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CYTOSKELETAL MECHANICS

This book presents a full spectrum of views on current approaches to modeling
cell mechanics. The authors of this book come from the biophysics, bioengi-
neering, and physical chemistry communities and each joins the discussion
with a unique perspective on biological systems. Consequently, the approaches
range from ¬nite element methods commonly used in continuum mechanics
to models of the cytoskeleton as a cross-linked polymer network to models of
glassy materials and gels. Studies re¬‚ect both the static, instantaneous nature
of the structure, as well as its dynamic nature due to polymerization and the full
array of biological processes. While it is unlikely that a single unifying approach
will evolve from this diversity, it is our hope that a better appreciation of the
various perspectives will lead to a highly coordinated approach to exploring the
essential problems and better discussions among investigators with differing
views.

Mohammad R. K. Mofrad is Assistant Professor of Bioengineering at the Uni-
versity of California, Berkeley, where he is also director of Berkeley Biome-
chanics Research Laboratory. After receiving his PhD from the University
of Toronto he was a post-doctoral Fellow at Harvard Medical School and a
principal research scientist at the Massachusetts Institute of Technology.

Roger D. Kamm is the Germeshausen Professor of Mechanical and Biological
Engineering in the Department of Mechanical Engineering and the Biological
Engineering Division at the Massachusetts Institute of Technology.
Cytoskeletal Mechanics
MODELS AND MEASUREMENTS


Edited by
MOHAMMAD R. K. MOFRAD
University of California, Berkeley

ROGER D. KAMM
Massachusetts Institute of Technology
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521846370

© Cambridge University Press 2006


This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.

First published in print format 2006

978-0-511-24934-1 eBook (EBL)
isbn-13
isbn-10 0-511-24934-9 eBook (EBL)

978-0-521-84637-0 hardback
isbn-13
0-521-84637-4 hardback
isbn-10




Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents




List of Contributors page vii
Preface ix

1 Introduction, with the biological basis for cell mechanics 1
Roger D. Kamm and Mohammad R. K. Mofrad

2 Experimental measurements of intracellular mechanics 18
Paul Janmey and Christoph Schmidt

3 The cytoskeleton as a soft glassy material 50
Jeffrey Fredberg and Ben Fabry

4 Continuum elastic or viscoelastic models for the cell 71
Mohammed R. K. Mofrad, Helene Karcher, and Roger D. Kamm

5 Multiphasic models of cell mechanics 84
Farshid Guilak, Mansoor A. Haider, Lori A. Setton,
Tod A. Laursen, and Frank P. T. Baaijens

6 Models of cytoskeletal mechanics based on tensegrity 103
Dimitrije Stamenovic
´

7 Cells, gels, and mechanics 129
Gerald H. Pollack

8 Polymer-based models of cytoskeletal networks 152
F. C. MacKintosh

9 Cell dynamics and the actin cytoskeleton 170
James L. McGrath and C. Forbes Dewey, Jr.

10 Active cellular protrusion: continuum theories and models 204
Marc Herant and Micah Dembo

11 Summary 225
Mohammad R. K. Mofrad and Roger D. Kamm

Index 231
v
Contributors




¦®« °. . ©® °µ¬ ®
Department of Biomedical Engineering Institute for Medicine and Engineering
Eindhoven University of Technology University of Pennsylvania

© ¤ § ¤. «
Department of Biomedical Engineering Department of Mechanical Engineering
Boston University and Biological Engineering
Division
. ¦ ¤·, . Massachusetts Institute of Technology
Department of Mechanical Engineering
and Biological Engineering Division ¬® «
Massachusetts Institute of Technology Biological Engineering Division
Massachusetts Institute of
® ¦
Technology
School of Public Health
Harvard University
¤ . ¬µ®
Department of Civil and Environmental
¦¦ ¦¤§
Engineering
School of Public Health
Duke University
Harvard University

¦. . «©®
¦©¤ §µ©¬«
Division of Physics and Astronomy
Department of Surgery
Vrije Universiteit
Duke University Medical Center

 ¬. §
® . ©¤
Department of Biomedical Engineering
Department of Mathematics
University of Rochester
North Carolina State University

 ® ¤ . «. ¦¤
Department of Biomedical Engineering Department of Bioengineering
Boston University University of California, Berkeley


vii
viii Contributors

§¬¤ . °¬¬« ¬© . ®
Department of Bioengineering Department of Biomedical Engineering
University of Washington Duke University

©° ©¤ ¤©©© ®©´

Institute for Medicine and Engineering Department of Biomedical Engineering
University of Pennsylvania Boston University
Preface




Although the importance of the cytoskeleton in fundamental cellular processes such
as migration, mechanotransduction, and shape stability have long been appreciated,
no single theoretical or conceptual model has emerged to become universally ac-
cepted. Instead, a collection of structural models has been proposed, each backed
by compelling experimental data and each with its own proponents. As a result, a
consensus has not yet been reached on a single description, and the debate continues.
One reason for the diversity of opinion is that the cytoskeleton plays numerous
roles and it has been examined from a variety of perspectives. Some biophysicists
see the cytoskeleton as a cross-linked, branched polymer and have extended previous
models for polymeric chains to describe the actin cytoskeleton. Structural engineers
have drawn upon approaches that either treat the ¬lamentous matrix as a continuum,
above some critical length scale, or as a collection of struts or beams that resist
deformation by the bending stiffness of each element. Others observe the similarity
between the cell and large-scale structures whose mechanical integrity is derived
from the balance between elements in tension and others in compression. And still
others see the cytoskeleton as a gel, which utilizes the potential for phase transition
to accomplish some of its dynamic processes. Underlying all of this complexity is the
knowledge that the cell is alive and is constantly changing its properties, actively, as
a consequence of many environmental factors. The ultimate truth, if indeed there is a
single explanation for all the observed phenomena, likely lies somewhere among the
existing theories.
As with the diversity of models, a variety of experimental approaches have been
devised to probe the structural characteristics of a cell. And as with the models,
different experimental approaches often lead to different ¬ndings, often due to the
fact that interpretation of the data relies on use of one or another of the theories.
But more than that, different experiments often probe the cell at very different length
scales, and this is bound to lead to variations depending on whether the measurement
is in¬‚uenced by local structures such as the adhesion complexes that bind a bead to
the cell.
We began this project with the intent of presenting in a single text the many and
varied ways in which the cytoskeleton is viewed, in the hope that such a collection
would spur on new experiments to test the theories, or the development of new theories
ix
x Preface

themselves. We viewed this as an ongoing debate, where one of the leading proponents
of each viewpoint could present their most compelling arguments in support of their
model, so that members of the larger scienti¬c community could form their own
opinions.
As such, this was intended to be a monograph that captured the current state of a
rapidly moving ¬eld. Since we began this project, however, it has been suggested that
this book could ¬ll a void in the area of cytoskeletal mechanics and might be useful
as a text for courses taught speci¬cally on the mechanics of a cell, or more broadly in
courses that cover a range of topics in biomechanics. In either case, our hope is that
this presentation might prove stimulating and educational to engineers, physicists,
and biologists wishing to expand their understanding of the critical importance of
mechanics in cell function, and the various ways in which it might be understood.
Finally, we wish to express our deepest gratitude to Peter Gordon and his colleagues
at Cambridge University Press, who provided us with the encouragement, technical
assistance, and overall guidance that were essential to the ultimate success of this
endeavor. In addition, we would like to acknowledge Peter Katsirubas at Techbooks,
who steered us through the ¬nal stages of editing.
1 Introduction, with the biological basis
for cell mechanics

Roger D. Kamm and Mohammad R. K. Mofrad




Introduction
All living things, despite their profound diversity, share a common architectural build-
ing block: the cell. Cells are the basic functional units of life, yet are themselves
comprised of numerous components with distinct mechanical characteristics. To per-
form their various functions, cells undergo or control a host of intra- and extracellular
events, many of which involve mechanical phenomena or that may be guided by the
forces experienced by the cell. The subject of cell mechanics encompasses a wide
range of essential cellular processes, ranging from macroscopic events like the main-
tenance of cell shape, cell motility, adhesion, and deformation to microscopic events
such as how cells sense mechanical signals and transduce them into a cascade of
biochemical signals ultimately leading to a host of biological responses. One goal
of the study of cell mechanics is to describe and evaluate mechanical properties of
cells and cellular structures and the mechanical interactions between cells and their
environment.
The ¬eld of cell mechanics recently has undergone rapid development with partic-
ular attention to the rheology of the cytoskeleton and the reconstituted gels of some of
the major cytoskeletal components “ actin ¬laments, intermediate ¬laments, micro-
tubules, and their cross-linking proteins “ that collectively are responsible for the main
structural properties and motilities of the cell. Another area of intense investigation is
the mechanical interaction of the cell with its surroundings and how this interaction
causes changes in cell morphology and biological signaling that ultimately lead to
functional adaptation or pathological conditions.
A wide range of computational models exists for cytoskeletal mechanics, ranging
from ¬nite element-based continuum models for cell deformation to actin ¬lament-
based models for cell motility. Numerous experimental techniques have also been
developed to quantify cytoskeletal mechanics, typically involving a mechanical per-
turbation of the cell in the form of either an imposed deformation or force and obser-
vation of the static and dynamic responses of the cell. These experimental measure-
ments, along with new computational approaches, have given rise to several theories
for describing the mechanics of living cells, modeling the cytoskeleton as a sim-
ple mechanical elastic, viscoelastic, or poro-viscoelastic continuum, a porous gel or

