high-frequency viscoelasticity of crosslinked actin filament networks measured by diffusing wave...

Andre PalmerJingyuan XuDenis Wirtz

High-frequency viscoelasticity of crosslinkedactin filament networks measuredby diffusing wave spectroscopy

Received: 20 January 1998Accepted: 12 February 1998

A. Palmer · J. Xu · D. Wirtz (✉)Department of Chemical EngineeringThe Johns Hopkins University3400 North Charles St.Baltimore, MD 21218USAE-mail: [email protected]

Abstract We study the short-timerelaxation dynamics of crosslinkedand uncrosslinked networks ofsemi-flexible polymers using diffus-ing wave spectroscopy (DWS). Thenetworks consist of concentrated so-lutions of actin filaments, cross-linked with increasing amounts ofa-actinin. Actin filaments (F-actin)are long semi-flexible polymerswith a contour length 1–100lm anda persistence length of 5–15lm; a-actinin is a small 200 kDa homodi-mer with two actin-binding sites.Using the large bandwidth of DWS,we measure the mean-square-dis-placement of 0.96lm diameter mi-crospheres imbedded in the polymernetwork, from which we extract thefrequency-dependent viscoelasticmoduli via a generalized Langevinequation. DWS measurements yield,in a single measurement, viscoelas-tic moduli at frequencies up to

105 Hz, almost three decades higherin frequency than probed by con-ventional mechanical rheology. Ourmeasurements show that the magni-tude of the small-frequency plateaumodulus of F-actin is greatly en-hanced in the presence ofa-actinin,and that the frequency dependenceof the viscoelastic moduli is muchstronger at intermediate frequencies.However, the frequency-dependenceof loss and storage moduli becomesimilar for both crosslinked and un-crosslinked networks at large fre-quencies,G' (x)!G'' (x)!x0.75 ± 0.08. Thishigh-frequency behavior is due tothe small-amplitude, large-frequencylateral fluctuations of actin filamentsbetween entanglements.

Key words Actin – a-actinin –diffusing wave spectroscopy –semi flexible polymer

Introduction

The short-time dynamics of polymers in entangled poly-mer solutions has long been the subject of research in the-oretical polymer physics (Rouse, 1953; Zimm, 1954; Doiand Edwards, 1989). However, the limited bandwidth ofcurrent mechanical rheometers and spectroscopic techni-ques such as fluorescence recovery after photobleaching(FRAP) and forced-Rayleigh scattering (FRS) has not al-lowed one to test these theories directly. One of the fewapproaches to extract high-frequency dynamical proper-ties of polymer systems such as loss and storage moduli

has involved in using the time-temperature superposi-tion (Ferry, 1980). By decreasing the temperature of apolymer melt or solution, both the terminal relaxationtime and entanglement time describing the onset of entan-glements can be conveniently increased and the relaxa-tion times become measurable by rheometric techni-ques. However, polymer blends and solutions may under-go phase transitions upon a temperature change. Theproximity to a phase boundary line can dramatically af-fect the dynamics of relaxation since large length scalecorrelations dominate the physics of relaxation in near-critical polymer systems, not the small length scale mo-

Rheol Acta 37:97–106 (1998)© Steinkopff Verlag 1998 ORIGINAL CONTRIBUTION

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tion of individual polymers. Furthermore, many polymerssuch as protein macromolecular assemblies degrade rap-idly upon temperature changes, therefore the time-tem-perature superposition cannot be conducted unambigu-ously with these systems.

The advent of diffusing wave spectroscopy (Weitzand Pine, 1993) and particle-tracking microrheology(Mason et al., 1997; Mason et al., 1997; Gittes et al.,1997; Schnurr et al., 1997) have allowed to monitor thehigh-frequency viscoelastic moduli of polymer gels andentangled solutions. These techniques probe viscoelasticmoduli over an extended frequency range in a singleexperiment at a fixed temperature. These techniquesmonitor the thermally excited motion of microspheresimbedded in a polymer solution. Using a Langevin de-scription of the motion of a microsphere in a viscoelas-tic fluid, one can extract viscoelastic moduli (Masonand Weitz, 1995; Mason et al., 1997) and creep compli-ance (this work) from the measured mean square dis-placement. Using DWS, the mean square displacementof the probing microsphere can be monitored with na-nosecond resolution; using particle-tracking microrheol-ogy, the mean square displacement can be monitoredwith sub-millisecond resolution. As a result, the me-chanical properties of polymer networks and solutionscan be measured at frequencies as high as 106 rad/s, al-most three orders of magnitude larger than probed byconventional mechanical rheometers. From these high-frequency measurements, we can gain a new insightinto the mechanism of early dynamics of stress relaxa-tion in entangled networks and an understanding of thetype of local macromolecular motion which producesstress relaxation at short times. This new insight allowsus to test current models of polymer motion at shortlength scales and short time scales.

The polymer motion inside an entangled polymer so-lution at small times, or equivalently at large frequen-cies, is typically described as that of a free polymer.Therefore in the case of flexible polymers, Zimm andRouse models of dynamics in dilute solutions havebeen used to describe the relaxation at time scales smal-ler than the entanglement timese (Rouse, 1953; Zimm,1956). At times longer thanse, the effects of entangle-ments become important and the polymer motion is as-sumed to be confined inside a tube; reptation dominatesthe mechanism of relaxation (Doi and Edwards, 1989).The decoupling between reptation at long times andRouse-like dynamics at short times has not been for-mally tested with non-invasive techniques in entangledpolymers (Doi and Edwards, 1989). The limited band-width of mechanical rheometers has also prevented asystematic study of the effect of backbone rigidity anddegree of crosslinking on the early dynamics of relaxa-tion in polymer solutions.

