differential interferometric particle tracking on the subnanometer- and submillisecond-scale

Differential interferometric particletracking on the subnanometer- and

submillisecond-scale

Dennis Müller,1,∗ Dieter R. Klopfenstein2 and Rainer G. Ulbrich1

1IV. Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1,37077 Göttingen, Germany

2Drittes Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1,37077 Göttingen, Germany

∗[email protected]

Abstract: We describe an interferometric method to measure the move-ment of a subwavelength probe particle relative to an immobilized referenceparticle with high spatial (Δx = 0.9nm) and temporal (Δt = 200μs) resolu-tion. The differential method eliminates microscope stage drift. An uprightmicroscope is equipped with laser dark field illumination (λ0 = 532nm,P0 = 30mW ) and a compact modified Mach-Zehnder interferometer ismounted on the camera exit of the microscope, where the beams ofscattered light of both particles are combined. The resulting interferogramsprovide in two channels subnanometer information about the motion ofthe probe particle relative to the reference particle. The interferogramsare probed with two avalanche photodiodes. We applied this method tomeasuring the movement of kinesin along microtubules and were able toresolve the generic 8-nm steps at high ATP concentrations without externalforces.

© 2013 Optical Society of America

OCIS codes: (180.0180) Microscopy; (180.3170) Interference microscopy.

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3425–3427 (1987).2. W. Denk and W. W. Webb, “Optical measurement of picometer displacements of transparent microscopic ob-

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using an optical trap.” Biophys. J. 70, 1813–1822 (1996).4. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-

hand: single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).5. J. Gelles, B. J. Schnapp, and M. P. Sheetz, “Tracking kinesin-driven movements with nanometre-scale precision,”

Nature 331, 450–453 (1988).6. M. Nishiyama, E. Muto, Y. Inoue, T. Yanagida, and H. Higuchi, “Substeps within the 8-nm step of the atpase

cycle of single kinesin molecules,” Nat. Cell Biol. 3, 425–428 (2001).7. K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical

trapping interferometry,” Nature 365, 721–727 (1993).8. E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, “Direct observation of base-

pair stepping by rna polymerase,” Nature 438, 460–465 (2005).9. O. Otto, F. Czerwinski, J. L. Gornall, G. Stober, L. B. Oddershede, R. Seidel, and U. F. Keyser, “Real-time

particle tracking at 10,000 fps using optical fiber illumination,” Opt. Express 18, 22722–22733 (2010).10. R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluores-

cent probes,” Biophys. J. 82, 2775–2783 (2002).

#181821 - $15.00 USD Received 19 Dec 2012; revised 15 Feb 2013; accepted 21 Feb 2013; published 18 Mar 2013(C) 2013 OSA 25 March 2013 / Vol. 21, No. 6 / OPTICS EXPRESS 7362

11. V. Jacobsen, P. Stoller, C. Brunner, V. Vogel, and V. Sandoghdar, “Interferometric optical detection and trackingof very small gold nanoparticles at a water-glass interface,” Opt. Express 14, 405–414 (2006).

12. A. R. Carter, G. M. King, T. A. Ulrich, W. Halsey, D. Alchenberger, and T. T. Perkins, “Stabilization of an opticalmicroscope to 0.1 nm in three dimensions,” Appl. Opt. 46, 421–427 (2007).

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15. M. Born and E. Wolf, Principles of Optics (Pergamon Press New York, 1980).16. Z. Wang, S. Khan, and M. P. Sheetz, “Single cytoplasmic dynein molecule movements: Characterization and

comparison with kinesin,” Biophys. J. 69, 2011–2023 (1995).17. M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd

