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CHAPTER 30 Extracting the Mechanical Properties of Microtubules from Thermal Fluctuation Measurements on an Attached Tracer Particle Katja M. Taute * , Francesco Pampaloni , and Ernst-Ludwig Florin * * Center for Nonlinear Dynamics, University of Texas at Austin, Austin, Texas 78712 Cell Biology and Biophysics Unit, European Molecular Biology Laboratory, 69117 Heidelberg, Germany Abstract I. Introduction II. Rationale III. Materials IV. Methods A. Tubulin Polymerization B. Substrate Preparation C. Microscopy D. Experiments E. Data Analysis V. Discussion VI. Summary Acknowledgments References Abstract The mechanical properties of microtubules have been the subject of intense study during recent decades because of their importance to the many cell functions that they are involved in. Observations of microtubule thermal fluctuations have proven to be a reliable method to extract mechanical properties because they provide intrinsic calibration. While analysis of the entire microtubule shape is limited by METHODS IN CELL BIOLOGY, VOL. 95 978-0-12-374815-7 Copyright Ó 2010 Elsevier Inc. All rights reserved. 601 DOI: 10.1016/S0091-679X(10)95030-9
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Page 1: Chapter 30 - Extracting the Mechanical Properties of ...

CHAPTER 30

Extracting the Mechanical Propertiesof Microtubules from Thermal FluctuationMeasurements on an Attached TracerParticle

Katja M. Taute*, Francesco Pampaloni†, and Ernst-LudwigFlorin**Center for Nonlinear Dynamics, University of Texas at Austin, Austin, Texas 78712

†Cell Biology and Biophysics Unit, European Molecular Biology Laboratory, 69117 Heidelberg, Germany

AbstractI. IntroductionII. RationaleIII. MaterialsIV. Methods

A. Tubulin PolymerizationB. Substrate PreparationC. MicroscopyD. ExperimentsE. Data Analysis

V. DiscussionVI. Summary

AcknowledgmentsReferences

Abstract

The mechanical properties of microtubules have been the subject of intense studyduring recent decades because of their importance to the many cell functions thatthey are involved in. Observations of microtubule thermal fluctuations have provento be a reliable method to extract mechanical properties because they provideintrinsic calibration. While analysis of the entire microtubule shape is limited by

METHODS IN CELL BIOLOGY, VOL. 95 978-0-12-374815-7Copyright � 2010 Elsevier Inc. All rights reserved. 601 DOI: 10.1016/S0091-679X(10)95030-9

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spatial resolution to very long microtubules, we show that even for short micro-tubules, one can obtain high-precision fluctuation information from one point alongthe contour by the use of tracer particles attached to the microtubule. The informa-tion is sufficient to extract key mechanical parameters such as stiffness and firstmode relaxation time. In this article, we discuss sample preparation as well asmeasurements and data analysis.

I. Introduction

Microtubules are involved in a wide range of cell functions, many of whichimpose requirements on the microtubules’ mechanical properties. Long axonalmicrotubules need to be rigid enough to serve as reasonably straight highways formotor proteins. Microtubules in the cytoplasm need to be not only flexible to notbreak under the strong, localized bending forces applied by molecular motors, butalso stiff to maintain directional cues. Short microtubules in the mitotic and meioticspindle need to be flexible to efficiently sample space when searching for chromo-somes to capture, and yet maintain sufficient rigidity to be able to facilitate theircongression on the spindle equator. Hence it is not surprising that, during the lastdecades, considerable effort has been invested into determining the mechanicalproperties of these truly multifunctional filaments.

Measurements on microtubule mechanics have been performed using both activeand passive methods. Active methods involve applying a controlled force to thefilament and measuring its response. Examples include bending by hydrodynamicflow (Dye et al., 1993) or by optical traps (Kurachi et al., 1995) and indentation byatomic force microscopy (AFM) (Kis et al., 2002). These methods require a well-calibrated manipulation device such as an AFM cantilever, an optical trap, or a flowof known strength in addition to a technique for high-precision measurements of theinflicted deformation.

