We describe an interferometric method to measure the movement of a subwavelength probe particle relative to an immobilized reference particle with high spatial (Δx = 0.9nm) and temporal (Δt = 200μs) resolution. The differential method eliminates microscope stage drift. An upright microscope is equipped with laser dark field illumination (λ0 = 532nm, P0 = 30mW) and a compact modified Mach-Zehnder interferometer is mounted on the camera exit of the microscope, where the beams of scattered light of both particles are combined. The resulting interferograms provide in two channels subnanometer information about the motion of the probe particle relative to the reference particle. The interferograms are probed with two avalanche photodiodes. We applied this method to measuring the movement of kinesin along microtubules and were able to resolve the generic 8-nm steps at high ATP concentrations without external forces.
© 2013 OSA
Single-particle tracking experiments are widely used to monitor dynamical processes on length scales from millimeter to subnanometer. Especially in biology, where relevant processes on the molecular level demand high spatial and temporal resolution, numerous optical methods have emerged. Either fluorescent, phase shifting, or light scattering markers have been used to visualize the movement of molecular specimens. The advancement of various tracking techniques has provided spatial precision far below the diffraction limit of visible light [1–3]. This has revealed insights into the mechanics of single biological molecules [4–6]. For instance, it has been directly observed that the molecular motor kinesin moves along microtubules with 8-nm steps . With big experimental effort it was even possible to resolve the 0.37-nm steps of RNA polymerase moving along DNA . High temporal resolution can be achieved either with a high-speed camera  or, which is much more cost-efficient, by using only few detector channels (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 the specimen itself. These issues can be moderated with more or less experimental effort. The finite photon flux, especially a constraint for fluorescent markers, limits the number of photons per frame, which in turn poses a fundamental limitation for the localization precision . This requires an appropriate choice of the sample rate but still results in a compromise between spatial and temporal resolution. In case of scattering particles there is no fundamental limitation of 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 as small as 10 nm have been tracked in a time window of about one microsecond . When a bigger light scattering marker is attached to a biological specimen, the flexible linkage often allows a relatively free motion of the marker within a confined volume. Thermal motion hence deteriorates the precision of localization of the specimen. The use of optical tweezers allows application and measurement of forces as well as the attenuation of the Brownian motion. Pointing fluctuations of the trapping laser beam due to random air currents have been suppressed by enclosing all optical components in a sealed box with helium gas . Microscope stage drift can be compensated by using markers in the object plane, whose motion is measured and subsequently substracted from the measured motion of the specimen. Other methods are to track a fixed marker in the object plane and then actively stabilize the microscope stage  or to decouple the whole setup from thermal fluctuations, which cause drift, by fully automating and placing it in a separate room .
We present a new method, which we call differential interferometric particle tracking. It combines a relatively simple experimental setup with high spatial and temporal resolution and inherent elimination of microscope stage drift. Differential interferometric particle tracking is based on interference between the scattered light of a probe particle and a fixed reference particle. An upright microscope has been equipped with laser dark field illumination, and a compact interferometer has been mounted on top of the camera exit of the microscope. Two avalanche photodiodes suffice to detect the interference signal, which provides information about one degree of freedom of the particle movement. Furthermore, this detection scheme is insensitive to laser power fluctuations and pointing fluctuations of the illuminating beam. To demonstrate the performance of this method we have applied it to the protein complex kinesin microtubule. The kinesin is labelled with a polystyrene bead with diameter 0.5μm. We could clearly resolve steps with a size of 8 nm without the use of an optical tweezer .
The basic idea of our tracking method is to overlay the beams of scattered light of a fixed reference particle and a motile probe particle in such a way that the resulting interference pattern yields precise information about the motion of both particles relative to each other. For this purpose 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 the vicinity of the object. Subwavelength particles in the object plane scatter in a first-order approximation spherical waves collected in the aperture angle θ of the objective lense (Olympus LM Plan FL 50×/0.50 BD, working distance: 10.6 mm). The beams of the reference particle at position R and the probe particle at position P are seperated from each other in the intermediate image plane of the microscope by mirror M1 and are each directed into one arm of the modified Mach-Zehnder interferometer. Both beams are overlayed with each other at the beam splitter BS. 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 at the positions P′ and R′ as well as the direction of propagation of both beams coincide, independent of the real particle separation s. Then, behind BS, the wavefronts of both beams match and both interference patterns are homogeneous spots without fringes. Their total radiation power is a function of the phase difference Φ between both beams. The convex lenses L1 and L2 behind each exit facet of the beam splitter create further intermediate image planes where the pinholes PH1 and PH2 select the overlay of the probe particle and the reference particle. The setup thus blocks stray light and scattered light from other particles in the object plane. The pinholes cut out a circular area corresponding to a diameter of 5 μm in the object plane, so that the minimum separation distance between both particles in the object plane should have this size. In order to adjust the positions of PH1 and PH2 a camera can be swivelled in the optical path behind each pinhole. In this way also the shape of the interferograms can be monitored, which is important for the adjustment of the interferometer. The avalanche photodiodes APD1 and APD2 behind each pinhole detect the total radiation power of both interferograms.
