1171 Cellular Nano 35. Cellular Nanomechanics · blocks: the cytoskeleton, cell membrane, nucleus, adhesive complexes, and motor proteins. Next, the various methods used to probe

1171

Cellular Nano35. Cellular Nanomechanics

Roger Kamm, Jan Lammerding, Mohammad Mofrad

Numerous applications of nanotechnology havebeen developed to probe the unique mechani-cal properties of cells. In addition, since biologicalmaterials exhibit such a wide spectrum of prop-erties, they offer new concepts for nonbiologicalbiomimetic applications. In this chapter, the vis-coelastic properties of a cell and its subcellularcompartments are described. First, a qualita-tive picture is presented of the relevant buildingblocks: the cytoskeleton, cell membrane, nucleus,adhesive complexes, and motor proteins. Next, thevarious methods used to probe cellular and sub-cellular mechanics are described, and some of thequantitative results presented. These measure-ments are then discussed in the context of severaltheories and computational methods that havebeen proposed to help interpret the measurementsand provide nanomechanical insight into their ori-gin. Finally, current understanding is summarizedin the context of directions for future research.

35.1 Overview.............................................. 117135.1.1 The Importance of Cell Mechanics

in Biology and Medicine ............... 117135.1.2 Examples Drawn from Biology

and Pathophysiology .................... 1172

35.2 Structural Components of a Cell ............. 117335.2.1 Membranes ................................. 117335.2.2 Cytoskeleton ................................ 117435.2.3 Nucleus....................................... 117735.2.4 Cell Contractility and Motor Proteins117835.2.5 Adhesion Complexes ..................... 1179

35.3 Experimental Methods .......................... 117935.3.1 Methods of Force Application......... 117935.3.2 Rheological Properties .................. 118335.3.3 Active Force Generation ................ 118435.3.4 Biological Responses .................... 118435.3.5 Nonlinear Effects .......................... 118435.3.6Homogeneity and Anisotropy......... 1184

35.4 Theoreticaland Computational Descriptions ............ 118535.4.1 Continuum Models ....................... 118535.4.2 Biopolymer Models ....................... 118735.4.3 Cellular Solids .............................. 118735.4.4Tensegrity ................................... 1188

35.5 Mechanics of Subcellular Structures ........ 118835.5.1 Cell–Cell and Cell–Matrix Adhesions118835.5.2 Cell Membranes ........................... 119035.5.3 Cell Nuclei ................................... 119235.5.4Mechanosensing Proteins .............. 1194

35.6 Current Understanding and Future Needs1196

References .................................................. 1196

35.1 Overview

35.1.1 The Importance of Cell Mechanicsin Biology and Medicine

All living things, despite their profound diversity, sharea common architectural building block: the cell. Cellsare the basic functional units of life, yet they are them-selves comprised of numerous components with distinctmechanical characteristics. To perform their variousfunctions, cells undergo or control a large range of intra-

and extracellular events, many of which involve me-chanical phenomena or may be guided by the forcesexperienced by the cell. These micro- and nanomechan-ical phenomena range from macroscopic events such asthe maintenance of cell shape, motility, adhesion, anddeformation, to microscopic events such as how cellssense mechanical signals and transduce them into a cas-cade of biochemical signals, ultimately leading to a hostof biological responses.

PartD

35

1172 Part D Bio-/Nanotribology and Bio-/Nanomechanics

Cell mechanics plays a major role in biology andphysiology. The ability of a cell to perform its func-tion often depends on its shape, and shape is maintainedthrough structural stiffness. In the blood circulation sys-tem, red blood cells, or erythrocytes, exist in the formof biconcave disks that are easily deformed to helpfacilitate their flow through the microcirculation andhave a relatively large surface-to-area ratio to enhancegas exchange. White cells, or leucocytes, are spherical,enabling them to roll along the vascular endotheliumbefore adhering and migrating into the tissue. Becausetheir diameter is larger than some of the capillariesthey pass through, leucocytes maintain excess mem-brane in the form of microvilli so that they can elongateat constant volume and not obstruct the microcircula-tion. Airway epithelial cells are covered with a bed ofcilia, finger-like cell extensions that propel mucus alongthe airways of the lung. Lastly, the cytoskeletal struc-ture in muscle cells is specifically organized to activelygenerate forces and to sustain large strains. In each ofthese examples, the internal structure of the cell alongwith the cell membrane provide the structural integrityto maintain the particular shape needed by the cell to ac-complish its function, although the specific componentsof the structure are highly variable and diverse.

Cell mechanics also plays an important role incell migration. Migration is critical during early de-velopment, but also in fully differentiated organism,e.g., in wound repair when cells from the surrounding,undamaged tissue migrate into the wound, in angiogen-esis (i. e., the generation of new blood vessels), or incombating infection when cells of the immune systemtransmigrate from the vascular system across the vesselwall and into the infected tissues.

Cell migration processes occurs in several stagesthat include:

1. Protrusion, the extension of the cell at the leadingedge in the direction of movement

2. Adhesion of the protrusion to the surrounding sub-strate or matrix

3. Contraction of the cell that transmits a force fromthese protrusions at the leading edge to the cellbody, pulling it forward

4. Release of the attachments at the rear, allowing netforward movement of the cell to occur [35.1–4]

Cellular mechanics and dynamics are critical mod-ulators at each of these steps.

The importance of cell mechanics in biology ismost apparent in mechanotransduction, i. e., the abilityof the cell to sense and respond to externally applied

forces. Most cells are able to sense when a physi-cal force is applied to the cell, and respond througha variety of biological pathways leading to such di-verse effects as changes in membrane channel activity,up- or downregulation of gene expression, alterationsin protein synthesis, and altered cell morphology. Thesignaling cascades that become activated as a conse-quence of mechanical stress have generally been wellcharacterized. However, the initiating process by whichcells convert the applied force into a biochemical sig-nal is much more poorly understood, and only recentlyhave researchers begun to unravel some of these funda-mental mechanisms. Some studies have suggested thata change in membrane fluidity acts to increase recep-tor mobility, leading to enhanced receptor clusteringand signal initiation. Stretch-activated ion channels orstrain-induced activation of G proteins represent othermeans of mechanotransduction. Similarly, disruption ofmicrotubules or conformational changes of cytoskele-tal proteins that alter their binding affinities have beenproposed as cellular mechanosensors. Yet others havefocused on the role of the glycocalyx, a layer ofcarbohydrate-rich proteins on the cell surface, in theresponse of endothelial cells to fluid shear stress. An-other potential mechanism is that forced deformationswithin the nucleus could directly alter transcriptionor transcription factor accessibility. Constrained au-tocrine signaling is yet another mechanism whereby thestrength of autocrine signaling is regulated by changesin the volume of extracellular compartments into whichthe receptor ligands are shed. Changing this volumeby mechanical deformation of the tissues can increasethe level of autocrine signaling [35.5]. Finally, othershave proposed conformational changes in intracellularproteins in the force transmission pathway connectingthe extracellular matrix with the cytoskeleton throughfocal adhesions (FAs) as the main mechanotransduc-tion mechanism. While all or a subset of these theoriesmay contribute to mechanotransduction, little direct ev-idence has been presented in their support. For reviewsof this topic, see [35.6–12].

35.1.2 Examples Drawn from Biologyand Pathophysiology

Many studies during the past two decades have shedlight on a wide range of cellular responses to mechan-ical stimulation. It is now widely accepted that stressesexperienced in vivo are instrumental in a wide spec-trum of pathologies. One of the first diseases found tobe linked to cellular stress was atherosclerosis, where

PartD

35.1

Cellular Nanomechanics 35.2 Structural Components of a Cell 1173

it was demonstrated that hemodynamic shear influ-ences endothelial function, and that conditions of lowor oscillatory shear stress are conducive to the forma-tion and growth of atherosclerotic lesions. Even beforethen, the role of mechanical stress on bone growthand healing was widely recognized, and since then,many other stress-influenced cell functions have beenidentified. Mechanotransduction of muscle cells is piv-otal to exercise-induced muscle growth (hypertrophy),

and defects – caused for example by mutations inmechanosensitive proteins – can result in muscular dys-trophies. Asthma is another particularly salient examplewhere the epithelial cells lining the airways are sub-jected to stresses as the pulmonary airways constrictduring breathing, and airway wall remodeling. Furtherknowledge of the mechanisms by which cells respondto such forces could enhance our understanding of thesediseases.

35.2 Structural Components of a Cell

Aside from providing the outer boundary of a celland nucleus, membranes enclose numerous intracellu-lar structures as well, and the discussion in this sectionpertains to all of these.

35.2.1 Membranes

The term membrane generally refers to the phospholipidbilayer and the proteins associated with it. The phospho-lipids contained in the membrane are arranged in twolayers, or leaflets, with their hydrophobic tails pointinginward and their hydrophilic heads outward. Together,they constitute a bilayer ≈ 6 nm in thickness (Fig. 35.1).

Outside

Inside

Oligosaccharide

Integral proteinPeripheral proteins

Leaflets

Hydrophilic polar headsFatty acid tails

Fig. 35.1 A model of the lipid bilayer, showing the hy-drophilic heads (polar head groups) on the exterior surfacesand the hydrophobic tails (fatty acid tails) pointing in. Alsoshown are examples of transmembrane or integral mem-brane proteins. The total thickness of the bilayer is ≈ 6 nm(after [35.13])

In addition to the phospholipids, the membrane con-tains glycolipids and cholesterol. While the amountof glycolipids is small, constituting only ≈ 2% of thetotal lipid content, cholesterol is a major membraneconstituent, ≈ 20% by weight, a value that remainsquite constant among different cell types. The specificmembrane composition is critical for determining mem-brane structural integrity; for example, both the bendingstiffness and the viscosity of the lipid bilayer arestrongly dependent on the cholesterol content. Cellu-lar membranes also contain many membrane-associatedproteins, which account for ≈ 50% of the membraneby weight but, because of their relatively large molecu-lar weight, only ≈ 1–2% of the number of membranemolecules. Membrane proteins serve a variety of func-tions, ranging from signaling to transport of ions andother molecules across the membrane to cell–cell andcell–matrix adhesion.

The plasma membrane has associated macromolec-ular structures on both intra- and extracellular sides,giving rise to a three-layer composite construction. Onthe intracellular side, the membrane is physically at-tached to a cortex or the cytoskeleton. The cortex isa dense, filamentous structure that lends stiffness to themembrane, and can also interact with various trans-membrane proteins, often impeding their free diffusioneither by steric interactions or direct chemical bond-ing. In some cells, the cortex is simply a region ofdense cytoskeletal matrix in the vicinity of the bilayer.In others, it exhibits a distinctly different structure orcomposition; for example, erythrocytes possess a cortexcomprised of a network of spectrin tetramers linked byactin filaments. This network is attached to the mem-brane by ankyrin and the integral membrane proteinband 3. This spectrin network accounts for much of thebending stiffness exhibited by the red cell membrane.

PartD

35.2


Most cells are coated by a glycocalyx, which hasbeen shown to extend as far as 0.5 μm from thesurface of endothelial cells, where it forms a com-pressible barrier separating circulating erythrocytes andleucocytes from the endothelial membrane. The gly-cocalyx is comprised of short oligosaccharide chains,glycoproteins, glycolipids, and high-molecular-weightproteoglycans, all organized into an interconnected net-work with an overall negative charge. Although itsfunction is not completely understood, it apparentlyplays a role in macromolecular transport across the en-dothelium and is an important factor in the interactionbetween bloodborne cells and the endothelium. Stud-ies have demonstrated that the glycocalyx in a capillaryis readily compressed by a passing leukocyte, yet issufficiently rigid to prevent flowing erythrocytes fromapproaching the endothelial surface.

35.2.2 Cytoskeleton

The cytoskeleton is the network of biopolymers thatpermeates the cell and largely account for its structuralintegrity. At high magnification, this network appears tobe comprised of several distinct types of intertwined fil-aments with a variety of interconnections, as can be seenin the micrograph in Fig. 35.2. The apparent stiffnessof the network, as that of other fibrous materials, de-pends fundamentally upon the elastic properties of theconstituent fibers.

The cytoskeletal matrix is primarily comprised ofthree constituents, actin microfilaments (≈ 7–9 nm indiameter), microtubules (24 nm), and intermediate fila-ments (≈ 10 nm) (Table 35.1). These form a complexinterconnected network that exists in a state of constantflux, especially when the cell is dividing, migrating orundergoing other dynamic processes. All are polymersbuilt from protein subunits, held together by noncova-lent bonds. They also share the common feature thatthey cross-link, often with the aid of other proteins,

Table 35.1 The main constituents of the cytoskeleton and their mechanical properties. Note that the effective Young’smodulus, estimated using the diameter and bending stiffness values for actin filaments and microtubules, is approximatelythe value that would be predicted on the basis of van der Waals attraction between two surfaces [35.14]. Recall that thepersistence length and bending stiffness are related through the expression lp = Kb/kBT , and that Kb = EI = π/4a4 Efor a rod of circular cross-section with radius a

Diameter (2a) Persistence length lp Bending stiffness Kb Young’s modulus E

(nm) (μm) (N m2) (Pa)

Actin filaments 9–10 15 7 × 10−26 1.3–2.5 × 109

Microtubules 25 6000 2.6 × 10−23 1.9 × 109

Intermediate filaments 10 1–3 4–12 × 10−27 1–3 × 109

to form bundles and lattice networks. The higher-orderstructures formed by these polymers, as we will see, arecritical determinants of cytoskeletal elasticity and canvary substantially between different cells.

Actin MicrofilamentsActin filaments play an essential role in virtually alltypes of motility. Actin–myosin interactions are of ob-vious importance in muscle, but are also instrumental inthe migration and movement of nonmuscle cells. Actinpolymerization is thought to be one of the factors initi-ating cell migration through the formation of filopodiaor lamellipodia [35.15].

Actin is one of the most prevalent proteins foundin the cell, ranging in concentration from 1–10% byweight of total cell protein in nonmuscle cells, to10–20% in muscle. Molecular actin is comprised of375 amino acids (molecular weight 43 kDa) and isfound in at least six forms that differ from each otheronly slightly. Of these, four are found in muscle, theother two in nonmuscle cells. Actin exists either inglobular form (G-actin monomers) or filamentous form

Fig. 35.2 The cytoskeleton of a macrophage lamel-lipodium as seen by electron microscopy. The fibrousstructure is mainly comprised of actin filaments ( c© JohnHartiwick, http://expmed.bwh.harvard.edu)

PartD

35.2


(F-actin polymer), with the balance between the two be-ing a highly dynamic process that is finely regulated bya variety of different factors. F-actin is a long, flexiblefilament, ≈ 9–10 nm in diameter. Subunits (monomers)are organized into a double-stranded helix having struc-tural and functional polarity (pointed or negative, andbarbed or positive ends) and a half-pitch of ≈ 37 nm(Fig. 35.3a). At the barbed end, an adenosine triphos-phate (ATP) binding cleft is exposed, allowing forbinding of monomer and linear growth of the filament.ATP binding and hydrolysis play a critical role in regu-lating actin dynamics and controlling the length of theactin filament.

