35. Cellular Nanomechanics Nano Roger Kamm, Jan Lammerding, Mohammad Mofrad

1171 Cellular35. Cellular Nanomechanics Nano Roger Kamm, Jan Lammerding, Mohammad Mofrad

35.2 Structural Components of a Cell ...... 1173 Numerous applications of nanotechnology have 35.2.1 Membranes ...... 1173 been developed to probe the unique mechani- 35.2.2 Cytoskeleton...... 1174 cal properties of cells. In addition, since biological 35.2.3 Nucleus...... 1177 materials exhibit such a wide spectrum of prop- 35.2.4 Cell Contractility and Motor Proteins1178 erties, they offer new concepts for nonbiological 35.2.5 Adhesion Complexes...... 1179 biomimetic applications. In this chapter, the viscoelastic properties of a cell and its subcellular 35.3 Experimental Methods ...... 1179 compartments are described. First, a qualita- 35.3.1 Methods of Force Application...... 1179 tive picture is presented of the relevant building 35.3.2 Rheological Properties ...... 1183 35.3.3 Active Force Generation ...... 1184 blocks: the cytoskeleton, cell membrane, nucleus, 35.3.4 Biological Responses ...... 1184 adhesive complexes, and motor proteins. Next, the 35.3.5 Nonlinear Effects...... 1184 various methods used to probe cellular and sub- 35.3.6Homogeneity and Anisotropy...... 1184 cellular mechanics are described, and some of the quantitative results presented. These measure- 35.4 Theoretical ments are then discussed in the context of several and Computational Descriptions ...... 1185 theories and computational methods that have 35.4.1 Continuum Models ...... 1185 been proposed to help interpret the measurements 35.4.2 Biopolymer Models...... 1187 and provide nanomechanical insight into their ori- 35.4.3 Cellular Solids ...... 1187 gin. Finally, current understanding is summarized 35.4.4Tensegrity ...... 1188 in the context of directions for future research. 35.5 Mechanics of Subcellular Structures...... 1188 35.5.1 Cell–Cell and Cell–Matrix Adhesions1188 35.5.2 Cell Membranes ...... 1190 35.1 Overview...... 1171 35.5.3 Cell Nuclei ...... 1192 35.1.1 The Importance of Cell Mechanics 35.5.4Mechanosensing Proteins...... 1194 in Biology and Medicine ...... 1171 35.6 Current Understanding and Future Needs1196 35.1.2 Examples Drawn from Biology and Pathophysiology ...... 1172 References ...... 1196

35.1 Overview 35.1.1 The Importance of Cell Mechanics and extracellular events, many of which involve me- in Biology and Medicine chanical phenomena or may be guided by the forces

experienced by the cell. These micro- and nanomechan- D Part All living things, despite their profound diversity, share ical phenomena range from macroscopic events such as a common architectural building block: the cell. Cells the maintenance of cell shape, motility, adhesion, and

are the basic functional units of life, yet they are them- deformation, to microscopic events such as how cells 35 selves comprised of numerous components with distinct sense mechanical signals and transduce them into a cas- mechanical characteristics. To perform their various cade of biochemical signals, ultimately leading to a host functions, cells undergo or control a large range of intra- of biological responses. 1172 Part D Bio-/Nanotribology and Bio-/Nanomechanics

Cell mechanics plays a major role in biology and forces. Most cells are able to sense when a physi- physiology. The ability of a cell to perform its func- cal force is applied to the cell, and respond through tion often depends on its shape, and shape is maintained a variety of biological pathways leading to such di- through structural stiffness. In the blood circulation sys- verse effects as changes in membrane channel activity, tem, red blood cells, or erythrocytes, exist in the form up- or downregulation of gene expression, alterations of biconcave disks that are easily deformed to help in protein synthesis, and altered cell morphology. The facilitate their flow through the microcirculation and signaling cascades that become activated as a conse- have a relatively large surface-to-area ratio to enhance quence of mechanical stress have generally been well gas exchange. White cells, or leucocytes, are spherical, characterized. However, the initiating process by which enabling them to roll along the vascular endothelium cells convert the applied force into a biochemical sig- before adhering and migrating into the tissue. Because nal is much more poorly understood, and only recently their diameter is larger than some of the capillaries have researchers begun to unravel some of these funda- they pass through, leucocytes maintain excess mem- mental mechanisms. Some studies have suggested that brane in the form of microvilli so that they can elongate a change in membrane fluidity acts to increase recep- at constant volume and not obstruct the microcircula- tor mobility, leading to enhanced receptor clustering tion. Airway epithelial cells are covered with a bed of and signal initiation. Stretch-activated ion channels or cilia, finger-like cell extensions that propel mucus along strain-induced activation of G proteins represent other the airways of the lung. Lastly, the cytoskeletal struc- means of mechanotransduction. Similarly, disruption of ture in muscle cells is specifically organized to actively microtubules or conformational changes of cytoskele- generate forces and to sustain large strains. In each of tal proteins that alter their binding affinities have been these examples, the internal structure of the cell along proposed as cellular mechanosensors. Yet others have with the cell membrane provide the structural integrity focused on the role of the glycocalyx, a layer of to maintain the particular shape needed by the cell to ac- carbohydrate-rich proteins on the cell surface, in the complish its function, although the specific components response of endothelial cells to fluid shear stress. An- of the structure are highly variable and diverse. other potential mechanism is that forced deformations Cell mechanics also plays an important role in within the nucleus could directly alter transcription cell migration. Migration is critical during early de- or transcription factor accessibility. Constrained au- velopment, but also in fully differentiated organism, tocrine signaling is yet another mechanism whereby the e.g., in wound repair when cells from the surrounding, strength of autocrine signaling is regulated by changes undamaged tissue migrate into the wound, in angiogen- in the volume of extracellular compartments into which esis (i. e., the generation of new blood vessels), or in the receptor ligands are shed. Changing this volume combating infection when cells of the immune system by mechanical deformation of the tissues can increase transmigrate from the vascular system across the vessel the level of autocrine signaling [35.5]. Finally, others wall and into the infected tissues. have proposed conformational changes in intracellular Cell migration processes occurs in several stages proteins in the force transmission pathway connecting that include: the extracellular matrix with the cytoskeleton through focal adhesions (FAs) as the main mechanotransduc- 1. Protrusion, the extension of the cell at the leading tion mechanism. While all or a subset of these theories edge in the direction of movement may contribute to mechanotransduction, little direct ev- 2. Adhesion of the protrusion to the surrounding sub- idence has been presented in their support. For reviews strate or matrix of this topic, see [35.6–12]. 3. Contraction of the cell that transmits a force from these protrusions at the leading edge to the cell 35.1.2 Examples Drawn from Biology body, pulling it forward and Pathophysiology 4. Release of the attachments at the rear, allowing net atD Part forward movement of the cell to occur [35.1–4] Many studies during the past two decades have shed Cellular mechanics and dynamics are critical mod- light on a wide range of cellular responses to mechan-

35.1 ulators at each of these steps. ical stimulation. It is now widely accepted that stresses The importance of cell mechanics in biology is experienced in vivo are instrumental in a wide spec- most apparent in mechanotransduction, i. e., the ability trum of pathologies. One of the first diseases found to of the cell to sense and respond to externally applied be linked to cellular stress was atherosclerosis, where Cellular Nanomechanics 35.2 Structural Components of a Cell 1173 it was demonstrated that hemodynamic shear influ- and defects – caused for example by mutations in ences endothelial function, and that conditions of low mechanosensitive proteins – can result in muscular dys- or oscillatory shear stress are conducive to the forma- trophies. Asthma is another particularly salient example tion and growth of atherosclerotic lesions. Even before where the epithelial cells lining the airways are sub- then, the role of mechanical stress on bone growth jected to stresses as the pulmonary airways constrict and healing was widely recognized, and since then, during breathing, and airway wall remodeling. Further many other stress-influenced cell functions have been knowledge of the mechanisms by which cells respond identified. Mechanotransduction of muscle cells is piv- to such forces could enhance our understanding of these otal to exercise-induced muscle growth (hypertrophy), diseases.

35.2 Structural Components of a Cell Aside from providing the outer boundary of a cell In addition to the phospholipids, the membrane con- and nucleus, membranes enclose numerous intracellu- tains glycolipids and cholesterol. While the amount lar structures as well, and the discussion in this section of glycolipids is small, constituting only ≈ 2% of the pertains to all of these. total lipid content, cholesterol is a major membrane constituent, ≈ 20% by weight, a value that remains 35.2.1 Membranes quite constant among different cell types. The specific membrane composition is critical for determining mem- The term membrane generally refers to the phospholipid brane structural integrity; for example, both the bending bilayer and the proteins associated with it. The phospho- stiffness and the viscosity of the lipid bilayer are lipids contained in the membrane are arranged in two strongly dependent on the cholesterol content. Cellu- layers, or leaflets, with their hydrophobic tails pointing lar membranes also contain many membrane-associated inward and their hydrophilic heads outward. Together, proteins, which account for ≈ 50% of the membrane they constitute a bilayer ≈ 6 nm in thickness (Fig. 35.1). by weight but, because of their relatively large molecular weight, only ≈ 1–2% of the number of membrane molecules. Membrane proteins serve a variety of func- Outside tions, ranging from signaling to transport of ions and Oligosaccharide other molecules across the membrane to cell–cell and Peripheral proteins cell–matrix adhesion. Integral protein The plasma membrane has associated macromolecular structures on both intra- and extracellular sides, giving rise to a three-layer composite construction. On Leaflets the intracellular side, the membrane is physically attached to a cortex or the cytoskeleton. The cortex is a dense, filamentous structure that lends stiffness to the membrane, and can also interact with various transmembrane proteins, often impeding their free diffusion either by steric interactions or direct chemical bonding. In some cells, the cortex is simply a region of Fatty acid tails Hydrophilic polar heads dense cytoskeletal matrix in the vicinity of the bilayer. Inside In others, it exhibits a distinctly different structure or

Fig. 35.1 A model of the lipid bilayer, showing the hy- composition; for example, erythrocytes possess a cortex D Part drophilic heads (polar head groups) on the exterior surfaces comprised of a network of spectrin tetramers linked by and the hydrophobic tails (fatty acid tails) pointing in. Also actin ﬁlaments. This network is attached to the mem-

shown are examples of transmembrane or integral membrane by ankyrin and the integral membrane protein 35.2 brane proteins. The total thickness of the bilayer is ≈ 6nm band 3. This spectrin network accounts for much of the (after [35.13]) bending stiffness exhibited by the red cell membrane. 1174 Part D Bio-/Nanotribology and Bio-/Nanomechanics

