<<

. 8
( 8)



stress contribution to the network momentum equation (Eq. 10.6) is integrated over
the range δ of network“membrane interaction. So, if we allow a generous average of
6 k B T interaction energy per actin monomer within range of the membrane, we have

= 3 — 106 θn n i n j dyn cm’2 ,
M
(10.29)
ij

where n is the unit normal vector to the membrane. If we would like the pseudopod
to extend a few µm within about a minute, we need the ¬‚ow velocity of cytoskeleton
to be of order 0.2 µm s’1 . By Eqs. 10.17 and 10.29, this means that δ ∼ 0.5 µm.
Fig. 10-5 shows the outline of the cell together with the network velocity ¬eld
sixty seconds into a two-dimensional cylindrical symmetry simulation starting from
a round cell. The ¬‚ow clearly changes direction right at the pseudopod surface with
the disjoining membrane“cytoskeleton force driving out the membrane while causing
a centripetal ¬‚ow of cytoskeleton into the cell.
The observant reader has probably noted that in this simulation, the frontal surface
of the pseudopod has a curiously uniform velocity. This is because the additional
220 M. Herant and M. Dembo




Fig. 10-5. Simulation Protrusion driven by membrane “ cytoskeleton repulsion. Length of the pseu-
dopod is ∼6 µm, pseudopod velocities are ∼0.1 µm s’1 .



velocity constraint of “no-shear” was introduced at the tip of the pseudopod. In
the absence of such a prescription, membrane“network repulsion can drive severe
rippling instabilities whereby the membrane grows folds that rapidly lengthen in a
way reminiscent of the Rayleigh-Taylor instability (heavy over light ¬‚uid). From
a biological point of view, the “no-shear” condition that we had to impose could
point to rigid transverse cross-linking of ¬laments at the leading edge that prevent
sliding. This instability also provides a potential mechanism for the formation of
microvilli.


Protrusion driven by cytoskeletal swelling
The mechanics of cytoskeletal swelling as a driver of protrusion is similar to that
of cytoskeletal“membrane repulsion. A region with increased cytoskeletal density
appears due to enhanced polymerization in a compartment close to the leading edge
of the pseudopod. Under repulsive self-interaction, the dense clump of cytoskeleton
swells, drawing in cytosol like an expanding sponge draws water. This is the volume
¬‚ow that accounts for the growth of the pseudopod. Finally, expansion into the cell
is balanced by viscous resistance of the underlying cytoplasm. This braces outward
expansion, which “ while it does not have to work against viscosity of an external
medium (assumed to be inviscid) “ does have to work against cortical tension as new
cellular area is created by the growth of the pseudopod.
Following the discussion of network swelling and in parallel with the simulation
of cytoskeletal“membrane repulsion, we assume a contribution of 6 k B T per actin
monomer to the cytoskeletal swelling stress density

= 3 — 106 θn dyn cm’2 .
n
(10.30)

Fig. 10-6 shows the outline of the cell together with the network velocity ¬eld
sixty seconds into the simulation, starting from a round cell. As in the simula-
tion of protrusion driven by cytoskeletal“membrane repulsion, there is an obvious
Active cellular protrusion: continuum theories and models 221




Fig. 10-6. Protrusion driven by cytoskeletal swelling. Length of the pseudopod is ∼ 6 µm, pseudopod
velocities are ∼ 0.1 µm s’1 .


retrograde ¬‚ow of cytoskeleton. However, one will note that the center of expan-
sion is slightly behind the leading edge, so that there also exists a small region of
forward cytoskeletal ¬‚ow at the front of the pseudopod. This is in contrast with
protrusion from cytoskeletal“membrane repulsion where the ¬‚ow of cytoskeleton
is retrograde all the way to the tip of the pseudopod and then changes to forward
motion at the membrane. (In reality, this occurs at the “gap” region depicted in
Fig. 10-2.)


Discussion
It is reassuring that choices of physico-biological parameters (such as viscosity or
interaction energies) that seem by and large reasonable can lead to a plausible mech-
anism for the protrusion of a pseudopod. The morphology and extension velocity
of the pseudopod is within range of the experimental data obtained for the neu-
trophil (Zhelev et al., 1996). In addition, retrograde ¬‚ow similar to that observed in
lamellipodial protrusion (for example, see Theriot and Mitchison, 1992) is evident.
There are, however, some dif¬culties with each model. In the case of the membrane“
cytoskeleton repulsion model, the range of interaction had to be set to 0.5 µm to
obtain suf¬cient elongation velocity. This is a large distance compared to a monomer,
but small compared to the persistence length of a ¬lament (for instance, see Kovar
and Pollard, 2004) which implies that the stress ¬eld would most likely have to be
stored in the large scale strain energy from deformation of the cytoskeleton (Herant
et al., 2006). In the case of the swelling model, it is dif¬cult to build up clumps of cy-
toskeleton of signi¬cant density and size (as are observed) without prompt smoothing
and dissipation by expansion. Although none of those caveats are model killers in the
sense that more complicated explanations can be invoked to rescue them, they hint at
more mechanical complexity than is presented in the rather simple approach followed
here.
A separate but nonetheless important issue is that of discrimination between models.
As Figs. 10-5 and 10-6 make clear, each model can produce a similar pseudopod.
222 M. Herant and M. Dembo

However, the cytoskeleton velocity ¬eld is different in a way that is intrinsic to the
mechanism of protrusion invoked in each case. In the swelling model, there exists a
stagnation point (or center of expansion) within the cell across which the cytoskeletal
¬‚ow goes from centripetal to centrifugal. The distance of the stagnation point from
the leading edge depends on the force resisting protrusion (for instance in the limit
of a hard wall, the stagnation point is at the membrane), but in certain conditions (see
Herant et al., 2003 for an example), it can lie far back from the leading edge. This is not
the case in the membrane“cytoskeleton repulsion model, where the stagnation point
lies in the “gap” region (Fig. 10-2) very near the membrane. Such distinction may
be a way to experimentally establish the distinction between swelling and repulsion
models of protrusion.
Finally, it would have been possible to compute models of alternative protrusion
theories such as those in Fig. 10.4 using the RIF formalism. We do not do so here
because these other theories are not currently favored.


Conclusions
For reasons of brevity, we have not included computational examples of protrusion
models that fail to produce cell-like behavior. The fact that these tend to make only
rare appearances in the literature can be misleading: in reality good models with
good parameters are needles in haystacks (for example, see Drury and Dembo,
2001). This is mostly due to the fact that the spaces of inputs (model speci¬ca-
tions, such as viscosity) and outputs (model behavior, such as cellular shape) that
need to be explored are both extremely large, as expected in complex biological
systems.
In general, our experience has been that in hindsight, it is not dif¬cult to discover
why it is that a particular model with a particular choice of parameters does not lead
to the behavior one was hoping for. On the other hand, we have also found that a priori
expectations about the kind of results a given model will produce are rarely precisely
matched by a numerical simulation. As a result, the process of constructing mechanical
models of cell behavior is one of iteration during which one re¬nes one™s experience
and intuition by running many numerical experiments. It is also our experience that
this process often leads to deeper insights about the fundamental mechanisms at play
in certain cellular events such as protrusions.
With this in mind, caution is in order when dealing with cartoon descriptions of
the mechanics of living cells. As compelling as a mechanical diagram in the form of
rods, ropes, pulleys, and motors working together may be, the actual implementation
of such models within a quantitative model may not lead to what one was expecting!
Thus, in a Popperian way, one of the principal bene¬ts of a rigorous framework for
cell mechanics is the ability to falsify invalid theories (Popper, 1968). This is why,
although such frameworks can be dif¬cult to develop and use, they are necessary to
reach a real understanding of the mechanics of living cells.
We thank Juliet Lee as well as the editors for comments on an early version of this
chapter. This work was supported by Whitaker biomedical engineering research grant
RG-02-0714 to MH and NIH grant RO1-GM 61806 to MD.
Active cellular protrusion: continuum theories and models 223

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11 Summary
Mohammad R.K. Mofrad and Roger D. Kamm




