Download - Compliance of actin filament networks measured by particle-tracking microrheology and diffusing wave spectroscopy

Jingyuan XuVirgile ViasnoffDenis Wirtz

Compliance of actin filament networksmeasured by particle-tracking microrheologyand diffusing wave spectroscopy

Received: 3 April 1998Accepted: 26 May 1998

J. Xu · V. ViasnoffDepartment of Chemical EngineeringThe John Hopkins University3400 North Charles StreetBaltimore, MD 21218USA

D. Wirtz (✉)Chemical EngineeringG.W.C. Whiting School of EngineeringThe John Hopkins University3400 North Charles StreetBaltimore, MD 21218-2694USAe-mail: [email protected]

Abstract We monitor the time-de-pendent shear compliance of a solu-tion of semi-flexible polymers,using diffusing wave spectroscopy(DWS) and video-enhanced single-particle-tracking (SPT) microrheol-ogy. These two techniques use thesmall thermally excited motion ofprobing microspheres to interrogatethe local properties of polymer solu-tions. The solutions consist of net-works of actin filaments which arelong semi-flexible polymers. We es-tablish a relationship between themean square displacement (MSD) ofmicrospheres imbedded in the solu-tion and the time-dependent creepcompliance of the solution,hDr 2(t)i=(kBT/pa)J(t). Here,J(t) isthe creep compliance,hDr 2(t)i is themean-square displacement, anda isthe radius of the microspherechosen to be larger than the meshsize of the polymer network. DWSallows us to measure mean squaredisplacements with microsecondtemporal resolution and A˚ ngstromspatial resolution. At short times,the mean square displacement of a0.96lm diameter sphere in a con-centrated actin solution displayssub-diffusion.hDr 2(t)i!t�, with acharacteristic exponent

�=0.78±0.05, which reflects the fi-nite rigidity of actin. At long times,the MSD reaches a plateau, with amagnitude that decreases with con-centration. The creep compliance isshown to be a weak function ofpolymer concentration and scales asJp!c–1.2±0.3. This exponent is cor-rectly described by a recent modeldescribing the viscoelasticity ofsemi-flexible polymer solutions. TheDWS and video-enhanced SPT mea-surements of the compliance plateauagree quantitatively with compliancemeasured independently using clas-sical mechanical rheometry for aviscous oil and for a solution offlexible polymers. This paper ex-tends the use of DWS and single-particle-tracking to probe the localmechanical properties of polymernetworks, shows for the first timethe proportionality between meansquare displacement and local creepcompliance, and therefore presents anew, direct way to extract the visco-elastic properties of polymer sys-tems and complex fluids.

Key words Actin – rheology –diffusing wave spectroscopy

Rheol Acta 37:387–398 (1998)© Steinkopff Verlag 1998 ORIGINAL CONTRIBUTION

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Introduction

The rheology of semi-flexible polymers is more com-plex than that of flexible and rigid polymers (de Gen-nes, 1979; Doi and Edwards, 1989; Ferry, 1980). Thedescription of semi-dilute and concentrated solutions ofsemi-flexible polymers is greatly complicated by thepresence of an additional intermediate lenght scale, out-side the monomer size and the polymer contour length:the persistence length. If the persistence lenght, whichdescribes the finite rigidity of a polymer, is of the orderof magnitude of the contour lengthL and much largerthan the diameter, the polymer is semi-flexible (Doi andEdwards, 1989).

Filamentous actin (F-actin), the polymerized form ofthe globular four-lobed protein actin (G-actin), formslong semi-flexible polymers with a diameterd=7 nm(Alberts et al., 1994). The persistence length of F-actin,lp, has been measured using different techniques, in-cluding dynamic light scattering, video-microscopy, andelectron microscopy (Gittes et al., 1993; Isambert et al.,1995; Muller et al., 1991). Most of these approacheshave confirmed that F-actin is a semi-flexible polymer,i.e. d�lp≤L. Furthermore, the rigidity and contourlength of F-actin can be finely controlled by regulatingproteins, such as�-actinin (crosslinking proteins), gel-solin (capping and severing proteins), and actophorin(severing proteins) (Alberts et al., 1994; Isambert et al.,1995). The relatively good control of the intrinsic prop-erties of F-actin makes it an ideal polymer to investi-gate the linear viscoelasticity of concentrated solutionsof semi-flexible polymers.

Due to its importance in providing non-muscle cellswith structural rigidity, the rheology of actin has beenstudied extensively (Alberts et al., 1994). However, un-til recently, these rheological studies have primarilyfocused on the magnitude of the plateau modulus thatactin solutions display at small frequencies (Haskell etal., 1994; Janmey et al., 1988, 1990, 1994; Sato et al.,1987; Tempel et al., 1996; Wachsstock et al., 1993;Wachsstock et al., 1994; Xu et al., 1998c; Zaner andHartwig, 1988). The magnitude of this plateau modulusand, therefore, the nature of the viscoelastic propertiesof actin solutions, has, however been the subject ofmuch debate. This debate has been recently resolved byXu et al. (1998a), who unambiguously showed thatcareless preparation and storage of actin can dramati-cally increase the small-frequency elastic modulus ofthe solution. We also recently showed how smallamounts of crosslinking proteins can greatly increasethe elastic modulus, especially at large strains (Palmeret al., 1998). Both papers showed the absolute necessityof using fresh, non-frozen actin, uncontaminated bycrosslinking proteins. In this paper, we use recent ad-vances in actin preparation, which eliminate crosslink-ing and capping proteins.