The interference pattern in the X-Y-plane behind each of the beam splitter exit facets can be seen as the superposition of two spherical waves each limited by the aperture angle θ′ = 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 is the 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 line is parallel to the x′-axis, the situation is similar to Young’s double slit experiment . For a small θ′ the intensity of both interference patterns can be approximated by cosine functions:
The relation between ds and dΦ can be derived from a consideration of the local phase fronts of the illuminating laser light field in the object plane (Fig. 3). The crucial quantity is the distance λx between two subsequent wavefronts in x-direction. Although the laser beam with vacuum wavelength λ0 passes materials with different refractive indices and hence changes its wavelength, 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 perpendicular to the beam axis, one has λx = λ0/ sin(α). A displacement ds of the probe particle in x-direction relative to the reference particle causes a phase shift in the interferogram
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 be constant, only its phase Φ changes according to Eq. (3). Then the current signals delivered by the APDs are functions of Φ:13].
However, the phase retrieval becomes defective when the current signals exceed an extremal value (Φ = 0, Φ = π, ...), because the output of the arccosine is limited to values in the interval 0...π. For instance, a current signal corresponding to Φ slightly below π cannot be distinguished from that corresponding to Φ slightly above π. As a consequence, the sign of the direction of the retrieved particle movement dŝ/dt is undetermined and changes when a Φ-value of an integer multiple of π is exceeded. There, the retrieved particle position ŝ of a linear movement has an extremal value. Furthermore, the phase detection (6) is based on the assumption that the fringe spacing of the interference pattern does not change significantly. But it does so if the probe particle moves a longer distance. Then, the amplitude and background of the current signals change and the phase retrieval becomes biased. In order to identify the limits of the covered distance in which the phase detection is sufficiently accurate, we have simulated the development of the interference patterns for a moving probe particle. For this purpose the interference patterns were computed as the superposition of two spherical waves with aperture angle θ′ 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 Φ̂ and the covered distance ŝ via (6) and (3). ŝ− s is the bias. As a rule of thumb we find that within a covered distance of s < 200 nm, the bias due to the change of the fringe spacing is below 1 nm. Caution must be used in the vicinity of the extremal values of the current signal. There, until approximately 25 nm before ŝ reaches an extremal value, the bias is below 1 nm.
We applied differential interferometric particle tracking to measuring the movement of a kinesin (Nkin 433) along microtubules. Motility assays were performed in flow chambers assembled from a microscope slide, two parallel strips of double-sided tape, and a DETA-coated microscope cover slip on top. Gold nanospheres with diameter 200 nm (BBInternational), which serve as reference particles, are randomly distributed and immobilized on the cover slip surface. The chamber is successively flushed with a mix of microtubules, casein and polystyrene beads (diameter 500 nm, Kisker Biotech) with kinesin linked to their surface. The microtubules strongly attach to the cover glass surface due to the DETA coating. They are alligned parallel 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 that the polystyrene beads will hardly stick to it. The kinesin linked to the protein G-coated bead surface by a anti-His6 tag antibody binds with AMP-PNP to a microtubule. Finally, after the interferometer has been adjusted for a selected pair of reference and probe particle, ATP at a concentration of 2.5 mM is pipetted into the flow chamber and after roughly 20 s the kinesin starts walking along the microtubules.
