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CHAPTER 19

Mechanical Response of

Cytoskeletal Networks

Margaret L. Gardel,* Karen E. Kasza,†CliVord P. Brangwynne,†

Jiayu Liu,‡and David A. Weitz†,‡

*Department of Physics and Institute for Biophysical Dynamics

University of Chicago, Illinois 60637

†School of Engineering and Applied Sciences

Harvard University

Cambridge, Massachusetts 02143

‡Department of Physics

Harvard University

Cambridge, Massachusetts 02143

Abstract

I. Introduction

II. Rheology

A. Frequency-Dependent Viscoelasticity

B. Stress-Dependent Elasticity

C. EVect of Measurement Length Scale

III. Cross-Linked F-Actin Networks

A. Biophysical Properties of F-Actin and Actin Cross-linking Proteins

B. Rheology of Rigidly Cross-Linked F-Actin Networks

C. Physiologically Cross-Linked F-Actin Networks

IV. EVects of Microtubules in Composite F-Actin Networks

A. Thermal Fluctuation Approaches

B. In Vitro MT Networks

C. Mechanics of Microtubules in Cells

V. Intermediate Filament Networks

A. Introduction

B. Mechanics of IFs

C. Mechanics of Networks

VI. Conclusions and Outlook

References

METHODS IN CELL BIOLOGY, VOL. 89

Copyright 2008, Elsevier Inc. All rights reserved.

0091-679X/08 $35.00

487

DOI: 10.1016/S0091-679X(08)00619-5

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Abstract

The cellular cytoskeleton is a dynamic network of filamentous proteins, consist-

ing of filamentous actin (F-actin), microtubules, and intermediate filaments. How-

ever, these networks are not simple linear, elastic solids; they can exhibit highly

nonlinear elasticity and athermal dynamics driven by ATP-dependent processes.

To build quantitative mechanical models describing complex cellular behaviors, it

is necessary to understand the underlying physical principles that regulate force

transmission and dynamics within these networks. In this chapter, we review our

current understanding of the physics of networks of cytoskeletal proteins formed

in vitro. We introduce rheology, the technique used to measure mechanical re-

sponse. We discuss our current understanding of the mechanical response of

F-actin networks, and how the biophysical properties of F-actin and actin cross-

linking proteins can dramatically impact the network mechanical response. We

discuss how incorporating dynamic and rigid microtubules into F-actin networks

can aVect the contours of growing microtubules and composite network rigidity.

Finally, we discuss the mechanical behaviors of intermediate filaments.

I. Introduction

Many aspects of cellular physiology rely on the ability to control mechanical

forces across the cell. For example, cells must be able to maintain their shape when

subjected to external shear stresses, such as forces exerted by blood flow in the

vasculature. During cell migration and division, forces generated within the cell are

required to drive morphogenic changes with extremely high spatial and temporal

precision. Moreover, adherent cells also generate force on their surrounding

environment; cellular force generation is required in remodeling of extracellular

matrix and tissue morphogenesis.

This varied mechanical behavior of cells is determined, to a large degree, by

networks of filamentous proteins called the cytoskeleton. Although we have the

tools to identify the proteins in these cytoskeletal networks and study their struc-

ture and their biochemical and biophysical properties, we still lack an understand-

ing of the biophysical properties of dynamic, multiprotein assemblies. This

knowledge of the biophysical properties of assemblies of cytoskeletal proteins is

necessary to link our knowledge of single molecules to whole cell physiology.

However, a complete understanding of the mechanical behavior of the dynamic

cytoskeleton is far from complete.

One approach is to develop techniques to measure mechanical properties of the

cytoskeleton in living cells (Bicek et al., 2007; Brangwynne et al., 2007a; Crocker

and HoVman, 2007; Kasza et al., 2007; Panorchan et al., 2007; Radmacher, 2007).

Such techniques will be critical in delineating the role of cytoskeletal elasticity in

dynamic cellular processes. However, because of the complexity of the living

cytoskeleton, it would be impossible to elucidate the physical origins of this cyto-

skeletal elasticity from live cell measurements in isolation. Thus, a complementary

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Margaret L. Gardel et al.

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approach is to study the behaviors of reconstituted networks of cytoskeletal pro-

teins in vitro. These measurements enable precise control over network parameters,

which is critical to develop predictive physical models. Mechanical measurements

of reconstituted cytoskeletal networks have revealed a rich and varied mechanical

response and have required the development of qualitatively new experimental

tools and physical models to describe physical behaviors of these protein networks.