1
2 R. D. Kamm and M. R. K. Mofrad

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 relevant scales and biomechanical issues of interest,
it may appear to the uninitiated that various authors are describing entirely different
structures. Yet depending on the test conditions or length scale of the measurement,
identical cells may be viewed quite differently: as either a continuum or a matrix
with ¬ne microstructure; as ¬‚uid-like or elastic; as a static structure; or as one with
dynamically changing properties. This resembles the old Rumi tale about various
people gathered in a dark room touching different parts of an elephant, each coming
up with a different theory on what indeed that object was. Light reveals the whole
object to prove the unity in diversity.
The objective of this book is to bring together diverse points of view regarding cell
mechanics, to contrast and compare these models, and to attempt to offer a uni¬ed
approach to the cell while addressing apparently irreconcilable differences. As with
many rapidly evolving ¬elds, there are con¬‚icting points of view. We have sought in
this book to capture the broad spectrum of opinions found in the literature and present
them to you, the reader, so that you can draw your own conclusions.
In this Introduction we will lay the groundwork for subsequent chapters by provid-
ing some essential background information on the environment surrounding a cell,
the molecular building blocks used to impart structural strength to the cell, and the
importance of cell mechanics in biological function. As one would expect, diverse
cell types exhibit diverse structure and nature has come up with a variety of ways in
which to convey structural integrity.


The role of cell mechanics in biological function
This topic could constitute an entire book in itself, so it is necessary to place some
constraints on our discussion. In this text, we focus primarily on eukaryotic cells of
animals. One exception to this is the red blood cell, or erythrocyte, which contains
no nucleus but which has been the prototypical cell for many mechanical studies
over the years. Also, while many of the chapters are restricted to issues relating
to the mechanics or dynamics of a cell as a material with properties that are time
invariant, it is important to recognize that cells are living, changing entities with the
capability to alter their mechanical properties in response to external stimuli. Many
of the biological functions of cells for which mechanics is central are active processes
for which the mechanics and biology are intrinsically linked. This is re¬‚ected in many
of the examples that follow and it is the speci¬c focus of Chapter 10.


Maintenance of cell shape
In many cases, the ability of a cell to perform its function depends on its shape,
and shape is maintained through structural stiffness. In the circulation, erythrocytes
exist in the form of biconcave disks that are easily deformed to help facilitate their
¬‚ow through the microcirculation and have a relatively large surface-to-area ratio to
enhance gas exchange. White cells, or leucocytes, are spherical, enabling them to roll
Introduction, with the biological basis for cell mechanics 3




Fig. 1-1. Some selected examples of cell morphology. (A) Neuron, with long projections (dendrites
and axons) that can extend a distance of 10 s of centimeters and form connections for communication
with other cells. (B) Cardiac myocyte, showing the striations associated with the individual sarcom-
eres of the contractile apparatus. (C) Various cells found in the arterial wall. Endothelial cells line the
vascular system, with a ¬‚attened, “pancake-like” morphology; neutrophils circulate in the blood until
recruited by chemoattractants to transmigrate into the tissue and convert to macrophages; ¬broblasts
function as the “factories” for the extracellular matrix; and smooth muscle cells contribute to vessel
contractility and ¬‚ow control.



along the vascular endothelium before adhering and migrating into the tissue. Because
their diameter is larger than some of the capillaries they pass through, leucocytes
maintain excess membrane in the form of microvilli so they can elongate at constant
volume and not obstruct the microcirculation. Neuronal cells extend long processes
along which signals are conducted. Airway epithelial cells are covered with a bed
of cilia, ¬nger-like cell extensions that propel mucus along the airways of the lung.
Some of the varieties of cell type are shown in Fig. 1-1. In each example, the internal
4 R. D. Kamm and M. R. K. Mofrad




Fig. 1-2. The processes contributing to cell migration: protrusion, adhesion, contraction, and rear
release. These steps can proceed in random order or simultaneously, but they all need to be operative
for cell migration to take place.


structure of the cell, along with the cell membrane, provides the structural integrity
that maintains the particular shape needed by the cell to accomplish its function,
although the speci¬c components of the structure are highly variable and diverse.


Cell migration
Many cells migrate, certainly during development (as the organism grows its vari-
ous parts), but also at maturity for purposes of wound repair (when cells from the
surrounding undamaged tissue migrate into the wound and renew the tissues) and in
combating infection (when cells of the immune system transmigrate from the vascu-
lar system across the vessel wall and into the infected tissues). Migration is also an
essential feature in cancer metastasis and during angiogenesis, the generation of new
vessels.
Descriptions of cell migration depict a process that occurs in several stages: protru-
sion, the extension of the cell at the leading edge in the direction of movement; adhe-
sion of the protrusion to the surrounding substrate or matrix; contraction of the cell that
transmits a force from these protrusions at the leading edge to the cell body, pulling
it forward; and release of the attachments at the rear, allowing net forward move-
ment of the cell to occur (see, for example, DiMilla, Barbee et al., 1991; Horwitz and
Webb, 2003; Friedl, Hegerfeldt et al., 2004; Christopher and Guan, 2000; and Fig. 1-2).
These events might occur sequentially, with the cellular protrusions “ called either
¬lopodia (“¬nger-like”) or lamellapodia (“sheet-like”) projections “ occurring as dis-
crete events: suddenly reaching forward, extending from the main body of the cell, or
more gradually and simultaneously, much like the progressive advance of a spreading
pool of viscous syrup down an inclined surface. While it is well known that cells sense
biochemical cues such as gradients in chemotactic agents, they can also apparently
sense their physical environment, because their direction of migration can be in¬‚u-
enced by variations in the stiffness of the substrate to which they adhere. Whatever
the mode of migration, however, the central role of cell mechanics, both its passive
stiffness and its active contractility, is obvious.
Introduction, with the biological basis for cell mechanics 5




Fig. 1-3. Hair cells found in the inner ear transduce sound via the stereocilia that project from their
apical surface. As the stereocilia bundle moves in response to ¬‚uid oscillations in the cochlea, tension
in the tip link (a ¬ne ¬lament connecting the tip of one stereocilium to the side of another) increases,
opening an ion channel to initiate the electrochemical response.



Mechanosensing
Nowhere is the importance of biology in cell mechanics more evident than in the ability
of the cell to sense and respond to externally applied forces. Many “ perhaps all “
cells are able to sense when a physical force is applied to them. They respond through
a variety of biological pathways that lead to such diverse consequences as changes in
membrane channel activity, up- or down-regulation of gene expression, alterations in
protein synthesis, or altered cell morphology. An elegant example of this process can
be found in the sensory cells of the inner-ear, called hair cells, which transduce the
mechanical vibration of the inner ear ¬‚uid into an electrical signal that propagates to
the brain (Hamill and Martinac, 2001; Hudspeth, 2001; Hudspeth, Choe et al., 2000).
By a remarkably clever design (Fig. 1-3), the stereocilia that extend from the apical
surface of the cells form bundles. The individual stereocilia that comprise a bundle
are able to slide relative to one another when the bundle is pushed one way or the other,
but some are connected through what is termed a “tip link” “ nothing more than a ¬ne
¬lament that connects the tip of one stereocilium to the side of another, the tension in
which is modulated by an adaption motor that moves along the internal actin ¬laments
and is tethered to the ion channel. As the neighboring ¬laments slide with respect to
6 R. D. Kamm and M. R. K. Mofrad

