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0143-8166/$ - se

doi:10.1016/j.op

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Optics and Lasers in Engineering 45 (2007) 450–457

Continuous and time-shared multiple optical tweezers forthe study of single motor proteins

Marco Capitanio�, Riccardo Cicchi, Francesco Saverio Pavone

L.E.N.S., Universita degli Studi di Firenze, Via Nello Carrara 1, 50019 Sesto Fiorentino, Firenze, Italy

Available online 2 August 2006

Abstract

We present a comparison between continuous (CW) and time-shared (TS) multiple optical tweezers applied to the study of the

interaction between a single motor protein (myosin) with its track (an actin filament). In the experimental assay, named ‘‘three-bead

assay’’, a single actin filament is stretched between two beads trapped in a CW or in a TS double trap. The actin filament is presented to a

single myosin molecule lying on a third bead attached to the coverslide. The CW double trap is obtained by splitting a single laser source

into two orthogonally polarized beams, while the TS one is obtained by rapidly scanning a single laser beam with acusto-optic

modulators. When using the CW traps, position detection of the left or right bead is obtained by means of a single quadrant detector

photodiode (QDP) placed in the back focal plane of the condenser and selected with a polarizer; when using the TS traps, the position of

multiple traps with the same QDP can be collected independently using triggered and synchronized generation and acquisition. The two

techniques are thoroughly analysed and compared, evidencing advantages and disadvantages of each one.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Position measurement; Polymer; Microscopy; Detection

1. Introduction

During the last decade, optical tweezers have become apowerful tool for manipulating single biomolecules and forinvestigating the mechanic and kinetic properties ofproteins and biopolymers.

Since their development [1], optical tweezers haveundergone many technical advances. Among these, multi-ple optical tweezers have been proved to be especiallyuseful in experiments where polymers need to be manipu-lated at the single-molecule level [2].

Multiple optical tweezers can be divided into two majorclasses: time-shared (TS) and continuous (CW). The firstclass is obtained by rapidly moving a single optical trapbetween different positions in the sample plane; if the timetaken to scan the different trap positions is much smallerthan the diffusion time of the trapped particles, the laserbeam works as stable multiple optical tweezers. Fasterscans imply that more traps can be generated simulta-

e front matter r 2006 Elsevier Ltd. All rights reserved.

tlaseng.2005.02.011

ing author. Tel.: +39055 4572476; fax: +39 055 4572451.

ess: capitan@lens.unifi.it (M. Capitanio).

neously and/or that more efficient trapping can beachieved. TS traps have been obtained by deflecting thelaser beam using galvano mirrors [3], piezoelectric mirrors[4], or acousto-optic modulators (AOMs) [5–7]; dependingon the technique used, scanning rates can reach respec-tively 10–50, 1–2, and 10–200 kHz. Traps generation isusually computer-controlled so that the number of traps,their position, and stiffness can all be modified in real time.TS traps are also easy to be constructed and aligned, sinceonly one laser beam is needed.CW multiple optical tweezers are obtained by simply

dividing a beam into two or more optical paths and thenrecombining the beams before the objective [8]; alterna-tively, two or more laser sources can be combined together[6]. This approach is simple to be realized when two trapsare needed, but becomes more complicated when morethan two traps are required.A double optical tweezers assay largely used to study

interactions between a single myosin motor and a singleactin filament is the ‘‘three-bead assay’’, developed byFiner et al. in 1994 [8]. In this assay, double opticaltweezers are used to catch and stretch an actin filament

ARTICLE IN PRESSM. Capitanio et al. / Optics and Lasers in Engineering 45 (2007) 450–457 451

between two trapped beads, creating a configurationusually named ‘‘dumbbell’’. The actin filament is presentedto a third bead stuck to the coverslide surface, which onaverage carries one myosin molecule. Interactions betweenactin and myosin are detected from noise reduction in thebead position signal and from data analysis the mechanicand kinetic properties of the molecule are derived [9].