Being microscopic objects, however, microtubules are subject to considerablethermal fluctuations which can act as sources of noise in active measurements.Solvent molecules in the surrounding solution are in constant Brownian motionand inflict random forces on the filament. As a result of these forces, the shape of amicrotubule is not static but fluctuates constantly. In order to limit the influence ofthese fluctuations on an active measurement, it is often necessary to inflict deforma-tions much stronger than those induced by the thermal forces. Very strong deforma-tions may however restrict the measurements to ranges of very high force.

Passive methods for mechanical measurements in turn make use of the presence ofthermal fluctuations as a well-calibrated source of bending forces. The thermal forcesthemselves replace the manipulation device used in active methods, and the resultingchanges in the filament’s shape are measured. The bending fluctuations of a filamentare best described in terms of their modes, which are basic shapes that the overallshape can be decomposed into. Figure 1 shows an example of a random microtubuleshape and its decomposition into modes. Although the thermal forces are stochastic,their statistical properties are known exactly as they can be described by statisticalmechanics. So while a single measurement of the thermally induced bending of amicrotubule is meaningless, a large set of such measurements can capture the fullstatistical range of bending fluctuations. Combined with biopolymer models, one can

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then extract very accurate information on the mechanical properties of the filament.The stronger the average deformation in shape produced by thermal forces, the softerthe filament must be. Additional information is encoded in the speed of the fluctua-tions. The stiffer a microtubule, the faster it will relax from a bent shape to its straightequilibrium shape. In turn, viscous drag, caused by hydrodynamic friction with thesurrounding buffer solution, will slow down the relaxation. Thus, the timescales ofthe fluctuations, called relaxation times, give information on both the stiffness andthe friction.

If the microtubule is imaged directly for measuring its shape (Gittes et al., 1993;Janson and Dogterom, 2004; Mizushima-Sugano et al., 1983), the spatial precisionobtained is generally on the order of tens of nanometers. Because the width of thebending fluctuations decreases with decreasing contour length, shape analysis tech-niques are limited by resolution to microtubules that are tens of micrometers long.Cell functions such as mitosis and meiosis however often involve much shortermicrotubules. As various studies have indicated that the stiffness of microtubules isdependent on their length (Kis et al., 2002; Kurachi et al., 1995; Pampaloni et al.,2006; Taute et al., 2008), there is an urgent need for high-precision techniques thatallow for passive measurements on short microtubules.

While shape analysis techniques yield the most complete picture of the micro-tubule’s fluctuations, it is actually possible to extract key mechanical parameters withfar less information. Bending fluctuation measurements from a single point along thecontour are actually sufficient to extract relaxation time and stiffness at least for thedominant mode. Using an attached fluorescent bead as a tracer particle, positioninformation for one point on the contour can be obtained with a precision in therange of a few nanometers. Utilizing this method, microtubules of lengths down to1 µm become accessible for measurements of mechanical properties.

Mode(B) (C)

(A)

1

2

3

Fig. 1 An example shape of a microtubule free in solution (A) and its decomposition (C) intocontributions from the first three modes (B). The full shape of the microtubule (black solid line) is thesum of the contributions from the individual modes. The first mode (dashed line) has the largest amplitude.(See Plate no. 45 in the Color Plate Section.)

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In this article, we describe and discuss the application of this technique for theexample of taxol-stabilized microtubules. The procedures can however easily bemodified for other sample preparations.

II. Rationale

Figure 2 shows a schematic of our assay. The microtubule is grafted to a substrateat one end while the other end is free to move. A fluorescent bead is bound at oneposition along the contour, and its position serves as a tracer of the thermal fluctua-tions of the microtubule which are observed from below. The width of the positionfluctuations of the bead then presents a measure of the stiffness of the microtubule.

Grafting the microtubule, rather than observing it free in solution, provides a fixedreference frame so that position measurements for the tracer always report micro-tubule fluctuations and not diffusion in the solution. We achieve covalent bonding ofmicrotubules to a substrate by using a standard thiol bonding procedure on modifiedgold surfaces (Frey and Corn, 1996; Patel et al., 1997).