The F-actin filaments can further organize into qua-ternary structures such as bundles or a lattice networkwith the aid of actin binding proteins (ABP). The bun-dles, also referred to as stress fibers, are closely packedparallel arrays of filaments, connected to each other byseveral members of the ABP family (e.g., α-actinin,fascin, and scruin). Stress fibers can vary in size, butare typically several hundred nanometers in diameter.Actin stress fibers tend to form when the cell requiresadditional strength, such as in endothelial cells, in re-sponse to an elevated shear stress, or in migratingfibroblasts. These fibers often concentrate around andattach to focal adhesion sites, and are therefore criticalto cell adhesion. Actin networks consist of an inter-connected matrix of F-actin filaments, the junctions ofwhich are often seen to be nearly orthogonal. At leasttwo distinct types of network are observed – corti-cal (membrane-associated and more planar in nature)and non-membrane-associated, which possess a moreisotropic three-dimensional structure. The formation ofbundles and networks is facilitated by a variety ofcross-linking proteins such as filamin, which forms a V-shaped polymer that connects two actin filaments nearlyat right angles.

The elastic properties of actin filaments have beenmeasured in a variety of ways: by axial stretch [35.16],twisting [35.17, 18], and bending [35.19]. By allmethods, single actin filaments were found to ex-hibit a Young’s modulus in the range of (1.3–2.5) ×109 N/m2. This range compares favorably with thatmeasured for silk and collagen [35.20] and is alsoroughly consistent with predictions based on van derWaals bonding between surfaces [35.14].

Intermediate FilamentsIntermediate filaments, which form ≈ 10 nm-diameterfibers, are much less studied than actin and not as wellcharacterized. Perhaps this is due to the fact that there

is no single molecular constituent that comprises the in-termediate filaments (IFs); instead, there are more than50 different IF genes that have been identified.

Intermediate filaments constitute ≈ 1% of total pro-tein in most cells, but can account for up to 85%in cells such as epidermal keratinocytes and neu-rons [35.21]. While they come in many varieties, theyshare a common structural organization. All have a largecentral α-helical rod domain flanked by amino- andcarboxy-terminal domains. Assembly occurs by the for-mation of dimers into a coiled coil structure. Thenthe dimers assemble in a staggered antiparallel ar-ray to form tetramers that connect end-to-end to formapolar protofilaments. These protofilaments assembleinto a rope-like structure containing ≈ 8 subunits each(Fig. 35.3c) [35.21]. Although intermediate filamentsare more stable than microfilaments, they can be modi-fied by phosphorylation. Intermediate filaments exhibita lower bending stiffness than either microfilaments ormicrotubules, as evidenced by their persistence lengthof only 1–3 μm. Linker proteins such as bullous pem-phigoid antigen 1 (BPAG1) and plectin contain bothactin and IF binding domains, providing a means bywhich these networks can be linked. Evidence also ex-ists for plectin binding to microtubules.

Intermediate filaments are often found surroundingthe nucleus and extending outward to the plasma mem-brane. In epithelial cells, keratin filaments connect tothe plasma membrane at desmosomes and hemidesmo-somes and help them to withstand mechanical stress.

MicrotubulesMicrotubules are important in determining cell shape,and they play a critical role in separating chromosomesduring mitosis. Microtubules are central to the motionof cilia and flagella. Compared with either microfila-ments or intermediate filaments, microtubules are rigidstructures, but exist in a dynamic equilibrium, much asdo microfilaments. Microtubules take the form of hol-low cylinders with ≈ 25 nm outer diameter and 14 nminner diameter. The tubular structures are comprisedof tubulin, a globular dimer consisting of two 55 kDapolypeptides, α- and β-tubulin. The dimers polymerizeto form microtubules that consist of 13 linear protofil-aments forming a hollow-cored cylinder (Fig. 35.3b).The filaments are polar, having a rapidly growing endand a slowly growing end, mediated by hydrolysis ofguanosine triphosphate (GTP) after polymerization. Ifhydrolysis occurs too quickly, before new GTP-boundtubulin can bind to the end, the microtubule might dis-assemble. Depending on the rate of hydrolysis and rate

PartD

35.2


14 nm 25 nm

a)

c)

b)

Depolymerization

G-actin

F-actin

Polymerization

Polypeptide N C

N

N

C

NC

C

C N

10 nm

Head

Coiled-coil

Tail

Dimer

Tetramer

Protofilament

Filament

Negative orpointed end

Positive orbarbed end

7–9 nm

α-Tubulin

�-Tubulin

Fig. 35.3 (a) Schematic showing the polymerization of G-actin monomers to form F-actin. (b) α- and β-tubulin organizedinto microtubules. (c) Organizational structure of an intermediate filament

of GDP-bound tubulin addition to the end, the micro-tubule can either grow or shrink [35.22]. In fact, free

ends tend to alternate between periods of steady growthand disassembly in a stochastic manner.

PartD

35.2


During interphase, microtubules tend to be anchoredat their negative ends to the centrosome, located near thenucleus. From there they extend to all parts of the cell,suggesting a strong role in maintaining the structuralintegrity of the cell and providing a cellular highwaysystem for cargo transported by motor proteins such asthe kinesins and dyneins. In many of these situations,the microtubule is thought to play a structural role thatrelies on it having a high bending stiffness, which wasfound to be on the order of 2.6 × 10−23 N m2 [35.20].

35.2.3 Nucleus

The nucleus is the distinguishing feature of eukary-otic cells and directs and controls deoxyribonucleic acid(DNA) replication, ribonucleic acid (RNA) transcrip-tion and processing, and ribosome assembly [35.23].Most eukaryotic cells contain a single nucleus. How-ever, some specialized cells (e.g., red blood cells)become anucleate during their maturation, while othercells such as skeletal and cardiac muscle cells can be-come multinucleated due to cell fusion. With a diameterin the range ≈ 5–20 μm, the nucleus is the largestcellular organelle. The nucleus is separated from the cy-toplasm by the nuclear envelope, which consists of twolipid bilayers, the inner and outer nuclear membrane,and the underlying nuclear lamina, a dense protein net-work consisting mostly of lamin proteins that controlthe nuclear shape, size, and stability (Fig. 35.4). Theouter nuclear membrane is continuous with the endo-plasmic reticulum and connects to the inner nuclearmembrane at the nuclear pores, thus enclosing the per-inuclear space. Nuclei typically contain a few thousandnuclear pores, comprised of hundreds of proteins thatform the nuclear pore complex that controls transportbetween the nucleus and the cytoplasm [35.24].

The nuclear interior contains the packaged DNAin the form of chromatin as well as diverse intranu-clear compartments referred to as subnuclear bodiesthat include nucleoli, Cajal bodies, and promyelocyticleukemia bodies (PML) bodies. These subnuclear bod-ies are not surrounded by membranes but self-organizethrough processes only incompletely understood. Chro-matin can be organized into two distinct forms, thedense and transcriptionally silent heterochromatin oftenlocated at the nuclear periphery, and the gene-rich andtranscriptionally active euchromatin. Chromatin is com-prised of 30 nm fibers that arise from regular wrappingof DNA around histone octamers to form nucleosomesresembling beads on a string and subsequent furthercompaction facilitated through the linker histone H1.

This process allows the ≈ 2 m of human DNA to bepackaged into a nucleus a few micrometers in diam-eter. The chromatin fibers in turn form higher-orderstructures such as euchromatin and heterochromatin,which can be dynamically regulated by biochemicalmodification (e.g., by methylation, acetylation or phos-phorylation) of histone proteins and DNA. Furthermore,chromatin fibers from single chromosomes often formdistinct and nonoverlapping chromosome territories. Inaddition to chromatin and nuclear bodies, the nuclearinterior also contains several structural proteins, includ-ing nuclear actins, myosin, spectrin, and nucleoplasmiclamins A and C [35.25–29]. However, it remains un-clear what intranuclear structures (often referred to asnuclear matrix) these proteins form within the nuclearinterior and how such structures could contribute to nu-clear processes such as transcription.

Importantly, the nucleus cannot be viewed in isola-tion from the surrounding cytoskeleton. The molecularmechanism by which the nucleus is connect to thecytoskeleton has puzzled researchers for many years,as it was unclear how forces required for nuclear po-sitioning and anchoring could be transmitted acrossthe 50 nm-wide perinuclear space. Recent work inCaenorhabditis elegans, Drosophila melanogaster, andmammalian cells led to the discovery of two new fami-

Cajal body

PML body

Nucleolus

Euchro-matin

Endoplasmicreticulum

Outer nuclear membraneInner nuclear membrane

Heterochromatin

Nuclear pore

Nuclear lamina

Fig. 35.4 Schematic drawing of a mammalian cell nucleus. The nu-cleus is surrounded by the inner and outer nuclear membranes andthe underlying nuclear lamina. Nuclear pores allow transport be-tween the nuclear interior and the cytoplasm. The nuclear interiorconsists of densely packed heterochromatin, mostly located at thenuclear periphery, the transcriptionally active euchromatin, and sev-eral intranuclear structures such as Cajal and PML bodies and thenucleoli. Not shown are intranuclear structures formed by laminsand other proteins that constitute the nuclear matrix

PartD

35.2


lies of nuclear envelope proteins that are ideally suitedto transmit forces from the cytoskeleton across the nu-clear envelope to the nuclear interior [35.30–39]. Thesefindings have led to the current model of nuclear–cytoskeletal coupling (Fig. 35.5), in which large nesprinisoforms located on the outer nuclear membrane can

Plectin

Actin filament

Intermediatefilament

Nesprin 1/2 Nuclear porecomplex

Lamins

Chromatin

Nesprin 3

SUN 1/2

Fig. 35.5 Nuclear cytoskeletal coupling. Schematic drawing of thecurrent model of nuclear cytoskeletal coupling in mammalian cells.The nesprin 1 and nesprin 2 giant isoforms contain an N-terminalactin-binding domain that can interact with cytoskeletal actin fil-aments. Shorter isoforms of nesprin 1 and 2 do not contain anactin-binding domain, but it is thought that the large spectrin repeatregions (brown and grey spheres) can interact with other cytoskele-tal elements. Nesprin 3 can directly bind to plectin, which can as-sociate with intermediate filaments. Nesprins can interacts with theinner nuclear membrane proteins SUN1 and SUN2 across the per-inuclear space through their C-terminal KASH (Klarsicht, ANC-1,Syne Homology)-domain. SUN proteins can in turn bind to the nu-clear lamina, nuclear pore complexes, and possibly other, yet to beidentified, proteins at the inner nuclear membrane. These proteinsinteract with chromatin and intranuclear proteins, thus completingthe physical link between the cytoskeleton and the nuclear interior

bind to cytoskeletal F-actin and intermediate filaments.At the same time, nesprins physically interact across theperinuclear space with Sad1p/UNC-84 (SUN) proteins,which are located at the inner nuclear membrane. There,SUN proteins can bind to lamins, chromatin, and otheras-yet unknown nuclear envelope proteins, thus creat-ing a physical link between the cytoskeleton and thenucleus [35.32]. Due to this intricate coupling betweenthe nucleus and the cytoskeleton, defects in nuclear en-velope proteins can have direct effects on cytoskeletalstructure and mechanics. For example, fibroblasts lack-ing the nuclear envelope proteins lamin A and C havereduced cytoskeletal stiffness and disturbed actin, vi-mentin, and microtubule organization [35.40–42], andmutations in nesprins, similar to lamins, can result inmuscular dystrophies [35.43].

35.2.4 Cell Contractility and Motor Proteins

All muscle cells use the molecular motor comprised ofactin and myosin to produce active contraction. Theseare arranged in a well-defined structure, the sarcom-ere, and the regularity of the sarcomeres gives rise tothe characteristic striated pattern seen in skeletal musclecells and cardiac myocytes. Importantly, even nonmus-cle cells contain contractile machinery, which they usefor a variety of functions such as maintaining celltension, changing cell shape, and cell migration. Promi-nent bundles of actin filaments, called stress fibers, arecontractile and house myosin filaments. The macro-molecular organization of stress fibers bares similaritiesto sarcomeres – actin filaments are in parallel arrange-ment with the filament polarity alternating alongside;laterally a space of ≈ 10 nm is maintained that situ-ates parallel bipolar myosin II filaments in between.The myosin head moves toward the positive pole ofthe actin filaments during a power stroke, producinga net contractility for an alternating-polarity arrange-ment. Stress fibers (like sarcomeres) have a troupe ofproteins, e.g., α-actinin, filamin, troponin, caldesmon,and tropomyosin, which control the arrangement ofactin and myosin, and regulate their interaction. The for-mation and strengthening of stress fibers is intricatelylinked to the mechanism by which cells respond to theirforce environs. First, stress fibers are known to formbetween points in the cell where actin myosin contrac-tility is resisted. Typically these are hot-spots of proteinactivity known as focal adhesions, where the actin cy-toskeleton gets anchored to transmembrane integrins,which are anchored to the matrix proteins. The focaladhesion also includes a cascade of proteins that relay

PartD

35.2

Cellular Nanomechanics 35.3 Experimental Methods 1179

biochemical signals, regulate its strengthening, and or-ganize the cytoskeleton for the growth of stress fibers(like cross-linking actin and recruiting myosin).

35.2.5 Adhesion Complexes

Adhesion complexes are collections of proteins forminga physical linkage between cytoskeleton and extracel-lular matrix (ECM) and between cells. A cell can usefocal adhesions to gain traction on the ECM during theprocess of spreading and migration. The assembly offocal adhesions is a dynamical process that is closelyregulated by the mechanical and chemical cues thatthe cell experiences. Both intracellular and extracellularmechanical stresses transmitted through focal adhesionsare important in the formation of a focal adhesion com-plex, whereas the release of stress results in the turnoverof a focal adhesion. The myosin-mediated contractile

force transmitted to the ECM and the tension applied tothe adhesion complex is necessary for promoting focaladhesion development. In contrast, disruption of myosinactivity effectively inhibits the formation of focal ad-hesions. In the absence of myosin contractile force,externally applied mechanical forces can also promotethe formation of focal adhesions. This force-regulatedfocal adhesion assembly allows a cell to probe themechanical stiffness of its surroundings and respond ac-cordingly, for example, by migrating in the directionof increasing substrate stiffness. Adhesion complexesalso play an important role in helping tissues form byholding the cells together. The cells of most tissuesare bound directly via cell–cell junctions. Cell–cell ad-hesion complexes are found in many different types,depending on tissue and cell type, and serve in bothmechanical coupling of cells as well as intercellulartransport.

35.3 Experimental Methods

35.3.1 Methods of Force Application

Measuring cellular or subcellular biomechanics oftenrequires the application of precisely controlled forcesto single or multiple cells and quantification of the in-duced deformation, although some techniques rely ondetecting forces generated by the cells or on observ-ing particles within the cytoplasm subjected to thermalmotion. Consequently, experimental methods can be di-vided into active (Table 35.2, Fig. 35.6) and passivetechniques. In the active techniques, applied forces aregenerally in the same range as physiological forces act-ing at the cellular and molecular level (Table 35.3), andinduced displacements are on the nanometer or microm-eter scale. For most methods – whether they are activeor passive – the cellular deformations are detected basedon computer-based image analysis of bright-field orfluorescence microscopy images, but some techniquesalso apply quadrant photodiode detectors for faster andhigher-resolution tracking of microspheres embedded inthe cytoplasm.