In this paper, we study the effect of transient crosslink-ing on the short-time motion of actin filaments in a

“tightly entangled” network (Morse, 1997). Actin fila-ments are long semi-flexible polymers (Alberts et al.,1994), with an extremely long persistence lengthlp&5–15lm compared to their diameterd&1 mm(Gittes et al., 1993; Isambert et al., 1995). At large con-centrations, actin filaments form a “tightly entangled net-work” because the persistence length of F-actin is muchlarger than the entanglement lengthle. The entanglementlength le&1 lm can be defined as the distance betweencollisions of a polymer with the confining walls of thetube (Morse, 1997). The tightly entangled regime, whichcorresponds to contour length densitiesq� 1=l2

p � 4:410–3lm–2, is intermediate between the loosely entangledregime and the nematic regime. According to the Onsagercriteria, the nematic regime corresponds to concentrationsq>qnem&10/Ld&150lm–2. The loosely entangled re-gime corresponds to concentrations large enough thatpolymer are entangled, but smaller than 1/l2p. In this re-gime, the entanglement length is much larger than thepersistence length�le � lp�: Most semi-flexible poly-mers are not rigid enough and long enough to displaythis intermediate regime, which is widest when the ratiolp/d is the largest. Therefore, filamentous actin consti-tutes one of the few polymers which allow us to probethe intermediate “tightly-entangled” regime of semi-flex-ible polymers.

One can envision that, at time scales smaller than acharacteristic timese, the fast small length scale bend-ing modes of wavelengthk< le dominate the (trans-verse) motion of the semi-flexible polymer. One canfurther assume that, when the polymer is not cross-linked and at time scales larger thanse, the motion ofthe semi-flexible polymer becomes hindered by the tubeformed by the surrounding polymers and can only relaxvia reptation. When the polymer is crosslinked, repta-tion becomes progressively prohibited and only theshort-time relaxation of subsections between entangle-ments is allowed.

As a dynamical crosslinker of actin, we usea-acti-nin, a small 200 kDa protein with two actin-bindingsites. In the tightly entanglement regime for whichlp� le, actin filaments are rigid between entangle-ments, therefore little contour length fluctuation relaxeslongitudinally. Therefore, in the case of permanentcrosslinks and for mechanical rheometers which have alimited bandwidth, loss and storage moduli are feature-less, i.e. G0�x� and G00�x� are independent of fre-quency. In the present case, the crosslinking proteina-actinin has a finite lifetime of binding to actin; hencecrosslinking ofa-actinin to F-actin is transient. Our newDWS measurements show that the magnitude of thesmall-frequency plateau modulus of actin networks isgreatly enhanced in the presence ofa-actinin, and thatthe frequency dependence of the viscoelastic moduli ismuch stronger at intermediate frequencies. The fre-quency-dependence of loss and storage moduli become

98 Rheologica Acta, Vol. 37, No. 2 (1998)© Steinkopff Verlag 1998

similar for both crosslinked and uncrosslinked networksat large frequencies,G' (x)!G'' (x)!x0.75 ± 0.08.

Methods and systems

Preparation of actin anda-actinin

Actin was extracted from rabbit skeletal muscle acetonepowder by the method of Spudich and Watt (1971).The resulting actin was then gel filtered on SephacrylS-300 HR instead of Sephadex G-150 (MacLean-Fletch-er and Pollard, 1980). The purified actin was stored asCa2+-actin in continuous dialysis at 48C against dailychanged buffer G (0.2 mM ATP, 0.5 mM DTT, 0.1 mmCaCl2, 1 mM NaAzide, and 2 mM tris-Cl, pH 8.0). Thefinal actin concentration was determined by ultravioletabsorbance at 290 nm, using an extinction coefficient of2.66×104 M–1 cm–1, and a cell path length of 1 cm.Actin filaments were generated by adding 0.1 volumeof 10-x KME (500 mM KCl, 10 mM MgCl2, 10 mMEGTA, 100 mM imidazole, pH 7.0) polymerizing saltbuffer solution to 0.9 volume of G-acin in buffer G.The actin used for all experiments came from the lastfraction of the actin peak obtained by one gel filtration.Alpha-actinin from Acanthamoeba Castellaniiis puri-fied as described in Wachstock et al. (1993, 1994) andwas a gift of Drs. Mullins, Kelleher, and Pollard. Thisa-actinin was further gel-filtrated on Sephacryl S-300HR in equilibrium with Buffer G. SDS electrophoresisshowed that the purity of thea-actinin was greater than90%.

Diffusing wave spectroscopy

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 sphericaloptical 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 ofvary 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. The meansquare displacement of the probing spheres is extractedfrom g2(t)–1 measurements via a root-search algorithmfollowing classical techniques (Weitz and Pine, 1993).