In order to probe the dynamics of actin motion inconcentrated solutions, we use two complementaryapproaches: mechanical rheometry and particle-trackingmicrorheometry. Particle-tracking microrheology, whichinvolves either diffusing wave spectroscopy (Mason etal., 1997b; Mason and Weitz , 1995; Palmer et al.,1998) or single particle-tracking microrheology (Gitteset al., 1997; Mason et al., 1997a,b; Schnurr et al.,1997), is based on the high-resolution measurement ofthe mean square displacement (MSD) of microspheresimbedded in the polymer solution to be probed. DWS,as developed by Mason and Pine (Mason and Weitz,1995) spectroscopically monitors the thermally excitedmotion of many spheres mixed with the polymer solu-tion; particle-tracking microrheology as developed byKuo, Wirtz, and coworkers (Mason et al., 1997a,b; Pal-mer et al., 1998) monitors the two-dimensional motionof a single microsphere in thetemporaldomain. Thusfar, different analytical schemes have been used to ex-tract the elastic and loss moduli of the solution (Masonet al., 1997b; Mason and Weitz, 1995; Schnurr et al.,1997). These schemes involve multiple transformationsteps of the real space measurements of the MSD or thespectrum of the MSD, to the Laplace space calculationof a viscoelastic modulus, to the use of Kramers-Kronigrelations, and finally via complex transformation to theelastic and loss moduli. Typically, these multiple-stepschemes, while powerful in generating linear viscoelas-tic moduli, truncate several temporal decades at smalland large time scales. This paper proposes a much moredirect interpretation of MSD measurements: we showthat the measured MSD is simply proportional to thetime-dependent shear creep compliance, which containsa great deal of the interesting rheological information.

In order to demonstrate the effectiveness of ourapproach, we independently measure the creep compli-anceJ(t) of actin solutions using particle-tracking mi-crorheometry and mechanical rheometry. We observeexcellent agreement between these independent mea-surements. As a further control, we measure the creepcompliance of a viscous oil and of a semi-dilute solu-tion of flexible polymers, polyethylene oxide in water,and obtain excellent agreement.

Particle-tracking measurements by DWS offer amuch extended temporal range especially at short timescales, smaller than 10–5 s, which cannot be probed bymechanical rheometry in one measurement. At theseshort times, we find thatJ(t) increases with an effectivepower law of time, J(t)!t� with �=0.78±0.05, inagreement with a recent model by Morse (Morse,1997), which predicts�=3/4. This exponent reflectsthe finite rigidity of F-actin and the resulting liquid-likebehavior of the F-actin networks at short times. We alsofind that the short-time compliance scales with the in-verse of the square root of actin concentration,J!c–0.50

for t<10–3 s. At times between 10–1 and 104 s as

388 Rheologica Acta, Vol. 37, No. 4 (1998)© Steinkopff Verlag 1998

probed by video-enhanced SPT and mechanical rheome-try, J(t) reaches a “plateau”Jp, which corresponds to aslowed increase of the creep at long times. For those in-termediate time scales, the creep has a concentration-de-pendent magnitude which scales asJp!c–b withb=1.3±0.2. Here again, this exponent is in qualitativeagreement with Morse’s model (Morse, 1997), whichdescribes the long-time scale viscoelasticity of solutionsof semi-flexible polymers. Morse’s model is based onthe reptation model of de Gennes (de Gennes, 1979)and Doi and Edwards (Doi and Edwards, 1989), whichassumes that the shear stress relaxes when polymerscan escape the tube inside which each polymer is con-fined by surrounding polymers. Morse’s model predictsb=1.4, as opposed tob=2.2 for solutions of flexiblepolymers, and assumes that the elasticity of a semi-flex-ible polymer network results from the forces preventingtransverse distortion of the polymer tube conformation(Morse, 1997).

Experimental techniques

Actin preparation

Actin is extracted from rabbit skeletal muscle acetonepowder by the method of Spudich and Watt (Spudichand Watt, 1971). The resulting actin is gel filtered onSephacryl S-300 HR instead of Sephadex G-150 (Mac-Lean-Fletcher and Pollard, 1980). The purified actin isstored as Ca2+-actin in continuous dialysis at 48Cagainst daily changed buffer G (0.2 mM ATP, 0.5 mMDTT, 0.1 mM CaCl2, 1 mM NaAzide, and 2 mM Tris-Cl, pH 8.0). The final actin concentration is determinedby ultra-violet absorbance at 290 nm, using an extinc-tion coefficient of 2.66×104 M–1 cm–1, and a cell pathlength of 1 cm. Ca2+ and Mg2+ actin filaments are gen-erated by adding 0.1 volume of 10×KMC (500 mMKCl, 10 mM MgCl2, 1 mM CaCl2, 20 mM Tris-Cl, pH8) and 10-x KME (500 mM KCl, 10 mM MgCl2,10 mM EGTA, 100 mM imidazole, pH 7.0) polymeriz-ing salt buffer solution to 0.9 volume of G-actin in buf-fer G, respectively. The actin used for all experimentscomes from the last fraction of the actin peak obtainedby one gel filtration.

Diffusing wave spectroscopy (DWS)and video-enhanced single-particle-tracking rheology(SPT)

The beam from an Ar+-ion laser operating in the single-line-frequency mode at a wavelength of 514 nm is fo-cused and incident upon a flat scattering cell whichcontains the polymerized actin solution and spherical

optical probes. The light multiply-scattered from the so-lution is collected by two photomultiplier tubes (PMT)via a single-mode optical fiber with a collimator lens ofvery narrow angle of acceptance at its front end and abeamsplitter at its back end. The outputs of the PMTsare directed to a correlator working in the pseudo cross-correlation mode to generate the autocorrelation func-tion g2(t)–1 from which quiescent rheological proper-ties of the actin solutions can be calculated (Palmer etal., 1998).