The primary objective of this book was to bring together various points of view
regarding cell mechanics, contrasting and comparing these diverse perspectives. This
¬nal short chapter summarizes the various models discussed in an attempt to identify
commonalities as well as any irreconcilable differences.
A wide range of computational and phenomenological models were described for
cytoskeletal mechanics, ranging from continuum models for cell deformation and
mechanical stress to actin-¬lament-based models for cell motility. A concise review
was also presented (Chapter 2) of numerous experimental techniques, which typically
aim to quantify cytoskeletal mechanics by exerting some sort of perturbation on the
cell and examining its static and dynamic responses. These experimental observations
along with computational approaches have given rise to several often contradictory
theories for describing the mechanics of living cells, modeling the cytoskeleton as a
simple mechanical elastic, viscoelastic, or poroviscoelastic continuum, a porous gel, a
soft glassy material, or a tensegrity (tension integrity) network incorporating discrete
structural elements that bear compression.
With such remarkable disparity among these models, largely due to the diversity of
scales and biomechanical issues of interest, it may appear to the uninitiated that various
authors are describing entirely different cells. Yet depending on the test conditions
or length scales of interest, identical cells may be viewed so differently as either a
continuum or as a discrete collection of structural elements.
Experimental data are accumulating, and promising methods have been proposed
to describe cell rheology. While there has been some convergence toward a range of
values for the cytoskeletal shear modulus, the range remains large, spanning several
orders of magnitude. This suggests either disparities in the measurement methods,
considerable variability between cells or between cell types, or differences in the
methods employed to interpret the data. A unique aspect of cellular mechanics is that
active as well as passive characteristics need to be considered.
A variety of different approaches have been described to simulate cell or cytoskele-
tal stiffness. Likely there is not a single “correct” model; rather, one model may prove
useful under certain circumstances while another model may be better suited in others.
In part, the model of choice will depend on the length scale of interest. Cells contain
a microarchitecture comprised of ¬laments ranging down to ∼10 nm in diameter
225
226 M.R.K. Mofrad and R.D. Kamm

with separation distances of ∼100 nm. When considering whole-cell deformations,
a continuum description may be appropriate; when the force probe is on the scale of
an AFM tip, then details of the ¬lament organization are almost certainly critical.
As a practical matter, it is important to determine what constitutive law best ¬ts
the observed structural behavior. While a linear elastic or even linear viscoelastic
material description is suf¬cient to mimic certain observations, other more complex
descriptions will almost certainly be needed to encompass a range of excitation fre-
quencies and large deformations. These are just now being identi¬ed. There seems
to be a growing consensus that the constitutive behavior of a cell corresponds to that
of a soft glassy material (see Chapter 3) even though the underlying basis for this
behavior is not yet clearly understood. Albeit lacking a fundamental understanding,
these measurements and the relative simplicity of the generalized form that they ex-
hibit provide at least two critical new insights. First is that the cell responds as though
the relaxation times are distributed according to a power law, suggesting many relax-
ation processes at low frequencies but progressively fewer as frequency is increased.
Second, cytoskeletal stiffness and friction or viscosity are interrelated, in that the
same underlying principles likely govern both. Both stiffness and friction appear to
be governed by a single parameter, the “effective temperature,” that re¬‚ects the extent
to which the material is solid-like or ¬‚uid-like. Bursac et al. (2005) speculate that
this might relate to a process in which the cytoskeleton is “trapped” in a collection of
energy wells but can occasionally “escape” utilizing, for example, either thermal or
chemical (such as, ATP-derived) energy. In this connection, the effective temperature
might be a measure of molecular agitation, re¬‚ecting the relative ability to escape.
As appealing as these ideas might be, however, they remain to be fully demonstrated,
and so remain intriguing speculation.
As Chapter 2 points out, while there appears to be some degree of convergence
regarding the values and frequency dependence of viscoelastic parameters for the
cytoskeleton, the results obtained remain somewhat dependent upon the method used
to probe the cell. In publications as recent as this past year, values for cytoskeletal
stiffness ranging from ∼20 Pa (Tseng et al., 2004) to 1.1 MPa (Marquez et al., 2005)
have appeared, and the bases for these discrepancies still need to be resolved. In
particular, as most (but not all) of the data on which the soft glassy material model
is based are obtained from one measurement method (magnetic twisting cytometry),
one still needs to exercise caution in making broad generalizations.
While some of the models appear quite disparate, there are some signi¬cant sim-
ilarities. The cellular solids and biopolymer (Chapter 8) theories differ in terms of
how the individual elements in the structure resist deformation, with the cellular solids
model considering these to be beams subject to bending, and the biopolymer theory
treating them as entropic chains that lose con¬gurational entropy as the material is
stretched. Recent studies (Gardel et al., 2004) are beginning to reconcile these dif-
ferences and, perhaps not surprisingly, are ¬nding that both descriptions might apply
depending upon the concentrations of actin and cross-linkers and the state of stress
in the material. Neither of these models, however, can be readily connected to the
observed behavior as a soft glassy material.
Another microstructural model is based on the concept of tensegrity (Chapter 6),
and is most closely related to the cellular solids model in that cytoskeletal structure
Summary 227

is de¬ned by an interconnected network of elastic elements. The key distinction here,
though, is that stiffness is conferred not by the stiffness of the individual elements,
but rather primarily by the baseline stresses they support. These stresses are imposed
either by cell adhesions to extracellular structures or by internal members such as
microtubules that are in compression. In either case, the elastic properties of the
elements take on secondary importance, provided they are suf¬ciently stiff to undergo
relatively small changes in length under normal stress.
In a sense, the continuum descriptions (Chapters 4, 5 and 10) are for the most
part independent of behavior at the microstructural level, and simply make use of
constitutive laws that can either be based on experiments or derived directly from
one of these microstructural models. Consequently, while the continuum models can
be useful in describing how deformations or stresses distribute throughout the cell,
they provide no information on the deformations at the microscale (that is, within
the individual elements of the matrix), and are entirely dependent on information
contained in the constitutive relation.
Although this one text could not possibly capture all the work being done on cell
mechanics in that it represents a broad spectrum of these activities, it should immedi-
ately become clear that one fruitful direction for future research is in the modeling of
dynamic processes “ cell migration, phagocytosis and division. In fact, with only a few
exceptions (notably the work described in Chapters 7, 9, and 10) the cell is treated as a
traditional engineering material, meaning one with properties that are time invariant.
Cells, on the other hand, are highly dynamic in that their cytoskeletal structures are
constantly changing in response to a variety of external stimuli including, especially,
external forces. Consequently, each time we probe a cell to measure its mechanical
properties, we may alter those same properties. One exception to this statement is the
use of the Brownian motions of intracellular structures to infer stiffness, but these
measurements are still being re¬ned; as currently implemented, they are subject to
some degree of uncertainty. Still, this represents an important direction for research,
and we are sure to see re¬nements and wider use of these nonintrusive methods in
the future.
While advances in cell mechanics are considerable, many open questions still
remain. Mechanotransduction, the active response of living cells to mechanical signals
remains an active area of investigation. It is well known that living cells respond to
mechanical stimulation in a variety of ways that affect nearly every aspect of their
function. Such responses can range from changes in cell morphology to activation of
signaling cascades to changes in cell phenotype. Mechanotransduction is an essential
function of the cell, controlling its growth, proliferation, protein synthesis, and gene
expression.
Despite the wide relevance and central importance of mechanically induced cellular
response, the mechanisms for sensation and transduction of mechanical stimuli into
biochemical signals are still largely unknown. What we know is that living cells
can sense mechanical stimuli. Forces applied to a cell or physical cues from the
extracellular environment can elicit a wide range of biochemical responses that affect
the cell™s phenotype in health and disease.
Various mechanisms have been proposed to explain this phenomenon. They in-
clude: changes in membrane ¬‚uidity that act to increase receptor mobility and
228 M.R.K. Mofrad and R.D. Kamm

