Annals of Biomedical Engineering,Vol. 30, pp. 546–554, 2002 0090-6964/2002/30~4!/546/9/$15.00Printed in the USA. All rights reserved. Copyright © 2002 Biomedical Engineering Society

Finite Element Analysis of Imposing Femtonewton Forceswith Micropipette Aspiration

JIN-YU SHAO

Department of Biomedical Engineering, Washington University, Saint Louis, MO

(Received 2 January 2002; accepted 4 March 2002)

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Abstract—A novel technique of imposing femtonewton forcewith micropipette aspiration@i.e., the extended micropipettaspiration technique~EMAT!# is proposed, and an axisymmeric finite element analysis of this technique is provided. TEMAT is experimentally based upon a micropipette manipution system and is theoretically based upon hydrodynamAny spherical object such as a human neutrophil or a labead can be employed as the force transducer, so cell–interactions can be directly studied. Our computational analshows that femtonewton forces can indeed be imposed.force magnitude is sensitive to the radius of the micropipeand the micropipette-transducer distance, but it is muchsensitive to other parameters including the radius of the traducer, the substrate curvature, and the thickness of the micpette wall. Combining the EMAT and the previously developmicropipette aspiration technique will allow us to imposeunprecedented range of forces, from a few femtonewtonsfew hundred piconewtons on single molecules or recepligand bonds. ©2002 Biomedical Engineering Societ@DOI: 10.1114/1.1476017#

Keywords—Molecular biomechanics, Receptor-ligand interations, Mechanotransduction, Piconewton force, Computatiofluid dynamics.

INTRODUCTION

Rupturing a receptor-ligand bond, deforming a singmolecule, and transmitting a mechanical signal througsingle protein all require forces at a piconewton or fetonewton level. Thus, imposing piconewton or femtonwton forces is critical to the study of receptor-liganinteractions,12 mechanotransduction,22 and proteinrheology.9,11,14,17,18,21Piconewton forces can be imposewith a variety of techniques that have been developedthe last two decades. These techniques include optweezers,2 the atomic force microscope~AFM!,4 themagnetic force apparatus,17,22 the flow chamber assay,1

Address all correspondence to Jin-Yu Shao, PhD, DepartmenBiomedical Engineering, Washington University, Campus Box 10517 Lopata Hall, One Brookings Drive, St. Louis, MO 63130-489Electronic mail: shao@biomed.wustl.edu

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the microneedle technique,8,23 the biomembrane forceprobe ~BFP!,5 and the micropipette aspiration techniqu~MAT !.15

The range of forces that optical tweezers can measis fairly limited, i.e., from ;1 to ;50 pN with a reso-lution of ;0.1 pN.10 The AFM can measure forces from;10 pN to several nanonewtons, but combining it wother microscopic techniques is difficult for the preseAlthough forces from 0.01 to 10 pN can be imposwith the magnetic force apparatus, such system is dcult to calibrate because paramagnetic beads thatthe same under a microscope might have different sceptibility to a magnetic field.18 The BFP can measurewide range of forces up to;1000 pN, but it cannotresolve forces smaller than 0.5 pN.5 With the micron-eedle technique, which is similar to the AFM, forcessmall as 1 pN can be measured.8,19,23 Optical tweezers,the AFM, the microneedle technique, and the BFP arebased upon measuring the deflection of a cantilever beor a latex bead. As such, they all require the use oforeign object as the force transducer. Although pulliforces can be directly imposed on spherical cells inflow chamber assay, simplified geometry usually hasbe assumed in the force calculation. With the MAT,wide range of forces as large as 100 nN can be impodirectly on a spherical object such as a cell or a bea15

However, at present, forces below 5 pN cannot beposed with this technique. Here, in light of the fact thboth optical tweezers and the magnetic force apparaare calibrated with the principles of hydrodynamics, wpropose a novel technique of imposing femtonewtforces that is directly based upon hydrodynamics, aprovide a detailed axisymmetric analysis with finite ement methods.