In this chapter, we review our current understanding of the biophysical properties

of networks of cytoskeletal proteins formed in vitro. In Section II, we discuss

rheology measurements and the importance of several parameters in interpretation

of these results. In Section III, we discuss the rheology of F-actin networks, high-

lighting how small changes in network composition can qualitatively change the

mechanical response. In Section IV, the eVects of incorporating dynamic micro-

tubules in composite F-actin networks will be discussed. Finally, in Section V, we

will discuss the mechanics of intermediate filament (IF) networks.

II. Rheology

Rheology is the study of how materials deform and flow in response to externally

applied force. In a simple elastic solid, such as a rubber band, applied forces are

stored in material deformation, or strain. The constant of proportionality between

the stress, force per unit area, and the strain, deformation per unit length, is called

the elastic modulus. The geometry of the measurement defines the area and length

scale used to determine stress and strain. Several diVerent kinds of elastic moduli

can be defined according to the direction of the applied force (Fig. 1). The tensile

Young’s modulus, E

tensile elasticity

Bulk modulus

Compressional modulus

Bending modulus, k

Shear modulus, G

Fig. 1

mechanical properties; the light gray shape,indicating the sampleafter deformation,is overlaidonto the

black shape, indicating the sample before deformation. The Young’s modulus, or tensile elasticity, is the

deformation in response to an applied tension whereas the bulk (compressional) modulus measures

material response to compression. The bending modulus measures resistance to bending of a rod along

its length and, finally, the shear modulus measures the response of a material to a shear deformation.

Schematics showing the direction of the applied stress in several common measurements of

19. Mechanical Response of Cytoskeletal Networks

489

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elasticity, or Young’s modulus, is determined by the measurement of extension of a

material under tension along a given axis. In contrast, the bulk modulus is a

measure of the deformation under a certain compression. The bending modulus

of a slender rod measures the object resistance to bending along its length. And,

finally, the shear elastic modulus describes object deformation resulting from a

shear, volume-preserving stress (Fig. 2). For a simple elastic solid, a steady shear

s(w)

g(w)

Δs(w)

s0

Δg(w)

x

Δx

Term

Strain

Stress

FrequencyFrequency of applied + measured

waveforms: g (w) =g sin(wt), s(w)=

ssin(wt)

Constant external stress applied to sample

during measurement

Prestress

Phase Shift

G?

G??

K?

K??

Shear moduli:

s0=0

Elastic (storage)

modulus

Viscous (loss)

modulus

s0>0

Differential elastic

modulus

Differential loss

modulus

A

h

A

Symbol

g

s

s0

w

d

Units

None

Pascal (Pa)

Pascal (Pa)

Time−1

Degrees

Pascal (Pa)

Pascal (Pa)

Pascal (Pa)

Pascal (Pa)

Definition

; sample deformation

Height (h)

x

Area (A)

Force

d(w)= tan−1 (G??(w)/G? (w))

d =0°, elastic solid; d=90°, fluid

G?(w) = s(w)/g(w) cos(d(w))

G??(w) = s(w)/g(w) sin(d(w))

K?(w) = Δs(w)/Δg(w) cos(Δd(w))

K?(w) = Δs(w)/Δg(w) cos(Δd(w))

Fig. 2

shear elastic modulus, G0(o), and shear viscous modulus, G00(o), an oscillatory shear stress, s(o), is

applied to the material and the resultant oscillatory strain, g(o) is measured. The frequency, o, is varied

to probe mechanical response over a range of timescales. (Right) To measure how the stiVness varies as

a function of external stress, a constant stress, s0, is applied and a small oscillatory stress, (Ds(o)), is

superposed to measure a diVerential elastic and viscous loss modulus.

This schematic defines many of the rheology terms used in this chapter. (Left) To measure the

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Margaret L. Gardel et al.

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stressresults inaconstantstrain.Incontrast,forasimplefluid,suchaswater, shear

forces result in a constant flow or rate of change of strain. The constant of

proportionality between the stress and strain rate, _ g, is called the viscosity, ?.

To date, most rheological measurements of cytoskeletal networks have been that

of the shear elastic andviscous modulus. Mechanical measurements of shear elastic

and viscous response over a range of frequencies and strain amplitudes are possible

with commercially available rheometers. Recent developments in rheometer tech-

nology now provide the capability of mechanical measurements with as little as

100 ml sample volume, a tenfold decrease in sample volume from previous genera-

tion instruments. Recently developed microrheological techniques now also pro-

vide measurement of compressional modulus (Chaudhuri et al., 2007). Reviews of

microrheological techniques can be found in Crocker and HoVman (2007), Kasza

et al. (2007), Panorchan et al. (2007), Radmacher (2007), and Weihs et al. (2006).