Most cells are coated by a glycocalyx, which has to form bundles and lattice networks. The higher-order been shown to extend as far as 0.5 μm from the structures formed by these polymers, as we will see, are surface of endothelial cells, where it forms a com- critical determinants of cytoskeletal elasticity and can pressible barrier separating circulating erythrocytes and vary substantially between different cells. leucocytes from the endothelial membrane. The glycocalyx is comprised of short oligosaccharide chains, Actin Microfilaments glycoproteins, glycolipids, and high-molecular-weight Actin filaments play an essential role in virtually all proteoglycans, all organized into an interconnected net- types of motility. Actin–myosin interactions are of ob- work with an overall negative charge. Although its vious importance in muscle, but are also instrumental in function is not completely understood, it apparently the migration and movement of nonmuscle cells. Actin plays a role in macromolecular transport across the en- polymerization is thought to be one of the factors initi- dothelium and is an important factor in the interaction ating cell migration through the formation of filopodia between bloodborne cells and the endothelium. Stud- or lamellipodia [35.15]. ies have demonstrated that the glycocalyx in a capillary Actin is one of the most prevalent proteins found is readily compressed by a passing leukocyte, yet is in the cell, ranging in concentration from 1–10% by sufficiently rigid to prevent flowing erythrocytes from weight of total cell protein in nonmuscle cells, to approaching the endothelial surface. 10–20% in muscle. Molecular actin is comprised of 375 amino acids (molecular weight 43 kDa) and is 35.2.2 Cytoskeleton found in at least six forms that differ from each other only slightly. Of these, four are found in muscle, the The cytoskeleton is the network of biopolymers that other two in nonmuscle cells. Actin exists either in permeates the cell and largely account for its structural globular form (G-actin monomers) or filamentous form integrity. At high magnification, this network appears to be comprised of several distinct types of intertwined filaments with a variety of interconnections, as can be seen in the micrograph in Fig. 35.2. The apparent stiffness of the network, as that of other fibrous materials, depends fundamentally upon the elastic properties of the constituent fibers. The cytoskeletal matrix is primarily comprised of three constituents, actin microfilaments (≈ 7–9nm in diameter), microtubules (24 nm), and intermediate filaments (≈ 10 nm) (Table 35.1). These form a complex interconnected network that exists in a state of constant flux, especially when the cell is dividing, migrating or undergoing other dynamic processes. All are polymers Fig. 35.2 The cytoskeleton of a macrophage lamel- built from protein subunits, held together by noncova- lipodium as seen by electron microscopy. The fibrous lent bonds. They also share the common feature that structure is mainly comprised of actin filaments (c John they cross-link, often with the aid of other proteins, Hartiwick, http://expmed.bwh.harvard.edu)

Table 35.1 The main constituents of the cytoskeleton and their mechanical properties. Note that the effective Young’s modulus, estimated using the diameter and bending stiffness values for actin ﬁlaments and microtubules, is approximately the value that would be predicted on the basis of van der Waals attraction between two surfaces [35.14]. Recall that the 4 persistence length and bending stiffness are related through the expression lp = Kb/kBT,andthatKb = EI = π/4a E

atD Part for 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) 35.2 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 Cellular Nanomechanics 35.2 Structural Components of a Cell 1175

(F-actin polymer), with the balance between the two be- is no single molecular constituent that comprises the in- ing a highly dynamic process that is finely regulated by termediate filaments (IFs); instead, there are more than a variety of different factors. F-actin is a long, flexible 50 different IF genes that have been identified. filament, ≈ 9–10 nm in diameter. Subunits (monomers) Intermediate filaments constitute ≈ 1% of total pro- are organized into a double-stranded helix having struc- tein in most cells, but can account for up to 85% tural and functional polarity (pointed or negative, and in cells such as epidermal keratinocytes and neu- barbed or positive ends) and a half-pitch of ≈ 37 nm rons [35.21]. While they come in many varieties, they (Fig. 35.3a). At the barbed end, an adenosine triphos- share a common structural organization. All have a large phate (ATP) binding cleft is exposed, allowing for central α-helical rod domain flanked by amino- and binding of monomer and linear growth of the filament. carboxy-terminal domains. Assembly occurs by the for- ATP binding and hydrolysis play a critical role in regu- mation of dimers into a coiled coil structure. Then lating actin dynamics and controlling the length of the the dimers assemble in a staggered antiparallel ar- actin filament. ray to form tetramers that connect end-to-end to form The F-actin filaments can further organize into qua- apolar protofilaments. These protofilaments assemble ternary structures such as bundles or a lattice network into a rope-like structure containing ≈ 8 subunits each with the aid of actin binding proteins (ABP). The bun- (Fig. 35.3c) [35.21]. Although intermediate filaments dles, also referred to as stress fibers, are closely packed are more stable than microfilaments, they can be modi- parallel arrays of filaments, connected to each other by fied by phosphorylation. Intermediate filaments exhibit several members of the ABP family (e.g., α-actinin, a lower bending stiffness than either microfilaments or fascin, and scruin). Stress fibers can vary in size, but microtubules, as evidenced by their persistence length are typically several hundred nanometers in diameter. of only 1–3 μm. Linker proteins such as bullous pem- Actin stress fibers tend to form when the cell requires phigoid antigen 1 (BPAG1) and plectin contain both additional strength, such as in endothelial cells, in re- actin and IF binding domains, providing a means by sponse to an elevated shear stress, or in migrating which these networks can be linked. Evidence also ex- fibroblasts. These fibers often concentrate around and ists for plectin binding to microtubules. attach to focal adhesion sites, and are therefore critical Intermediate filaments are often found surrounding to cell adhesion. Actin networks consist of an inter- the nucleus and extending outward to the plasma mem- connected matrix of F-actin filaments, the junctions of brane. In epithelial cells, keratin filaments connect to which are often seen to be nearly orthogonal. At least the plasma membrane at desmosomes and hemidesmo- two distinct types of network are observed – corti- somes and help them to withstand mechanical stress. cal (membrane-associated and more planar in nature) and non-membrane-associated, which possess a more Microtubules isotropic three-dimensional structure. The formation of Microtubules are important in determining cell shape, bundles and networks is facilitated by a variety of and they play a critical role in separating chromosomes cross-linking proteins such as filamin, which forms a V- during mitosis. Microtubules are central to the motion shaped polymer that connects two actin filaments nearly of cilia and flagella. Compared with either microfila- at right angles. ments or intermediate filaments, microtubules are rigid The elastic properties of actin filaments have been structures, but exist in a dynamic equilibrium, much as measured in a variety of ways: by axial stretch [35.16], do microfilaments. Microtubules take the form of hol- twisting [35.17, 18], and bending [35.19]. By all low cylinders with ≈ 25 nm outer diameter and 14 nm methods, single actin filaments were found to ex- inner diameter. The tubular structures are comprised hibit a Young’s modulus in the range of (1.3–2.5) × of tubulin, a globular dimer consisting of two 55 kDa 109 N/m2. This range compares favorably with that polypeptides, α-andβ-tubulin. The dimers polymerize measured for silk and collagen [35.20] and is also to form microtubules that consist of 13 linear protofil-

roughly consistent with predictions based on van der aments forming a hollow-cored cylinder (Fig. 35.3b). D Part Waals bonding between surfaces [35.14]. The ﬁlaments are polar, having a rapidly growing end and a slowly growing end, mediated by hydrolysis of

Intermediate Filaments guanosine triphosphate (GTP) after polymerization. If 35.2 Intermediate ﬁlaments, which form ≈ 10 nm-diameter hydrolysis occurs too quickly, before new GTP-bound ﬁbers, are much less studied than actin and not as well tubulin can bind to the end, the microtubule might dis- characterized. Perhaps this is due to the fact that there assemble. Depending on the rate of hydrolysis and rate 1176 Part D Bio-/Nanotribology and Bio-/Nanomechanics

a) b) Negative or Depolymerization pointed end

14 nm 25 nm

F-actin

G-actin

7–9 nm -Tubulin α-Tubulin Polymerization Positive or barbed end

c) Polypeptide N C

Head Tail

Coiled-coil C N Dimer N C

N C Tetramer CN

Protofilament

10 nm

Filament atD Part

Fig. 35.3 (a) Schematic showing the polymerization of G-actin monomers to form F-actin. (b) α-andβ-tubulin organized

35.2 into microtubules. (c) Organizational structure of an intermediate ﬁlament

of GDP-bound tubulin addition to the end, the micro- ends tend to alternate between periods of steady growth tubule can either grow or shrink [35.22]. In fact, free and disassembly in a stochastic manner. Cellular Nanomechanics 35.2 Structural Components of a Cell 1177

During interphase, microtubules tend to be anchored This process allows the ≈ 2 m of human DNA to be at their negative ends to the centrosome, located near the packaged into a nucleus a few micrometers in diam- nucleus. From there they extend to all parts of the cell, eter. The chromatin fibers in turn form higher-order suggesting a strong role in maintaining the structural structures such as euchromatin and heterochromatin, integrity of the cell and providing a cellular highway which can be dynamically regulated by biochemical system for cargo transported by motor proteins such as modification (e.g., by methylation, acetylation or phos- the kinesins and dyneins. In many of these situations, phorylation) of histone proteins and DNA. Furthermore, the microtubule is thought to play a structural role that chromatin fibers from single chromosomes often form relies on it having a high bending stiffness, which was distinct and nonoverlapping chromosome territories. In found to be on the order of 2.6×10−23 Nm2 [35.20]. addition to chromatin and nuclear bodies, the nuclear interior also contains several structural proteins, includ- 35.2.3 Nucleus ing nuclear actins, myosin, spectrin, and nucleoplasmic lamins A and C [35.25–29]. However, it remains un- The nucleus is the distinguishing feature of eukary- clear what intranuclear structures (often referred to as otic cells and directs and controls deoxyribonucleic acid nuclear matrix) these proteins form within the nuclear (DNA) replication, ribonucleic acid (RNA) transcrip- interior and how such structures could contribute to nu- tion and processing, and ribosome assembly [35.23]. clear processes such as transcription. Most eukaryotic cells contain a single nucleus. How- Importantly, the nucleus cannot be viewed in isola- ever, some specialized cells (e.g., red blood cells) tion from the surrounding cytoskeleton. The molecular become anucleate during their maturation, while other mechanism by which the nucleus is connect to the cells such as skeletal and cardiac muscle cells can be- cytoskeleton has puzzled researchers for many years, come multinucleated due to cell fusion. With a diameter as it was unclear how forces required for nuclear po- in the range ≈ 5–20μm, the nucleus is the largest sitioning and anchoring could be transmitted across cellular organelle. The nucleus is separated from the cy- the 50 nm-wide perinuclear space. Recent work in toplasm by the nuclear envelope, which consists of two Caenorhabditis elegans, Drosophila melanogaster,and lipid bilayers, the inner and outer nuclear membrane, mammalian cells led to the discovery of two new fami- and the underlying nuclear lamina, a dense protein network consisting mostly of lamin proteins that control the nuclear shape, size, and stability (Fig. 35.4). The Cajal body outer nuclear membrane is continuous with the endoplasmic reticulum and connects to the inner nuclear membrane at the nuclear pores, thus enclosing the per- PML body inuclear space. Nuclei typically contain a few thousand nuclear pores, comprised of hundreds of proteins that Nucleolus form the nuclear pore complex that controls transport between the nucleus and the cytoplasm [35.24]. Endoplasmic The nuclear interior contains the packaged DNA Euchro- reticulum in the form of chromatin as well as diverse intranu- matin clear compartments referred to as subnuclear bodies Heterochromatin that include nucleoli, Cajal bodies, and promyelocytic Nuclear pore Outer nuclear membrane leukemia bodies (PML) bodies. These subnuclear bod- Nuclear lamina Inner nuclear membrane ies are not surrounded by membranes but self-organize through processes only incompletely understood. Chro- Fig. 35.4 Schematic drawing of a mammalian cell nucleus. The nu- matin can be organized into two distinct forms, the cleus is surrounded by the inner and outer nuclear membranes and

dense and transcriptionally silent heterochromatin often the underlying nuclear lamina. Nuclear pores allow transport be- D Part located at the nuclear periphery, and the gene-rich and tween the nuclear interior and the cytoplasm. The nuclear interior transcriptionally active euchromatin. Chromatin is com- consists of densely packed heterochromatin, mostly located at the

prised of 30 nm ﬁbers that arise from regular wrapping nuclear periphery, the transcriptionally active euchromatin, and sev- 35.2 of DNA around histone octamers to form nucleosomes eral intranuclear structures such as Cajal and PML bodies and the resembling beads on a string and subsequent further nucleoli. Not shown are intranuclear structures formed by lamins compaction facilitated through the linker histone H1. and other proteins that constitute the nuclear matrix 1178 Part D Bio-/Nanotribology and Bio-/Nanomechanics

lies of nuclear envelope proteins that are ideally suited bind to cytoskeletal F-actin and intermediate filaments. to transmit forces from the cytoskeleton across the nu- At the same time, nesprins physically interact across the clear envelope to the nuclear interior [35.30–39]. These perinuclear space with Sad1p/UNC-84 (SUN) proteins, findings have led to the current model of nuclear– which are located at the inner nuclear membrane. There, cytoskeletal coupling (Fig. 35.5), in which large nesprin SUN proteins can bind to lamins, chromatin, and other isoforms located on the outer nuclear membrane can as-yet unknown nuclear envelope proteins, thus creat- ing a physical link between the cytoskeleton and the nucleus [35.32]. Due to this intricate coupling between the nucleus and the cytoskeleton, defects in nuclear envelope proteins can have direct effects on cytoskeletal structure and mechanics. For example, fibroblasts lacking the nuclear envelope proteins lamin A and C have reduced cytoskeletal stiffness and disturbed actin, vi- mentin, and microtubule organization [35.40–42], and mutations in nesprins, similar to lamins, can result in muscular dystrophies [35.43].