Using a raised substrate ensures that the microtubule can fluctuate freely in threedimensions. Proximity to a surface would increase the viscous drag that is acting onthe microtubule (Hunt et al., 1994) and would hence modify the dynamics of themicrotubule thermal fluctuations and make the interpretation of measurement datamore complex. To avoid this, we use an electron microscopy gold grid with parallelbars as a substrate. The free end of a microtubule projecting out from the top of a baris then at least 15 µm away from either surface of the sample chamber.

Coverslips

Goldsubstrate

Fluctuations

Objectivelens

Fig. 2 Schematic of the assay. One end of the microtubule is covalently grafted on the gold substrate.The free end fluctuates in three dimensions and is at least 15 µm away from either coverslip. A two-dimensional projection of the fluctuations is observed from below. The attached bead is used as a tracer ofthe position fluctuations. Lengths are measured from the edge of the substrate along the microtubulecontour. La denotes the length from the edge to the attached bead, and L is the length to the free tip. (SeePlate no. 46 in the Color Plate Section.)

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Binding of beads to microtubules can easily be achieved by using avidin-coatedbeads in conjunction with biotinilated microtubules. The size of the fluorescentbeads must be chosen so as to balance requirements for high fluorescence intensityand minimal additional hydrodynamic drag. If the drag coefficient of the bead,�bead=6��r, becomes larger than that of the microtubule, it will dominate the time-scale of the fluctuations. Here r is the radius of the bead and � the viscosity of thesurrounding solution. We find that 200 nm beads provide a satisfactory compromise(Taute et al., 2008).

The nature of the fluctuations of a semiflexible filament grafted at one end hasbeen described in detail by Wiggins et al. (1998), Benetatos and Frey (2003), andothers. The starting point most commonly used is the wormlike chain model (Kratkyand Porod, 1949; Saitô et al., 1967). In short, the filament undergoes transversefluctuations in two dimensions, in the focal plane and perpendicular to it. In our case,we will concentrate on the transverse fluctuations in the focal plane. Figure 3Ashows an example of a bending shape and its mode decomposition for a microtubulegrafted at one end.

The mean square displacement (MSD) of the fluctuations in this direction is givenby the sum of the MSDs of the individual modes. As shown in Fig. 3B, the MSDof each mode first increases with time and then plateaus at a saturation levelwhich equals twice the position variance of the mode (Granek, 1997; Kroy andFrey, 1997):

0 0.2 0.4 0.6 0.8 1s = La/L

Tra

nsve

rse

posi

tion

Full shape1st mode2nd mode3rd mode

Time

MS

D

τ2

τ2τ3

τ1

2V2

2V1

Full MSD1st mode MSD2nd mode MSD3rd mode MSD

2V3

2V2

(A) (B)

Fig. 3 Bending modes of a grafted microtubule. (A) An example of a mode decomposition for thetransverse fluctuations, given as a function of the normalized position along the contour, s= La/L, wheres= 0 corresponds to the position of the grafted end and s= 1 denotes the free tip of the microtubule, isshown. The full shape is given by the sum of the contributions from all the individual modes. For s> 0.5,the first mode presents a fair approximation of the full shape. If many measurements of bending amplitudesare obtained from one position s along the contour, the mean square displacement (MSD) can becomputed. (B) The theoretical MSD according to Eq. (1) for s= 0.6 is shown. V1 and V2 represent thetransverse position variance of the first and second modes, respectively. As V1/V2� 24, the first modedominates the amplitude, and the higher mode contributions are negligible. If the viscous drag is assumedto be the same for both modes, �1 is � 40 times larger than �2. (See Plate no. 47 in the Color PlateSection.)

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MSDðtÞ ¼Xn

2L3

lpq4nð1� e�t=�nÞW 2

n ðLa=LÞ ð1Þ

where La and L refer to the microtubule contour length from the edge of the substrateto the bead and the free end, respectively. The persistence length, lp, is the char-acteristic decay length of the tangent–tangent correlations along the filament contourand is directly proportional to the bending stiffness, �, via the temperature (Kratkyand Porod, 1949):

� ¼ lpkBT ð2ÞTheWn(La/L) are the spatial modes of the fluctuations which are biharmonic functions

with mode numbers of q1�1.875, q2� 4.694, q3� 7.855, and qn� (n�1/2)� for largen, for the given boundary conditions (Wiggins et al., 1998). The sum is taken over allmodes.