The proposed technique, which will be referred tothe extended micropipette aspiration technique~EMAT!hereinafter, is a natural extension of the MAT. Both tMAT and the EMAT are experimentally based uponmicropipette manipulation system.15 Their only differ-ence is the position of the transducer when forcesimposed. In the MAT, the force transducer~a spherical

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547Imposing Femtonewton Forces with Micropipettes

cell or bead, whose radius isRb! is placed inside acylindrical micropipette and moves toward and awfrom a substrate repetitively under alternating pressuinside the micropipette. When the transducer adherethe substrate, the force~F! can be calculated with15

F5pRp2DpS 12

Ut

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whereRp is the radius of the micropipette@Rp is slightlylarger thanRb and the change inF caused by this dif-ference has been neglected in Eq.~1!#, Dp is the aspira-tion pressure in the micropipette,Ut is the velocity ofthe adhered transducer, andU f is the neutral-buoyanvelocity of the free transducer. In the EMAT, once tspherical force transducer adheres to a substrate, thecropipette is withdrawn to a certain position, leaving tforce transducer outside the micropipette at a distancd.A pulling force on the transducer can then be imposedgenerating a fluid flow around the transducer withaspiration pressure inside the micropipette. The magtude of the force~f ! can be calculated numerically andifferent Dp or d will yield different force magnitudes. Ifa constant force is desired, the transducer can be plain a region where the force magnitude is not sensitived. If a variable force is desired, a variable pressure wsuffice. With a micropipette manipulation system, a wirange of pressures~from ;0.25 to;1000 pN/mm2! canbe applied inside a micropipette.13,15 We will show thatcombining the EMAT and the MAT will enable us timpose forces from a few femtonewtons to a few hudred nanonewtons.

Like the MAT, the EMAT does not need a separacalibration because the pressure inside the micropipis applied with a well-calibrated manometer. The forresolution of the EMAT relies uponRp , Rb , andDp asin the MAT. It also relies upond, the substrate curvatureand the thickness of the micropipette wall~Tp!. Althoughour analysis of these parameters presented later isfor the case of a spherical transducer, the sphericitythe transducer is not indispensable. As long as an obhas an axisymmetric geometry and its location canmanipulated with a micropipette, it can serve as the fotransducer although more complicated finite elemanalysis might be needed for the force calculation. Bthe MAT and the EMAT can be readily adopted by labratories with a micropipette manipulation system, as was those with a patch clamping device. Since the detamechanical properties of a single deoxyribose nucacid can be revealed only when femtonewton forcesimposed,17,18 the EMAT should have great potential imolecular rheology, single-bond mechanics, and singprotein mechanotransduction.

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METHODS

Proposed Experimental Method

The details of a micropipette manipulation system aavailable in the literature.15 Two main features of such asystem are:~1! single or multiple glass micropipettes thare capable of moving cells and other objects inexperimental chamber in three dimensions~these mi-cropipettes have diameters on the order of micromeand are controlled by micromanipulators!, and~2! preciseaspiration and expelling pressures that can be appinside the micropipettes. In this section, using the stuof the interaction between a protein-coated bead anhuman neutrophil as an example, we will describeexperimental procedure of the proposed EMAT.

First, a neutrophil~radius Rc! will be held with amicropipette that has a small aspiration pressure. Thenantibody- or ligand-coated latex bead will be aspiratinto another micropipette whose radius~Rp! is almost thesame as that of the bead~Rb!. The antibodies or ligandscoated on the bead will be chosen in such a way tthey can react with the receptors on the cell surfaThen, the bead will be placed close to the cell surfaand the pressure inside the micropipette containingbead will be zeroed. A precise aspiration pressure~Dp!can be applied inside this micropipette, causing the labead to move toward the left@Fig. 1~a!#. The setupshown in Fig. 1~a! positions the cell, the bead, and thmicropipettes in the same focal plane and forms an asymmetric configuration.@Please note that Figs. 1~a!,1~b!, and 1~c! were created from two other micrographto illustrate our experimental design.# At this point, anexpelling pressure that is greater than the aspiration psure will be superimposed to drive the bead to the riand bring the bead into contact with the cell surfacea certain period of time@Fig. 1~b!#. Afterward, the rightmicropipette will be withdrawn a certain distance@Fig.1~c!#. Then, the expelling pressure superimposed insthe left micropipette will be removed, so that only thaspiration pressure remains. If there is no adhesiontween the bead and the cell, the bead will move freback to the left micropipette and eventually be suckinto it. This whole process will then be repeated untiladhesion event occurs between the bead and the cethe bead adheres to the cell, the bead will remain stionary and experience a pulling force~f !. This pullingforce cannot be calculated with an analytical formula liEq. ~1!. However, it can be calculated numerically.