Myosin is a family of motor proteins almost ubiquitousin eukaryotic cells. The chemical energy contained in anATP molecule is converted during a chemo-mechanicalcycle in which the myosin motor produces its move-ment (working stroke). In particular, myosin II is anon-processive molecular motor responsible for musclecontraction. During the last decade, its properties at thesingle-molecule level have been investigated using both CW[8–13] and TS [14,15] double optical tweezers. The sameexperimental assay has been used to study cardiac V1 andV3 myosin [16], smooth muscle myosin [17,18], myosin I[19], and myosin V [20,21].

We have built a multiple optical tweezers setup that canwork both as CW and TS double optical tweezers. CWtraps are obtained by splitting an infrared laser source intotwo orthogonally polarized beams, while TS traps areobtained by using two crossed AOMs. Position detectionof the CW or TS tweezers is achieved using a singlequadrant detector photodiode (QDP) placed in the back-focal plane of the condenser. Separate detection of the TStraps is possible using an acquisition board synchronizedand triggered to the generation signal, as explained in thenext section.

We have compared the two techniques in a three-beadassay. We show that, using CW traps, the stability of thesystem is within 1 nm and the working stroke of themolecule can be determined with sub-nanometer accuracy,along with its kinetic properties. On the other hand, in theTS configuration, the dumbbell oscillates at the switchingfrequency with amplitude that depends on the trapsstiffness and position and on the switching frequency. Thisoscillation does not change the kinetics of the moleculesignificantly, but must be taken into account whenevaluating the working stroke using back focal planedetection.

2. The experimental apparatus

The experimental apparatus (Fig. 1) was set up around acustom made optical microscope. The mechanical structurewas designed in order to obtain high mechanical stabilityand to allow using custom optics. Two linear manualtranslators (M-014 Physik Instrumente) and a piezoelectricstage (P-527.2 CL Physik Instrumente), allow gross (25mmstroke) and fine (1 nm resolution) movements of the samplein the xy plane, respectively. The objective (Nikon Plan-Apo 60X, NA 1.2, WD 0.2mm, water immersion) ismounted on a piezoelectric translator that provides fine z

movements (P-721.20 Physik Instrumente, 100 mm stroke,1 nm minimum displacement). The condenser (Olympus

U-AAC, Aplanat, Achromat, NA 1.4, oil immersion) isused both to illuminate the sample and to collect theforward scattered laser light for position detection.The apparatus is provided with three light sources. A

halogen lamp (Schott KL 1500 LCD, 150W) suppliesillumination for bright-field microscopy. The sample imageis acquired on two different CCD cameras. One of them(CCD 200X, Hamamatsu C3077) is used for wide-fieldmagnification, while the other (CCD 2000X, Ganz ZC-F11C3) collects a high-magnification image that serves asthe basis for an optical feedback system, which preventsthermal drifts by stabilizing the sample position within1 nm [12,22].A laser beam at 532 nm wavelength, coming from a

Nd:YAG duplicated laser (Coherent, VERDI V-10),provides illumination for fluorescence microscopy. Thefluorescence image of the sample is projected onto anintensified CCD camera (Hamamatsu C2400).Multiple optical tweezers were integrated within the

optical and fluorescence microscope. The trapping beam,coming from a Nd:YAG laser (Spectra-Physics MillenniaIR, l ¼ 1064 nm), passes through an optical isolator (OI)to prevent back-reflections of the trapping light into thelaser cavity and through a telescope to collimate andexpand the beam. A l/2 waveplate and a polarizing beamsplitter (PBS) divide the beam into two beams withmutually orthogonal polarizations: the undeflected beamis used for the TS multiple optical tweezers, while bothones are used for the CW double optical tweezers.

2.1. TS multiple optical tweezers

The undeflected beam passes through two crossedAOMs (A & A DTS-XY 250), placed in a conjugate pointof the aperture diaphragm of the objective. The modulatorsare driven by a two-channel digital synthesizer (DDS, A &A AA.DDS.XX) that supplies two sinusoidal signals withfrequencies between 60 and 90MHz and amplitudes from 0to 30V. These signals drive the first-order Bragg diffractedbeam emerging from the two AOMs. The beam angulardeflection is proportional to the frequency, while the beamintensity is proportional to the squared amplitude of thedriving signal. The beam angular deflection from theAOMs is converted into a linear displacement of theoptical trap in the focal plane of the objective. Byalternatively switching the signals between two frequencies,we caused the optical trap to switch between two positionswithin the focal plane of the objective. Moreover, byalternatively switching the signal amplitude between twovalues, we independently regulated the stiffness of the twooptical traps. The rising time of the DDS is fewnanoseconds, while the bandwidth of the AOMs limitsthe time needed to deflect the beam to �5 ms. This is fastenough to realize TS double optical tweezers. In fact, if theswitching time is much less than the diffusion time of thetrapped particle (about 10ms for a 1 mm diameter sphere to