lead to enhanced receptor clustering and signal initiation (Haidekker et al., 2000);
stretch-activated ion channels (Hamil and Martinac, 2001); mechanical disruption
of microtubules (Odde and co-workers, 1999); and forced deformations within
the nucleus (Maniotis et al., 1997). Constrained autocrine signaling, whereby the
strength of autocrine signaling is regulated by changes in the volume of extracellular
compartments into which the receptor ligands are shed, is yet another mechanism
(Tschumperlin et al., 2004). Changing this volume by mechanical deformation of the
tissues can increase the level of autocrine signaling.
Finally, others have proposed conformational changes in intracellular proteins
along the force-transmission pathway, connecting the extracellular matrix with the
cytoskeleton through focal adhesions, as the main mechanotransduction mechanism
(see Kamm and Kaazempur-Mofrad, 2004 for a review). In particular, the hypothe-
sis that links mechanotransduction phenomena to mechanically induced alterations
in the molecular conformation of proteins has been gaining increased support. For
example, certain proteins that reside in ˜closed™ conformation can be mechanically
triggered to reveal cryptic binding sites. Similarly, small conformational changes may
also change binding af¬nity or enzyme activity. For example, when protein binding
occurs through hydrophobic site interactions, a conformational change could modify
this function and potentially disrupt it totally. Force transmission from the extracellu-
lar matrix to the cell interior occurs through a chain of proteins, located in the focal
adhesion sites, that are comprised of an integrin“extracellular matrix protein bond
(primarily vitronectin and ¬bronectin), integrin-associated proteins on the intracellu-
lar side (paxillin, talin, vinculin, and others), and proteins linking the focal adhesion
complex to the cytoskeleton. Stresses transmitted through adhesion receptors and
distributed throughout the cell could cause conformational changes in individual
force-transmitting proteins, any of which would be a candidate for force transduction
into a biochemical signal. The process by which changes in protein conformation
give rise to protein clustering at a focal adhesion or initiate intracellular signaling,
however, remains largely unknown (Geiger et al., 2001).
External stresses imposed on the cell are transmitted through the cytoskeleton to
remote locations within the cell. To understand these stress distributions requires
knowledge of cytoskeletal rheology, as governed by the structural proteins, actin ¬l-
aments, microtubules, and intermediate ¬laments. For example, a simpli¬ed picture
can be painted of the cytoskeletal rheology that is limited to actin ¬laments and actin
cross-linking proteins living in a dynamic equilibrium. These cross-links constantly
form and unbind at rates that are largely in¬‚uenced by the forces borne by the individ-
ual molecules. Cytoskeletal rheology would then be determined at the molecular scale
by the mechanics and binding kinetics of the actin cross-linking proteins, as well as
by the actin matrix itself (Gardel et al., 2004). To understand the phenomena related
to mechanotransduction in living cells and their cytoskeletal rheology, the mechanics
and chemistry of single molecules that form the biological signaling pathways that
act in concert with the mechanics must be examined.
Another largely open question in the ¬eld of cytoskeletal mechanics is related to the
cell migration and motility that is essential in a variety of biological processes in health
(such as embryonic development, angiogenesis, and wound healing) or disease (as in
cancer metastasis). As discussed in Chapter 9 and 10, the process of cell motility or
Summary 229

migration consists of several steps involving multiple mechanobiological signals and
events starting with the leading edge protrusion, formation of new adhesion plaques
at the front edge, followed by contraction of the cell and the release of adhesions at
the rear (see Li et al., 2005 for a recent review). A host of mechanical and biochemical
factors, namely extracellular matrix cues, chemoattractant concentration gradients,
substrate rigidity, and other mechanical signals, in¬‚uence these processes. Many
unanswered questions remain in understanding the signaling molecules that play a
key role in cell migration, and how they are regulated both in time and 3D space. It
is largely unknown how a cell actively controls the traction force at a focal adhesion
or how this force varies with time during the cell migration.
To understand the mechanobiology of the cell requires a multiscale/multiphysics
view of how externally applied stresses or traction forces are transmitted through
focal adhesion receptors and distributed throughout the cell, leading subsequently
to conformational changes that occur in individual mechanosensing proteins that in
turn lead to increased enzymatic activity or altered binding af¬nities. This presents
both a challenge and an opportunity for further research into the intrinsically coupled
mechanobiological phenomena that eventually determine the macroscopic behavior
and function of the cell.
Because no one method has emerged as clearly superior in describing the mechanics
and biology of the cell across all cell types and physical conditions, this might re¬‚ect
the need for new approaches and ideas. We hope that this monograph has inspired new
researchers with fresh ideas directed toward that goal. Perhaps the biggest question
that still remains is whether it is at all possible to construct a single model that
is universally applicable and can be used to describe all types of cell mechanical
behavior.


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Index




Acetylated low-density lipoprotein, 58 in cell migration, 75, 108, 210
Acoustic microscopy, 42“43 and cell stiffness, 10, 225
Actin crosslinking in, 166, 228
cortical, 174 and cytochalisin-D, 57, 111
depolymerization (See depolymerization DLS measurement of, 40
of actin) in the ECM, 173
diffusion coef¬cient of, 192 in load transfer, 114
dynamics, 180“188 in muscle contraction, 108, 142“143,
elastic modulus of, 178 145“146
mechanics of, 177“180 myosin, interactions with, 117, 124
role of, 170“176 persistence length of, 38, 107, 178
swelling stress values, 215 polymerization (See polymerization of actin)
F-Actin power-law relationships, 58
about, 153, 154, 172 and prestress, 104
acoustic signals in, 42“43 structure of, 179
architecture, regulation of, 175 in tensegrity models, 106, 107, 117
and Arp 2/3, 186 Young™s modulus, 172
±-actinin
ATP hydrolysis rate, 182
in cytosol conversion, 206 about, 11
discovery of, 181 in focal adhesions, 14
persistence length of, 155, 167 Actin monomers
shear response in, 163 about, 171
stiffness measurements of, 161, 165 in cell protrusion, 210
G-Actin diffusion coef¬cient of, 171, 191
about, 153, 154 recycling of, 187, 191“193
assembly of, 182, 183“185 Actin networks
in cell protrusion, 210 cell membranes, dynamics near, 212
cellular concentration of, 192 compression in, 179
discovery of, 181 crosslinking in, 166, 171, 228
and pro¬lin, 185, 188 cytochalisin-D and, 57, 111, 181, 204
recycling of, 192, 193 differential modulus in, 166
Actin binding proteins (ABPs) elastic moduli in, 167, 178, 227
about, 11, 14 shear stress in, 179
in actin dynamics regulation, 183“188 stiffness in, 178
Actin cortex, 153 swelling stress and, 179, 215
Actin ¬laments volume equations, 211
about, 12, 13, 73, 107, 153, 154, 171, 175 Young™s modulus in, 179
in AFM, 25 Action at a distance effect, 106, 110“112,
assembly of, 185“187 (See also polymerization 114
of actin) Actoclampin model, 193“194
bending of, 161, 178 ADF/co¬lin, 183“185, 191


231
232 Index

Adherent cells Biphasic properties, measurement of, 92“94, 95,
cortical membrane model for, 115, 118 96
protrusion models for, 218“222 Boltzmann™s constant, 155, 210, 214
Adhesion de¬ned, 4 Bone
Af¬ne network model deformation measurement in, 28“29
about, 116, 164 mechanosensing in, 19
shear modulus of, 164 repair of, 28
uniform strain in, 166 stiffness in, 8
Af¬ne strain approximation, 117 stress, response to, 6
Af¬nity in phase transitions, 138 Bouchard™s theory of glasses, 63
AFM. See atomic force microscopy (AFM) Boundary conditions
Aggregate modulus in osteoarthritis, 95 in cell protrusion, 208
Airways, pulmonary, 8 in continuum models, 75, 76
Alginate capsules, acoustic signals in, 42“43 in force extension relationships, 157
Amoeba, 75, 204 Bowen™s theory of incompressible mixtures, 86
Anisotropy in cell behavior, 98 Boyle van™t Hoff relation and osmolality, 89
Apoptosis, 188 Brownian motion
Arc length, short ¬laments, 157 in cytoskeletal ¬laments, 160“162, 167, 227
Area expansion modulus de¬ned, 10 and mean square displacements, 161
Arp 2/3 complex measurement of, 37, 81, 154, 155
in actin ¬lament assembly, 185“187 Brownian ratchet model, 193“194, 209, 213
in cell locomotion, 189 Buckling. See microtubules, buckling in
in the ECM, 173
Arterial wall cells, 3 Cable-and-strut tensegrity model
Atherosclerosis, 171 about, 106, 118, 121
Atomic force microscopy (AFM) tension-compression synergy in, 108
and continuum models, 81 of viscoelasticity, 123
in deformation measurement, 25“26, 43, 72, Cadherins, binding af¬nity in, 9, 228
73 Calcium, release of, 113, 138
shear/loss moduli and, 58 Cancer cells
strain distributions, 74 melanoma, deformation measurement in, 20
Autocrine signaling, 228 power-law relationships, 58, 61
Capping protein, 187
Bead diameter, 116 Cardiac tissue
Bead displacements, measurement of, 112 contraction in, 9
Bead pulling, cell response to, 118 Cartilage
Bead rotation and prestress, 115. See also rotation behavior of, describing, 87“88
Bending deformation measurement in, 93, 96“98
3D, 156 examination of, 171
of actin ¬laments, 161, 178 mechanical properties, measurement of, 85,
energy in force extension relationships, 157 93
moduli, 13, 155 mechanosensing in, 19
stiffness (See stiffness, bending) momentum balance equations in, 86
temperature and, 154“156 stiffness in, 8
and wavelength, 161 in vivo state, characterization of, 94
Binding af¬nity, changes in, 6, 228, 229 Caspases, 188
Biopolymers. See also polymers Cauchy-Green tensor in viscoelasticity
about, 154“162 measurements, 88
modeling, 226 Cauchy stress tensors in momentum balance laws,
persistence lengths of, 155 86
stiffness measurements of, 161, 165 Caulerpa spp., 131
Biphasic elastic formulations and creep response, Cell division, 25
90 Cell indentation. See indentation;
Biphasic models microindentation
of cell mechanics, 85“88 Cell locomotion. See also migration of cells
continuum viscoelastic, 84 about, 152, 153
linear isotropic, equations for, 87 actin™s role in, 171, 193“195
time dependencies and, 88 Arp 2/3 complex in, 189
of viscoelasticity, 87“88, 98 leukocytes, 189, 195
Index 233