Actin is polymerized in situ for 12–14 h before mea-surement by loading the scattering cell with a solutionof monomeric actin mixed with the polymerizing saltsolution and a dilute suspension of monodisperse latexmicrospheres (Duke Scientific Corp.) of diameter0.96lm at a volume fraction of 0.01. The scatteringcell is then immediately capped tightly. We verified byboth time-resolved static light scattering and time-re-solved mechanical rheology that G-actin was fully poly-merized into F-actin at all concentrations presented inthis paper before 12 h. Using static light scattering, weverified that more than 95% of the scattering intensityin the transmission geometry was due to the micro-spheres, less than 5% due to the actin filament networkitself. We also verified using mechanical rheology thatthe added latex beads did not affect the rheology ofpolymerized actin, representing less than 5–7% of themagnitude ofG'(x) and G''(x) at all actin concentra-tions used in this paper. All DWS measurements areconducted at a temperature ofT=238C.

In addition to DWS, we conduct video-enhanced sin-gle-particle-tracking (SPT) microrheology measure-ments. Briefly (Mason et al., 1997a), an extremelydilute suspension of 0.96lm diameter microspheres ismixed with an unpolymerized G-actin solution. Poly-merizing salt is added, which promotes the self-assem-bly of G-actin into long semi-flexible F-actin polymers.After approximately 10 h of curing, an elastic F-actinnetwork forms. The position of a single bead is trackedwith 5–10 nm resolution by monitoring the center-of-mass displacements of the two-dimensional light inten-sity of the Airy figure of the microsphere. The rate ofdata acquisition for video-enhanced SPT is slightlysmaller than the video rate of image acquisition. There-fore, the maximum frequency of viscoelastic moduliwhich can be probed by that technique is about 5 Hz.For each system, we monitor between 15 and 30 micro-spheres, we conduct five independent measurements oneach bead, and on three different samples of actin ateach actin concentration. Further details about ourvideo-microscopy based, single particle-tracking micro-rheometer (SPT) are given in Mason et al. 1997a).

Advantages of particle-tracking microrheometry(DWS and video-enhanced SPT) include the possibly togenerate viscoelastic moduli without subjecting thespecimen to shear, which can induce bundling of the

389J. Xu et al.Compliance of actin filament networks

polymers. Video-enhanced SPT also allows to extractviscoelastic moduli that are anisotropic and require ex-tremely small sample volumes&20ll, as opposed to&1 ml for a regular rheometer and for the DWS instru-ment. Furthermore, video-enhanced SPT is relativelysimple to implement. Note, however, that DWS gener-ates viscoelastic moduli over an extremely large rangeof frequencies, 2 to 5 decades wider than probed byclassical rheometry and video-enhanced SPT.

Mechanical rheometry

In order to compare our optical measurements withclassical mechanical measurements, we employ a stress-controlled mechanical rheometer (Rheometrics andHaake) equipped with a 40 or 50 mm diameter coneand plate geometry and a strain-controlled rheometer(Rheometrics) with rapid control of the stress. To pre-vent possible evaporation effects, the cone and platetools are enclosed in a custom-made vapor trap andsealed at the edges with an hydrophobic oil. The tem-perature of the sample is fixed atT=238C to within0.18C. The G-actin solution is placed between the coneand plate tools and allowed to polymerize in the pres-ence of polymerizing salt for 12–14 h, prior to the mea-surements. The linear and nonlinear viscoelastic proper-ties of actin solutions, PEO solutions, are also found tobe unaffected by the presence of the probing micro-spheres.

Theory

Derivation and interpretation of the relationbetween mean-square displacementand creep complicance

In this section, we establish a relationship between themean square displacement of a microsphere imbeddedin a viscoelastic fluid and the shear creep compliance ofthat fluid. The local viscoelastic modulus of a fluid andthe mean square displacement of a microsphere sus-pended in that fluid can be relatedvia (Mason andWeitz, 1995)

~G�s� � s

6pa

�6kBT

s2hD~r2�s�i ÿms�� kBT

pashD~r2�s�i ;

�1�Here,kB is Boltzmann’s constant,T is the temperatureof the sample,a is the radius of the microsphere,m isits mass,s is the Laplace frequency, and~ representsthe unilateral Laplace transformation defined as~X�s� � L�X�t�� R10 X�t� exp�ÿst�dt. Equation (1)relates the unilateral Laplace transform of the stress re-

laxation modulusGr�t�, ~G�s� � s ~Gr�s�, to the Laplacetransform of the mean-square displacementhDr2�t�i.While Eq. (1) is model-dependent, it is interesting tonote that it yields the correct results for the extremecases of a purely viscous liquid and a purely elasticsolid. For a viscous liquid, hD~r2�s�i � 6D=s2� kBT=pgas2 and ~G�s� � gs, whereg is the constantviscosity of the liquid; thereforeG0�x� � 0 andG00�x� � gsx. Instead, for an elastic solid,hD~r2�s�i � r20 and ~G�s� � G0 � kBT=par20, whereG0 isthe elastic modulus; therefore,G0�x� � G0 andG00�x� � 0.