A. Frequency-Dependent Viscoelasticity

In general, the rheological behaviors of cytoskeletal polymer networks display

characteristics of both elastic solids and viscous fluids and, thus, are viscoelastic.

To characterize the linear viscoelastic response, small amplitude, oscillatory shear

strain, g sin(ot), is applied and the resultant oscillatory stress, s sin(otþd), is

measured , where d is the phase shift of the measured stress and is 0 < d < p/2.

(Figure 2 describes much of the terminology used in this chapter.) The in-phase

component of the stress response determines the shear elastic modulus,

G

the material. The out-of-phase response measures the viscous loss modulus,

G

in the material. In general, G

Thus, materials that behave solid-like at certain frequencies may behave liquid-like

at diVerent frequencies; measurements of the frequency-dependent moduli of

solutions of flexible polymers (polyethylene oxide) and the biopolymer, filamen-

tous actin (F-actin) are shown in Fig. 3A. The solution of flexible polymers (black

symbols) is predominately viscous, and the viscous modulus (open symbols) dom-

inates over the elastic modulus (filled symbols) over the entire frequency range. In

contrast, the solution of F-actin filaments (gray symbols, Fig. 3A) is dominated by

the viscous modulus at frequencies higher than 0.1 Hz but becomes dominated by

the elastic modulus at lower frequencies. Thus, it is critical to make measurements

over an extended frequency range to ascertain critical relaxation times in the

sample. Moreover, frequency-dependent dynamics should be carefully considered

in establishing mechanical models.

The measurements shown in Fig. 3A are measurements of linear elastic and

viscous moduli. In the linear regime, the stress and the strain are linearly dependent

and, since the moduli are the ratio between these quantities, the measured moduli

are independent of the magnitude of applied stress or strain. For flexible polymers,

the moduli can remain linear up to extremely high (>100%) strains. (Consider

0ðoÞ ¼ ðs=gÞcosðdðoÞÞ, and is a measure of how mechanical energy is stored in

00ðoÞ ¼ ðs=gÞsinðdðoÞÞ, and is a measure of how mechanical energy is dissipated

0and G

00are frequency-dependent measurements.

19. Mechanical Response of Cytoskeletal Networks

491

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extending a rubber band; the force required to extend it a certain distance

will remain linear up to several times its original length.) However, for many

biopolymer networks, the linear elastic regime can be quite small (<10%). To

confirm you are measuring linear elastic properties, it is recommended that you

make measurements at two diVerent levels of stress and confirm you measure

identical frequency-dependent behaviors.

B. Stress-Dependent Elasticity

The mechanical response of cytoskeletal networks can be highly nonlinear such

that the elastic properties are critically dependent on the stress that is applied to the

network. When the elasticity increases with increasing applied stress or strain,

materials are said to ‘‘stress-stiVen’’ or ‘‘strain-stiVen’’ (Fig. 3B). In contrast, if

the elasticity decreases with increased stress, the material is said to ‘‘stress-soften’’

or, likewise, ‘‘strain-soften’’ (Fig. 3B).

Stress-stiVening behavior has been observed for many cytoskeletal networks, for

example, F-actin networks cross-linked with a variety of actin-binding proteins

(Gardel et al., 2004a, 2006b; MacKintosh et al., 1995; Storm et al., 2005; Xu et al.,

2000) and intermediate filament networks (Storm et al., 2005). In this nonlinear

regime, F-actin networks compress in the direction normal to that of the shear and

exert negative normal stress (Janmey et al., 2007). The origins of stress-stiVening

can occur in nonlinearities in elasticity of individual actin filaments or reorganiza-

tion of the network under applied stress.

Not all reconstituted cytoskeletal networks exhibit stress stiVening under shear.

Some show stress weakening: the modulus decreases as the applied stress increases.

This is usually found in networks that are weakly connected. For example, pure

F-actin solutions, weakly cross-linked actin networks (Gardel et al., 2004a; Xu

G? (Pa)

s (Pa)

B

10−3

10−2

10−1

w (Hz)

100

101

10−2

10−1

100

101

102

100

101

102

101

100

10−1

G?, G?? (Pa)

A

G?

G??

Fig. 3

network of F-actin (gray symbols) and solution of flexible polymers (black symbols) illustrating the

frequency dependence of these parameters (B) Measurement of G0as a function of applied stress for a

network that stress stiVens (top, gray squares) and stress weakens (bottom, black squares).

(A) Frequency-dependent elastic (filled symbols) and viscous (open symbols) moduli of a

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Margaret L. Gardel et al.

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et al., 1998), and pure microtubule networks (Lin et al., 2007) all show stress-

softening behavior. Under compression, branched, dendritic networks of F-actin

are also shown to reversibly stress soften at high loads (Chaudhuri et al., 2007).