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1. Introduction

Single-particle tracking experiments are widely used to monitor dynamical processes on lengthscales from millimeter to subnanometer. Especially in biology, where relevant processes onthe molecular level demand high spatial and temporal resolution, numerous optical methodshave emerged. Either fluorescent, phase shifting, or light scattering markers have been used tovisualize the movement of molecular specimens. The advancement of various tracking tech-niques has provided spatial precision far below the diffraction limit of visible light [1–3]. Thishas revealed insights into the mechanics of single biological molecules [4–6]. For instance, ithas been directly observed that the molecular motor kinesin moves along microtubules with8-nm steps [7]. With big experimental effort it was even possible to resolve the 0.37-nm stepsof RNA polymerase moving along DNA [8]. High temporal resolution can be achieved eitherwith a high-speed camera [9] or, which is much more cost-efficient, by using only few detectorchannels (e.g. quadrant photodiodes) with high detection bandwidth.

In practice, the spatial and temporal resolution is limited by factors like finite photon flux,or mechanical instabilities of the optical setup (e.g. microscope stage drift), as well as of thespecimen itself. These issues can be moderated with more or less experimental effort. The finitephoton flux, especially a constraint for fluorescent markers, limits the number of photons perframe, which in turn poses a fundamental limitation for the localization precision [10]. Thisrequires an appropriate choice of the sample rate but still results in a compromise between spa-tial and temporal resolution. In case of scattering particles there is no fundamental limitationof the photon flux since the scattered power is proportional to the intensity of the illumination.Although the scattered power strongly depends on the size of the particle, gold particles assmall as 10 nm have been tracked in a time window of about one microsecond [11]. When abigger light scattering marker is attached to a biological specimen, the flexible linkage oftenallows a relatively free motion of the marker within a confined volume. Thermal motion hencedeteriorates the precision of localization of the specimen. The use of optical tweezers allowsapplication and measurement of forces as well as the attenuation of the Brownian motion. Point-ing fluctuations of the trapping laser beam due to random air currents have been suppressed byenclosing all optical components in a sealed box with helium gas [8]. Microscope stage driftcan be compensated by using markers in the object plane, whose motion is measured and sub-sequently substracted from the measured motion of the specimen. Other methods are to tracka fixed marker in the object plane and then actively stabilize the microscope stage [12] or todecouple the whole setup from thermal fluctuations, which cause drift, by fully automating andplacing it in a separate room [13].


We present a new method, which we call differential interferometric particle tracking. Itcombines a relatively simple experimental setup with high spatial and temporal resolution andinherent elimination of microscope stage drift. Differential interferometric particle trackingis based on interference between the scattered light of a probe particle and a fixed referenceparticle. An upright microscope has been equipped with laser dark field illumination, and acompact interferometer has been mounted on top of the camera exit of the microscope. Twoavalanche photodiodes suffice to detect the interference signal, which provides informationabout one degree of freedom of the particle movement. Furthermore, this detection scheme isinsensitive to laser power fluctuations and pointing fluctuations of the illuminating beam. Todemonstrate the performance of this method we have applied it to the protein complex kinesinmicrotubule. The kinesin is labelled with a polystyrene bead with diameter 0.5μm. We couldclearly resolve steps with a size of 8 nm without the use of an optical tweezer [14].