one another, tension is developed in the tip link, generating a force at the point where
the ¬lament connects to the side of the stereocilium. This force acts to change the
conformation of a transmembrane protein that acts as an ion channel, causing it to open
and allowing the transient entry of calcium ions. This ¬‚ux of positive ions initiates
the electrical signal that eventually reaches the brain and is perceived as sound.
Although the details of force transmission to the ion channel in the case of hair-
cell excitation are not known, another mechanosensitive ion channel, the Mechano-
sensitive channel of Large conductance (MscL) has been studied extensively (Chang,
Spencer et al., 1998; Hamill and Martinac, 2001), and molecular dynamic simulation
has been used to show how stresses in the cell membrane act directly on the channel
and cause it to change its conductance (Gullingsrud, Kosztin et al., 2001).
This is but one example of the many ways a cell can physically “feel” its surround-
ings. Other mechanisms are only now being explored, but include: (1) conformational
changes in intracellular proteins due to the transmission of external forces to the cell
interior, leading to changes in reaction rates through a change in binding af¬nity;
(2) changes in the viscosity of the cell membrane, altering the rate of diffusion of
transmembrane proteins and consequently their reaction rates; and (3) direct transmis-
sion of force to the nucleus and to the chromatin contained inside, affecting expression
of speci¬c genes. These other mechanisms are less well understood than mechanosen-
sitive channels, and it is likely that other mechanisms exist as well that have not yet
been identi¬ed (for reviews of this topic, see Bao and Suresh, 2003; Chen, Tan et al.,
2004; Huang, Kamm et al., 2004; Davies, 2002; Ingber, 1998; Shyy and Chien, 2002;
Janmey and Weitz, 2004).
Although the detailed mechanisms remain ill-de¬ned, the consequences of force
applied to cells are well documented (see, for example, Dewey, Bussolari et al., 1981;
Lehoux and Tedgui, 2003; Davies, 1995; McCormick, Frye et al., 2003; Gimbrone,
Topper et al., 2000). Various forms of force application “ whether transmitted via
cell membrane adhesion proteins (such as the heterodimeric integrin family) or by
the effects of ¬‚uid shear stress, transmitted either directly to the cell membrane or via
the surface glycocalyx that coats the endothelial surface “ elicit a biological response
(see Fig. 1-4). Known responses to force can be observed in a matter of seconds, as in
the case of channel activation, but can continue for hours after the initiating event, as
for example changes in gene expression, protein synthesis, or morphological changes.
Various signaling pathways that mediate these cellular responses have been identi¬ed
and have been extensively reviewed (Davies, 2002; Hamill and Martinac, 2001; Malek
and Izumo, 1994; Gimbrone, Topper et al., 2000).


Stress responses and the role of mechanical forces in disease
One reason for the strong interest in mechanosensation and the signaling pathways
that become activated is that physical forces have been found to be instrumental in the
process by which tissues remodel themselves in response to stress. Bone, for example,
is known to respond to such changes in internal stress levels as occur following fracture
or during prolonged exposure to microgravity. Many cells have shown that they can
both sense and respond to a mechanical stimulus. While many of these responses
appear designed to help the cell resist large deformations and possible structural
Introduction, with the biological basis for cell mechanics 7




Fig. 1-4. Forces experienced by the endothelial lining of a blood vessel and the various pathways of
force transmission, via receptor complexes, the glycocalyx, and the cytoskeleton even reaching the
nucleus, cell-cell adhesions, and cell-matrix adhesions. Any of these locations is a potential site at
which mechanical force can be transduced into a biochemical signal.


damage, others have an undesirable outcome, including atherosclerosis, arthritis, and
pulmonary hypertension; there exists an extensive literature on each of these topics.


Active cell contraction
One important subset of cells primarily exists for the purpose of generating force. Cell
types for which this is true include vascular smooth muscle cells, cardiac myocytes,
and skeletal muscle cells. While the force-generating structures may differ in detail,
the mechanisms of force generation have much in common. All muscle cells use
the molecular motor comprised of actin and myosin to produce active contraction.
These motor proteins are arranged in a well-de¬ned structure, the sarcomere, and the
regularity of the sarcomeres gives rise to the characteristic striated pattern seen clearly
in skeletal muscle cells and cardiac myocytes (Fig. 1-5). Even nonmuscle cells contain
contractile machinery, however, used for a variety of functions such as maintaining
a resting level of cell tension, changing cell shape, or in cell migration. Most cells
are capable of migration; in many, this capability only expresses itself when the cell
is stimulated. For example, neutrophils are quiescent while in the circulation but
become one of the most highly mobile migratory cells in the body when activated by
signals emanating from a local infection.


Structural anatomy of a cell
Cells are biologically active, and their structure often re¬‚ects or responds to their
physical environment. This is perhaps the primary distinction between traditional
mechanics and the mechanics of biological materials. This is a fundamental difference
8 R. D. Kamm and M. R. K. Mofrad




Fig. 1-5. Cardiac myocytes in culture showing the internal striations corresponding to the individual
sarcomeres used for contraction. Courtesy of Jan Lammerding.


from inert materials and it must be kept in mind as we progress through the various
descriptions found in this book. A second important distinction from most engineering
materials is that thermal ¬‚uctuations often need to be considered, as these in¬‚uence
both the biochemical processes that lead to intracellular remodeling but also directly
in¬‚uence the elastic characteristics of the membrane and the biological ¬laments that
comprise the cytoskeleton.
Cells often do not constitute the primary structural elements of the tissue in which
they reside. For example, in either bone or cartilage, the mechanical stiffness of the
resident cells are inconsequential in terms of their contribution to the modulus of the
tissue, and their deformation is dictated almost entirely by that of the surrounding
matrix “ collagen, and hydroxyapatite in the case of bone, and a mix of collagen and
proteoglycans with a high negative charge density in the case of cartilage. The role
of cells in these tissues is not structural, yet through the mechanisms discussed above,
cells are essential in regulating the composition and organization of the structures
contained in the extracellular regions that determine the tissue™s elasticity and strength
through the cellular response to stress.
In other tissues, the structural role of the resident cells is much more direct and
signi¬cant. Obviously, in muscle, the contractile force generated and the modulus
of the tissue, either in the contracted or the relaxed state, are dominated by cellular
activity. In other tissues, such as arterial wall or pulmonary airways, for example,
collagen and elastin ¬laments in the extracellular matrix normally balance the bulk of
Introduction, with the biological basis for cell mechanics 9

Table 1-1. Major families of adhesion molecules. (E)-extracellular; (I) intracellular

Family Location and/or function Ligands recognized
Integrins Focal adhesions, (E) ¬bronectin, collagen,
hemi-desmosomes, leukocyte laminin, immunoglobulins,
(“spreading”) adhesion, (I) actin ¬laments
primarily focal adhesions to
matrix but also in some cell-cell
adhesions
Selectins Circulating cells and endothelial Carbohydrates
cells, “rolling” adhesion
Ig superfamily Important in immune Integrins, homophillic
(immunoglobulin) response
Cadherens Adherens junctions, desmosomes (E) homophillic, (I) actin
¬laments, intermediate ¬laments
Transmembrane Fibroblasts, epithelial cells (E) collagen, ¬bronectin
proteoglycans (I) actin ¬laments, heterophillic

the stress. During activation of the smooth muscle, however, stress shifts from these
extracellular constituents to the contractile cells, and the vessel constricts to a diameter
much smaller than that associated with the passive wall stiffness. In the case of cardiac
tissue, the contractile cells, or myocytes (Fig. 1-5), constitute a large fraction of the
tissue volume and are primarily responsible for the stresses and deformations of the
myocardium that are time varying through the cardiac cycle.

The extracellular matrix and its attachment to cells
Contrary to the situation in most cell mechanics experiments in vitro, where forces
might be applied directly to the cells via tethered beads, a micropipette, an AFM probe,
or ¬‚uid shear stress, forces in vivo are often transmitted to the cell via the extracellular
matrix (ECM), which shares in the load-supporting function. Many cell membrane
receptors contain extracellular domains that bind to the various proteins of the ECM.
For example, members of the integrin family can bind to ¬bronectin, vitronectin,
collagen, and laminin. Intracellular domains of these same proteins bind directly
(or indirectly, through other membrane-associated proteins) to the cytoskeleton. The
number and variety of linking proteins is quite remarkable, as described in detail in a
recent review (Geiger and Bershadsky, 2002). Other adhesion molecules bind to the
ECM, basement membrane, neighboring cells, or cells suspended in ¬‚owing blood.
Adhesion molecules can be either homophillic (binding to other identical molecules)
or heterophillic (Table 1-1). Of these transmembrane molecules (both proteins and
proteoglycans) many attach directly to the cytoskeleton, which often exhibits a denser,
more rigid structure in the vicinity of an adhesion site.