Active MeasurementsAtomic Force Microscopy. Atomic force microscopy(AFM) – only developed in 1986 – is now routinelyused in cell biology to image cells, measure cytoskele-tal stiffness, and quantify single-molecule interactions.In an atomic force microscope, a small tip attached to

Table 35.2 Methods of force application

Method Typical force range

Atomic force microscopy 10 pN–100 nN

Microindenter 1–100 nN

Microplate stretcher 1–100 nN

Magnetic bead microrheology 10 pN–1 nN

(twisting)

Magnetic bead microrheology 100 pN–10 nN

(pulling)

Optical traps 1–500 pN

Micropipette aspiration 1–100 nN

Substrate strain 1–30% strain

Shear flow 1–100 Pa

MEMS devices 0.5–1500 nN

Table 35.3 Typical force ranges in cell biology

Biological force Force range

Force generated by motor ≈ 1–10 pN

proteins (e.g., kinesin, myosin)

Force transmitted ≈ 1–200 pN

by protein–protein interactions (rate dependent)

Force required for (partial) protein ≈ 100 pN

unfolding

Force generated by migrating ≈ 1 nN–10 μN

or contracting cells

PartD

35.3


a) f)

b) g)

c) h)

d) i)

e) j)

Fig. 35.6a–j Experimental methods for cellular force ap-plication. Overview of techniques to apply preciselycontrolled forces or deformations to cells for active cellularbiomechanics measurements: (a) atomic force microscopy,(b) microindentation or cell poking, (c) parallel microplatestretcher, (d) pulling magnetic bead microrheology (single-pole magnetic trap), (e) optical trap, (f) micropipetteaspiration of adherent cell, (g) substrate strain experi-ments, (h) flow chamber for fluid shear-stress application,(i) micromachined device, in which the cell is plated ontomicroscopic platforms that are then moved apart, (j) cellplated on micropillars that deform (i. e., deflect) under thecellular traction force. This passive technique can be mod-ified by embedding small magnetic particles in some of thepillars, which can then be actively manipulated to applya highly localized force onto the basal cell surface

a flexible cantilever is controlled with (sub-)nanometerprecision to carefully probe (i. e., indent) the cell sur-face. Deflection of the cantilever can then be used toinfer the indentation depth and calculate the appliedforce. The tip of the cantilever typically has the shapeof a pyramid (with a tip radius of ≈ 20 nm), but someapplications use small polystyrene beads (0.1–2 μmdiameter) attached to the cantilever or tipless probesto provide a larger contact area with the cell. The me-

chanical properties of the probed cell (e.g., Young’smodulus or shear modulus) are inferred from the force–indentation curves, often assuming a Hertz model forlinear elastic, isotropic material [35.44], although sev-eral modifications have been proposed to account forexample for the finite thickness of thin cell extensionssuch as lamellipodia [35.45]. These theoretical modelsoften provide a surprisingly good fit to the experimen-tal data, despite the fact that cells are nonisotropic,nonhomogeneous, and nonlinear elastic materials. Onebenefit of the atomic force microscope is that the us-able force range spans several orders of magnitude(Table 35.2), depending on the spring constant of the se-lected cantilever. Therefore, AFM can also be used forsingle-molecule measurements, for example, to mea-sure bond strength. In this case, the AFM probe isfunctionalized with low concentrations of the protein ofinterest and then brought into contact with the appro-priate binding partner immobilized on a rigid surface.After binding occurs, the AFM tip is then carefully re-tracted until the molecules dissociate, which results ina sudden drop of force. Rupture forces at different ve-locities can be directly inferred from AFM data.

Microindenter and Microplate Stretcher. Thesecustom-made devices are closely related to the AFMprinciple. In the case of the microindenter, a small(≈ 10–100 μm-diameter) probe is used to poke singlecells while measuring induced cytoskeletal and nucleardeformations under a fluorescence microscope. The ap-plied force is measured with a sensitive force transducerattached to the indenter. For the microplate stretcher,cells are placed between a rigid, piezo-controlled plateand a thin, flexible plate and allowed to adhere to bothplates. The slides are then slowly moved apart, thusstretching the cell between the plates while imaging theexperiment under a microscope. Here, the induced de-flection of the thin plate is used to infer the appliedforce. The forces that can be achieved with these toolsare higher than that of a typical atomic force microscopeand are sufficient to induce significant deformations ofan entire cell.

Magnetic Twisting Cytometry. In this technique, me-chanical measurements are based on the displacementsof small (≈ 0.2–5 μm diameter), ferromagnetic orparamagnetic beads attached to the cell surface andsubjected to a magnetic force. Early versions of thistechnique [35.46, 47] used two orthogonal magneticfields, one to magnetize the ferromagnetic particles witha brief, intense pulse, and the second one to induce

PartD

35.3


a twisting, magnetic torque to the beads. The inducedbead rotation is then measured by a change in theorientation of the induced magnetic field or through mi-croscopic observations. The latter approach offers theadvantage that it can detect rotation and displacementof individual beads, so that loosely attached beads thatmight rotate freely can be excluded from the overallmeasurements. In a related variation of this technique,often referred to as magnetic trap or tweezers, paramag-netic beads on the cell surface or inside the cytoplasmare manipulated by a single- or multipole electromag-net controlled through a computer. The force acting ona paramagnetic bead inside a magnetic field is given bythe equation F = μ0χV∇(H · H), where F is the mag-netic force, μ0 is the permeability constant, χ is thevolume susceptibility, V is the bead volume, and H isthe external magnetic field strength. Thus, the appliedforce increases with increasing field strength and in-creasing field gradient. For a single-pole magnetic trap,the magnetic field decays rapidly with increasing dis-tance from the tip of the pole. This results in a steepgradient and large forces near the tip that exponentiallydecay away from the tip, requiring careful calibrationof the magnetic trap. One of the limitations of single-pole magnetic traps is the unidirectional force direction,i. e., forces can only be exerted in the direction towardsthe magnetic trap (pulling), but this can be overcomeby using multipole magnetic traps. Another limitation isthat the bead localization on the cell is random, so caremust be taken to only compare results from cells withsimilar bead positions (e.g., on the nucleus, the nuclearperiphery or the lamellipodia). Lastly, the induced beadrotation and displacement are strongly dependent onthe bead attachment angle, i. e., how deep the magneticbead is embedded in the cell surface, requiring carefulcontrols or confirmation by confocal three-dimensional(3-D) reconstruction.

Optical Traps/Tweezers. This technique is similar tothe magnetic trap experiments, as the induced displace-ment of microscopic beads (50–1000 nm) attached tothe cell surface or inside the cytoplasm subjected toa precisely controlled force is used to determine themechanical properties of the cytoskeleton. The majordifference is that, in an optical trap, a focused laserbeam is used to position and displace a bead with highrefractive index on the cell, allowing precise bead ma-nipulation in all directions. The optical trap acts as anelastic spring with a tunable spring constant, so theforce applied to the bead can also be controlled withhigh precision (Chap. 32). Another advantage of the

optical trap system is that multiple beads can be inde-pendently controlled by splitting the laser beam, so thatcells can, for example, be stretched between two beadsthat are slowly moved apart. The biggest limitation ofusing optical traps in cell mechanics experiments is thatthe maximal force level is limited to < 1 nN, as largerforces would require higher laser power that could ex-cessively heat the cell. While the small forces generatedby an optical trap are ideally suited for single-moleculestudies and are sufficient for experiments that measure,for example, membrane tethers, they are often too smallto induce large-scale cellular deformations, especiallywhen probing stiffer cells such as myocytes.

Micropipette Aspiration. In these experiments, sin-gle adherent or suspended cells are partially aspiratedinto a micropipette with a ≈ 2–10 μm-diameter open-ing by applying precisely controlled suction pressure(typically 100–10 000 Pa). The aspirated cell is imagedon a microscope and cellular deformations such as theaspirated tongue lengths are computed from the cellgeometry. In a technique called fluorescent/confocal-imaged microdeformation, fluorescent labeling of spe-cific intracellular components such as the nucleus, thenuclear lamina or nucleoli is used to provide additionalinformation on the subcellular deformations duringmicropipette aspiration [35.48–51]. One potential limi-tation of the micropipette aspiration technique is that theinterpretation of the experiments is not always straight-forward. Analytical or computational models are oftennecessary to derive material properties from the ge-ometric measurements of the aspirated cells and theapplied pressure, and the underlying assumptions mayat times be difficult to validate. However, this techniquehas been proven very useful when studying cells withrelatively homogeneous structural organization such asred blood cells or neutrophils [35.49, 52–54].

Substrate Strain. Unlike the other techniques, these ex-periments do not apply controlled forces to single cells,but instead use carefully controlled strain applicationto induce deformation in cells plated on a flexible sub-strate. Generally, cells are plated on transparent, elasticsilicone membranes coated with extracellular matrixproteins and subjected to uniaxial or biaxial strain. De-pending on the particular experiment, the strain can beheld constant or varied over time (e.g., cyclic strain ap-plication). Strain levels normally do not exceed 30% inorder to avoid damaging the cells, but the exact levelsare cell-type dependent. Cells are imaged under a mi-croscope before, during, and after strain application,

PartD

35.3


and small markers or fluorescently labeled componentsof the cells are used to calculate intracellular deforma-tions and applied membrane strain. Since the cell isfirmly attached to the extracellular matrix on the sil-icone membrane through cell surface receptors whichare connected to the cytoskeleton, the cytoskeletonwill experience strain levels comparable to the appliedsubstrate strain, whereas the stiffer nucleus typically de-forms significantly less [35.41, 55, 56]. The advantagesof this technique are that force can be applied to sev-eral cells at once and that the strain application closelyresembles physiological mechanical stress, e.g., in mus-cle cells or endothelial cells subjected to blood vesselexpansion. The major limitations are that this techniqueis only suitable for adherent cells and that the appliedforces cannot be determined directly.

Shear Flow. The most commonly used devices for shearstress application are the cone and plate rheometer andflow (or perfusion) chambers. Cone and plate rheome-ters allow precise control over the applied shear stress,but are generally not equipped to image cells duringshear stress application, making it difficult to visual-ize induced cellular deformations. On the other hand,flow chambers are routinely used to study cellular de-formations under shear stress [35.57] or to investigatecellular responses to shear stimulation such as calciuminflux or cytoskeletal remodeling. Parallel-plate flowchambers are made of transparent glass slides sepa-rated by a thin spacer/gasket (the channel height is often≈ 100 μm) and can thus apply precisely controlled fluidshear stress to a cell monolayer plated on the bottomslide while simultaneously imaging the cells on a mi-croscope. The shear stress at the cell surface can becalculated for a Newtonian fluid in the parallel plategeometry as τ = 6Qμ/wh2, where τ is the fluid shearstress at the wall (and cell surface), Q is the flow rate,μ is the viscosity, w is the channel width, and h is thechannel height. The shear stress can thus be adjusted tophysiological shear stress levels (≈ 0.1–10 Pa) by al-tering the height of the flow chamber or modulating theflow rate, and experiments can be carried out with ei-ther constant or pulsatile flow. However, the actual shearstress at the cell surface might deviate from the pre-dicted wall shear stress, as small variations in the cellheight and topology can cause local variations, whichmight in fact contribute to the alignment of endothelialcells to the flow direction.

Micromachined Devices. Within the last decade,microfabrication has enabled the design of numer-

ous custom-designed devices to measure cellularmechanics, including microelectromechanical systems(MEMS). Typically, these devices contain an actuatorto apply precisely controlled forces or deformations tosingle cells and a force sensor, often comprised of anelement with known spring constant that deforms underthe applied load. In some cases, these two componentscan be combined into a single element. One advantageof micromachined devices is the ability to directly con-trol force application and sense the applied forces withhigh precision without requiring elaborate assumptionsof cellular structure and mechanical properties. Also, byappropriately tuning the geometry of the force sensor,relatively high forces (up to 1500 nN) can be measured,allowing measurements of forces required to detach ad-herent cells from the substrate. The major disadvantagesare the still relatively high costs and the need for spe-cialized equipment in the fabrication process.

Passive MeasurementsIn contrast to the active measurements, these experi-ments measure forces generated by the cells themselvesor quantify the random motion of particles embed-ded in the cytoplasm subjected to thermal fluctuations.Typical examples of passive measurement techniquesinclude particle tracking of beads in the cytoplasmand traction force microscopy. In passive bead mi-crorheology, small (0.1–2 μm diameter) beads areinjected into the cytoplasm or taken up by endocyto-sis and are then tracked with high spatial and temporalresolution with a laser beam and quadrant photode-tector. The complex cytoplasmic shear modulus G(s)can then be computed from the unilateral Laplacetransform of the measured mean-squared displacement〈Δr2(s)〉 using a generalized Stokes–Einstein equationas G(s) = kBT/(πas〈Δr2(s)〉), where s is the Laplacefrequency, kB is the Boltzmann constant, T is the tem-perature, and a is the bead radius [35.58]. Traction forcemicroscopy was originally based on wrinkles gener-ated in thin silicone sheets by the contraction of cellsplated on top of these sheets, but has subsequently beenrefined for more quantitative force determination byplating cells on polyacrylamide gels with fluorescentbeads embedded in the gel. The mechanical stiffnessof the polyacrylamide gel can be tuned based on thecross-linker concentration, and microscopic measure-ments of the induced displacement of beads near thesurface can be converted into cellular forces exerted onthe gel by elastic theory [35.59]. Most recently, a newgeneration of traction force microscopy has emergedin which cells are coated on a fine grid of micro-

PartD

35.3


fabricated micropillars made of polydimethylsiloxane(PDMS) [35.60]. In this case, the stiffness of micropil-lars can be controlled by adjusting the cross-sectionalarea or length of the micropillars, and the cellular forceexerted on each micropillar can be measured based onthe deflection of the pillar using beam bending theory.This technique offers the advantage that force appli-cation is localized to individual micropillars, and thatthe geometry and stiffness can be independently ad-justed.

35.3.2 Rheological Properties

The rheological properties of the cell, such as its vis-coelastic properties and its diffusion parameters, are keyto the cell’s ability to accomplish its diverse functionsin health and disease. A wide range of computationaland phenomenological models as well as experimen-tal techniques have been proposed over the past twodecades to describe the cell, giving rise to several, of-ten contradictory, theories for describing the rheologyof the cytoskeleton. The highly heterogeneous struc-ture of the cytoskeleton, coupled with a small linearresponse regime [35.62] and active dynamics and con-tinuously remodeling, present a major challenge forquantitative measurements and descriptions of its rhe-ology [35.63].

Nonetheless, a wide range of computational modelsexist for cytoskeletal rheology and mechanics, rang-ing from continuum to discrete descriptions of thecytoskeleton (see reviews in [35.64]). A major chal-lenge in cytoskeletal rheology and mechanics is howto relate experimental observations to theoretical andphenomenological models. Much effort has recently fo-cused on the interesting rheological behavior of cells,measured by one of the methods described above. Mostoften, cellular viscoelastic properties are expressed interms of the complex shear modulus (see ViscoelasticSolid/Poroelastic Solid), the real part G ′ indicating theelastic component, and the imaginary part G ′′ the vis-cous component. While each measurement method hasits drawbacks, perhaps the most comprehensive datahave been obtained through magnetic twisting cytom-etry (Fig. 35.6). These measurements have shown thata cell exhibits a relatively simple power-law behaviorover much of the frequency domain, with [35.61]

G ′ + iG ′′ = G0

(ω

ω0

)x−1

(1+ iη)Γ (2− x)

× cos[π

2(x −1)

]+ iωμ , (35.1)

10–2 10–1 100 101 102 103

G (Pa)

Frequency (Hz)

105

104

103

102

G'G''

Fig. 35.7 Typical experimental results for the frequency-dependent shear moduli obtained by magnetic twistingcytometry (after [35.61])

where Γ is the Gamma function, μ is the vis-cosity, and η = tan((x − 1)π/2)G0, ω0, and x areparameters of the model. All but x, however, havebeen found to be nearly constant in all experi-ments. This turns out to be similar to the behaviorof soft glassy materials, although the fundamentalbasis for this behavior remains a topic of some de-bate.