ARTICLE IN PRESS

Fig. 1. A schematic drawing of the experimental setup. DDS: direct digital

synthetizer, AOM: acousto-optic modulator, PBS: polarizing beam

splitter, OI: optical isolator, QDP: quadrant detector photodiode.

M. Capitanio et al. / Optics and Lasers in Engineering 45 (2007) 450–457452

diffuse 100 nm) [23], the beam recovers to the originalposition before the particle escapes from the trap.

The objective focus the beam into the specimen, thuscreating the optical tweezers, and the light scattered by thebead is collected by the condenser. An interference patternbetween forward scattered and non-scattered light isprojected onto a QDP (3D Position Detector, UDTDLS-20) placed in a plane conjugated to the back-focalplane of the condenser. The position of barycenter of thelight intensity is proportional to the displacement of thetrapped bead for small bead displacements (typically300 nm from the trap center for 1 mm diameter beads).This well-known method [24] allows 3D position trackingof the trapped object with a resolution of about 1 nm in1ms, enough to resolve Brownian fluctuations of micro-metric beads immersed in a biological solution.

The QDP and its electronics provide three signalsproportional to the bead displacement along the threespatial directions and these signals are collected by a 12 bita/d acquiring board (DATA ACQUISITION, NationalInstruments PCI-6024E, 200 kHz maximum samplingfrequency).

Our detection system allows detection of multiple trapsthanks to the synchronization between traps switching anddata acquisition. A digital output board, equipped with aDSP, (DATA GENERATION, ViewPoint DIO-128,128 bits divided on four ports, maximum output scan rate500 kHz), provides signals for the DDS (19 bits for thefrequency and 8 bits for the amplitude for each of the twoDDS channels). A common clock drives both the generat-ing and the acquiring board, and generation/acquisitionare triggered by a common signal (‘‘trig line’’ in Fig. 1). Ingeneral, the acquisition frequency is set to a multipleinteger n of the generation frequency and acquired data arecomposed by alternating data blocks containing n pointsfor each trap. Odd blocks correspond to the position of thefirst trap, while even blocks correspond to the position ofthe second one. The two boards are delivered withLabView drivers and synchronized generation/acquisitionis controlled by custom software realized using the Lab-View development environment. When the software isexecuted, it first generates two synchronized square wavesto switch both the frequency and the amplitude of theacoustic wave in the AOM. Then, it enters into two loopsrunning in parallel: the first one reads the amplitude andfrequency values set by the operator and reinitializes thegeneration board to the new values; the second one readsacquired data and separates and displays alternating datablocks corresponding to the left and right bead positionsignals.

Limits in the generation rate arise from the limitedbandwidth of the generating board (�500 kHz) and of theAOM (�200 kHz). However, the bigger limit of theapparatus has been found to be the acquisition rate,caused by the low bandwidth of the QDP. In fact, silicondetectors show a slower response in the infrared region ofthe frequencies spectrum with respect to the visible region

[25]. The main reason is the higher penetration depth of1064 nm radiation with respect to visible light. TheNd:YAG laser radiation penetrates into the bulk materialand a significant fraction of the electron–hole couples aregenerated outside the depletion region. These electron–holecouples slowly migrate towards the depletion region andare then accelerated by the electric field. This phenomenoncauses a low cutting frequency in the detector responsethat, in our case, has been measured to be about 10 kHz(see Fig. 2A).Fig. 2B shows 1ms of the x position signal acquired at