Transmission of force to the cytoskeleton and the role
of the lipid bilayer
Cells are separated from the external environment by a thin lipid bilayer consisting of a
rich mix of phospholipids, glycolipids, cholesterol, and a vast array of transmembrane
10 R. D. Kamm and M. R. K. Mofrad

proteins that constitute about 50 percent of the membrane by weight but only 1 to
2 percent of the total number of molecules residing in the membrane. Phospholipids,
which are the most abundant, are amphipathic, having a hydrophilic part residing on
the outside surface of the bilayer and a hydrophobic part on the inside. Some of the
proteins serve as ion channels, others as a pathway for transmembrane signaling. Still
others provide a structural bridge across the membrane, allowing for direct adhesion
between the internal cytoskeleton and the extracellular matrix. Together, these are
commonly referred to as integral membrane proteins. Roughly half of these integral
proteins are able to freely diffuse within the membrane, while the rest are anchored
to the cytoskeleton.
In addition to its role in communicating stress and biochemical signals into the
cell, the membrane also serves a barrier function, isolating the cell interior from its
extracellular environment and maintaining the appropriate biochemical conditions
within for critical cell functions. By itself, the bilayer generally contributes little to
the overall stiffness of the cell, except in situations in which the membrane becomes
taut, as might occur due to osmotic swelling. In general, the bilayer can be thought
of as a two-dimensional ¬‚uid within which the numerous integral membrane proteins
diffuse, a concept ¬rst introduced in 1972 by Singer and Nicolson as the ¬‚uid mosaic
model (Singer and Nicolson, 1972). The bilayer maintains a nearly constant thick-
ness of about 6 nm under stress, and exhibits an area-expansion modulus, de¬ned
as the in-plane tension divided by the fractional area change, of about 0.1“1.0 N/m
(for pure lipid bilayers) or 0.45 N/m (for a red blood cell) (Waugh and Evans, 1979).
Rupture strength, in terms of the maximum tension that the membrane can withstand,
lies in the range of 0.01“0.02 N/m, for a red blood cell and a lipid vesicle, respectively
(Mohandas and Evans, 1994). Values for membrane and cortex bending stiffness re-
ported in the literature (for example, ∼ 2 ’ 4 — 10’19 N·m for the red blood cell mem-
brane (Strey, Peterson et al., 1995; Scheffer, Bitler et al., 2001), and 1 ’ 2 — 10’18 N·m
for neutrophils (Zhelev, Needham et al., 1994), are not much larger than that for pure
lipid bilayers (Evans and Rawicz, 1990), despite the fact that they include the effects
of the membrane-associated cortex of cytoskeletal ¬laments, primarily spectrin in
the case of erythrocytes and actin for leukocytes. When subjected to in-plane shear
stresses, pure lipid bilayers exhibit a negligible shear modulus, whereas red blood cells
have a shear modulus of about 10’6 N·s/m (Evans and Rawicz, 1990). Forces can
be transmitted to the membrane via transmembrane proteins or proteins that extend
only partially through the bilayer. When tethered to an external bead, for example,
the latter can transmit normal forces; when forces are applied tangent to the bilayer,
the protein can be dragged along, experiencing primarily a viscous resistance unless
it is tethered to the cytoskeleton. Many proteins project some distance into the cell,
so their motion is impeded even if they are not bound to the cytoskeleton due to steric
interactions with the membrane-associated cytoskeleton.


Intracellular structures
In this text we primarily address the properties of a generic cell, without explic-
itly recognizing the distinctions, often quite marked, between different cell types.
It is important, however, to recognize several different intracellular structures that
Introduction, with the biological basis for cell mechanics 11

in¬‚uence the material properties of the cell that may, at times, need to be taken into
account in modeling. Many cells (leucocytes, erythrocytes, and epithelial cells, for
example) contain a relatively dense structure adjacent to the cell membrane called the
cortex, with little by way of an internal network. In erythrocytes, this cortex contains
another ¬lamentous protein, spectrin, and largely accounts for the shape rigidity of
the cell. Many epithelial cells, such as those found in the intestine or lining the pul-
monary airways, also contain projections (called microvili in the intestine and cilia in
the lung) that extend from their apical surface. Cilia, in particular, are instrumental in
the transport of mucus along the airway tree and have a well-de¬ned internal structure,
primarily due to microtubules, that imparts considerable rigidity.
Of the various internal structures, the nucleus is perhaps the most signi¬cant, from
both a biological and a structural perspective. We know relatively little about the
mechanical properties of the nucleus, but some recent studies have begun to probe
nuclear mechanics, considering the separate contributions of the nuclear envelope,
consisting of two lipid bilayers and a nuclear lamina, and the nucleoplasm, consisting
largely of chromatin (Dahl, Kahn et al., 2004; Dahl, Engler et al., 2005).
Migrating cells have a rather unique structure, but again are quite variable from
cell type to cell type. In general, the leading edge of the cell sends out protrusions,
either lamellipodia or ¬lopodia, that are rich in actin and highly cross-linked. The
dynamics of actin polymerization and depolymerization is critical to migration and is
the focus of much recent investigation (see, for example Chapter 9 and Bindschadler,
Dewey, and McGrath, 2004). Active contraction of the network due to actin-myosin
interactions also plays a central role and provides the necessary propulsive force.
Actin ¬laments form by polymerization of globular, monomeric actin (G-actin)
into a twisted strand of ¬lamentous actin (F-actin) 7“9 nm in diameter with structural
polarity having a barbed end and a pointed end. Monomers consist of 375 amino acids
with a molecular weight of 43 kDa. ATP can bind to the barbed end, which allows
for monomer addition and ¬lament growth, while depolymerization occurs preferen-
tially at the pointed end (Fig. 1-6A). Filament growth and organization is regulated
by many factors, including ionic concentrations and a variety of capping, binding,
branching, and severing proteins. From actin ¬laments, tertiary structures such as
¬ber bundles, termed “stress ¬bers,” or a three-dimensional lattice-like network can
be formed through the action of various actin-binding proteins (ABPs). Some ex-
amples of ABPs are ¬mbrin and ±-actinin, both instrumental in the formation of
stress ¬bers or bundles of actin ¬laments, and ¬lamin, which connects ¬laments into
a three-dimensional space-¬lling matrix or gel with ¬laments joined at nearly a right
angle. Recent rheological studies of reconstituted actin gels containing various con-
centrations of ABPs (see Chapter 2, or Tseng, An et al., 2004) have illustrated the
rich complexities of even such simple systems and have also provided new insights
into the nature of such matrices.
The importance of actin ¬laments is re¬‚ected in the fact that actin constitutes from 1
to 10 percent of all the protein in most cells, and is present at even higher concentrations
in muscle cells. Actin is thought to be the primary structural component of most
cells; it responds rapidly and dramatically to external forces and is also instrumental
in the formation of leading-edge protrusions during cell migration. As the data in
Table 1-2 illustrate, actin ¬laments measured by a variety of techniques (Yasuda,
12 R. D. Kamm and M. R. K. Mofrad




Fig. 1-6. Filaments that constitute the cytoskeleton. (A) Actin ¬laments. (B) Microtubules.
(C) Intermediate ¬laments.



Miyata et al., 1996; Tsuda, Yasutake et al., 1996; Higuchi and Goldman, 1995) are stiff,
having a persistence length of several microns, and an effective Young™s modulus,
determined from its bending stiffness and radius of 1 ’ 3 —109 Pa, comparable to
that of polystyrene (3 — 109 Pa) and nearly equal to that of bone (9 —109 Pa).
Microtubules constitute a second major constituent of the cytoskeleton. These are
polymerized ¬laments constructed from monomers of ±- and β-tubulin in a helical
Introduction, with the biological basis for cell mechanics 13

Table 1-2. Elastic properties of cytoskeletal ¬laments

Diameter, Persistence Bending stiffness, Young™s
K B (Nm2 )
2a (nm) length, l p (µm) modulus, E (Pa)
7 — 10’26 1.3“2.5 — 109
Actin ¬lament 6“8 15
2.6 — 10’23 1.9 — 109
Microtubule 25 6000
4 — 10’27
∼1 1 — 109
Intermediate ¬lament 10
The elastic properties of actin ¬laments and microtubules are approximately consistent with a prediction
based on the force of van der Waals attraction between two surfaces (J. Howard, 2001). Persistence length
(l p ) and bending stiffness (K B ) are related through the expression l p = K B /kB T . Bending stiffness and
Young™s modulus (E) are related through the expression K B = E I = π a 4 E for a solid rod of circular
4
cross-section with radius a, and I = π (ao ’ ai4 ) for a hollow cylinder with inside and outside radii ai
4
4
and ao , respectively.



arrangement, both 55 kDa polypeptides, that organize into a small, hollow cylinder
(Fig. 1-6B). The ¬laments have an outer diameter of about 25 nm and exhibit a
high bending stiffness, even greater than that of an actin ¬lament (Table 1-2) with
a persistence length of about 6 mm (Gittes, Mickey et al., 1993). Tubular structures
tend to be more resistant to bending than solid cylinders with the same amount of
material per unit length, and this combined with the larger radius accounts for the
high bending stiffness of microtubules despite having an effective Young™s modulus
similar to that of actin. Because of their high bending stiffness, they are especially
useful in the formation of long slender structures such as cilia and ¬‚agella. They also
provide the network along which chromosomes are transported during cell division.
Microtubules are highly dynamic, even more so than actin, undergoing constant
polymerization and depolymerization, so that the half-life of a microtubule is typically
only a few minutes. (Mitchison and Kirschner, 1984). Growth is asymmetric, as with
actin, with polymerization typically occurring rapidly at one end and more slowly
at the other, and turnover is generally quite rapid; the half-life of a microtubule is
typically on the order of minutes.
Intermediate ¬laments (IFs) constitute a superfamily of proteins containing more
than ¬fty different members. They have in common a structure consisting of a cen-
tral ±-helical domain of over 300 residues that forms a coiled coil. The dimers then
assemble into a staggered array to form tetramers that connect end-to-end, forming
proto¬laments (Fig. 1-6C). These in turn bundle into ropelike structures, each con-
taining about eight proto¬laments with a persistence length of about 1 µm (Mucke,
Kreplak et al., 2004). Aside from these differences in structure, intermediate ¬la-
ments differ from micro¬laments and microtubules in terms of their long-term sta-
bility and high resistance to solubility in salts. Also, unlike polymerization of other
cytoskeletal ¬laments, intermediate ¬laments form without the need for GTP or ATP
hydrolysis.
In recent experiments, intermediate ¬laments have been labeled with a ¬‚uorescent
marker and used to map the strain ¬eld within the cell (Helmke, Thakker et al., 2001).
This is facilitated by the tendency for IFs to be present throughout the entire cell at a
suf¬ciently high concentration that they can serve as ¬ducial markers.
Of course these are but a few of the numerous proteins that contribute to the mechan-
ical properties of a cell. The ones mentioned above “ actin ¬laments, microtubules,
14 R. D. Kamm and M. R. K. Mofrad