The data in Fig. 35.7 [35.61] apply to small defor-mations in the linear regime. However, linearity persistsup to surprisingly large deformations. Much of whatwe know about nonlinear effects has been obtainedfrom experiments on reconstituted gels, typically ofactin with one or more actin cross-linking proteins.From these experiments, the following observationshave been made:

1. Linearity persists up to ≈ 30% strain.2. Values of G ′ and G ′′ in this linear regime are of-

ten orders of magnitude lower than are measured incells.

3. Strain stiffening is observed at strains > 30%, fol-lowed by a precipitous drop in G ′ at strains of≈ 70%, presumably indicating rupture or unfoldingof the cross-linkers.

All of these issues continue to be actively studied in thehope of generating a comprehensive understanding ofcytoskeletal rheology.

PartD

35.3


35.3.3 Active Force Generation

The cytoskeleton is an active structure that maintainscell shape and facilitates its motion. The cytoskele-tal network is in a nonequilibrium state that drivesmotor proteins as force-generating features in cells.The cytoskeleton is activated by these molecularmotors, which are nanometer-sized force-generatingproteins, e.g., myosin. To better understand the na-ture of such active force-generation systems, re-searchers have developed simplified models of theactive cytoskeleton by mixing actin with actin bind-ing proteins (e.g., filamin and α-actinin) and molecularmotor proteins myosins [35.65]. Such a reconsti-tuted synthetic cytoskeleton exhibits local contractionsreminiscent of living cells. Tension generated bycontraction can lead to drastic increase of the cy-toskeletal stiffness. Such models demonstrated thata remarkably simple system, with just three compo-nents (myosin, actin, and ATP), can reproduce keyphenomena also observed in far more complex livingcells [35.66].

35.3.4 Biological Responses

The cellular responses to mechanical stimuli takeplace over several time scales and range from intra-cellular signaling to changes in cellular morphologyand function. The fastest responses occur within sec-onds to minutes of stimulation and include changesin intracellular ion concentrations (especially Ca2+)through opening of mechanosensitive ion channelsand activation of cellular signaling pathways suchas nuclear factor-κB (NF-κB), phosphatidylinositol-3-kinase (PI3K)/protein kinase B (Akt), protein kinase C(PKC), Rho family GTPases, and mitogen-activatedprotein kinases (MAPKs) [35.67]. These first stepsdo not require synthesis of new proteins, but involvemodification (e.g., phosphorylation) and intracellulartranslocation of existing proteins. These initial eventsoften trigger activation of mechanosensitive immedi-ate early genes such as egr-1, c-fos, c-jun or c-mycencoding transcription factors that turn on additionaldownstream genes. These later response genes ofteninclude proteins involved in cellular structure (e.g.,actin, myosin) or extracellular matrix remodeling. How-ever, cytoskeletal remodeling in response to mechanicalstress or strain can start even before synthesis of

new proteins as early signaling pathways can alsomediate polymerization and depolymerization of cy-toskeletal structures. For example, endothelial cellsexposed to fluid shear stress begin to align withthe flow direction within 20 min, and the process iscompleted within 24 h. The multifaceted response tomechanical stimulation can effectively mediate sev-eral cellular functions, including DNA and RNAsynthesis, hypertrophy (increase in cell size), prolif-eration, apoptosis, migration, and extracellular matrixremodeling.

35.3.5 Nonlinear Effects

As described earlier, cells are composed of intricatenetworks of filamentous structures, called the cy-toskeleton. These networks exhibit unique propertiesincluding relatively large shear moduli, strong signa-tures of nonlinear response in which, for example, theshear modulus can increase drastically under modeststrains [35.68,69]. Models of semiflexible polymer net-works have emerged to describe these unique propertiesand the dynamics of the cytoskeletal networks. Thesemodels involve a semiflexible description of the con-stituent actin filaments [35.70]. Quantitative models ofthe nonlinear behavior of the cytoskeletal network areessential for understanding the complex, dynamic, andnonlinear behavior of cells, and ultimately the tissuesand organs.

35.3.6 Homogeneity and Anisotropy

Cells are inhomogeneous structures composed of vari-ous intracellular components, including the lipid bilayermembrane that encases the cell, the actin cortex, whichis a dense actin network providing stability for thecell membrane, cytoskeletal networks, and the nucleus,which itself comprises various important substructures.The cytoskeleton is primarily responsible for the struc-tural integrity and stiffness exhibited by a cell. It iscomposed of a system of highly entangled protein fil-aments that permeate the microfluidic space of thecytosol. The major components of the cytoskeletalnetwork, i. e., the actin filaments, intermediate fila-ments, microtubules, and their cross-linking proteinsoffer an exquisite microenvironment which is highlyinhomogeneous and anisotropic in its structure andgeometry.

PartD

35.3

Cellular Nanomechanics 35.4 Theoretical and Computational Descriptions 1185

35.4 Theoretical and Computational Descriptions

As is evident from the discussion above, the cytoskele-ton is a complex biopolymer network with varyingdegrees of connectivity, existing in a state of dynamicequilibrium. The dynamic state arises from the ongoingpolymerization and depolymerization of the constituentfilaments and the changing density of cross-links be-tween filaments of the same or different family. Thispicture is complicated further by the milieu of otherintracellular constituents that may or may not affectstructural properties exhibited by the cell.

Our objective in this section is to come to a betterappreciation of how the elastic properties and geomet-ric arrangement of the constituent filaments give rise tothe material properties observed by the various experi-mental methods just described. These experiments haveshown that the elastic modulus of cells can vary con-siderably from one cell type to another. Cells of theepidermis, for example, require greater structural in-tegrity than the red blood cells subjected to the relativelylow shear stresses in the blood. Even within a given celltype, the elastic properties can change. Skeletal muscle,for example, changes its modulus by over an order ofmagnitude within a small fraction of a second. Othercells change too, but more often over a longer timeperiod, in response, for example, to changes in theirmechanical environment.

According to the various measurements that havebeen made, cells seem to range in shear modulus inthe range ≈ 10–10 000 Pa, and therefore exhibit a stiff-ness somewhat lower than collagen gels (or commongelatins) at low concentration or relaxed skeletal mus-cle. This wide range of moduli probably says moreabout differences in the models used as a basis to in-fer the shear modulus from the data than it does aboutreal cell-to-cell variations. At best, these numbers in-ferred from experiment should be viewed as measuresof an effective stiffness, and comparisons between dif-ferent measurement methods and interpretation shouldbe made with caution. Other biological materials (e.g.,bone, wood) exhibit a much higher modulus, but notbecause of their cellular content. Rather, their high stiff-ness is due to the calcification in bone and the collagenmatrix found in wood and most plants. In tissues, too,the stiffness we measure is more often associated withthe extracellular matrix with its elastin and collagen,than the resident cells. Even within the cytoskeleton, in-dividual filaments (i. e., F-actin, intermediate filaments,and microtubules) exhibit moduli much greater than themeasured bulk modulus of the cytoskeleton. In the next

section we explore different approaches that have beendeveloped to relate the properties of the individual fibersto those of the assembled network.

35.4.1 Continuum Models

Elastic SolidIn several of the experiments described earlier, the as-sumption of a homogenous and isotropic elastic solidis used to infer a value for an effective Young’s modu-lus for the cell. It is not our intention here to providea comprehensive description of elasticity theory, butrather to outline some of the basics. For a full deriva-tion of the governing equations, the reader is referred toany of a number of excellent textbooks [35.71]. Assum-ing equilibrium conditions (i. e., all forces balance eachother) and a material that can be modeled as a Hookean(i. e., linear) elastic solid, the constitutive equations re-lating the mechanical stress and strain can be writtenin tensor notation as τij = Cijklεkl , where the summa-tion convention is used. Here τij are the elements ofthe stress tensor, εkl are the elements of the strain ten-sor, and Cijkl is the 81-element coefficient matrix. Inthe case of an isotropic material, the coefficient matrixreduces to just two independent elastic constants, withthe constitutive equation now reduced to the followingsimplified form:

τij = 2Geij +λεkkδij , (35.2)

where λ and G are the Lamé constants with G beingtermed the shear modulus. Recall that this expressioncan also be written in the inverted form

εij =[

(1+ν)

E

]τij −

( ν

E

)τkkδij . (35.3)

Here the two material constants are now given as ν,the Poisson’s ratio, and E, the Young’s modulus. Whenusing these to solve a particular problem, a set of appro-priate boundary conditions also needs to be posed. Forexample, in the case of measurements by an indentationprobe, the displacement of the surface in contact withthe probe might be given. In addition, it would be nec-essary to specify that the opposite side of the cell is heldfixed, and that the unsupported sides and the region onthe upper surface not in contact with the probe have zeroapplied stress. Given these, the equations above providea complete solution.

The use of the above equations presumes that theresponse of the cytoskeleton can be represented by an

PartD

35.4


a

l

l

l

a) b) c)

Fig. 35.8a–c Cell mechanics models. (a) Schematic of a network of biopolymers consisting of a fiber matrix with cross-links. (b) Unit cell used in the cellular solids model (after [35.72]). (c) Tensegrity structure showing the balance betweenelements in compression (cylinders) and others in tension (lines) (after [35.73])

elastic continuum lacking any discernable microstruc-ture. The validity of this clearly depends upon the lengthscale of interest in the problem, and is a serious concernin the context of cytoskeletal mechanics where the typ-ical length scale (spacing distance between filaments)may be on the order of 10–100 nm. This becomes com-parable to the linear dimension of the region of interestin some of the experimental procedures (e.g., cell pok-ing, AFM measurements) and needs to be kept in mind.Furthermore, while this is a convenient approach for an-alyzing how cells deform under loading, it provides noinsight into the relationship between the macroscopicproperties that are measured and the elastic propertiesof the constituent matrix elements.

Viscoelastic Solid/Poroelastic SolidA viscoelastic material is to an elastic material asa spring and dashpot network is to a system contain-ing only springs. In both the discrete and continuoussystems, the instantaneous displacement or strain isa function of the stress history. Similarly, the instanta-neous stress depends on the strain history. This simplerealization, in combination with the assumption that allfunctions describing the material properties are contin-uous, leads to a set of generalized constitutive equationsfor a viscoelastic solid.

Many of the experiments performed on cells havebeen interpreted in the context of either an elastic orviscoelastic model. It might be argued, however, thatlike bone, cartilage, and many other biological tissues,it may be more appropriate to view the cytoskele-

ton as a poroelastic material. While attempts to applya poroelastic model to studies of cell mechanics are justbeginning, it remains to be seen whether a viscoelas-tic or poroelastic description is more appropriate. Fora description of poroelastic theory, the reader is referredto [35.74].

Oscillatory Simple Shear. A useful example of vis-coelastic behavior derives from the case in which theupper surface of a sample is being oscillated sinu-soidally in the plane of the surface so that the shearstrain satisfies ε21 = ε∗

21 sin ωt and the material experi-ences an oscillatory shear stress. In this case, the stressin the viscoelastic material will also vary sinusoidally,but out of phase with the strain. The relationshipbetween stress and strain is often described by introduc-ing the complex shear modulus (as introduced above),G∗ = G ′ + iG ′′, satisfying the following expressions

τ21(t) = ε∗21(G ′ sin ωt + G ′′ cos ωt) , (35.4)

τ21(t) = τ∗21 sin(ωt +φ)

= τ∗21 cos φ sin ωt + τ∗

21 sin φ cos ωt , (35.5)

so that G ′, G ′′, and φ, the phase lag, are related to theamplitudes of strain and stress through the expressions

G ′ = ω

∞∫0

G(t′) sin ωt′ dt′ =(

τ∗21

ε∗21

)cos φ , (35.6)

G ′′ = ω

∞∫0

G(t′) cos ωt′ dt′ =(

τ∗21

ε∗21

)sin φ . (35.7)

PartD

35.4

Cellular Nanomechanics 35.4 Theoretical and Computational Descriptions 1187

Note that G ′ represents the component of stress in phasewith the imposed strain and G ′′ the out-of-phase com-ponent. For a purely elastic material (in which strain andstress are in phase), G ′ is simply the shear modulus Gand G ′′ is zero. On the other hand, it can be shown thatfor a viscous Newtonian fluid, G ′ is zero, and G ′′ = μω,where μ is the Newtonian viscosity.

It should be noted that these results could also havebeen obtained by expressing the time-varying strain asa complex quantity. In so doing, the shear modulus iscomplex, and G ′ and G ′′ are, respectively, the real andimaginary parts of the stress–strain ratio.

35.4.2 Biopolymer Models

Cytoskeletal networks can be viewed as a polymer gelin which the matrix is considered to consist of relativelystraight segments connecting junctions where the fila-ments are either chemically cross-linked, or effectivelyso due to entanglements (Fig. 35.8a). Using conceptsfrom polymer physics [35.75] the force required tochange the length of one segment of a polymer filament(F-actin, for example) of length Le by an amount δ canbe expressed as

FT ≈ kBTl2p

L4e

δ ≈ K2b

kBTL4eδ , (35.8)

where kB is Boltzmann’s constant, T is temperature,Kb is the bending stiffness of the filament, lp is thepersistence length, and Le is the distance between thepoints where the tension is applied (or, the distance be-tween points of entanglement or cross-linking betweennetwork filaments). This expression arises from a con-sideration of the curvature of the polymer resulting fromBrownian or thermal fluctuations, and is based on theassumption that thermal energy is equally partitionedamong the different modes of oscillation. The poly-mer filaments are therefore assumed to be bent priorto the application of stress. Externally imposed forceseither increase or decrease the end-to-end length ofthese filaments, and the deformation produced dependsboth on the intrinsic bending stiffness of the filamentsand on their initial degree of curvature due to thermalfluctuations.

For a network comprised of such filaments in whichthe distance separating points at which physical bondsexist between the filaments or regions of entanglementis Le, the change in filament length between bonds dueto a shear strain θ is δ ∝ θLe. Making use of the factthat the number of filaments per unit area parallel to thesurface on which the stress is applied scales inversely

with the square of the characteristic mesh spacing ξ , theshear stress required to produce a strain θ is given by

τ ∝ FT(number of filaments)

(unit area)∝ kBTl2

pθ(L3

eξ2) (35.9)

from which a shear modulus of the (elastic) network Gncan be determined as the ratio of stress to strain

Gn ∝ τ

θ∝ kBTl2

p(L3

eξ2) . (35.10)

The scaling of Gn clearly depends on both the distancebetween cross-links or entanglements and the meshsize. As the concentration of polymer increases, ξ de-creases and so likely does Le, at least in terms of thedegree of entanglement. As the concentration of actinbinding protein (ABP) increases, Le will decrease. Themesh spacing ξ also depends on the monomer con-centration or solid fraction Φ (monomer volume/totalvolume). Assuming that the filaments are stiff andhomogeneously dispersed through the medium, this re-lationship can be expressed as ξ ∝ a/Φ1/2, where a isthe monomer or filaments radius. Substituting yields

Gn ∝ kBTl2pΦ(

L3ea2

) . (35.11)

In the limit of a highly cross-linked network, in whichcase Le ≈ ξ , this leads to

Gn ∝ kBTl2p

(Φ

a2

)5/2

. (35.12)

Alternatively, we can express this in terms of the densityof cross-links ρc. Note that, if ρc is the number of cross-links per unit volume, it should vary inversely with thevolume associated with each bond or entanglement, i. e.,as L−3

e . Combining this with the expressions (35.10–35.12) gives

Gn ∝ kBTl2pΦρc

a2. (35.13)

The Young’s modulus of the network En can be ob-tained by a similar procedure, and can be shown to scalein the same manner as Gn.

