the former, the struts directly contribute to the structure™s resistance to shape
distortion, whereas in the latter this contribution does not exist.
The shear modulus (G) of the cable and strut model can be also obtained using the
af¬ne network approach, as in the case of the tensed cable net model. It was found
(Stamenovi´ , 2005) that
c
G = 0.8(P ’ PQ ) + 0.2(BP + B Q PQ ) (6.4)
where P is the prestress carried by the cables and PQ is the portion of P balanced
by the struts, B ≡ (dF/dL)/(F/L) is the nondimensional cable stiffness and B Q ≡
(dQ/dl )/(Q/l ) is the nondimensional strut stiffness. The difference P ’ PQ repre-
sents the portion of P transmitted to and balanced by the substrate and is denoted
by PS (see Fig. 6-2b). It is this PS that can be directly measured using the traction
microscopy technique (Wang et al., 2002).
It was shown in the section on tensed cable nets that B = 2.4. The quantity B Q is
determined based on the buckling behavior of microtubules (Stamenovi´ et al., 2002a).
c
It is found that B Q ≈ 0.54. By substituting this value and B = 2.4 into Eq. 6.4 and
taking into account that in well-spread smooth muscle cells, microtubules balance
on average ∼14 percent of PS , that is, PQ = 0.14PS (Stamenovi´ et al., 2002a), it
c
is obtained that G = 1.19P, which is close to the experimental data of G = 1.04P
(Fig. 6-4).
It is noteworthy that if PQ = 0, for example, in a case where microtubules are
disrupted, Eq. 6.4 reduces to Eq. 6.3. If disruption of microtubules would not affect
P, then according to Eqs. 6.3 and 6.4, for a given P, the shear modulus G would be
∼8 percent lower in the case of intact microtubules than in the case of disrupted
microtubules. In reality, such conditions in cells are hard to achieve. An experimental
Models of cytoskeletal mechanics based on tensegrity 119


(a) (b)




Fig. 6-7. Six-strut tensegrity model in the round (a) and spread (b) con¬gurations anchored to the
substrate. Anchoring nodes A1 , A2 and A3 (round) and A1 , A2 , A3 , B1 , B2 , and B3 (spread) are
indicated by solid triangles. Pulling force F (thick arrow) is applied at node D1 . Reprinted with
permission from Coughlin and Stamenovi´ , 1998.
c



condition that comes close to this occurs in airway smooth muscle cells stimulated
by a saturated dose of histamine (10 µM). In those cells the level of prestress
was maintained constant prior to and after disruption of microtubules by colchicine
(Wang et al., 2001; 2002). It was found that disruption of microtubules causes a small
(∼10 percent), but not signi¬cant, increase in cell stiffness (Stamenovi´ et al., 2002b),
c
which is close to the predicted value of ∼8 percent. On the other hand, in nonstimu-
lated endothelial cells, disruption of microtubules causes a signi¬cant (∼20 percent)
decrease in cell stiffness (Wang et al., 1993; Wang, 1998), which is opposite from
the model prediction. A possible reason for this decrease in stiffness in endothelial
cells is that in the absence of compression-supporting microtubules, cytoskeletal pre-
stress in those cells decreased, and consequently the cytoskeletal lattice became more
compliant.
Most of the criticism for the cable net model also applies to the cable-and-strut
model. However, the ability of the model to predict the G vs. P relationship as well as
the mechanical role of cytoskeleton-based microtubules such that they are consistent
with corresponding experimental data, suggests that the model has captured the basic
mechanisms by which the cytoskeleton resists shape distortion.
Consider next an application of a so-called six-strut tensegrity model to study
the effect of cell spreading on cell deformability (Coughlin and Stamenovi´ , 1998).
c
This particular model has been frequently used in studies of cytoskeletal mechanics
(Ingber, 1993; Stamenovi´ et al., 1996; Coughlin and Stamenovi´ , 1998; Volokh
c c
et al., 2000; Wang and Stamenovi´ , 2000; Wendling et al., 1999). It is comprised
c
of six struts interconnected with twenty-four cables (see Fig. 6-7). Although this
model represents a gross oversimpli¬cation of cytoskeletal architecture, surprisingly
it has provided good predictions and simulations of various mechanical behaviors
observed in living cells, suggesting that it embodies key mechanisms that determine
cytoskeleton mechanics.
In the six-strut tensegrity model, the struts are viewed as slender bars that support
no lateral load. Initially, the cables are under tension balanced entirely by compression
120 D. Stamenovic
´