Fig. 1-7. A small sampling of the proteins found in a focal adhesion complex (FAC). Forces are
typically transmitted from the extracellular matrix (for example, ¬bronectin), via the integral mem-
brane adhesion receptors (±’ and β’integrins), various membrane-associated proteins (focal adhe-
sion kinase (FAK), paxillin (Pax), talin, Crk-associated substrate (CAS)), to actin-binding proteins
(±-actinin) that link the FAC to the cytoskeleton. Adapted from Geiger and Bershadsky 2002.



and intermediate ¬laments “ are primarily associated with the cytoskeleton, but even
within the cytoskeletal network are found numerous linking proteins (ABPs consti-
tuting one family) that in¬‚uence the strength and integrity of the resulting matrix. In
addition to these are the molecular constituents of the cell membrane, nuclear mem-
brane, and all the organelles and other intracellular bodies that in¬‚uence the overall
mechanical response of a cell. In fact, intracellular structure should be noted for its
complexity, as can be seen in Fig. 1-7, which shows just a small subset of the numerous
proteins that link the extracellular matrix and the cytoskeleton. Any of these consti-
tutes a pathway for transmitting force across the cell membrane, between the proteins
found in the adhesion complexes, and through the cytoskeletal network. To the extent
that a particular protein is located along the force transmission pathway, not only does
it play a role in transmitting stress, but it also represents a candidate for mechanosens-
ing due to the conformational changes that arise from the transmission of force.
Active contraction is another fundamental feature of the cytoskeleton that in¬‚u-
ences its structural properties. While this is an obvious characteristic of the various
types of muscle cell, most cells contain contractile machinery, and even in their resting
Introduction, with the biological basis for cell mechanics 15

state can exert a force on their surroundings. Forces have been measured in resting
¬broblasts, for example, where intracellular tension gives rise to stresses in the fo-
cal adhesions of the cell adherent to a ¬‚exible two-dimensional substrate of about
5 nN/µm2 , or 5 kPa (Balaban, Schwarz et al., 2001). In experiments with various cell
types grown in a three-dimensional gel such as collagen, the cells actively contract
the matrix by more than 50 percent (Sieminski, Hebbel et al., 2004). These contractile
forces are associated with intracellular molecular motors such as those in the myosin
family.


Overview
This book presents a full spectrum of views on current approaches to modeling cell
mechanics. In part, this diversity of opinion stems from the different backgrounds of
contributors to the ¬eld. Indeed, the authors of this book come from the biophysics,
bioengineering, and physical chemistry communities, and each joins the discussion
with a unique perspective on biological systems. Consequently, the approaches range
from ¬nite element methods commonly used in continuum mechanics to models of
the cytoskeleton as a cross-linked polymer network to models of soft glassy materials
and gels. Studies re¬‚ect both the static, instantaneous nature of the structure as well
as its dynamic nature due to polymerization and the full array of biological processes.
It is unlikely that a single unifying approach will evolve from this diversity, in part
because of the complexity of the phenomena underlying the mechanical properties of
the cell. It is our hope, however, that a better appreciation of the various perspectives
will lead to a more highly coordinated approach to the essential problems and might
facilitate discussions among investigators with differing views.
Perhaps the most important purpose of this monograph is to stimulate new ideas
and approaches. Because no single method has emerged as clearly superior, this might
re¬‚ect the need for approaches not yet envisaged. That much of the work presented
here derives from publications over the past several years reinforces the notion that
cell mechanics is a rapidly evolving ¬eld. The next decade will likely yield further
advances not yet foreseen.


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Introduction, with the biological basis for cell mechanics 17

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2 Experimental measurements
of intracellular mechanics

Paul Janmey and Christoph Schmidt




Novel methods to measure the viscoelasticity of soft materials and new theories
ABSTRACT:
relating these measurements to the underlying molecular structures have the potential to rev-
olutionize our understanding of complex viscoelastic materials like cytoplasm. Much of the
progress in this ¬eld has been in methods to apply piconewton forces and to detect motions
over distances of nanometers, thus performing mechanical manipulations on the scale of single
macromolecules and measuring the viscoelastic properties of volumes as small as fractions
of a cell. Exogenous forces ranging from pN to nN are applied by optical traps, magnetic
beads, glass needles, and atomic force microscope cantilevers, while deformations on a scale
of nanometers to microns are measured by de¬‚ection of lasers onto optical detectors or by high
resolution light microscopy.
Complementary to the use of external forces to probe material properties of the cell are
analyses of the thermal motion of refractile particles such as internal vesicles or submicron-sized
beads imbedded within the cell. Measurements of local viscoelastic parameters are essential for
mapping the properties of small but heterogeneous materials like cytoplasm; some methods,
most notably atomic force microscopy and optical tracking methods, enable high-resolution
mapping of the cell™s viscoelasticity.
A signi¬cant challenge in this ¬eld is to relate experimental and theoretical results derived
from systems on a molecular scale to similar measurements on a macroscopic scale, for example
from tissues, cell extracts, or puri¬ed polymer systems, and thus provide a self-consistent set
of experimental methods that span many decades in time and length scales. At present, the
new methods of nanoscale rheology often yield results that differ from bulk measurements by
an order of magnitude. Such discrepancies are not a trivial result of experimental inaccuracy,
but result from physical effects that only currently are being recognized and solved. This
chapter will summarize some recent advances in methodology and provide examples where
experimental results may motivate new theoretical insights into both cell biology and material
science.



Introduction
The mechanical properties of cells have been matters of study and debate for cen-
turies. Because cells perform a variety of mechanical processes, such as locomotion,
secretion, and cell division, mechanical properties are relevant for biological function.
Certain cells, such as plant cells and bacteria, have a hard cell wall that dominates
18
Experimental measurements of intracellular mechanics 19

the mechanics, whereas most other cells have a soft membrane and their mechan-
ical properties are determined largely by an internal protein polymer network, the
cytoskeleton. Early observations of single cells by microscopy showed regions of
cytoplasm that were devoid of particles undergoing Brownian motion, and therefore
were presumed to be “glassy” (see Chapter 3) or in some sense solid (Stossel, 1990).
The interior of the cell, variously called the protoplasm, the ectoplasm, or more gen-
erally the cytoplasm, was shown to have both viscous and elastic features. A variety
of methods were designed to measure these properties quantitatively.


Forces to which cells are exposed in a biological context
The range of stresses (force per area) to which different tissues are naturally exposed is
large. Cytoskeletal structures have evolved accordingly and are not only responsible
for passively providing material strength. They are also intimately involved in the
sensing of external forces and the cellular responses to those forces. How cells respond
to mechanical stress depends not only on speci¬c molecular sensors and signaling
pathways but also on their internal mechanical properties or rheologic parameters,
because these material properties determine how the cell deforms when subjected to
force (Janmey and Weitz, 2004).
It is likely that different structures and mechanisms are responsible for different
forms of mechanical sensing. For example, cartilage typically experiences stresses on
the order of 20 MPa, and the individual chondrocytes within it alter their expression of
glycosaminoglycans and other constituents as they deform in response to such large
forces (Grodzinsky et al., 2000). Bone and the osteocytes within it respond to similarly
large stresses (Ehrlich and Lanyon, 2002), although the stress to which a cell imbedded
within the bone matrix is directly exposed is not always clear. At the other extreme,
endothelial cells undergo a wide range of morphological and transcriptional changes in
response to shear stresses less than 1 Pa (Dewey et al., 1981), and neutrophils activate
in response to similar or even smaller shear stresses (Fukuda and Schmid-Schonbein,
2003). Not only the magnitude but the geometry and time course of mechanical
perturbations are critical to elicit speci¬c cellular effects. Some tissues like tendons
or skeletal muscle experience or generate mainly uniaxial forces and deformations,
while others, such as the cells lining blood vessels, normally experience shear stresses
due to ¬‚uid ¬‚ow. These cells often respond to changes in stress or to oscillatory stress
patterns rather than to a speci¬c magnitude of stress (Bacabac et al., 2004; Davies
et al., 1986; Florian et al., 2003; Ohura et al., 2003; Turner et al., 1995). Many cells,
including the cells lining blood vessels and epithelial cells in the lung, experience
large-area-dilation forces, and in these settings both the magnitude and the temporal
characteristics of the force are critical to cell response (Waters et al., 2002).


