feedback-tracking microrheology in ... - science advancestin solutions], we applied...

SC I ENCE ADVANCES | R E S EARCH ART I C L E

B IOMECHAN ICS

1Department of Physics Graduate School of Sciences Kyushu University 744 MotookaNishi-ku Fukuoka 819-0395 Japan 2Third Institute of Physics Faculty of PhysicsUniversity of Goumlttingen Friedrich-Hund-Platz 1 37077 Goumlttingen GermanyThese authors contributed equally to this workdaggerPresent address Department of Physics Duke University Durham NC 27708 USADaggerCorresponding author Email mizunophyskyushu-uacjp

Nishizawa et al Sci Adv 20173 e1700318 29 September 2017

Copyright copy 2017

The Authors some

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American Association

for the Advancement

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Feedback-tracking microrheology in living cellsKenji Nishizawa1 Marcel Bremerich1 Heev Ayade1 Christoph F Schmidt2dagger

Takayuki Ariga1 Daisuke Mizuno1Dagger

Living cells are composed of active materials in which forces are generated by the energy derived from me-tabolism Forces and structures self-organize to shape the cell and drive its dynamic functions Understandingthe out-of-equilibrium mechanics is challenging because constituent materials the cytoskeleton and the cytosol areextraordinarily heterogeneous and their physical properties are strongly affected by the internally generated forcesWe have analyzed dynamics inside two types of eukaryotic cells fibroblasts and epithelial-like HeLa cells withsimultaneous active and passive microrheology using laser interferometry and optical trapping technology Wedeveloped a method to track microscopic probes stably in cells in the presence of vigorous cytoplasmic fluctuationsby using smooth three-dimensional (3D) feedback of a piezo-actuated sample stage To interpret the data we pres-ent a theory that adapts the fluctuation-dissipation theorem (FDT) to out-of-equilibrium systems that are subjectedto positional feedback which introduces an additional nonequilibrium effect We discuss the interplay betweenmaterial properties and nonthermal force fluctuations in the living cells that we quantify through the violationsof the FDT In adherent fibroblasts we observed a well-known polymer network viscoelastic response where thecomplex shear modulus scales as Gordm (minusiw)34 In the more 3D confluent epithelial cells we found glassy mechanicswith G ordm (minusiw)12 that we attribute to glassy dynamics in the cytosol The glassy state in living cells showscharacteristics that appear distinct from classical glasses and unique to nonequilibrium materials that are activatedby molecular motors

hm

on M

arch 19 2020ttpadvancessciencem

agorg

INTRODUCTIONThe complex interior of a biological cell is an active system because it ismaintained away from thermodynamic equilibrium by internal energydissipation and in particular by a variety of force-generating machi-neries (1 2) Two primary components determine the mechanics of acell the cytoskeleton a network of semiflexible protein polymers andthe cytoplasm a concentrated solution of biological macromoleculesand organelles These components govern the dynamics of cellfunctions and have characteristics of soft condensed materials (3 4)For decades semiflexible polymers and soft glassy materials have beenof central interest in materials science because of their complexmechanical properties (5ndash10) Both types of materials in cells alter theirproperties drastically in response to external perturbations cytoskeletalnetworks typically stiffen under motor-generated mechanical stresses(5ndash8) whereas the glassy cytoplasm can be fluidized under mechanicalload (9 10) In cells metabolic activity generates chemical nonequili-brium which powers mechanoenzymes (motor proteins) Thefunctions of these mechanoenzymes are in turn influenced by theirmechanical interactions with the surrounding materials (11 12) Thedynamics of cellular constituents under these conditions obviously donot display equilibrium statistics When investigating the mechanics ofthe interior of a living cell it is therefore essential to clearly quantify thenonequilibrium activity

Mechanical properties of macroscopic materials are typicallymeasured by rheometers that apply a stress and measure the strain re-sponse or vice versa A corresponding approach inside a cell ischallenging because dimensions of cells are typically on the order of tensof micrometers The method of choice in this case is microrheology

(MR) in whichmicrometer-sized tracer beads are tracked andmanipu-lated to probe the viscoelastic properties of their surroundings (13)Standard options of implementing MR include the passive tracking ofbead fluctuations [passiveMR (PMR)] (14ndash17) and activemanipulationof the beads by known forces [activeMR (AMR)] (18ndash20) The standardprocedure used to calculate a materialrsquos viscoelasticity based onmeasured bead fluctuations (PMR) relies on the fluctuation-dissipationtheorem (FDT) (16 21) which is only valid in thermodynamic equilib-rium However mechanoenzymes in cells can create nonthermal fluc-tuations greatly in excess of thermal fluctuations (1 2 22) In such asituation AMR can still be used to measure the mechanical responseproperties of the nonequilibrium system directly Knowing themechanical properties of the system then makes it possible to estimatethe amplitude of the purely thermal fluctuations (18 21) Therefore thecombination of AMR and PMR allows one to separately quantifythermal and nonthermal fluctuations The extent to which the FDT isviolated is a novel metric that indicates how far the system is from ther-modynamic equilibrium owing to the dissipation of the internally trans-duced or externally injected mechanical energy (22ndash24) PMR in livingcells has been performed for instance by recording videos of embeddedprobe particles (25 26) Although technically simple videoMR is lackingin both spatial and temporal resolution (27) and one cannot performAMR AMR in cells has rarely been conducted (28) because the strongmotor-driven fluctuations make it difficult to exert a well-controlledforce on the beads for a time that is sufficiently long to obtain a completespectrum of response

In previous in vitro (noncell) studies simultaneous AMR and PMRwere performed using optical trapping and laser interferometry tech-nology (2 18 24) Two laser beams were focused on the probe particleOnewas used to apply a sinusoidal force (optical trapping) and the oth-er was used to detect the beadrsquos position precisely relative to the laserfocus (laser interferometry) There are several problems with applyingthis technology in living cells In laser interferometry the probe positionis detected relative to the symmetry axis of the probe laser The detectionrange is limited The distance between the probe and the laser focus

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must remain sufficiently smaller than the radius of either the beamwaist(~02 mm) or the probe particle (29) In metabolically active cells aweakly trapped probe typically rapidly moves out of this range duringobservations A strong optical trap could possiblymaintain the probe inthe trap but large stresses would build up locally as a result of internalactivities It would then be impossible to determine the linear responsemechanics of cells because cellular materials (cytoskeleton and cyto-plasm) exhibit strongly nonlinear viscoelasticities (5 7 30) Increasingthe radius of the beamwaist extends the detection range (29) but reducesthe laser diffraction efficiency and therefore the signal-to-noise ratio

Here we introduce a new method to solve this problem We imple-mented three-dimensional (3D) feedback to smoothly reposition thepiezo-actuated sample stage and the stage holding the objective lens tomaintain the probe bead in the detection range The total beadmotionwhich is the core observable for both AMR and PMR can be obtainedby summing the low-frequency stage movement and the high-frequencymotion of the probe relative to the laser focus as determined by inter-ferometry Feedback-implementedMR successfully solved not only theproblem of stably tracking probe beads in vigorously fluctuating envir-onments but also another problem with conventional optical trappingMR namely that the low-frequency response (and fluctuations) of theprobe are suppressed by the optical trapping potential (18 31) whichmakes it practically impossible tomeasure slowdynamics This problemdisappears in feedback-implemented MR Because the feedback-controlled stage follows the slow motion of probe beads and always re-tains the beads in the laser focus the bead motions at low frequenciesare not affected by the optical trapping force At higher frequenciesthough the optical trap exerts a time-varying force on the probe parti-cle In other words the system is already driven out of equilibrium bythe optical trapping force Via the feedback mechanism this drivingforce is correlated with the thermal forces We provide a theoreticalprocedure that can be used to apply the FDT to correct for these artifi-cial nonequilibrium effects due to feedback

After validating the procedure by performing control experiments inequilibrium materials [polyacrylamide (PAAm) gels and entangled ac-tin solutions] we applied feedback-enhanced AMR and PMR incultured eukaryotic cells At low frequencies we observed a significantviolation of the FDT The nonthermal fluctuations were several ordersof magnitude larger than the thermal fluctuations These fluctuationsreflect more or less correlated activity of motors at intermediate fre-quencies (24) Fluctuations typically become random at longer timescales such that themean squared displacement scales linearlywith time(32 33) Because the dominating driving forces for these random fluc-tuations are also nonthermal as demonstrated by the breaking of theFDT this motion is what has been referred to as ldquoactive diffusionrdquo orldquoactive stirringrdquo (22 32 34) On the other hand the high-frequency re-sponse in cells satisfies the FDT and exhibits a power-law behaviorconsistent with that predicted for semiflexible polymer gels in equilib-rium (7 18) and colloidal glasses (35 36) Through mechanicalexperiments with isolated adherent fibroblast cells (which wereflattened and in the flat part of the cell abundant in actin cytoskeleton)we found widely distributed elastic properties of such cells as expectedwhich displayed the well-known viscoelastic behavior Gordm (minusiw)34 athigh frequencies In further experiments we found that confluent sheetsof epithelium-like HeLa cells showed highly reproducible viscoelasticityand a power-law behavior Gordm (minusiw)12 that is distinct from that ex-hibited by semiflexible polymers but is typically observed for soft glassymaterials (35 36) We observed that the high-frequency shear modulusincreased exponentially with cytosol concentration To explain these


observations we suggest that one can model the HeLa cell epitheliumas an actively stirred glass where the heterogeneous dynamics that typ-ically appear in inactive glasses can continuously relax

RESULTSTheoretical foundation of feedback MRConventional AMR without feedback is explained in detail in note S1(18) to provide the background for MR under feedback control Theexperimental setup for feedback MR is shown in Fig 1A and its tech-nical details are given in Materials and Methods When the position ofthe sample stage is controlled by feedback the total displacement u ofthe probe in the sample is given by u = uQPD + ustage where uQPD is thedistance between the probe particle and the focus of the probe laser andustage is the displacement of the piezo stage as shown in Fig 1B Theoptical trapping force applied to the probe particle by both the driveand probe lasers is given by kdL exp(minusiwt) minus (kd + kp)uQPD where Lis the amplitude of the drive laser oscillation and kd and kp refer tothe trap stiffnesses of the drive and probe lasers respectively The Lan-gevin equation for the probe particle under feedback control is

kpuQPDethtTHORN thorn intt

infingetht tprimeTHORN _uethtprimeTHORNdtprimefrac14 kdethLeiwt uQPDethtTHORNTHORN thorn zethtTHORN thorn f ethtTHORN eth1THORN

where _uethtTHORNdenotes the velocity of the probe in the coordinate system ofthe sample medium that is stationary with respect to the feedback-controlled piezo stage but moves in the lab frame g(t) is the friction