35.4.3 Cellular Solids

The theory of cellular solids was developed for thepurpose of relating the macromechanical propertiesof low-density cellular materials to their microstruc-tural characteristics. The approach used is based on

PartD

35.4


the concept that the material can be modeled as be-ing comprised of many unit cells, one representationof which is shown in Fig. 35.8b. When a cellular solidis stressed under tension or compression, the fibersact like struts and beams that deform under stress.The unit cell model in Fig. 35.8b has struts or fiberelements of length L and cross-section of radius a.The relative density of the material is defined by thevolume fraction of solid material Φ. This is calcu-lated as the solid volume contained within a unit cell(∝ a2 L) divided by the total unit cell volume (∝ L3)or Φ ∝ (a/L)2. Beam theory gives the deflection δ ofa beam of length L subject to a force F acting at itsmidpoint as

δ ∝ FL3

Ef I, (35.14)

where Ef is the stiffness of the beam constitutivematerial and I is the beam’s second moment ofarea. The moment of area for a beam of thick-ness a is given by I ∝ a4. The stress τ is theforce per unit area, or τ ∝ F/L2. The strain is re-lated to beam deflection δ by ε ∝ δ/L . Using these,combined with the expressions above, the networkYoung’s modulus or elastic modulus can be expressedas

En = τ

ε= c1 Ef I

L4, (35.15)

where c1 is a constant of proportionality, or

En

Ef= c1Φ

2 . (35.16)

Data from a wide range of materials and cell geometriesgive a value for c1 of ≈ 1 [35.72]. A similar anal-ysis for cellular materials subjected to shear stressesresults in an expression for the network shear modulusGn [35.72]:

Gn

Ef∝ c2Φ

2 , (35.17)

where c2 ≈ 3/8. If the material is linearly elastic andisotropic, these values lead to a value of 1/3 for thePoisson ratio ν.

35.4.4 Tensegrity

Networks can also derive their structural integrity froman interaction between members that are in compressionand members in tension. Some familiar large-scale ex-amples include the circus tents and the geodesic dome.In the case of the circus tent, the rigidity of the structureis due to the balance between the tent poles in compres-sion and the ropes anchored to the ground in tension.The rigidity of the structure, in this case, is related tothe elastic characteristics of the tension elements.

Ingber [35.76] first proposed that the cytoskeletonbehaves like a tensegrity structure with the microtubulesacting in compression and the F-actin microfilamentsacting in tension. In support of this concept, micro-tubules have been shown to be capable of supportingcompressive loads and the F-actin network exhibitsbehavior at junctions consistent with their being intension. Intermediate filaments may also be involved,although their contributions at this stage are unclear.

Analysis of a fully three-dimensional tensegrity net-work begins, as in the case of the cellular solids model,with a unit cell consisting of an interconnected systemof elements in tension in balance with other elementsin compression. When the three-dimensional tenseg-rity network of Fig. 35.8c is used, analysis shows thatprestress plays an important role. Intuitively, it is notsurprising that networks with greater pretension in theelastic members should exhibit greater resistance to de-formation. The effect of prestress increases the networkYoung’s modulus for small strains, and only in the limitof infinite prestress does En become independent of pre-stress. Wang and co-workers [35.77] have shown thatthe cytoskeleton exhibits this same tendency, becom-ing increasingly stiff when, for example, the cell passesfrom a spherical (low prestress) to flattened (high pre-stress) state.

35.5 Mechanics of Subcellular Structures

35.5.1 Cell–Cell and Cell–Matrix Adhesions

Cells can adhere to their surroundings by either non-specific or receptor–ligand (specific) bonding. Thoughboth mechanisms are likely active in most situations,

receptor–ligand bonding is the stronger of the two, bya considerable margin, and is therefore the most rel-evant mechanically. In situations for which either thereceptors or their ligands are not present, however, suchas might be the case in certain in vitro experiments,

PartD

35.5

Cellular Nanomechanics 35.5 Mechanics of Subcellular Structures 1189

nonspecific binding can be important as well, and canproduce a net attractive force per unit area of ≈ 100 Paat a separation distance of ≈ 25 nm.

Models for Receptor-Mediated AdhesionCell–cell adhesion has obvious similarities to adhesionof cells to substrates, the surrounding extracellular ma-trix, and other cells, so a similar approach can be used.This requires, in addition, consideration of such factorsas the distribution, type, and density of receptor–ligandbonds or potential bonds, and the elastic properties ofthe structures to which they are anchored. On the intra-cellular side, this involves the series of couplings thatlink the receptor to the cell. In the simplest case, thismight simply be a link to the lipid bilayer if the receptorhas no intracellular connections. More typically, espe-cially for couplings with a structural role, it involvesa series of proteins ultimately linking the receptor to thecytoskeleton.

The typical setting in vivo is one in which cellsadhere to other cells or to the extracellular matrix.Adhesions are more easily probed, however, throughin vitro experiments, where cell adhesion more oftenoccurs to an artificial substrate mediated through oneof several extracellular proteins that are used to coatthe surface. Generally, either collagen or fibronectin isused. Cells are often adhered to these substrates, but toproduce a more controlled environment, rigid beads aresometimes coated with the appropriate receptors so thatone specific receptor–ligand interaction can be probed.While these systems are useful as models of certain ad-hesion phenomena, it is important to recognize in theinterpretation of these experiments that, when bound toa rigid substrate or bead, the binding proteins cannotfreely diffuse, as they would in a more natural envi-ronment. In particular, the formation of focal adhesionswould not occur in bead–substrate experiments becausethe receptors would be constrained from aggregating.

It is useful to consider a single adhesion bond, forexample, one linking the actin matrix of the cytoskele-ton to a β1 integrin, and the integrin receptor bindingto the extracellular matrix beyond the cell membrane. Ifthe bond is stressed, as for example if the cell experi-ences a force relative to the ECM, it will at first stretchan amount dictated by the level of force in the bondand the stiffness of the complex. Each bond, as wellas each protein in the bond complex, can be thoughtof as having a certain stiffness, giving rise to a picturein which several springs are considered connected inseries. Forces acting on the adhesion complex are trans-mitted via this series of bonded proteins between the

cytoskeleton (CSK) and the ECM, producing local de-formations and stresses in the corresponding matricesthat decay with distance from the adhesion site. On theintracellular side, these forces are transmitted via a com-plex involving vinculin, α-actinin, paxillin, and talin.Attachment to the extracellular matrix is mediated by anarginine–glycine–aspartic acid (RGD) sequence (in fi-bronectin for example), which in turn, has binding sitesfor collagen and fibrin. If the force is sufficiently strong,above a certain threshold value F0 say, and applied fora sufficiently long time, the bond might be severed (de-couple). Typical values of this threshold force lie in therange of 10–100 pN [35.78], but depend, as well, on therate at which the force is increased. Under more rapidforce application, the threshold is high compared withwhen the force is increased slowly. This simply reflectsthe fact that detachment is a stochastic process and that,at any given level of force, there exists a finite proba-bility of detachment, and this probability increases withtime.

We need also to consider that, in equilibrium, a frac-tion Φ = K (1 + K )−1 (where K is the equilibriumconstant) of the receptors will be bound at any giventime, and that the receptor–ligand complexes are con-tinually cycling between the bound and unbound states,characterized by their respective rate constants. Ona larger scale, numerous adhesion sites typically act inparallel, each contributing some amount to the total ad-hesion force. This gives rise to an average local stress τ

that can be thought of as the product of the average forceper bond times the bond density. This stress is typicallynonuniform. In the adhesion of a cell to a flat sub-strate, the bonds near the periphery of the contact zonedetermine the strength of the cell against detachment,and when the cell is subjected to a detaching force, thestresses are concentrated in this peripheral region. In thevicinity of the edge of adhesion, several factors will in-fluence the stress distribution, including the mechanicalproperties of the membrane (in particular the bendingstiffness Kb), the receptor density, the spring constantsof the bond, and the bond strength.

The interplay between receptor–ligand bonding andmembrane stiffness also determines whether a cell willspread over a surface or peel away from it under a giventension. Membrane bending stiffness acts to maintainthe membrane and substrate within close proximity nearthe edge of contact. Spreading can occur only if theirseparation distance is within the extended length of thereceptor ligand bond over a distance comparable to thelinear spacing between neighboring bonds. Under hightension, the surfaces rapidly diverge and spreading is

PartD

35.5


prevented; if the tension is sufficiently low, receptor–ligand bonding can occur and spreading can occur.

Transient Cell AdhesionConcepts of transient adhesion and release are of par-ticular importance in the case of leukocyte rolling,adhesion, and transmigration across the endotheliallayer of a blood vessel. In addition, cell migrationthrough extracellular matrix or along a substrate re-quires the ability of the cell to both form and releaseadhesions. One problem that has received consider-able attention is the interaction between an adherentor rolling leukocyte and the adhesion receptors onthe endothelium. These studies are complicated bythe three-dimensional nature of the flow, the compli-ance of the interacting surfaces, and receptor–liganddynamics. In a series of recent studies, Hammer andcoworkers [35.79] have addressed several of these is-sues, computing the viscous force and torque actingupon a sphere near to a planar wall from the mobil-ity matrix [35.80] and using the Bell model of receptorbinding as described above [35.81], but neglecting theeffects of cell deformation.

A Monte Carlo method is employed at each timestep in the calculation to determine bond formation orbreakage. In these studies, the nature of the leukocyte–endothelial interaction is characterized as firm adhesion,rolling adhesion, bimodal adhesion or no adhesion, andmapped as a function of the two parameters of theBell model. Most literature values for the Bell modelparameters for a variety of selectin receptors (knownto be instrumental in leukocyte rolling) correspondto the rolling adhesion regime for typical shear rateof 100 s−1. An interesting, but counterintuitive, resultfrom this study was the observation that, as the bondstiffness was increased from the value used in most cal-culations (100 dyn/cm), adhesiveness decreased, whichthe authors attribute to reduced deflection for a givenlevel of force, leading subsequently to more rapid dis-sociation.

35.5.2 Cell Membranes

For the purpose of analysis, we treat the cell mem-brane as a homogeneous two-dimensional (2-D) plateor sheet completely enclosing the cytoplasm. The mem-brane referred to here can be thought of either as thelipid bilayer by itself, or more typically, as the bilayerplus the associated cortex of cytoskeletal filaments andthe glycocalyx on the extracellular surface. In addi-tion, though not explicitly recognized in the analysis,

transmembrane proteins and their attachments to the in-tracellular and extracellular milieu are included in termsof their influence on the continuum properties of themodel. Were it not for these, the membrane would ex-hibit little resistance to shear deformation.

In qualitative terms, the lipid bilayer can be thoughtof as a two-dimensional fluid, within which the indi-vidual lipid molecules, or other molecules embeddedin the membrane, are relatively free to move about bydiffusion or directed motion. Phospholipid moleculesin either of the two layers resist being pulled apart,however, so each layer is highly inextensible. This alsocontributes to the bending stiffness, which is low in ab-solute terms but high for such a thin layer, since bendingrequires one layer to expand while the other is com-pressed. By contrast, the two layers readily slide relativeto each other. These qualitative notions are put in morequantitative terms in the next section.

Types of DeformationAny deformation of the membrane can be thought of,in general terms, as a superposition of several simplerdeformations. For small strains in which linearization isappropriate, the principle of superposition is rigorouslyvalid. For larger strains, however, linear theory breaksdown and superposition can only be used as a rough,qualitative guide in visualizing combined influences.Here we present the three primary types of deformation:pure extension, pure bending, and pure shear.

Pure Extension. In discussing the extensional stiffnessof the membrane, we need to distinguish the behavior atlow tension from that at high tension. As an extensionalstress is first applied at the edges of a lipid bilayer,the projected or apparent membrane area first increaseswhile the actual or true membrane area remains con-stant. This results from the suppression of out-of-planeundulations. Forces acting to resist membrane flatten-ing originate from entropic effects analogous to thoseseen in the case of a flexible polymer as its end-to-enddistance is increased – many more membrane configu-rations exist with undulations compared with the single,perfectly flat state. Only when these undulations havebeen eliminated does the true membrane area, propor-tional to the surface area per molecule group, begin toincrease, and this is associated with a relatively abruptincrease in extensional stiffness. For now, we consideronly the stiffness of a flat membrane and leave the dis-cussion of undulations of entropic origin to a later pointin the chapter. Hence we initially neglect all entropiceffects, or equivalently, consider the membrane to be at

PartD

35.5


σ2

σ1 σ1 σ1σ1

σ2

σ2

σ2

AA0

x2

x1

x3M1 M1

Neutral plane

x1

a) b) c)

Fig. 35.9a–c Membrane mechanics. (a) A membrane, initially of area A0, subjected to a uniform extensional stressalong its edges (σ1 = σ2), causing an increase in area to A. (b) Application of equal moments M1 at the two edges ofa membrane sheet. (c) Shear deformations produced by the application of nonequal surface tensions on the boundaries ofa membrane sheet. The inner square experiences shear strains as shown by the arrows

zero temperature. Based in this assumption, consider aninfinitesimal plate initially of area A0 = L2

0 that is de-formed by a uniform normal stress τ11 = σ1 = τ22 = σ2applied to its edges (Fig. 35.9a) to a new area A. Theexpressions relating stress and strain in two dimensions,and in the absence of stresses normal to the x1–x2 plane,can be written as

σα = E

1−ν2(εα +νεβ) , (35.18)

where the length of one edge is Lα = L0(1+ εα); notethat the subscripts α, β are used here rather than i, j todistinguish stresses and strains in two dimensions fromthose, more generally, in three. Thus, whereas i and jcan be either 1, 2 or 3, α and β are restricted to be-ing either 1 or 2. When this stress is uniform in theplane of the membrane (the x1–x2 plane) it can be re-placed, without loss of generality, by a surface tensionNα (force per unit length) defined as σαh, where h is thethickness of the membrane. These can be combined inthe case when N1 = N2 = constant and, consequently,ε1 = ε2 = ε to give

N = Eh

1−νε . (35.19)

In terms of a plate stretched uniformly in both direc-tions, we can define the areal strain as

ΔA

A0= A − A0

A0= L2

0(1+ ε)2 − L20

L20

∼= 2ε , (35.20)

where the last approximation is appropriate for smallstrains. By combining this result with (35.19), we candefine the area expansion modulus Ke (units of N/m)using

N = Eh

2(1−ν)

ΔA

A0≡ Ke

ΔA

A0. (35.21)

Note that, although we used the continuum structuralequations in our analysis, the final result can also beviewed as simply the definition of the area expansionmodulus and applies regardless of whether or not themembrane can be modeled as a continuum.

Experimental measurements of Ke lie in the rangeof 0.1–1 N/m for various types of lipid bilayers and≈ 0.45 N/m (450 dyn/cm) for red blood cell mem-branes [35.82]. These numbers suggest that cell mem-branes are quite resistant to extension and, for thatreason, are often treated as inextensible. This discussionneglects the effects of thermal fluctuations in the mem-brane that give rise to a much more compliant behaviorat the smallest areal strains. When surface stress is suf-ficient to smooth out most thermal fluctuations, the cellor vesicle will exhibit the large moduli given here.