Now, we relate the creep compliance to the visco-elastic modulus. The shear stress and shear strain are re-lated to one another as

s�t� � ÿZ t0

Gr�tÿ t0� _c�t0�dt0 �2�

c�t� �Z t0

J�tÿ t0� _s�t0�dt0 �3�

where _s � ds=dt and _c � dc=dt are the rates of changeof the shear stress and shear strain,Gr�t� is the linearstress relaxation modulus, andJ�t� is the linear creepcompliance. Physically,Gr�t� is the stress generated bya step strainc � c0H�t�, whereH�t� is a Heavisidefunction andJ�t� is the strain resulting from a step ofstresss � s0H�t�. For instance, for an elastic solid

Gr�t� � G0 and J�t� � 1

G0�4�

For a purely viscous fluid of viscosityg,

Gr�t� � gd�t� and J�t� � t

gH�t� �5�

Unilateral Laplace transformation of Eqs. (2) and (3)yields simple relationships between the transforms~J�s�,~G�s�, and~c�s�:

~s�s� � s ~Gr�s�~c�s� � ~G�s�~c�s� �6�~c�s� � s~J�s�~s�s� : �7�

Hence,

s~J�s� ~G�s� � 1 ; �8�which establishes a relation between the Laplace trans-forms of the creep compliance and of the stress relaxa-tion modulus. Since s ~G�s�hD~r2�s�i � kBT=pa (seeEq. (1)), we can directly relatehD~r2�s�i to ~J�s� as

hD~r2�s�i � kBTpa

~J�s�or


hDr2�t�i � kBTpa

J�t� �9�

using J�0� � 0 for a viscoelastic fluid. This last equa-tion is the central relationship of this paper, which wetest experimentally by measuring independentlyhDr2�t�i and J�t� by particle-tracking and by mechani-cal rheometry, respectively. Equation (9) neglects iner-tial effects, which become important at times scalest <10–6 s. But here again, Eq. (9) yields the correct re-sults for the extreme cases of a purely viscous fluid anda purely elastic solid. For a viscous liquid,hDr2�t�i� 6DtH�t� � �kBTpga�tH�t� and J�t� � �t=g�H�t�,which verifies Eq. (5). For an elastic solid,hD~r2�s�i � r20 andJ�t� � 1=G0, whereG0 � kBT=par20,which verifies Eq. (4).

This simple relation between mean-square displace-ment and the creep compliance can be further inter-preted as follows. A microsphere of radiusa imbeddedin a complex fluid is subject to an average thermalBrownian force of amplitudekBT=a. The resultingstress on the surrounding fluid is proportional tokBT=a

3. The strain is therefore proportional to thecreep. Since the strain is the divergence of the displace-ment, it is proportional to hDr2�t�i=a2. Therefore,hDr2�t�i=a2 / J�t�kBT=a3.

Equation (8) can also be projected into the temporaldomain:

J�0�Gr�t� �Z t0

dJ

dt0�tÿ t0�Gr�t0�dt0 � H�t� �10�

Using J�0� � 0, this last relation can be rewritten as aVolterra equationZ t

0

J�tÿ t0�Gr�t0�dt0 � tH�t� �11�

Since J�t� is proportional tohDr2�t�i, one could usethis last integral relation to numerically extract thestress relaxation modulus from optically measured meansquare displacements.

Results and discussion

We present measurements of the shear compliance ofactin filament networks obtained by single particle-tracking (SPT), diffusing wave spectroscopy (DWS),and mechanical rheometry. These measurements achievethree objectives: to gain a new insight into the fast dy-namics of motion of individual filaments at short times,which is dominated by bending fluctuations, to furthertest Morse’s model of rheology of concentrated solu-tions of semi-flexible polymers, and to establish a fun-

damental relationship between mean square displace-ment as measured by DWS and SPT and compliance asmeasured by mechanical rheometry.

DWS and video-enhanced SPT measurementsof mean-square displacement

The mean-square displacement of a microsphere im-bedded in a polymer solution can be measured in twoways. The mean square displacement is either measureddirectly from the two-dimensional trajectories of a sin-gle microsphere or indirectly from the autocorrelationfunction of the light multiply scattered by an ensembleof microspheres. The former approach involves the useof a two-dimensional single-particle-tracking technique(Mason et al., 1997b; Palmer et al., 1998). The latterapproach involves the use of diffusing wave spectro-scopy (Mason et al., 1997b; Mason and Weitz, 1995;Palmer et al., 1998). This paper presents mean squaredisplacement measurements using the two techniques.However, DWS has a bandwidth superior to video-enhanced particle-tracking. Here, we present video-enhanced particle-tracking measurements to demonstratethat a cheaper microscope can provide for a relativelyquantitative measurement of the creep at small frequen-cies. For both techniques, we use 0.96lm diameter mi-crospheres, which are mixed with the G-actin solutionat small volume fractions. Salt, which promotes G-actinself-assembly, is added to the mixture and mean squaredisplacements are collected after 12–14 h, the time ittakes for the elasticity of the F-actin network to reach asteady state, as verified by mechanical rheometry.