Cell mechanics. See also cells, dynamics network elasticity in, 163“165
about, 1, 84 power-law relationships, 58
biphasic models of, 85“88 secretion in, 139“140
changes, measurements of, 59 shape
measurements of, 52“54, 56“57 and behavior, 104
role of, 2 deformation of, 74, 183
Cell membranes. See also lipid bilayer membrane maintenance of, 2“3, 171
about, 170, 172 and prestress, 109, 118, 121, 124
actin network dynamics near, 212 and stress/strain, 98
in diffusion dynamics, 130“132 spreading of (See cell spreading)
disruption of, 130, 131 stress/ion concentration, characterization of,
¬‚uidity changes, response to, 228 89
force per unit area equation, 211 stress/strain behavior in, 109
force transmission in, 80 structures in, 10“15, 152
and magnetic ¬eld gradients, 34“35 surface indentation in deformation
mechanical tension applied to, 26 measurement, 22“26
modeling, 77, 80, 173, 227 suspended, modeling, 118
protrusion of, 189, 205“209 temperature ¬‚uctuations in, 8, 36, 123, 165
Cell pokers in deformation measurement, 22“23, water retention in, 132“134
118 Cell spreading
Cells. See also individual cell type by name deformity and, 119
about, 129, 170 and prestress, 113, 114, 121, 124
adherent, 115, 118, 218“222 and stiffness, 120
anatomy, structural, 7“9 Cellular tensegrity model
arterial wall, 3 about, 104, 106
behavior experimental data, 107
anisotropy in, 98 principles of, 104, 114, 129
describing, 84“85, 95 Chains, force extension in, 156“159
diffusion in, 129 Chondrocytes
regulation of, 104 biphasic properties, modeling, 96
stress/strain, 108“109 creep response in, 90
biphasic properties, measurement of, 95 examination of, 171
cancer, power-law relationships, 58, 61 mechanical properties, measurement of, 85, 93
crawling, actin dynamics in, 188“195 modeling, 77, 87, 90
dynamics (See also cell mechanics) volume changes, calculation of, 92
about, 135“138 Chondrons, 93
modeling, 227 Cilia, 11
tensegrity and, 121“124 Colchicines, 113, 119
energy potential, 135 Collagen
environment binding af¬nity in, 9, 228
interactions with, 152 in the ECM, 173
prediction of, 98 in the PCM, 93
sensing, 5“6 power-law relationships, 58
function, 124, 134“135 stiffness measurements of, 165
as gels, 132“135 Compression
incompressible, Poisson ratio in, 92 in actin networks, 179
loading, response to, 87“89, 98 in cytoskeletal ¬laments, 164, 227
mammalian, formin in, 186 in microplates, 28
mechanical properties in microtubules, 107, 112“113, 114, 118
measurement of, 50, 51“62, 84, 92, 115 and prestress, 104, 118
regulation of, 98, 105, 117, 124, 152 and tensed cables, 116
representation of, 206 Computation domains, geometry of, 77“78
mechanosensing in, 5“6, 14, 19, 28 Conductance, changes in, 6
membrane-wounded state in, 131 Cones, indentation depth in, 24
migration of (See cell locomotion; migration of Connectin, 141
cells) Connecting ¬laments in muscle contraction, 142,
modeling of, 71“72, 76“78, 81 146
multiphasic properties, measurement of, 89, 94, Constant phase model. See structural damping
96 equation
234 Index

Constitutive laws Cortical membrane, 205, 210. See also cell
in continuum models, 227 membranes
Darcy™s law in, 87 Cortical membrane model, 115“116, 117
hyperelastic, 98 Courant condition, 217“218
Lam´ coef¬cients in, 87
e Creep response
multiphasic about, 84, 85, 98 modeling, 87, 90, 123
in protrusion, 209, 217“218 in the PCM, 93
and stress-strain relationships, 72 and shear moduli, 35
and structural behavior, 226 in vascular endothelium, 172
Continuum mechanics in cell representation, Crk-associated substrate, 14
206 Cross-bridges in muscle contraction, 110, 145“146
Continuum viscoelastic models Crosslinking
about, 71“72, 81, 227 in actin networks, 166, 171, 228
anisotropy in, 98 and effective modulus, 180
biphasic/triphasic, 84 in polymers, 165, 180
limitations of, 81, 98 in proteins, 153, 163“165
principles of, 76, 78 shear modulus of, 164“165, 180
purpose of, 72“75 in smooth muscle, 110
stress in, 104 in tensegrity models, 106, 124
Contraction of muscles Crossover frequency. See frequency
about, 4, 7, 8, 140“147 Curvature, measurement of, 155, 156, 157
actin ¬laments in, 108, 142“143, 145“146 Cytochalisin-D
airways, pulmonary, 8 and the actin network, 57, 111, 181, 204
cardiac tissue, 9 and cell migration, 34
in cell migration, 108 shear/loss moduli and, 56“57
connecting ¬laments in, 142, 146 Cytoindentation in cell modulus measurement,
connectin in, 141 94“96
cross-bridges in, 110, 145“146 Cytometry techniques, 109. See also individual
in the cytoskeleton, 14 technique by name
epithelial cells, 108 Cytoplasm
¬broblasts, 14, 107, 113 about, 85, 170, 205
and force/stiffness, 66, 68 dynamics, 205
force transmission and, 66, 68 energy potential, 135
in HASM, 65, 109, 110 as gel, 132
immunoglobulins in, 141 tearing and strain, 114
inchworm mechanism of, 143“147 viscosity, measurement of, 20“21
in lamellipodia, 108 volume ¬‚ow, 206
microtubules and, 108, 112 Cytoskeleton
modeling, 81, 227 about, 62, 153, 170, 206
modulation of, 122 deformability, mechanisms of, 117
myo¬brils in, 144 density, 204“205, 209
myosin-based, 191 environment, interactions with, 172“174, 176
myosin in, 141 ¬laments
phase transitions in, 129, 146 about, 154, 167
power-law relationships, 123 assembly of, 124
power-law rheology, 123 behaviors, 159, 164
and prestress, 104, 112, 114 Brownian motion in, 160“162, 167, 227
prestress in, 104, 112, 114 compression in, 164
sarcomeres in, 7, 8, 144 dynamics of, 160“162, 167
and stiffness, 66, 68, 104, 111, 114 elastic properties of, 13
stiffness and, 66, 68, 104, 111, 114 imaging, 161
temperature and, 158“159 loading, response to, 156“159
thermal ¬‚uctuations in, 158“159 nonlinear responses in, 167
Convergence-elongation theory, 191 orientation, calculation of, 157
Cortex relaxation rates of, 161
about, 11 rotation in, 164
modeling, 78, 80, 227 stretching of, 164
shear moduli in, 35, 153 force transmission in, 9“10, 14, 67, 111, 116
Cortical layer, tensile force in, 116 mechanics, ¬eld of, 1“2
Index 235