In the nonlinear elastic regime, large amplitude oscillatory measurements are

inaccurate, as the response waveforms are not sinusoidal (Xu et al., 2000). To

accurately measure the frequency-dependent nonlinear mechanical response, a

static prestress can be applied to the network, and the linear, diVerential elastic

modulus, K

superposed oscillatory stress (Gardel et al., 2004a,b; Fig. 2, right). However, if a

material remodels and the strain changes with time when imposed by a constant

external stress alternative, nonoscillatory rheology measurements may be

necessary.

0, and loss modulus, K

00are determined from the response to a small,

C. EVect of Measurement Length Scale

Due to the inherent rigidity of cytoskeletal polymers, cytoskeletal networks

formed in vitro are structured at micrometer length scales. The mechanical re-

sponse of cytoskeletal networks can depend on the length scale at which the

measurement is taken (Gardel et al., 2003; Liu et al., 2006). Conventional rhe-

ometers measure average mechanical response of a material at length scales

>100 mm. By contrast, microrheological techniques can be used to measure me-

chanical response at micrometer length scales; however, interpretations of these

measurements are not usually straightforward for cytoskeletal networks structured

at micrometer length scales (Gardel et al., 2003; Valentine et al., 2004; Wong et al.,

2004). Direct visualization of the deformations of filaments such as F-actin and

microtubules (Bicek et al., 2007; Brangwynne et al., 2007a) can also be used to

calculate local stresses (see Section IV).

III. Cross-Linked F-Actin Networks

A. Biophysical Properties of F-Actin and Actin Cross-linking Proteins

1. Actin Filaments

Actin is the most abundant protein found in eukaryotic cells. It comprises 10% of

the total protein mass in muscle cells and up to 5% in nonmuscle cells (Lodish et al.,

1999). Globular actin (G-actin) polymerizes to form F-actin with a diameter, d, of

5 nm and contour lengths, Lc, up to 20 mm (Fig. 4). The extensional modulus, or

Young’s modulus, E, of F-actin is approximately 109Pa, similar to that of plexiglass

(Kojima et al., 1994). However, due to the nanometer-scale filament diameter, the

bending modulus, k0? Ed4, is quite soft. The ratio of k0to thermal energy, kBT,

defines a length scale called the persistence length, ‘p? k0=kBT. This is the length

overwhichvectorstangenttothefilamentcontourbecomeuncorrelatedbytheeVects

of thermally driven bending fluctuations. For F-actin, ‘p? 8 ? 17mm, (Gittes et al.,

19. Mechanical Response of Cytoskeletal Networks

493

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1993; Ott et al., 1993) and, thus, is semiflexible at micrometer length scales with a

persistence length intermediate to that of DNA, ‘p? 0:05 mm, and microtubules,

‘p? 1000 mm.

Transverse fluctuations driven by thermal energy (T > 0) also result in contrac-

tion of the end-to-end length of the polymer, L, such that L < Lc(Fig. 4). In the

linear regime, applied tensile force, t, to the end of the filament results in extension,

dL, of the filament such that: t ? ½k2=ðkTL4Þ? ? ðdLÞ (MacKintosh et al., 1995).

This constant of proportionality, k2=ðkTL4Þ, defines a spring constant that arises

from purely thermal eVects, which seek to maximize entropy by maximizing the

number of available configurations of the polymer. The distribution and number

of available configurations depends on the length, L, of the polymer such that the

spring constant will decrease simply by increasing filament length. However, as

L ! Lc, the entropic spring constant diverges such that the force-extension rela-

tionship is highly nonlinear (Bustamante et al., 1994; Fixman and Kovac, 1973;

Liu and Pollack, 2002). At high extension, the tensile force diverges nonlinearly

with increasing extension such that: t ? 1=ðLc? LÞ2. Thus, the force-extension

relationship depends sensitively on the magnitude of extension.

The elastic properties of actin filaments are also sensitive to binding proteins and

molecules. For instance phalloidin andjasplakinolide,twosmall molecules thatstabi-

lize F-actin enhance F-actin stiVness (Isambert et al., 1995; Visegrady et al., 2004).

It has been shown that a member of the formin family of actin-binding and nucleator

proteins, mDia1, decreases the stiVness of actin filaments (Bugyi et al., 2006).