The experimental procedure described earlier canmodified slightly to accommodate other needs

~a! The neutrophil in Fig. 1 can be replaced by aother type of cell or substrate~e.g., a flat surfacecoated with proteins or cultured cells!.

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~b! The bead~i.e., the force transducer! in Fig. 1 canbe replaced by any spherical cell that doesadhere to the micropipette wall.

~c! The bead in Fig. 1 can also be replaced byspherical bead that has a radius larger the radiuthe left micropipette. In this case, the bead wapproach the cell surface being held by the lmicropipette withDp, instead of being driven bythe expelling pressure. Once the bead and theare in contact for a certain period of time, an epelling pressure can be superimposed inside themicropipette to release the bead to the cell surfa

FIGURE 1. A sequence of micrographs showing how a neu-trophil is probed by a protein-coated bead with the EMAT.The block arrow shows the direction of the fluid flow in theleft micropipette. „a… The bead is moving toward the cellsurface. „b… The bead contacts the cell for a certain period oftime. „c… The bead is pulled by a femtonewton force. Alsoshown in „c… is the cylindrical coordinate system used in thecomputational analysis.

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Moreover, any combination of these three modificatiocan be made simultaneously, making the EMAT veversatile.

Theoretical Formulation

In this section, we describe how to calculatef withFIDAP, a finite element analysis software packagecomputational fluid dynamics~Fluent Incorporated,Lebanon, NH!. The medium in the experimental chambcan be treated as an isotropic, homogeneous, and incpressible fluid because it is usually a dilute solutionsalts and proteins. The micropipettes, the bead, andcell are arranged on a straight line~Fig. 1! and theshortest dimension of the experimental chamber, whis in the shape of a rectangular parallelepiped, is usu2000mm. Therefore, the fluid flow generated byDp canbe treated as an unbounded axisymmetric problewhich is governed by the continuity equation and tNavier–Stokes equations~the body force is neglected!:

¹"v50, ~2!

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wherev is the velocity,r is the density of medium,p isthe static pressure, andm is the medium viscosity. Acylindrical coordinate system is chosen with the originthe center of the opening of the left micropipetteshown in Fig. 1~c!. For an unbounded problem, a largfluid domain must be used. In this case, it was found tthe following dimensions were large enough:Lp520 mmand H515 mm ~Fig. 2!. As shown in electron micro-scopic studies,20 sharp corners with right angles at thedges of the micropipette opening were used. Due tosmall scale of this problem, it is convenient to use mcron, second, and milligram~equivalent to pN•s2/mm! asthe base units for length, time, and mass. Therefore,viscosity and the density of the medium were chosenbe 1023 pN•s/mm2 ~equivalent to 1023 N•s/m2! and1029 pN •s2/mm4 ~equivalent to 1029 mg/mm3!.

The entry length of the fluid flow into a cylindricatube at zero Reynolds number is about 1.3 times the tradius.6 Here, the Reynolds number is 231025 for amaximum fluid velocity of 10mm/s in the left micropi-pette and a tube radius of 2mm. At such a low Reynoldsnumber, the flow will reach the steady state with a chacteristic time of;1025 s,3 so only the steady statsolution needs to be considered for the force calculatiThe tube length of 20mm is long enough for the flow todevelop into the Poiseuille profile. Thus, at the outletthe left micropipette~the left end shown in Fig. 2 wherez52Lp!, the following velocity is given:

549Imposing Femtonewton Forces with Micropipettes

FIGURE 2. The chosen computational domain and the mesh generated in FIDAP for a particular case where L pÄ20 mm, HÄ15 mm, RpÄ2 mm, RbÄ2 mm, RcÄ4 mm, dÄ2 mm, and TpÄ1 mm. L p is the length of a segment of the left micropipette, H isthe height of the computational domain, Rp is the radius of the left micropipette, Rb is the bead radius, Rc is the cell radius, dis the distance between the bead and the opening of the left micropipette, and Tp is the thickness of the micropipette wall. Theouter radius of the right micropipette is 2 mm. The size of the entire computational domain is 40 mmÃ15 mm.