(a) (b)

Fig. 6-8. (a) Data for stiffness vs. applied stress in round and spread cultured endothelial cells
measured by magnetic twisting cytometry; points means ± SE (n = 3 wells, 20,000 cells/well).
Both con¬gurations exhibit stress-hardening behavior with greater hardening in the spread than in
the round con¬guration. (Adapted with permission from Wang and Ingber (1994).) (b) Simulations
of stiffness vs. applied force (F) in spread and round con¬gurations of the six-strut model (Fig. 6-7)
are qualitatively consistent with the data in panel (a). The force is given in the unit of force and
the stiffness in the unit of force/length. Adapted with permission from Coughlin and Stamenovi´ , c
1998.


of the struts. The structure is then attached to a rigid substrate at three nodes through
frictionless ball-joint connections (Fig. 6-7a). The initial force distribution within the
structure is not affected by this attachment. This is referred to as a ˜round con¬g-
uration.™ To mimic cell spreading, three additional nodes are also anchored to the
substrate (Fig. 6-7b). This is referred to as a ˜spread con¬guration.™ As a consequence
of spreading, force distribution is altered from the one in the round con¬guration.
Tension in the cables is now partly balanced by the struts and partly by reaction forces
at the anchoring nodes. In both spread and round con¬gurations, a vertical pulling
force (F) is applied at a node distal from the substrate (Fig. 6-7). The corresponding
vertical displacement ( x) is calculated and the structural stiffness as G = F/ x.
Two cases were considered, one where struts are rigid and cables linearly elastic, and
the other where both struts and cables are elastic and struts buckle under compression.
Here we present results from the case with rigid struts; corresponding results obtained
with buckling struts are qualitatively similar (Coughlin and Stamenovi´ , 1998). The
c
model predicts that stiffness increases with spreading (Fig. 6-8b). The reason is that
tension (prestress) in the cables increases with spreading. The model also predicts
approximately linear stress-hardening behavior and predicts that this dependence is
greater in the spread than in the round con¬guration (Fig. 6-8b). All these predictions
are consistent (Fig. 6-8a) with the corresponding behavior in round and in spread
endothelial cells (Wang and Ingber, 1994). Further attachments of the nodes to the
substrate, that is, further spreading, would gradually eliminate the struts from the
force balance scheme and their role will be taken over by the substrate.
Models of cytoskeletal mechanics based on tensegrity 121

Taken together, the above results indicate that the cable-and-strut model provides
a good and plausible description of cytoskeletal mechanics. It reiterates the central
role of cytoskeletal prestress in cell deformability. The cable-and-strut model also
reveals the potential contribution of microtubules; they balance a fraction of the
prestress, and their deformability (buckling) contributes to the overall deformability
of the cytoskeleton. This contribution decreases as cell spreading increases.
In all of the above considerations, intermediate ¬laments are viewed only as a
stabilizing support during buckling of microtubules. To investigate their contribution
to cytoskeletal mechanics as stress-bearing members, elastic cables that connect the
nodes of the six-strut tensegrity model with its geometric center are added to the model
(Wang and Stamenovi´ , 2000). This was based on the observed role of intermediate
c
¬laments as “guy wires” between the cell surface and the nucleus (Maniotis et al.,
1997). It was shown that by including those cable members in the six-strut model, the
model can account for the observed difference in the stress-strain behavior measured
by magnetic twisting cytometry between normal cells and cells in which intermediate
¬laments were inhibited (Wang and Stamenovi´ , 2000).
c

Summary
Mathematical descriptions of standard tensegrity models of cellular mechanics pro-
vide insight into how the cytoskeletal prestress determines cell deformability. Three
key mechanisms through which the prestress secures shape stability of the cytoskele-
ton are changes in spacing, orientation, and length of structural members of the
cytoskeleton. Importantly, these mechanisms are not tied to the manner by which the
cytoskeletal prestress is balanced. This, in turn, implies that the close association
between cell stiffness and the cytoskeletal prestress is a common characteristic of all
prestressed structures. Quantitatively, however, this relationship does depend on the
architectural organization of the cytoskeletal lattice, including the manner in which
the prestress is balanced. The cable-and-strut model shows that in highly spread cells,
where virtually the entire prestress is balanced by the substrate, the contribution of
microtubules to deformability of the cytoskeleton is negligible. In less-spread cells,
however, where the contribution of internal compression members to balancing the
prestress increases at the expense of the substrate, deformability of microtubules im-
portantly contributes to the overall lattice deformability. Thus, which of the three
models would be appropriate to describe mechanical behavior of a cell would depend
upon the cell type and the extent of cell spreading.