2. Actin Cross-Linking Proteins

In the cytoskeleton, the local microstructure and connectivity of F-actin is

controlled by actin-binding proteins (Kreis and Vale, 1999). These binding pro-

teins control the organization of F-actin into mesh-like gels, branched dendritic

T=0

L=Lc

T>0

dL

F

L

Fig. 4

forces(T¼0),asemiflexiblepolymerappearsasarod,withthefullpolymercontourlength,Lc,identical

to the shortest distance between the ends of the polymer, L. However, thermally induced transverse

bendingfluctuations(T>0)leadtocontractionofLsuchthatL<Lc.Anappliedtensileforce,F,extends

the filament by a length, dL, and, because Lcis constant, this reduces the amplitude of the thermally

induced bending fluctuations, giving rise to a force-extension relation that is entropic in origin.

(Left) Electron micrograph of F-actin. Scale bar is 1 mm. (Right) In the absence of thermal

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Margaret L. Gardel et al.

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networks, or parallel bundles, and it is these large-scale cytoskeletal structures that

determine force transmission at the cellular level. Some proteins, such as fimbrin

and a-actinin, are small and tend to organize actin filaments into bundles, whereas

others, like filamin and spectrin, tend to organize F-actin into more network-like

structures.

The cross-linking proteins found inside most cells are quite diVerent from simple

rigid, permanent cross-links in two important ways. Most physiological cross-links

are dynamic, with finite binding aYnities to actin filaments that results in the

disassociation of cross-links from F-actin over timescales relevant for cellular

remodeling. Moreover, physiological cross-links have a compliance that depends

ontheir detailed molecularstructure and determines networkmechanical response.

Thus, not surprisingly, the kinetics and mechanics of F-actin-binding proteins can

have a significant impact on the mechanical response of cytoskeletal networks.

Typical F-actin cross-linking proteins are dynamic; they have characteristic on

and oV rates that are on the order of seconds to tens of seconds. The cross-linking

protein a-actinin, which is commonly found in contractile F-actin bundles, is a

dumb-bell shaped dimer with F-actin-binding domains spaced approximately

30 nm apart. Typical dissociation constants for a-actinin are on the order of

Kd¼ 1 mM and dissociation rates are on the order of 1 s?1, but vary between

diVerent isoforms (Wachsstock et al., 1993), with temperature (Tempel et al., 1996)

and the mechanical force exerted on the cross-link (Lieleg and Bausch, 2007).

Physiologically relevant cross-links cannot be thought of simply as completely

rigid structural elements; they can, in fact, contribute significantly to network

compliance. Filamin proteins found in humans are quite large dimers of two

280-kDa polypeptide chains, each consisting of 1 actin-binding domain, 24

b-sheet repeats forming 2 rod domains, and 2 unstructured ‘‘hinge’’ sequences

(Stossel et al., 2001). The contour length of the dimer is approximately 150 nm,

making it one of the larger actin cross-links in the cell (Fig. 5A). Unlike many other

0

Force (pN)

0

100

200

300

200 nm

200100

Extension (nm)

300 400

AB

Fig. 5

extension curve for a filamin A molecule measured by atomic force microscopy. The characteristic

sawtooth pattern is associated with unfolding events of b-sheet domains in the molecule (with permis-

sion, Furuike et al., 2001).

(A) Electron micrographs of filamin A dimer (with permission, Stossel et al., 2001). (B) Force-

19. Mechanical Response of Cytoskeletal Networks

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cross-linking proteins that dimerize parallel to each other in order to form a small

rod, the filamin molecules dimerize such that they form a V-shape with actin-

binding domains at the end of each arm. This geometry is thought to allow filamin

molecules to preferentially cross-link actin filaments orthogonally and to form

strong networks even at low concentrations.

The compliance of a single filamin molecule can be probed with atomic force

microscopy force-extension measurements. Initial results suggest that for forces

less than 50–100 pN, a single filamin A molecule can be modeled as a worm-like

chain; for larger forces, reversible unfolding of b-sheet repeats occurs, leading to a

large increase in cross-link contour length (Furuike et al., 2001; Fig. 5B). It is

important to note that forces reported for these types of unfolding measurements

are rate dependent; the longer a force is applied to the molecule, the lower the

threshold force required for the conformational change.

One additional class of binding proteins is molecular motors such as myosin.

The conformation change of the molecule as it undergoes ATP hydrolysis can

generate pico-Newton scale forces within the F-actin network or bundle. These

forces can generate filament motion, such as observed in F-actin sliding within the

contraction of a sarcomere. These actively generated forces can significantly

change the mechanical properties and the structure of the cytoskeletal network

in which they are embedded (Bendix et al., 2008).

B. Rheology of Rigidly Cross-Linked F-Actin Networks

Although the importance of understanding mechanical response of cytoskeletal

networks has been appreciated for several decades, predictive physical models to

describe the full range of mechanical response observed in these networks have

proven elusive. This has been, in part, due to the large sample volumes required by

conventional rheology (1–2 ml per measurement) and the inability to purify suY-

cient quantities of protein with adequate purity to perform in vitro measurements.