mechanotransduction in, 227, 228 stress/strain displacement, linear models of,
modeling of, 71, 77, 80, 81, 114“121, 124 80
momentum equations for, 208 Young™s modulus of. See Young™s modulus of
particle movement in, 34 elasticity
prestress in, 107“108, 114, 117, 124 (See also Elastic moduli
prestress) in actin networks, 167, 178, 227
stiffness values, 226 in cell dynamics, 122, 166
structure of, 176, 183 and frequency, 180
velocity ¬eld in, 222 in MTC, 54“55, 57, 58
volume equations for, 219 power-law relationships, 59“60
Cytosol, 206, 207, 210. See also cytoplasm Elastic propulsion theory, 194, 195
Elastic ratchet model, 193“194
Darcy™s law in constitutive law, 87 Electroporation, 130
DBcAMP, 56“57, 61 Endothelial cells
Deformation measurement. See rheology actin cytoskeleton in, 107, 117
Deformations action at a distance effect in, 110“112
cell shape, 74, 183 deformation measurement in, 28
non-af¬ne, 164 examination of, 171, 172
Depolymerization of actin, 177, 181“183, 189, 192 focal adhesions in, 107
Dictyostelium, deformation measurement in, 20 ¬‚uid shear stress in, 177
Differential modulus in actin networks, 166 mechanical properties of, 172, 173
Diffusing wave spectroscopy (DWS), 40“41 microtubules in, 108, 119
Diffusion modeling, 78, 80“81, 90
in cell behavior, 129, 140 power-law relationships, 58
coef¬cients shear/loss moduli in, 58
actin monomers, 171, 191 stiffness in, 119, 120
extracting, 41“42, 171 stress response in, 19, 116, 120
in deformation measurement, 21 Endothelial monolayer, 171
force-driven, 213 Energy
and mean square displacements, 161 bending, in force extension relationships,
membrane pumps in, 130 157
paradigm, problems with, 130“132 elastic and microtubule disruption, 112
rate, changes in, 6 hyperelastic strain energy function, 98
Discrete models, stress in, 105, 109 potential in cells, 135
Disease temperature and, 226
mechanotransduction and, 228 wells in SGR, 64
PCM response to, 93 Entanglement length in semi-¬‚exible polymers,
stress response to, 6 162
DNA Entropy in cell response, 157, 163, 165, 226
force extension relationships in, 157 Environment
persistence length of, 155 and cell locomotion, 152
stiffness measurements of, 161 cytoskeletal interactions with, 172“174, 176
Donnan osmotic pressure relation de¬ned, 89 and phase transitions, 137, 139
Drug treatment prediction of in cells, 98
and actin polymerization, 110 sensing by cells, 5“6, 14, 19, 28
and cell migration, 34 Enzyme activity, changes in, 228, 229
shear/loss moduli and, 57 Epithelial cells
Dynamic light scattering (DLS), 39“40 about, 3
Dynamic moduli, calculation of, 95 and cellular insult, 131
contraction in, 108
Effective modulus and crosslinking, 180 membrane-wounded state in, 131
Elastic energy and microtubule disruption, 112 modeling, 79“81
Elasticity. See also viscoelasticity power-law relationships, 58, 61
biphasic elastic formulations and creep shear/loss moduli in, 25, 58
response, 90 structure of, 11
of cytoskeletal ¬laments, 13 Equilibrium moduli
Hookean, shear modulus of, 116, 123 calculation of, 95
network, in cells, 163“165 in endothelial cells, 171
solutions, response to, 162 in protrusion, 219
236 Index

Erythrocytes FLMP, power-law relationships, 61
about, 2 Fluctuation dissipation theorem in SGR, 65
deformation measurement in, 26, 40, 73, 74 Fluid ¬‚ow
microcirculation, 74 and cell shape, 98
modeling of, 71, 75, 78“79, 90, 118 deformation measurement, methods in,
movement of, 34 28“29
optical trapping of, 32 multiphasic/triphasic models, 88“89, 96
rupture strength in, 10 and stress, 67 (See also shear stress)
stiffness of, 11 velocity in constitutive law, 87
structure of, 11, 174 Fluid mosaic model, 10
Euler-Lagrange scheme, 217“218 Fluorescence correlation spectroscopy (FCS),
Extracellular matrix (ECM) 41“42
about, 9, 93, 173 Fluorescence microscopy, 154, 174
behavior of, describing, 87“88, 98 Focal adhesion complex (FAC)
chondrocyte properties in, 92, 96 about, 14, 173
interactions with, 84, 96“98, 124 in force transmission, 80, 114, 229
mechanotransduction in, 228 formation of, 112
prestress in, 104, 107, 116 mechanotransduction in, 227, 228
production of, 28 prestress in, 107
Focal adhesion kinase (FAK), 14
Falling sphere method, 20“21 Force balance in cell protrusion, 205
Ferri/ferromagnetic particles, deformation Force extension relationships
measurements in, 34, 35, 51“52, 112, about, 157, 159
115 bending energy in, 157
Fibrin, stiffness measurements of, 165 boundary conditions in, 157
Fibrinogen in endothelial cells, 172 in chains, 156“159
Fibroblasts in DNA, 157
about, 3, 181 Force transmission
actin dynamics in, 189, 190 in cell membranes, 80
action at a distance effect in, 111 in cell migration, 75, 205
contraction in, 14, 107, 113 cellular exposure to, 19, 228
force transmission in, 80 in chains, 156“159
gelsolin-null, 187 and contraction, 66, 68
microtubules in, 108 and crosslinking, 110, 111
migration in, 75, 191 to the cytoskeleton, 9“10, 14, 67, 111, 116
modeling, 79“81, 90, 227 focal adhesions in, 80, 114, 229
power-law relationships, 58 in ¬broblasts, 80, 107
shear/loss moduli in, 25, 35, 58 and focal adhesion sites, 80
viscosity measurement in, 28 in the glycocalyx, 3, 7
Fibronectin and hysteresivity, 66, 68
action at a distance effect, modeling, 110“112 modeling, 72, 80, 82, 227
binding af¬nity in, 9, 228 in the nucleus, 111
power-law relationships, 58 in particles, 30“31
viscosity measurement in, 28 small particles, 30“31
Fick™s law, 192 in solvent displacement, 132, 134
Ficoll, 192 and strain, 166
Filamin-A, role of, 175“176 in tensegrity models, 106, 113
Filaments, cytoskeletal. See cytoskeleton, time dependencies and, 67
¬laments vs. stiffness, 120
Filamin, 11 Formins in actin ¬lament assembly, 185“186
Filopodia, 4, 189, 191 Frequency
Fimbrin, 11 and elastic moduli, 180
Finite element methods (FEM) in indentation studies, 95
in cell behavior modeling, 90, 92, 93, 98 in MTC measurements, 53, 56
in microcirculation studies, 74 in polymer solutions, 163, 166
in multiphase models, 96“98 in power-law relationships, 61, 124
in protrusion modeling, 217“218 and shear/loss moduli, 54“55, 57, 58, 163
in stress/strain evaluation, 80“81 and stiffness, 163
in viscoelasticity measurements, 87, 88 and viscoelasticity, 36, 123“124
Index 237