Viscoelastic moduli of crosslinked and uncrosslinkedactin filament networks are calculated from measuredmean-square displacements via a generalized Langevin

equation of diffusion of a microsphere in a viscoelasticfluid (Mason and Weitz, 1995). Briefly, the local visco-elastic modulus of a fluid and the mean square displace-ment of a microsphere suspended in that fluid are re-lated via3

~G�s� � s

6pa

�6kBT

s2hD~r2�s�i ÿms

�� kBT

pashD~r2�s�i ; �1�

wherekB 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~ represents theunilateral Laplace transformation defined as~X�s� � L�X�t�� R10 X�t�exp�ÿst�dt. Equation (1) re-lates 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. InEq. (1), the inertia term (the second term in the brack-ets) is neglected because it corresponds to very highfrequencies, of the order 106 Hz. The knowledge of~G�s� is typically sufficient to characterize the rheologi-cal behavior of a polymer solution. However, in orderto present our optical measurements in a more familiarfashion, we use the analytic continuation between thereal function ~G�s� and the complex functionG��x� � G0�x� � iG00�x�. We extract storage and lossmoduli by replacings by ix in ~G�s� and by extractingreal and imaginary parts from the resulting imaginaryfunction ~G�s � ix�, respectively. Equation (1) yieldsthe correct results for the extreme cases of a purely vis-cous liquid and purely elastic solid. For a viscousliquid, hD~r2�s�i � 6D=s2 � kBT=pgas2 and ~G�s� � gswhereg is the constant viscosity of the liquid; thereforeG0�x� � 0 and G00�x� � gsx. Instead, for an elasticsolid, hD~r2�s�i � r2

0 and ~G�s� � G0 � kBT=par20 where

G0 is the elastic modulus; therefore,G0�x� � G0 andG00�x� � 0. A viscoelastic fluid such as an actin gel hasboth non-vanishing loss and storage moduli.

Actin is polymerized in situ for 12 h before measure-ment by loading the scattering cell with a solution ofmonomeric actin mixed with the polymerizing salt solu-tion and a dilute suspension of monodisperse latex mi-crospheres (Duke Scientific Corp.) of radius 0.48lm ata volume fraction of 0.01. The scattering cell is thenimmediately tightly capped. We verified by both time-resolved static light scattering and time-resolved me-chanical rheology that G-actin was fully polymerizedinto F-actin at all concentrations presented in this paperbefore 24 h. Using static light scattering, we verifiedthat more than 99% of the scattering intensity in thetransmission geometry was due to the microspheres,less than 1% due to the actin filament network itself.We also verified using mechanical rheology that theadded latex beads did not affect the rheology of poly-merized actin, representing less than 5% of the magni-tude of G0�x� and G00�x� at all actin concentrations

99A. Palmer et al.High-frequency viscoelasticity of crosslinked actin filament networks measured

used in this paper. All DWS measurements are con-ducted at a temperature ofT=238C.

Results and discussion

Diffusion of a microsphere in a crosslinked actinnetwork

The frequency-dependent viscoelastic moduli of cross-linked and uncrosslinked actin filament networks arededuced from the time-dependent mean-square displace-ments of 0.96lm diameter microspheres imbedded inthe F-actin solutions. Measurements of the mean-squaredisplacements and associated diffusion coefficients inthe temporal domain provide us with a new physical in-sight into the dynamics of relaxation of crosslinked anduncrosslinked actin gels complementary to the insightgained from rheological measurements. The measure-ments of mean square displacements, unlike the visco-elastic moduli calculated using Eq. (1), are model-inde-pendent.

Mean square displacement

The mean square displacementhDr2�t�i is extractedfrom the measured autocorrelation functiong2(t)–1 ofthe light multiple scattered by the microspheres mixedwith the actin solution. Using an efficient custom root-searching program, we calculatehDr2�t�i with nanose-cond temporal resolution and nanometer spatial resolu-tion. Figure 1 shows the correlation functiong2(t)–1 for

concentrated solutions of pure actin filaments and con-centrated solutions of actin filaments crosslinked withincreasing amounts of crosslinkers. We observe that thepresence of crosslinkers in F-actin solutions retard therelaxation of the intensity autocorrelation function. Thiseffect is accentuated for increasing amounts of crosslin-kers. At short times, the correlation function undergoes(approximately) exponential decay for both crosslinkedand uncrosslinked actin networks, as shown in the insetof Fig. 1, which showsg2(t)–1 decreasing linearly withtime in a log-linear plot.

Figure 2 displays the mean square displacement cal-culated from the measurements ofg2(t)–1. For pure ac-tin and at early times, we observe that the mean squaredisplacement varies with time ashDr2�t�i / ta witha=0.75±0.05. Therefore even at the earliest timesprobed by DWS,t<10–5 s, we observe that the motionof 0.96lm microspheres in actin filament networks issub-diffusive with an exponenta smaller than 1. An ex-ponenta=1 would correspond to the free Brownian dif-fusion of the microspheres in the suspending solution.For actin crosslinked witha-actinin, we observe thatwhile the displacements of the same microspheres aremuch smaller in amplitude, the temporal dependence ofthe mean square displacement is similar to that ob-served with uncrosslinked actin networks. For cross-linked actin networks, the exponent representing thevariation of hDr2�t�i with time seems to decreaseslightly from a=0.75 toa=0.68 for increasing amountsof crosslinkers. This slight decrease of the exponent canbe due to the fact that the fitting of the mean square