Tensegrity and cellular dynamics
In previous sections it was shown how tensegrity-based models could account for
static elastic behavior of cells. However, cells are known to exhibit time- and rate-of-
deformation-dependent viscoelastic behavior (Petersen et al., 1982; Evans and Yeung,
1989; Sato et al., 1990; Wang and Ingber, 1994; Bausch et al., 1998; Fabry et al.,
2001). Because in their natural habitat cells are often exposed to dynamic loads (for
example, pulsatile blood ¬‚ow in vascular endothelial cells, periodic stretching of the
extracellular matrix in various pulmonary adherent cells), their viscoelastic properties
122 D. Stamenovic
´


(a)




(b)




Fig. 6-9. (a) For a given frequency of loading (ω), the storage (elastic) modulus (G ) increases with
increasing cytoskeletal contractile prestress (P) at all frequencies. (b) The loss (viscous) modulus
(G ) also increased with P at all frequencies. Cell contractility was modulated by histamine and iso-
proterenol. P was measured by traction cytometry and G and G by magnetic oscillatory cytometry.
Data are means ± SE. Adapted with permission from Stamenovi´ et al., 2004.
c



are important determinants of their mechanical behavior. As the tensegrity-based
models have provided a reasonably good description of elastic behavior of adherent
cells, it is of considerable interest to investigate whether these models can be extended
to describe viscoelastic cell behavior.
Recent oscillatory measurements on cultured airway smooth muscle cells indicate
that the cytoskeletal prestress may play an important role in determining cell dynamics.
It was found (see Fig. 6-9) that the cell dynamic modulus (G — ) is systematically
altered in response to modulations of cell contractility; at a given frequency, the
real and imaginary components of G — ’ the storage (elastic) modulus (G ) and loss
(viscous) modulus (G ), respectively “ increase with increasing contractile prestress
P(Stamenovi´ et al., 2002b; 2004). These prestress-dependences of G and G suggest
c
the possibility that cells may utilize similar mechanisms to resist dynamical loads as
they do in the case of static loads. Whereas it is clear how geometrical rearrangements
of cytoskeletal ¬laments may come into play in determining the dependence of G
on P, it is not that obvious how they could explain the dependence of G on P. A
possible explanation for the latter is as follows. In a purely elastic prestressed structure
Models of cytoskeletal mechanics based on tensegrity 123

that is subjected to a harmonic strain excitation, all three mechanisms are in phase
with the applied strain as long as the structural response is approximately linear and
inertial effects are negligible. Consequently, G ≡ 0. However, in a structure affected
by linear damping, the three mechanisms may not all be in phase with the applied
strain. As these mechanisms depend on P, phase lags associated with each of them
will also depend on P. Consequently, G = 0 and depends on P. The mathematical
description of this argument is as follows.
There have been several attempts to model cell viscoelastic behavior using the
cable-and-strut model. Ca˜ adas et al. (2002) and Sultan et al. (2004) used the six-
n
strut tensegrity model (Fig. 6-7a) with viscoelastic Voigt elements instead of elastic
cables and with rigid struts to study the creep and the oscillatory responses of the
cell, respectively. Their models predicted prestress-dependent viscoelastic properties
that are qualitatively consistent with experiments. Sultan et al. (2004) also attempted
to quantitatively match model predictions with experimental data. They showed that
with a suitable choice of model parameters one can provide a very good quantitative
correspondence to the observed dependences of G and G on P (Fig. 6-9). How-
ever, this could be accomplished only with a very high degree of inhomogeneity in
model parameters (variation of several orders of magnitudes), which is not physically
realistic.
The speci¬c issue of time- and rate-of-deformation-dependence in explaining the
viscoelastic behavior of cells is covered in Chapters 3, 4 and 5 of this book. However,
it will be addressed brie¬‚y here in the context of the tensegrity idea. A growing body
of evidence indicates that the oscillatory response of various cell types follows a
weak power-law dependence on frequency, ωk where 0 ¤ k ¤ 1, over several orders
of magnitude of ω (Goldmann and Ezzel, 1996; Fabry et al., 2001; Alcaraz et al.,
2003). In the limits when k = 0, rheological behavior is Hookean elastic solid-like,
and when k = 1 it is Newtonian viscous ¬‚uid-like. A power-law behavior implies the
absence of an internal time scale in the structure. Thus, it rules out the Voigt model,
the Maxwell model, the standard linear solid model, and other models with a discrete
number of time constants (see for example, Sato et al., 1990; Baush et al., 1998). The
power-law behavior observed in cells persists even after cell contractility is altered.
The only parameter that changes is the power-law exponent k; in contracted airway
smooth muscle cells k decreases, whereas in relaxed cells it increases relative to the
baseline (Fabry et al., 2001; Stamenovi´ et al., 2004). Based on these observations,
c
an empirical relationship between k and P has been established (Stamenovi´ et al., c
2004). It was found that k decreases approximately logarithmically with increasing P.
This result suggests that the cytoskeletal contractile stress regulates the transition
between solid-like and ¬‚uid-like cell behavior.
The observed relationship between k and P appears not to be an a priori predic-
tion of the tensegrity-based models. Rather, it depends on rheological properties of
individual structural members and is rooted in the dynamics (thermal ¬‚uctuations)
of molecules of the cytoskeleton. These dynamics can lead to a power-law behavior
of the entire network (Suki et al., 1994). It is feasible, however, that tensile force car-
ried by prestressed cytoskeletal ¬laments may in¬‚uence their molecular dynamics,
which in turn may explain why P affects the exponent k in the power-law behavior of
cells (Stamenovi´ et al., 2004). This has yet to be shown. Another possibility is that
c
124 D. Stamenovic
´