Methods to measure intracellular rheology by macrorheology,
diffusion, and sedimentation
The experimental designs to measure cytoplasmic (micro)rheology have to overcome
three major challenges: the small size of the cell; the heterogeneous structure of the
cell interior; and the active remodeling of the cytoplasm that occurs both constitutively,
20 P. Janmey and C. Schmidt

as part of the resting metabolic state, and directly, in response to the application of
forces necessary to perform the rheologic measurement. The more strongly the cell is
perturbed in an effort to measure its mechanical state, the more it reacts biochemically
to change that state (Glogauer et al., 1997). Furthermore it is important to distinguish
linear response to small strains from nonlinear response to larger strains. Structural
cellular materials typically have a very small range of linear response (on the order of
10 percent) and beyond that react nonlinearly, for example by strain hardening or shear
thinning or both in sequence. To overcome these problems a number of experimental
methods have been devised.


Whole cell aggregates
The simplest and in some sense crudest method to measure intracellular mechanics
is to use standard rheologic instruments to obtain stress/strain relations on a macro-
scopic sample containing many cells, but in which a single cell type is arranged in a
regular pattern. Perhaps the most successful application of this method has been the
study of muscle ¬bers, in which actin-myosin-based ¬bers are arranged in parallel and
attached longitudinally, allowing an inference of single cell quantities directly from
the properties of the macroscopic sample. One example of the validity of the assump-
tions that go into such measurements is the excellent agreement of single molecule
measurements of the force-elongation relation for titin molecules with macroscopic
compliance measurements of muscle ¬bers where the restoring force derives mainly
from a large number of such molecules working in series and in parallel (Kellermayer
et al., 1997). Another simple application of this method is the measurement of close-
packed sedimented samples of a single cell type, with the assumption that during
measurement, the deformation is related to the deformation of the cell interior rather
than to the sliding of cells past each other. Such measurement have, for example, shown
the effects of single actin-binding protein mutations in Dictyostelium (Eichinger
et al., 1996) and melanoma cells (Cunningham et al., 1992). These simple mea-
surements have the serious disadvantage that properties of a single cell require as-
sumptions or veri¬cation of how the cells attach to each other, and in most cases the
contributions of membrane deformation cannot be separated from those of the cell
interior or the extracellular matrix.


Sedimentation of particles
To overcome the problems inherent in the measurement of macroscopic samples, a
variety of elegant solutions have been devised. Generally, in order to resolve varying
viscoelastic properties within a system, one has to use probes of a size comparable
to or smaller than the inhomogeneities. Such microscopic probes can be fashioned
in different ways. One of the simplest and oldest methods to measure cytoplasmic
viscosity relies on observations of diffusion or sedimentation of intracellular gran-
ules with higher speci¬c gravity through the cytoplasmic continuum. Generally these
measurements were performed on relatively large cells containing colored or re-
tractile particles easily visible in the microscope. Such measurements (reviewed in
Heilbrunn, 1952; Heilbrunn, 1956) are among the earliest to obtain values similar to
Experimental measurements of intracellular mechanics 21

those measured currently, but they are limited to specialized cells and cannot measure
elasticity in addition to viscosity.
Sedimentation measurements are done by a variety of elegant methods. One of the
earliest such studies (Heilbronn, 1914) observed the rate of falling of starch grains
within a bean cell and compared the rate of sedimentation of the same starch particles
puri¬ed from the cells in ¬‚uids of known densities and viscosities measured by con-
ventional viscometers, to obtain a value of 8 mPa.s for cytoplasmic viscosity. These
measurements were an early application of a falling-sphere method commonly used
in macroscopic rheometry (Rockwell et al., 1984). The viscosity of the cytoplasm in
this application was determined by relation to calibrated liquids; because the starch
particles are relatively uniform and could be puri¬ed from the cell, inaccuracies as-
sociated with measurement of their small size were avoided. More generally, any
gravity-dependent velocity V of a particle of radius r and density σ in the cytoplasm
of density ρ could be used to measure cytoplasmic viscosity · by use of the relation
2g (σ ’ ρ) r 2
V= , (2.1)

in which g is the gravitational acceleration. Without centrifugation internal organelles
rarely sediment, but a suf¬ciently large density difference between an internal particle
and the surrounding cytoplasm could be created by injecting a small droplet of inert oil
into a large cell, like a muscle ¬ber (Reiser, 1949) to obtain values of 29 mPa.s from the
rate at which the drop rose in the cytoplasm. Alternatively, internal organelles could be
made to sediment by known gravitational forces in a centrifuge, and in various cells “
including oocytes, amoebas, and slime molds “ cytoplasmic viscosities between 2
and 20 mPa.s have been commonly reported, although some much higher values
greater than 1 Pa.s were also observed (reviewed in Heilbrunn, 1956). The large
differences in viscosity were presumed to arise from experimental differences in the
sedimentation rates, because these early studies also showed that cytoplasm was a
highly non-Newtonian ¬‚uid and that the apparent viscosity strongly decreased with
increasing shear rate.


Diffusion
Measurements of viscosity by diffusion (Heilbrunn, 1956) have been done by ¬rst
centrifuging a large cell, such as a sea urchin egg or an amoeba, with sedimentation
forces typically between 100 and 5000 times that of gravity, suf¬cient for internal
organelles to get concentrated at the bottom while the cell remains intact. Then the
displacement of a single particle of radius r in one direction x(t) is monitored, and
the cytoplasmic viscosity is measured from the Stokes-Einstein relation:
kB T
x 2 (t) = , (2.2)
3π·r
where the brackets denote ensemble averaging, k B is the Boltzmann constant, and T is
temperature. Such measurements, dating from at least the 1920s (decades before video
microscopy and image processing) showed that the viscosity within sea urchin egg
cytoplasm was 4 mPa.s (4 cP), only four times higher than that of water, but that other
22 P. Janmey and C. Schmidt

cells exhibited much higher internal viscosities. Three other important features of
intracellular material properties were evident from these studies. First, it was shown
that the apparent viscosity of the protoplasm depended strongly on ¬‚ow rates, as
varied, for example, by changing the sedimentation force in the centrifuge. Second,
viscous ¬‚ow of internal organelles could generally be measured only deeper inside the
cell, away from the periphery, where an elastic cortical layer could be distinguished
from the more liquid cell interior. Third, the cellular viscosity was often strongly
temperature dependent.


Mechanical indentation of the cell surface

Glass microneedles
Glass needles can be made thin enough to apply to a cell measured forces large
enough to deform it but small enough that the cell is not damaged. An early use of
such needles was to pull on individual cultured neurons (Bray, 1984); these studies
showed how such point forces could be used to initiate neurite extension. Improved
instrumentation and methods allowed an accurate estimate of the forces needed to ini-
tiate these changes. The method (Heidemann and Buxbaum, 1994; Heidemann et al.,
1999) begins with calibration of the bending constant of a wire needle essentially by
hanging a weight from the end of a thin metal wire and determining its spring constant
from de¬‚ection of the loaded end by the relation:
FL 3
y(L) = (2.3)
3EI
where y(L) is the displacement of the end of a wire of length L , F is the force due
to the weight, E is the material™s Young™s modulus of elasticity, and I = πr 4 /4 is the
second moment of inertia of the rod of radius r.
The product EI is a constant for each rod; in practice the ¬rst calibrated rod is used
to provide a known (smaller) force to a thinner, usually glass, rod, to calibrate that
rod, and repeat the process until a rod is calibrated that can provide nN or smaller
forces depending on its length and radius.