35.2.4 Cell Contractility and Motor Proteins

All muscle cells use the molecular motor comprised of actin and myosin to produce active contraction. These are arranged in a well-defined structure, the sarcom- ere, and the regularity of the sarcomeres gives rise to the characteristic striated pattern seen in skeletal muscle cells and cardiac myocytes. Importantly, even nonmuscle cells contain contractile machinery, which they use for a variety of functions such as maintaining cell tension, changing cell shape, and cell migration. Promi- nent bundles of actin filaments, called stress fibers, are Nesprin 1/2Plectin Nuclear pore complex contractile and house myosin filaments. The macro- Actin molecular organization of stress fibers bares similarities Nesprin 3 filament Lamins to sarcomeres – actin filaments are in parallel arrange- SUN 1/2 Intermediate Chromatin ment with the filament polarity alternating alongside; filament laterally a space of ≈ 10 nm is maintained that situ- ates parallel bipolar myosin II filaments in between. Fig. 35.5 Nuclear cytoskeletal coupling. Schematic drawing of the The myosin head moves toward the positive pole of current model of nuclear cytoskeletal coupling in mammalian cells. the actin filaments during a power stroke, producing The nesprin 1 and nesprin 2 giant isoforms contain an N-terminal a net contractility for an alternating-polarity arrange- actin-binding domain that can interact with cytoskeletal actin fil- ment. Stress fibers (like sarcomeres) have a troupe of aments. Shorter isoforms of nesprin 1 and 2 do not contain an proteins, e.g., α-actinin, filamin, troponin, caldesmon, actin-binding domain, but it is thought that the large spectrin repeat and tropomyosin, which control the arrangement of regions (brown and grey spheres) can interact with other cytoskele- actin and myosin, and regulate their interaction. The for- tal elements. Nesprin 3 can directly bind to plectin, which can as- mation and strengthening of stress fibers is intricately sociate with intermediate filaments. Nesprins can interacts with the linked to the mechanism by which cells respond to their

atD Part inner nuclear membrane proteins SUN1 and SUN2 across the per- force environs. First, stress ﬁbers are known to form inuclear space through their C-terminal KASH (Klarsicht, ANC-1, between points in the cell where actin myosin contrac- Syne Homology)-domain. SUN proteins can in turn bind to the nu- tility is resisted. Typically these are hot-spots of protein

35.2 clear lamina, nuclear pore complexes, and possibly other, yet to be activity known as focal adhesions, where the actin cy- identiﬁed, proteins at the inner nuclear membrane. These proteins toskeleton gets anchored to transmembrane integrins, interact with chromatin and intranuclear proteins, thus completing which are anchored to the matrix proteins. The focal the physical link between the cytoskeleton and the nuclear interior adhesion also includes a cascade of proteins that relay Cellular Nanomechanics 35.3 Experimental Methods 1179 biochemical signals, regulate its strengthening, and or- force transmitted to the ECM and the tension applied to ganize the cytoskeleton for the growth of stress ﬁbers the adhesion complex is necessary for promoting focal (like cross-linking actin and recruiting myosin). adhesion development. In contrast, disruption of myosin activity effectively inhibits the formation of focal ad- 35.2.5 Adhesion Complexes hesions. In the absence of myosin contractile force, externally applied mechanical forces can also promote Adhesion complexes are collections of proteins forming the formation of focal adhesions. This force-regulated a physical linkage between cytoskeleton and extracel- focal adhesion assembly allows a cell to probe the lular matrix (ECM) and between cells. A cell can use mechanical stiffness of its surroundings and respond ac- focal adhesions to gain traction on the ECM during the cordingly, for example, by migrating in the direction process of spreading and migration. The assembly of of increasing substrate stiffness. Adhesion complexes focal adhesions is a dynamical process that is closely also play an important role in helping tissues form by regulated by the mechanical and chemical cues that holding the cells together. The cells of most tissues the cell experiences. Both intracellular and extracellular are bound directly via cell–cell junctions. Cell–cell ad- mechanical stresses transmitted through focal adhesions hesion complexes are found in many different types, are important in the formation of a focal adhesion com- depending on tissue and cell type, and serve in both plex, whereas the release of stress results in the turnover mechanical coupling of cells as well as intercellular of a focal adhesion. The myosin-mediated contractile transport.

35.3 Experimental Methods 35.3.1 Methods of Force Application Table 35.2 Methods of force application Method Typical force range Measuring cellular or subcellular biomechanics often requires the application of precisely controlled forces Atomic force microscopy 10 pN–100 nN to single or multiple cells and quantification of the in- Microindenter 1–100 nN duced deformation, although some techniques rely on Microplate stretcher 1–100 nN detecting forces generated by the cells or on observ- Magnetic bead microrheology 10 pN–1 nN ing particles within the cytoplasm subjected to thermal (twisting) motion. Consequently, experimental methods can be di- Magnetic bead microrheology 100 pN–10 nN vided into active (Table 35.2,Fig.35.6) and passive (pulling) techniques. In the active techniques, applied forces are Optical traps 1–500 pN generally in the same range as physiological forces act- Micropipette aspiration 1–100 nN ing at the cellular and molecular level (Table 35.3), and Substrate strain 1–30%strain induced displacements are on the nanometer or microm- Shear flow 1–100 Pa eter scale. For most methods – whether they are active . or passive – the cellular deformations are detected based MEMS devices 0 5–1500 nN on computer-based image analysis of bright-field or Table 35.3 Typical force ranges in cell biology fluorescence microscopy images, but some techniques also apply quadrant photodiode detectors for faster and Biological force Force range higher-resolution tracking of microspheres embedded in Force generated by motor ≈ 1–10 pN the cytoplasm. proteins (e.g., kinesin, myosin)

Force transmitted ≈ 1–200 pN D Part Active Measurements by protein–protein interactions (rate dependent) Atomic Force Microscopy. Atomic force microscopy Force required for (partial) protein ≈ 100 pN

(AFM) – only developed in 1986 – is now routinely 35.3 unfolding used in cell biology to image cells, measure cytoskele- ≈ μ tal stiffness, and quantify single-molecule interactions. Force generated by migrating 1 nN–10 N In an atomic force microscope, a small tip attached to or contracting cells 1180 Part D Bio-/Nanotribology and Bio-/Nanomechanics

chanical properties of the probed cell (e.g., Young’s a) f) modulus or shear modulus) are inferred from the force– indentation curves, often assuming a Hertz model for linear elastic, isotropic material [35.44], although several modifications have been proposed to account for example for the finite thickness of thin cell extensions b) g) such as lamellipodia [35.45]. These theoretical models often provide a surprisingly good fit to the experimental data, despite the fact that cells are nonisotropic, nonhomogeneous, and nonlinear elastic materials. One benefit of the atomic force microscope is that the us- c) h) 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 for single-molecule measurements, for example, to mea- d) i) sure bond strength. In this case, the AFM probe is functionalized with low concentrations of the protein of interest and then brought into contact with the appropriate binding partner immobilized on a rigid surface. After binding occurs, the AFM tip is then carefully re- e) j) tracted until the molecules dissociate, which results in a sudden drop of force. Rupture forces at different ve- locities can be directly inferred from AFM data.

Microindenter and Microplate Stretcher. These Fig. 35.6a–j Experimental methods for cellular force ap- custom-made devices are closely related to the AFM plication. Overview of techniques to apply precisely principle. In the case of the microindenter, a small controlled forces or deformations to cells for active cellular (≈ 10–100 μm-diameter) probe is used to poke single biomechanics measurements: (a) atomic force microscopy, cells while measuring induced cytoskeletal and nuclear (b) microindentation or cell poking, (c) parallel microplate deformations under a fluorescence microscope. The ap- stretcher, (d) pulling magnetic bead microrheology (single- plied force is measured with a sensitive force transducer pole magnetic trap), (e) optical trap, (f) micropipette attached to the indenter. For the microplate stretcher, aspiration of adherent cell, (g) substrate strain experi- cells are placed between a rigid, piezo-controlled plate ments, (h) flow chamber for fluid shear-stress application, and a thin, flexible plate and allowed to adhere to both (i) micromachined device, in which the cell is plated onto plates. The slides are then slowly moved apart, thus microscopic platforms that are then moved apart, (j) cell stretching the cell between the plates while imaging the plated on micropillars that deform (i. e., deflect) under the experiment under a microscope. Here, the induced de- cellular traction force. This passive technique can be mod- flection of the thin plate is used to infer the applied ified by embedding small magnetic particles in some of the force. The forces that can be achieved with these tools pillars, which can then be actively manipulated to apply are higher than that of a typical atomic force microscope a highly localized force onto the basal cell surface and are sufficient to induce significant deformations of an entire cell. a flexible cantilever is controlled with (sub-)nanometer precision to carefully probe (i. e., indent) the cell sur- Magnetic Twisting Cytometry. In this technique, me-

atD Part face. Deflection of the cantilever can then be used to chanical measurements are based on the displacements infer the indentation depth and calculate the applied of small (≈ 0.2–5μm diameter), ferromagnetic or force. The tip of the cantilever typically has the shape paramagnetic beads attached to the cell surface and ≈ 35.3 of a pyramid (with a tip radius of 20 nm), but some subjected to a magnetic force. Early versions of this applications use small polystyrene beads (0.1–2μm technique [35.46, 47] used two orthogonal magnetic diameter) attached to the cantilever or tipless probes fields, one to magnetize the ferromagnetic particles with to provide a larger contact area with the cell. The me- a brief, intense pulse, and the second one to induce Cellular Nanomechanics 35.3 Experimental Methods 1181 a twisting, magnetic torque to the beads. The induced optical trap system is that multiple beads can be inde- bead rotation is then measured by a change in the pendently controlled by splitting the laser beam, so that orientation of the induced magnetic field or through mi- cells can, for example, be stretched between two beads croscopic observations. The latter approach offers the that are slowly moved apart. The biggest limitation of advantage that it can detect rotation and displacement using optical traps in cell mechanics experiments is that of individual beads, so that loosely attached beads that the maximal force level is limited to < 1 nN, as larger might rotate freely can be excluded from the overall forces would require higher laser power that could ex- measurements. In a related variation of this technique, cessively heat the cell. While the small forces generated often referred to as magnetic trap or tweezers, paramag- by an optical trap are ideally suited for single-molecule netic beads on the cell surface or inside the cytoplasm studies and are sufficient for experiments that measure, are manipulated by a single- or multipole electromag- for example, membrane tethers, they are often too small net controlled through a computer. The force acting on to induce large-scale cellular deformations, especially a paramagnetic bead inside a magnetic field is given by when probing stiffer cells such as myocytes. the equation F = μ0χV∇(H · H), where F is the magnetic force, μ0 is the permeability constant, χ is the Micropipette Aspiration. In these experiments, sin- volume susceptibility, V is the bead volume, and H is gle adherent or suspended cells are partially aspirated the external magnetic field strength. Thus, the applied into a micropipette with a ≈ 2–10μm-diameter open- force increases with increasing field strength and in- ing by applying precisely controlled suction pressure creasing field gradient. For a single-pole magnetic trap, (typically 100–10 000 Pa). The aspirated cell is imaged the magnetic field decays rapidly with increasing dis- on a microscope and cellular deformations such as the tance from the tip of the pole. This results in a steep aspirated tongue lengths are computed from the cell gradient and large forces near the tip that exponentially geometry. In a technique called fluorescent/confocal- decay away from the tip, requiring careful calibration imaged microdeformation, fluorescent labeling of spe- of the magnetic trap. One of the limitations of single- cific intracellular components such as the nucleus, the pole magnetic traps is the unidirectional force direction, nuclear lamina or nucleoli is used to provide additional i. e., forces can only be exerted in the direction towards information on the subcellular deformations during the magnetic trap (pulling), but this can be overcome micropipette aspiration [35.48–51]. One potential limi- by using multipole magnetic traps. Another limitation is tation of the micropipette aspiration technique is that the that the bead localization on the cell is random, so care interpretation of the experiments is not always straight- must be taken to only compare results from cells with forward. Analytical or computational models are often similar bead positions (e.g., on the nucleus, the nuclear necessary to derive material properties from the ge- periphery or the lamellipodia). Lastly, the induced bead ometric measurements of the aspirated cells and the rotation and displacement are strongly dependent on applied pressure, and the underlying assumptions may the bead attachment angle, i. e., how deep the magnetic at times be difficult to validate. However, this technique bead is embedded in the cell surface, requiring careful has been proven very useful when studying cells with controls or confirmation by confocal three-dimensional relatively homogeneous structural organization such as (3-D) reconstruction. red blood cells or neutrophils [35.49, 52–54].