50 kHz while the laser beam was synchronously switched at10 kHz between two different positions along the x-axis. Inthis case, the position of the first trap alternates every n ¼ 5points with the position of the second trap. The positionsignal does not switch instantaneously from one level to theother because of the limited bandwidth of the detector andreach a steady-state value only in correspondence of thelast sample of each block of five points. For this reason,10 kHz is the maximum switching frequency that we canreach with the QDP used: in order to have a good andindependent measurement of the position of each of thetwo beads, the acquisition has been performed at a higherfrequency (50 kHz or more) and only the steady-state value(the nth sample point) of the signal has been considered asthe bead position. Alternatively, switching was set at a ratemuch lower than the bandwidth of the detector (2 kHz):in this case, one point could be acquired for each point

ARTICLE IN PRESS

10 100 1000 100001E-8

1E-7

1E-6

1E-5

1E-4

pow

er s

pect

rum

(V

2 /H

z)

frequency (Hz)0 10 15 20 25 30 35 40 45

-4

-3

-2

-1

0

1

2

3

4

sign

al (

V)

samples(A) (B)

5

Fig. 2. (A) Continuous grey line: power spectrum of the position signal acquired with 1mm diameter bead trapped by the optical tweezers (100 s

acquisition, 20 kHz sampling rate). Dashed black line: Lorentzian fit of the acquired data. It is clearly distinguishable the divergence between experimental

data and the Lorentzian curve moving towards high frequencies. This effect is due to the cutting frequency of the quadrant detector. (B) 1ms of the x

position signal acquired at 50 kHz while the laser beam was synchronously switched at 10 kHz between two different positions.

Fig. 3. Motion of the dumbbell in a TS trap. (A) Position of the dumbbell

at the instant when the left trap is switched on and the right trap is

switched off. (B) Position of the dumbbell at the instant when the right

trap is switched on and the left trap is switched off.

M. Capitanio et al. / Optics and Lasers in Engineering 45 (2007) 450–457 453

generated to independently measure the position of eachbead.

2.2. CW double tweezers apparatus

The CW double optical tweezers are formed by twoorthogonally polarized beams. The undeflected beampasses through the AOM and through a second PBS,where it is recombined with the deflected beam. In this way,we obtain two independent orthogonally polarized opticaltraps in the objective focal plane, one with fixed positionand the other movable by the AOMs.

A polarizer was placed in front of the QDP in order toacquire the position of one trap at a time. A 901 rotation ofthe polarizer allows detecting the other trap.

3. Dumbbell dynamic in CW and TS optical tweezers

In this section, the motion of the dumbbell in a CW andin a TS optical tweezers are compared. The two traps aresupposed to have the same stiffness kt/2; the combinedstiffness of the double trap (kt�0.05 pN/nm) is chosen to bemuch smaller than the myosin stiffness (km�1 pN/nm) [26].The actin filament can be considered almost inextensible[27], its stiffness being ka�44 pN/nm for a 1 mm longfilament. The links between the beads and the filament canreach 1–3 pN/nm when the tension on the dumbbellexceeds �3 pN [9]. Under these conditions, the dumbbellcan be considered rigid compared to the trap stiffness.

In a CW double trap, the motion of the dumbbell isdescribed in good approximation by a Langevin equationof motion, where the initial term is neglected:

gdx

dtþ ktxðtÞ ¼ FBðtÞ, (1)

where FB(t) is a net random Brownian force acting on bothbeads and the drag force arises mostly from the beads (gE2(6pZr) for beads of radius r immersed in a fluid of viscosityZ). In this case, the motion corresponds to that of a

harmonic oscillator in a thermal bath and the distributionof the dumbbell position is Gaussian distribution centredaround x ¼ 0.For a TS double trap, the force acting on the dumbbell

continuously switches from the left to the right trap, with aswitching time D. In stationary conditions, the dumbbelloscillates with a frequency 1/D. We indicate xL

on the averagedisplacement of the left bead with respect to the center ofthe left trap, at the instant when the left trap is switchedon (and the right trap is switched off, see Fig. 3A).Analogously, xR

on is the average displacement of the rightbead with respect to the center of the right trap, at theinstant when the right trap is switched on (and the left trapis switched off, see Fig. 3B); xL

off and xRoff are analogously

defined. Taking into account the symmetry of the system,in stationary conditions we have xR

on¼ �xL

on and xRoff¼

�xLoff. Moreover,

xonL � xoff

R ¼ D� a, (2)