molecules of the cytoskeleton exhibit highly nonhomogeneous properties that would
lead to a wide distribution of time constants, and thereby to a power-law behavior, as
shown by Sultan et al. (2004).
In summary, the basic mechanisms of the tensegrity model can explain the depen-
dence of cell viscoelastic properties on the cytoskeletal prestress. These mechanisms
cannot completely explain the frequency response of cells, however, which conforms
to a power-law. This power-law behavior seems to be primarily determined by rheol-
ogy of individual cytoskeletal ¬laments and their own dynamics (thermal ¬‚uctuations,
and so forth), rather than by structural dynamics of the cytoskeleton.


Conclusion
This chapter has shown that the tensegrity model is a useful approach for studying
mechanics of living cells starting from ¬rst principles. This approach elucidates how
simple structural models naturally come to express many seemingly complex behav-
iors observed in cells. This does not preclude the numerous chemically and genetically
mediated mechanisms (such as, cytoskeletal remodeling, acto-myosin motor kinet-
ics, cross-linking) that are known to regulate cytoskeletal ¬lament assembly and force
generation. Rather, it elucidates a higher level of organization in which these events
function and may be regulated.
Taken together, results presented in this chapter can be summarized as follows.
First, the cytoskeletal prestress is a key determinant of cell deformability. This fea-
ture is consistent with all forms of cellular tensegrity models: the cortical membrane
model; the cable net model; and the cable-and-strut model. As a consequence, cell
stiffness increases with increasing prestress in nearly direct proportion. Second, de-
pending on the cell type and the extent of cell spreading, one may invoke accordingly
different types of tensegrity models in order to describe the effect of the prestress on
cellular mechanics. Clearly, various types of ad hoc models unrelated to tensegrity
may also provide very useful descriptions of cell mechanical behavior under certain
experimental conditions (compare Theret et al., 1988; Sato et al., 1990; Forgacs, 1995;
Satcher and Dewey, 1996; Bausch et al., 1998; Fabry et al., 2001). However, the stud-
ies described here show that the current formulation of the cellular tensegrity model,
although highly simpli¬ed, embodies many of the key behaviors of cells. Third, the
tensegrity model can explain some aspects of cell viscoelastic behavior, but not all.
The behavior appears to be primarily related to rheology and molecular dynamics
of individual cytoskeletal ¬laments. Nevertheless, the observed relationship between
viscoelastic properties of the cell and cytoskeletal prestress suggests that rheology of
individual ¬laments may be modulated by the prestress through the mechanisms of
tensegrity. This is a subject of future studies that will show whether the tensegrity
model is useful only in describing and understanding static elastic behavior of cells,
or whether it is also useful for describing and understanding cell dynamic viscoelastic
behavior.
A long-term goal is to use the tensegrity idea as a mathematical framework to help
understand and predict how mechanical and chemical signals interplay to regulate
cell function as well as gene expression. In addition, this model may reveal how
cytoskeletal structure, prestress, and the extracellular matrix come into play in the
Models of cytoskeletal mechanics based on tensegrity 125

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