The relaxation times of the individual modes are given by the following equation(Aragón and Pecora, 1985):

�n ¼ �L4

�q4nð3Þ

The term � is the drag per unit length acting on the filament. As the relaxationtimes scale with the inverse fourth power of the mode number, the higher modes inEq. (1) saturate very quickly (compare Fig. 3B). Because the mode amplitudesfollow the same scaling, the first mode strongly dominates the amplitudes of thefluctuations as long as the position of the bead is closer to the free end than to thesubstrate, that is, La/L> 0.5. For smaller ratios of La/L, the W2

n(La/L) contributionfrom the second mode becomes increasingly important. Figure 3B shows the relativecontributions for an example ratio of s= La/L= 0.6.

In summary, as long as La/L> 0.5, the transverse MSD is well approximated bythe first mode. The saturation level, which corresponds to 2 times the variance of thefirst mode fluctuations, is determined by the stiffness of the filament. The timescaleof saturation, the relaxation time � , is determined by an interplay of stiffness andfriction. The saturation level and relaxation time hence present two independentways of assessing the mechanical properties of the filament.

III. Materials

Table I lists the reagents required for the assay described in this article. The mainbuffer solution used is BRB80 (80mM PIPES, 1mM EGTA, 2mM MgCl2, pH 6.8,with KOH). Only deionized water with a minimum resistivity of 18M�cm wasused.

The following aliquots were prepared:

• Guanosine triphosphate (GTP) is dissolved in water at 100mM and aliquoted in1 µl units for storage at –80°C.

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• Experiment-sized aliquots of unlabeled and biotinilated tubulin are prepared bysuspending the lyophilized protein in ice-cold 1mM GTP/BRB80 and snap-freezing them in liquid nitrogen for storage at –80°C. Volumes andconcentrations are 10 µl at 10mg/ml for the unlabeled and 2 µl at 5mg/ml forthe biotinilated tubulin.

• Paclitaxel (taxol) is dissolved in anhydrous dimethyl sulfoxide (DMSO) at 10mMand frozen in 1 µl units in 1.5ml centrifuge tubes for storage at –80°C.

IV. Methods

A. Tubulin Polymerization

A 1mM solution of GTP in BRB80 is prepared by 100-fold dilution of 1 GTPaliquot in BRB80 and stored on ice. For the polymerization of microtubules, onealiquot of each of the unlabeled and the biotinilated tubulin is quickly thawed at37°C and then stored on ice. One vial of 20 µg of rhodamine tubulin is dissolved in4 µl of ice-cold 1mM GTP/BRB80, and the three solutions mixed to give final ratiosof 10:2:1 for the unlabeled, rhodamine-labeled, and biotinilated tubulin. The solutionis then diluted with ice-cold 1mM GTP/BRB80 to give a total tubulin concentrationaround 4mg/ml and a total volume of 30 µl. At this point, the solution is subjected toa cold spin in order to remove aggregates of denatured protein. Centrifugation isperformed using a precooled TLA-100 rotor in a Beckman TL100 ultracentrifuge at

Table IList of Reagents Needed

Name Supplier Product code

Unlabeled tubulin C TL238Rhodamine tubulin C TL331MBiotinilated tubulin C T333PIPES free acid SA P6757EGTA SA E43781M MgCl2 SA M1028KOH pellets F P250-500GTP SA G5884Taxol SA T1912DMSO SA 276855Ammonia 30% F A669S500Hydrogen peroxide 30% F H32510011-MUA SA 450561NHS SA 130672EDC SA E7750D-Glucose F D16500Glucose oxidase SA G2133Catalase SA C40Hemoglobin SA H2500

C, Cytoskeleton Inc., Denver, CO; SA, Sigma-Aldrich Inc., St. Louis, MO; F, Fisher Scientific, HanoverPark, IL; DMSO, dimethyl sulfoxide; GTP, guanosine triphosphate; 11-MUA, 11-mercapto undecanoic acid;NHS, N-hydroxysuccinimide; EDC, N-(3-Dimethylaminopropyl)-N0-ethylcarbodiimide hydrochloride.

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2°C, 350,000 g for 5min. The supernatant is transferred to a 37°C water bath for 20–40min of polymerization.