Optical Traps/Tweezers. This technique is similar to Substrate Strain. Unlike the other techniques, these ex- the magnetic trap experiments, as the induced displace- periments do not apply controlled forces to single cells, ment of microscopic beads (50–1000 nm) attached to but instead use carefully controlled strain application the cell surface or inside the cytoplasm subjected to to induce deformation in cells plated on a ﬂexible sub- a precisely controlled force is used to determine the strate. Generally, cells are plated on transparent, elastic mechanical properties of the cytoskeleton. The major silicone membranes coated with extracellular matrix

difference is that, in an optical trap, a focused laser proteins and subjected to uniaxial or biaxial strain. De- D Part beam is used to position and displace a bead with high pending on the particular experiment, the strain can be refractive index on the cell, allowing precise bead ma- held constant or varied over time (e.g., cyclic strain ap-

nipulation in all directions. The optical trap acts as an plication). Strain levels normally do not exceed 30% in 35.3 elastic spring with a tunable spring constant, so the order to avoid damaging the cells, but the exact levels force applied to the bead can also be controlled with are cell-type dependent. Cells are imaged under a mi- high precision (Chap. 32). Another advantage of the croscope before, during, and after strain application, 1182 Part D Bio-/Nanotribology and Bio-/Nanomechanics

and small markers or fluorescently labeled components ous custom-designed devices to measure cellular of the cells are used to calculate intracellular deforma- mechanics, including microelectromechanical systems tions and applied membrane strain. Since the cell is (MEMS). Typically, these devices contain an actuator firmly attached to the extracellular matrix on the sil- to apply precisely controlled forces or deformations to icone membrane through cell surface receptors which single cells and a force sensor, often comprised of an are connected to the cytoskeleton, the cytoskeleton element with known spring constant that deforms under will experience strain levels comparable to the applied the applied load. In some cases, these two components substrate strain, whereas the stiffer nucleus typically de- can be combined into a single element. One advantage forms significantly less [35.41, 55, 56]. The advantages of micromachined devices is the ability to directly con- of this technique are that force can be applied to sev- trol force application and sense the applied forces with eral cells at once and that the strain application closely high precision without requiring elaborate assumptions resembles physiological mechanical stress, e.g., in mus- of cellular structure and mechanical properties. Also, by cle cells or endothelial cells subjected to blood vessel appropriately tuning the geometry of the force sensor, expansion. The major limitations are that this technique relatively high forces (up to 1500 nN) can be measured, is only suitable for adherent cells and that the applied allowing measurements of forces required to detach ad- forces cannot be determined directly. herent cells from the substrate. The major disadvantages are the still relatively high costs and the need for spe- Shear Flow. The most commonly used devices for shear cialized equipment in the fabrication process. stress application are the cone and plate rheometer and flow (or perfusion) chambers. Cone and plate rheome- Passive Measurements ters allow precise control over the applied shear stress, In contrast to the active measurements, these experi- but are generally not equipped to image cells during ments measure forces generated by the cells themselves shear stress application, making it difficult to visual- or quantify the random motion of particles embed- ize induced cellular deformations. On the other hand, ded in the cytoplasm subjected to thermal fluctuations. flow chambers are routinely used to study cellular de- Typical examples of passive measurement techniques formations under shear stress [35.57] or to investigate include particle tracking of beads in the cytoplasm cellular responses to shear stimulation such as calcium and traction force microscopy. In passive bead mi- influx or cytoskeletal remodeling. Parallel-plate flow crorheology, small (0.1–2μm diameter) beads are chambers are made of transparent glass slides sepa- injected into the cytoplasm or taken up by endocyto- rated by a thin spacer/gasket (the channel height is often sis and are then tracked with high spatial and temporal ≈ 100 μm) and can thus apply precisely controlled fluid resolution with a laser beam and quadrant photode- shear stress to a cell monolayer plated on the bottom tector. The complex cytoplasmic shear modulus G(s) slide while simultaneously imaging the cells on a mi- can then be computed from the unilateral Laplace croscope. The shear stress at the cell surface can be transform of the measured mean-squared displacement calculated for a Newtonian fluid in the parallel plate Δr2(s) using a generalized Stokes–Einstein equation 2 2 geometry as τ = 6Qμ/wh , where τ is the fluid shear as G(s) = kBT/(πasΔr (s)), where s is the Laplace stress at the wall (and cell surface), Q is the flow rate, frequency, kB is the Boltzmann constant, T is the tem- μ is the viscosity, w is the channel width, and h is the perature, and a is the bead radius [35.58]. Traction force channel height. The shear stress can thus be adjusted to microscopy was originally based on wrinkles gener- physiological shear stress levels (≈ 0.1–10Pa) by al- ated in thin silicone sheets by the contraction of cells tering the height of the flow chamber or modulating the plated on top of these sheets, but has subsequently been flow rate, and experiments can be carried out with ei- refined for more quantitative force determination by ther constant or pulsatile flow. However, the actual shear plating cells on polyacrylamide gels with fluorescent stress at the cell surface might deviate from the pre- beads embedded in the gel. The mechanical stiffness

atD Part dicted wall shear stress, as small variations in the cell of the polyacrylamide gel can be tuned based on the height and topology can cause local variations, which cross-linker concentration, and microscopic measure- might in fact contribute to the alignment of endothelial ments of the induced displacement of beads near the

35.3 cells to the flow direction. surface can be converted into cellular forces exerted on the gel by elastic theory [35.59]. Most recently, a new Micromachined Devices. Within the last decade, generation of traction force microscopy has emerged microfabrication has enabled the design of numer- in which cells are coated on a fine grid of micro- Cellular Nanomechanics 35.3 Experimental Methods 1183 fabricated micropillars made of polydimethylsiloxane G (Pa) (PDMS) [35.60]. In this case, the stiffness of micropil- 105 lars can be controlled by adjusting the cross-sectional G' area or length of the micropillars, and the cellular force G'' exerted on each micropillar can be measured based on the deflection of the pillar using beam bending theory. This technique offers the advantage that force appli- 104 cation is localized to individual micropillars, and that the geometry and stiffness can be independently adjusted.

3 35.3.2 Rheological Properties 10

The rheological properties of the cell, such as its viscoelastic properties and its diffusion parameters, are key to the cell’s ability to accomplish its diverse functions 102 in health and disease. A wide range of computational 10–2 10–1 100 101 102 103 and phenomenological models as well as experimen- Frequency (Hz) tal techniques have been proposed over the past two Fig. 35.7 Typical experimental results for the frequency- decades to describe the cell, giving rise to several, of- dependent shear moduli obtained by magnetic twisting ten contradictory, theories for describing the rheology cytometry (after [35.61]) of the cytoskeleton. The highly heterogeneous structure of the cytoskeleton, coupled with a small linear where Γ is the Gamma function, μ is the vis- response regime [35.62] and active dynamics and con- cosity, and η = tan((x − 1)π/2)G0, ω0,andx are tinuously remodeling, present a major challenge for parameters of the model. All but x,however,have quantitative measurements and descriptions of its rhe- been found to be nearly constant in all experi- ology [35.63]. ments. This turns out to be similar to the behavior Nonetheless, a wide range of computational models of soft glassy materials, although the fundamental exist for cytoskeletal rheology and mechanics, rang- basis for this behavior remains a topic of some de- ing from continuum to discrete descriptions of the bate. cytoskeleton (see reviews in [35.64]). A major chal- The data in Fig. 35.7 [35.61]applytosmalldefor- lenge in cytoskeletal rheology and mechanics is how mations in the linear regime. However, linearity persists to relate experimental observations to theoretical and up to surprisingly large deformations. Much of what phenomenological models. Much effort has recently fo- we know about nonlinear effects has been obtained cused on the interesting rheological behavior of cells, from experiments on reconstituted gels, typically of measured by one of the methods described above. Most actin with one or more actin cross-linking proteins. often, cellular viscoelastic properties are expressed in From these experiments, the following observations terms of the complex shear modulus (see Viscoelastic have been made: Solid/Poroelastic Solid), the real part G indicating the elastic component, and the imaginary part G the vis- 1. Linearity persists up to ≈ 30% strain. cous component. While each measurement method has 2. Values of G and G in this linear regime are of- its drawbacks, perhaps the most comprehensive data ten orders of magnitude lower than are measured in have been obtained through magnetic twisting cytom- cells. etry (Fig. 35.6). These measurements have shown that 3. Strain stiffening is observed at strains > 30%, fol- a cell exhibits a relatively simple power-law behavior lowed by a precipitous drop in G at strains of D Part over much of the frequency domain, with [35.61] ≈ 70%, presumably indicating rupture or unfolding x−1 of the cross-linkers. ω G + G = G + η Γ − x 35.3 i 0 ω (1 i ) (2 ) 0 All of these issues continue to be actively studied in the π hope of generating a comprehensive understanding of ×cos (x − 1) + iωμ , (35.1) 2 cytoskeletal rheology. 1184 Part D Bio-/Nanotribology and Bio-/Nanomechanics