Galerkin ¬nite element scheme, 217“218 Integral membrane proteins, binding af¬nity in, 10
Gel dynamics Integrins
motion and, 138“147 in actin binding, 171, 176
principles of, 129 action at a distance effect, modeling, 110“112,
shear modulus of, 180 114
Gelsolin, 143, 187“188 binding af¬nity in, 9, 228
Gene expression in focal adhesions, 14
and cellular insult, 131 modeling, 79, 115, 118
changes in, 6, 152 and prestress, 128
mechanotransduction and, 227 Intermediate ¬laments
regulation of, 124 about, 13
Glass microneedles in deformation measurement, and prestress, 104, 109, 114
22 role of, 121, 171
Glycocalyx, 3, 7 and stiffness, 114
Goblet cells, 139 stiffness measurements of, 165
GP1b±, 176 in tensegrity models, 106, 113“114
GTPase, 176 Intervetebral discs, osmotic loading in, 98
Intracellular tomography technique
Hair cells, mechanosensing in, 5 action at a distance effect in, 112
Heart disease, 171 shear disturbance, modeling, 111
Hertz relation, 24 Ions
Histamine channels, 228 (See also calcium, release of;
power-law relationships, 61 potassium pump activity; sodium pump
and prestress, 110, 111, 119, 122 activity)
shear/loss moduli of, 57, 58 concentration, characterization of, 89
Hookean elasticity, shear modulus of, 116, 123 multiphasic/triphasic models, 88“89
Hooke™s Law, 22 partitioning in cell function, 134“135
Human airway smooth muscle (HASM) Isoproterenol and cell stiffness, 111, 122
action at a distance effect in, 112 Isotropy
cell dynamics in, 115, 122“123 in modeling, 71, 88, 227
contraction in, 65, 109, 110 in polymer-gel phase transitions, 138, 147
malleability of, 65
MTC measurements of, 54“57 Kelvin model, 90
power-law relationships, 60, 61 Keratocyte lamellipodia, 189, 191, 192
stress/strain behavior in, 109
work of traction in, 110, 111 Lam´ coef¬cients in constitutive law,
e
Hydrostatic pressure protrusion, 215“216 Lamellipodia
Hyperelastic strain energy function and about, 189
anisotropy, 98 actin depolymerization in, 192
Hypertonic permeability model of protrusion, 215, contraction in, 108
216 modeling, 191, 227
Hysteresivity protrusion of, 221
and force transmission, 66, 68 Laminin, binding af¬nity in, 9, 228
in power-law relationships, 60, 61 Langevin equation, 160
time dependencies and, 67 Latrunculin A, 59
Hysteretic damping law. See structural damping Leukocytes
equation about, 2“3
deformation measurements in, 43
Immunoglobulins, 9 locomotion of, 189, 195
Inchworm mechanism of muscle contraction, modeling, 78“79, 227
143“147 stiffness in, 11
Indentation. See also microindentation structure of, 11
in cell modulus measurement, 94“96 Linear damping in cell dynamics, 123
cones, depth of, 24 Linear elastic models of stress/strain
frequency in, 95 displacement, 80
spheres, depth in, 24 Linear isotropic biphasic models, equations for, 87
and stiffness, 24 Linear momentum conservation in continuum
surface, in deformation measurement, 22“26, 72 models, 75, 205
Inner ear cells, mechanosensing in, 5 Linear solid model, 123
238 Index

Lipid bilayer membrane, 9“10, 89. See also cell Micropipettes in cell modeling, 90, 110“112,
membranes 131
Lipid vesicles, rupture strength in, 10 Microplates, shearing/compression method in
Listeria monocytogenes, 193, 212 deformation measurement, 28
Loads Microscopy. See individual technique by name
and cell stiffness, 10, 108“109 Microtubules
in discrete models, 109 about, 12, 13, 107, 154
¬lament response to, 156“159, 165 in AFM, 25
long-distance transfer of, 113, 114 buckling in, 107, 108, 112“114, 118
in tensegrity models, 106, 113 disruption, cellular response to, 228
work of traction transfer and, 113 in endothelial cells, 108
Loss moduli and G-actin, 192, 193
in cell dynamics, 122 in load transfer, 113, 114
in MTC, 54“55, 56 modeling, 155, 167, 227
power-law relationships, 58, 59“60 persistence length of, 107, 155
and prestress, 109, 113, 119, 121
Macrophages, 58, 61 role of, 171
Magnetic methods in deformation measurement, in tensegrity models, 106
32“36 Migration of cells. See also cell locomotion
Magnetic tweezers, 34 about, 4, 7
Magnetic twisting cytometry (MTC) actin ¬laments in, 75, 108, 210
about, 51“52 anatomical structures in, 11
in the cortical membrane model, 115, 116 contraction in, 108
in stiffness measurement, 120 and drug treatment, 34
Magnetocytometry, 72, 79, 111 force transmission in, 75, 205
Mass conservation in protrusion, 206 modeling, 75, 81, 228
Mast cells, 139 Mitosis, measurement of, 25
Maxwell model, 123 Mixture momentum equation in triphasic models
Maxwell viscoelastic ¬‚uids, modeling, 78“79 de¬ned, 89
Mean square displacements, 65, 161 Molecular ratchet model, 194“195, 213
Mechanosensing in cells, 5“6, 14, 19, 28. See also Momentum balance laws, solid/¬‚uid phases,
environment 86
Mechanosensitive channel of large conductance Momentum conservation in cell protrusion, 205,
(MscL) de¬ned, 5 207“208, 217“218
Mechanotransduction, 227 Momentum exchange vector de¬ned, 86
Melanoma cells, deformation measurement in, Motion
20 3D, 156
Membrane pumps in diffusion dynamics, 130 and gel dynamics, 138“147
Mesh boundary in protrusion modeling, 217“218 longitudinal dynamics, calculation of, 162
Mesh size in semi-¬‚exible polymers, 162 planar, 155“156
Mesoscopic model, 193 subdiffusive, 161
MG63 osteoblast cells, modeling of, 95 transverse equations of, 160, 161
Mica surfaces, solvent displacement on, 132, 134 MTC. See magnetic twisting cytometry (MTC)
Microbeads Multiphasic constitutive laws. See constitutive
force transmission in, 72, 80 laws, multiphasic
stress/strain displacements in, 80 Multiphasic/triphasic models
Microcirculation and cell shape, 74 about, 88“89
Microelectrodes, 130 applications of, 96
Microindentation cell environment prediction, constitutive
cell multiphasic properties modeling, 94“96 models in, 98
in deformation measurement, 72, 73 continuum viscoelastic, 84
Micropipette aspiration solids, 88“89
about, 90 triphasic continuum mixture models, 88, 89
of chondrocytes, 77, 87, 93 Muscle contraction. See contraction of muscles
and continuum models, 73, 76“78 Muscle ¬bers, deformation measurement in, 20.
in deformation measurement, 26, 43, 71, 72, 73 See also human airway smooth muscle
in erythrocyte modeling, 118 (HASM); myocytes, cardiac; smooth muscle
in viscoelasticity modeling, 78“79 cells; striated muscle
Index 239

Mutual compliance de¬ned, 38 and intermediate ¬bers, 113
Myocytes, cardiac modeling, 77, 112, 227
about, 3 prestress in, 117
contraction in, 7, 9 stiffness in, 28
membrane-wounded state in, 131
patch removal from, 131 One-particle method, 37“38
shear/loss moduli in, 25, 58 Optical bead pulling, 75
Myo¬brils in muscle contraction, 144 Optical microscopy, 141, 174
Myosin Optical stretcher, 42
actin ¬laments, interactions with, 117, 124 Optical traps in deformation measurement,
in cell protrusion, 204, 209, 215, 217 30“31
inhibitors and G-actin, 192, 193 Optical tweezers, 32, 74
in muscle contraction, 141 Oscillatory responses, modeling, 123
and reptation, 143 Osmolality
Myosin light chain kinase cell response to, 98
binding dynamics of, 66 modeling, 88, 89
cross-bridge kinetics in, 110 Osmometers, 89
phosphorylation of, 113 Osteoarthritis, PCM measurements in, 93
Osteoblast cells, modeling of, 95
Nematocyst vesicles, 140 Osteocytes, 28
Network elasticity in cells, 163“165
Network-membrane interaction theories of Particles
protrusion, 209“213 ferri/ferromagnetic, deformation measurements
Network-network interaction theories of in, 34, 35, 51“52, 112, 115
protrusion, 209, 213“215 force transmission, 30“31
Network phase, momentum equations for, methods, 37“39
207“208 movement in the cytoskeleton, 34
Neurites, extension of, 108 sedimentation of in rheology, 20“21
Neuronal cells superparamagnetic, deformation measurements
about, 3 in, 34
action at a distance effect in, 111 twisting of by magnetic forces, 35“36
membrane disruption in, 131 Passive microrheology, 36
Neutrophils Patch-clamp method, 130
about, 3 Paxilin, 14, 228
gelsolin in, 187 PCM. See pericellular matrix (PCM)
microcirculation in, 74 Pericellular matrix (PCM)
modeling of, 71, 73, 78“79, 90 biphasic properties of, 92“94
power-law relationships, 58, 61 role of, 93
protrusion in, 218 in stress/strain patterns, 96
stiffness in, 10 viscoelastic response, modeling, 93
stress response in, 19 Persistence lengths
Newtonian viscosity of actin, 38, 107, 178
modeling, 78“79, 123 and bending stiffness, 13, 154, 157, 158
in MTC measurements, 54, 56, 57 of biopolymers, 155
Nocodazole and cell migration, 34 and ¬lament behavior, 159, 167
Noise, 53, 63 of ¬lamentous proteins, 156
Noise temperature of microtubules, 107, 155
about, 50 in semi-¬‚exible polymers, 162
in power-law relationships, 61, 68 Phagocytosis, 34
in SGR theory, 64, 67 Phase transitions
time dependencies and, 67 about, 129, 140
Non-af¬ne deformations, 164 af¬nity in, 138
Nucleic acids, 132 in cell dynamics, 135“138
Nucleus environment and, 137, 139
about, 11 isotropy in, 138, 147
in cytosol conversion, 206 motion and, 138“147
disruption, cellular response to, 228 in muscle contraction, 146
force transmission in, 111 in polymers, 137, 138
240 Index