Meanwhile, the rotor was prewarmed to 37°C. To prepare a 20 µM solution oftaxol in BRB80, 500 µl of BRB80 as well as an aliquot of taxol is warmed at 37°Cfor a few minutes, before being mixed vigorously by pipetting and vortexing. A shortspin of 2min at 5,000 g is performed in a small tabletop centrifuge to removepotential taxol precipitate, and the supernatant retained at room temperature.

In order to prevent competitive binding of free tubulin to the substrate, themicrotubules are pelleted and resuspended after polymerization. Centrifugation isperformed with the same equipment as before, but for 10min using settings of 35°Cand 110,000 g. The supernatant is carefully removed and the pellet washed withwarm BRB80 before being resuspended in 50 µl of the taxol/BRB80 solution. Pipettips need to be cut for all solutions containing microtubules to avoid breaking themby shear.

B. Substrate Preparation

Parallel bar gold grids for electron microscopy can be acquired commercially(G1016A, Ted Pella Inc., Redding, CA). These grids have an outer diameter of 3.05mm, are 14–18 µm thick, and feature parallel bars with a pitch of �60 µm and awidth of �12 µm. The grids are glued onto round 15mm coverslips (Menzel Gläser,Germany) using a chemically inert silicone glue (Elastosil N10, Wacker Chemie,Germany) and left to dry for at least several hours. Prior to use, the grids are cleanedby immersing them in a solution of 5% ammonia and 5% hydrogen peroxide in waterfor 15min and successively rinsing them with copious amounts of water and ethanol.They are then immersed overnight in a freshly prepared 5mM solution of 11-mercapto undecanoic acid (11-MUA) in ethanol to facilitate formation of a mono-layer on the gold surface. We perform all these procedures in custom-built teflonwells with a volume of �1ml.

The next day, the gold grids are rinsed with copious amounts of ethanol, driedwith pure nitrogen gas, and quickly assembled into a flow chamber. The flowchamber consists of a 24� 32mm coverslip (Menzel Gläser, Germany) with spacersmade of parafilm that the round coverslip is inverted onto, with the grid facing down(see Fig. 4). The grid is oriented such that its bars are perpendicular to the direction

Goldgrid

Sideview

Fig. 4 Sample chamber in top view and side view. Slices of parafilm serve as a spacer between arectangular and a round coverslip. A gold grid is glued to the inner side of the round coverslip. (See Plateno. 48 in the Color Plate Section.)

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of flow in the chamber. The sample chamber is heated briefly on a hot plate andcompressed slightly for the parafilm to seal. The volume of the sample chamber is onthe order of �10 µl, and solutions can easily be exchanged by blotting with labtissue. In order to avoid bubble formation, it can be useful to first fill the chamberwith ethanol before flushing it with BRB80.

The monolayer on the gold surface is then activated by flushing in a freshlyprepared solution of 100mM N-hydroxysuccinimide (NHS) and 100mMN-(3-Dimethylaminopropyl)-N0-ethylcarbodiimide hydrochloride (EDC) in BRB80.In order to prevent the sample chamber from drying out, it should be stored in ahumid environment between sample preparation steps, for example inside a petri dishcontaining some lab tissues soaked in water. After 30min of activation, the chamberneeds to be thoroughly flushed with BRB80 before 20–40µl of a 10–100� dilutedmicrotubule solution is flushed in. Dilution of the microtubule solution is done intaxol/BRB80. The microtubule solution is incubated for 30–60min in order to allowtime for covalent bond formation between microtubules and the gold substrate. In themeantime, a 400� diluted solution of 200 nm neutravidine-coated, yellow-greenfluorescent beads (F-8774, Molecular Probes, Invitrogen, Carlsbad, CA) in BRB80is sonicated for 20min to be flushed in after microtubule incubation. The beads arenow incubated for 30–60min with the microtubules in order to facilitate bond forma-tion between the biotin on the microtubules and the neutravidine on the beads.