35.3.3 Active Force Generation new proteins as early signaling pathways can also mediate polymerization and depolymerization of cy- The cytoskeleton is an active structure that maintains toskeletal structures. For example, endothelial cells cell shape and facilitates its motion. The cytoskele- exposed to fluid shear stress begin to align with tal network is in a nonequilibrium state that drives the flow direction within 20 min, and the process is motor proteins as force-generating features in cells. completed within 24 h. The multifaceted response to The cytoskeleton is activated by these molecular mechanical stimulation can effectively mediate sev- motors, which are nanometer-sized force-generating eral cellular functions, including DNA and RNA proteins, e.g., myosin. To better understand the na- synthesis, hypertrophy (increase in cell size), prolif- ture of such active force-generation systems, re- eration, apoptosis, migration, and extracellular matrix searchers have developed simplified models of the remodeling. active cytoskeleton by mixing actin with actin binding proteins (e.g., filamin and α-actinin) and molecular 35.3.5 Nonlinear Effects motor proteins myosins [35.65]. Such a reconstituted synthetic cytoskeleton exhibits local contractions As described earlier, cells are composed of intricate reminiscent of living cells. Tension generated by networks of filamentous structures, called the cy- contraction can lead to drastic increase of the cytoskeleton. These networks exhibit unique properties toskeletal stiffness. Such models demonstrated that including relatively large shear moduli, strong signa- a remarkably simple system, with just three compo- tures of nonlinear response in which, for example, the nents (myosin, actin, and ATP), can reproduce key shear modulus can increase drastically under modest phenomena also observed in far more complex living strains [35.68,69]. Models of semiflexible polymer net- cells [35.66]. works have emerged to describe these unique properties and the dynamics of the cytoskeletal networks. These 35.3.4 Biological Responses models involve a semiflexible description of the constituent actin filaments [35.70]. Quantitative models of The cellular responses to mechanical stimuli take the nonlinear behavior of the cytoskeletal network are place over several time scales and range from intra- essential for understanding the complex, dynamic, and cellular signaling to changes in cellular morphology nonlinear behavior of cells, and ultimately the tissues and function. The fastest responses occur within sec- and organs. onds to minutes of stimulation and include changes in intracellular ion concentrations (especially Ca2+) 35.3.6 Homogeneity and Anisotropy through opening of mechanosensitive ion channels and activation of cellular signaling pathways such Cells are inhomogeneous structures composed of vari- as nuclear factor-κB(NF-κB), phosphatidylinositol-3- ous intracellular components, including the lipid bilayer kinase (PI3K)/protein kinase B (Akt), protein kinase C membrane that encases the cell, the actin cortex, which (PKC), Rho family GTPases, and mitogen-activated is a dense actin network providing stability for the protein kinases (MAPKs) [35.67]. These first steps cell membrane, cytoskeletal networks, and the nucleus, do not require synthesis of new proteins, but involve which itself comprises various important substructures. modification (e.g., phosphorylation) and intracellular The cytoskeleton is primarily responsible for the struc- translocation of existing proteins. These initial events tural integrity and stiffness exhibited by a cell. It is often trigger activation of mechanosensitive immedi- composed of a system of highly entangled protein fil- ate early genes such as egr-1, c-fos, c-jun or c-myc aments that permeate the microfluidic space of the encoding transcription factors that turn on additional cytosol. The major components of the cytoskeletal downstream genes. These later response genes often network, i. e., the actin filaments, intermediate fila-

atD Part include proteins involved in cellular structure (e.g., ments, microtubules, and their cross-linking proteins actin, myosin) or extracellular matrix remodeling. How- offer an exquisite microenvironment which is highly ever, cytoskeletal remodeling in response to mechanical inhomogeneous and anisotropic in its structure and

35.3 stress or strain can start even before synthesis of geometry. Cellular Nanomechanics 35.4 Theoretical and Computational Descriptions 1185

35.4 Theoretical and Computational Descriptions As is evident from the discussion above, the cytoskele- section we explore different approaches that have been ton is a complex biopolymer network with varying developed to relate the properties of the individual fibers degrees of connectivity, existing in a state of dynamic to those of the assembled network. equilibrium. The dynamic state arises from the ongoing polymerization and depolymerization of the constituent 35.4.1 Continuum Models filaments and the changing density of cross-links between filaments of the same or different family. This Elastic Solid picture is complicated further by the milieu of other In several of the experiments described earlier, the as- intracellular constituents that may or may not affect sumption of a homogenous and isotropic elastic solid structural properties exhibited by the cell. is used to infer a value for an effective Young’s modu- Ourobjectiveinthissectionistocometoabetter lus for the cell. It is not our intention here to provide appreciation of how the elastic properties and geomet- a comprehensive description of elasticity theory, but ric arrangement of the constituent filaments give rise to rather to outline some of the basics. For a full deriva- the material properties observed by the various experi- tion of the governing equations, the reader is referred to mental methods just described. These experiments have any of a number of excellent textbooks [35.71]. Assum- shown that the elastic modulus of cells can vary con- ing equilibrium conditions (i. e., all forces balance each siderably from one cell type to another. Cells of the other) and a material that can be modeled as a Hookean epidermis, for example, require greater structural in- (i. e., linear) elastic solid, the constitutive equations re- tegrity than the red blood cells subjected to the relatively lating the mechanical stress and strain can be written low shear stresses in the blood. Even within a given cell in tensor notation as τij = Cijklεkl, where the summa- type, the elastic properties can change. Skeletal muscle, tion convention is used. Here τij are the elements of for example, changes its modulus by over an order of the stress tensor, εkl are the elements of the strain ten- magnitude within a small fraction of a second. Other sor, and Cijkl is the 81-element coefficient matrix. In cells change too, but more often over a longer time the case of an isotropic material, the coefficient matrix period, in response, for example, to changes in their reduces to just two independent elastic constants, with mechanical environment. the constitutive equation now reduced to the following According to the various measurements that have simplified form: been made, cells seem to range in shear modulus in τ = 2Ge + λε δ , (35.2) the range ≈ 10–10 000 Pa, and therefore exhibit a stiff- ij ij kk ij ness somewhat lower than collagen gels (or common where λ and G are the Lamé constants with G being gelatins) at low concentration or relaxed skeletal mus- termed the shear modulus. Recall that this expression cle. This wide range of moduli probably says more can also be written in the inverted form about differences in the models used as a basis to in- (1 + ν) ν fer the shear modulus from the data than it does about εij = τij − τkkδij . (35.3) real cell-to-cell variations. At best, these numbers in- E E ferred from experiment should be viewed as measures Here the two material constants are now given as ν, of an effective stiffness, and comparisons between dif- the Poisson’s ratio, and E, the Young’s modulus. When ferent measurement methods and interpretation should using these to solve a particular problem, a set of appro- be made with caution. Other biological materials (e.g., priate boundary conditions also needs to be posed. For bone, wood) exhibit a much higher modulus, but not example, in the case of measurements by an indentation because of their cellular content. Rather, their high stiff- probe, the displacement of the surface in contact with ness is due to the calcification in bone and the collagen the probe might be given. In addition, it would be nec-

matrix found in wood and most plants. In tissues, too, essary to specify that the opposite side of the cell is held D Part the stiffness we measure is more often associated with ﬁxed, and that the unsupported sides and the region on the extracellular matrix with its elastin and collagen, the upper surface not in contact with the probe have zero

than the resident cells. Even within the cytoskeleton, in- applied stress. Given these, the equations above provide 35.4 dividual ﬁlaments (i. e., F-actin, intermediate ﬁlaments, a complete solution. and microtubules) exhibit moduli much greater than the The use of the above equations presumes that the measured bulk modulus of the cytoskeleton. In the next response of the cytoskeleton can be represented by an 1186 Part D Bio-/Nanotribology and Bio-/Nanomechanics

a) b)a 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 between elements in compression (cylinders) and others in tension (lines) (after [35.73])

elastic continuum lacking any discernable microstruc- ton as a poroelastic material. While attempts to apply ture. The validity of this clearly depends upon the length a poroelastic model to studies of cell mechanics are just scale of interest in the problem, and is a serious concern beginning, it remains to be seen whether a viscoelas- in the context of cytoskeletal mechanics where the typ- tic or poroelastic description is more appropriate. For ical length scale (spacing distance between ﬁlaments) a description of poroelastic theory, the reader is referred may be on the order of 10–100 nm. This becomes com- to [35.74]. parable to the linear dimension of the region of interest in some of the experimental procedures (e.g., cell pok- Oscillatory Simple Shear. A useful example of vis- ing, AFM measurements) and needs to be kept in mind. coelastic behavior derives from the case in which the Furthermore, while this is a convenient approach for an- upper surface of a sample is being oscillated sinu- alyzing how cells deform under loading, it provides no soidally in the plane of the surface so that the shear ε = ε∗ ω insight into the relationship between the macroscopic strain satisﬁes 21 21 sin t and the material experi- properties that are measured and the elastic properties ences an oscillatory shear stress. In this case, the stress of the constituent matrix elements. in the viscoelastic material will also vary sinusoidally, but out of phase with the strain. The relationship Viscoelastic Solid/Poroelastic Solid between stress and strain is often described by introduc- A viscoelastic material is to an elastic material as ing the complex shear modulus (as introduced above), a spring and dashpot network is to a system contain- G∗ = G + iG, satisfying the following expressions ing only springs. In both the discrete and continuous τ = ε∗ ω + ω , 21(t) 21(G sin t G cos t) (35.4) systems, the instantaneous displacement or strain is τ = τ∗ ω + φ a function of the stress history. Similarly, the instanta- 21(t) 21 sin( t ) = τ∗ φ ω + τ∗ φ ω , neous stress depends on the strain history. This simple 21 cos sin t 21 sin cos t (35.5) realization, in combination with the assumption that all so that G, G,andφ, the phase lag, are related to the functions describing the material properties are contin- amplitudes of strain and stress through the expressions

atD Part uous, leads to a set of generalized constitutive equations ∞ ∗ for a viscoelastic solid. τ G = ω G(t ) sin ωt dt = 21 cos φ, (35.6) Many of the experiments performed on cells have ε∗ 21 0 35.4 been interpreted in the context of either an elastic or ∞ viscoelastic model. It might be argued, however, that ∗ τ G = ω G t ωt t = 21 φ. (35.7) like bone, cartilage, and many other biological tissues, ( )cos d ε∗ sin 21 it may be more appropriate to view the cytoskele- 0 Cellular Nanomechanics 35.4 Theoretical and Computational Descriptions 1187

Note that G represents the component of stress in phase with the square of the characteristic mesh spacing ξ,the with the imposed strain and G the out-of-phase com- shear stress required to produce a strain θ is given by ponent. For a purely elastic material (in which strain and 2θ (number of filaments) kBTlp stress are in phase), G is simply the shear modulus G τ ∝ F ∝ (35.9) T 3ξ2 and G is zero. On the other hand, it can be shown that (unit area) Le for a viscous Newtonian fluid, G is zero, and G = μω, where μ is the Newtonian viscosity. from which a shear modulus of the (elastic) network Gn It should be noted that these results could also have can be determined as the ratio of stress to strain been obtained by expressing the time-varying strain as τ k Tl 2 ∝ ∝ B p . a complex quantity. In so doing, the shear modulus is Gn (35.10) θ L3ξ2 complex, and G and G are, respectively, the real and e imaginary parts of the stress–strain ratio. The scaling of Gn clearly depends on both the distance between cross-links or entanglements and the mesh 35.4.2 Biopolymer Models size. As the concentration of polymer increases, ξ de- creases and so likely does Le, at least in terms of the Cytoskeletal networks can be viewed as a polymer gel degree of entanglement. As the concentration of actin in which the matrix is considered to consist of relatively binding protein (ABP) increases, Le will decrease. The straight segments connecting junctions where the fila- mesh spacing ξ also depends on the monomer con- ments are either chemically cross-linked, or effectively centration or solid fraction Φ (monomer volume/total so due to entanglements (Fig. 35.8a). Using concepts volume). Assuming that the filaments are stiff and from polymer physics [35.75] the force required to homogeneously dispersed through the medium, this re- change the length of one segment of a polymer filament lationship can be expressed as ξ ∝ a/Φ1/2, where a is (F-actin, for example) of length Le by an amount δ can the monomer or filaments radius. Substituting yields be expressed as k Tl 2Φ 2 2 ∝ B p . kBTlp K Gn (35.11) F ≈ δ ≈ b δ, (35.8) L3a2 T 4 4 e Le kBTLe In the limit of a highly cross-linked network, in which where kB is Boltzmann’s constant, T is temperature, case Le ≈ ξ,thisleadsto Kb is the bending stiffness of the filament, lp is the / persistence length, and Le is the distance between the Φ 5 2 points where the tension is applied (or, the distance be- G ∝ k Tl 2 . (35.12) n B p a2 tween points of entanglement or cross-linking between network filaments). This expression arises from a con- Alternatively, we can express this in terms of the density sideration of the curvature of the polymer resulting from of cross-links ρc. Note that, if ρc is the number of cross- Brownian or thermal fluctuations, and is based on the links per unit volume, it should vary inversely with the assumption that thermal energy is equally partitioned volume associated with each bond or entanglement, i. e., −3 among the different modes of oscillation. The poly- as Le . Combining this with the expressions (35.10– mer filaments are therefore assumed to be bent prior 35.12)gives to the application of stress. Externally imposed forces Φρc either increase or decrease the end-to-end length of G ∝ k Tl 2 . (35.13) n B p 2 these filaments, and the deformation produced depends a both on the intrinsic bending stiffness of the filaments The Young’s modulus of the network En can be ob- and on their initial degree of curvature due to thermal tained by a similar procedure, and can be shown to scale fluctuations. in the same manner as Gn.