2. Experiment

The basic idea of our tracking method is to overlay the beams of scattered light of a fixed ref-erence particle and a motile probe particle in such a way that the resulting interference patternyields precise information about the motion of both particles relative to each other. For thispurpose an upright microscope is equipped with a laser light source (λ0 = 532nm, P0 = 30mW )for dark field illumination (Fig. 1). The laser beam comes from the side and is focused in thevicinity of the object. Subwavelength particles in the object plane scatter in a first-order ap-proximation spherical waves collected in the aperture angle θ of the objective lense (OlympusLM Plan FL 50×/0.50 BD, working distance: 10.6 mm). The beams of the reference particle atposition R and the probe particle at position P are seperated from each other in the intermediateimage plane of the microscope by mirror M1 and are each directed into one arm of the modifiedMach-Zehnder interferometer. Both beams are overlayed with each other at the beam splitterBS. Two inversely phased interferograms are formed, one at each exit facet of the beam splitter.The mirrors M1 and M2 of the interferometer can be adjusted such that both virtual images atthe positions P′ and R′ as well as the direction of propagation of both beams coincide, indepen-dent of the real particle separation s. Then, behind BS, the wavefronts of both beams match andboth interference patterns are homogeneous spots without fringes. Their total radiation power isa function of the phase difference Φ between both beams. The convex lenses L1 and L2 behindeach exit facet of the beam splitter create further intermediate image planes where the pinholesPH1 and PH2 select the overlay of the probe particle and the reference particle. The setup thusblocks stray light and scattered light from other particles in the object plane. The pinholes cutout a circular area corresponding to a diameter of 5 μm in the object plane, so that the minimumseparation distance between both particles in the object plane should have this size. In order toadjust the positions of PH1 and PH2 a camera can be swivelled in the optical path behind eachpinhole. In this way also the shape of the interferograms can be monitored, which is importantfor the adjustment of the interferometer. The avalanche photodiodes APD1 and APD2 behindeach pinhole detect the total radiation power of both interferograms.

The interference pattern in the X-Y -plane behind each of the beam splitter exit facetscan be seen as the superposition of two spherical waves each limited by the aperture angleθ ′ = arctan(tan(θ)/M) originating at the virtual images P′ and R′ in the x′-y′-plane (Fig. 2).M = 50 is the magnification of the microscope. θ ′ limits the extent of the interference pattern.Each beam creates separately a circular spot with radius ρ = z′0 tan(θ ′) in the X-Y -plane. z′0 isthe distance between the virtual image plane and the X-Y -plane. If the separation s′ between P′and R′ is sufficiently small, both spots largly overlap in a circular area with radius ρ . If the lineP′R′ is parallel to the x′-axis, the situation is similar to Young’s double slit experiment [15]. For


RP

R’P’

objective lenselaser

tubus lense

M3APD2

L2

L1interference plane (X-Y)

APD1

PH1

PH2BS

M2

M1 intermediateimage plane

object plane (x-y)

virtual image plane (x’-y’)

{{

microscope

interferometer

Fig. 1. Experimental setup for differential interferometric particle tracking. A laser beam(λ0 = 532nm, P0 = 30mW ) illuminates the object in dark field configuration. The inter-mediate image plane is bisected by mirror M1 so that the beams of scattered light of thereference particle R and the probe particle P can be overlayed with each other at beam split-ter BS. The total radiation power of the resulting interferograms is detected by avalanchephotodiodes APD1 and APD2. L1, L2: Convex lenses. PH1, PH2: Pinholes.

a small θ ′ the intensity of both interference patterns can be approximated by cosine functions:

I1(X ,Y ) =A1

2(1+ cos(KX +Φ(s)))+B1, (1)

I2(X ,Y ) =A2

2(1− cos(KX +Φ(s)))+B2. (2)

A1 and A2 are the amplitude intensities of the interferograms, B1 and B2 are the backgroundintensities. The fringe spacing 2π/K = λ0z′0/s′ is determined by the seperation s′ between thevirtual images and can be set arbitrarily by adjusting the mirrors M1 and M2. The phase Φ ofthe interferogram equals the phase difference between both spherical waves at their origins P′and R′ at a particular time. This in turn reflects the phase difference of the illuminating laserbeam at the particle positions P and R in the object plane at a particular time, which is relatedto the separation s between both particles, plus a constant value resulting from a difference ofoptical path of the interferometer arms. Hence, a phase change dΦ in the interferogram is a very


P’ R’ x’

z’-ρ ρ

z0’

s’

θ’

I(X)

XX-Y-plane

x’-y’-plane

Fig. 2. Detail of the interferometer in Fig. 1. The appearence of the interferogram can beseen as the superposition of two spherical waves originating at the virtual particle positionsP′ and R′. The resulting interference pattern is for small aperture angles θ ′ in first-orderapproximation a cosine function with a fringe spacing depending on the separation s′ be-tween P′ and R′. The phase of the cosine function equals the phase difference between bothspherical waves at their origins at a particular time.

sensitive measure for a movement ds of the particles relative to each other. Furthermore, Φ islargely independent of combined movements of the reference particle and the probe particle,like microscope stage drift or motion of the illuminating laser beam relative to the object.