Fig. 1 Autocorrelation function of the light intensity multiply scat-tered by the 0.96lm diameter microspheres imbedded in a 24lMactin network in the presence or in the absence of crosslinkers. Therelaxation of the correlation function is observed to be retarded by thepresence of crosslinkers. The mean square displacement of the prob-ing microspheres is extracted from these measurements and shown inFig. 3

Fig. 2 Mean square displacement of 0.96lm microspheres of pure-actin filament networks. At short-timeshDr2�t�i / t0:75. At longtimes, the microspheres’ motion becomes hindered by the tight en-tangled network. Mean square displacement of the same microspheresin the same actin network, but crosslinked with increasing amounts ofcrosslinking proteins,a-actinin. The onset of the large-time plateauoccurs earlier than for uncrosslinked networks. However, the time-dependence of the mean square displacement is identical to that ob-served in uncrosslinked actin networks

displacement curves is not obtained in the power-lawregime. As shown in Fig. 2, the power law regime oc-curs past a characteristic time, which decreases with in-creasing concentration of crosslinkers.

The thermally excited motion of 0.96lm micro-spheres in a 24lM actin solution is sub-diffusive be-cause its size is much smaller than the mesh size of thenetwork. At short times and due to its size, the probingmicrospheres cannot percolate through the network; in-stead, they are elastically trapped by the mesh. Asshown by Fig. 2, the magnitude of the displacements isalways much smaller than the diameter of the micro-sphere and smaller than the mesh size of the actin net-work. Also, the relaxation time of the individual actinfilaments is huge (sD&103–104 s). Hence, over theprobed temporal range, the individual filaments sur-rounding the microspheres do not fully disengage fromtheir tube. A microrheological signature of full relaxa-tion is a re-increase ofhDr2�t�i after the plateau at in-termediate times. Our mean square displacements arenot conducted over long enough periods of time to cap-ture this final relaxation.

It is interesting to invoke the proportionality betweenthe mean square displacement and the shear creep com-plianceJ(t),

J �t� � pa

kBThDr2�t�i �2�

This relation is demonstrated in the appendix. Equa-tion (1) directly relates the local displacement of a mi-crosphere imbedded in a polymer network to the localmechanical properties of that network. The shear creepcompliance of a network corresponds to the deforma-tion of that network normalized by the applied stress.Here, the force generated by the thermal motion of themicrosphere in the surrounding network, or equivalentlygenerated by the thermal motion of the network on theprobing microsphere, is small since it is of thermal ori-gin and is of the orderkBT=a. If the fluid were purelyviscous,J �t� � t=g / t; if the fluid were purely solid-like J �t� � J0 / t0. Therefore, an exponenta � 0:75indicates that an actin network dissipates energy morethan it stores energy at short time scales. The inset inFig. 2 shows the creep compliance for actin networks inthe presence and in the absence of crosslinkers. OurDWS measurements suggest that while the creep com-pliance in an uncrosslinked network is much smallerthan in an uncrosslinked network, the rate of increaseof the creep compliance is similar for crosslinked anduncrosslinked actin networks at short times.

At longer times, the displacement of the microspherein the uncrosslinked actin network is slowed. The mi-crosphere becomes progressively trapped by the tight,elastic actin network. The origin of this trapping is en-tropic as verified by analytical centrifugation and gelelectrophoresis (not shown here), which showed no

chemical binding of the microsphere with actin. At longtimes, the displacement of the same microsphere in acrosslinked network remains smaller than for an uncros-slinked actin network. However, the rate of change ofthe mean square displacement is unexpectedly strongerfor increasing amounts of crosslinkers. We shall see be-low that this effect is due to the finite lifetime of bind-ing of a-actinin to F-actin. This indicates that the mech-anism of stress relaxation becomes similar for cross-linked and uncrosslinked actin networks at short times.

Diffusion coefficient

The displacement of a microsphere in an entangled ac-tin network can be further analyzed by extracting thetime-dependent diffusion coefficient from the meansquare displacement measurements. We define the time-dependent diffusion coefficient as

D�t� � hDr2�t�i=6t �3�Figure 3 displaysD (t) for various actin concentrationand a-actinin concentrations. We find that the diffusioncoefficient of the microsphere in the actin network issmaller than the bare diffusion coefficient of the samemicrosphere in the same buffer solution viscous fluid,given by D0 � kBT=6pgs a � 0:45 lm2/s, whereg2 � 1 cP is the “microviscosity” of the buffer solution.Of course, since the microsphere’s motion of subdiffu-sive, the tracer diffusion coefficient becomes rapidlymuch smaller thanD0 at times t>10–5 s. Figure 3shows that, as expected from the data in Fig. 2, the dif-fusion coefficient of the microsphere is always smallerfor a crosslinked actin network, due to the hindered dif-fusion of actin filaments in the presence of crosslinkers.Figure 3 also shows that the diffusion of a 0.96lmmicrosphere in a crosslinked or uncrosslinked actin net-work cannot be represented by the constant Browniandiffusion coefficient in a fluid of effective macroscopicviscosity g � 1 P, the macroscopic viscosity of 24lMactin solution. The diffusion coefficient would be aconstant equal toD� kBT=6pga � 4:510ÿ3 lm2/s. Thediffusion coefficient of a 0.96lm microsphere dependsstrongly on time, which reflects not only the viscoelas-tic nature of the fluid, but also its multi-componentstructure.