For a network comprised of such ﬁlaments in which D Part the distance separating points at which physical bonds 35.4.3 Cellular Solids exist between the ﬁlaments or regions of entanglement

is Le, the change in ﬁlament length between bonds due The theory of cellular solids was developed for the 35.4 to a shear strain θ is δ ∝ θLe. Making use of the fact purpose of relating the macromechanical properties that the number of ﬁlaments per unit area parallel to the of low-density cellular materials to their microstruc- surface on which the stress is applied scales inversely tural characteristics. The approach used is based on 1188 Part D Bio-/Nanotribology and Bio-/Nanomechanics

the concept that the material can be modeled as be- where c2 ≈ 3/8. If the material is linearly elastic and ing comprised of many unit cells, one representation isotropic, these values lead to a value of 1/3forthe of which is shown in Fig. 35.8b. When a cellular solid Poisson ratio ν. is stressed under tension or compression, the fibers act like struts and beams that deform under stress. 35.4.4 Tensegrity The unit cell model in Fig. 35.8b has struts or fiber elements of length L and cross-section of radius a. Networks can also derive their structural integrity from The relative density of the material is defined by the an interaction between members that are in compression volume fraction of solid material Φ. This is calcu- and members in tension. Some familiar large-scale ex- lated as the solid volume contained within a unit cell amples include the circus tents and the geodesic dome. (∝ a2 L) divided by the total unit cell volume (∝ L3) In the case of the circus tent, the rigidity of the structure or Φ ∝ (a/L)2. Beam theory gives the deflection δ of is due to the balance between the tent poles in compres- a beam of length L subject to a force F acting at its sion and the ropes anchored to the ground in tension. midpoint as The rigidity of the structure, in this case, is related to the elastic characteristics of the tension elements. FL3 δ ∝ , (35.14) Ingber [35.76] first proposed that the cytoskeleton Ef I behaves like a tensegrity structure with the microtubules acting in compression and the F-actin microfilaments where Ef is the stiffness of the beam constitutive material and I is the beam’s second moment of acting in tension. In support of this concept, micro- area. The moment of area for a beam of thick- tubules have been shown to be capable of supporting ness a is given by I ∝ a4. The stress τ is the compressive loads and the F-actin network exhibits force per unit area, or τ ∝ F/L2. The strain is re- behavior at junctions consistent with their being in lated to beam deflection δ by ε ∝ δ/L. Using these, tension. Intermediate filaments may also be involved, combined with the expressions above, the network although their contributions at this stage are unclear. Young’s modulus or elastic modulus can be expressed Analysis of a fully three-dimensional tensegrity net- as work begins, as in the case of the cellular solids model, with a unit cell consisting of an interconnected system τ c1 Ef I of elements in tension in balance with other elements En = = , (35.15) ε L4 in compression. When the three-dimensional tensegrity network of Fig. 35.8c is used, analysis shows that where c1 is a constant of proportionality, or prestress plays an important role. Intuitively, it is not En 2 surprising that networks with greater pretension in the = c1Φ . (35.16) Ef elastic members should exhibit greater resistance to deformation. The effect of prestress increases the network Data from a wide range of materials and cell geometries Young’s modulus for small strains, and only in the limit give a value for c of ≈ 1 [35.72]. A similar anal- 1 of infinite prestress does E become independent of pre- ysis for cellular materials subjected to shear stresses n stress. Wang and co-workers [35.77]haveshownthat results in an expression for the network shear modulus the cytoskeleton exhibits this same tendency, becom- G [35.72]: n ing increasingly stiff when, for example, the cell passes Gn 2 from a spherical (low prestress) to flattened (high pre- ∝ c2Φ , (35.17) Ef stress) state.

atD Part 35.5 Mechanics of Subcellular Structures 35.5.1 Cell–Cell and Cell–Matrix Adhesions receptor–ligand bonding is the stronger of the two, by

35.5 a considerable margin, and is therefore the most rel- Cells can adhere to their surroundings by either non- evant mechanically. In situations for which either the specific or receptor–ligand (specific) bonding. Though receptors or their ligands are not present, however, such both mechanisms are likely active in most situations, as might be the case in certain in vitro experiments, Cellular Nanomechanics 35.5 Mechanics of Subcellular Structures 1189 nonspecific binding can be important as well, and can cytoskeleton (CSK) and the ECM, producing local de- produce a net attractive force per unit area of ≈ 100 Pa formations and stresses in the corresponding matrices at a separation distance of ≈ 25 nm. that decay with distance from the adhesion site. On the intracellular side, these forces are transmitted via a com- Models for Receptor-Mediated Adhesion plex involving vinculin, α-actinin, paxillin, and talin. Cell–cell adhesion has obvious similarities to adhesion Attachment to the extracellular matrix is mediated by an of cells to substrates, the surrounding extracellular ma- arginine–glycine–aspartic acid (RGD) sequence (in fi- trix, and other cells, so a similar approach can be used. bronectin for example), which in turn, has binding sites This requires, in addition, consideration of such factors for collagen and fibrin. If the force is sufficiently strong, as the distribution, type, and density of receptor–ligand above a certain threshold value F0 say, and applied for bonds or potential bonds, and the elastic properties of a sufficiently long time, the bond might be severed (de- the structures to which they are anchored. On the intra- couple). Typical values of this threshold force lie in the cellular side, this involves the series of couplings that range of 10–100 pN [35.78], but depend, as well, on the link the receptor to the cell. In the simplest case, this rate at which the force is increased. Under more rapid might simply be a link to the lipid bilayer if the receptor force application, the threshold is high compared with has no intracellular connections. More typically, espe- when the force is increased slowly. This simply reflects cially for couplings with a structural role, it involves the fact that detachment is a stochastic process and that, a series of proteins ultimately linking the receptor to the at any given level of force, there exists a finite proba- cytoskeleton. bility of detachment, and this probability increases with The typical setting in vivo is one in which cells time. adhere to other cells or to the extracellular matrix. We need also to consider that, in equilibrium, a frac- Adhesions are more easily probed, however, through tion Φ = K(1 + K)−1 (where K is the equilibrium in vitro experiments, where cell adhesion more often constant) of the receptors will be bound at any given occurs to an artificial substrate mediated through one time, and that the receptor–ligand complexes are con- of several extracellular proteins that are used to coat tinually cycling between the bound and unbound states, the surface. Generally, either collagen or fibronectin is characterized by their respective rate constants. On used. Cells are often adhered to these substrates, but to a larger scale, numerous adhesion sites typically act in produce a more controlled environment, rigid beads are parallel, each contributing some amount to the total ad- sometimes coated with the appropriate receptors so that hesion force. This gives rise to an average local stress τ one specific receptor–ligand interaction can be probed. that can be thought of as the product of the average force While these systems are useful as models of certain ad- per bond times the bond density. This stress is typically hesion phenomena, it is important to recognize in the nonuniform. In the adhesion of a cell to a flat sub- interpretation of these experiments that, when bound to strate, the bonds near the periphery of the contact zone a rigid substrate or bead, the binding proteins cannot determine the strength of the cell against detachment, freely diffuse, as they would in a more natural envi- and when the cell is subjected to a detaching force, the ronment. In particular, the formation of focal adhesions stresses are concentrated in this peripheral region. In the would not occur in bead–substrate experiments because vicinity of the edge of adhesion, several factors will in- the receptors would be constrained from aggregating. fluence the stress distribution, including the mechanical It is useful to consider a single adhesion bond, for properties of the membrane (in particular the bending example, one linking the actin matrix of the cytoskele- stiffness Kb), the receptor density, the spring constants ton to a β1 integrin, and the integrin receptor binding of the bond, and the bond strength. to the extracellular matrix beyond the cell membrane. If The interplay between receptor–ligand bonding and the bond is stressed, as for example if the cell experi- membrane stiffness also determines whether a cell will ences a force relative to the ECM, it will at first stretch spread over a surface or peel away from it under a given

an amount dictated by the level of force in the bond tension. Membrane bending stiffness acts to maintain D Part and the stiffness of the complex. Each bond, as well the membrane and substrate within close proximity near as each protein in the bond complex, can be thought the edge of contact. Spreading can occur only if their

of as having a certain stiffness, giving rise to a picture separation distance is within the extended length of the 35.5 in which several springs are considered connected in receptor ligand bond over a distance comparable to the series. Forces acting on the adhesion complex are trans- linear spacing between neighboring bonds. Under high mitted via this series of bonded proteins between the tension, the surfaces rapidly diverge and spreading is 1190 Part D Bio-/Nanotribology and Bio-/Nanomechanics

prevented; if the tension is sufficiently low, receptor– transmembrane proteins and their attachments to the in- ligand bonding can occur and spreading can occur. tracellular and extracellular milieu are included in terms of their influence on the continuum properties of the Transient Cell Adhesion model. Were it not for these, the membrane would ex- Concepts of transient adhesion and release are of par- hibit little resistance to shear deformation. ticular importance in the case of leukocyte rolling, In qualitative terms, the lipid bilayer can be thought adhesion, and transmigration across the endothelial of as a two-dimensional fluid, within which the indi- layer of a blood vessel. In addition, cell migration vidual lipid molecules, or other molecules embedded through extracellular matrix or along a substrate re- in the membrane, are relatively free to move about by quires the ability of the cell to both form and release diffusion or directed motion. Phospholipid molecules adhesions. One problem that has received consider- in either of the two layers resist being pulled apart, able attention is the interaction between an adherent however, so each layer is highly inextensible. This also or rolling leukocyte and the adhesion receptors on contributes to the bending stiffness, which is low in ab- the endothelium. These studies are complicated by solute terms but high for such a thin layer, since bending the three-dimensional nature of the flow, the compli- requires one layer to expand while the other is com- ance of the interacting surfaces, and receptor–ligand pressed. By contrast, the two layers readily slide relative dynamics. In a series of recent studies, Hammer and to each other. These qualitative notions are put in more coworkers [35.79] have addressed several of these is- quantitative terms in the next section. sues, computing the viscous force and torque acting upon a sphere near to a planar wall from the mobil- Types of Deformation ity matrix [35.80] and using the Bell model of receptor Any deformation of the membrane can be thought of, binding as described above [35.81], but neglecting the in general terms, as a superposition of several simpler effects of cell deformation. deformations. For small strains in which linearization is A Monte Carlo method is employed at each time appropriate, the principle of superposition is rigorously step in the calculation to determine bond formation or valid. For larger strains, however, linear theory breaks breakage. In these studies, the nature of the leukocyte– down and superposition can only be used as a rough, endothelial interaction is characterized as firm adhesion, qualitative guide in visualizing combined influences. rolling adhesion, bimodal adhesion or no adhesion, and Here we present the three primary types of deformation: mapped as a function of the two parameters of the pure extension, pure bending, and pure shear. Bell model. Most literature values for the Bell model parameters for a variety of selectin receptors (known Pure Extension. In discussing the extensional stiffness to be instrumental in leukocyte rolling) correspond of the membrane, we need to distinguish the behavior at to the rolling adhesion regime for typical shear rate low tension from that at high tension. As an extensional of 100 s−1. An interesting, but counterintuitive, result stress is first applied at the edges of a lipid bilayer, from this study was the observation that, as the bond the projected or apparent membrane area first increases stiffness was increased from the value used in most cal- while the actual or true membrane area remains con- culations (100 dyn/cm), adhesiveness decreased,which stant. This results from the suppression of out-of-plane the authors attribute to reduced deflection for a given undulations. Forces acting to resist membrane flatten- level of force, leading subsequently to more rapid dis- ing originate from entropic effects analogous to those sociation. seen in the case of a flexible polymer as its end-to-end distance is increased – many more membrane configu- 35.5.2 Cell Membranes rations exist with undulations compared with the single, perfectly flat state. Only when these undulations have For the purpose of analysis, we treat the cell mem- been eliminated does the true membrane area, propor-

atD Part brane as a homogeneous two-dimensional (2-D) plate tional to the surface area per molecule group, begin to or sheet completely enclosing the cytoplasm. The mem- increase, and this is associated with a relatively abrupt brane referred to here can be thought of either as the increase in extensional stiffness. For now, we consider