Fig 1 Experimental setup for feedback MR (A) Optical trapndashbased AMR setupwith 3D feedback controls A probe particle in a cell is trapped and oscillated bythe drive laser (pink) QPD signals from the probe laser (cyan) are fed into PIDcontrollers which regulate piezo stage motion in x and y directions such thatthe trapped probe particle is maintained near the optical trap center Thebright-field image of the probe is simultaneously projected onto a charge-coupled device (CCD) camera [green thick line light-emitting diode (LED) lightsource is omitted] The image is analyzed on a personal computer (PC) to adjustthe z position of the objective lens through a z direction piezo scanner (B) Sche-matics for bead displacements and laser positions Positions of fixed probe laser(830 nm cyan) and acousto-optic deflector (AOD)ndashsteered drive laser (1064 nmpink) are shown by solid and dashed vertical lines respectively The drive laser isdrawn displaced to the right of the probe laser The pale particle indicates theprobe starting position

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function and z(t) and f (t) are the thermal and nonthermal forcesrespectively The ensemble average of Eq 1 yields the frequency-dependent response which is written as

languethwTHORNrangFB frac14 languQPDethwTHORNrangthorn langustageethwTHORNrangfrac14 aethwTHORNfFethwTHORN ethkd thorn kpTHORNlanguQPDethwTHORNrangg eth2THORN

where aethwTHORN frac14 aprimeethwTHORN thorn iaPrimeethwTHORN frac14 1=frac12 iw~gethwTHORN is the probersquos re-sponse function without an optical trap In Eq 2 we have introducedan apparent driving force FethtTHORN frac14 kdLeiwt equiv FethwTHORNeiwt Here andhereafter the superscript ldquoFBrdquo indicates the total displacement whenthe position of the piezo stage is feedback-controlled ldquo~rdquo and ldquo^rdquodenote the Fourier-transformed function and the amplitude of thesinusoidal signal respectively and angled brackets denote the statis-tical or time average

ForMR under feedback control the output voltageV(t)=uQPDC ofthe quadrant photodiode (QPD) that detects the probe laser deflec-tion is fed to the ldquoMeasurerdquo input of an analog proportional-integral-derivative (PID) controller (Fig 2) HereV(t) is the output voltage ofthe QPD and C is the calibration factor for the displacement responseof the QPD The target value ofV(t) is the set point s(t) On the basis ofthe error signal V(t) minus s(t) the PID controller generates a control signal

eethtTHORN frac14 PethV sTHORN thorn IintethV sTHORNdt thorn D ddt ethV sTHORN where P I and D

are the respective weight factors for the three modes of feedback Herethese feedback parameters were set so that only the second (integral)term is used to control the piezo stage (P = D = 0 I ne 0) The set point

s(t) was set to 0 leading to the simple formula e(t) = IintVdt Integratingthe signal implies low-pass filtering and guarantees that the piezo stagecan be controlled stably below its resonance frequency We can thenassume that ustage responds to e(t) with a proportionality constantB equiv ustage(t)e(t) Thus

ustageethtTHORN frac14 BeethtTHORN frac14 BICintuQPDdt frac14 1

tintuQPDdt eth3THORN


where t equiv CBI is the overall response time of our feedback-trackingsystem Fourier-transforming u = uQPD + ustage and Eq 3 yields

~u frac14 ~uQPD thorn ~ustage frac14 eth1 iwtTHORN~ustage frac14 eth1 1iwt

THORN~uQPD eth4THORN

The same relation holds for the sinusoidal responses ucirc(w) ucircQPD(w)and ucircstage(w) by changing ~ to ^ The stage displacement ustage(t) =Be(t) in response to the probe motion u(t) is low-passndashfiltered asũstage(w)ũ(w) = 1(1 minus iwt) The response time t of the measurementsystem must be significantly longer than the response time of the piezo-actuated stage which is approximately ~1 ms as seen in Fig 3B Theupper limit for useful low-pass filtering is set by the dynamics of the sys-tem and the detection range of the laser focus The probe bead should notmove more than about halfway through the detection range during thefilter time Therefore t was set typically between tens and hundreds ofmilliseconds (see fig S1C)

We define the total response of the probe movement to the oscilla-tory force applied by the drive laser under feedback control asAFB equiv languethwTHORNrangFB=FethwTHORN Using this definition and substituting Eq 4 intoEq 2 the following relationship between AFB(w) and a(w) is obtained

AFBethwTHORN frac14 languethwTHORNrangFBFethwTHORN frac14 aethwTHORN

1thorn baethwTHORN eth5THORN

where b equiv kt(1 minus 1iwt) is the parameter describing the correction un-der feedback and kt is the sum of the trap stiffnesses kt equiv kd + kp Theintrinsic response function a(w) can then be obtained from AFB(w)From a(w) the complex shear modulus G(w) = Gprime(w) + iGPrime(w) ofthe surrounding medium is obtained via the generalized Stokes relation(13 37)

GethwTHORN frac14 16paaethwTHORN eth6THORN

where a is the radius of the probe particle

Feedback AMR in thermal equilibriumMR experiments with feedback were first tested in aqueous polymernetworks in thermodynamic equilibrium using tightly cross-linkedPAAmgels andpolystyrenebeadswith2a=2mmdiameter asprobesAs illustratedin the signal diagram in Fig 2 the electric signalsV(t)ordm uQPD and e(t)ordmustage could be measured directly Therefore we define corresponding re-sponsesAQPDethwTHORN equiv languQPDethwTHORNrang=FethwTHORNand AstageethwTHORN equiv langustageethwTHORNrang=FethwTHORNas shown in Eqs 10 and 12 respectively in Materials and MethodsFigure 3A and fig S2 depict the values of AQPD (closed circles) andAstage (open squares) that weremeasured using feedbackAMR in PAAmThe spring constant for the effective potential that the probe particleexperienced from the surrounding matrix was calculated from the shearmodulus as 6p|G|a (ge50 times 10minus6 Nm) which is considerably larger thantheoptical trap stiffness kt = kp+kd =68times10

minus6Nm In this case becausethe effect of kt on the motion of the probe was negligible a showed littledifference fromAFB =AQPD +Astage In Fig 3A we present the values ofa estimated from AQPD and Astage using Eqs 11 and 13 as given inMaterials and Methods The rapid displacement of the probe ispredominantly given by uQPD (u ~ uQPD ≫ ustage) and thereforea ~ AQPD at high frequencies On the other hand the slow displacement

Fig 2 Diagram of signal flow in AMR under feedback control For AMRfeedback control of the piezo stage (red solid lines) is performed under sinusoidalforce application to the probe particle with the drive laser (blue dashed lines) InPMR the feedback loop is run without force application V(t) andor e(t) ismeasured with a lock-in amplifier (AMR) or directly digitized and recorded by aPC (PMR)

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is mostly given by the feedback-controlled displacement of the piezostage (u ~ ustage ≫ uQPD) such that a ~ Astage at low frequenciesThe crossover frequency between these limiting behaviors that is shownin Fig 3A is consistent with the estimated system response (note S2)fc = 1(2tp) ~ 4 Hz

Figure 3B presents the viscoelastic shear modulus of the PAAmgel GQPD and Gstage were estimated from AQPD and Astage usingEqs 6 11 and 13 The inertial effect due to the mass of the probeparticle and the matrix surrounding it which becomes prominent athigh frequencies (gt10 kHz) was taken into account following theprocedure described elsewhere (7 18) Reliable values were not ob-tained for Gstage at frequencies higher than 100 Hz Otherwise thevalues of GQPD and Gstage are mostly consistent The solid and brokencurves are the fits of a function that assumed a low-frequency elasticplateau in addition to a high-frequency power law G(w) = g0 + g1(minusiw)

12following the classical Rouse model for cross-linked networks com-posed of flexible polymers (38)

Feedback PMR and artificial breaking of FDTNext we discuss feedback PMR that is probe fluctuations underfeedback control for a sample in thermodynamic equilibrium To per-form feedback PMR we turned off either the drive laser (kd = 0) or itsspatial oscillation (L = 0) In both cases the probe laser was used to de-tect displacements The feedback nowmanipulated the stage position tocancel the thermal displacements of the probe The Langevin equationfor the probe movements is then

ktuQPDethtTHORN thorn intt

infingetht tprimeTHORN _uethtprimeTHORNdtprime frac14 zethtTHORN eth7THORN

The Fourier transform of Eq 7 and the relation between ũ and ũQPD viathe PID feedback (Eq 3) lead to a linear relation between u and thethermally fluctuating force z (Fig 4A) ~u frac14 AFBethwTHORN~z Note that thisrelation is similar to the response to the externally applied force thatis given in Eq 5 Even if there are no nonthermal forces generated inthe sample that is even if f(t) = 0 the probe particle is subjected to thenonthermally fluctuating optical-trapping force because of the stage


feedback Suppose that the responseAFB and corresponding fluctuationslang|ũ(w)|2rangFB are measured without knowing whether an active feedbackloop is present In this case it is tempting to expect that the FDT of thefirst kind would hold as langj~uethwTHORNj2rangFB frac14 2kBTAPrime

FBethwTHORN=w (16 21) To in-vestigate this assumption we performedMR under feedback to estimatelang|ũ(w)|2rangFB and APrime

FB independently using a material at thermal equilib-rium namely an entangled actin network prepared at a concentration of1mgml in an aqueous buffer Probe particles were trappedwith kt=kp=48 times 10minus6 Nm and their fluctuations were recorded with and withoutfeedbackWhile lang|ũ(w)|2rangFBwasmeasured directlyAPrime

FBethwTHORNwas obtainedby substituting a(w) kt and t in Eq 5 with their independentlymeasured values The violation of the FDT was evidenced by the cleardisagreement between wlang|ũ(w)|2rangFB and 2kBTAPrime

FBethwTHORN (see fig S3)This violation of the FDT breaking is reasonable because the system

was artificially driven out of equilibrium as a result of the imposedfeedback as schematically shown in Fig 4A Expressing the FDT aslangj~uethwTHORNj2rangFB frac14 2kBTAPrime

FBethwTHORN=w is therefore not correct Fluctuationsunder feedback control are described by rewriting the Fourier transformof Eq 7 as