Phospholipids, binding af¬nity in, 10, 228 myosin light chain kinase, 58, 61
Phosphorylation of myosin light chain kinase, 113 neutrophils, 58, 61
Plasma membrane. See Cell membranes noise temperature in, 61, 68
Platelets, gelsolin in, 187 relaxation rates, 226
Poisson™s ratio RGD peptide, 58
in chondrocytes, 92 shear/loss moduli, 58, 59“60
in constitutive law, 87 in smooth muscle cells, 123
in incompressible cells, 92 stiffness, normalized, in, 60, 61
in osteoarthritis, 94, 96 structural damping equation in, 61
Polyacrylamide gel substrate, prestress urokinase, 58
transmission in, 107, 110 vitronectin, 58
Polymerization of actin Pressure in momentum conservation, 207
about, 175, 177, 206 Prestress
and cell rigidity, 180, 204“205 about, 104
in cytoskeleton production, 207 actin ¬laments and, 104
discovery of, 181“183 balancing of, 110, 111, 113, 116, 118,
by drug treatment, 110 121
in ¬broblasts, 107, 189 and bead rotation, 115
in protrusion, 193“195, 209“217 cell dynamics and, 122“123
regulation of, 187 and cell shape, 109, 118, 121, 124
stiffness and, 180, 204“205 cell spreading and, 113, 114, 121, 124
Polymerization zone, 211 colchicine and, 119
Polymers. See also Biopolymers compression and, 104, 118
crosslinking in, 165, 180 contractile, 104, 112, 113
deformation measurements in, 40, 51“62 differential modulus in, 166
hydrogel dynamics, 137 in the ECM, 104, 107, 116
modeling, 155“156, 227 in focal adhesions, 107
nonlinear responses in, 167 histamine and, 110, 111, 119, 122
phase transitions in, 129, 137, 138 integrins and, 107
semi-¬‚exible and intermediate ¬laments, 104, 109, 114
entanglement length in, 162 measurement of, 109, 111, 113, 118
solutions of, 163, 164, 167 microtubules and, 109, 113, 119, 121
solutions, frequency in, 163, 166 and model response, 118
stiffness, mechanical in, 154, 156 in the nucleus, 117
Polysaccharides, 132 and stiffness, 9, 105, 109“111, 118, 124
Porous solid model of actin ¬lament structure, structures, 105
179 transmission in polyacrylamide gel substrates,
Post-buckling equilibrium theory of Euler, 113 107, 110
Potassium pump activity, 134. See also ions and viscoelasticity, 123
Power-law rheology Probes, 34, 37
acetylated low-density lipoprotein, 58 Pro¬lin, 185, 188
actin ¬laments, 58 Proteins
cancer cells, 58, 61 about, 132
collagen, 58 crosslinking in, 153
in continuum modeling, 81 ¬lamentous, 156
contraction, 123 interactions, dynamics of, 50
data normalization method, 60“62 mechanotransduction in, 227, 228
DBcAMP, 61 modeling, 71, 227
elastic moduli, 59“60 Proteoglycans, binding af¬nity in, 9, 228
endothelial cells, 58 Protopod dynamics, measurement of, 75
epithelial cells, 58, 61 Protrusion
¬broblasts, 58, 107 about, 4, 204“205, 218“222
¬bronectin, 58 actin-based, 193“195, 209“217
FLMP, 61 in amoeba, 204
frequency, 123“124 boundary conditions in, 208
HASMs, 61, 60 cell, viscosity in, 212, 219
histamine in, 61 of cell membranes, 189, 205“209
hysteresivity in, 60, 61 constitutive equations in, 209
macrophages, 58, 61 constitutive laws in, 209, 217“218
Index 241

cytoskeletal theories of, 209“217 optical tweezers in, 74
hydrostatic pressure, 215“216 in polymers, 40, 51“62
hypertonic permeability model of, 215“216 probe motion in, 34
mass conservation in, 206 sedimentation of particles in, 20“21
momentum conservation in, 205, 207“208, spectrin in, 32, 73
217“218 in superparamagnetic particles, 34
myosin in, 141, 204, 209, 215, 217 whole cell aggregates in, 20
repulsion-driven, 219“220, 222 RIF. See reactive interpenetrative ¬‚ows (RIF)
shearing motor model of, 215, 217 Rotation
swelling-driven, 220“221 in cytoskeletal ¬laments, 164, 227
swelling model of, 220“221 and prestress, 115
Pseudopodia, 189, 205, 216, 218“222 and torque, 36, 115
Round con¬guration de¬ned, 120
Quartz and water absorption, 132 Rupture strength in cells, 10

Radial strain distributions via AFM, 74 Sarcomeres
Ratchet model, 193 about, 141
Rate of deformation and viscoelasticity, 123“124 in contraction, 7, 8, 144
Reactive interpenetrative ¬‚ows (RIF) experimental data, 144
about, 204 Scanning probe microscopy, 94“96
numerical implementation of, 217“218 Secretion, 139“140
principles of, 129, 205“209 Sedimentation of particles in deformation
Red blood cells. See erythrocytes measurement, 20“21
Relaxation rates Selectins, binding af¬nity in, 9, 228
of cytoskeletal ¬laments, 161 Semi-¬‚exible chains. See biopolymers; polymers
power-law relationships, 226 SGR. See soft glassy rheology (SGR)
Release de¬ned, 4 Shear disturbance, modeling, 111
Reptation, 142, 143, 144 Shearing motor model of protrusion, 215, 217
RGD peptide, power-law relationships, 58 Shear moduli
Rheology. See also passive microrheology; actin cortex, 153
power-law rheology; soft glassy rheology cable-and-strut model, 118
(SGR) creep response and, 35
about, 18, 19“22, 74 in crosslinking, 164“165
atomic force microscopy in, 25“26, 43, 72, cytoskeletal, 225
73 de¬ned, 11
in bone, 28“29 ¬broblasts, 80, 107
in cartilage, 93, 96“98 gels, 180
cell pokers in, 22“23, 118 Hookean elasticity, 116
Dictyostelium, 20 in MTC, 54“55, 57, 58
diffusion in, 21 in the one-particle method, 37“38
in endothelial cells, 28 power-law relationships, 58“60
in erythrocytes, 26, 40, 73, 74 in semi-¬‚exible polymers, 163, 164
in ferri/ferromagnetic particles, 34, 35, 51“52, tensed cable model, 117
112, 115 Shear stress. See also stress
in ¬‚uid ¬‚ow, 28“29 about, 116, 165
glass microneedles in, 22 in actin networks, 179
in leukocytes, 43 ¬‚uid, 172, 176
magnetic methods in, 32“36 nonlinear, 167
magnetocytometry in, 72, 79, 111 Shigella spp., 193“194
in melanoma, 20 Signals, collecting, 41“42
in melanoma cells, 20 Silicon rubber substrate, prestress transmission in,
microindentation in, 73 107
micropipette aspiration, 26, 43, 71, 73 Six-strut tensegrity model, 115, 119
microplates, shearing/compression method, 28 Skeletal muscle cells, contraction in, 7
multiphasic models of, 96 Smooth muscle cells
in muscle ¬bers, 20 (See also human airway about, 3
smooth muscle (HASM); smooth muscle actin polymerization in, 110
cells; striated muscle) contraction in, 7
optical traps in, 30“31 cross-bridges in, 110
242 Index