R

α

P(t1) P(t2)

xλ

xλ

0λ

dss(t2)

s(t1)

x

z

water

air

glass cover slip

laser

Fig. 3. Phase fronts of the illuminating laser beam in the object plane. The different refrac-tive indices of air, glass, and water do not affect the distance λx = λ0/sin(α) between twoconsecutive wavefronts in x-direction. The angle between the laser beam and the normal tothe object plane is α = 80◦. From geometrical considerations a relation can be obtained be-tween the displacement ds of the probe particle relative to the reference particle and a shiftof the phase difference dΦ of the illuminating laser beam at both particles at a particulartime.

The relation between ds and dΦ can be derived from a consideration of the local phasefronts of the illuminating laser light field in the object plane (Fig. 3). The crucial quantity is thedistance λx between two subsequent wavefronts in x-direction. Although the laser beam withvacuum wavelength λ0 passes materials with different refractive indices and hence changes itswavelength, this does not affect λx. The illuminating laser beam and the normal to the object


plane enclose an angle of α = 80◦. Close to the beam axis, where the wavefronts are perpen-dicular to the beam axis, one has λx = λ0/sin(α). A displacement ds of the probe particle inx-direction relative to the reference particle causes a phase shift in the interferogram

dΦ =2πλx

ds =2π sin(α)

λ0ds. (3)

For α = 80◦ and λ0 = 532 nm we obtain ds/dΦ = 86nm/rad.The total radiation power of the interferograms is each measured with an APD. For small

particle displacements ds we can assume the fringe spacing 2π/K of the interferogram to beconstant, only its phase Φ changes according to Eq. (3). Then the current signals delivered bythe APDs are functions of Φ:

S1(Φ) =∫

aperture

I1(X ,Y,Φ)dXdY = χ(t)(a1

2(1+ cos(Φ))+b1

), (4)

S2(Φ) =∫

aperture

I2(X ,Y,Φ)dXdY = χ(t)(a2

2(1− cos(Φ))+b2

). (5)

χ(t) is a dimensionless factor describing laser power fluctuations. a1 and a2 are the amplitudesof the current signals, b1 and b2 are the background signals. The amplitudes and backgroundsignals depend on the aperture (determined by θ ′) and the fringe spacing of the interferograms.If the interferometer is adjusted such that the virtual particle positions P′ and R′ coincide (s′ =0), the interference patterns are homogeneous without fringes (K = 0) as described above. Then,the amplitudes a1 and a2 are maximal and the background signals b1 and b2 are minimal. Thephase can be retrieved from both measured signals by

Φ = arccos

(2

S1(b2 +a2/2)−S2(b1 +a1/2)S1a2 +S2a1

). (6)

The signal parameters a1, a2, b1 and b2 must be known. They are obtained with a precedingcalibration measurement where a piezo actuator in the interferometer moves mirror M2 backand forth, so that the minimum and maximum values of the current signals can be determined.Note, that this phase retrieval is insensitive to laser power fluctuations. It takes advantage of thefact that this disturbance affects both detection channels at the same time. Hence, laser powerfluctuations are eliminated effectively from the result and no active stabilization of the laserpower is required [13].