After the incubation, the flow chamber is flushed with BRB80 several times toremove all unbound microtubules and beads. As fluorescent microtubules are subjectto phototoxicity and can break under illumination (Guo et al., 2006; Vigers et al.,1988), an oxygen scavenging system needs to be employed. We use a solutionconsisting of 0.3mg/ml glucose oxidase, 0.3mg/ml catalase, 30mM glucose, andhemoglobin diluted 100 times from a saturated solution in BRB80 to flush into thesample chamber before sealing it with valap (vaseline, lanolin, and paraffin at 1:1:1w/w/w). The sample is now mounted with vacuum grease on a custom-built sampleholder for an inverted widefield fluorescence microscope.

C. Microscopy

All experiments are performed on a Zeiss Axiovert 10 inverted microscopemodified for high mechanical stability. In order to minimize movement of the sampleand the objective lens relative to one another, the original revolver and stage wereremoved and replaced by a solid metal stage that holds both the sample and theobjective lens. The focus can be fine-tuned within a range of 100 µm using a piezopositioner (F-100, Mad City Labs, Madison, WI) which is attached to our stage andmoves the objective lens. As there are tens of micrometers of solution between thecoverslip and the microtubule to be observed, a water rather than oil immersion lens(UplanSApo 60� W, Olympus, Center Valley, PA) has to be used in order to limitoptical aberrations (Hell et al., 1993). Fluorescence excitation is provided by anX-cite 120 light source (EXFO Life Sciences, Ontario, Canada). The filter sets usedare optimized for yellow-green and rhodamine fluorescence, respectively. Whilewith the yellow-green filter set only the beads were excited, the rhodamine filterset allowed for simultaneous observation of beads and microtubules. In order tominimize the number of optical elements in the light path, our high-sensitivity CCDcamera (Sensicam QE, PCO, Germany) was placed at the bottom port.

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As the mechanical stability of the setup is paramount to the success of thistechnique, it is advisable to test it before attempting experiments. A simple way oftesting the mechanical stability and thus the resolution that can be achieved is toperform long-term imaging and subsequent particle tracking of immobilized fluor-escent beads. Immobilization can easily be achieved by drying a dilute solution ofbeads onto a clean coverslip or by mixing them into a hot liquid agarose solution(concentration > 1% w/w), which is then flowed into a sample chamber for solidi-fication. The changes in position of the immobilized beads report on movement ofthe sample relative to the objective lens. If several thousand frames are recorded overthe course of several minutes, the data should give a good estimate of the noiseencountered during experiments. After correcting for linear drift, the standard devia-tion of the position measurements for any one bead should not be more than 10 nmfor our technique to yield quantitative results for the entire range of microtubulelengths. However, a precision of up to a few tens of nanometers is still sufficient toobtain qualitative information on long microtubules.

D. Experiments

Because the beads stick to the gold bars, it is easy to recognize the structure of thegrid using the yellow-green filter set. Beads attached to fluctuating microtubules cannow be identified by looking for beads that are moving between the bars. Using therhodamine filter, the geometry of the arrangement can then be inspected, andindividual grafted microtubules with a single bead bound to their contour can beselected. A few hundred frames are taken using this filter to allow for lengthmeasurements. Figure 5 shows a snapshot of such a microtubule taken in therhodamine filter.

Then, using the yellow-green filter, at least 104 images of only the bead’sfluctuations are recorded. Limiting the region of interest to the range of the bead’sfluctuations allows for frame rates above video rate. As fluctuations become muchmore rapid with decreasing microtubule length, integration times should be adjustedsuch as to capture the dynamics, while keeping in mind the trade-off betweentemporal and spatial resolution. Short integration times, while increasing framerates and temporal resolution, decrease the number of photons captured per imageand can hence decrease the attainable spatial resolution. In practice, integration timesdown to 15 ms yield enough photons to achieve a good resolution in the regime ofshort microtubules. For longer microtubules, longer integration times can help to

10 µm

Fig. 5 Fluorescence image of a grafted microtubule with an attached bead as seen in the rhodaminefilter. The gold bar is on the left side.

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obtain enough signal to localize the bead even when its fluctuations temporarilymove it away from the focal plane. One well-prepared sample typically yields aboutfive measurements.