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where Umax is the maximum fluid velocity in the lefmicropipette. An alternative boundary condition atz52Lp is the pressure boundary condition. Although tpressure atz52Lp can be calculated with Poiseuillelaw if Dp, Rp , and the length of the whole left microppette is known, the real length of the micropipette canbe used because the micropipette has a tapered shInstead, the equivalent length of the micropipette mbe used and its determination requiresUmax to beknown.15 Consequently, the velocity boundary conditiois more convenient atz52Lp . Another advantage ousing the velocity boundary condition is that the bounary condition would remain the same no matter whvalue we use forLp in the computation.

On the surfaces of the cell, the bead, and the micropettes, the no-slip boundary condition is applied:

v50. ~5!

At the far-field boundaries of the chosen computatiodomain ~Fig. 2!, the zero tangential and normal stressare assigned

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Numerical Computation

Combined with Eqs.~4!–~7!, Eqs. ~2! and ~3! can besolved with FIDAP. In FIDAP, four modules are employed for the following four steps of computation:~1!FI–GEN or GAMBIT for mesh generation and smooting, ~2! FIPREP for problem definition and boundacondition application,~3! FISOLV for solution, and~4!FIPOST for postprocessing.

For this work, FI–GEN was used for mesh genetion. First, all the edges were meshed manually so tthe desired mesh density could be achieved in sevselected regions. For example, it was expected thatchange in velocity and pressure would be more dramaround the opening of the left micropipette and the lpole of the bead, so finer meshes were generated in tregions and three boundary layers each with a thicknof 0.002mm were applied. Finer meshes were also geerated at the outlet of the left micropipette so that Eq.~4!could be satisfied more rigorously. Then, the fluid dmain was paved with four-node linear quadrilateral ements and a projection size of 0.1. Finally, the mesh wrenumbered to improve the efficiency of computatioFor the typical geometry shown in Fig. 2, 21291 ements were generated.

The algebraic equations resulting from the finite ement discretization were solved with the segregasolver, which is appropriate for large two-dimensionproblems and most three-dimensional problemsFIDAP. With this solver, each conservation equation wsolved in a sequential manner and the hybrid relaxatscheme was employed. The pressure was solved wi

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mixed and discontinuous approach, where the preswas discretized as a constant in each element andtributed an additional degree of freedom to the systemunknowns to be solved for. The iteration would be tminated when the following convergence criterion wsatisfied:

iX i2X i 21iiX i i

<DTOL, ~8!

whereX is the solution vector,i is the iteration step, andthe default value for DTOL~the tolerance for the relativedifference in velocity or pressure between two consetive iteration steps! is 0.001. The normi•i is defined asthe root mean square norm summed over all the eqtions of our model and is calculated separately for edegree of freedom. All these options can be controlledFIPREP.

For the case shown in Fig. 2, it took;90 min of CPUtime ~33 iterations! to obtain the convergent solution oa networked UNIX workstation~Sun Ultra10, 360 MHz!.Then f was calculated with the STRPRINT commandFIPOST. With this command, stress and pressure canintegrated on any defined surface. The velocity field,pressure contour, and the shear stress profile can aplotted in FIPOST.

The FIDAP script for the computation described heis available from the author upon request.

Numerical Tests

First, the classical problem of an unbounded unidirtional flow over a sphere~radius 1mm! was solved andthe drag force on the sphere was calculated. This calated force is consistent with the theoretical predictiobtained from the Stokes formula. Second, if a largfluid domain, a longer segment of the left micropipetand a finer mesh~twice as large or dense! were used forthe case shown in Fig. 2,f was less than 1% differentMoreover, the following modifications did not yield anf that was more than 1% different:~1! designating theopening and the inner wall of the left micropipettethree- or four-layered boundary edges,~2! altering theprojection size of the paving algorithm to 0.05,~3! in-creasing the right micropipette length to 20mm, and~4!choosing 0.0001 as the value for DTOL.