However, the phase retrieval becomes defective when the current signals exceed an extremalvalue (Φ = 0,Φ = π , ...), because the output of the arccosine is limited to values in the interval0...π . For instance, a current signal corresponding to Φ slightly below π cannot be distinguishedfrom that corresponding to Φ slightly above π . As a consequence, the sign of the direction ofthe retrieved particle movement ds/dt is undetermined and changes when a Φ-value of an in-teger multiple of π is exceeded. There, the retrieved particle position s of a linear movementhas an extremal value. Furthermore, the phase detection (6) is based on the assumption thatthe fringe spacing of the interference pattern does not change significantly. But it does so ifthe probe particle moves a longer distance. Then, the amplitude and background of the currentsignals change and the phase retrieval becomes biased. In order to identify the limits of thecovered distance in which the phase detection is sufficiently accurate, we have simulated thedevelopment of the interference patterns for a moving probe particle. For this purpose the inter-ference patterns were computed as the superposition of two spherical waves with aperture angleθ ′ and distance s′, starting with homogeneous interference patterns (K = 0) at s′ = 0. The total


radiation power of each interferogram serves to retrieve the phase Φ and the covered distances via (6) and (3). s− s is the bias. As a rule of thumb we find that within a covered distanceof s < 200 nm, the bias due to the change of the fringe spacing is below 1 nm. Caution mustbe used in the vicinity of the extremal values of the current signal. There, until approximately25 nm before s reaches an extremal value, the bias is below 1 nm.

We applied differential interferometric particle tracking to measuring the movement of a ki-nesin (Nkin 433) along microtubules. Motility assays were performed in flow chambers assem-bled from a microscope slide, two parallel strips of double-sided tape, and a DETA-coated mi-croscope cover slip on top. Gold nanospheres with diameter 200 nm (BBInternational), whichserve as reference particles, are randomly distributed and immobilized on the cover slip sur-face. The chamber is successively flushed with a mix of microtubules, casein and polystyrenebeads (diameter 500 nm, Kisker Biotech) with kinesin linked to their surface. The microtubulesstrongly attach to the cover glass surface due to the DETA coating. They are alligned paral-lel to the flow direction, which is made to coincide with the x-direction in the object plane.Once the microtubules are fixed, the remaining cover slip surface is covered with casein so thatthe polystyrene beads will hardly stick to it. The kinesin linked to the protein G-coated beadsurface by a anti-His6 tag antibody binds with AMP-PNP to a microtubule. Finally, after theinterferometer has been adjusted for a selected pair of reference and probe particle, ATP at aconcentration of 2.5 mM is pipetted into the flow chamber and after roughly 20 s the kinesinstarts walking along the microtubules.

3. Results

3.1. Verification of the phase-distance relation

Fig. 4. Experimental verification of the phase-distance relation in Eq. (3). Focussed (A) andslightly defocussed (B) image of an immobilized pair of gold nanospheres with diameter200 nm in a flow chamber. (C) Gaussian functions (blue curve) are fit to the intensityprofile (black dots) of the focussed image in order to obtain the particle distance s. (D)The intensity profile of the defocussed image serves to determine the phase difference Φbetween the waves of scattered light from both particles by fitting a cosine function to thecentral part. (E) s plotted against Φ for several particle pairs (circles) compared with thecomputed relation from Eq. (3) (solid straight line).

In order to verify the relation between a displacement ds of the probe particle relative tothe reference particle and the corresponding phase shift of the interferograms dΦ (Eq. (3)), aflow chamber with immobilized gold nanospheres is prepared as described above and flooded


with water. Pairs of gold particles with a distance of roughly 2 μm which are arranged in x-direction are searched for. A camera is positioned in place of mirror M1 in the intermediateimage plane of the microscope and a focussed and a slightly defocussed image of each pair istaken (Figs. 4(A) and 4(B)). The defocussed image shows interference fringes as the blurredspots of both particles overlap. By summing up the pixel values of each pixel column, an inten-sity profile is obtained for each image. The particle distance s is determined with an accuracy ofapproximately 20 nm by fitting Gaussian functions to the intensity profile of a focussed image(Fig. 4(C)). A cosine function is fitted to the central part of the intensity profile of the corre-sponding defocussed image (Fig. 4(D)) in order to obtain the phase difference Φ between theinterfering waves (compare with Fig. 2). In Fig. 4(E), Φ is plotted against s for several particlepairs (circles) together with the phase-distance relation computed with Eq. (3) with startingvalue Φ(s = 0) = 0 (solid straight line). Since the cosine fit delivers phase values only fromthe interval 0...2π , a multiple integer of 2π is added to the measured phase for better compa-rability with the computed values. This integer is gained from the measured particle distances: The phase is supposed to repeat after every 540 nm of increased particle distance, thereforef loor(s/540 nm)·2π is added to the measured Φ. Fig. 4(E) shows that the measured and thecomputed relation between phase and particle distance match very well.