Selection of a probing microsphere follows strictcriteria. The microsphere cannot be smaller than themesh size of the actin network because the relationshipbetween the local displacements of a microsphere in afluid as described by Eq. (1) and the macroscopic visco-elasticity of that fluid is valid as long as the size of thebead is much larger than the mesh size of the network.This is because the Langevin equation describing thethermally-excited motion of the bead and underlyingEq. (1) is valid in the continuous limit for which thenetwork-generating viscoelasticity forms a continuumfor the motion of the bead. Also, the hydrodynamic in-teractions which scales asrÿ1 ought to encompass asufficiently large number of entanglements, which is thecase when the network is tight or when the bead islarge. The bead cannot be too large, however. If thebead is much larger than the mesh, it sediments, its mo-tion is anisotropic and is governed by the macroscopicviscosity of the suspending fluid. Also, the bead cannotbind chemically to the polymer network because itslong-time, large-length-scale displacements would be ar-tificially hindered, which could generate spurious re-sults in the calculation of the creep compliance and the


viscoelastic moduli. Using analytical centrifugation andgel electrophoresis, we checked that latex beads andactin do not bind to one another. Finally, in the case ofDWS, the concentration of microspheres should belarge enough to allow for the limit of multiple scatter-ing to be reached, but not too large since hydrodynamicinteractions between probing microspheres can causebiased motion, which is not taken into account inEq. (1).

Figure 1 displays the mean square displacement of0.96lm diameter microspheres measured by diffusingwave spectroscopy. Over the limited range of timescales probed by video-enhanced SPT, we observe afair agreement between the two particle-tracking mea-surements. We measure mean square displacementsover an extended range of actin concentrations. For allactin concentrations tested, the mean square displace-ment undergoes a sequence of dynamical transitionswhich are directly related to the short-time and long-time dynamics of the actin filaments surrounding theprobing microsphere. At short times and over morethan two temporal decades, until a crossover timese themean square displacement grows with a characteristicpower law of time:

hDr2�t�i / t � � � 0:78� 0:10 for t < se �13�This last result implies that the motion of a microspherein a tight F-actin network is subdiffusive (� < 1) evenat the shortest probed time scales, i.e. down to 10–6 s.

The exponent� reflects the finite, local rigidity ofactin filaments. A recent model by Morse (1997)predicts that the viscoelastic modulus of a solution ofsemi-flexible polymer displays a power law behavior athigh-frequencies asjG��x�j / x3=4 which, according toEq. (1), predicts� � 3=4, in excellent agreement withour high resolution MSD. Also, an exponent� � 1=2is predicted by the Rouse model in the case of aflexible-polymer network for which hydrodynamicinteractions are ignored (Doi and Edwards, 1989),which further confirms our interpretation of�. Thisexponent can also be interpreted as the phase shiftbetween loss and storage moduli at high frequencies. Inthe power law regime at large frequencies or small timescales, ifG��x� / �ix��, then G00=G0 � tan�p�=2�.For instance, if the fluid surrounding the microsphere ispurely viscous, G0 � 0 and � � 1, if the fluid ispurely elastic, G00 � 0 and � � 0. Therefore,� alsoreflects the viscoelastic nature of the fluid at highfrequencies, determining if dissipation or elasticitydominates at short time scales. Since the power lawbehavior hDr2�t�i / t3=4 extends over more than twodecades, the viscoelastic modulus has a power lawbehavior over an extended frequency range asG��x� / �ix�3=4. As a result, the loss modulus dom-inates the storage modulus for a solution of semi-flexible polymers at large frequencies, which has beenverified recently by two research groups (Gittes et al.,1997; Palmer et al., 1998).

The concentration dependence of the mean-squaredisplacement allows to gain insight into the nature ofthe interactions between the probing bead and thepolymer network at short times. If we rescale the short-time displacement at different actin concentrations by�c/24lM)1/2, choosing the 24lM solution as standard,the mean-square displacements at other concentrationsoverlap for the entire concentration range, i.e. both forthe isotropic phase and the anisotropic phase (seeFig. 2). Therefore, at short time scales, the mean squaredisplacement scales as

hDr2�t�i / cÿ1=2 t� se �14�with actin concentration. This result is unexpected sinceone would anticipate the viscoelastic modulus to be lin-early dependent on concentration at large frequencies,or equivalentlyhDr2�t�i / cÿ1.

A possible explanation for this scaling law, given byMaggs (1997), is that the displacement of the micro-spheres at short time scales is dominated by hydrody-namic interactions. The total dissipation of the bead isorder � g�m2=n2��a2nh� since the hydrodynamic flowcreated by the moving bead is screened into the solu-tion, and is non-negligible up to a distancenh into thesample from the bead. Here,g is the solvent viscosity,mis the velocity of the bead, andnh is of the order of the


Fig. 1 Mean square displacements of 0.965lm diameter latex micro-spheres imbedded in F-actin solutions of increasing concentrations asmeasured by diffusing wave spectroscopy. Even at the shortest timescales, the motion of the microspheres is sub-diffusive,hDr2�t�i / t �with �<1 because the microspheres’ motion is elastically hindered bythe dense actin network, which has a mesh size smaller than the beadsize. The exponent� is �=0.78±0.05, and is independent of actinconcentration

mesh sizen. Since nh � n / cÿ1=2 for a solution ofsemi-flexible polymers for which lp � �n; le� andhDr2�t�i is approximately inversely proportional to thefriction, then the friction is proportional toc1=2, in qual-itative agreement with our observations. However, thisreasoning is approximate here since the motion of thebead is not diffusive even at the shortest time scalesprobed by DWS; therefore one cannot associate acon-stant friction coefficient to the bead.

As shown in Fig. 1, at intermediate time scales andover a wide temporal range, the exponent� decreasesprogressively to values much smaller than 3/4. This cor-responds to the progressive elastic trapping of the beadby the cage formed by the surrounding actin polymers.As verified by video-microscopy, electron microscopy,and analytical centrifugation (not shown here), the trap-ping or caging of the bead by the network is entropicin origin because the bead does not bind to actin.