E. Data Analysis

Two-dimensional (2D) position data can be obtained from the time series usingany high-precision single particle tracking algorithm. Typically, a Gaussian profile isfitted to the bead’s point-spread function to obtain its center for every image.Theoretical limits for the precision of such a procedure are discussed in Thompsonet al. (2002). In practice, however, the long-term mechanical stability of the setup isthe limiting factor. With our current setup, we achieve a precision of �3 nm.

The position data are then rotated to allow for an easy decomposition of thetransverse and the much narrower longitudinal position fluctuations. This can beaccomplished iteratively by repeated fitting of a straight line into the 2D positiondistribution and subsequent rotation of the coordinate system such that the slopebecomes zero. An example of position data obtained in this way in shown in Fig. 6.

In theory, the stiffness of the microtubule can be extracted from the width of thetransverse fluctuations (Gholami et al., 2006). In reality, however, the measuredwidth may be different from the true width due to low-pass filtering, incomplete

−600 −400 −200 0 200 400 600

−200

−100

0

100

Transverse position (nm)

long

.pos

ition

(nm

)

−500 0 5000

200

400

600

800

Transverse position (nm)

Cou

nts

−150 −100 −50 00

500

1000

1500

Longitudinal position (nm)

Cou

nts

(A)

(B) (C)

Fig. 6 Position data from a time series of 15,000 frames of a bead attached to the end of a microtubuleof length L= 3.9 µm. (A) Position distribution. The transverse fluctuations have a Gaussian distribution(B) and are much wider than the asymmetric longitudinal fluctuations (C).

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sampling, or localization errors. While the first effect can often be corrected for in thedata analysis, the latter two effects must be minimized by good experimentalpractices.

Sampling not only refers to the total number of data points taken, but must takeinto account how many of the data points are correlated. The relaxation time of thelowest mode provides the correlation time of the time series. We find that in order toobtain a sufficient number of independent data points, the time series needs to cover�103 relaxation times. As these relaxation times are not known a priori, the lengthof the time series required can be difficult to estimate and is best established byexperimentation.

Low-pass filtering is a result of the finite integration time used to obtain animage. The bead’s position will be slightly blurred because it is moving duringthe integration. The measured center of its position will always be biasedtoward the maximum of the position distribution, thereby leading to a mea-sured width of the distribution that is smaller than the true width (Wong andHalvorsen, 2006). The smaller the relaxation time compared to the integrationtime, the larger the effect. If the relaxation time is known, however, the low-pass filtering effect can be corrected for. In practice, this is only necessary inorder to obtain quantitative information on microtubules as short as a fewmicrometers.

One way of obtaining dynamical information such as relaxation times is tocompute the MSD as a function of time. Error bars on averages of finite timeseries with correlations can be difficult to determine but are often necessary inorder to be able to perform accurate fits. A recipe for computing error estimates hasbeen provided by Flyvbjerg and Petersen (1989). As shown in Eq. (1) and Fig. 3,the MSD is the sum of the MSDs for all the different modes, each of whichsaturates exponentially. Because relaxation times decrease rapidly with increasingmode number, the higher modes will be subject to much stronger low-pass filteringand hence their measured amplitudes will be even smaller than their true ampli-tudes. If the frame rate is optimized for obtaining information on the first mode, thehigher modes will generally already be saturated at the first MSD data point andhence at most represent a constant offset to the MSD data (compare Fig. 3).Equation (1) should then reduce to

MSDðtÞ ¼ offsetþ 2V1ð1� e�t=�Þ ð4Þ

where V1= (L3/lpq14)W1

2(La/L) represents the position variance of the first mode.Because now only one mode can be seen, �1 has been replaced by �. A fit ofEq. (4) is generally sufficient whenever the relaxation time is many timeslarger than the integration time and low-pass filtering does not play a role. Fortaxol-stabilized microtubules measured at video rate, this is generally the caseif they have a length of �10 µm or more. While Eq. (4) still allows forqualitative comparison of shorter microtubules of equal length, quantitativeresults in this length regime require a low-pass filtering correction to be takeninto account.