This high resistance to area change is in large partdue to the energy penalty associated with exposingthe hydrophobic core of the membrane to water thatoccurs as the spacing between individual amphiphilicmolecules is increased (for a detailed description ofbilayer structure and thermodynamics, see [35.83]).Continuing to increase extensional stress, the lipid bi-layer eventually ruptures, but at very small extensionalstrains, in the vicinity of 2–3% [35.84]. Note that a bi-layer in a lipid vesicle, for example, stretches primarilyby increasing the area per molecule since recruitment ofadditional material to the membrane occurs very slowly.

Using these expressions, the surface tension atwhich the membrane would rupture can be com-puted to be ≈ 0.06 N/m if we use a value near thehigher end of the observed range Ke = 1 N/m. Cellsalso often exhibit an intrinsic surface tension. Re-ported values are small, however, lying in the range of≈ 10−5 –10−4 N/m [35.85].

PartD

35.5


Pure Bending. By contrast, lipid bilayers exhibit verylow bending stiffness, so low that it is often neglectedin models of membrane mechanics. It can be impor-tant in certain situations, however, and is essential, forexample, in analyzing the thermal fluctuations of vesi-cles. Bending stiffness arises from the same type ofmolecular interactions that cause extensional stiffness.When an initially flat bilayer is bent, the hydrophilichead groups on the outside of the bend move furtherapart while on the inside, intermolecular spacing de-creases; both represent departures from the equilibrium,unstressed state and require energy.

Returning to our simple continuum plate model,consider a bending moment applied to the two ends,causing the plate to curve slightly (Fig. 35.9b). If thebending is due to moments applied at the two ends aboutthe x2-axis, then the bending moment per unit length isrelated to the deflection by

Mα = − Et3

12(1−ν2)

(∂2u3

∂x2α

)= −Kb

(∂2u3

∂x2α

),

(35.22)

where Kb is termed the bending stiffness having units ofNm. Implicit in this expression are the assumptions thatthere exists a mid-plane (the neutral plane) on which thein-plane stress and strain are both zero, and that straightlines perpendicular to this mid-plane remain straightand normal to this surface after deformation.

Typical values for the bending stiffness Kb lie in therange of 10−19 N m (10−12 dyn cm) for a red blood cellor lipid bilayers [35.84]. This value is larger, on the or-der of (1–2) × 10−18 N m [35.86], for other cell types(e.g., neutrophils, endothelial cells) that possess a moreextensive cortex.

Pure Shear. Shear deformations arise when a mem-brane is stretched in one direction by a surface tensionN1 (units of force/length) while the lateral surface con-tracts under a lesser tension N2, at constant surfacearea and in the absence of bending (Fig. 35.9c). Sur-faces oriented at 45◦ to the boundaries experience pureshear stresses of magnitude (N1 − N2)/2. When sub-jected to shear stresses in the plane of the membrane,a pure lipid bilayer behaves essentially as a liquid. Itexhibits a membrane viscosity in that it poses a resist-ing force proportional to the rate of shear deformation,but only a small shear modulus to static shear defor-mations. It is not clear, in fact, whether or not purelipid bilayers exhibit a nonzero shear modulus. Typicalcell membranes do exhibit a shear modulus, however,

largely due to the cortex of cytoskeletal filaments thatlie on the intracellular side of the membrane. In a redblood cell this matrix, as discussed above, consistsof interconnected filamentous spectrin and actin withattachments to the membrane via ankyrin. The equa-tions relating shear force per unit length of membraneto shear deformation (Hooke’s law) can be expressedas N12 = τ12h = 2Gε12 = Ksε12, where we define themembrane shear modulus Ks in units of N/m. Typi-cal values lie in the range (6–9) × 10−6 N/m for a redblood cell membrane. A pure lipid bilayer exhibits a vis-cous resistance to shear deformations characterized bya shear viscosity of ≈ 10−6 N s/m [35.53].

35.5.3 Cell Nuclei

Measurements based on micropipette aspiration, atomicforce microscopy, and substrate strain experiments sug-gest that the nucleus is ≈ 2–10 times stiffer than thesurrounding cytoskeleton [35.55, 87–89]. What makesthe nucleus so hard to deform? This question may inpart depend on the particular mechanical load appli-cation. In cellular compression experiments, the highlycondensed chromatin provides a large resistance to nu-clear compression. In contrast, the nuclear lamina canact as an elastic molecular shock absorber [35.90] thatseems ideally suited to sustain large strain when the cellor nucleus is stretched. This idea is supported by mi-cropipette aspiration experiments on isolated nuclei thatreveal that, in condensed nuclei, chromatin is the ma-jor contributor to nuclear stiffness, whereas the nuclearlamina carries much of the applied mechanical load inswollen nuclei that closely resemble nuclei in intactcells [35.48]. Similarly, condensed nuclei were foundto be much stiffer than swollen nuclei [35.48]. Basedon the nuclear structure, the mechanical contributionsof the nucleus can be separated into three intercon-nected components: the nuclear membranes, the nuclearlamina, and the nuclear interior, which is mostly com-prised of chromatin but could also contain a nuclearmatrix of lamins, actins, and other proteins. Each ofthese structural components has its own physical prop-erties. The specific contributions have been addressedin experiments that induced osmotic swelling of iso-lated nuclei by varying the buffer salt concentrationto separate the nuclear envelope from the underlyingchromatin [35.48, 90]. Additional insights have comefrom micropipette aspiration experiments, in which spe-cific components of the nucleus (e.g., the lamina, thenuclear membrane or the chromatin) are fluorescentlylabeled and monitored during aspiration. These experi-

PartD

35.5


ments have revealed that the nuclear interior behaves asa compressible, viscoelastic gel and can be compactedby more than 60% during micropipette aspiration be-fore becoming resistant to further compression [35.51].The reduction in nuclear volume during micropipetteaspiration results in buckling of the nuclear envelopeoutside the pipette [35.48, 51], further demonstratingtight physical attachment between the nuclear enve-lope and the chromatin interior. Nucleoli and otherintranuclear structures deform along with the nuclear in-terior in these experiments [35.50, 51], but might haveslightly different mechanical stiffness. The viscoelas-tic nature of the chromatin also appears responsiblefor the persistent plastic deformations of nuclei seenafter micropipette aspiration of isolated nuclei [35.50,91]. In contrast to the gel-like nuclear interior, thenuclear lamina itself behaves as a two-dimensionalelastic shell. This can be illustrated when monitoringthe fluorescence intensity of green fluorescent protein(GFP)–lamin A incorporated into the nuclear laminaduring micropipette aspiration experiments. While thefluorescence intensity is very uniform in the nonaspi-rated section of the nucleus, the fluorescence intensitydecays exponentially inside the aspirated segment, asthe nuclear lamina is stretched under the aspiration pres-sure. Such a fluorescence intensity profile can be wellfitted by the predicted strain profile of a solid elasticshell [35.51, 92]. Unlike the nuclear lamina, the nu-clear membranes have fluid-like characteristics, i. e.,they do not show any induced strain during micropipetteaspiration as they can flow in response to pressure appli-cation [35.51]. Further illustrating the fluid-like natureis the fact that small membrane vesicles continue toform blebs in the aspirated section of the nucleus aslong as the suction pressure persists [35.51]. Conse-quently, the nuclear membrane cannot carry much ofthe mechanical load, leaving the nuclear interior andthe nuclear lamina as the major contributors to nuclearstiffness.

In addition to the various structural components,the differentiation state of a cell can also modulate the

Table 35.4 Reported values for nuclear stiffness. (∗ Power-law rheology, not single spring constant; the nuclear envelopeexpansion modulus was measured to be ≈ 325 mN/m for these cells, after [35.48])

Cell type Experimental method Nuclear stiffness Reference

Chondrocytes Micropipette aspiration 1–2 kPa [35.88]

Bovine endothelial cells Parallel-plate compression 5–8 kPa [35.87]

Monkey kidney epithelial cells AFM and micropipette aspiration ≈ 1–10 kPa∗ [35.48]

Mouse fibroblasts Intranuclear particle tracking 18 Pa [35.93]

Human cervical cancer cells (HeLa) Magnetic tweezers 250 Pa [35.94]

mechanical properties of its nucleus. A recent studythat probed nuclei from human embryonic stem cellsand differentiated cells found that nuclei from embry-onic stem cells were significantly more deformablethan nuclei from differentiated cells and became upto sixfold stiffer relative to the cytoplasm during dif-ferentiation [35.50]. While some of this increase innuclear stiffness could be caused by changes in chro-matin structure and organization, much of it can beattributed to a stiffer lamina through increased expres-sion of lamins A and C, as independent experimentshave shown that lamins A and C are the main contribu-tors to nuclear stiffness [35.95].

Overall, micropipette aspiration has been the mostwidely used technique to probe nuclear mechanics, al-though cellular strain experiments, AFM, microbeadrheology, and parallel-plate compression experimentshave also been successfully applied to measure nu-clear stiffness (Table 35.4). While some groups havedescribed continuous creep of aspirated nuclei into themicropipette [35.48, 50, 90, 96], others [35.51, 88, 92]found that the nucleus stabilized after an equilibrationperiod of ≈ 10 s. These differences may be due to vari-ations in the experimental setup or the particular celltype studied. Depending on the experimental techniqueused, various mathematical models have been appliedto describe the induced nuclear deformations. For mi-cropipette aspiration experiments, the creep complianceis often inferred from the following equation:

J(t, ΔP) = Φ4π

3

L

D

1

ΔP, (35.23)

where ΔP is the applied pressure difference, L is theaspirated tongue length, and D is the micropipette diam-eter, and the dimensionless parameter Φ (value = 2.1)accounts for the effects of the micropipette geome-try [35.97]. The challenge in finding analytical expres-sion for the mechanical behavior of the nucleus is thatthe nucleus is comprised of various, structurally verydifferent domains, as discussed before. Finite-elementmodeling can address some of these complex interac-

PartD

35.5


tions [35.98], but often has to make assumptions onthe specific material parameters that are difficult toverify. An alternative approach is to directly comparenuclear deformations under identical experimental con-ditions, e.g., the aspirated lengths of the nucleus duringa constant applied pressure gradient in micropipette as-piration experiments or the induced nuclear strain undera given applied substrate strain in cell strain experi-ments [35.99].

35.5.4 Mechanosensing Proteins

While the complex biochemical pathways have been ex-plored in considerable detail by biologists during thepast several decades, work has only recently begun tounderstand the mechanical intracellular pathways forthe propagation of force through the cell. For exam-ple, Geiger and Bershadsky [35.100] have mapped thecomplex and interconnected pathways that can be traced

FibronectinFibronectin

CASCASCAS

��αα

��

FAKFAKFAKFAK

TalinTalin

Actin filamentsActin filaments

VinculinVinculin PaxPax

αα

SrcSrc

α-Actininα-Actinin

Fig. 35.10 A small sampling of the proteins found in a focaladhesion complex. Forces are typically transmitted from the extra-cellular matrix (e.g., fibronectin), via the integral membrane adhe-sion receptors (α- and β-integrins), various membrane-associatedproteins (focal adhesion kinase (FAK), paxillin (Pax), talin, Crk-associated substrate (CAS)), to actin-binding proteins (α-actinin)that link the focal adhesion complex to the cytoskeleton (af-ter [35.100], with permission)

from the extracellular domain to the cytoskeletal ma-trix within the confines of a focal adhesion, a smallportion of which is depicted in Fig. 35.10. Forcestransmitted to the cell, either by tethering to the ex-tracellular matrix or via ligand-coated beads in variousin vitro experiments, are transmitted via the integrins(or other transmembrane receptors) through a collectionof membrane-associated proteins, ultimately linking upwith the cytoskeleton and propagated throughout thecell. Many of the proteins that form this pathway areknown signaling molecules and several have been im-plicated in mechanotransduction. The process by whichchanges in protein conformation give rise to proteinclustering at a focal adhesion or initiate intracellularsignaling, however, remains largely unknown. This hascalled for close examination of the individual proteinsor protein complexes that transmit the forces through-out the cell and their interconnections. Consider oneexample pathway from Fig. 35.11. It can be seen thatintegrins bind to talin, which in turn connects withF-actin either directly or via vinculin. Talin and inte-grin also bind to focal adhesion kinase (FAK). Thisforce pathway alone gives rise to several possibilitiesfor mechanotransduction.

Even in the absence of biochemical signals, me-chanical stimulation can still elicit the formation ofviable focal adhesions [35.101]. These focal adhesionsare produced as a result of direct molecular confor-mational changes in focal adhesion-forming molecules.Force-induced conformational changes in talin, vin-culin, and focal adhesion kinase (FAK) initiate focaladhesion formation by anchoring integrin moleculesto actin filaments and recruiting other focal adhesion-forming molecules to the site of interaction. Othermechanosensing molecules such as filamin and α-actinin stabilize actin filaments through mechanobrac-ing and making scaffolds. The conformational changesin the focal adhesion-initiating molecules and theactin filament scaffold bracing molecules demonstratethe mechanisms of molecular mechanotransduction.In the following, we will briefly review some re-cent work in molecular dynamics simulation of somemechanosensing proteins, namely talin interaction withvinculin [35.102, 103], focal adhesion kinase interac-tion with paxillin [35.104], and molecular mechanicsof actin binding proteins α-actinin [35.105] and fil-amin [35.106].

TalinTalin interacts with the membrane-bound integrin mol-ecule at its head domain, and with vinculin in its tail

PartD

35.5


a)

Cell membrane

Talin

PIP2

F-actin

Inactivevinculin

Activatedvinculin

Activated α/�-integrin dimer

b)

Fig. 35.11 (a) Model of the initial adhesion consisting of integrin–talin–F-actin linkage. Vinculin is present in the cytosolin an inactive, autoinhibitory conformation, and tensile force is applied to the integrin dimer from the outside of the cellmembrane. (b) Transmitted force through the linkage alters the talin configuration, and recruits vinculin to reinforce theinitial adhesion linkage (PIP2 – phosphatidylinositol 4,5-bisphosphate)

domain. It has been implicated as a site of molecularactivation by external force. The forces acting on thecells from the ECM are transmitted from the integrinmolecules to the talin head and then to all regions ofthe talin tail (Fig. 35.11). The talin tail has 11 vinculinbinding sites (VBS), some of which are on amphipathicalpha helixes and are buried in the hydrophobic coreof the talin tail. VBS1 is one of the cryptic bindingsites buried in the hydrophobic core of the TAL5 regionof talin’s tail. TAL5 has several surface polar residuesthat are sites of hydrogen bonding with other secondarystructures in the rod domain. Transduced external forcesfrom the ECM are applied to the TAL5 domain viathese residues [35.107]. In order to simulate the forcetransduction through the cryptic VBS1 of TAL5, Leeet al. [35.103] applied forces averaging 21.2 pN at theresidues near the N-terminal near the talin head whileconstraining fixed the polar residues at the C-terminal,which interact with the adjacent secondary structure do-main. The external locally applied force caused TAL5to elongate, H1 to move in the direction of the pullingforce, H1 to apply torque to H4 via its hydrogen bonds,and H4 to rotate, exposing the VBS1 (Fig. 35.12). Onceexposed to solvent the VBS1 is activated and can inter-act with vinculin. The force-induced activation of VBSsites in the talin tail together with force-induced acti-vation of vinculin can anchor the stimulated integrinmolecules to the actin cytoskeleton and thereby initiatefocal adhesions.