~uethwTHORN frac14 a1thorn kta

eth~z thorn kt~ustageTHORN frac14 AethwTHORNeth~z thorn kt~ustageTHORN eth8THORN

HereA(w) is the response of the optically trapped probe in the absenceof feedback which can be expressed as A(w) = a(w)[1 + kta(w)] [seenote S1 (18 31)] Equation 8 indicates that ũ(w) is driven by ~z and theoptical trapping force due to the movement of the piezo stage ktũstage(the green arrow in Fig 4A) The actual optical force applied to theprobe is minusktuQPD although ndashktu = ndashkt(uQPD + ustage) would have beenapplied to the probe (see notes S1 and S3) were it not for the feedbackTherefore the difference ktustage is the driving force due to the feedbackThe obvious consequence of the feedback is that the optical trappingforce ktuQPD(t) is strongly correlated with z(t) lang|ũ(w)|2rangFB is thereforenot equal to the sum of the second moments for thermal and non-thermal fluctuations In our previous studies that have reported theFDT violations in nonequilibrium systems (1 2 18 24) these corre-lations between thermal and nonthermal forces were assumed to notexist However supposing linear response the statistical properties

Fig 3 Feedback AMR in equilibrium sample (A) Real parts of responsefunctions of probes embedded in a cross-linked PAAm gel measured underfeedback control Open squares and filled circles indicate responses measuredwith QPD and stage respectively Solid and dashed curves show the estimatedmaterial response aprime of the gel as estimated from Aprime

QPD and Aprimestage respectively

(B) Complex shear modulus measured with AMR under feedback control (symbols)and fit of G(w) = 19 + 059(minusiw)12 (solid and dashed curves) Circles and trianglesare real and imaginary parts of the modulus respectively Filled and opensymbols correspond to moduli obtained using stage response and QPD responserespectively

Fig 4 Feedback PMR and conventional PMR (A) Diagram of signal flow inPMR under feedback control AFB is the apparent response function in the timedomain that relates (thermal) fluctuating force z(tprime) and total probe displacementu(t) under feedback uQPD and ustage are correlated via the PID controller A is theresponse function of the optically trapped probe in the absence of feedback ktustagerepresents the reduction of the optical trapping force due to feedback (B) Powerspectral densities (PSDs) of displacement fluctuations of the probe embeddedin entangled actin solution (1 mgml) measured with PMR Open triangles andthe dashed curve show PSDs directly measured with and without feedback respec-tively Open circles and the solid curve show lang|ũ(w)|2rangno trap calculated from lang|ũ(w)|2rangFB

using Eq 9 and PSDs calculated from lang|ũ(w)|2rangtrap using the conventional PMRmethod respectively

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of z should not be affected by the application of the additional forcektustage Thus z can be described using the FDT of the second kind as

langj~zethwTHORNj2rang frac14 2kBTRefrac12 1=iwAethwTHORN frac14 2kBTRefrac12 1=iwaethwTHORN T h e

PSD of the total displacement under feedback is given bylangj~uethwTHORNj2rangFB frac14jAFBethwTHORNj2langj~zethwTHORNj2rang frac14 j1 =eth1thorn baTHORNj22kBTaPrime=w By independentlymeasuring a it is possible to calculate the PSD in the absence of theoptical trap potential as

langj~uethwTHORNj2rangno trap frac14 j1thorn baethwTHORNj2langj~uethwTHORNj2rangFB eth9THORN

Here and hereafter the superscript ldquono traprdquo denotes fluctuationsmeasured in the absence of the optical trap

The PSDs in the presence and absence of the optical trap lang|ũ(w)|2rangtrapand lang|ũ(w)|2rangno trap weremeasured by conventional PMR and are shownas broken and solid curves respectively in Fig 4B In conventional opticaltrapndashbased PMR experiments the probe fluctuations are suppressed bythe optical trap below the frequency at which |ktA(w)| ~ 1 is reached InFig 4B this crossover is observed at around 100 Hz and lang|ũ(w)|2rangno trap

and lang|ũ(w)|2rangtrap are distinct at lower frequencies In feedback PMRthe total fluctuations of the probe u(t) [=uQPD(t) + ustage(t)] can bereconstructed and the power spectrum lang|ũ(w)|2rangFB (red triangles inFig 4B) shows additional features In this case another crossover be-tween lang|ũ(w)|2rangFB and lang|ũ(w)|2rangtrap is observed at fc = 12pt ~ 10Hz Be-

low this frequency lang|ũ(w)|2rangFB deviates from lang|ũ(w)|2rangtrap and

approaches lang|ũ(w)|2rangno trap because the slow motions of the probewhich the feedback follows are less affected by the optical trap Byobtaining a by conventional PMR(note S3)we calculated the right-handside of Eq 9 (blue circles in Fig 4B) Note that kt and t which werenecessary to evaluate b were obtained by independent calibration(seeMaterials andMethods and note S2) The exact agreement between

lang|ũ(w)|2rangno trap obtained using conventional and feedbackPMR(the sol-

id curve and circles in Fig 4B) validates our analysis procedure

Intracellular mechanics measured with feedback MRAfter testing the proposed method in equilibrium systems we per-formed MR experiments with feedback in living cells Melamine parti-cles with a high refractive index (and a diameter of 2a = 068 mm) werefirst incorporated into NIH-3T3 fibroblast cells via phagocytosis Thetotal probe displacement u(t) = uQPD(t) + ustage(t) is shown in Fig 5AThe feedback maintains the probe in the detection range (~100 nm)for hours which is sufficiently long to complete AMR experimentswhereas the total fluctuations become larger than the detection rangein less than 30 s The frequency dependence of G(w) which wascalculated via Eqs 6 and 11 is shown in Fig 5B Because the refractiveindex profile of the cells is more or less inhomogeneous and distinctfrom the refractive index of the aqueous solvent kp and kd weremeasured in situ for each particle in the cells following the proceduredescribed in note S4 Because the fluctuations and responses of probesingested by phagocytosis into cells vary significantly statistical aver-aging smears out characteristic features of each spectrum such as relaxa-tion times and power-law exponents Therefore a typical result thatwas fitted by the simple sum of two independent power laws G(w) ~140(iw)012 + (iw)074 is shown by the solid curves in Fig 5B Howevernote that the exponents and the prefactors vary fromcell to cell andwith


the position of probe particles in the cells In particular noncoatedprobes phagocytosed into cells are likely surrounded by plasma mem-branes that can interact with rigid intracellular constituents Dependingon the intracellular structures to which a probe binds responses willbroadly vary This opens an interesting option for future experimentswhere one could specifically target specific cell compartments such asthe nucleus the endoplasmic reticulum or the Golgi apparatus bycoating the beads with for example antibodies

As a first step in the study of intracellular mechanics we insteadchose another approachWe coated the surfaces of the probe beadswithpolyethylene glycol (PEG) strands (39) and then bombarded cells withthe probe beads using a gene gun (40) In aqueous environments PEGcoating generally passivates probe surfaces The hydrophilic PEG createsrepulsive interactions with both hydrophilic and hydrophobic biopoly-mers and prevents sticking to cytoskeletal elements and other proteinsin cells (41)We prepared an epithelium-like sheet of confluent HeLacells and used melamine particles (with a diameter of 2a = 1 mm)incorporated at the center between the cell membrane and the nucleus

Fig 5 Material properties of fibroblasts and HeLa cells measured withfeedback MR (A) 2D trajectory of a probe in a cell measured under feedbackThe total period of observation was ~30 min (B) Shear modulus measured in theperipheral region of an adherent fibroblast which is rich in actin cytoskeletonSolid and dashed curves are fits of an empirical model describing low-frequencyrelaxations and high-frequency power-law behavior of a semiflexible polymernetwork as G(w) ~ 140(iw)012 + (iw)074 (C) Shear modulus measured in confluentHeLa cells (n = 9 cells cultured in different dishes) Bars indicate the unbiasedestimate of the SD Power-law form of viscoelasticities G(w) ordm (minusiw)05 indicatesa glassy response of (actin networkndashpoor) cytosol (D) G0 for osmotically com-pressed HeLa cells (pink diamonds) plotted against macromolecular concentra-tion relative to isotonic condition (f = 1) Exponential dependence onmacromolecule concentrations f (solid line exponential dashed curve lineardependence) is typical of strong glass formers rather than polymer networksThe asterisk marks G0 for the cell extract taken from HeLa cells (021 gml)

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as probes With this preparation we found that the distribution of themechanical properties measured was narrow showing quantitativelysimilar behaviors among different HeLa cells (n = 9 Fig 5C) Unlikeisolated fibroblasts Gprime and GPrime in the confluent epithelium vary as asimple power law G(w) = G0(minusiw)

05 in a wide range of frequencies(Fig 5C and fig S4) The concentration dependence of G0 helps toexplain this unexpected power-law exponent in living epithelium as ex-plained in detail in Discussion Because high-frequency fluctuations incells are purely thermal as will be discussed below we obtained the con-centration dependence of the prefactorG0 with feedback PMR To varyintracellular concentration we osmotically compressed (or dilated) thecells by adding sucrose (or water) to the culture mediumG0 was foundto be exponentially dependent on the concentration of macromoleculesin HeLa cells (Fig 5D) The same frequency dependence G(w) =G0(minusiw)

05 was also found in control experiments with HeLa cell ex-tracts at a close to physiological concentration The resulting G0 forthe cell extracts is shown by the asterisk in Fig 5D

Feedback PMR and FDT violation in living cellsWe now compare feedback PMR and AMR results that were obtainedsimultaneously in NIH-3T3 cells Figure 6A shows displacements uQPDand ustage in the x direction We evaluated the PSDs of the probe

displacement signals such as lang|ũ(w)|2rangFB lang|ũQPD(w)|2rang and lang|ũstage(w)|2rang (Fig 6B) We observed that lang|ũ(w)|2rangFB was obtained by simply

summing lang|ũQPD(w)|2rang and lang|ũstage(w)|2rang This relationship stems fromthe fact that ũQPD and ũstage are correlated with a phase difference of p2because ũstage = minusũQPDiwt as shown in Eq 3 The same relation wasalso observed in a sample in thermodynamic equilibrium (fig S5) Theresponse time of the feedback-tracking system in these experiments forNIH-3T3 cells t = 0145 s was measured independently as illustratedin fig S1A This response time corresponds to a crossover between

lang|ũQPD(w)|2rang and lang|ũstage(w)|2rang at fc ~ 11 Hz The same probe particle

was also tracked without feedback and the resulting lang|ũ(w)|2rangtrap isshown in Fig 6B Without feedback the bandwidth of lang|ũ(w)|2rangtrap islimited at low frequencies because the particle could not be tracked forlonger than several tens of seconds in any of the trials In the measuredfrequency range lang|ũ(w)|2rangtrap and lang|ũ(w)|2rangFB are consistent