Improvement in the torque sensitivity of commercially available rheometers as well

as the establishment of bacteria and insect cell expression systems for protein

expression has overcome many of these diYculties.

In the last several years, much progress has been made in understanding the

elastic response of F-actin filaments cross-linked into networks by very rigid,

nondynamic linkers. This class of cross-linkers greatly simplifies the interpreta-

tions of the rheology in two distinct ways. When the cross-linkers are more rigid

than F-actin filaments, then the mechanical response of the composite network is

predominately determined by deformations of the softer F-actin filaments; in this

case, the cross-linkers serve to determine the architecture of the network. When

cross-linkers have a very high binding aYnity and remain bound to F-actin

over long times (>minutes), then we do not have to consider the additional time-

scales associated with cross-linking binding aYnity, which can lead to network

remodeling under external stress.

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Margaret L. Gardel et al.

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Two realizations of this are cross-linking through avidin–biotin cross-links

(MacKintosh et al., 1995) and the actin-binding protein, scruin (Gardel et al.,

2004a; Shin et al., 2004). In these networks, network compliance is due to the

semiflexibility of individual F-actin filaments. Such a network can be considered to

have an average distance between actin filaments, or mesh size, x ? 1=

distance between cross-links, ‘cwhere ‘c> x for homogeneous networks.

ffiffiffiffiffi

cA

p

with a

1. Network Elasticity and Microscopic Deformation

In order to establish an understanding of the elastic properties of a material, it is

required to know how it will deform in response to an external shear stress.

For semiflexible polymers, such as F-actin, strain energy can be stored either in

filament bending or in stretching. These elastic constants depend on the length

of filament segment that is being deformed, for instance, ‘cfor a homogeneous

cross-linked F-actin network. Recent theoretical work has shown that the

deformation of F-actin networks under an external shear stress is dominated by

stretching of filaments in the limit of high cross-link and F-actin concentration

and long filament lengths (Head et al., 2003a,b). Here, the deformations in the

network are self-similar at all length scales, or aYne (Fig. 6). In contrast, in

the limit of low cross-link and F-actin concentration and short F-actin

lengths, deformations imposed by the external shear stress result in filament

bending and nonaYne deformation throughout the network (Das et al., 2007;

Nonaffine

Affine

Solution

Log(c)

Log(L)

Affine

mechanical

Affine

entropic

Nonaffine

Fig. 6

network is indicated by slender black rods that is confined between two parallel plates indicated by dark

gray rods. The direction of shear at the macroscopic level is indicated by the arrow with the open

arrowhead, whereas filled arrows indicate direction of microscopic deformations within the sample. In

nonaYne deformations, the directions of deformation within the sample are not similar to each other or

to the direction of macroscopic shear; this type of deformation is realized in very sparse networks. In

aYne deformation, the direction of macroscopic deformation is highly self-similar to the directions of

microscopic deformation within the sample; this type of deformation is realized in highly concentrated

polymer networks. (Right) A sketch of the various elastic regimes in terms of molecular weight L and

polymer concentration c. The solid line represents where network rigidity first appears at the macro-

scopic level. For aYne deformation, elastic response can arise both from the filament stretching of

entropically derived bending fluctuations or from the Young’s modulus of individual filaments.

(Left) Schematics indicating diVerence between aYne and nonaYne deformations. A fibrous

19. Mechanical Response of Cytoskeletal Networks

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Head et al., 2003a,b; Fig. 6). These predictions have been confirmed in experiments

by visualizing the deformations of F-actin networks during application of shear

deformation using confocal microscopy (Liu et al., 2007) where nonaYnity is

calculated as the deviation of network deformations after shear from the assumed

aYne positions; these experiments confirmed that weakly cross-linked F-actin

networks exhibited nonaYne deformations, whereas deformations of strongly

cross-linked networks were more aYne.