An alternative mechanism was recently proposed byHytönen and Vogel [35.108] which involves the unfold-

ing of the rod domain in order to expose the vinculinbinding site VBS1. Using steered molecular dynamics,Hytönen and Vogel demonstrated that the talin rod canbe fragmented into three helix subbundles, which is fol-lowed by the sequential exposure of vinculin-bindinghelices to water. The unfolding of a vinculin-binding he-lix into a completely stretched polypeptide might theninhibit further binding of vinculin.

α-ActininOne major actin filament cross-linker is α-actinin.α-Actinin is a cross-linking molecule that produces

L622

b)a)

VBS1 VBS1

Fig. 35.12a,b With external force applied to the polarresidues in the TAL5 region of the talin rod, domain activa-tion of the VBS1 is achieved through rotation of the buriedVBS residues to the surface of the molecule. (a) TAL5region before application of external force. The VBS1 isburied in the hydrophobic patch. (b) After the externalforce is applied, TAL5 undergoes a conformational changeand the VBS residues are exposed to the surface. In thisrepresentation, movement of the VBS to the surface isshown by the movement of the arrow (edge of VBS) pastthe dotted lines

PartD

35.5


a scaffold for parallel actin filaments, such as those instress fibers or in muscle z-disk formations. α-Actinin isan antiparallel homodimer with three major domains, anactin binding domain, a calmodulin homology domain,and a central rod domain. In order to act as a molecu-lar scaffold, cross-linking actin filaments in a stressedcytoskeletal network α-actinin must have a semiflex-ible structure. By exploring the natural frequenciesof α-actinin and its response to external forces, Goljiet al. [35.105] characterized the flexible and rigid re-gions of α-actinin and have demonstrated the molecularconformational changes that underlie the semiflexiblecross-linking and mechanical bracing of α-actinin.

MicrotubulesAs yet another mechanosensing protein system one canname the microtubule filaments [35.109]. The micro-tubule cytoskeleton, which is comprised of individualmicrotubules typically nucleated from a microtubuleorganizing center near the nucleus, usually forms a ra-dial network that radiates outward to the cell periphery.

These microtubules are inherently polarized due totheir head-to-tail assembly, and this allows for proces-sive movement of cargo via motor proteins that movespecifically towards either the plus or the minus end.Microtubular filaments resist mechanical deformationand facilitate transport of intracellular cargo via cou-pling to molecular motors. These filaments are typicallyin compression, effectively balancing the tensile forcesexhibited by the actin–myosin system [35.76].

Stretch-Sensitive Ion ChannelsAnother class of mechanosensing proteins are themechanosensitive ion channels, e.g., the mechanosen-sitive channel of large conductance (MscL), which hasbeen studied extensively [35.110, 111]; molecular dy-namic simulation has been used to show how stressesin the cell membrane act directly on the channel andcause it to change its conductance [35.112]. Thesechannels have emerged as another potential gatewaysfor the cell to sense and respond to environmentalstimuli.

35.6 Current Understanding and Future Needs

Progress is rapidly being made on both the experimen-tal and computational fronts to further understand thenanomechanics of the cell. Using either AFM or op-tical tweezers, controlled force applications in the pNrange and displacement measures in the nm range arealready within current capabilities. Optical traps wererecently used in a clever experimental assay systemto measure the rupture force of a complex formed byan actin-binding protein (namely, filamin or α-actinin)linking two actin filaments [35.113]. At the same time,single-molecule fluorescence measurements provide theopportunity to monitor single binding events. On thecomputational side, technical and technological barri-ers are being slowly surmounted. Molecular dynamics

can be applied to proteins to predict their conforma-tional change under force, and docking simulationsprovide a means for determining binding affinities indifferent conformational states. The barriers to progresslie primarily in our lack of atomistic models withadequate resolution for those proteins of greatest inter-est, located in the force transmission pathway. Thesetend to be difficult to crystallize, so few structures areavailable. Moreover, due to their size, simulations arecomputationally intensive, especially if the presence ofwater molecules is included explicitly. Despite theseconstraints, some progress can be made by using sub-domains of the proteins of interest, provided that theirfunctionality can be demonstrated.

References

35.1 P.A. DiMilla, K. Barbee, D.A. Lauffenburger: Math-ematical model for the effects of adhesion andmechanics on cell migration speed, Biophys. J.60(1), 15–37 (1991)

35.2 R. Horwitz, D. Webb: Cell migration, Curr. Biol.13(19), R756–R759 (2003)

35.3 P. Friedl: Prespecification and plasticity: shiftingmechanisms of cell migration, Curr. Opin. Cell Biol.16(1), 14–23 (2004)

35.4 R.A. Christopher, J.L. Guan: To move or not: Howa cell responds, Int. J. Mol. Med. 5(6), 575–581(2000)

PartD

35

Cellular Nanomechanics References 1197

35.5 D.J. Tschumperlin, G. Dai, I.V. Maly, T. Kikuchi,L.H. Laiho, A.K. McVittie, K.J. Haley, C.M. Lilly,P.T. So, D.A. Lauffenburger, R.D. Kamm, J.M. Dra-zen: Mechanotransduction through growth-factorshedding into the extracellular space, Nature429(6987), 83–86 (2004)

35.6 D.E. Ingber: Cellular basis of mechanotransduction,Biol. Bull. 194(3), 323–325 (1998), discussion 325–327

35.7 P.F. Davies: Multiple signaling pathways in flow-mediated endothelial mechanotransduction: PYK-ing the right location, Arterioscler. Thromb. Vasc.Biol. 22(11), 1755–1757 (2002)

35.8 J.Y. Shyy, S. Chien: Role of integrins in endothelialmechanosensing of shear stress, Circ. Res. 91(9),769–775 (2002)

35.9 H. Huang, R.D. Kamm, R.T. Lee: Cell mechanicsand mechanotransduction: pathways, probes, andphysiology, Am. J. Physiol. Cell Physiol. 287(1), C1–C11 (2004)

35.10 P.A. Janmey, D.A. Weitz: Dealing with mechanics:mechanisms of force transduction in cells, TrendsBiochem. Sci. 29(7), 364–370 (2004)

35.11 V. Vogel, M. Sheetz: Local force and geometry sens-ing regulate cell functions, Nat. Rev. Mol. Cell Biol.7(4), 265–275 (2006)

35.12 M.R.K. Mofrad, R.D. Kamm (Eds.): Cellular Mechan-otransduction: Diverse Perspectives from Moleculesto Tissues (Cambridge Univ. Press, Cambridge 2010)

35.13 H. Lodish, A. Berk, P. Matsudaira, C.A. Kaiser,M. Krieger, M.P. Scott, L. Zipursky, J. Darnell: Mo-lecular Cell Biology, 5th edn. (Freeman, New York2000)

35.14 J. Howard: Mechanics of Motor Proteins and theCytoskeleton (Sinauer, Sunderland 2001) pp. 288–289

35.15 T.M. Svitkina, G.G. Borisy: Arp2/3 complex and actindepolymerizing factor/cofilin in dendritic organi-zation and treadmilling of actin filament array inlamellipodia, J. Cell Biol. 145(5), 1009–1026 (1999)

35.16 H. Higuchi, Y.E. Goldman: Sliding distance per ATPmolecule hydrolyzed by myosin heads during iso-tonic shortening of skinned muscle fibers, Biophys.J. 69(4), 1491–1507 (1995)

35.17 Y. Tsuda, H. Yasutake, A. Ishijima, T. Yanagida:Torsional rigidity of single actin filaments andactin-actin bond breaking force under torsionmeasured directly by in vitro micromanipulation,Proc. Natl. Acad. Sci. USA 93(23), 12937–12942 (1996)

35.18 R. Yasuda, H. Miyata, K. Kinosita Jr.: Direct mea-surement of the torsional rigidity of single actinfilaments, J. Mol. Biol. 263(2), 227–236 (1996)

35.19 H. Isambert, P. Venier, A.C. Maggs, A. Fattoum,R. Kassab, D. Pantaloni, M.F. Carlier: Flexibilityof actin filaments derived from thermal fluctua-tions. Effect of bound nucleotide, phalloidin, andmuscle regulatory proteins, J. Biol. Chem. 270(19),11437–11444 (1995)

35.20 F. Gittes, B. Mickey, J. Nettleton, J. Howard: Flex-ural rigidity of microtubules and actin filamentsmeasured from thermal fluctuations in shape,J. Cell Biol. 120(4), 923–934 (1993)

35.21 E. Fuchs, D.W. Cleveland: A structural scaffoldingof intermediate filaments in health and disease,Science 279(5350), 514–519 (1998)

35.22 M. Kirschner, E. Schulze: Morphogenesis and thecontrol of microtubule dynamics in cells, J. Cell Sci.Suppl. 5, 293–310 (1986)

35.23 M. Cohen, K.K. Lee, K.L. Wilson, Y. Gruenbaum:Transcriptional repression, apoptosis, human dis-ease and the functional evolution of the nuclearlamina, Trends Biochem. Sci. 26(1), 41–47 (2001)

35.24 F. Alber, S. Dokudovskaya, L.M. Veenhoff,W. Zhang, J. Kipper, D. Devos, A. Suprapto,O. Karni-Schmidt, R. Williams, B.T. Chait, A. Sali,M.P. Rout: The molecular architecture of the nu-clear pore complex, Nature 450(7170), 695–701(2007)

35.25 J.L.V. Broers, H.J. Kuijpers, C. Östlund, H.J. Worman,J. Endert, F.C.S. Ramaekers: Both lamin A and laminC mutations cause lamina instability as well as lossof internal nuclear lamin organization, Exp. CellRes. 304(2), 582–592 (2005)

35.26 T. Pederson, U. Aebi: Nuclear actin extends, withno contraction in sight, Mol. Biol. Cell 16(11), 5055–5960 (2005)

35.27 W.A. Hofmann, T. Johnson, M. Klapczynski,J.-L. Fan, P. de Lanerolle: From transcription totransport: Emerging roles for nuclear myosin I,Biochem. Cell Biol. 84(4), 418–426 (2006)

35.28 W.A. Hofmann, L. Stojiljkovic, B. Fuchsova,G.M. Vargas, E. Mavrommatis, V. Philimonenko,K. Kysela, J.A. Goodrich, J.L. Lessard, T.J. Hope,P. Hozak, P. de Lanerolle: Actin is part ofpre-initiation complexes and is necessary for tran-scription by RNA polymerase II, Nat. Cell Biol. 6(11),1094–1101 (2004)

35.29 K.G. Young, R. Kothary: Spectrin repeat proteins inthe nucleus, Bioessays 27(2), 144–152 (2005)

35.30 V.C. Padmakumar, S. Abraham, S. Braune,A.A. Noegel, B. Tunggal, I. Karakesisoglou, E. Ko-renbaum: Enaptin, a giant actin-binding protein,is an element of the nuclear membrane and theactin cytoskeleton, Exp. Cell Res. 295(2), 330–339(2004)

35.31 Q. Zhang, C.D. Ragnauth, J.N. Skepper, N.F. Worth,D.T. Warren, R.G. Roberts, P.L. Weissberg, J.A. El-lis, C.M. Shanahan: Nesprin-2 is a multi-isomericprotein that binds lamin and emerin at the nu-clear envelope and forms a subcellular networkin skeletal muscle, J. Cell Sci. 118(4), 673–687(2005)

35.32 M. Crisp, Q. Liu, K. Roux, J.B. Rattner, C. Shanahan,B. Burke, P.D. Stahl, D. Hodzic: Coupling of thenucleus and cytoplasm: Role of the LINC complex,J. Cell Biol. 172(1), 41–53 (2006)

PartD

35


35.33 T. Libotte, H. Zaim, S. Abraham, V.C. Padmakumar,M. Schneide, W. Lu, M. Munck, C. Hutchi-son, M. Wehnert, B. Fahrenkrog, U. Saude,U. Aebi, A.A. Noegel, I. Karakesisoglou: LaminA/C-dependent localization of Nesprin-2, a giantscaffolder at the nuclear envelope, Mol. Biol. Cell16(7), 3411–3424 (2005)

35.34 V.C. Padmakumar, T. Libotte, W. Lu, H. Zaim,S. Abraham, A.A. Noegel, J. Gotzmann, R. Foisner,I. Karakesisoglou: The inner nuclear membraneprotein Sun1 mediates the anchorage of Nesprin-2to the nuclear envelope, J. Cell Sci. 118(15), 3419–3430 (2005)

35.35 D.T. Warren, Q. Zhang, P.L. Weissberg, C.M. Shana-han: Nesprins: Intracellular scaffolds that maintaincell architecture and coordinate cell function?, Ex-pert Rev. Mol. Med. 7(11), 1–15 (2005)

35.36 H.J. Worman, G.G. Gundersen: Here come the SUNs:A nucleocytoskeletal missing link, Trends Cell Biol.16(2), 67–69 (2006)

35.37 F. Haque, D.J. Lloyd, D.T. Smallwood, C.L. Dent,C.M. Shanahan, A.M. Fry, R.C. Trembath, S. Shack-leton: SUN1 interacts with nuclear lamin A andcytoplasmic nesprins to provide a physical con-nection between the nuclear lamina and thecytoskeleton, Mol. Cell Biol. 26(10), 3738–3751(2006)

35.38 G.C. Pare, J.L. Easlick, J.M. Mislow, E.M. McNally,M.S. Kapiloff: Nesprin-1α contributes to the tar-geting of mAKAP to the cardiac myocyte nuclearenvelope, Exp. Cell Res. 303(2), 388–399 (2005)

35.39 S. Münter, J. Enninga, R. Vazquez-Martinez, E. Del-barre, B. David-Watine, U. Nehrbass, S.L. Shorte:Actin polymerisation at the cytoplasmic face ofeukaryotic nuclei, BMC Cell Biol. 7, 23 (2006)

35.40 J.L.V. Broers, E.A.G. Peeters, H.J.H. Kuijpers, J. En-dert, C.V.C. Bouten, C.W.J. Oomens, F.P.T. Baaijens,F.C.S. Ramaekers: Decreased mechanical stiff-ness in LMNA–/– cells is caused by defectivenucleo-cytoskeletal integrity: implications for thedevelopment of laminopathies, Hum. Mol. Genet.13(21), 2567–2580 (2004)

35.41 J. Lammerding, P.C. Schulze, T. Takahashi, S. Koz-lov, T. Sullivan, R.D. Kamm, C.L. Stewart, R.T. Lee:Lamin A/C deficiency causes defective nuclearmechanics and mechanotransduction, J. Clin. In-vestig. 113(3), 370–378 (2004)

35.42 J.S.H. Lee, C.M. Hale, P. Panorchan, S.B. Khatau,J.P. George, Y. Tseng, C.L. Stewart, D. Hodzic,D. Wirtz: Nuclear lamin A/C deficiency inducesdefects in cell mechanics, polarization, and mi-gration, Biophys. J. 93(7), 2542–2552 (2007)

35.43 Q. Zhang, C. Bethmann, N.F. Worth, J.D. Davies,C. Wasner, A. Feuer, C.D. Ragnauth, Q. Yi, J.A. Mel-lad, D.T. Warren, M.A. Wheeler, J.A. Ellis, J.N. Skep-per, M. Vorgerd, B. Schlotter-Weigel, P.L. Weiss-berg, R.G. Roberts, M. Wehnert, C.M. Shanahan:Nesprin-1 and -2 are involved in the pathogene-

sis of Emery Dreifuss muscular dystrophy and arecritical for nuclear envelope integrity, Hum. Mol.Genet. 16(23), 2816–2833 (2007)

35.44 M. Radmacher, J.P. Cleveland, M. Fritz, H.G. Hans-ma, P.K. Hansma: Mapping interaction forces withthe atomic force microscope, Biophys. J. 66(6),2159–2165 (1994)