The imaginary parts aPrime of the response functions that were obtainedwith AMR and PMR using the expression wlang|ũ(w)|2rangno trap2kBT are


shown in Fig 7A The agreement of these functions at frequencies higherthan ~10 Hz indicates that the FDT is satisfied here whereas the obviousdiscrepancy demonstrates the breaking of the FDT at lower frequencies(1 2 18 24) The PSD of the nonthermal fluctuations was obtained usingEq 15 by subtracting the thermal fluctuations from the total fluctua-tions The resulting functionwlangj~uethwTHORNj2rangnonthermal

=2kBT is shown by thesolid curve in Fig 7A Attempts to deduce nonthermal forces fromlangj~uethwTHORNj2rangnonthermal

have been reported earlier (1 18 22) and repeatedrecently (28 42 43) In these studies the probe motion was described

using the Langevin equation intt

infingetht tprimeTHORN _uethtprimeTHORNdtprime frac14 zethtTHORN thorn f ethtTHORN Assuming that z(t) and f(t) are not correlated taking the Fourier trans-form of the statistical average immediately gives langj~uethwTHORNj2rangnonthermal frac14jaethwTHORNj2langj~f ethwTHORNj2rang

In general viscoelastic materials thermal fluctuations exhibit power-law scalingwlangj~uethwTHORNj2rangnonthermal

ordm wn where the exponent v is limitedto the rangeminus1le nle 1 The exponent for Brownianmotion or simplediffusion is n = minus1 This limitation does not necessarily hold for non-thermal fluctuations The inset in Fig 7A shows the power-law expo-

nents of the nonthermal fluctuations wlangj~uethwTHORNj2rangnonthermalthat were

calculated at each frequency using d log wlang|ũ(w)|2rangnonthermald log w

At frequencies 01 Hz lt f lt 10 Hz exponents smaller than minus1 wereobserved exceeding the lower limit of v for equilibrium systems Thisis likely due to the directed transport of the probe as seen in the 2Dprobe trajectory (Fig 5A) Transport direction memory becomes ev-ident in the calculation of the autocorrelation function for the velocitydirection n

rarrethtTHORN (Fig 7B) The nonzero autocorrelation confirms that themotion is directed but the direction memory was lost at around 1 s forthis sample Directed transport sometimes called ldquosuper-diffusionrdquo isanother indication of an out-of-equilibrium situation and typicallyresults from motor protein activity in cells

DISCUSSIONIn cells molecular motors such as myosins generate forces that inducenonthermal fluctuations that are tracked by our probe particles Non-muscle cytoplasmic myosins bind to actin and progressively build upstress in the network with correlation times on the order of seconds

Fig 7 FDT violation and out-of-equilibrium fluctuations in a fibroblast cell(A) Imaginary part of the complex response function measured inside a fibroblastusing feedback AMR compared to the corresponding function wlang|ũ(w)|2rangno trap2kBT measured with PMR Curves agree at high frequencies and clearly differ at fre-quencies less than 10 Hz The solid curve shows purely nonthermal fluctuations calculatedusing Eq 15 Solid lines show power laws for comparison Inset Local power-law exponentof nonthermal fluctuations calculated using d logwlang|ũ(w)|2rangnonthermald logw Crossoverbetween different scaling regimes can be observed (B) Autocorrelation function ofvelocity direction

Fig 6 Displacements of melamine resin probe bead (2a = 680 nm) incultured fibroblast cell (NIH-3T3) (A) QPD and control signals for piezo stageas output by PID controller were recorded and calibrated to obtain displacementsuQPD (green) and ustage (blue) respectively Total displacement u(t) (red) was ob-tained by summing these quantities (B) The displacement PSD measured usingQPD with conventional PMR (black curve) agrees with that obtained underfeedback by summing uQPD and ustage

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Myosins then stochastically unbind from the actin filaments and there-by instantaneously release stress stored in the elastic network (2 6 44)These motor-induced fluctuations have been studied in reconstitutedactinmyosin gels where the abrupt stress release causes directedmotionof probes (fig S6A) (2 44) Considering the standard relation betweenstep response and the frequency response of a material the correlationtime t1 of the directed motion can be estimated from the angular fre-quency w1 (~1t1) where Gprime(w1) = GPrime(w1) (see fig S6B and the discus-sion thereof) The abrupt release of stress corresponds to a step in f(t) inEq 1 and therefore on averagelangj~f ethwTHORNj2rangordmw2 in Eq 15 if such eventsoccur in an uncorrelated manner (45) The nonthermal fluctuationsderived frommeasuring both lang|ũ(w)|2rangnonthermal and a(w) (18 24) wereconsistently explained by inserting this power-law dependency into Eq15 langj~uethwTHORNj2rangnonthermal frac14 jaethwTHORNj2langj~f ethwTHORNj2rang The same discussion wasspeculated to hold for nonthermal fluctuations in living cells (28)A viscoelastic response function in the cell of the form |a(w)|2 ordm1|G(w)|2 ordm wminus036 at frequencies less than 10 Hz (Fig 5B) would

lead to wlangj~uethwTHORNj2rangnonthermalordmwjaethwTHORNj2langj~f ethwTHORNj2rangordmw136 which was

close to the observed relationship wlang|ũ(w)|2rangnonthermal ordm wminus14 How-

ever note that the actual correlation time of the directed motions ob-served in cells (gt1 s) is much longer than the estimate t1 le 1 msbased on the abrupt stress-release model (see the arrow in Fig 5B forthe estimate of w1) The assumption of uncorrelated motor activity isthus likely to neglect important aspects of intracellular activity A morerealistic interpretation of the directed fluctuations (super-diffusion) ob-served in cells is that the probes are driven by collective and correlatedforce generation The collective action of force generators likely persist-ently stirs the cytoplasm on relatively long time scales so that thememory of force is not lost immediately after a single motor detachesfrom the cytoskeleton At longer time scales (w2p lt 01 Hz) thedisplacement PSDs show power-law behavior (Fig 7A inset) similar tothat of simple diffusion in a viscous medium wlang|ũ(w)|2rangno trap2kBT =aPrime(w)ordm wminus1 (22 31 33) However the slope of minus1 here does not implythermal diffusion because the probe is driven by nonthermal forces asproven by the breaking of the FDT Probes moving in cells lose theirvelocity memory on the same time scale (ge 1 s) The slope of minus1 is thusdue to the Markovian nature of probe displacements driven by cells

The high-frequency shear modulus of the fibroblast cellordm (minusiw)074

observed in Fig 5B is consistent with theoretical predictionsordm (minusiw)34

for networks of semiflexible polymers such as the cytoskeleton (46 47)Isolated fibroblast cells that are sparsely cultured on glass coverslipsacquire flattened shapes adhere well to the substrate and grow F-actinndashrich cross-linked cytoskeletal structures (48) which should re-semble pure actin networks in their response characteristics At lowfrequencies mechanical response is predominantly elastic (Gprime gtgt GPrime)but shows slow relaxation with small power-law exponents around02plusmn01 (3 4 49) A similar behavior has been observed in various ran-dom networks of polymers (7 18) and has been explained by nonaffinerelaxations (7) in glassy worm-like chains (10 50) These theoreticalmodels have also successfully explained the mechanical responses ofcells to forces that were applied from the outside (1 3 4 49) Our studyreveals that the same general physics appears to apply to the viscoelasticresponse inside of cytoskeleton-rich living cells

HeLa cells in confluent layers in contrast to well-spread fibroblastsare laterally confined and polarized They are roughly as tall as they arewide and the actin cytoskeleton is mostly confined to the membranecortex fromwhich our probe particleswere sufficiently separated (51 52)The response measured in different HeLa cells showed a surprisingly


narrow distribution especially at high frequencies We could clearlyshow that the power-law form of the shear modulus G(w) ~ G0(minusiw)

05

(Fig 5C) was distinct from the two regimesG(w)ordm (minusiw)02 plusmn 01 andG(w)ordm (minusiw)075 observed in surface-adherent fibroblasts (solid lines inFig 5B) The power-law dependence [G(w) ordm G0(minusiw)

05] is a well-known characteristic of glassy suspensions of colloids with nonstickysurfaces such as emulsions foams and swollenmicrogels (35 36) Thusthemechanics of the interior of living confluent epithelial cells that issparse in cytoskeletal network structure are likely better modeled as asoft glassy material (3 10) Note that a model for soft glassy materialswas also used earlier to explain the results of low-frequency cell mechanicsexperiments (3) In that case aG(w)ordm (minusiw)02 plusmn 01 power lawwas foundfor the slow response of the actin cytoskeleton caused by cross-linkconstraint release Here we report the expected three-fourths scalingG(w)ordm (minusiw)075 at higher frequencies when probing the actin cyto-skeleton in flattened fibroblasts This is a very different physical situa-tion from the one we describe here for the inside of the epithelial cellswhere we see in the same frequency range a quite different responseG(w) ordm G0(minusiw)

05 in the highly crowded glassy but not network-likecytoplasm

A power-law dependence G(w) ordm G0(minusiw)05 has also been ob-

served in networks of flexible polymers for which the Rouse model ap-plies (2 38 46) The concentration dependence of the shear modulusprefactor (G0) provides a way to distinguish between a flexible polymernetwork and colloidal glass For a Rouse network G0 is proportionalto the polymer concentration (38 46) In contrast close to the glasstransition of glass-forming materials viscoelastic moduli grow morerapidly than linear increase (53) To probe concentration dependencewe osmotically compressed or expanded cells that were initially in iso-tonic conditions and measured the shear modulus as a function of rel-ative intracellularmacromolecular concentration (Fig 5D) ThemeasuredG0 increased exponentially with themacromolecular concentration f inHeLa cells which is inconsistent with Rouse polymer networks (38 46)The more plausible explanation for this dependence is thus that themechanics inside confluent HeLa cells are predominantly determinedby the glassy cytosol that is tightly packed with biomolecular colloidsand approaches a jamming transition (54)