ARTICLE IN PRESSM. Capitanio et al. / Optics and Lasers in Engineering 45 (2007) 450–457454

where D is the traps displacement and a is the beadsdisplacement (which are supposed to be constant duringthe dumbbell oscillation). The motion of the left beadduring the time D when the left trap is switched on isdescribed by the equation

gdxL

dtþ

kt

2xLðtÞ ¼ 0, (3)

where we have neglected the thermal force, since we aremainly interested in the average position of the bead. FromEq. (3), xLðtÞ ¼ xon

L eð�kt=2gÞt andxoffL ¼ xon

L eð�kt=2gÞD ¼ �xoffR . From Eq. (2) we thus obtain

xonL ¼ �xon

R ¼D� a

1þ eð�kt=2gÞD,

xoffL ¼ �xoff

R ¼D� a

1þ eðkt=2gÞD,

xonL � xoff

L

�� �� ¼ xonR � xoff

R

�� �� ¼ ðD� aÞ1� eð�kt=2gÞD

1þ eð�kt=2gÞD. ð4Þ

The last equation represents the amplitude of thedumbbell oscillation. Once the bead diameter, the actinfilament length, and the trap stiffness have been fixed, thedumbbell oscillation depends only on the displacement D

and switching time D. The oscillation amplitude tends tozero for D�a (null pretensioning on the filament) or forD52g/kt (switching time much smaller than the relaxationtime-constant of the dumbbell). However, the displacementbetween the two traps must be set to a value high enough toexert an average tension on the filament 43 pN to obtain‘‘rigid’’ actin-bead links; with kt�0.05 pN/nm, this condi-tion corresponds to D�a4240 nm. The relaxation time-constant of the dumbbell is, using 1 mm diameter beads andthe same trap stiffness, about 0.8ms. Fig. 4 shows thebehaviour of the oscillation amplitude as a function of theswitching time D, with D�a ¼ 240 nm.

The oscillation amplitude grows almost linearly fromzero and reaches a saturation value D�a for a switching

00 1 2 3 4 5 6 7 8

50

100

150

200

250

|xon

- x

off|

(nm

)

|xon

- x

off|

(nm

)

Δ (ms)

0.02 0.04 0.06 0.08 0.100

2

4

6

8

10

12

14

Δ (ms)

Fig. 4. Amplitude of the dumbbell oscillation versus switching time in a

TS trap. Displacement of the traps relative to the actin filament D�a is set

to 240 nm.

time of few milliseconds. The inset of Fig. 4 shows that forthe smaller switching time that can be reached with oursetup (D ¼ 10�4 s), the oscillation amplitude is �14 nm.This value is high with respect to the typical conforma-tional changes of the myosin molecule (5–6 nm), butis still of the order of the fluctuations imposed by

Brownian forcesffiffiffiffiffiffiffiffiffihx2i

p¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBT=kt

p� 9 nm

� �. Only when

Do10�5 s, the oscillation amplitude becomes of the orderof the nanometer, which corresponds to the typicalposition stability of our apparatus [22]. It is worth nothingthat, as explained in the previous section, in order to obtainthe smaller switching time (10�4), acquisition is performedat a rate 5 times faster than generation (50 kHz acquisitionvs. 10 kHz generation) and only the last sample acquiredbefore switching to the other trap is considered for positiondetection. Since the dumbbell is oscillating during theacquisition, the acquired sample corresponds to theposition of the bead at the end of the oscillation, i. e.xRoff¼ �xL

off.On the other hand, when trap-switching is performed at

a slower rate (2 kHz), acquisition and generation rates canbe set to the same frequency, so that the average position ofthe bead during the oscillation is sampled. However, due tothe slower switching time, the oscillation of the dumbbellbecomes much wider (about 70 nm, see Fig. 4 and Eq. (4)).This wider oscillation can introduce additional effects, suchas variable tensioning on the filament during the dumbbelloscillation, variable stiffness of the actin-bead links,broadening of the position noise and of the error thataffects the measurement of the working stroke.