35.45 R.E. Mahaffy, S. Park, E. Gerde, J. Käs, C.K. Shih:Quantitative analysis of the viscoelastic propertiesof thin regions of fibroblasts using atomic forcemicroscopy, Biophys. J. 86(3), 1777–1793 (2004)

35.46 N. Wang, J.P. Butler, D.E. Ingber: Mechanotrans-duction across the cell surface and through thecytoskeleton, Science 260, 1124–1127 (1993)

35.47 N. Wang, D.E. Ingber: Control of cytoskeletal me-chanics by extracellular matrix, cell shape, andmechanical tension, Biophys. J. 66(6), 2181–2189(1994)

35.48 K.N. Dahl, A.J. Engler, J.D. Pajerowski, D.E. Discher:Power-law rheology of isolated nuclei with defor-mation mapping of nuclear substructures, Biophys.J. 89(4), 2855–2864 (2005)

35.49 D.E. Discher, N. Mohandas, E.A. Evans: Molecularmaps of red cell deformation: hidden elasticity andin situ connectivity, Science 266(5187), 1032–1035(1994)

35.50 J.D. Pajerowski, K.N. Dahl, F.L. Zhong, P.J. Sammak,D.E. Discher: Physical plasticity of the nucleus instem cell differentiation, Proc. Natl. Acad. Sci. USA104(40), 15619–15624 (2007)

35.51 A.C. Rowat, J. Lammerding, J.H. Ipsen: Mechani-cal properties of the cell nucleus and the effect ofemerin deficiency, Biophys. J. 91(12), 4649–4664(2006)

35.52 E. Evans, B. Kukan: Passive material behaviorof granulocytes based on large deformation andrecovery after deformation tests, Blood 64(5),1028–1035 (1984)

35.53 E.A. Evans: Bending elastic modulus of red bloodcell membrane derived from buckling instability inmicropipet aspiration tests, Biophys. J. 43(1), 27–30(1983)

35.54 E.A. Evans, R.M. Hochmuth: Membrane viscoelas-ticity, Biophys. J. 16(1), 1–11 (1976)

35.55 N. Caille, Y. Tardy, J.J. Meister: Assessment of strainfield in endothelial cells subjected to uniaxial de-formation of their substrate, Ann. Biomed. Eng.26(3), 409–416 (1998)

35.56 V.L. Verstraeten, J.Y. Ji, K.S. Cummings, R.T. Lee,J. Lammerding: Increased mechanosensitivity andnuclear stiffness in Hutchinson-Gilford progeriacells: effects of farnesyltransferase inhibitors, Ag-ing Cell 7(3), 383–393 (2008)

35.57 N. Mohandas, R.M. Hochmuth, E.E. Spaeth: Adhe-sion of red cells to foreign surfaces in the presenceof flow, J. Biomed. Mater. Res. 8(2), 119–136 (1974)

35.58 T.G. Mason, K. Ganesan, D. Wirtz, J.H. van Zan-ten, S.C. Kuo: Particle tracking microrheology of

PartD

35

Cellular Nanomechanics References 1199

complex fluids, Phys. Rev. Lett. 79(17), 3282–3285(1977)

35.59 T. Oliver, M. Dembo, K. Jacobson: Traction forces inlocomoting cells, Cell Motil. Cytoskelet. 31(3), 225–240 (1995)

35.60 J.L. Tan, J. Tien, D.M. Pirone, D.S. Gray, K. Bhadri-raju, C.S. Chen: Cells lying on a bed of micronee-dles: An approach to isolate mechanical force, Proc.Natl. Acad. Sci. USA 100(4), 1484–1489 (2003)

35.61 B. Fabry, G.N. Maksym, J.P. Butler, M. Glogauer,D. Navajas, N.A. Taback, E.J. Millet, J.J. Fredberg:Time scale and other invariants of integrative me-chanical behavior in living cells, Phys. Rev. E 68(4),041914 (2003)

35.62 M. Glogauer, P. Arora, G. Yao, I. Sokholov, J. Fer-rier, C.A. McCulloch: Calcium ions and tyrosinephosphorylation interact coordinately with actinto regulate cytoprotective responses to stretching,J. Cell Sci. 110(1), 11–21 (1977)

35.63 P.A. Janmey: Mechanical properties of cytoskeletalpolymers, Curr. Opin. Cell Biol. 3(1), 4–11 (1991)

35.64 M.R.K. Mofrad, N.A. Abdul-Rahim, H. Karcher,P.J. Mack, B. Yap, R.D. Kamm: Exploring themolecular basis for mechanosensation, signaltransduction, and cytoskeletal remodeling, ActaBiomater. 1(3), 281–293 (2005)

35.65 D. Mizuno, C. Tardin, C.F. Schmidt, F.C. MacKintosh:Nonequilibrium mechanics of active cytoskeletalnetworks, Science 315(5810), 370–373 (2007)

35.66 F.C. MacKintosh, A.J. Levine: Nonequilibrium me-chanics and dynamics of motor-activated gels,Phys. Rev. Lett. 100(1), 018104 (2008)

35.67 J. Lammerding, R.D. Kamm, R.T. Lee: Mechan-otransduction in cardiac myocytes, Ann. NY Acad.Sci. 1015, 53–70 (2004)

35.68 M.L. Gardel, J.H. Shin, F.C. MacKintosh, L. Mahade-van, P.A. Matsudaira, D.A. Weitz: Scaling of F-actinnetwork rheology to probe single filament elastic-ity and dynamics, Phys. Rev. Lett. 93(18), 188102(2004)

35.69 K.E. Kasza, A.C. Rowat, J. Liu, T.E. Angelini,C.P. Brangwynne, G.H. Koenderink, D.A. Weitz: Thecell as a material, Curr. Opin. Cell Biol. 19(1), 101–107(2007)

35.70 C. Storm, J.J. Pastore, F.C. MacKintosh, T.C. Luben-sky, P.A. Janmey: Nonlinear elasticity in biologicalgels, Nature 435(7039), 191–194 (2005)

35.71 J.D. Humphrey, S. DeLange: An Introduction toBiomechanics (Springer, Berlin Heidelberg 2004)

35.72 L.J. Gibson, M.F. Ashby: Cellular Solids: Structureand Properties (Cambridge Univ. Press, Cambridge1999)

35.73 D. Stamenovic, J. Fredberg, N. Wang, J. Butler,D. Ingber: A microstructural approach to cytoskele-tal mechanics based on tensegrity, J. Theor. Biol.181, 125–136 (1996)

35.74 R.C. Lee, E.H. Frank, A.J. Grodzinsky, D.K. Roylance:Oscillatory compressional behavior of articular

cartilage and its associated electromechanicalproperties, J. Biomech. Eng. 103(4), 280–292 (1981)

35.75 H. Isambert, A.C. Maggs: Dynamics and rheologyof actin soltions, Macromolecules 29, 1036–1040(1996)

35.76 D.E. Ingber: Cellular tensegrity: Defining new rulesof biological design that govern the cytoskeleton,J. Cell Sci. 104(3), 613–627 (1993)

35.77 N. Wang, I.M. Tolic-Nørrelykke, J. Chen, S.M. Mi-jailovich, J.P. Butler, J.J. Fredberg, D. Stamenovic:Cell prestress. I. Stiffness and prestress are closelyassociated in adherent contractile cells, Am. J.Physiol. Cell Physiol. 282(3), C606–C616 (2002)

35.78 E. Evans, K. Ritchie: Dynamic strength of molecularadhesion bonds, Biophys. J. 72(4), 1541–1555 (1977)

35.79 K.C. Chang, D.F. Tees, D.A. Hammer: The state di-agram for cell adhesion under flow: Leukocyterolling and firm adhesion, Proc. Natl. Acad. Sci.USA 97(21), 11262–11267 (2000)

35.80 D.A. Hammer, S.M. Apte: Simulation of cell rollingand adhesion on surfaces in shear flow: generalresults and analysis of selectin-mediated neu-trophil adhesion, Biophys. J. 63(1), 35–57 (1992)

35.81 G.I. Bell: Models for the specific adhesion of cellsto cells, Science 200, 618–627 (1978)

35.82 R. Waugh, E.A. Evans: Thermoelasticity of red bloodcell membrane, Biophys. J. 26(1), 115–131 (1979)

35.83 J.N. Israelachvili: Intermolecular and Surface Forces(Academic, San Diego 1992)

35.84 N. Mohandas, E. Evans: Mechanical properties ofthe red cell membrane in relation to molecularstructure and genetic defects, Annu. Rev. Biophys.Biomol. Struct. 23, 787–818 (1994)

35.85 E.D. Sheets, R. Simson, K. Jacobson: New insightsinto membrane dynamics from the analysis ofcell surface interactions by physical methods, Curr.Opin. Cell Biol. 7(5), 707–714 (1995)

35.86 D.V. Zhelev, D. Needham, R.M. Hochmuth: Role ofthe membrane cortex in neutrophil deformation insmall pipets, Biophys. J. 67(2), 696–705. (1994)

35.87 N. Caille, O. Thoumine, Y. Tardy, J.-J. Meister:Contribution of the nucleus to the mechanicalproperties of endothelial cells, J. Biomech. 35(2),177–187 (2002)

35.88 F. Guilak, J.R. Tedrow, R. Burgkart: Viscoelasticproperties of the cell nucleus, Biochem. Biophys.Res. Commun. 269(3), 781–786 (2000)

35.89 A.C. Rowat, J. Lammerding, H. Herrmann, U. Ueli:Towards an integrated understanding of the struc-ture and mechanics of the cell nucleus, Bioessays30(3), 226–236 (2008)

35.90 K.N. Dahl, S.M. Kahn, K.L. Wilson, D.E. Discher:The nuclear envelope lamina network has elasticityand a compressibility limit suggestive of a molecu-lar shock absorber, J. Cell Sci. 117(20), 4779–4786(2004)

35.91 S. Deguchi, K. Maeda, T. Ohashi, M. Sato: Flow-induced hardening of endothelial nucleus as an

PartD

35


intracellular stress-bearing organelle, J. Biomech.38(9), 1751–1759 (2005)

35.92 A.C. Rowat, L.J. Foster, M.M. Nielsen, M. Weiss,J.H. Ipsen: Characterization of the elastic proper-ties of the nuclear envelope, J. R. Soc. Interface 2,63–69 (2005)

35.93 Y. Tseng, J.S.H. Lee, T.P. Kole, I. Jiang, D. Wirtz:Micro-organization and visco-elasticity of theinterphase nucleus revealed by particle nanotrack-ing, J. Cell Sci. 117(10), 2159–2167 (2004)

35.94 A.H.B. de Vries, B. Krenn, R. van Driel, V. Sub-ramaniam, J.S. Kanger: Direct observation ofnanomechanical properties of chromatin in livingcells, Nano Lett. 7(5), 1424–1427 (2007)

35.95 J. Lammerding, L.G. Fong, J.Y. Ji, K. Reue,C.L. Stewart, S.G. Young, R.T. Lee: Lamins A andC but not lamin B1 regulate nuclear mechanics,J. Biol. Chem. 281(35), 25768–25780 (2006)

35.96 K.N. Dahl, P. Scaffidi, M.F. Islam, A.G. Yodh,K.L. Wilson, T. Misteli: Distinct structural andmechanical properties of the nuclear lam-ina in Hutchinson-Gilford progeria syndrome,Proc. Natl. Acad. Sci. USA 103(27), 10271–10276(2006)

35.97 D.P. Theret, M.J. Levesque, M. Sato, R.M. Nerem,L.T. Wheeler: The application of a homogeneoushalf-space model in the analysis of endothelialcell micropipette measurements, J. Biomech. Eng.110, 190–199 (1998)

35.98 A. Vaziri, M.R. Mofrad: Mechanics and deformationof the nucleus in micropipette aspiration experi-ment, J. Biomech. 40(9), 2053–2062 (2007)

35.99 J. Lammerding, K.N. Dahl, D.E. Discher, R.D. Kamm:Nuclear mechanics and methods, Methods CellBiol. 83, 269–294 (2007)

35.100 B. Geiger, A. Bershadsky: Exploring the neighbor-hood: Adhesion-coupled cell mechanosensors, Cell110(2), 139–142 (2002)

35.101 M.P. Sheetz: Cell control by membrane–cytoskele-ton adhesion, Nat. Rev. Mol. Cell Biol. 2(5), 392–396(2001)

35.102 S.E. Lee, S. Chunsrivirot, R.D. Kamm, M.R.K. Mofrad:Molecular dynamics study of talin-vinculin bind-ing, Biophys. J. 95(4), 2027–2036 (2008)

35.103 S.E. Lee, R.D. Kamm, M.R. Mofrad: Force-inducedactivation of talin and its possible role in focal ad-

hesion mechanotransduction, J. Biomech. 40(9),2096–2106 (2007)

35.104 M.R.K. Mofrad, J. GoljiJ, N.A. Abdul-Rahim,R.D. Kamm: Force-induced unfolding of the fo-cal adhesion targeting domain and the influenceof paxillin binding, Mech. Chem. Biosyst. 1(4), 253–265 (2004)

35.105 J. Golji, R. Collins, M.R. Mofrad: Molecular me-chanics of the α-actinin rod domain: Bending,torsional, and extensional behavior, PLoS Comput.Biol. 5(5), e1000389 (2009)

35.106 K.S. Kolahi, M.R. Mofrad: Molecular mechanics offilamin’s rod domain, Biophys. J. 94(3), 1075–1083(2008)

35.107 B. Patel, A.R. Gingras, A.A. Bobkov, L.M. Fujimoto,M. Zhang, R.C. Liddington, D. Mazzeo, J. Emsley,G.C.K. Roberts, I.L. Barsukov, D.R. Critchley: Theactivity of the vinculin binding sites in talin isinfluenced by the stability of the helical bundlesthat make up the talin rod, J. Biol. Chem. 281(11),7458–7467 (2006)

35.108 V.P. Hytönen, V. Vogel: How force might acti-vate talin’s vinculin binding sites: SMD revealsa structural mechanism, PLoS Comput. Biol. 4(2),E24 (2008)

35.109 D.J. Odde, L. Ma, A.H. Briggs, A. DeMarco,M.W. Kirschner: Microtubule bending and break-ing in living fibroblast cells, J. Cell Sci. 112(19),3283–3288 (1999)

35.110 G. Chang, R.H. Spencer, A.T. Lee, M.T. Barclay,D.C. Rees: Structure of the MscL homolog fromMycobacterium tuberculosis: A gated mechanosen-sitive ion channel, Science 282(5397), 2220–2226(1998)

35.111 O.P. Hamill, B. Martinac: Molecular basis ofmechanotransduction in living cells, Physiol. Rev.81(2), 685–740 (2001)

35.112 J. Gullingsrud, D. Kosztin, K. Schulten: Structuraldeterminants of MscL gating studied by moleculardynamics simulations, Biophys. J. 80(5), 2074–2081(2001)

35.113 J.M. Ferrer, H. Lee, J. Chen, B. Pelz, F. Nakamura,R.D. Kamm, M.J. Lang: Measuring molecular rup-ture forces between single actin filaments andactin-binding proteins, Proc. Natl. Acad. Sci. USA105(27), 9221–9226 (2008)

PartD

35

Documents

1171 Cellular Nano 35. Cellular Nanomechanics · blocks: the cytoskeleton, cell membrane, nucleus, adhesive complexes, and motor proteins. Next, the various methods used to probe