Diffusion coefficientand velocity autocorrelation function

Mean square displacements can be further analyzed interms of a time-dependent diffusion coefficient. Fig-ure 3a shows the diffusion coefficient defined as

D�t� � hDr2�t�i=6t �15�and calculated from the data in Fig. 1. We observe thatthe motion of a microsphere cannot be described by asingle diffusion coefficient, but by a diffusion coeffi-cient which varies with time. At short times and corre-sponding small length scales, the microspheres experi-ence the small amplitude, high-frequency lateral fluctua-tions of the subsections of the actin filaments betweenentanglements. At long times and long length scales,

the microsphere becomes elastically trapped by the ac-tin mesh.

The associated diffusion coefficientD of the probingmicrosphere is non-constant and spans almost fivedecades. The diffusion coefficient varies fromD � 10ÿ6 lm2/s for large actin concentrations and longtimes toD � 0:3 lm2/s for small actin concentrationsand short times (see Fig. 3A). The former value corre-sponds to near-arrest of the microsphere by the elasticactin network: the microsphere probes the elasticity ofthe actin mesh at long times. The latter is close to butsmaller than the diffusion coefficient of the same micro-sphere in water of viscosity 1 cP, which is equal toD0 � kBT=6pga � 0:46 lm2/s. If the bead size were


Fig. 2 Overlap of the small-time scale mean-square displacementmeasurements rescaled by (c/24lM)1/2

Fig. 3 A Time-dependent diffusion coefficient of a microsphere im-bedded in an actin solution. The diffusion coefficient is defined byD�t� � hDr2�t�i=6t and calculated from the data in Fig. 1.B Diffu-sion coefficientD as a function of the average displacement of themicrosphere

��hDr2ip

much smaller than the mesh size, actin would not hin-der its diffusion andD would beD0. When the beadsize is much larger than the mesh size (which is thecase in the present work), actin rapidly hinders thebead’s motion as shown in Fig. 3A.

To establish the relation between the diffusion coeffi-cient and the local displacement of microsphere, weplot D as a function of

��hDr2ip(see Fig. 3B). This fig-

ure shows two distinct dynamical regions. First, a slownear-plateau at short length scales, then a sharp down-ward transition past a characteristic crossover lengthscale. This crossover length scale decreases from&20to 2 nm when actin concentration is increased from 24to 164lM. Over the same concentration range, thiscrossover length scale is always much smaller than theaverage mesh size of the actin network, which variesfrom &150 to 60 nm over the concentration range in-vestigated here.

The values ofD that we measured by DWS can becompared with values obtained using classical dynamiclight scattering by Newman et al. (Newman et al.,1989). These authors measured the diffusion coefficientof latex spheres imbedded in actin solutions and ob-tained values rangingD=D0 � 1ÿ 0:2 for 270 nm ra-dius spheres in actin solutions of concentrations rangingc=1-22lM. The magnitude of these values agree withour optical measurements at least if we useD opticallymeasured at very short times and small actin concentra-tions. But, as shown in Fig. 3, the diffusion of latexspheres in concentrated F-actin solutions is not welldescribed by a single diffusion coefficient since theirtransport becomes subdiffusive at times as short as10–4 s. Indeed, we obtain values that are 105 to 106

smaller thanD0 at long times and large actin concentra-

tions. Moreover, the interpretation of dynamic lightscattering data at high actin concentrations becomes dif-ficult (if not incorrect) because of the multiply-scatterednature of the light intensity.

The displacement of a microsphere in a solution ofsemi-flexible polymers can be further studied by calcu-lating the velocity autocorrelation functionhm�t�m�0�i,as shown in Fig. 4. The functionhm�t�m�0�i is definedas

hm�t�m�0�i � ÿ dD�t�dt

�16�

Figure 4 shows the extreme sensitivity of DWS, sincehm�t�m�0�i varies more than 14 decades over the probedtemporal range. The autocorrelation functionhm�t�m�0�idecays via three distinct temporal regions. At shorttimes, the velocity autocorrelation function decays as apower law of time over two decades:

hm�t�m�0�i / tÿ1:75�0:10 t� se : �17�The “long-time tail” is different from that observedin dilute colloidal suspensions for whichhm�t�m�0�i / tÿ3=2 at short times (Weitz and Pine,1993). At intermediate times, we also observe a powerlaw behavior with a somewhat smaller exponent,hm�t�m�0�i / tÿ0:97�0:15, over more than three decades.At long times, the autocorrelation functions decayagain rapidly with an effective power lawhm�t�m�0�i / tÿ1:75�0:10, also over more than threedecades. We can qualitatively associatehm�t�m�0�i to thedissipation of the microsphere by interaction with themesh, orvice versathe dissipation due to the lateralfluctuations of subsections of the polymers betweenentanglements. Dissipation is proportional tom2 whichscales astÿ1:75, suggesting that the dissipation or lossmodulus is large at short time scales, scaling as


Fig. 4 Velocity autocorrelation functionhm�t�m�0�i of a microspherein an actin solution

Fig. 5 Evolution of the creep compliance for a 100lM F-actin solu-tion as measured using DWS (closed circles), single particle-trackingmicrorheology (closed squares) and mechanical rheometry (open sym-bols)

G00 / x0:75 with frequency, which confirms the resultsof Palmer et al. (1998).