Small frequency viscoelastic moduli

The mean square displacement of a microsphere in apolymer network is also directly related to the bulk vis-coelastic moduli of that network. These moduli includethe frequency-dependent storage modulusG0�x�, theloss modulusG00�x�, and amplitude of the viscoelastic


modulusjG��x�j � ��G02�x� �G002�x�p

. As detailed inother papers, the transformation from mean square dis-placements to viscoelastic moduli can be achieved viatwo different, but related schemes (Mason and Weitz,1995; Mason et al., 1997; Gittes et al., 1997).

Figure 4 shows DWS measurements of the amplitudeof the viscoelastic modulusjG��x�j of an actin networkin the presence and in the absence of crosslinking pro-

teins calculated from the measured mean square dis-placements in Fig. 2. One of the most striking differ-ences between crosslinked and uncrosslinked actin net-works is the frequency dependence ofjG��x�j. In theabsence of crosslinkers, the viscoelastic modulus of anF-actin network is relatively independent of frequencyat frequencies up to 100 rad/s. In the presence of cross-linkers, the viscoelastic modulus becomes a strong func-tion of frequency at frequencies as low as 10–3 rad/s.These optical measurements are in agreement with rheo-logical measurements by Wachstock et al. (1993, 1994),at least at small frequencies. The viscoelastic behaviorof crosslinked and uncrosslinked actin networks is simi-lar at very small and very large frequencies. As sug-gested by Sato et al. (1989), the strong dependence ofthe viscoelastic moduli with frequency at intermediatefrequencies can be attributed to the dynamics of attach-ment ofa-actinin with F-actin.

One can describe the dynamic attachment ofAcanthamoebaa-actinin (AC) with actin (A) as an asso-ciation reaction of the type (Wachstock et al., 1993,1994; Tempel et al., 1996).

A�AC k�ÿ! A �AC; �4�kÿ

ÿwhere k+ and k– are the association and dissociationrates, respectively. Wachstock et al. (1993) measuredk+=1 lM–1 s–1, k–=5.2 s–1, and Kd=k–/k+=0.19lM.Therefore, the lifetime of a binding event betweena-ac-tinin and F-actin is 1/k–=0.19 s. Unlike the more famil-iar case of chemical gels, for which crosslinks are per-manent, the lifetime of binding ofa-actinin to F-actin isrelatively short. As a result, viscoelastic moduli are fre-quency dependent as easily observed within the fre-quency range probed by optical rheometry and mechan-ical rheometry. At very small frequencies or long timescales, most crosslinkers are statistically detached fromthe actin polymers and the crosslinked actin networkbehaves as if it were uncrosslinked. This interpretationis supported by Fig. 4: the magnitude ofjG��x�j be-comes similar for crosslinked and uncrosslinked actinnetworks. At larger frequencies or shorter time scales,most crosslinkers are attached to the filaments, whichslows down the relaxation of the stress at short times,or equivalently, increases the viscoelastic moduli atlarge frequencies. However, as shown by Fig. 4, we donot observe a sharp increase of the viscoelastic modulusat the frequency corresponding to the inverse of thelifetime of the crosslink. The increase ofjG��x�j withfrequency is observed to occur even at frequenciesmuch smaller thank–. This result suggests that in orderfor the stress to relax in a crosslinked network, thecrosslinks need to relax somewhat cooperatively, whichdecelerates the stress relaxation. Furthermore,jG��x�jis observed to remain a strong function of frequency atfrequencies much larger thank–. By analogy with the


Fig. 3 Time-dependent diffusion coefficient,D�t� � hDr2�t�i=6t, ofmicrospheres imbedded in crosslinked and uncrosslinked actinnetworks for increasinga-actinin concentration. The microsphere’sradius a is much larger than the actin network mesh sizea� n � 0:15 lm for c=24lM. At times as short as 10–6 s, the trans-port of the microsphere is sub-diffusive. The corresponding diffusioncoefficient is smaller than that of the same microsphere in water.Same conditions as in Fig. 2

Fig. 4 Magnitude of the frequency dependence viscoelastic modulusof actin networks, crosslinked with increasing amounts ofa-actininmeasured by DWS. At large frequencies,jG��x�j / x0:75 for thelow-concentration isotropic phase, the large-concentration nematicphase, and the crosslinked forms of actin networks. This exponentreflects the finite rigidity of actin filaments and is due to the high-frequency lateral fluctuations of the actin filaments between entangle-ments

classical rheological behavior of chemical gels, whichdisplay a large elastic plateau over an extended fre-quency range, one could have expected a rather fre-quency-independent profile forjG��x�j at frequenciesx>k–. At very large frequencies,x� kÿ, the frequencydependence and the magnitude ofjG��x�j become simi-lar again to the case of an uncrosslinked actin network.We shall further discuss the high-frequency regime be-low.

Note that these optical measurements have the im-portant advantage over classical mechanical measure-ments in that they avoid possible shear-induced orienta-tion of the network. As shown in a separate paper(Wirtz et al., 1998), the presence of crosslinkers candramatically reduce the range of strain values for whichone can measure the rheological properties of actin net-works in the linear regime. In these papers, we studiedthe effect of strain in steady and oscillatory shear onthe viscoelastic moduli of crosslinked and uncrosslinkednetworks. We observed that in the presence of crosslin-kers, strains as small as 2% can generate strain-harden-ing. This strain hardening behavior subsists over asmall strain range. Past a strain of 10%, crosslinked ac-tin networks yield rapidly. Diffusing wave spectroscopydoes not involve any forced shearing of the probed ac-tin networks. Therefore, DWS measurements can effec-tively probe the rheology of crosslinked actin gels inthe linear regime, which is difficult to achieve by classi-cal mechanical rheometry.