It has been speculated that crowding in cells containing high concen-trations of macromolecules (proteins RNA and organelles) affects cellmechanics (55 56) The exponential dependence of G0 on the macro-molecule concentration is reminiscent of that of strong glass formersmade of soft colloids (57) It was shown that the viscoelasticity ofextracted cytosol increases much more rapidly than the cytosol of theliving cell namely with a super-exponential dependence on f In addi-tion the absolute value ofG0 of extracted cytosol which lacks nonequi-librium stress fluctuations is far larger than that we see in living cells(see the asterisk in Fig 5D) This comparison leads us to speculate thatthe out-of-equilibrium activity in the cell might be responsible for boththe weaker viscoelastic response and the exponential (as opposed tosuper-exponential) concentration dependence Super-exponential de-pendency is a typical consequence of jamming in tightly packed rigidcolloids Frustration of local arrangements leads to inhomogeneous dis-tributions of residual forces and stresses in the sample which can onlyrelax through rare cooperative rearrangements Therefore the expo-nential increase of viscosity might be explained by active stirring incells (2) that can release and homogenize the locally frozen stressesActive stirring should also make the living HeLa cells respond morehomogeneously than expected explaining the surprisingly narrowdistribution of measured viscoelasticities (fig S4)

7 of 12


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The mechanical response of many biomaterials is not well under-stood even in thermodynamic equilibrium In cells mechanical re-sponses are often further modulated by cell activity Motor-generatedforces for instance stiffen cytoskeletal networks in vitro and in vivountil the network structure is broken by excessive stress (2 6 18 42)Cytoskeletal networks in cells are also thought to undergo sol-gel tran-sitions when self-organized contractile stresses drive cytoplasmic flows(58 59) Our feedback MR experiments in epithelial-like HeLa cellsrevealed that the living cytoplasm exhibits glass-like behavior Thisprovides a critical role for active stress generation in the cell The cyto-plasm can be fluidized by activity just as colloidal glasses and granularmaterials are under mechanical loads or external perturbations (54 60)It will be crucial to understand in detail how the nonthermal fluctua-tions in cells determine themechanical properties of living cells (22 24)A precise quantitative analysis of the out-of-equilibrium mechanics incells as can be performed with the approach presented here will be es-sential to further investigate the fascinating interplay between cellularmechanics and nonequilibrium force generation

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MATERIALS AND METHODSConventional AMR and PMRThe details of the experimental setup for conventionalMRusing opticaltraps and laser interferometry without feedback control are given in de-tail elsewhere (18) (see notes S1 and S3) Briefly the setup consists oftwo optical traps generated by lasers of two differentwavelengths (Fig 1A and B) In the AMR study one of the lasers (l = 1064 nm NdYVO4Compass Coherent) was operated as the drive laser and was deflectedby an AOD (model DTSX-400-1064 AA Opto-Electronic SA) whichwas sinusoidally driven by a voltage-controlled oscillator (model AADRF 40 Opto-Electronic SA) The other laser (l = 830 nmIQ1C140 Power Technology) was used as the probe laser remainedstationary during operation and was used to detect the position of thesame probe particle by back-focal-plane laser interferometry (14 15 28)Angular intensity shifts of the probe laser light due to the displacementof the probe particle were detected with a QPD (SPOT9D OSI Opto-electronicsAS) The output signal from theQPDwhichwas synchronouswith the oscillations was measured using a lock-in amplifier (Model7225 Signal Recovery) For the PMR experiments the oscillation ofthe drive laser was turned off and the thermal andor nonthermal spon-taneous motions of the probe particles were detected by laser interfer-ometry The output signal from the QPDwas digitized at 100 kHz withanAD board (AD24DSA board PCI-4462 National Instruments) andrecorded on a PC

3D feedbackSlow movements of probe particles were compensated for in threedimensions by feedback loops controlling the position of the samplestage (in the x and y directions) and the objective lens (in the z direc-tion) As shown in Fig 1A the in-plane (x and y) displacements of theprobe particle were measured using laser interferometry as describedabove The voltage signals for both the x and y displacements that wereoutput from the QPD were fed into two analog PID controllers(SIM960 100 kHz StanfordResearch Systems Inc) with the ldquoset pointrdquoinputs grounded using BNC grounding caps The controller outputswere connected to the 2Dpiezo stage [SFS-120XY(WA) SIGMAKOKI]on which the sample chamber was mounted For slow probe particlemovements this feedback loop repositioned the piezo stage such thatthe bead always remained near the center of the laser focus (Fig 8A)


Wedenote the piezo stage displacement asustage The voltages output bytheQPDandPID controllers [V(t) and e(t) respectively] were recordedon a PC with a 24-bit AD board (NI-PCI-4462 National Instruments)and were separately calibrated to uQPD and ustage respectively (as de-scribed in the next section) The total displacement of the probe beadin the sample chamber was then obtained as uequiv uQPD + ustage (Fig 1B)

Slow out-of-plane (z direction) movements of a probe particlewhich would affect the stiffness of the optical trap and the sensitivityof the QPD output to the in-plane (x and y) displacements werecompensated for by feedback control of the z position of the objectivelens (Fig 1A) Out-of-plane movements of a probe particle were de-tected by analyzing the patterns in the bead images that were projectedonto a CCD camera (WAT-902BWatec) Themicroscope images of thebeads were collected with an image-capture board (IMAQ PCI-1408National Instruments) and analyzed in real time by custom-writtensoftware (LabVIEW 86 National Instruments) First the center-of-mass position of each probe particle was obtained on the basis of thepattern in the corresponding particle image The radial brightnessdistribution of the centro-symmetric pattern was then calculatedas schematically shown in Fig 8B (bottom) Because of the diffrac-tion and interference of the LED illumination (white NVSW119BTAudio-Q) the image center was dark (bright) when the microscopefocus was slightly above (below) the particle The PID feedback forthe z position of the objective lens was set up such that brightness inthe center and at the fringe of the particle pattern remained equal Asoftware-based PID loop was created on a PC equipped with a signalgenerator (NI PXI 6221 National Instruments) that supplied a real-timevoltage signal that was fed to the z axis piezo scanner that held the objec-tive (PIFOC 72Z1 Physik Instrumente GmbH amp Co KG) The particlepattern was therefore maintained as shown in the leftmost images in thethree panels in Fig 8B when the z axis feedback was active

Calibration of laser interferometry displacement detectionand trap stiffness in cellsThe probe displacement uQPD(t) and the applied optical trapping forceF(t) were obtained by calibrating the output voltage V(t) of QPDs forboth the probe and the drive lasers For dilute samples with a refractiveindex nearly identical to that of the solvent (for example actin in watery

Fig 8 3D feedback (A) Typical voltage time series recorded by the QPD with orwithout feedback obtained with a probe embedded in a fibroblast cell Strongfluctuations of the QPD signal (top) are suppressed when feedback is active(bottom) (B) The radial distribution of probe image brightness is used forfeedback control of the z position of the objective lens Top images show probepatterns and bottom images schematically show typical intensity profiles in radialcoordinates z position is controlled such that the brightness at the fringe and inthe center of the pattern remains equal (dotted line)

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buffer or PAAm gels) the trap stiffness (the effective spring constant)k equiv F(t)uQPD and the calibration factor for the displacement responseof the QPD Cequiv uQPD(t)V(t) were obtained by analyzing the Brownianmotion of the probe beads in pure solvent (14 16)

This conventional calibrationmethod is however not valid in livingcells The refractive index in cells depends on particulars such as the celltype osmotic pressure of culture media location in the cell and the de-termination method Measured values for most animal cells are ncell ~138 plusmn 002 larger than that of water (nwater = 133) (61ndash66) Here thetrap stiffness kcell and the calibration factor Ccell in the cells were esti-mated using the following three-step procedure (i) With the laserfocused on the bead the piezo stage was oscillated with feedback at~05 Hz in the x or y direction to oscillate the probe bead sinusoidallythrough the focus of the low-power superimposed laser focus By re-cording the video images and theQPDoutput from the low-power lasersimultaneously we obtained Ccell by directly comparing the amplitudesofu(t) andV(t) This procedurewas carried out for every particle whichwe observed in cells and eliminates the errors due to for example in-homogeneities in cells and distributions of probe properties (ii) TheBrownian motion of probe particles was measured in an aqueous mix-ture containing 40 weight (wt ) glycerol which has a refractiveindex similar to that of cells (67) Probe particles were trapped at aheight above the glass-medium interface that was close to that of theexperiments in cells to reduce the effect of the spherical aberrationandor the effect of the optical interference near the coverslip surface(68ndash70) Using the known viscosity of the mixture of ~22 times 10minus3 Pasat 37degC the power spectrum of the Brownian motion provided thevalues for both kg and Cg in the solution (iii) Because both k and 1C(diffraction efficiency) are proportional to the effective polarizability ofthe probe an approximately inverse relation between k andC holds (seefig S7) (29 71) When the laser power and its profile are equal kcell cantherefore be obtained using kcell = kgCgCcell (see note S4) Although thetrap force calibration does not eliminate the effect of probe-sizedistribution polydispersity (lt5) is small and only marginally affectsour experiment (16)

Feedback AMR from measured voltage signalsIn the actual AMRmeasurements (Fig 2) the measured electric signalswere V(t) and e(t) neither of which is directly proportional to the totaldisplacement u(t) The lock-in amplifier converts these signals and pro-vides outputs langV ethwTHORNrang and langeethwTHORNrang Theoretically the intrinsic responsefunction a(w) and the shear elastic modulus G(w) can be derived fromeither of these outputs if the calibration factors B and C as well as thecorrection parameters kt and t are known Therefore there are two pro-cedures to perform AMR in the feedback mode based on the measuredsignals The first option is to use the output of the QPD langV ethwTHORNrang frac14languQPDethwTHORNrang=C Here we define the apparent response function AQPD(w)as the ratio of langucircQPD(w)rang to the applied force FethwTHORN By substitutingEq 4 into Eq 2 to eliminate ucircstage we obtain

AQPDethwTHORNequiv languQPDethwTHORNrangFethwTHORN frac14 1

1thorn baethwTHORN sdotiwtaethwTHORN1 iwt

eth10THORN

Thus a(w) is obtained from the measured langV ethwTHORNrang as

aethwTHORN frac14 ktbsdot

AQPDethwTHORN1 ktAQPDethwTHORN frac14 eth1 iwtTHORNClangV ethwTHORNrang

iwtethFethwTHORN ktClangV ethwTHORNrangTHORN eth11THORN

where the relation AQPDethwTHORN frac14 ClangV ethwTHORNrang=FethwTHORN has been applied