Creep compliance calculated from MSD measurementsand measured by mechanical rheometry

We now directly test Eq. (9), which relates the meansquare displacementhDr2�t�i of a microsphere in afluid to the shear complianceJ�t� of that fluid. We cal-culate J�t� from our high-resolution mean square dis-placement measurements and compare these calculatedvalues to J�t� measured by a stress-controlled rheo-

meter. Figures 5–7 demonstrate the remarkable agree-ment between our optical and mechanical measurementsof J�t� for systems as diverse as a viscous oil, solutionsof flexible polymers, and solutions of semi-flexiblepolymers. Because of the limited overlap of the two op-tical and mechanical approaches, this agreement can betested at intermediate time scales only.

This agreement supports our new interpretation ofthe mean-square displacement, supports the use of parti-cle-tracking rheometry to probe the local compliance ofa polymer system or complex fluids, and eliminates theextra transformations for the calculation of the visco-elastic moduli. Indeed, our creep compliance measure-ments, being proportional to the “raw” data generatedby DWS and particle-tracking, do not crop small andlarge frequencies out of calculated values of the visco-elastic moduli. Figures 5–7 also show the complemen-tarity of particle-tracking measurements and classicalmechanical measurements. Very short time scales are in-accessible to classical mechanical rheometry.Vice ver-sa, very long time scales are inaccessible to opticalrheometry and particle-tracking rheometry.

Figure 5 displays the creep compliance of a 100lMactin solution. Creep compliance is measured using


Fig. 6 Evolution of the mean square displacement and of the creepcompliance of solutions of flexible polymers and a purely viscousfluid. A Mean square displacement and diffusion coefficient of0.96lm diameter microspheres in a 1 P viscous oil.B Creep compli-ance measured by DWS (closed circles) and mechanical rheometry(open circles) of a 1 P viscous oil. The inverse of the slope yields aviscosity of 1.03 P.C Mean square displacements and correspondingtime-dependent diffusion coefficients in three PEO/water solutions inthe semi-dilute regime measured by DWS. The mean square displace-ment follows a power law increase with time,hDr2�t�i / t0:67 PEO-water solutions.D Creep compliance of PEO solutions

both particle-tracking micro-rheology and classical me-chanical rheometry. We observe excellent agreementover the temporal range probed by mechanical rheome-try. Of course, since mean-square displacement andcreep compliance are proportional, the shape of theJ�t�curve is identical. Therefore, the discussion given aboveregarding the short- and long-time behavior ofhDr2�t�iholds for J�t�; including the concentration dependenceof J�t� at short time scales.

The present work is primarily concerned withsolutions of semi-flexible polymers, but in order tofurther test Eq. (9), we conduct two “control” experi-ments. The first one involves a purely viscous fluid, aviscous of fixed viscosityg=1.03 P. Figure 6 displaysthe mean-square displacements measured by DWS, thediffusion coefficient, and the corresponding creepcompliance. We first verify that the motion of themicrosphere is purely diffusive since the log-log plot inFig. 6A yields a linear increase of the mean squaredisplacement with a slope of�=1.03±0.02. We expectthis result since the motion of a Brownian sphere in aviscous fluid should be purely diffusive andhDr2�t�i=�kBT=pag�tH�t� at long times. The diffusion coeffi-cient of a 0.96lm diameter microsphere in a 1.03 Pviscous oil is predicted to beD0 � kBT=6pga=4.4×10–3 lm2/s, which we measure using DWS as shownin Fig. 6A. Finally, according to Eq. (9), we expectJ�t� � �t=g�H�t�, which we obtain as shown inFig. 6 B. The slope in Fig. 6b yields a bulk viscosityg=1.03±0.05 P, which is the correct viscosity of the oil.This result shows that particle-tracking measurementsare not limited to viscoelastic fluids such as F-actin so-lutions, but can probe purely viscous fluids. This agree-ment also confirms that our new approach is quantita-tive.

The second control experiment involves the measure-ment of the creep compliance of a viscoelastic solutionof flexible polymer, an aqueous solution of polyethy-lene oxide. Here again, we conduct DWS measurementsand classical creep compliance measurements. Fig-ure 6c displays the mean square displacement and thediffusion coefficient; Fig. 6d displays calculated andmeasured creep compliance. The same level of agree-ment as for the actin solutions and the viscous oil isfound for the semi-dilute solution of flexible polymer.

The agreement of DWS and video-enhanced SPTmeasurements with mechanical measurements at longtimes can be considered surprising. As pointed out byGittes et al. (1997), the viscous solvent moves togetherwith the probed network at frequencies larger thanxc � Gn2=ga2, whereG is the modulus of the actinnetwork, n is the mesh size, andg is the viscosity ofthe buffer. This last expression is obtained by thebalance between viscous and elastic force actin on themicrosphere. The formalism presented in the Theorysection describes the Brownian dynamics of a micro-

sphere at timest < tc � xÿ1c . We estimate this charac-teristic time to be ordertc �1–0.03 s. Therefore, theobserved agreement between our mechanical measure-ments, which are limited to times longer than 0.007 s,and our optical measurements at timest > tc is some-what unexpected.