Smooth muscle cells (Contd.) frequency and, 163
HASM (See human airway smooth muscle indentation and, 24
(HASM)) and intermediate ¬laments, 114
membrane disruption in, 131 isoproterenol and, 111
membrane-wounded state in, 131 measurement of, 161
microtubules in, 118 mechanical in polymers, 154, 156
MTC measurement of, 51“52, 56“57 and microtubule disruption, 119
power-law relationships in, 123 modeling, 77, 227
shear/loss moduli in, 25, 58 in neutrophils, 10
time dependencies in, 67 normalized, in power-law relationships, 60,
Sodium pump activity, 134. See also ions 61
Softening in tensegrity architecture, 109 and prestress, 105, 109“111, 119, 124
Soft glassy materials (SGMs), 62 shear, 116
Soft glassy rheology (SGR) simulation of, 225
biological insights from, 65, 226 static model of, 64
energy wells in, 64 structural, 120
¬‚uctuation dissipation theorem in, 65 and temperature, 177
principles of, 51, 62“65, 129 tensed cable model, 117
structural damping equation in, 64 tensile, 107, 116
Soft tissue behavior, modeling, 98 and tension, 158
Solids time dependencies and, 67
modeling, 155, 226, 227 and viscosity, 226
multiphasic/triphasic models, 88“89 vs. applied stress, 120, 167, 227
stress/strain behavior in, 109 Storage modulus. See elastic moduli
viscoelastic interactions of, 87, 90 Strain. See also stress
Sollich™s Theory of SGMs, 63 in actin networks, 179
Solutions, 162, 163. See also biopolymers; AFM indentation distributions, 74
polymers and bead displacement, 80
Solvent-network drag, 207 in continuum models, 104
Solvent phase, momentum equations for, cytoplasmic tearing and, 114
207“208 evaluation of, 74, 76“78, 80“81
Solvents multiphasic models of, 96
cell response to, 139 network response to, 164, 166
displacement of, 132, 134 patterns
in mass conservation, 207 identi¬cation of, 81
momentum conservation in, 205, 207 prediction of, 81
Speckle microscopy, 192 Stress. See also strain
Spectrin in actin networks, 179, 215
and cell stiffness, 11 applied, vs. stiffness, 120, 167, 227
in deformation measurement, 32, 72, 73 and bead displacement, 80
lattice, modeling, 118 cell response to, 19, 67, 104, 116, 156“159
Spheres, 24, 74 and constitutive law, 72
Spread con¬guration de¬ned, 120 differential modulus in, 167
Stereocilia, 5 ¬bers about, 11
Stiffness ¬‚uids, shear, 75
and actin polymerization, 180, 204“205 hardening, prediction of, 120
bending multiphasic models of, 96
in actin networks, 178 network response to, 164, 166
in the cortical layer, 116 patterns
in microtubules, 13, 154 evaluation of, 72, 74, 76“78, 80“81
and persistence length, 13, 154, 157, 158 identi¬cation of, 81, 229
relations of, 13, 155“156 prediction of, 81
Young™s modulus, 13, 155“156 time dependence of, 96
in bone, 8 response in disease, 6
cable-and-strut model, 118 restoring, mechanism of, 105
and cell load, 108“109 and stiffness, 114
and cell spreading, 120 and strain ¬eld, 29, 166
contraction and, 66, 68, 104, 111, 114 swelling, 164, 166, 213, 215
and crosslinking, 110, 166 and tension, 165
Index 243

Stress ¬bers Thermal concepts. See temperature
about, 174, 181 Thick ¬laments in muscle contraction, 142,
generation of, 186 145“146
Stress-supported structures, 105 Thin ¬laments in muscle contraction, 142, 144,
Stretching of cytoskeletal ¬laments, 164 146
Striated muscle, contraction in, 65 Thrombin in microtubule buckling, 108
Structural damping equation B-Thymosin, 188
in MTC, 55“56 Time dependencies
in power-law relationships, 61 and biphasic models, 88
and soft glassy rheology, 64 in cell behavior, 82
Substrates and prestress, 113, 118 ¬broblasts, 80, 107
Superparamagnetic particles, deformation modeling, 67, 88, 227
measurements in, 34 smooth muscle, 67
Surface indentation in deformation measurement, and stress patterns, 96
22“26 viscoelasticity, 123“124
Suspended cells, modeling, 118 Tip link, 5
Swelling Titin in muscle contraction, 141
model of protrusion, 220“221 Torque
stress and network dynamics, 179, 215 (See also measurements of, 52“54
stress, swelling) and rotation, 36, 115
Swinging cross-bridge mechanism model, 141 Traction, work of. See work of traction
Traction cytometry technique in prestress
Talin measurements, 109, 111, 113, 118
in actin binding, 171 Traction microscopy and elastic energy, 112
in the ECM, 173 Treadmilling, 182, 204
in FACs, 14 Triphasic models
Tangential strain distributions via AFM, 74 about, 88“89
Tangent vectors, 155, 156 applications of, 96
Temperature cell environment prediction, constitutive
and contraction, 158“159 models in, 98
and energy, 226 continuum mixture models, 88, 89
¬lament bending and, 154“156 continuum viscoelastic, 84
¬‚uctuations solids, 88“89
in cells, 8, 36, 123, 165 Tropomyosin, 191
imaging, 161 Trypsin and FACs, 107
and motion, measurement of, 18, 81 Two-particle methods, 38“39
noise (See noise temperature)
and SGMs, 63 Upper convected Maxwell model, 88
shear/loss moduli and, 54“55 Urokinase power-law relationships, 58
and stiffness, 177
viscosity and, 21 Vector tangents, 155, 156
and volume in actin networks, 211 Vertical displacement, calculation of, 120
Tensed cable nets model, 116 Vertical strain distributions, 74, 75
Tensegrity Vesicles, 10, 139, 147
and cellular dynamics, 121“124 Vimentin in stress mapping, 29, 113
de¬ned, 105“106, 226 Vinculin, 228
Tensegrity architecture, 104, 105, 109 Viscoelasticity
Tensegrity methods. See also cellular tensegrity biphasic models of, 87“88, 98
model cable-and-strut models of, 123
about, 85, 98, 104, 124, 125 Cauchy-Green tensor, 88
mathematical models of, 114“121 and creep response, 90, 93
Tensile stiffness, 107, 116 ¬nite element methods, 87, 88
Tension and frequency, 36, 123“124
cortical, 205, 216, 218 measurement of, 18, 28, 33, 36, 78“79, 90, 205
and shear stress, 165, 166 mechanical basis of, 85
short ¬laments, 159 modeling, 78“79, 124, 227
Tension-compression synergy in the prestress and, 123
cable-and-strut tensegrity model, 108 rate of deformation and, 123“124
Tethered ratchet model, 194 time dependencies of, 123“124
244 Index

Viscosity Wavelength
in cell protrusion, 212, 219 drag coef¬cient and, 160
and cytoskeletal density, 209 and thermal properties, 158,
of cytosol, 207 161
measurement of, 20“21, 26, 28 Whole cell aggregates in deformation
modeling, 77, 227 measurement, 20
Newtonian (See Newtonian viscosity) Work of traction
and stiffness, 226 in HASMs, 110, 111
temperature and, 21 transfer of, 113, 229
Viscous modulus. See loss moduli Worm-like chain model, 155“156
Vitronectin
binding af¬nity in, 9, 228 Yeast and Arp 2/3, 186
power-law relationships, 58 Young™s modulus
Voigt model, 123 actin ¬laments, 172
in actin networks, 179
WASp/Scar family proteins, 186, 189 and bending stiffness, 13, 155“156
Water calculation of, 90
bonding in, 134 in constitutive law, 87
diffusion coef¬cient of, 171 in elastic body cell model, 26
in polymer-gel phase transitions, 138 in osteoarthritis, 93
retention in cells, 132“134 in the PCM, 93

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