3.2. Spatial resolution and drift elimination

The spatial resolution of a motility assay as described above is mainly limited by three errorsources: instrumentation error, movement of the microtubules, and thermal motion of the beadsresulting from flexibility of the bead-microtubule linkage [16]. Instrumentation error includingmechanical instabilities of the experimental setup as well as electronic noise, can be evaluatedby adsorbing beads on a dry surface. The laser power is adapted such that the signal amplitudesand backgrounds are similar to those of motility assays. Measurements with a bandwidth of5 kHz and a measuring time of 60 s show a nearly drift free particle separation s with a standarddeviation of only 0.9 nm (Fig. 5, orange curve). In order to quantify the drift, the data has beensmoothed with a moving average with window width 1 s (black curve). The filtered data coversa range of only 0.4 nm, which is a measure for the microscope stage drift within one minute.This demonstrates the power of the drift elimination which is inherent in our tracking method.Typical values for microscope stage drift are roughly 5 nm/sec [16].

3.3. Kinesin steps

Figure 6 shows two distinct tracks of bead movement driven by kinesin (blue and red) and thetrack of a stationary bead sticking to the cover glass surface of a flooded flow chamber withoutkinesin (black). The tracks were recorded with a sample rate of 40 kHz. However, for the sakeof noise reduction, the plotted curves represent data binned and averaged in intervals of 100samples. In Fig. 6(A) the starting of a bead movement at t = 25 s is displayed. t = 0 representsthe beginning of the process of flushing the flow chamber with ATP, which takes roughly 5 s.Due to this interval we can rule out the possibility that the observed movement is caused by thestream of the flushing process. The measured path reveals several steps and plateaus. A chosensection with length 1 s, indicated by a black rectangle, is plotted in Fig. 6(B) more in detail(red) together with another kinesin track (blue) and the track of a stationary bead sticking tothe cover glass surface (black). Both kinesin paths differ strongly in noise and velocity. In thered curve one 8-nm step and another 16-nm step, which possibly consists of two consecutive8-nm steps, can be clearly recognized (marked by arrows). This corresponds to an averagevelocity of 24 nm/sec. The standard deviation of the unbinned data in the straight section (t =29.2 s...29.5 s) is 3.8 nm, which is nearly the same for the track of the stationary bead. Theblue curve displayes the starting of another bead movement after ATP has been injected into


Fig. 5. Demonstration of the instrumentation error and drift elimination: Reference andprobe particle are both immobilized on a dry glass surface. Measurement over 60 s at abandwidth of 5 kHz. The reconstructed particle separation s− s0 (orange curve, s0 is themean value) has a standard deviation of 0.9 nm. The black curve represents the filtered datawhich has been smoothed with a moving average with window width 1s. The filtered datacovers a range of 0.4 nm.

Fig. 6. (A) Starting of a kinesin-driven bead movement (at t = 25 s). ATP has been flushedinto the flow channel at t = 0. (B) More detailed depiction of the section indicated in (A)by the black rectangle (red curve) together with another starting of a kinesin-driven beadmovement (blue) and of a bead stuck to the glass surface of the flow chamber (black). Inthe red curve three 8-nm steps can be clearly recognized (marked by arrows). The blue andthe black curve are shifted in s-direction. Horizontal lines are spaced at 8-nm intervals.

the flow chamber. The kinesin starts moving at t = 29.8 s. The average velocity is then roughly250 nm/sec. Single steps can not be resolved since the track is superimposed by much largernoise. The maximal peak-to-peak displacement in the unbinned data in the straight section(t = 29.2 s...29.8 s) is roughly 200 nm. This is consistent with further findings and agrees withthe range one would expect from geometrical considerations for constrained Brownian motionof a bead with diameter 500 nm attached to an immobile microtubule on a substrate with zero-length linkage when the attachment point can move freely on the microtubule [17].