35.5 lipid bilayer by itself, or more typically, as the bilayer only the stiffness of a ﬂat membrane and leave the dis- plus the associated cortex of cytoskeletal ﬁlaments and cussion of undulations of entropic origin to a later point the glycocalyx on the extracellular surface. In addi- in the chapter. Hence we initially neglect all entropic tion, though not explicitly recognized in the analysis, effects, or equivalently, consider the membrane to be at Cellular Nanomechanics 35.5 Mechanics of Subcellular Structures 1191

a) b) c) σ2 σ2

A0 M1 M1 σ1 σ1 x3 σ1 σ1 Neutral plane x2

x1 σ2 x1 σ2

Fig. 35.9a–c Membrane mechanics. (a) A membrane, initially of area A0, subjected to a uniform extensional stress along its edges (σ1 = σ2), causing an increase in area to A. (b) Application of equal moments M1 at the two edges of a membrane sheet. (c) Shear deformations produced by the application of nonequal surface tensions on the boundaries of a membrane sheet. The inner square experiences shear strains as shown by the arrows zero temperature. Based in this assumption, consider an Note that, although we used the continuum structural = 2 infinitesimal plate initially of area A0 L0 that is de- equations in our analysis, the final result can also be formed by a uniform normal stress τ11 = σ1 = τ22 = σ2 viewed as simply the definition of the area expansion applied to its edges (Fig. 35.9a) to a new area A. The modulus and applies regardless of whether or not the expressions relating stress and strain in two dimensions, membrane can be modeled as a continuum. and in the absence of stresses normal to the x1–x2 plane, Experimental measurements of Ke lie in the range can be written as of 0.1–1N/m for various types of lipid bilayers and E ≈ 0.45 N/m (450 dyn/cm) for red blood cell mem- σα = (εα + νεβ) , (35.18) 1 − ν2 branes [35.82]. These numbers suggest that cell membranes are quite resistant to extension and, for that where the length of one edge is Lα = L0(1 + εα); note that the subscripts α, β are used here rather than i, j to reason, are often treated as inextensible. This discussion distinguish stresses and strains in two dimensions from neglects the effects of thermal fluctuations in the mem- those, more generally, in three. Thus, whereas i and j brane that give rise to a much more compliant behavior can be either 1, 2 or 3, α and β are restricted to be- at the smallest areal strains. When surface stress is suf- ing either 1 or 2. When this stress is uniform in the ficient to smooth out most thermal fluctuations, the cell or vesicle will exhibit the large moduli given here. plane of the membrane (the x1–x2 plane) it can be re- placed, without loss of generality, by a surface tension This high resistance to area change is in large part Nα (force per unit length) defined as σαh, where h is the due to the energy penalty associated with exposing thickness of the membrane. These can be combined in the hydrophobic core of the membrane to water that occurs as the spacing between individual amphiphilic the case when N1 = N2 = constant and, consequently, molecules is increased (for a detailed description of ε1 = ε2 = ε to give bilayer structure and thermodynamics, see [35.83]). Eh N = ε. (35.19) Continuing to increase extensional stress, the lipid bi- 1 − ν layer eventually ruptures, but at very small extensional In terms of a plate stretched uniformly in both direc- strains, in the vicinity of 2–3% [35.84]. Note that a bi- tions, we can define the areal strain as layer in a lipid vesicle, for example, stretches primarily 2 + ε 2 − 2 by increasing the area per molecule since recruitment of ΔA A − A0 L (1 ) L ∼ = = 0 0 = ε, (35.20) 2 2 additional material to the membrane occurs very slowly. A0 A0 L0 Using these expressions, the surface tension at D Part where the last approximation is appropriate for small which the membrane would rupture can be com- strains. By combining this result with (35.19), we can puted to be ≈ 0.06 N/m if we use a value near the define the area expansion modulus K (units of N/m) = / e higher end of the observed range Ke 1N m. Cells 35.5 using also often exhibit an intrinsic surface tension. Re- Δ Δ = Eh A ≡ A . ported values are small, however, lying in the range of N Ke (35.21) ≈ −5 −4 / 2(1 − ν) A0 A0 10 –10 N m [35.85]. 1192 Part D Bio-/Nanotribology and Bio-/Nanomechanics

Pure Bending. By contrast, lipid bilayers exhibit very largely due to the cortex of cytoskeletal filaments that low bending stiffness, so low that it is often neglected lie on the intracellular side of the membrane. In a red in models of membrane mechanics. It can be impor- blood cell this matrix, as discussed above, consists tant in certain situations, however, and is essential, for of interconnected filamentous spectrin and actin with example, in analyzing the thermal fluctuations of vesi- attachments to the membrane via ankyrin. The equa- cles. Bending stiffness arises from the same type of tions relating shear force per unit length of membrane molecular interactions that cause extensional stiffness. to shear deformation (Hooke’s law) can be expressed When an initially flat bilayer is bent, the hydrophilic as N12 = τ12h = 2Gε12 = Ksε12, where we define the head groups on the outside of the bend move further membrane shear modulus Ks in units of N/m. Typi- apart while on the inside, intermolecular spacing de- cal values lie in the range (6–9) × 10−6 N/m for a red creases; both represent departures from the equilibrium, blood cell membrane. A pure lipid bilayer exhibits a vis- unstressed state and require energy. cous resistance to shear deformations characterized by Returning to our simple continuum plate model, a shear viscosity of ≈ 10−6 Ns/m [35.53]. consider a bending moment applied to the two ends, causing the plate to curve slightly (Fig. 35.9b). If the 35.5.3 Cell Nuclei bending is due to moments applied at the two ends about the x2-axis, then the bending moment per unit length is Measurements based on micropipette aspiration, atomic related to the deflection by force microscopy, and substrate strain experiments suggest that the nucleus is ≈ 2–10 times stiffer than the 3 2 2 Et ∂ u3 ∂ u3 surrounding cytoskeleton [35.55, 87–89]. What makes Mα =− =−K , 2 2 b 2 12(1 − ν ) ∂xα ∂xα the nucleus so hard to deform? This question may in (35.22) part depend on the particular mechanical load application. In cellular compression experiments, the highly where Kb is termed the bending stiffness having units of condensed chromatin provides a large resistance to nu- Nm. Implicit in this expression are the assumptions that clear compression. In contrast, the nuclear lamina can there exists a mid-plane (the neutral plane)onwhichthe act as an elastic molecular shock absorber [35.90]that in-plane stress and strain are both zero, and that straight seems ideally suited to sustain large strain when the cell lines perpendicular to this mid-plane remain straight or nucleus is stretched. This idea is supported by mi- and normal to this surface after deformation. cropipette aspiration experiments on isolated nuclei that Typical values for the bending stiffness Kb lie in the reveal that, in condensed nuclei, chromatin is the ma- range of 10−19 Nm(10−12 dyn cm) for a red blood cell jor contributor to nuclear stiffness, whereas the nuclear or lipid bilayers [35.84]. This value is larger, on the or- lamina carries much of the applied mechanical load in derof(1–2)×10−18 N m [35.86], for other cell types swollen nuclei that closely resemble nuclei in intact (e.g., neutrophils, endothelial cells) that possess a more cells [35.48]. Similarly, condensed nuclei were found extensive cortex. to be much stiffer than swollen nuclei [35.48]. Based on the nuclear structure, the mechanical contributions Pure Shear. Shear deformations arise when a mem- of the nucleus can be separated into three intercon- brane is stretched in one direction by a surface tension nected components: the nuclear membranes, the nuclear N1 (units of force/length) while the lateral surface con- lamina, and the nuclear interior, which is mostly com- tracts under a lesser tension N2, at constant surface prised of chromatin but could also contain a nuclear area and in the absence of bending (Fig. 35.9c). Sur- matrix of lamins, actins, and other proteins. Each of faces oriented at 45◦ to the boundaries experience pure these structural components has its own physical prop- shear stresses of magnitude (N1 − N2)/2. When sub- erties. The specific contributions have been addressed jected to shear stresses in the plane of the membrane, in experiments that induced osmotic swelling of iso-

atD Part a pure lipid bilayer behaves essentially as a liquid. It lated nuclei by varying the buffer salt concentration exhibits a membrane viscosity in that it poses a resist- to separate the nuclear envelope from the underlying ing force proportional to the rate of shear deformation, chromatin [35.48, 90]. Additional insights have come

35.5 but only a small shear modulus to static shear defor- from micropipette aspiration experiments, in which spe- mations. It is not clear, in fact, whether or not pure cific components of the nucleus (e.g., the lamina, the lipid bilayers exhibit a nonzero shear modulus. Typical nuclear membrane or the chromatin) are fluorescently cell membranes do exhibit a shear modulus, however, labeled and monitored during aspiration. These experi- Cellular Nanomechanics 35.5 Mechanics of Subcellular Structures 1193 ments have revealed that the nuclear interior behaves as mechanical properties of its nucleus. A recent study a compressible, viscoelastic gel and can be compacted that probed nuclei from human embryonic stem cells by more than 60% during micropipette aspiration be- and differentiated cells found that nuclei from embry- fore becoming resistant to further compression [35.51]. onic stem cells were significantly more deformable The reduction in nuclear volume during micropipette than nuclei from differentiated cells and became up aspiration results in buckling of the nuclear envelope to sixfold stiffer relative to the cytoplasm during dif- outside the pipette [35.48, 51], further demonstrating ferentiation [35.50]. While some of this increase in tight physical attachment between the nuclear enve- nuclear stiffness could be caused by changes in chro- lope and the chromatin interior. Nucleoli and other matin structure and organization, much of it can be intranuclear structures deform along with the nuclear in- attributed to a stiffer lamina through increased expres- terior in these experiments [35.50, 51], but might have sion of lamins A and C, as independent experiments slightly different mechanical stiffness. The viscoelas- have shown that lamins A and C are the main contribu- tic nature of the chromatin also appears responsible tors to nuclear stiffness [35.95]. for the persistent plastic deformations of nuclei seen Overall, micropipette aspiration has been the most after micropipette aspiration of isolated nuclei [35.50, widely used technique to probe nuclear mechanics, al- 91]. In contrast to the gel-like nuclear interior, the though cellular strain experiments, AFM, microbead nuclear lamina itself behaves as a two-dimensional rheology, and parallel-plate compression experiments elastic shell. This can be illustrated when monitoring have also been successfully applied to measure nu- the fluorescence intensity of green fluorescent protein clear stiffness (Table 35.4). While some groups have (GFP)–lamin A incorporated into the nuclear lamina described continuous creep of aspirated nuclei into the during micropipette aspiration experiments. While the micropipette [35.48, 50, 90, 96], others [35.51, 88, 92] fluorescence intensity is very uniform in the nonaspi- found that the nucleus stabilized after an equilibration rated section of the nucleus, the fluorescence intensity period of ≈ 10 s. These differences may be due to vari- decays exponentially inside the aspirated segment, as ations in the experimental setup or the particular cell the nuclear lamina is stretched under the aspiration pres- type studied. Depending on the experimental technique sure. Such a fluorescence intensity profile can be well used, various mathematical models have been applied fitted by the predicted strain profile of a solid elastic to describe the induced nuclear deformations. For mi- shell [35.51, 92]. Unlike the nuclear lamina, the nu- cropipette aspiration experiments, the creep compliance clear membranes have fluid-like characteristics, i. e., is often inferred from the following equation: they do not show any induced strain during micropipette 4π L 1 aspiration as they can flow in response to pressure appli- J t, ΔP = Φ , (35.23) ( ) Δ cation [35.51]. Further illustrating the fluid-like nature 3 D P is the fact that small membrane vesicles continue to where ΔP is the applied pressure difference, L is the form blebs in the aspirated section of the nucleus as aspirated tongue length, and D is the micropipette diam- long as the suction pressure persists [35.51]. Conse- eter, and the dimensionless parameter Φ (value = 2.1) quently, the nuclear membrane cannot carry much of accounts for the effects of the micropipette geome- the mechanical load, leaving the nuclear interior and try [35.97]. The challenge in finding analytical expres- the nuclear lamina as the major contributors to nuclear sion for the mechanical behavior of the nucleus is that stiffness. the nucleus is comprised of various, structurally very In addition to the various structural components, different domains, as discussed before. Finite-element the differentiation state of a cell can also modulate the modeling can address some of these complex interac-