4. CW and TS three-bead assay

We have compared CW and TS recordings of theposition signals when a single myosin molecule wasinteracting with a single actin filament. Preparation ofproteins and sample cell has been performed by standardmethods [28]. During a single acquisition, 100 s of theposition signal from one or both beads were recorded,depending if CW or TS tweezers were utilized, respectively.During acquisition, the displacement between the actinfilament and the myosin molecule was stabilized by meansof a feedback system, obtaining �0.8 nm RMS noise alongthe three spatial directions. The feedback system calculatesthe position of the bead stuck to the coverslide and drivesthe piezoelectric translators in order to compensate thermaldrifts [22]. All recordings were performed while moving themyosin molecule 72 nm along the actin filament withconstant velocity, in order to average the distribution ofbinding probability along the filament and to obtain ameasurement of the real working stroke [12,22]. Since trapstiffness was chosen much smaller than the myosinstiffness, attachment of the molecule to the actin filamentresulted in an increased rigidity of the dumbbell and in areduction of the Brownian noise and position variance.Experimental data was thus analysed by fitting the signal

ARTICLE IN PRESS

-20 -10 0 10 20 300

100

200

300

400

num

ber

of e

vent

s

mean position value (nm)

unbound state

bound state

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-60

-40

-20

0

20

40

60

disp

lace

men

t (nm

)

time (s)

Left ch.Right ch.

workingstroke

U B U B U B U B U UB

(A) (B)

Fig. 5. (A) The working stroke of the molecule is obtained from the difference between the mean position value of bound and unbound states. Data are

taken from a 100 s position recording containing 1213 interactions, while a single myosin molecule is interacting with the actin filament in presence of 5mM[ATP]. (B) Position signals from both beads while the myosin molecule is interacting with the actin filament. Bound and unbound states are labelled,

respectively, ‘‘B’’ and ‘‘U’’. When the position noise in the unbound state is set to a common average value for left and right channels, the bound state

levels are shifted by some nanometers.

M. Capitanio et al. / Optics and Lasers in Engineering 45 (2007) 450–457 455

variance with a two-state function (high variance ¼ un-bound state, low variance ¼ bound state) using a variancehidden-Markov algorithm [9]. From statistics on lifetimesand displacements of bound and unbound states wetherefore obtained kinetics of attachment and detachmentof the molecule and its working stroke, respectively.

Experiments performed on skeletal myosin II using theCW double trap showed results that agree with literature.We found that the mean position of bound and unboundstates were Gaussian distributed around different valuesand from the difference between the two averages weobtained the working stroke (see Fig. 5A). Working strokestypically in the range of 5–670.5 nm were found. Lifetimesof bound states were exponentially distributed with a meanlifetime ton, which depended on ATP concentration. With[ATP] ¼ 10 mM we found ton ¼ 2174ms.

Experiments performed using the TS double trap(10 kHz switching frequency) similar results, with meanpositions of bound and unbound states Gaussian distrib-uted around different values and lifetimes of bound statesexponentially distributed with ton ¼ 2475ms at 10 mMATP. On the other hand, the value of the working strokemeasured on the left and right channels was typicallydifferent by some nanometers and varied depending on thetension applied to the dumbbell. This unexpected beha-viour can be understood using the model developed in theprevious section. In the unbound state the dumbbelloscillates with amplitude of about 14 nm at 3 pN oftension, which increases with increasing tension. Weassume that, on average, the myosin molecule attaches toactin at the time D/2 and performs the working strokefaster than the time resolution of the apparatus. With thevalues of switching time, relaxation time and pretensioningconsidered in the precedent paragraph, the average point ofattachment of myosin coincides, in good approximation(5%), with the middle point of the dumbbell oscillation.The dumbbell in the bound state can oscillate much less

than in the unbound state because of the high stiffness ofthe molecule, but with a much faster time constant. Inparticular, the equation of motion of the left bead in thebound state is

gdxL

dtþ

kt

2xLðtÞ þ km½xLðtÞ � x0

L� ¼ 0,

where km is the myosin stiffness and x0L ¼ ðD� aÞ=2þ d is

the average point of attachment of myosin plus theworking stroke (d). The solution is similar to the oneobtained in the unbound state (Eq. (4)), but with arelaxation-time constant g