Cell poker
A pioneering effort to apply forces locally to the surface of live cells was the devel-
opment of the cell poker (Daily et al., 1984; Petersen et al., 1982). In this device,
shown schematically in Fig. 2-1, a cell is suspended in ¬‚uid from a glass coverslip on
an upright microscope, below which is a vertical glass needle attached at its opposite
end to a wire needle that is in turn coupled to a piezoelectric actuator that moves the
wire/needle assembly up and down. The vertical displacements of both ends of the
wire are measured optically, and a difference in the displacements of these points, x,
occurs because of resistance to moving the glass needle tip as it makes contact with
the cell. The force exerted by the tip F on the cell surface is determined by Hooke™s
law F = kx from the stiffness of the wire k, which can be calibrated by macroscopic
means such as the hanging of known weights from a speci¬ed length of wire. Using this
Experimental measurements of intracellular mechanics 23




Fig. 2-1. Schematic representation of the cell-poking apparatus. Positioning of the cell (C)
relative to the poker tip (T) is achieved by translating the top of the temperature control unit (TC)
or by rotating the holder on which the coverslip is mounted. The motor assembly can be translated
to ensure the tip is positioned in the ¬eld of view. W, steel wire; LPM, linear piezoelectric mo-
tor; MS and TS, optical sensors; MF, motor ¬‚ag; TF, tip ¬‚ag; MO, modulation contrast objective;
MC, matching condenser.


instrument, displacements less than 100 nm can be resolved corresponding to forces
less than 10 nN. A typical force vs. displacement curve from this instrument as shown
in Fig. 2-2 reveals a signi¬cant degree of both elasticity and unrecoverable deforma-
tion from plasticity or ¬‚ow of the cytoplasm. Such measurements have demonstrated
both a signi¬cant elastic response as well as a plastic deformation of the cell, and the
time course and magnitudes of these processes can be probed by varying the rate at
which the forces are applied. Because the tip is considerably smaller than the cross-
sectional area of the cell, local viscoelasticity could be probed at different regions
of the cell or as active motion or other responses are triggered. The earliest such
measurements revealed a large difference in relative stiffness over different areas of
the cell and a high degree of softening when actin-¬lament-disorganizing drugs like




Fig. 2-2. Force displacement curve as the cell poker tip ¬rst indents the cell (upper curve) and then
is lowered away from the cell contact (lower curve).
24 P. Janmey and C. Schmidt




Fig. 2-3. Cell poking with the tip of an atomic force microscope. Upper image: If a regular sharp
tip is used, inhomogeneities encountered on the nm scale of the tip radius are likely to make the
result dif¬cult to interpret. Lower image: Using a micrometer-sized bead attached to the tip, force
sensitivity is maintained while the cell response is averaged over a micrometer scale.


cytochalasin were applied. The measurements also showed that the apparent stiffness
of the cell increased as the amplitude of indentation increased. How this nonlinear
elastic response is related to the material properties of the cell is, however, not straight-
forward to deduce, because of a number of complicating effects, as the earliest such
studies pointed out.
For a homogeneous, semi-in¬nite elastic solid, given the geometry of the glass
needle tip and the force of indentation, the force-displacement curves are determined
by two material properties, the Young™s modulus and Poisson™s ratio, in a way that is
described by the Hertz relation. For a sphere the result for the force as a function of
indentation depth δ is (Hertz, 1882; Landau and Lifshitz, 1970):
4 E
Fsphere = R 1/2 δ 3/2 (2.4)
3 (1 ’ ν 2)

with the Young™s modulus E, Poisson ratio ν, and sphere radius R. For indentation
with a conical object, the result is:
π E
Fcone = tan(±) δ 2 (2.5)
2 (1 ’ ν 2)

with the cone opening angle ±.
The application of the Hertz model in relation to cell-poking measurements is,
however, often not meaningful for at least three reasons. First, the Hertz relation
is not valid if the cell thickness is not much greater than the degree of indentation.
Second, the cell cytoskeleton is in most cases far from being an isotropic homogeneous
material. And third, forces exerted on a cell typically initiate biochemical as well as
other active reactions. These issues have been extensively discussed both in terms of
the cell poker (Daily et al., 1984), and more recently in applications of the scanning
force microscope that operates on the same principle.
Experimental measurements of intracellular mechanics 25


Atomic force microscopy
A very sensitive local mechanical probe is provided by atomic force microscopy
(AFM). An AFM in an imaging mode works by scanning a sharp microfabricated
tip over a surface while simultaneously recording tip de¬‚ection. The de¬‚ection time
course is then converted into an image of the surface pro¬le (Binnig et al., 1986).
Imaging can be done in different modes “ contact mode (Dufrene, 2003), tapping
mode (Hansma et al., 1994), jumping mode (de Pablo et al., 1998) or others “ which
are usually designed to minimize damage to the sample or distortions of the surface
by the imaging method. When one wants to probe the mechanical properties of a
material surface, however, one can also use an AFM tip to exert precisely controlled
forces in selected locations and record the corresponding sample displacements. In
many ways this method is related to cell poking with larger probes, but it holds the
potential of better spatial and force resolution. The obvious limitation of the technique
is that manipulation can only occur through the accessible surface of a cell, that is
one cannot measure elastic moduli well inside the cell without an in¬‚uence of bound-
ary conditions. One can both indent cells or pull on cells when the tips are attached
strongly enough to the cell surface. The indentation approach has been used to test the
elastic properties of various types of cell. Initial studies have used conventional sharp
(radius of 10s of nm) tips and applied the Hertz model as described above (reviewed in
MacKintosh and Schmidt, 1999). The same caveats hold in this case as in the discus-
sion of other cell-poking experiments: the cell is not a homogeneous, isotropic, passive
elastic solid. The thin parts of cells, at the cell periphery in surface-attached cells, are
particularly interesting to study because they are crucial for cell motility but are usu-
ally too thin to apply the standard Hertz model. When using an AFM with a sharp tip,
the spatial inhomogeneity of cells “ for example the presence of bundles of actin (stress
¬bres), microtubules, and more “ is likely more of a problem, because spatial averag-
ing in the case of a larger probe tends to make the material look more homogeneous.
Results of initial experiments were thus rather qualitative, but differences between the
cell center and its periphery could be detected (Dvorak and Nagao, 1998). A problem
with quasi-static or low-frequency measurements is that the cell will react to forces
exerted on it and the response measured will not only re¬‚ect passive material prop-
erties, but also active cellular responses. AFM has also been used on cells in a high-
frequency mode, namely the tapping mode. It was observed that cells dynamically
stiffened when they were probed with a rapidly oscillating tip, as one would expect
(Putman et al., 1994).
A more quantitative technique has been developed more recently, using polystyrene
beads of carefully controlled radius attached to the AFM tips to contact cells (Mahaffy
et al., 2004; Mahaffy et al., 2000). This creates a well-de¬ned probe geometry and
provides another parameter, namely bead radius, to control for inhomogeneities. Val-
ues for zero-frequency shear moduli were between 1 and 2 kPa for the ¬broblast
cells studied. The probing was in this case also done with an oscillating tip, to
measure frequency-dependent viscoelastic response with a bandwidth of 50“300 Hz
and data were evaluated with an extended Hertz model valid for oscillating probes
(Mahaffy et al., 2000). A problem for determining the viscous part of the response is
the hydrodynamic drag on the rest of the cantilever that dominates and changes
26 P. Janmey and C. Schmidt

with decreasing distance from the surface and with tip-sample contact and is not easy
to correct for.
The Hertz model has been further modi¬ed to account for ¬nite sample thickness
and boundary conditions on the substrate (Mahaffy et al., 2004), which makes it
possible to estimate elastic constants also for the thin lamellipodia of cells, which
were found again to be between 1 and 2 kPa in ¬broblasts.


Mechanical tension applied to the cell membrane
Pulling on a cell membrane by controlled suction within a micropipette has been
an important tool to measure the viscosity and elastic response of cells to controlled
forces. The initial report of a cell elastimeter based on micropipette aspiration (Mitchi-
son and Swann, 1954) has guided many studies that have employed this method to
deform the membranes of a variety of cells, especially red blood cells, which lack
a three-dimensional cytoskeleton but have a continuous viscoelastic protein network
lining their outer membrane (Discher et al., 1994; Evans and Hochmuth, 1976). One
important advantage of this method is that the cell can either be suspended in solution
while bound to the micropipette or attached to a surface as the micropipette applies
negative pressure from the top. The ability to probe nonadherent cells has made mi-
cropipette aspiration a powerful method to probe the viscoelasticity of blood cells
including erythrocytes, leukocytes, and monocytes (Chien et al., 1984; Dong et al.,
1988; Richelme et al., 2000).
A typical micropipette aspiration system is shown in Fig. 2-4. Images of two red
blood cells partly pulled into a micropipette are shown in Fig. 2-5. Micropipette aspi-
ration provides measures of three quantities: the cortical tension in the cell membrane;
the cytoplasmic viscosity; and the cell elasticity. If the cell can be modeled as a liquid
drop with a cortical tension, as appears suitable to leukocytes under some conditions,
the cortical tension t is calculated from the pressure at which the aspirated part of the
cell forms a hemispherical cap within the pipette.
For a cell modeled as an elastic body, its Young™s modulus E is determined by the
relation

2π L p
P= φ,
E (2.6)
3 Rp

where P is the pressure difference inside and outside the pipette, L p is the length
of the cell pulled into the pipette with radius R p , and φ is a geometric constant with
a value around 2.1 (Evans and Yeung, 1989).
For liquid-like ¬‚ow of cells at pressures exceeding the cortical tension, the cyto-
plasmic viscosity is calculated from the relation

Rp p
·= , (2.7)
dLp
m(1 ’ R p /R)
dt


where · is the viscosity, R is the diameter of the cell outside the pipette, and m is a
constant with a value around 9 (Evans and Yeung, 1989).
Experimental measurements of intracellular mechanics 27




Fig. 2-4. Experimental study of cell response to mechanical forces. Cells are deposited on the stage
of an inverted microscope equipped with a video camera. The video output is connected to a digitizer
mounted on a desk computer. Cells are aspirated into micropipettes connected to a syringe mounted
on a syringe holder. Pressure is monitored with a sensor connected to the computer. Pressure and time
values are superimposed on live cell images before recording on videotapes for delayed analysis.
From Richelme et al., 2000.




