Mechanical Response of Cytoskeletal Networks

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  • CHAPTER 19

    Mechanical Response ofCytoskeletal Networks

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

    Jiayu Liu, and David A. Weitz,

    *Department of Physics and Institute for Biophysical DynamicsUniversity of Chicago, Illinois 60637

    School of Engineering and Applied SciencesHarvard UniversityCambridge, Massachusetts 02143

    Department of PhysicsHarvard UniversityCambridge, Massachusetts 02143

    AbstractI. IntroductionII. Rheology

    A. Frequency-Dependent ViscoelasticityB. Stress-Dependent ElasticityC. EVect of Measurement Length Scale

    III. Cross-Linked F-Actin NetworksA. Biophysical Properties of F-Actin and Actin Cross-linking ProteinsB. Rheology of Rigidly Cross-Linked F-Actin NetworksC. Physiologically Cross-Linked F-Actin Networks

    IV. EVects of Microtubules in Composite F-Actin NetworksA. Thermal Fluctuation ApproachesB. In Vitro MT NetworksC. Mechanics of Microtubules in Cells

    V. Intermediate Filament NetworksA. IntroductionB. Mechanics of IFsC. Mechanics of Networks

    VI. Conclusions and OutlookReferences

    METHODS IN CELL BIOLOGY, VOL. 89 0091-679X/08 $35.00Copyright 2008, Elsevier Inc. All rights reserved. 487 DOI: 10.1016/S0091-679X(08)00619-5

  • 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 highlynonlinear elasticity and athermal dynamics driven by ATP-dependent processes.To build quantitative mechanical models describing complex cellular behaviors, itis necessary to understand the underlying physical principles that regulate forcetransmission and dynamics within these networks. In this chapter, we review ourcurrent understanding of the physics of networks of cytoskeletal proteins formedin vitro. We introduce rheology, the technique used to measure mechanical re-sponse. We discuss our current understanding of the mechanical response ofF-actin networks, and how the biophysical properties of F-actin and actin cross-linking proteins can dramatically impact the network mechanical response. Wediscuss how incorporating dynamic and rigid microtubules into F-actin networkscan 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 mechanicalforces across the cell. For example, cells must be able to maintain their shape whensubjected to external shear stresses, such as forces exerted by blood flow in thevasculature. During cell migration and division, forces generated within the cell arerequired to drive morphogenic changes with extremely high spatial and temporalprecision. Moreover, adherent cells also generate force on their surroundingenvironment; cellular force generation is required in remodeling of extracellularmatrix and tissue morphogenesis.

    This varied mechanical behavior of cells is determined, to a large degree, bynetworks of filamentous proteins called the cytoskeleton. Although we have thetools 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. Thisknowledge of the biophysical properties of assemblies of cytoskeletal proteins isnecessary to link our knowledge of single molecules to whole cell physiology.However, a complete understanding of the mechanical behavior of the dynamiccytoskeleton is far from complete.

    One approach is to develop techniques to measure mechanical properties of thecytoskeleton in living cells (Bicek et al., 2007; Brangwynne et al., 2007a; Crockerand HoVman, 2007; Kasza et al., 2007; Panorchan et al., 2007; Radmacher, 2007).Such techniques will be critical in delineating the role of cytoskeletal elasticity indynamic cellular processes. However, because of the complexity of the livingcytoskeleton, it would be impossible to elucidate the physical origins of this cyto-skeletal elasticity from live cell measurements in isolation. Thus, a complementary

    488 Margaret L. Gardel et al.

  • 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 measurementsof reconstituted cytoskeletal networks have revealed a rich and varied mechanicalresponse and have required the development of qualitatively new experimentaltools and physical models to describe physical behaviors of these protein networks.In this chapter, we review our current understanding of the biophysical propertiesof networks of cytoskeletal proteins formed in vitro. In Section II, we discussrheology measurements and the importance of several parameters in interpretationof these results. In Section III, we discuss the rheology of F-actin networks, high-lighting how small changes in network composition can qualitatively change themechanical response. In Section IV, the eVects of incorporating dynamic micro-tubules in composite F-actin networks will be discussed. Finally, in Section V, wewill discuss the mechanics of intermediate filament (IF) networks.

    II. Rheology

    Rheology is the study of howmaterials deform and flow in response to externallyapplied force. In a simple elastic solid, such as a rubber band, applied forces arestored in material deformation, or strain. The constant of proportionality betweenthe stress, force per unit area, and the strain, deformation per unit length, is calledthe elastic modulus. The geometry of the measurement defines the area and lengthscale used to determine stress and strain. Several diVerent kinds of elastic modulican be defined according to the direction of the applied force (Fig. 1). The tensile

    Youngs modulus, Etensile elasticity

    Bulk modulusCompressional modulus

    Bending modulus, k Shear modulus, G

    Fig. 1 Schematics showing the direction of the applied stress in several common measurements ofmechanical properties; the light gray shape, indicating the sample after deformation, is overlaid onto the

    black shape, indicating the sample before deformation. The Youngs modulus, or tensile elasticity, is thedeformation 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.

    19. Mechanical Response of Cytoskeletal Networks 489

  • elasticity, or Youngs modulus, is determined by the measurement of extension of amaterial under tension along a given axis. In contrast, the bulk modulus is ameasure of the deformation under a certain compression. The bending modulusof a slender rod measures the object resistance to bending along its length. And,finally, the shear elastic modulus describes object deformation resulting from ashear, volume-preserving stress (Fig. 2). For a simple elastic solid, a steady shear

    s (w)g (w)s (w)


    g (w)

    x x

    Term Strain


    Frequency Frequency of applied + measured


    Phase Shift





    Shear moduli:s0= 0

    s0> 0




    Elastic (storage)modulus

    Viscous (loss)modulus

    Differential elasticmodulus

    Differential lossmodulus






    Units None

    Pascal (Pa)

    Pascal (Pa)



    Pascal (Pa)

    Pascal (Pa)

    Pascal (Pa)

    Pascal (Pa)


    Height (h)x

    Area (A)Force

    ; sample deformation

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

    d (w) = tan1 (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) = s (w)/g (w) cos(d (w))

    K!(w) = s (w)/g (w) cos(d (w))

    Constant external stress applied to sampleduring measurement

    Fig. 2 This schematic defines many of the rheology terms used in this chapter. (Left) To measure theshear elastic modulus, G0(o), and shear viscous modulus, G00(o), an oscillatory shear stress, s(o), isapplied to the material and the resultant oscillatory strain, g(o) is measured. The frequency, o, is variedto probe mechanical response over a range of timescales. (Right) To measure how the stiVness varies asa function of external stress, a constant stress, s0, is applied and a small oscillatory stress, (Ds(o)), issuperposed to measure a diVerential elastic and viscous loss modulus.

    490 Margaret L. Gardel et al.

  • stress results in a constant strain. In contrast, for a simple fluid, such as water, shearforces result in a constant flow or rate of change of strain. The constant ofproportionality 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 and viscous modulus. Mechanical measurements of shear elasticand viscous response over a range of frequencies and strain amplitudes are possiblewith commercially available rheometers. Recent developments in rheometer tech-nology now provide the capability of mechanical measurements with as little as100 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 ofmicrorheological techniq