�ðkm þ kt=2Þ � 20ms, which is

now much smaller than the faster switching time D that wecan reach in our apparatus (100 ms). Therefore, during thetime D, the dumbbell goes all through the end of theexponentially decaying motion. The full oscillation in thebound state is, in good approximation,

xonL � xoff

L

�� �� ¼ xonR � xoff

R

�� �� ¼ ðD� aÞkt

2km¼ 2

T

km� 6 nm,

where T ¼ ðD� aÞ=2� kt=2 is the average pretensioningexerted on the dumbbell. On the other hand, if we couldpush the switching frequency up to 100 kHz, the fulloscillation would reduce to about 1 nm, as in the unboundstate. When the switching rate is 10 kHz, the workingstroke on the two channels is measured from the differencebetween the position of the bead in the bound andunbound states at the end of the oscillation (xoff

L and xoffR ):

dL ¼xonL � xoff

L

�� ��U

2þ d�

xonL � xoff

L

�� ��B

2,

dR ¼ �xonR � xoff

R

�� ��U

2þ dþ

xonR � xoff

R

�� ��B

2,

where the subscripts U and B indicate that the positionof the bead is measured in the unbound and bound state,respectively. Therefore, the working stroke measured onthe two channels is different and depends on the amplitude

ARTICLE IN PRESSM. Capitanio et al. / Optics and Lasers in Engineering 45 (2007) 450–457456

of the oscillation in the two states. Superimposition ofposition recordings from both channels (see Fig. 5B)clearly shows that when the average unbound position forleft and right channels are centered on a common value, thebound state levels are systematically shifted by somenanometers. In this case, the working stroke of the myosinmolecule can be obtained from the average working strokemeasured on the two channels: d ¼ ðdL þ dRÞ=2:

5. Conclusions

We have presented an experimental apparatus that iscapable of working of both as CW or as TS double opticaltweezers. Position detection is achieved using a single QDPplaced in the back focal plane of the condenser. Whenusing the CW traps, left or right trap detection is selectedby means of a polarizer; when using the TS configuration,detection of multiple traps is achieved with a triggered andsynchronized generation/acquisition, during which n ac-quired points from the left trap position alternates with n

points from the right one. Due to limitations in the detectorbandwidth, the maximum switching rate is set to 10 kHz.

CW or TS double optical tweezers have been applied to athree-bead assay, in order to study interactions between asingle myosin molecule and an actin filament.

Using both configurations, we have been able to measuresingle interactions and to derive the kinetic (attachmentand detachment rates) and mechanic (working stroke)properties of the molecule. However, when using TS traps,the dumbbell oscillates at the switching frequency withamplitude that depends on the tensioning applied to thefilament, on the traps stiffness, and on the switchingfrequency. At 10 kHz switching rate, the oscillationamplitude is 414 nm in the unbound state and 46 nm inthe bound state which is not negligible with respect to theconformational changes of the myosin molecule (5–6 nm).Therefore, the oscillations of the dumbbell must be takeninto account when evaluating the working stroke. Thisproblem would be overcome if the switching rate could bepushed over 100 kHz, thus reducing the oscillationamplitude to about 1 nm. Another way would be tomeasure the average position of the dumbbell oscillationin between the dumbbell while it is oscillating; thedifference between the average position of the dumbbelloscillation in bound and unbound states would represent agood estimate of the working stroke. The average positioncould be measured, for example, using switching ratesmuch smaller than the detector bandwidth, or by means ofa detection system independent of the trapping laser, as inRef. [14].

Acknowledgements

The authors are grateful to M. Canepari, P. Cacciafesta,and R. Bottinelli for the biochemical preparations used in

the acto-myosin experiments. Work under Contract HPRI-CT-1999-00111 CE, partially supported by ‘‘SINPHYS’’PAIS 2002 project of INFM.

Note added in proof: During processing of this manuscript,novel silicon QDPs designed specifically for near-infraredlasers (1064 nm) have become available (UDT SPOT-15-YAG). We have verified that using these photodiodes,bandwidths up to 100 kHz can be easily reached.

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