Fig. 2-5. Aspiration of a ¬‚accid (a) and swollen (b) red blood cell into a pipette. The diameter of the
¬‚accid cell is approximately 8 µm and that of the swollen cell is about 6 µm. The scale bars indicate
5 µm. From Hochmuth, 2000.
28 P. Janmey and C. Schmidt




Fig. 2-6. Diagram for a device for compression of a cell between microplates. Variations of this
design also allow for imposition of shear deformation. From Caille et al., 2002.



Shearing and compression between microplates
For cells that normally adhere to surfaces, an elegant but technically challenging
method to measure viscoelasticity is by attaching them at both top and bottom to glass
surfaces that can be moved with respect to each other in compression, extension, or
shear (Thoumine et al., 1999). A schematic diagram of such a system is shown in
Fig. 2-6.
In this method a cell such as a ¬broblast that adheres tightly to glass surfaces coated
with adhesion proteins such as ¬bronectin is grown on a relatively rigid plate; a second,
¬‚exible plate is then placed on the top surface. Piezo-driven motors displace the rigid
plate a known distance to determine the strain, and the de¬‚ection of the ¬‚exible
microplate provides a measure of the stress imposed on the cell surface. Use of this
device to provide well-de¬ned strains with simultaneous imaging of internal structures
such as the nucleus provides a measure of the elastic modulus of ¬broblasts around
1000 Pa, consistent with measurements by AFM, and has shown that the stiffness of
the nucleus is approximately ten times greater than that of the cytoplasmic protein
networks (Caille et al., 2002; Thoumine and Ott, 1997). A recent re¬nement of the
microcantilever apparatus allows a cell in suspension to be captured by both upper
and lower plates nearly simultaneously and to measure the forces exerted by the cell
as it begins to spread on the glass surfaces (Desprat et al., 2005).


Fluid ¬‚ow
Cells have to withstand direct mechanical deformations through contact with other
cells or the environment, but some cells are also regularly exposed to ¬‚uid stresses,
such as vascular endothelial cells in the circulating system or certain bone cells
(osteocytes) within the bone matrix. Cells sense these stresses and their responses
are crucial for many regulatory processes. For example, in vascular endothelial cells,
mechanosensing is believed to control the production of protective extracellular matrix
(Barbee et al., 1995; Weinbaum et al., 2003); whereas in bone, mechanosensing is
at the basis of bone repair and adaptive restructuring processes (Burger and Klein-
Nulend, 1999; Wolff, 1986). Osteocytes have been studied in vitro after extraction from
the bone matrix in parallel plate ¬‚ow chambers (Fig. 2-7). Monolayers of osteocytes
coated onto one of the chamber surfaces were exposed to shear stress while the
Experimental measurements of intracellular mechanics 29




Fig. 2-7. Fluid ¬‚ow system to stimulate mechanosensitive bone cells, consisting of a culture chamber
containing the cells, a pulse generator controlling the ¬‚uid ¬‚ow, and ¬‚ow meters. The response of
the cells is either biochemically measured from the cells after the application of ¬‚ow (for example
prostaglandin release) or measured in the medium after ¬‚owing over the cells (for example nitric
oxide). From Klein-Nulend et al., 2003.


response was measured by detecting the amount of nitric oxide produced as a function
of ¬‚uid ¬‚ow rate (Bacabac et al., 2002; Rubin and Lanyon, 1984).
The strain ¬eld within individual surface-attached cells in response to shear ¬‚ow
has been mapped in bovine vascular endothelial cells with the help of endogenous
¬‚uorescent vimentin (Helmke et al., 2003; Helmke et al., 2001). It was found that
the spatial distribution of strain is rather inhomogeneous, and that strain is focused to
localized areas within the cells. The method can only measure strain and not stress.
The sites for mechanosensing might be those where strain is large if some large
distortion of the sensing element is required to create a signal, in other words, if
the sensor is “soft.” On the other hand, the sites for sensing might also be those
where stress is focused and where little strain occurs if the sensing element re-
quires a small distortion, or is “hard,” and functions by having a relatively high force
threshold.
Numerical simulations can be applied to both the cell and the ¬‚uid passing over it.
A combination of ¬nite element analysis and computational ¬‚uid dynamics has been
used to model the ¬‚ow across the surface of an adhering cell and to calculate the shear
stresses in different spots on the cell (Barbee et al., 1995; Charras and Horton, 2002).
This analysis provides a distribution of stress given a real (to some resolution) cell
shape, but without knowing the material inhomogeneities inside, the material had to
be assumed to be linear elastic and isotropic. The method was also applied to model
stress and strain distributions inside cells that were manipulated by AFM, magnetic
bead pulling or twisting, and substrate stretching, and proved useful to compare the
effects of the various ways of mechanical distortion.
30 P. Janmey and C. Schmidt




Fig. 2-8. Schematic diagram of an optical trap.


Optical traps
Optical traps (see Fig. 2-8) use a laser beam focused through a high-numerical aper-
ture microscope objective lens to three-dimensionally trap micron-sized refractile
particles, usually silica or latex beads (Ashkin, 1997; Svoboda and Block, 1994). The
force acting on the bead at a certain distance from the laser focus is in general very
dif¬cult to calculate because (1) a high-NA laser focus is not well approximated by a
Gaussian, and (2) a micron-sized refractive particle will substantially affect the light
¬eld. Approximations are possible for both small particles (Rayleigh limit) and large
particles (ray optics limit) with respect to the laser wave length. For a small particle,
the force can be subdivided into a “gradient force” pulling the particle towards the
laser focus and a scattering force pushing it along the propagation direction of the
laser (Ashkin, 1992). Assuming a Gaussian focus and a particle much smaller than
the laser wavelength, the gradient forces in radial and axial direction are (Agayan
et al., 2002):
w0
4
2r 2 2r 2
1
∝ ’± I0 z 2 ’ · exp ’ 2
axial
Fg (2.8)
w 4 (z) w 6 (z) w (z)
z0

w02
2r 2
∝ ’± I0r 4 · exp ’ 2
radial
Fg (2.9)
w (z) w (z)
and the scattering forces:
r 2 z2 ’ z0
w0 w0
2
2 2
2r 2
∝ ± I0 2 km 1 ’ ’ · exp ’ 2
Fsaxial
w (z) z 0 w 2 (z) w (z)
2 z2 + z2 2
0
(2.10)

w0 km r
2
2r 2
Fsradial ∝ ± I0 exp ’ 2 (2.11)
w 2 (z) R (z) w (z)
Experimental measurements of intracellular mechanics 31

with complex polarizability ± = ± + i± , laser intensity I0 , w0 the beam radius in the
focus, and w(z) = w0 1 + (z/z 0 ) 2 the beam radius near the focus; z 0 = πw0 /»m the
2

Rayleigh range, km = 2π/»m the wave vector, with »m the wavelength in the medium
with refractive index n m . (For details and prefactors see Agayan et al., 2002).
Stable trapping will only occur if the gradient force wins over the trapping force all
around the focus. Trap stability thus depends on the geometry of the applied ¬eld and
on properties of the trapped particle and the surrounding medium. The forces generally
depend on particle size and the relative index of refraction n = n p /n m , where n p and
n m are the indices of the particle and the medium, respectively, which is hidden in
the polarizability ± in Eqs. 2.8“2.11. In the geometrical optics regime, maximal trap
strength is particle-size-independent, but increases with n over some intermediate
range until, at larger values of n, the scattering force exceeds the gradient force. The
scattering force on a nonabsorbing Rayleigh particle of diameter d is proportional
to its scattering cross-section, thus the scattering force scales with the square of the
polarizability (volume) (Jackson, 1975), or as d 6 . The gradient force scales linearly
with polarizability (volume), that is, it has a d 3 -dependence (Ashkin et al., 1986;
Harada and Asakura, 1996).
A trappable bead can then be attached to the surface of a cell and can be used to
deform the cell membrane locally. The method has the advantage that no mechanical
access to the cells is necessary. Using beads of micron size furthermore makes it
possible to choose the site to be probed on the cell with relatively high resolution.
A disadvantage is that the forces that can be exerted are dif¬cult to increase beyond
about 100 pN, orders of magnitude smaller than can be achieved with micropipettes
or AFM tips. At high laser powers, local heating may not be negligible (Peterman
et al., 2003a). Force and displacement can be detected, however, with great accuracy,
sub-nm for the displacement and sub-pN for the force, using interferometric methods
(Gittes and Schmidt, 1998; Pralle et al., 1999). This makes the method well suited
to study linear response parameters of cells. Interferometric detection can also be
as fast as 10 µs, opening up another dimension in the study of cell viscoelasticity.
Focusing on different frequency regimes should make it, for example, possible to
differentiate between active, motor-driven responses and passive viscoelasticity. Such
an application of optical tweezers is closely related to laser-based microrheology,
which can also be applied inside the cells (see Passive Microrheology). We will here

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