High-frequency viscoelastic moduli

The extended frequency range probed by DWS allowsus to probe the early dynamics of relaxation of cross-linked and uncrosslinked polymers in a network. Theinset in Fig. 4, shows the viscoelastic modulus at fre-quenciesx>103 Hz. At those frequencies, the visco-elastic moduli of both crosslinked and uncrosslinked ac-tin networks are observed to increase with similar effec-tive power laws of frequency,jG��x�j / xa with0.75<a<0.8. However, as reflected by the time-depen-dence of the mean square displacement at short times,we observe a systematic decrease of the effective expo-nent from 0.80 to 0.75 when the ratio crosslinker con-centration is decreased from 2.4 to 1.8lM.

In this high frequency power law regime, the lossmodulus dominates the elastic modulus, which is a con-sequence of the fact thata>1/2. Indeed, the ratio be-tween the loss modulus and the storage modulus is larg-er than once sinceG00=G0 � tan �ap=2� or G00 � 2:41G0,independently of frequency. This result can be con-trasted to the more usual case of (smaller) syntheticflexible such as polystyrene in toluene and semi-flex-ible polymers such as PBLG in m-cresol for whichG'>G'' even at the high frequencies probed by particle

tracking microrheology (Mason et al., 1997; Gittes etal., 1997; Schnurr et al., 1997) and DWS (Weitz andPine, 1993). Of course, at extremely high frequencies,x>1 MHz, the dissipation from the solvent dominatesthe rheology of actin filament networks since it growswith frequency as

jG�j � G00 � g1x; �5�whereg1 is the high-frequency viscosity of the buffersolution. Our DWS measurements do not display thelinear dependence predicted by Eq. (5), as shown inFigs. 4 and 5. Therefore, the viscoelastic properties ofactin networks at the largest frequencies measured byDWS are dominated by the local dissipative motion ofthe actin filaments.

We can attempt to understand the dynamics of stressrelaxation in crosslinked actin networks at intermediateand large frequencies by comparing the density ofcrosslinkers with the density of potential crosslinkingpoints in the actin gel. For a 24lM actin solution, theaverage density of overlaps is&1/n3&300lm–3,where n&0.15lm is the average distance betweenoverlapping actin filaments (i.e. the mesh size). Due totheir small size and the presence of two F-actin bindingsites, the only effective way thata-actinin moleculescan bind to F-actin is to be located at a topologicaloverlap between filaments. Therefore, if ana-actininmolecule is bound to one actin filament, after sometime it will actively create an effective overlap with asecond filament. The number of overlaps per chain isapproximatelyL/n&140, whereL&20lm is the aver-age contour length of an actin filaments.L/n would re-present the maximum number of available sites alongan actin filament for crosslinking bya-actinin. Ignoringthe finite lifetime of the crosslinker, fora-actinin con-centrations between 1.8–2.4lM, the average density ofcrosslinkers is&1.1–1.4·103 lm–3. For a 24lM actinsolution, the contour length density of polymer isq=clANA%38.5lm–2, whereNA is the Avogadro num-ber andlA=2.75 nm is the curvilinear length of a G-ac-tin monomer (Alberts et al., 1994). Hence, the curvi-linear length between crosslinkers along each actin fila-ment is about 0.026–0.036lm. Therefore, 100% ofpolymer overlaps are associated with a crosslinker. Thisassumes that alla-actinin molecules crosslink actin fila-ments, which overestimates the number ofa-actininmolecules attached to F-actin since crosslinks have afinite lifetime. Also, whena-actinin concentration be-comes large, it has been observed that actin filamentsdo not form isotropic networks: actin filaments bundle(Wachstock et al. 1993, 1994).

At times much longer than the average lifetime of acrosslink, the number ofa-actinin molecules that arebound is reduced and the effective number of overlapsassociated with a crosslinker becomes smaller than100%. Since fewer crosslinkers are bound, the dynamics


of relaxation of crosslinked actin filaments becomes sim-ilar to the dynamics of relaxation in a uncrosslinked net-work at long time scales or large frequencies, as observedin Figs. 4 and 5. A similar interpretation of the data re-sults if we remember that the mean square displacementof the probing microspheres is the normalized creep com-pliance of the network under stress. At time longer thanthe lifetime of crosslinking ofa-actinin to F-actin, the de-formation or creep of a strained actin network becomesrelatively independent of the crosslinkers.