Concentration dependence of the compliance

Figure 7 displays the time-dependent compliance of ac-tin solutions for concentrations between 10 and 164lMas measured by DWS. Since the shear compliance isproportional to the mean square displacement, the de-scription and discussion of these measurements areidentical to that offered above for the mean-square dis-placements. Figures 8 and 9 display the value ofJ�t�as a function of actin concentration, at different sam-pling times. At short times, we find thatJ�t� is weaklydependent on actin concentration,

J / cÿ0:53 t� se �18�for t=10–5 s and for c <60lM. However, at longtimes, this concentration dependence becomes steeper,

J�0� / cÿ1:3�0:2 t� se �19�for t=102 s and for c <60lM. At these long timescales the creep and, correspondingly, the mean-squaredisplacement is not strongly dependent on time. On thesame figure, we show video-enhanced particle-trackingmicrorheological measurements of the plateau compli-ance. Figure 9 shows the excellent agreement betweenDWS and single-particle-tracking measurements. The


Fig. 7 Evolution of the shear creep compliance for F-actin solutionswith concentrations ranging from 10 to 164lM. A creep compliance“plateau” is reached faster and at a lower value for increasing actinconcentration, reflecting both the decreasing size of the actin meshand the corresponding increase of the elasticity of the network

concentration dependence of the plateau compliance foractin is much weaker than predicted for semi-dilute so-lutions of flexible polymers, for whichJ�0� / cÿ2:2 (Doiand Edwards, 1989).

Using Morse’s model of shear rheology of actinsolutions, we can attempt to explain the unusualconcentration dependence of the creep compliance.According to Morse’s model, the long-time shearmodulus of a solution of semi-flexible polymerscan be described by the classical expressionG�t�� G�0�Pp;odd�8=p2p2� exp�ÿp2t�=sD� of the de Gen-nes-Doi-Edwards model of reptation (Doi and Edwards,1989). Here,G�0� is the plateau modulus andsD is theterminal relaxation time of the polymer, i.e. the timenecessary for an actin filament to disengage from itsinitial tube. Therefore, the steady state complianceis predicted to be J�0� � R10 tG�t�dt=�R10 G�t�dt�2� 6=5G�0� (Doi and Edwards, 1989).

For actin solutions,G0p > G00p at long times, henceG�0� � G0p and the compliance is of the order of theinverse of the small-frequency elastic modulus,J�0� / 1=G0p. Morse predicts that the concentrationdependence of the elastic plateau modulus depends onconcentration asG0p / c7=5, which has recently beenverified experimentally by both optical and mechanicalrheometry (Xu et al., 1998b). Therefore,J�0� / cÿ7=5,in agreement with our mechanical, DWS, and particle-tracking measurements (see Fig. 9). Therefore, the ori-gin of the long-time creep is due to the reptation of thelong semi-flexible polymers (Morse, 1997). However,this argument is approximate and would be correct ifJ�t� were a true plateau and if a F-actin network werean elastic solid. Our recent work shows that, at equilib-rium, actin rheology is indeed solid-like, but with a

non-negligible dissipative component,G00 � (0.3–0.4)×G0 at small frequencies (Xu et al., 1998b). A moreformal expression, which specifically predicts the (time-dependent) shear creep compliance of a solution ofsemi-flexible polymers is therefore needed to compareour new data to a model.

Summary and conclusions

This paper predicts the proportionality between themean square displacement of a microsphere imbeddedin a fluid and the local linear creep compliance of thatfluid. We test this relation using mechanical rheometry,diffusing wave spectroscopy, and particle-trackingmicrorheometry on three different systems: a purely vis-cous oil, a viscoelastic solution of flexible polymers,and a viscoelastic solution of semi-flexible polymers.Mechanical rheometry measures the creep directly byapplying a small stress and by measuring the resultingstrain. This conventional measurement is macroscopic.Instead, our particle-tracking microrheology instrumentsmeasure the mean-square displacement resulting fromthe thermal energy imparted to a small microsphere sus-pended in the same system. This measurement is meso-scopic because it is conducted at length scales muchsmaller than the sample size and the actin filaments, butstill much larger than the mesh size and the smallsolvent molecules. We observe excellent agreement forthree different systems over the temporal range sharedby the mechanical and optical instruments.

These instruments are complementary: while me-chanical rheometry is best suited to probe the long-time


Fig. 8 Values ofJ�t� as a function of actin concentration at differentsampling times. Note the saturation of the creep compliance for actinconcentrations larger than&50lM, which corresponds to the onsetof a liquid crystalline ordering in the system

Fig. 9 Plateau shear compliance for F-actin solutions as a functionof actin concentrationc. The magnitude of the plateau is arbitrarilydefined asJp. The shear compliance plateau evaluated at 103 s in-creases with actin concentration asJp / cÿ1:3�0:2. This figure showsthe excellent agreement between DWS measurements (circles) andvideo-enhanced particle-tracking measurements (squares) of the con-centration-dependent creep

creep behavior of polymer solutions, DWS is bestsuited to probe short time scales, from 10–6 to 102 s.Video-based SPT is not as accurate as DWS, but usesmuch smaller amounts of sample and is straightforwardto implement.

In the case of a solution of semi-flexible polymers,we observe that the creep compliance scales asJ�t� / t0:78�0:05 at short times, which is a direct conse-quence of the rigidity of actin. At long times, the shearcompliance reaches a quasi-plateau, which decreaseswith concentration asJ / cÿ1:2�0:3. This concentration

dependence is also a direct consequence of the finiterigidity of actin. We are currently testing Eq. (9) onother polymer systems such as solutions of micro-tubules, which are rigid polymers, and colloidal sys-tems, such as clays.

Acknowledgments The authors acknowledge D. Morse and A.C.Maggs for helpful discussions, and thank A. Palmer, K. Rufener andT.G. Mason for their help in the DWS measurements. D.W. acknowl-edges financial support from the Whitaker Foundation and theNational Science Foundation, grants DMR 9623972 (CAREER), CTS9502810, and CTS 9625468.


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