RESULTS

Velocity, Pressure, and Streamline

Shown in Fig. 3 are the velocity field~a!, the pressurecontour ~b!, and the streamline pattern~c! calculatedwith FIDAP for the typical case shown in Fig. 2~Umax

510 mm/s!. Although it is difficult to see in Fig. 3~a!,the velocity reached and kept the Poiseuille pro

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shortly after the fluid entered the left micropipette. Ashown in Fig. 3~b!, the pressure drop occurred maininside the left micropipette, i.e., the existence of tcell-bead pair and the right micropipette does not pduce much resistance to the overall fluid flow. Thisconsistent with the finding that the bead does not drtically affect the free-motion of a buoyant spherical oject inside the left micropipette as long as the beadoutside the left micropipette.15 At the far-field boundariesof the fluid domain, both the velocity and the pressuare almost zero. The streamline pattern shown in F3~c! is typical of a low Reynolds number flow.

Pressure and Shear Rate on the Bead Surface

The pressure and the shear stress on the bead suboth contribute tof. Shown in Fig. 4 are the distributionof the pressure and the shear rate on the bead sur

The shear rate is defined asA2s̃: s̃, wheres̃ is the shearrate tensor. The pressure is the smallest at the left polthe bead surface@Fig. 4~a!#. On the back of the beadsurface~close to the right micropipette! where there isnot much flow activity, the pressure and the shear rare both almost zero. Since the radial and axial velocgradients are both zero at the left pole of the bead sface, the shear rate is also zero there. For this particcase,f is calculated to be 112 fN. IfUmax doubles,f andthe pressure atz52Lp will also double.

The Effects of d and Rc on f

The distance between the force transducer~i.e., thebead! and the opening of the left micropipette is a majparameter that influences the magnitude off. Figure 5shows the dependence off upon d for several sets ofmicropipette, bead, and cell radii. It is clear that, whendincreases,f decreases in a fashion similar to an exponetial decay. Asd becomes larger,f becomes less sensitivto d. With larger Rp or Rb , larger f can be imposed ifother parameters remain the same. If the velocity bouary condition shown in Eq.~4! was replaced by anequivalent pressure boundary condition, the result woremain the same. Due to the linear nature of low Renolds number flow problems, the magnitude off can beeasily varied by changingDp, which has a linear rela-tionship with Umax.

15 It appears that the value ofRc

affects f only slightly, since f changes little whenRc

changes from 4mm to infinity ~i.e., the flat substrate!, asshown in Fig. 5.

The Effects of Tp , Rp , and Rb on f

In addition tod andRc discussed above,Tp , Rp , andRb can affectf as shown in Fig. 6. Obviously, the mossensitive parameter among these three isRp . A 5% un-certainty inRp or Rb will cause a maximum of approxi

551Imposing Femtonewton Forces with Micropipettes

FIGURE 3. The velocity field „a…, the pressure contour „b…, and the streamline pattern „c… for the case shown in Fig. 2. The unitsfor velocity and pressure are mmÕs and pN Õmm2, respectively.

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mately 9.2% or 4.2% uncertainty inf for the data shownin Fig. 6, so bothRp and Rb need to be determinecarefully in the experiment.Tp affects f only slightly.Larger Tp likely increased the pressure drop aroundopening of the left micropipette, resulting in a largerf. Inthe case of our default parameters here, if Eqs.~6! and

~7! are replaced by the no-slip boundary condition atupper boundary of the fluid domain, the computed forwill be approximately 5.5% larger. However, if we increaseH to 20 mm, this difference will be only abou0.68%; if we decreaseH to 10 mm, this difference willjump to about 40%. Therefore, in the experiment, t

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micropipettes should be kept at least 15mm away fromthe bottom of the chamber to minimize the effect of tboundary on the force calculation.