The alternative procedure is to use the output of the PID controllerlangeethwTHORNrang frac14 langustageethwTHORNrang=B instead of that of the QPD Substituting Eq 4into Eq 2 to eliminate ucircQPD we obtain the following expression

AstageethwTHORNequivlangustageethwTHORNrang

FethwTHORN frac14 11thorn baethwTHORN sdot

aethwTHORN1 iwt

eth12THORN

where Astage(w) is defined as the response function for the position ofthe feedback-controlled piezo stage Thus a(w) can be estimated fromlangeethwTHORNrang as


iwtAstage

1thorn iwtktAstagefrac14 eth1 iwtTHORNBlangeethwTHORNrang

FethwTHORN thorn iwtktBlangeethwTHORNrangeth13THORN

As can be seen from Eqs 5 10 and 12AFB =AQPD +Astage This relationis the direct consequence of u = uQPD + ustage In both cases the shearviscoelastic modulus can be obtained from a(w) via Eq 6

Feedback PMR far from equilibriumSuppose that a nonthermal fluctuating force f(t) ne 0 is acting on theprobe particle and that f(t) is not correlated with the thermal fluctu-ating force z(t) The Fourier transform of eq S1 in note S1 aftersetting L = 0 yields lang|ũ(w)|2rangtrap for the case in which no feedbackis applied (that is the conventional PMR) The PSD can then be ob-tained by summing the thermal and nonthermal fluctuations

langj~uethwTHORNj2rangtrap frac14 jAethwTHORNj2 langj~zethwTHORNj2rang thorn langj~f ethwTHORNj2rangn o

The trap-corrected

fluctuations lang|ũ(w)|2rangno trap on the other hand can be expressed as

langj~uethwTHORNj2rangno trap frac14 jaethwTHORNj2 langj~zethwTHORNj2rangthorn langj~f ethwTHORNj2rangn o

eth14THORN

If a(w) is independently measured with AMR then the effects of the

optical trap in lang|ũ(w)|2rangtrap can be corrected using lang|ũ(w)|2rangno trap =

|1 + kta(w)|2lang|ũ(w)|2rangtrap = lang|ũ(w)|2rangtrap|1 minus ktA(w)|

2 (18)For the feedback PMR experiments the PSDof u= uQPD + ustage can

bewritten as langj~uethwTHORNj2rangFB frac14 jAFBethwTHORNj2 langj~zethwTHORNj2rangthorn langj~f ethwTHORNj2rangn o

using

the response function AFB(w) defined in Eq 5 in Results The fluctua-tions in the absence of optical trapping and feedback can then be esti-mated by independentlymeasuringa(w) viaAMR aswell as kt and t asshown in the experimental section and note S1 The resulting equationlang|ũ(w)|2rangno trap = |1 + ba(w)|2lang|ũ(w)|2rangFB is apparently equal to Eq 9 inResults However this time the FDT is not satisfied because of the ad-ditional fluctuationgeneratedby thenonthermal forcelangj~uethwTHORNj2rangnonthermal frac14jaethwTHORNj2langj~f ethwTHORNj2rangAssuming that the FDT is valid for the thermal part ofthe fluctuations in Eq 14 that is jaethwTHORNj2langj~zethwTHORNj2rang frac14 2kBTaPrime=w non-

thermal fluctuationslang|ũ(w)|2rangnonthermal occurring in far-from-equilibriumsystems such as living cells can then be related to the violation of theFDT using

langj~uethwTHORNj2rangnonthermal frac14 langj~uethwTHORNj2rangno trap 2kBTaPrime

w

frac14 jaethwTHORNj2langj~f ethwTHORNj2rang eth15THORNGel preparationGels consisting of 175 wt PAAm were prepared with 30 wt acryl-amidebis-acrylamide mixed solution (3751 Wako Chemicals Inc)

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polystyrene beads (Polysciences Inc 2a = 2 mm diameter) and 0033ammonium persulfate The solution was thoroughly degassed in a vac-uum and polymerization was initiated by adding 005 tetramethy-lethylenediamine The solution was loaded into a glass chamber andpolymerized at room temperature for several hours while the chamberwas gently rotated This thermal equilibrium sample was measuredby feedback AMR with the following calibration values kp = 94 times10minus7 Nm kd = 58 times 10minus6 Nm and t = 38 times 10minus2 s (I = 2 sminus1 C = 41 times10minus7 mV B = 54 times 10minus6 mV)

G-actin was prepared from rabbit skeletal muscle according to stan-dard protocols (72) and was stored at minus80degC in G-buffer [2 mM tris-Cl02 mM CaCl2 05 mM DTT and 02 mM ATP (pH 75)] G-actinwas diluted into F-buffer [1mMNa2ATP 2mMHepes 1mMEGTA2 mM MgCl2 and 50 mM KCl (pH 75)] to initiate actin polymeriza-tion An F-actin solution (1 mgml) including polystyrene beads (Poly-sciences Inc 2a = 1 mm diameter) was directly infused into samplechambers Polymerization occurred at room temperature for about30 min The thermal equilibrium sample was then measured byfeedback PMR with the following calibration values kt = kp = 48 times10minus6 Nm and t = 0024 s

Cell preparationMouse fibroblasts (NIH-3T3) were cultivated on fibronectin-coatedglass-bottom petri dishes in Dulbeccorsquos modified Eaglersquos medium withglucose (1 mgml) penicillin (100 Uml) streptomycin (01 mgml)and 10 fetal bovine serum at 37degC When the population of cells ina flask was semiconfluent the medium was replaced by a CO2-independent culture medium (Gibco) without serum containingmelamine resin particles (diameter 680 nm refractive index 168 poly-dispersity lt5 microParticles GmbH) as probe beads The disheswere then incubated for several hours until the beads were phagocy-tosed Excess beads were removed by washing the dishes withphosphate-buffered saline Finally a CO2-independent culturemediumwithout probe beads was added to perform the measurements Themechanical properties weremeasured by feedbackMRwith the followingcalibration values kp = 12 times 10minus5 Nm and kd = 51 times 10minus5 Nm

HeLa cells were seeded on fibronectin-coated glass-bottom petridishes in the same culture medium as above except that this mediumincluded amphotericin B (250 mgml) When the population of cells ina flask was confluent the medium was replaced by a CO2-independentculture medium (Wako) The probe particles [melamine particlescoated with PEG (39) diameter 2a = 1 mm refractive index 168 poly-dispersity lt5 microParticles GmbH] were introduced into the HeLacells using a gene gun (PDS-1000He Bio-Rad) PEG coating has beenwidely used to study biomaterials with MR to generally passivate probesurfaces In aqueous environments hydrophilic PEG acts as a polymerbrush and prevents sticking to other objects or to other molecules Be-cause this barrier is merely nanometers thick it does not prevent themicrometer-sized beads from hydrodynamically coupling to theirsurroundings (16 18 73) The dishes were then placed in an incubatorfor at least several hours until the cell membranes recovered Excessbeads were removed by washing the dishes with phosphate-bufferedsaline Finally a CO2-independent culture medium without serum wasadded to perform the measurements Macromolecular concentrationsin cells were varied by osmotic compression (55) Intracellular macro-molecular concentrationwas systematically increased by adding sucroseto the culture media thus increasing the osmotic pressure The volumefraction of intracellular macromolecules was estimated from the ionicstrength of themedium according to Ponderrsquos law (74) Themechanical


properties were measured by feedbackMRwith the following trap stiff-nesses kp ~ 1 times 10minus6 Nm and kd ~ 1 times 10minus5 Nm The temperatureincrease due to the drive laser (lt100 mW) will be at most 1degndash2degC atthe beam focus (75) To characterize cells that are more sensitive thanthe ones used in this study or to be able to perform time-lapse experi-ments over extended periods of time the laser power can be furtherreduced to ~10 mW (causing less than a ~02degC temperature increase)In addition it has been shown that laser wavelengths of 830 or 970 nmcause the least nonthermal damage in cells (76)

For comparison cytoplasmic extracts of HeLa cells (CC-01-40-25Cilbiotech) were used As purchased they contain most of the intra-cellular contents except the membrane fraction and nuclear compo-nents When we used these extracts cytoskeleton polymerizationwas inhibited by 01 mM cytochalasin B and molecules smaller than3 kDa were removed by filtration to inhibit metabolic activity

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at httpadvancessciencemagorgcgicontentfull39e1700318DC1

note S1 Conventional AMR without feedback control

note S2 Response time measurement of the feedback system

note S3 Conventional PMR in thermal equilibrium

note S4 Inverse proportionality between k and C

fig S1 Experimental test of feedback performance

fig S2 Imaginary part of the response functions of a probe in a PAAm gel

fig S3 Apparent violation of the FDT under feedback

fig S4 Complex shear modulus in HeLa cells measured with feedback AMR

fig S5 PSD obtained under feedback PMR in actin solution

fig S6 Fluctuation and mechanical property in actinmyosin active gel

fig S7 Inversely proportional relationship between trap stiffness and displacement responseused for calibration in cells

REFERENCES AND NOTES1 D Mizuno R Bacabac C Tardin D Head C F Schmidt High-resolution probing of

cellular force transmission Phys Rev Lett 102 168102 (2009)

2 D Mizuno C Tardin C F Schmidt F C MacKintosh Nonequilibrium mechanics of activecytoskeletal networks Science 315 370ndash373 (2007)

3 B Fabry G N Maksym J P Butler M Glogauer D Navajas J J Fredberg Scaling themicrorheology of living cells Phys Rev Lett 87 148102 (2001)

4 P Bursac G Lenormand B Fabry M Oliver D A Weitz V Viasnoff J P ButlerJ J Fredberg Cytoskeletal remodelling and slow dynamics in the living cell Nat Mater 4557ndash561 (2005)

5 M L Gardel J H Shin F C MacKintosh L Mahadevan P Matsudaira D A WeitzElastic behavior of cross-linked and bundled actin networks Science 304 1301ndash1305(2004)

6 G H Koenderink Z Dogic F Nakamura P M Bendix F C MacKintosh J H HartwigT P Stossel D A Weitz An active biopolymer network controlled by molecular motorsProc Natl Acad Sci USA 106 15192ndash15197 (2009)

7 D A Head E Ikebe A Nakamasu P Zhang L G Villaruz S Kinoshita S Ando D MizunoHigh-frequency affine mechanics and nonaffine relaxation in a model cytoskeletonPhys Rev E 89 042711 (2014)