At intermediate times, the finite lifetime of binding ofa-actinin becomes important. In an uncrosslinked actinnetwork, the time at which effects of entanglements be-come important is of the order ofse � fl4

e=kBTlp, wheref is the friction coefficient of actin filament per unitlength. According to Morses’ model (1997), for timest� se, the dynamics of relaxation is dominated by thebending modes or transverse fluctuations of actin atwavelengths smaller than the entanglement length. Sinceactin filaments are effectively rigid between entangle-ments �lp� �le; n��, then actin filaments cannot relaxby contour length diffusion, unlike flexible polymers.The dynamics of stress relaxation in a semi-flexible poly-mer solution are dominated by the lateral motion of thefilaments between entanglements. This motion becomesrapidly hindered for timest>se. The forces which pre-vent this small lateral motion generate a stress calledthe “tension” stress. In a crosslinked actin network, thenature of the dynamics of crosslinked actin networksshould become similar to the dynamics of uncrosslinkednetworks. This reasoning holds as long as the distance

between crosslinkers does not become much smallerthan the distance between overlaps and at short times.

The entanglement length, which is aboutle&0.5–1.0lm, is larger than the average distance between over-laps saturated by a crosslinker. This might explain whyeffects of crosslinking become observable in viscoelasticspectra at high frequencies, i.e., the power-law behavioris quickly shortened by the presence of crosslinkers.

The early onset of crosslinking effects is also observedin Fig. 5, which displays the loss and storage moduli as afunction of frequency for actin gels crosslinked with in-creasing amounts ofa-actinin. At smalla-actinin concen-trations, the storage modulus is relatively flat at small fre-quencies, and increases asG0�x� / x3=4 at large frequen-cies. At largera-actinin concentrations, the storage mod-ulus is a strong function of frequency and reaches apower-law behavior at large frequencies. Also the cross-over frequency at which loss and storage moduli becomeequal,G0�x� � G00�x�, decreases for increasinga-actininconcentrations.

We recently showed that, in the absence of crosslin-kers, the “curvature” stress, which stems from the slowreptation of actin in its tube, dominates the stress at smallfrequencies. At large frequencies and in the absence ofcrosslinkers, the stress is instead dominated by the ten-sion stress, which is characterized byGtens / x3=4

(Schnurr et al., 1997). The tension stress originates fromthe restricted bending motion of actin filaments betweenentanglements. We also observe ax3/4 dependence forG'and G'' at large frequencies in the presence of crosslin-kers, which is the first evidence that the tension stressalso dominates the stress for crosslinked actin networksat large frequencies. For crosslinked networks at smallfrequencies, one can suspect that, since reptation isgreatly prohibited, the curvature stress is diminished. Adirect clue for the absence of curvature stress in a cross-linked actin network is the steep frequency dependence ofG' at small and intermediate frequencies. By contrast,G'reaches a true plateau modulus for uncrosslinked actinnetworks. This plateau modulus was shown to vary withconcentration asG0p / c1:2�0:2 (Wirtz et al., 1997), whichis characteristic of a curvature stress. We are currentlystudying the effect of actin concentration of the magni-tude of G' and G'' over an extended frequency rangefor crosslinked actin networks. SinceGtens / c2:25 as op-posed toGcurve / c1:4 (Morse, 1997), that study couldhelp further identify the origin of the stress in a cross-linked actin network at small and large frequencies.

Summary and conclusions

We have presented DWS measurements of the short-time motion of semi-flexible polymers in a crosslinkedsolution of semi-flexible polymers. The system that we


Fig. 5 Loss and storage moduli of actin networks, crosslinked withincreasing amounts ofa-actinin measured by DWS. We observe thatthe crossover frequency at which the loss modulus becomes largerthan the elastic modulus is decreased for increasing crosslinker con-centrations. We also observe that, for crosslinked actin networks, theelastic modulus is more frequency dependent at intermediate frequen-cies

studied consists of actin filament networks crosslinkedwith Acanthamoebaa-actinin, a small protein with twoactin binding sites. Using the large bandwidth of diffus-ing wave spectroscopy, we measured the mean squaredisplacement of small microspheres imbedded in cross-linked and uncrosslinked actin networks over an ex-tended frequency range. From these mean square dis-placement measurements, we extracted the creep compli-ance and the viscoelastic moduli of the actin network. Weobserved that the scaling nature of crosslinked actin net-works is similar to that of uncrosslinked networks both atextremely small and extremely large frequencies. Atextremely small frequencies, which corresponds to timescales larger than the lifetime of a crosslinker, the motionof actin filaments is facilitated and resembles that of un-crosslinked actin filaments. However, the finite lifetimeof crosslinking renders the viscoelastic moduli frequencydependent at intermediate frequencies. At very largefrequencies, the modulus varies with frequency asjG��x�j / x3=4. This effective power law, which is dif-ferent from that predicted for flexible polymers but simi-lar to that predicted for uncrosslinked semi-flexible poly-mers, results from the finite rigidity of actin filaments. Weare currently studying the effect of actin concentration onthe rheology of crosslinked actin networks.

Acknowledgments The Authors thank D. Morse and Scot C. Kuofor insightful conversations and T.G. Mason and K. Rufener for theirhelp in the DWS measurements. We thank R.D. Mullins, J.F. Kelle-her, and T.D. Pollard for their generous gift ofAcanthamoebaa-acti-nin. This work was supported by the Whitaker Foundation and theNational Science Foundation, grants DMR 9623972 (CAREER), CTS9502810, and CTS 9625468.