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

Cell type Experimental method Nuclear stiffness Reference D Part Chondrocytes Micropipette aspiration 1–2kPa [35.88] Bovine endothelial cells Parallel-plate compression 5–8kPa [35.87] Monkey kidney epithelial cells AFM and micropipette aspiration ≈ 1–10 kPa∗ [35.48] 35.5 Mouse ﬁbroblasts Intranuclear particle tracking 18 Pa [35.93] Human cervical cancer cells (HeLa) Magnetic tweezers 250 Pa [35.94] 1194 Part D Bio-/Nanotribology and Bio-/Nanomechanics

tions [35.98], but often has to make assumptions on from the extracellular domain to the cytoskeletal ma- the specific material parameters that are difficult to trix within the confines of a focal adhesion, a small verify. An alternative approach is to directly compare portion of which is depicted in Fig. 35.10. Forces nuclear deformations under identical experimental con- transmitted to the cell, either by tethering to the ex- ditions, e.g., the aspirated lengths of the nucleus during tracellular matrix or via ligand-coated beads in various a constant applied pressure gradient in micropipette as- in vitro experiments, are transmitted via the integrins piration experiments or the induced nuclear strain under (or other transmembrane receptors) through a collection a given applied substrate strain in cell strain experi- of membrane-associated proteins, ultimately linking up ments [35.99]. with the cytoskeleton and propagated throughout the cell. Many of the proteins that form this pathway are 35.5.4 Mechanosensing Proteins known signaling molecules and several have been implicated in mechanotransduction. The process by which While the complex biochemical pathways have been ex- changes in protein conformation give rise to protein plored in considerable detail by biologists during the clustering at a focal adhesion or initiate intracellular past several decades, work has only recently begun to signaling, however, remains largely unknown. This has understand the mechanical intracellular pathways for called for close examination of the individual proteins the propagation of force through the cell. For exam- or protein complexes that transmit the forces through- ple, Geiger and Bershadsky [35.100] have mapped the out the cell and their interconnections. Consider one complex and interconnected pathways that can be traced example pathway from Fig. 35.11. It can be seen that integrins bind to talin, which in turn connects with F-actin either directly or via vinculin. Talin and inte- Fibronectin grin also bind to focal adhesion kinase (FAK). This force pathway alone gives rise to several possibilities for mechanotransduction. α Even in the absence of biochemical signals, mechanical stimulation can still elicit the formation of α viable focal adhesions [35.101]. These focal adhesions are produced as a result of direct molecular conformational changes in focal adhesion-forming molecules. CASCAS FAK Force-induced conformational changes in talin, vin- FAK Talin Src culin, and focal adhesion kinase (FAK) initiate focal Pax adhesion formation by anchoring integrin molecules Vinculin to actin filaments and recruiting other focal adhesion- α-Actinin forming molecules to the site of interaction. Other mechanosensing molecules such as filamin and α- actinin stabilize actin filaments through mechanobrac- ing and making scaffolds. The conformational changes Actin filaments in the focal adhesion-initiating molecules and the actin filament scaffold bracing molecules demonstrate the mechanisms of molecular mechanotransduction. In the following, we will briefly review some recent work in molecular dynamics simulation of some mechanosensing proteins, namely talin interaction with Fig. 35.10 A small sampling of the proteins found in a focal vinculin [35.102, 103], focal adhesion kinase interac-

atD Part adhesion complex. Forces are typically transmitted from the extra- tion with paxillin [35.104], and molecular mechanics cellular matrix (e.g., ﬁbronectin), via the integral membrane adhe- of actin binding proteins α-actinin [35.105]andﬁl- sion receptors (α-andβ-integrins), various membrane-associated amin [35.106].

35.5 proteins (focal adhesion kinase (FAK), paxillin (Pax), talin, Crk- associated substrate (CAS)), to actin-binding proteins (α-actinin) Talin that link the focal adhesion complex to the cytoskeleton (af- Talin interacts with the membrane-bound integrin mol- ter [35.100], with permission) ecule at its head domain, and with vinculin in its tail Cellular Nanomechanics 35.5 Mechanics of Subcellular Structures 1195

a) b)

Activated α/-integrin dimer Cell membrane

PIP2

Talin

Inactive vinculin

Activated F-actin vinculin

Fig. 35.11 (a) Model of the initial adhesion consisting of integrin–talin–F-actin linkage. Vinculin is present in the cytosol in an inactive, autoinhibitory conformation, and tensile force is applied to the integrin dimer from the outside of the cell membrane. (b) Transmitted force through the linkage alters the talin configuration, and recruits vinculin to reinforce the initial adhesion linkage (PIP2 – phosphatidylinositol 4,5-bisphosphate) domain. It has been implicated as a site of molecular ing of the rod domain in order to expose the vinculin activation by external force. The forces acting on the binding site VBS1. Using steered molecular dynamics, cells from the ECM are transmitted from the integrin Hytönen and Vogel demonstrated that the talin rod can molecules to the talin head and then to all regions of be fragmented into three helix subbundles, which is fol- the talin tail (Fig. 35.11). The talin tail has 11 vinculin lowed by the sequential exposure of vinculin-binding binding sites (VBS), some of which are on amphipathic helices to water. The unfolding of a vinculin-binding he- alpha helixes and are buried in the hydrophobic core lix into a completely stretched polypeptide might then of the talin tail. VBS1 is one of the cryptic binding inhibit further binding of vinculin. sites buried in the hydrophobic core of the TAL5 region of talin’s tail. TAL5 has several surface polar residues α-Actinin that are sites of hydrogen bonding with other secondary One major actin filament cross-linker is α-actinin. structures in the rod domain. Transduced external forces α-Actinin is a cross-linking molecule that produces from the ECM are applied to the TAL5 domain via these residues [35.107]. In order to simulate the force transduction through the cryptic VBS1 of TAL5, Lee a) b) et al. [35.103] applied forces averaging 21.2pNatthe residues near the N-terminal near the talin head while constraining fixed the polar residues at the C-terminal, which interact with the adjacent secondary structure domain. The external locally applied force caused TAL5 VBS1L622 VBS1 to elongate, H1 to move in the direction of the pulling Fig. 35.12a,b With external force applied to the polar force, H1 to apply torque to H4 via its hydrogen bonds, residues in the TAL5 region of the talin rod, domain activa- and H4 to rotate, exposing the VBS1 (Fig. 35.12). Once tion of the VBS1 is achieved through rotation of the buried exposed to solvent the VBS1 is activated and can inter- VBS residues to the surface of the molecule. (a) TAL5

act with vinculin. The force-induced activation of VBS region before application of external force. The VBS1 is D Part sites in the talin tail together with force-induced acti- buried in the hydrophobic patch. (b) After the external vation of vinculin can anchor the stimulated integrin force is applied, TAL5 undergoes a conformational change

molecules to the actin cytoskeleton and thereby initiate and the VBS residues are exposed to the surface. In this 35.5 focal adhesions. representation, movement of the VBS to the surface is An alternative mechanism was recently proposed by shown by the movement of the arrow (edge of VBS) past Hytönen and Vogel [35.108] which involves the unfold- the dotted lines 1196 Part D Bio-/Nanotribology and Bio-/Nanomechanics

a scaffold for parallel actin filaments, such as those in These microtubules are inherently polarized due to stress fibers or in muscle z-disk formations. α-Actinin is their head-to-tail assembly, and this allows for proces- an antiparallel homodimer with three major domains, an sive movement of cargo via motor proteins that move actin binding domain, a calmodulin homology domain, specifically towards either the plus or the minus end. and a central rod domain. In order to act as a molecu- Microtubular filaments resist mechanical deformation lar scaffold, cross-linking actin filaments in a stressed and facilitate transport of intracellular cargo via cou- cytoskeletal network α-actinin must have a semiflex- pling to molecular motors. These filaments are typically ible structure. By exploring the natural frequencies in compression, effectively balancing the tensile forces of α-actinin and its response to external forces, Golji exhibited by the actin–myosin system [35.76]. et al. [35.105] characterized the flexible and rigid regions of α-actinin and have demonstrated the molecular Stretch-Sensitive Ion Channels conformational changes that underlie the semiflexible Another class of mechanosensing proteins are the cross-linking and mechanical bracing of α-actinin. mechanosensitive ion channels, e.g., the mechanosensitive channel of large conductance (MscL), which has Microtubules been studied extensively [35.110, 111]; molecular dy- As yet another mechanosensing protein system one can namic simulation has been used to show how stresses name the microtubule filaments [35.109]. The micro- in the cell membrane act directly on the channel and tubule cytoskeleton, which is comprised of individual cause it to change its conductance [35.112]. These microtubules typically nucleated from a microtubule channels have emerged as another potential gateways organizing center near the nucleus, usually forms a ra- for the cell to sense and respond to environmental dial network that radiates outward to the cell periphery. stimuli.

35.6 Current Understanding and Future Needs Progress is rapidly being made on both the experimen- can be applied to proteins to predict their conforma- tal and computational fronts to further understand the tional change under force, and docking simulations nanomechanics of the cell. Using either AFM or op- provide a means for determining binding affinities in tical tweezers, controlled force applications in the pN different conformational states. The barriers to progress range and displacement measures in the nm range are lie primarily in our lack of atomistic models with already within current capabilities. Optical traps were adequate resolution for those proteins of greatest inter- recently used in a clever experimental assay system est, located in the force transmission pathway. These to measure the rupture force of a complex formed by tend to be difficult to crystallize, so few structures are an actin-binding protein (namely, filamin or α-actinin) available. Moreover, due to their size, simulations are linking two actin filaments [35.113]. At the same time, computationally intensive, especially if the presence of single-molecule fluorescence measurements provide the water molecules is included explicitly. Despite these opportunity to monitor single binding events. On the constraints, some progress can be made by using sub- computational side, technical and technological barri- domains of the proteins of interest, provided that their ers are being slowly surmounted. Molecular dynamics functionality can be demonstrated.

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