Low-pass filtering affects both the amplitude of the measured MSD and its overallshape. In order to extract true first mode relaxation times and amplitudes, we there-fore fit a corrected expression of the type given as follows:

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MSDðtÞ ¼ offsetþ 2V1 2�2

I2e�I=� � 1þ I

� �� sinh I

2�

� �I2�

!2

e�t=�

0@

1A ð5Þ

The expression in large brackets corresponds to (1�e�t/�) after low-pass filteringby fluorescence imaging with integration time I. Taking the limit I! 0, we retrieve(1�e�t/�). The derivation of the expression involves convolution of the true fluctua-tions with a function F(t,I)= 1 for |t|< I/2 and F(t, I)= 0 otherwise. The calculationis best performed in Fourier space and is an extension of the framework presented byWong and Halvorsen (Wong and Halvorsen, 2006) in the context of optical tweezers.Figure 7 shows an example of MSD data obtained as described for a situation wherethe relaxation time � is �2.5 times larger than the integration time I. Omitting low-pass filtering effects in the fit already yields a �10% difference in the resulting fitparameters. The smaller � /I, the larger the resulting effect.

V. Discussion

For taxol-stabilized microtubules, the technique described here yields reliable datain the length range of �1–40 µm. Longer microtubules may fluctuate out of focuswhich leads to errors when using 2D tracking algorithms. In this case, either a three-dimensional tracking technique would have to be employed (Speidel et al., 2003), orthe analysis of the data would have to be modified to accommodate for the bias ofdetecting only positions close to the focal plane. In addition, the fluctuations of long

0.05 0.1 0.15 0.2 0.3 0.40.025

0.03

0.035

0.04

0.045

0.05

0.055

0.065

Time (s)

MS

D (

µm2 )

DataFit of Eq. (5)Fit of Eq. (4)

Fig. 7 A log–log plot of mean square displacement data for the same microtubule as in Fig. 6. A framerate of 23.43Hz and integration time I of 25 ms were used. The dashed line is a weighted fit of Eq. (4) to theexperimental data, yielding a relaxation time of (69.5 ± 1.3) ms and an amplitude V1 of (0.0301± 0.0002)µm2. The solid line is a weighted fit according to Eq. (5) which takes into account low-pass filtering. It yieldsa relaxation time of (61.3 ± 3.5) ms and an amplitude V1 of (0.032 ± 0.001) µm

2.

30. Extracting the Mechanical Properties of Microtubules 613

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microtubules become so slow that the length of observations required to gatheradequate statistics becomes very long. Microtubules that are tens of micrometerslong may have relaxation times of above 10 s. Gathering 103 independent data pointsthus requires that observations be carried out over 103� 10 s= 104 s which corre-sponds to almost 3 h. Recording time series over a period of more than 1 h wouldrequire a shutter to be employed to prevent bleaching.

The fluctuations of shorter microtubules in turn become too small and too fast tobe resolved with this assay. Assuming a persistence length of 600 µm (Taute et al.,2008), a microtubule of length 1 µm exhibits fluctuations with a width of �20 nm atits end. Given that low-pass filtering will reduce the measured width even more,measurement errors may start to affect the data in this length regime. Microtubulesshorter than 1 µm may become accessible to measurements by increasing themechanical stability of the setup beyond �1 nm and by using high-speed detectors.

As these length limits are dictated by the size and speed of the fluctuations, theymay be different for microtubule preparations that lead to different stiffnesses. Thesample preparation procedure described here requires microtubules that are stable atleast for the time span of several hours which is required to prepare the sample andperform the measurements. The general technique of deducing the mechanicalproperties of grafted microtubules by employing a tracer particle to measure theirbending fluctuations is, however, applicable to any preparation.

VI. Summary

Without the need for a sophisticated manipulation device and by extracting anexact calibration from statistical mechanics, thermal fluctuation measurements com-bine simplicity with high-precision results. The use of tracer particles for bendingfluctuation measurements on microtubules presents an elegant solution for determin-ing their mechanical properties in a length range crucial to cell biology and notcovered by other passive measurement techniques.

Future improvements in mechanical stability and detection speed may increase theregime covered by this technique even into the submicron range of microtubulelengths.

AcknowledgmentsThe authors would like to thank Kathleen Hinko for careful proofreading of this manuscript. KMT

gratefully acknowledges support by the Daimler-Benz-Foundation. This research was funded by a grantfrom the National Science Foundation (CMMI-0728166).

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