2. Entropic Elasticity of F-Actin Networks

In networks of F-actin cross-linked with incompliant cross-links where shear

stress results in aYne deformations, the elastic response is dominated by stretching

of individual actin filaments. At the filament length scale, the strain, g, is propor-

tional to d=‘cwhere d is the extension of individual filaments and ‘cis the distance

between cross-links. The stress, s, can be considered as F/x2, where F is the force

applied to individual filaments and x is the mesh size of the network. Thus, we can

relate the force-extension of single filaments (Section III.A.1) to the network

elasticity. For networks structured at micrometer length scales, the spring constant

determined by entropic fluctuations determines the elastic response at small strains

such that:

G

0?s

g?

k2

kBTx2‘3

c

where the contour length is determined by the distance between cross-links and is

proportional to the entanglement length. Because the entropic spring constant is

highly sensitive to the contour length, this model predicts a sharp dependence of

the elastic stiVness with both the F-actin concentration, cA, and the ratio of cross-

links to actin monomers, R, such that:

G

0? c11=5

A

Rð6xþ15yÞ=5

where the exponent x characterizes how eYciently the cross-linker bundles F-actin

and y characterizes the cross-linking eYciency (Shin et al., 2004). The variation of

the elastic stiVness as a function of F-actin concentration has been observed

experimentally (Gardel et al., 2004a; MacKintosh et al., 1995; Fig. 7). The pro-

nounced dependence of the elastic stiVness observed as a function of polymer and

cross-link density is in sharp contrast to the weak dependence observed in net-

works of flexible polymers.

Densely cross-linked F-actin networks exhibit nonlinear elasticity at large stres-

ses and strains, where G

stress,smax, and strain, gmax, at which the network ‘‘breaks’’ (Fig. 2B). In this

system, the breaking stress is linearly proportional to the density of F-actin fila-

ments and suggests that individual F-actin ruptures (Gardel et al., 2004b). The

maximum strain is observed to vary such that gmax? ‘c? c?2=5

0increases as a function of stress until a maximum

A

and directly

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Margaret L. Gardel et al.

Page 13

reflects the change in contour length resulting from varying F-actin concentration

(Gardel et al., 2004a). Moreover, the qualitative form of the nonlinearity in the

stress–strain relationship at the network length scale is identical to divergence

observed in the force–extension relationship for single semiflexible polymers as

the extension approaches the polymer contour length (Gardel et al., 2004a,b).

Thus, the nonlinear strain-stiVening response of these F-actin networks at macro-

scopic length scales directly reflects the nonlinear stiVening of individual filaments.

3. Other Regimes of Elastic Response

As the concentration of cross-links or the filament persistence length increases,

the entropic spring constant to stretch semiflexible filaments will increase suY-

ciently such that the deformation of filaments is dominated by the Young’s

modulus of the filament. Here, the elasticity is still due to stretching individual

F-actin filaments, but thermal eVects do not play a role and the elastic response

of these networks is more similar to that of a dense network of macroscopic rods

(e.g., imagine a dense network of cross-linked pencils or spaghetti). Here, no

mechanism for significant stress stiVening at the scale of individual rods is estab-

lished. However, reorganization of these networks under applied stress may lead to

stress stiVening. Such a regime of elasticity may be observed in networks of highly

bundled F-actin filaments; such networks have not been observed experimentally.

In contrast, as the density of cross-links or filament persistence length decreases,

filamentswilltendtobend(andbuckle)underanexternalsheardeformation.Bending

deformations result in deformations that are not self-similar, or aYne, within the

network (Head et al., 2003a,b). Experimental measurements have shown an increase

300

G0 (Pa)

30.0

3.00

0.30

R

0.03

100

10−3

10−2

10−1

100

101

CA (mM)

Fig. 7

concentration, and cA, F-actin concentration. The range in colors corresponds to the magnitude of

the linear elastic modulus, G0(indicated by the heat scale) whereas the symbols denote networks that

exhibit stress stiVening (þ) or stress weakening (o) (with permission, Gardel et al., 2004).

State diagram of rigidly cross-linked F-actin networks over a range of R, the cross-link

19. Mechanical Response of Cytoskeletal Networks

499

Page 14

innonaYnedeformationsatlowcross-linkconcentrations(Liuetal.,2007)aswellas

an abrogation of stress-stiVening response (Gardel et al., 2004a). Instead, these net-

works soften under increasing strain and linear response is observed for strains as

large as one. For these networks, the linear elastic modulus is less sensitive to varia-

tionsincross-linkdensityandactinconcentration.Whileacompletecomparisonwith

theory is still required, it appears that in this regime, network elasticity is dominated

by filament bending, with nonlinear response due to buckling of single filaments

(Gardel et al., 2004a; Head et al., 2003a,b; Liu et al., 2007).

The rich variety of elastic response in even a model system of F-actin cross-

linked by rigid, nondynamic cross-links demonstrates the complexity involved with

building mechanical models of networks of cross-linked semiflexible polymers that

can exhibit both entropic and enthalpic contributions to the mechanical response.

