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

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<ul><li><p>Jingyuan XuVirgile ViasnoffDenis Wirtz</p><p>Compliance of actin filament networksmeasured by particle-tracking microrheologyand diffusing wave spectroscopy</p><p>Received: 3 April 1998Accepted: 26 May 1998</p><p>J. Xu V. ViasnoffDepartment of Chemical EngineeringThe John Hopkins University3400 North Charles StreetBaltimore, MD 21218USA</p><p>D. Wirtz ())Chemical EngineeringG.W. C. Whiting School of EngineeringThe John Hopkins University3400 North Charles StreetBaltimore, MD 21218-2694USAe-mail:</p><p>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, and a 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 Angstromspatial resolution. At short times,the mean square displacement of a0.96 lm diameter sphere in a con-centrated actin solution displayssub-diffusion. hDr 2(t)i!t, with acharacteristic exponent</p><p>=0.780.05, 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 1.20.3. 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.</p><p>Key words Actin rheology diffusing wave spectroscopy</p><p>Rheol Acta 37:387398 (1998) Steinkopff Verlag 1998 ORIGINAL CONTRIBUTION</p><p>RA904</p></li><li><p>Introduction</p><p>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 length L and much largerthan the diameter, the polymer is semi-flexible (Doi andEdwards, 1989).</p><p>Filamentous actin (F-actin), the polymerized form ofthe globular four-lobed protein actin (G-actin), formslong semi-flexible polymers with a diameter d=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. dlp 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.</p><p>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.</p><p>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 the temporal domain. 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.</p><p>In order to demonstrate the effectiveness of ourapproach, we independently measure the creep compli-ance J (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.</p><p>Particle-tracking measurements by DWS offer amuch extended temporal range especially at short timescales, smaller than 105 s, which cannot be probed bymechanical rheometry in one measurement. At theseshort times, we find that J (t) increases with an effectivepower law of time, J (t)!t with =0.780.05, 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!c0.50for t</p></li><li><p>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 as Jp!cb withb=1.30.2. Here again, this exponent is in qualitativeagreement with Morses model (Morse, 1997), whichdescribes the long-time scale viscoelasticity of solutionsof semi-flexible polymers. Morses 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. Morses model predictsb=1.4, as opposed to b=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).</p><p>Experimental techniques</p><p>Actin preparation</p><p>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 4 8Cagainst 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 2.66104 M1 cm1, and a cell pathlength of 1 cm. Ca2+ and Mg2+ actin filaments are gen-erated by adding 0.1 volume of 10KMC (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.</p><p>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</p><p>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).</p><p>Actin is polymerized in situ for 1214 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 diameter0.96 lm 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 57% of themagnitude of G'(x) and G''(x) at all actin concentra-tions used in this paper. All DWS measurements areconducted at a temperature of T= 23 8C.</p><p>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 0.96 lm 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 510 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).</p><p>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</p><p>389J. Xu et al.Compliance of actin filament networks</p></li><li><p>polymers. Video-enhanced SPT also allows to extractviscoelastic moduli that are anisotropic and require ex-tremely small sample volumes &amp;20 ll, as opposed to&amp;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.</p><p>Mechanical rheometry</p><p>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 at T=23 8C to within0.1 8C. The G-actin solution is placed between the coneand plate tools and allowed to polymerize in the pres-ence of polymerizing salt for 1214 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.</p><p>Theory</p><p>Derivation an...</p></li></ul>


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