In order to further examine the origin of this noise, we have recorded tracks of both kinesindriven beads before ATP has been flushed into the flow chamber. The power spectral densities


Fig. 7. Power spectral densities of the tracks from Fig. 6 (same colors) before ATP has beenadded. The orange curve is the power spectral density of the track plotted in Fig. 5. Blackcurve: Lorentzian fit of the measured spectrum (black dots).

of these two measurements as well as that of the stationary bead track are plotted in Fig. 7.The graphs can be well approximated by a Lorentzian, which describes the power spectraldensity of Brownian motion of a particle in a harmonic potential [18]. To illustrate this, themeasured power spectral density of the stationary bead (black dots) is plotted together with aLorentz fit (black solid curve). By contrast, the power spectral density of the track of beadson a dry surface (orange, based on the data plotted in Fig. 5) is largely uniform. We inferthat the noise in the tracks in Fig. 6 arises from constrained Brownian bead motion. The slowbead (red) is supposed to be confined by sticking to the cover glass surface whereas the fastbead (blue) is trapped to a volume delimited by the geometry and the flexibility of the linkagebetween bead and microtubule, which consists of the kinesin and the anti-His6 tag antibody.The steps in the red curve are possibly the generic 8-nm steps of a kinesin pulling a stuck beadwith a tensioned linkage between bead and microtubule. For frequencies greater than 300 Hzthe curves, except that of the dry assay, have a log-log slope of -2, which characterizes purelydiffusive motion in the local viscous environment. These sections of the spectra differ from eachother by a multiplicative factor. Its inverse reflects the viscous drag. Compared to the stationarybead, the viscous drag for the slow and the fast bead are decreased by a factor of ≈ 1.4 and≈ 15, respectively. Hence, the various confinements are reflected in different viscous drags.It is known from literature that viscous drag force causes a decrease of kinesin velocity [19].This explains the different observed velocities of the traces in Fig. 6(B). We are monitoring thestarting of displacement of the beads when ATP has been flushed into the channel. Reducedvelocity could be contributed by a residual AMP-PNP concentration in the solution. Also whencompared to microtubule gliding assays, kinesin’s average velocity under saturating conditionsis measured for gliding microtubules over a period of tens of seconds at steady state. In our casewe observed the initial bead movement over maximally 1 s interval, which might underestimatethe velocity over longer observation times.

4. Conclusion

We have introduced differential interferometric particle tracking, which provides high spatial(Δx = 0.9nm) and temporal (Δt = 200μs) resolution. The setup is relatively simple, since themethod eliminates microscope stage drift, pointing fluctuations of the illuminating laser beam


and laser power fluctuations inherently. Therefore there is no need to stabilize the setup actively.Only two single channel detectors are sufficient to provide a one-dimensional measurement ofparticle motion. We have applied this technique to measuring the movement of kinesin alongmicrotubules and were able to clearly recognize 8-nm steps. Although the sign of the directionof motion remains undetermined, this is not a problem for applications like tracking molecularmotors. Differential interferometric particle tracking works well for a covered distance of up to200 nm. The high temporal resolution enabled us to investigate the power spectral densities ofdifferent tracks. From this we could infer the local confinement of individual tracked beads.

Acknowledgments

This work was supported by the German Research Foundation (DFG) in the framework of SFB755 "Nanoscale Photonic Imaging".


differential interferometric particle tracking on the subnanometer- and submillisecond-scale

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