C. Physiologically Cross-Linked F-Actin Networks

F-actin networks formed with rigid, incompliant cross-links form a benchmark

to understanding the elastic response of cytoskeletal F-actin networks. However,

as discussed in Section III.A.2, physiological F-actin cross-linking proteins typi-

cally have a finite binding aYnity to F-actin and significant compliance. The extent

of F-actin-binding aYnity of the cross-linker determines a timescale over which

forces are eYciently transmitted through the F-actin/cross-link connection and

dramatically aVects how forces are transmitted and dissipated through the net-

work. When the cross-link that has comparable stiVness to that of an F-actin

filament, the network will elasticity will some superposition of the elastic response

of each element individually. Thus, the changes in the kinetics and mechanics of

individual cross-linking proteins can dramatically aVect the mechanical response

of the F-actin network.

1. EVects of Cross-Link Binding Kinetics: a-Actinin

The contribution of cross-link binding kinetics to network material properties

has been studied most explicitly in the a-actinin and fascin systems. The dynamic

nature of cytoskeletal cross-links means that networks formed with them are able

to reorganize and remodel, or look ‘‘fluid-like’’ at long times (Sato et al., 1987). In

particular, temperature has been used to systematically alter the binding aYnity of

a-actinin to F-actin, and the mechanics of the resulting network probed with bulk

rheology (Tempel et al., 1996; Xu et al., 1998). The key experimental observation is

that as temperature is increased from 8 to 25?C, the a-actinin cross-linked F-actin

networks become softer and more fluid-like. At 8?C, the networks are stiV, elastic

networks that look similar to networks cross-linked with rigid, static cross-links.

As the temperature is raised to 25?C, the network stiVness decreases by nearly a

factor of 10 and the network becomes more fluid-like.

There are a variety of eVects that could contribute to this behavior, including

changes to F-actin dynamics and the fraction of bound a-actinin cross-links.

However, these experiments found that the dominant eVect of increasing

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Margaret L. Gardel et al.

Page 15

temperature is to increase the rate of a-actinin unbinding from F-actin, implying

that as cross-link dissociation rates increase, the network becomes a more dynamic

structure that can relax stress. This suggests that if cells require cytoskeletal

structures to reorganize and remodel, it is important to have dynamic cross-link

proteins like a-actinin, not permanent ones like scruin. One interesting example

where cross-link binding kinetics has a strong biological consequence is in an

a-actinin-4 isoform having a point mutation that causes increased actin-binding

aYnity (Weins et al., 2005; Yao et al., 2004). This increased binding aYnity is

associated with cytoskeletal abnormalities in focal segmental glomerulosclerosis, a

lesion found in kidney disease that results from a range of disorders including

infection, diabetes, and hypertension.

Mechanical load can also have an eVect on cross-link binding kinetics. When

large shear stresses are applied to fascin cross-linked and bundled F-actin net-

works, network elasticity depends on the forced unbinding of cross-links in a

manner that depends on the rate at which stress is applied (Lieleg and Bausch,

2007). Although temperature is unlikely to be an important control parameter

in vivo, mechanical force on actin-binding proteins may regulate both mechanical

response of the network and organization of signaling within the cytoplasm.

However, it is unknown to what extent cross-link kinetics play a role in regulation

of mechanical stresses within live cells to enable rapid and local cytoskeletal

reorganization.

2. EVect of Cross-Link Compliance: Filamin A

Cross-link geometry and compliance can also contribute significantly to F-actin

network elasticity. Rigidly cross-linked networks have a well-defined elastic

plateau where the elastic modulus is orders of magnitude larger than the

viscous modulus, and energy is stored elastically in the network. In contrast,

networks formed from F-actin cross-linked with filamin A (FLNa) have an elastic

modulus that is only two or three times the viscous modulus, and the elastic

modulus decreases as a weak power law over timescales between a second and

thousands of seconds (Gardel et al., 2006a,b) (Fig. 8), similar to the timescale

dependenceoftheelasticityoflivingcells(Fabryetal.,2001).Moreover,incontrast

to the F-actin–scruin networks where the linear elastic modulus can be tuned over

severalordersofmagnitudebyvaryingcross-linkdensity,thelinearelasticmodulus

for F-actin–FLNa networks is only weakly dependent on the FLNa concentration

and is typically in the range of 0.1–1 Pa (Gardel et al., 2006a), less than tenfold

larger than for F-actin solutions formed without any cross-links.

Insight into how cross-link compliance can alter macroscopic mechanical

response can be gained from a recent experiment in which the total length of the

cross-link ddFLN, a filamin isoform from Dictyostelium discoideum, is systemati-

cally altered and the mechanics of the resulting network are probed using bulk

rheology (Wagner et al., 2006). In these networks, as the length of the cross-linker

is systematically increased, the stress transmission in networks becomes

19. Mechanical Response of Cytoskeletal Networks

501