Appendix

In this section, we establish a relationship between themean square displacement of a microsphere imbeddedin a viscoelastic fluid and the creep compliance of thatfluid. The shear stress and shear strain are related toone another as

s�t� � ÿZ t

0

Gr�tÿ t0� _c�t0�dt0 �A:1�

c�t� �Z t

0

J �tÿ t0� _s�t0�dt0 �A:2�

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� where H (t) is a Heavisidefunction andJ(t) is the strain resulting from a step ofstresss � s0H�t�. For instance, in an elastic solid

Gr�t� � G0 and J �t� � 1

G0�A:3�

For a purely viscous fluid of viscosityg,

Gr�t� � gd�t� and J �t� � t

gH�t� �A:4�

Unilateral Laplace transformation of Eqs. (A.1) and(A.2) yields simple relationships between the transforms~J �s�; ~G�s�; and ~c�s� :

~s�s� � s ~Gr�s�~c�s� � ~G�s�~c�s� �A:5�

~c�s� � s ~J �s�~s�s�: �A:6�Hence,

s ~J �s� ~G�s� � 1; �A:7�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 � kBT

pa~J �s�

or

hDr2�t�i � kBT

paJ �t� �A:8�

using J(0)=0. Equation (A.8) is approximate becauseEq. (1) has not been demonstrated regorously. Also,Eq. (A.8) neglects inertial effects, which become impor-tant at time scalest<10–6 s. Equation (A.8) yields thecorrect results for the extreme cases of a purely viscousfluid and a purely elastic solid. For a viscous liquid,hDr2�t�i � 6DtH�t� ��kBT=pga�tH�t� and J �t� ��t=g�H�t�, which verifies Eq. (A.4). For an elastic solid,hD~r2�s�i � r2

0 andJ �t� � 1=G0, whereG0 � kBT=par20,

which verifies Eq. (A.3).



References

Alberts B, Bray D, Lewis J, Rabb M, Ro-berts K, Watson J (1994) Molecular biol-ogy of the cell. 3rd Ed Garland Publish-ing

Doi M, Edwards SF (1989) The theory ofpolymer dynamics. Clarendon Press,Oxford

Ferry JD (1980) Viscoelastic properties ofpolymers. John Wiley and Sons, NewYork

Gittes F, Mickey B, Nettleton J, Howard JJ(1993) Flexural rigidity of microtubulesand actin filaments measured from ther-mal fluctuations in shape. J Cell Biol120:923–934

Gittes F, Schnurr B, Olmsted PD, MacintoshFC, Schmidt CF (1997) Microscopic vis-coelasticity: shar moduli of soft materialsdetermined from thermal fluctuations.Phys Rev Lett 79:3286–3289

Isambert H, Venier P, Maggs AC, Fattoum A,Kassab R, Pantaloni D, Carlier MF(1995) Flexibility of actin filaments de-rived from thermal fluctuations. J BiolChem 270:11437–11444

MacLean-Fletcher SD, Pollard TD (1980)Viscometric analysis of the gelation ofacanthamoeba extracts and purification oftwo gelation factors. J Cell Biol 85:414–428

Mason TG, Weitz DA (1995) Optical mea-surements of frequency-dependent linearviscoelastic moduli of complex fluids.Phys Rev Lett 74:1252–1255

Mason TG, Dhople A, Wirtz D (1997) Rhe-ology and microrheology of concentratedDNA solutions. In: Wirtz D, Halsey TC(eds) Statistical mechanics in physics andbiology. MRS Symp Proc 463:15–20

Mason TG, Ganesan K, van Zanten J, WirtzD, Kuo SC (1997) Particle-tracking mi-crorheology of complex fluids. Phys RevLett 79:3282–3285

Morse D (1997) preprintRouse PE (1953) A theory of the linear vis-

coelastic properties of dilute solutions ofcoiling polymers. J Chem Phys 21:1272–1276

Sato M, Schwarz WH, Pollard TD (1989)Dependence of the mechanical propertiesof actin/a-actinin gels on deformationrate. Nature (London) 325:828–830

Schnurr B, Gittes F, Olmsted PD, SchmidtCF, Macintosh FC (1997) In: Wirtz D,Halsey TC (eds) Local viscoelasticity ofbiopolymer solutions statistical mechanicsin physics and biology. MRS Symp Proc463:15–20

Spudich JA, Watt SJ (1971) The regulationof rabbit skeletal muscle contraction. JBiol Chem 246:4866–4871

Tempel M, Isenberg G, Sackmann E (1996)Temperature-induced sol-gel transitionand microgel formation in a-actinincross-linked actin networks: a rheologicalstudy. Phys Rev E 54:1802–1808

Wachstock D, Schwarz WH, Pollard TD(1993) Affinity of a-actinin for actin de-termines the structure and mechanicalproperties of actin filament gels. BiophysJ 65:205–214

Wachstock D, Schwarz WH, Pollard TD(1994) Crosslinker dynamics determinethe mechanical properties of actin gels.Biophys J 66:801–809

Weitz DA, Pine DJ (1993) In: Wyn B (ed)Dynamic light scattering. Oxford Univer-sity Press, Oxford

Wirtz et al. unpublished resultsXu J, Wirtz D, Pollard TD (1998) The

dynamics of crosslinking determine therheological properties of actin filamentnetworks. To appear in J Biol Chem

Zimm BH (1956) Dynamics of polymer mol-ecules in dilute solutions: Viscoelasticity,flow birefringence, and dielectric loss. JChem Phys 37:2547–2568

high-frequency viscoelasticity of crosslinked actin filament networks measured by diffusing wave...

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