DISCUSSION

We presented a comprehensive computational anaof the EMAT, a novel technique that can be usedimpose femtonewton forces. Combining the MAT athe EMAT will enable us to impose forces from a fefemtonewtons to hundreds of nanonewtons. Toknowledge, such a wide range of forces cannot beposed by any other technique alone. If a bead is usethe force transducer, bead–bead or bead-substrate iactions can be studied. Then the MAT and EMAT wbecome a complementary tool to other techniquesmeasuring minute forces. The advantage of the EMand the MAT over other techniques is that a sphericell such as a human neutrophil or a suspended cultucell can be used directly as the force transducer,cell–cell or cell-substrate interactions can be direcprobed.

FIGURE 4. The distributions of the pressure „a… and theshear rate „b… on the bead surface for the case shown in Fig.2. Here, s represents the longitudinal arc length on the beadsurface starting from the left pole.

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Like the BFP, two modes of the EMAT can be estalished: horizontal~Fig. 1! and vertical. In the verticalmode, the bead, the cell, and the micropipettesaligned vertically on a straight line and a horizontal mcroscope is required for carrying out the experiment. Tlowest force that the horizontal mode can impose isfected by the weight of the force transducer. If a postyrene bead with a 2mm radius is used as the forctransducer, the net weight of this bead in water wouldaround 0.02 pN, so forces smaller than 0.1 pN or 100imposed with the horizontal mode will deviate more th11° from the horizontal direction. However, the weigof the force transducer should not affect the directionthe force imposed with the vertical mode. If forcesmaller than 100 fN are needed, the vertical mode shobe established and the net weight of the transducerbe added or subtracted from the drag force,f, calculatedwith FIDAP.

FIGURE 5. The effects of d and Rc on f. The values of Rp andRb are shown in micrometers. Shown here with plus signsare the results when the neutrophil in Fig. 1 is replaced by alarge flat substrate „i.e., RcÄ`….

FIGURE 6. The effects of Tp , Rp , and Rd on f. The defaultparameters are L pÄ20 mm, HÄ15 mm, RpÄ2 mm, RbÄ4 mm,RcÄ4 mm, dÄ4 mm, and TpÄ1 mm.

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553Imposing Femtonewton Forces with Micropipettes

The magnitude off can be easily adjusted by usindifferent aspiration pressures. While a bead that is larthan the left micropipette shown in Fig. 1 can still bused as the force transducer, a bead that is smallerthe micropipette should be avoided because the posof the bead, hence, the position of adhesion, cannocontrolled well with the micropipette and large forcuncertainties will occur. The magnitude off is very sen-sitive to the radius of the left micropipette, which shoube determined with great caution. With computerized iage analysis, the micropipette diameter can be measwith at least a resolution of about 0.1mm,16 so theuncertainty inf caused by a micropipette that has a 8mmdiameter should be;2.1% ~Fig. 6!. If a bead with an 8mm diameter is used as the force transducer, a 0.1mmuncertainty in the diameter will contribute an approxmately 0.57% uncertainty inf. The uncertainty infcaused by the uncertainty ind depends on the stiffness othe transducer~k!. A 0.1 mm uncertainty ind will causean approximately 3% uncertainty inf for all the resultsshown in Fig. 5. Therefore, for a micropipette with amm radius and a bead with the same radius, the oveuncertainty caused by geometrical uncertainties shobe under 6%. Even though we can measured very ac-curately, Brownian motion of the transducer could cauan uncertainty in f, which can be approximated bAkkBT, wherekB is Boltzmann’s constant andT is thetemperature. For the data shown in Fig. 5, the smaluncertainty is 4.7% in the region where the transducethe stiffest and the largest uncertainty is 20% in tregion where the transducer is the softest.

If a bead is used as the force transducer of the EMthe single particle tracking technique can be usedmonitor the bead position with nanometer resolutio7

Therefore, when a receptor-ligand bond or a moleculestretched with the EMAT, the change in the length of tbond or molecule can be tracked very precisely. If a cis used as the force transducer of the EMAT, the sinparticle tracking technique can still be applied, but ttracking resolution will be compromised. Neverthelethe EMAT should give us a viable alternative for impoing femtonewton forces on single molecules or bonthus having great potential in single-bond mechanicssingle-protein rheology.

ACKNOWLEDGMENTS

This work was supported by the Barnes-Jewish Hpital Foundation and the Whitaker Foundation. The athors thank Dr. Taber and his wife for their careful reaing of the manuscript.

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