8 D A Head D Mizuno Local mechanical response in semiflexible polymer networkssubjected to an axisymmetric prestress Phys Rev E 88 022717 (2013)

9 T G Mason J Bibette D A Weitz Yielding and flow of monodisperse emulsionsJ Colloid Interface Sci 179 439ndash448 (1996)

10 P Sollich F Lequeux P Hebraud M E Cates Rheology of soft glassy materials Phys RevLett 78 2020ndash2023 (1997)

11 X Trepat L Deng S S An D Navajas D J Tschumperlin W T Gerthoffer J P ButlerJ J Fredberg Universal physical responses to stretch in the living cell Nature 447592ndash595 (2007)

12 R Krishnan C Y Park Y-C Lin J Mead R T Jaspers X Trepat G Lenormand D TambeA V Smolensky A H Knoll J P Butler J J Fredberg Reinforcement versus fluidization incytoskeletal mechanoresponsiveness PLOS ONE 4 e5486 (2009)

10 of 12


on March 19 2020


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13 T G Mason D A Weitz Optical measurements of frequency-dependent linearviscoelastic moduli of complex fluids Phys Rev Lett 74 1250ndash1253 (1995)

14 F Gittes C F Schmidt Signals and noise in micromechanical measurementsMethods Cell Biol 55 129ndash156 (1998)

15 T G Mason K Ganesan J H van Zanten D Wirtz S C Kuo Particle trackingmicrorheology of complex fluids Phys Rev Lett 79 3282ndash3285 (1997)

16 B Schnurr F Gittes F C MacKintosh C F Schmidt Determining microscopicviscoelasticity in flexible and semiflexible polymer networks from thermal fluctuationsMacromolecules 30 7781ndash7792 (1997)

17 J Xu A Palmer D Wirtz Rheology and microrheology of semiflexible polymer solutionsActin filament networks Macromolecules 31 6486ndash6492 (1998)

18 D Mizuno D A Head F C MacKintosh C F Schmidt Active and passivemicrorheology in equilibrium and nonequilibrium systems Macromolecules 417194ndash7202 (2008)

19 L A Hough H D Ou-Yang Correlated motions of two hydrodynamically coupledparticles confined in separate quadratic potential wells Phys Rev E 65 021906(2002)

20 D Mizuno Y Kimura R Hayakawa Electrophoretic microrheology in a dilute lamellarphase of a nonionic surfactant Phys Rev Lett 87 088104 (2001)

21 L D Landau E M Lifshieumltigraves L P Pitaevskiaeligi Statistical Physics (Pergamon Press ed 31980)

22 A W C Lau B D Hoffman A Davies J C Crocker T C Lubensky Microrheologystress fluctuations and active behavior of living cells Phys Rev Lett 91 198101(2003)

23 T Harada S-i Sasa Equality connecting energy dissipation with a violation of thefluctuation-response relation Phys Rev Lett 95 130602 (2005)

24 D A Head D Mizuno Nonlocal fluctuation correlations in active gels Phys Rev E 81041910 (2010)

25 B D Hoffman G Massiera K M Van Citters J C Crocker The consensus mechanics ofcultured mammalian cells Proc Natl Acad Sci USA 103 10259ndash10264 (2006)

26 Y Tseng T P Kole D Wirtz Micromechanical mapping of live cells by multiple-particle-tracking microrheology Biophys J 83 3162ndash3176 (2002)

27 R E Thompson D R Larson W W Webb Precise nanometer localization analysis forindividual fluorescent probes Biophys J 82 2775ndash2783 (2002)

28 M Guo A J Ehrlicher M H Jensen M Renz J R Moore R D GoldmanJ Lippincott-Schwartz F C Mackintosh D A Weitz Probing the stochastic motor-driven properties of the cytoplasm using force spectrum microscopy Cell 158822ndash832 (2014)

29 F Gittes C F Schmidt Interference model for back-focal-plane displacement detectionin optical tweezers Opt Lett 23 7ndash9 (1998)

30 C Storm J J Pastore F C MacKintosh T C Lubensky P A Janmey Nonlinear elasticityin biological gels Nature 435 191ndash194 (2005)

31 M Atakhorrami J I Sulkowska K M Addas G H Koenderink J X Tang A J LevineF C Mackintosh C F Schmidt Correlated fluctuations of microparticles in viscoelasticsolutions Quantitative measurement of material properties by microrheology in thepresence of optical traps Phys Rev E 73 061501 (2006)

32 M Guo A J Ehrlicher S Mahammad H Fabich M H Jensen J R Moore J J FredbergR D Goldman D A Weitz The role of vimentin intermediate filaments in cortical andcytoplasmic mechanics Biophys J 105 1562ndash1568 (2013)

33 M Roumlding M Guo D A Weitz M Rudemo A Saumlrkkauml Identifying directional persistencein intracellular particle motion using Hidden Markov Models Math Biosci 248 140ndash145(2014)

34 N Fakhri A D Wessel C Willms M Pasquali D R Klopfenstein F C MacKintoshC F Schmidt High-resolution mapping of intracellular fluctuations using carbonnanotubes Science 344 1031ndash1035 (2014)

35 R N Zia B J Landrum W B Russel A micro-mechanical study of coarsening andrheology of colloidal gels Cage building cage hopping and Smoluchowskirsquos ratchetJ Rheol 58 1121ndash1157 (2014)

36 A J Liu S Ramaswamy T G Mason H Gang D A Weitz Anomalous viscous loss inemulsions Phys Rev Lett 76 3017ndash3020 (1996)

37 F Gittes B Schnurr P D Olmsted F C MacKintosh C F Schmidt Microscopicviscoelasticity Shear moduli of soft materials determined from thermal fluctuationsPhys Rev Lett 79 3286ndash3289 (1997)

38 M Doi S F Edwards The Theory of Polymer Dynamics (Clarendon Press 1986)

39 J He J X Tang Surface adsorption and hopping cause probe-size-dependentmicrorheology of actin networks Phys Rev E 83 041902 (2011)

40 P Panorchan J S H Lee T P Kole Y Tseng D Wirtz Microrheology and ROCKsignaling of human endothelial cells embedded in a 3D matrix Biophys J 913499ndash3507 (2006)

41 A K Gupta S Wells Surface modified superparamagnetic nanoparticles for drugdelivery Preparation characterization and cytotoxicity studies IEEE Trans Nanobiosci 366ndash73 (2004)


42 F Schlosser F Rehfeldt C F Schmidt Force fluctuations in three-dimensional suspendedfibroblasts Philos Trans R Soc London Ser B 370 20140028 (2015)

43 H Turlier D A Fedosov B Audoly T Auth N S Gov C Sykes J-F Joanny G GompperT Betz Equilibrium physics breakdown reveals the active nature of red blood cellflickering Nat Phys 12 513ndash519 (2016)

44 T Toyota D A Head C F Schmidt D Mizuno Non-Gaussian athermal fluctuations inactive gels Soft Matter 7 3234ndash3239 (2011)

45 F C MacKintosh A J Levine Nonequilibrium mechanics and dynamics of motor-activated gels Phys Rev Lett 100 018104 (2008)

46 F Gittes F C MacKintosh Dynamic shear modulus of a semiflexible polymer networkPhys Rev E 58 R1241ndashR1244 (1998)

47 G H Koenderink M Atakhorrami F C MacKintosh C F Schmidt High-frequency stressrelaxation in semiflexible polymer solutions and networks Phys Rev Lett 96 138307(2006)

48 B Alberts A Johnson J Lewis D Morgan M Raff K Roberts P Walter Molecular Biologyof the Cell (Garland Publishing Inc ed 6 2015)

49 B Fabry G N Maksym J P Butler M Glogauer D Navajas N A Taback E J MilletJ J Fredberg Time scale and other invariants of integrative mechanical behavior in livingcells Phys Rev E 68 041914 (2003)

50 K Kroy J Glaser The glassy wormlike chain New J Phys 9 416 (2007)51 R W Mays K A Beck W J Nelson Organization and function of the cytoskeleton in

polarized epithelial cells A component of the protein sorting machinery Curr Opin CellBiol 6 16ndash24 (1994)

52 R Bacallao C Antony C Dotti E Karsenti E H Stelzer K Simons The subcellularorganization of Madin-Darby canine kidney-cells during the formation of a polarizedepithelium J Cell Biol 109 2817ndash2832 (1989)

53 G L Hunter E R Weeks The physics of the colloidal glass transition Rep Prog Phys 75066501 (2012)

54 B R Parry I V Surovtsev M T Cabeen C S OrsquoHem E R Dufresne C Jacobs-Wagner Thebacterial cytoplasm has glass-like properties and is fluidized by metabolic activity Cell156 183ndash194 (2014)

55 E H Zhou X Trepat C Y Park G Lenormand M N Oliver S M Mijailovich C HardinD A Weitz J P Butler J J Fredberg Universal behavior of the osmotically compressedcell and its analogy to the colloidal glass transition Proc Natl Acad Sci USA 10610632ndash10637 (2009)

56 G T Charras T J Mitchison L Mahadevan Animal cell hydraulics J Cell Sci 1223233ndash3241 (2009)

57 J Mattsson H M Wyss A Fernandez-Nieves K Miyazaki Z Hu D R ReichmanD A Weitz Soft colloids make strong glasses Nature 462 83ndash86 (2009)

58 Y Nishigami M Ichikawa T Kazama R Kobayashi T Shimmen K Yoshikawa S SonobeReconstruction of active regular motion in amoeba extract Dynamic cooperationbetween sol and gel states PLOS ONE 8 e70317 (2013)

59 T Nakagaki R D Guy Intelligent behaviors of amoeboid movement based on complexdynamics of soft matter Soft Matter 4 57ndash67 (2008)

60 A J Liu S R Nagel Nonlinear dynamics Jamming is not just cool any more Nature 39621ndash22 (1998)

61 K G Phillips S L Jacques O J T McCarty Measurement of single cell refractive indexdry mass volume and density using a transillumination microscope Phys Rev Lett 109118105 (2012)

62 N Lue G Popescu T Ikeda R R Dasari K Badizadegan M S Feld Live cell refractometryusing microfluidic devices Opt Lett 31 2759ndash2761 (2006)

63 M Sernetz A Thaer Immersion refractometry on living cells with the method ofrefractive index gradient Z Anal Chem 252 90ndash93 (1970)

64 R Barer S Joseph Refractometry of living cells 1 Basic principles Q J Microsc Sci 95399ndash423 (1954)

65 B Rappaz P Marquet E Cuche Y Emery C Depeursinge P J Magistretti Measurementof the integral refractive index and dynamic cell morphometry of living cells with digitalholographic microscopy Opt Express 13 9361ndash9373 (2005)

66 C L Curl C J Bellair T Harris B E Allman P J Harris A G Stewart A RobertsK A Nugent L M D Delbridge Refractive index measurement in viable cells usingquantitative phase-amplitude microscopy and confocal microscopy Cytometry A 65A88ndash92 (2005)

67 Glycerine Producers Association Physical Properties of Glycerine and Its Solutions(Glycerine Producersrsquo Association 1963)

68 K C Vermeulen G J L Wuite G J M Stienen C F Schmidt Optical trap stiffness in thepresence and absence of spherical aberrations Appl Opt 45 1812ndash1819 (2006)

69 K C Vermeulen J van Mameren G J M Stienen E J G Peterman G J L WuiteC F Schmidt Calibrating bead displacements in optical tweezers using acousto-opticdeflectors Rev Sci Instrum 77 013704 (2006)

70 K C Neuman S M Block Optical trapping Rev Sci Instrum 75 2787ndash2809 (2004)71 T Tlusty A Meller R Bar-Ziv Optical gradient forces of strongly localized fields Phys Rev

Lett 81 1738ndash1741 (1998)

11 of 12


72 L W Cunningham D W Frederiksen R B Vallee Structural and Contractile Proteins(Academic Press 1982)

73 A J Levine T C Lubensky Two-point microrheology and the electrostatic analogyPhys Rev E 65 011501 (2001)

74 H P Ting-Beall D Needham R M Hochmuth Volume and osmotic properties of humanneutrophils Blood 81 2774ndash2780 (1993)

75 E J G Peterman F Gittes C F Schmidt Laser-induced heating in optical trapsBiophys J 84 1308ndash1316 (2003)

76 K C Neuman E H Chadd G F Liou K Bergman S M Block Characterizationof photodamage to Escherichia coli in optical traps Biophys J 77 2856ndash2863(1999)

AcknowledgmentsFunding This work was supported by the Japan Society for the Promotion of ScienceKAKENHI (grant numbers JP15H01494 JP25127712 JP25103011 JP15H03710 andJP23684036 to DM grant numbers JP25870173 and JP15K05248 to TA and grant numberJP15J04464 to KN) KN was also supported by the Sasakawa Scientific Research Grant from


the Japan Science Society (grant number 26ndash219) CFS was supported by the DeutscheForschungsgemeinschaft Collaborative Research Center SFB 937 (Project A2) and theEuropean Research Council Advanced Grant FP7 ERC-2013-AdG Project 324 340528 Authorcontributions KN and MB conducted experiments and analysis HA provided ananalysis script DM designed the research DM KN MB TA and CFS discussed the resultsand wrote the manuscript Competing interests The authors declare that they have nocompeting interests Data and materials availability All data needed to evaluate theconclusions in the paper are present in the paper andor the Supplementary MaterialsAdditional data related to this paper may be requested from the authors

Submitted 31 January 2017Accepted 7 September 2017Published 29 September 2017101126sciadv1700318

Citation K Nishizawa M Bremerich H Ayade C F Schmidt T Ariga D Mizuno Feedback-tracking microrheology in living cells Sci Adv 3 e1700318 (2017)

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Feedback-tracking microrheology in living cellsKenji Nishizawa Marcel Bremerich Heev Ayade Christoph F Schmidt Takayuki Ariga and Daisuke Mizuno

DOI 101126sciadv1700318 (9) e17003183Sci Adv

ARTICLE TOOLS httpadvancessciencemagorgcontent39e1700318

MATERIALSSUPPLEMENTARY httpadvancessciencemagorgcontentsuppl2017092539e1700318DC1

REFERENCES

httpadvancessciencemagorgcontent39e1700318BIBLThis article cites 71 articles 9 of which you can access for free

PERMISSIONS httpwwwsciencemagorghelpreprints-and-permissions

Terms of ServiceUse of this article is subject to the

is a registered trademark of AAASScience AdvancesYork Avenue NW Washington DC 20005 The title (ISSN 2375-2548) is published by the American Association for the Advancement of Science 1200 NewScience Advances

License 40 (CC BY-NC)Science No claim to original US Government Works Distributed under a Creative Commons Attribution NonCommercial Copyright copy 2017 The Authors some rights reserved exclusive licensee American Association for the Advancement of

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must remain sufficiently smaller than the radius of either the beamwaist(~02 mm) or the probe particle (29) In metabolically active cells aweakly trapped probe typically rapidly moves out of this range duringobservations A strong optical trap could possiblymaintain the probe inthe trap but large stresses would build up locally as a result of internalactivities It would then be impossible to determine the linear responsemechanics of cells because cellular materials (cytoskeleton and cyto-plasm) exhibit strongly nonlinear viscoelasticities (5 7 30) Increasingthe radius of the beamwaist extends the detection range (29) but reducesthe laser diffraction efficiency and therefore the signal-to-noise ratio

Here we introduce a new method to solve this problem We imple-mented three-dimensional (3D) feedback to smoothly reposition thepiezo-actuated sample stage and the stage holding the objective lens tomaintain the probe bead in the detection range The total beadmotionwhich is the core observable for both AMR and PMR can be obtainedby summing the low-frequency stage movement and the high-frequencymotion of the probe relative to the laser focus as determined by inter-ferometry Feedback-implementedMR successfully solved not only theproblem of stably tracking probe beads in vigorously fluctuating envir-onments but also another problem with conventional optical trappingMR namely that the low-frequency response (and fluctuations) of theprobe are suppressed by the optical trapping potential (18 31) whichmakes it practically impossible tomeasure slowdynamics This problemdisappears in feedback-implemented MR Because the feedback-controlled stage follows the slow motion of probe beads and always re-tains the beads in the laser focus the bead motions at low frequenciesare not affected by the optical trapping force At higher frequenciesthough the optical trap exerts a time-varying force on the probe parti-cle In other words the system is already driven out of equilibrium bythe optical trapping force Via the feedback mechanism this drivingforce is correlated with the thermal forces We provide a theoreticalprocedure that can be used to apply the FDT to correct for these artifi-cial nonequilibrium effects due to feedback

After validating the procedure by performing control experiments inequilibrium materials [polyacrylamide (PAAm) gels and entangled ac-tin solutions] we applied feedback-enhanced AMR and PMR incultured eukaryotic cells At low frequencies we observed a significantviolation of the FDT The nonthermal fluctuations were several ordersof magnitude larger than the thermal fluctuations These fluctuationsreflect more or less correlated activity of motors at intermediate fre-quencies (24) Fluctuations typically become random at longer timescales such that themean squared displacement scales linearlywith time(32 33) Because the dominating driving forces for these random fluc-tuations are also nonthermal as demonstrated by the breaking of theFDT this motion is what has been referred to as ldquoactive diffusionrdquo orldquoactive stirringrdquo (22 32 34) On the other hand the high-frequency re-sponse in cells satisfies the FDT and exhibits a power-law behaviorconsistent with that predicted for semiflexible polymer gels in equilib-rium (7 18) and colloidal glasses (35 36) Through mechanicalexperiments with isolated adherent fibroblast cells (which wereflattened and in the flat part of the cell abundant in actin cytoskeleton)we found widely distributed elastic properties of such cells as expectedwhich displayed the well-known viscoelastic behavior Gordm (minusiw)34 athigh frequencies In further experiments we found that confluent sheetsof epithelium-like HeLa cells showed highly reproducible viscoelasticityand a power-law behavior Gordm (minusiw)12 that is distinct from that ex-hibited by semiflexible polymers but is typically observed for soft glassymaterials (35 36) We observed that the high-frequency shear modulusincreased exponentially with cytosol concentration To explain these


observations we suggest that one can model the HeLa cell epitheliumas an actively stirred glass where the heterogeneous dynamics that typ-ically appear in inactive glasses can continuously relax

RESULTSTheoretical foundation of feedback MRConventional AMR without feedback is explained in detail in note S1(18) to provide the background for MR under feedback control Theexperimental setup for feedback MR is shown in Fig 1A and its tech-nical details are given in Materials and Methods When the position ofthe sample stage is controlled by feedback the total displacement u ofthe probe in the sample is given by u = uQPD + ustage where uQPD is thedistance between the probe particle and the focus of the probe laser andustage is the displacement of the piezo stage as shown in Fig 1B Theoptical trapping force applied to the probe particle by both the driveand probe lasers is given by kdL exp(minusiwt) minus (kd + kp)uQPD where Lis the amplitude of the drive laser oscillation and kd and kp refer tothe trap stiffnesses of the drive and probe lasers respectively The Lan-gevin equation for the probe particle under feedback control is

kpuQPDethtTHORN thorn intt

infingetht tprimeTHORN _uethtprimeTHORNdtprimefrac14 kdethLeiwt uQPDethtTHORNTHORN thorn zethtTHORN thorn f ethtTHORN eth1THORN

where _uethtTHORNdenotes the velocity of the probe in the coordinate system ofthe sample medium that is stationary with respect to the feedback-controlled piezo stage but moves in the lab frame g(t) is the friction

Fig 1 Experimental setup for feedback MR (A) Optical trapndashbased AMR setupwith 3D feedback controls A probe particle in a cell is trapped and oscillated bythe drive laser (pink) QPD signals from the probe laser (cyan) are fed into PIDcontrollers which regulate piezo stage motion in x and y directions such thatthe trapped probe particle is maintained near the optical trap center Thebright-field image of the probe is simultaneously projected onto a charge-coupled device (CCD) camera [green thick line light-emitting diode (LED) lightsource is omitted] The image is analyzed on a personal computer (PC) to adjustthe z position of the objective lens through a z direction piezo scanner (B) Sche-matics for bead displacements and laser positions Positions of fixed probe laser(830 nm cyan) and acousto-optic deflector (AOD)ndashsteered drive laser (1064 nmpink) are shown by solid and dashed vertical lines respectively The drive laser isdrawn displaced to the right of the probe laser The pale particle indicates theprobe starting position

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eth10THORN










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The trap-corrected



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Lett 81 1738ndash1741 (1998)

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REFERENCES






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Lett 81 1738ndash1741 (1998)

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REFERENCES






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REFERENCES






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feedback-tracking microrheology in ... - science advancestin solutions], we applied...

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