3.1. Verification of the phase-distance relation
In order to verify the relation between a displacement ds of the probe particle relative to the reference particle and the corresponding phase shift of the interferograms dΦ (Eq. (3)), a flow 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 intermediate image plane of the microscope and a focussed and a slightly defocussed image of each pair is taken (Figs. 4(A) and 4(B)). The defocussed image shows interference fringes as the blurred spots of both particles overlap. By summing up the pixel values of each pixel column, an intensity profile is obtained for each image. The particle distance s is determined with an accuracy of approximately 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 corresponding defocussed image (Fig. 4(D)) in order to obtain the phase difference Φ between the interfering waves (compare with Fig. 2). In Fig. 4(E), Φ is plotted against s for several particle pairs (circles) together with the phase-distance relation computed with Eq. (3) with starting value Φ(s = 0) = 0 (solid straight line). Since the cosine fit delivers phase values only from the interval 0...2π, a multiple integer of 2π is added to the measured phase for better comparability with the computed values. This integer is gained from the measured particle distance s: The phase is supposed to repeat after every 540 nm of increased particle distance, therefore floor(s/540 nm)·2π is added to the measured Φ. Fig. 4(E) shows that the measured and the computed 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 error sources: instrumentation error, movement of the microtubules, and thermal motion of the beads resulting from flexibility of the bead-microtubule linkage . Instrumentation error including mechanical instabilities of the experimental setup as well as electronic noise, can be evaluated by adsorbing beads on a dry surface. The laser power is adapted such that the signal amplitudes and backgrounds are similar to those of motility assays. Measurements with a bandwidth of 5 kHz and a measuring time of 60 s show a nearly drift free particle separation s with a standard deviation of only 0.9 nm (Fig. 5, orange curve). In order to quantify the drift, the data has been smoothed with a moving average with window width 1 s (black curve). The filtered data covers a 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 .
3.3. Kinesin steps
Figure 6 shows two distinct tracks of bead movement driven by kinesin (blue and red) and the track of a stationary bead sticking to the cover glass surface of a flooded flow chamber without kinesin (black). The tracks were recorded with a sample rate of 40 kHz. However, for the sake of noise reduction, the plotted curves represent data binned and averaged in intervals of 100 samples. In Fig. 6(A) the starting of a bead movement at t = 25 s is displayed. t = 0 represents the 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 the stream of the flushing process. The measured path reveals several steps and plateaus. A chosen section 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 to the cover glass surface (black). Both kinesin paths differ strongly in noise and velocity. In the red curve one 8-nm step and another 16-nm step, which possibly consists of two consecutive 8-nm steps, can be clearly recognized (marked by arrows). This corresponds to an average velocity 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. The blue curve displayes the starting of another bead movement after ATP has been injected into the flow chamber. The kinesin starts moving at t = 29.8 s. The average velocity is then roughly 250 nm/sec. Single steps can not be resolved since the track is superimposed by much larger noise. 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 with the range one would expect from geometrical considerations for constrained Brownian motion of 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 .
In order to further examine the origin of this noise, we have recorded tracks of both kinesin driven beads before ATP has been flushed into the flow chamber. The power spectral densities 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 spectral density of Brownian motion of a particle in a harmonic potential . To illustrate this, the measured power spectral density of the stationary bead (black dots) is plotted together with a Lorentz fit (black solid curve). By contrast, the power spectral density of the track of beads on a dry surface (orange, based on the data plotted in Fig. 5) is largely uniform. We infer that the noise in the tracks in Fig. 6 arises from constrained Brownian bead motion. The slow bead (red) is supposed to be confined by sticking to the cover glass surface whereas the fast bead (blue) is trapped to a volume delimited by the geometry and the flexibility of the linkage between 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 bead with a tensioned linkage between bead and microtubule. For frequencies greater than 300 Hz the curves, except that of the dry assay, have a log-log slope of −2, which characterizes purely diffusive motion in the local viscous environment. These sections of the spectra differ from each other by a multiplicative factor. Its inverse reflects the viscous drag. Compared to the stationary bead, 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 . This explains the different observed velocities of the traces in Fig. 6(B). We are monitoring the starting of displacement of the beads when ATP has been flushed into the channel. Reduced velocity could be contributed by a residual AMP-PNP concentration in the solution. Also when compared to microtubule gliding assays, kinesin’s average velocity under saturating conditions is measured for gliding microtubules over a period of tens of seconds at steady state. In our case we observed the initial bead movement over maximally 1 s interval, which might underestimate the velocity over longer observation times.
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 the method 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 of particle motion. We have applied this technique to measuring the movement of kinesin along microtubules and were able to clearly recognize 8-nm steps. Although the sign of the direction of motion remains undetermined, this is not a problem for applications like tracking molecular motors. Differential interferometric particle tracking works well for a covered distance of up to 200 nm. The high temporal resolution enabled us to investigate the power spectral densities of different tracks. From this we could infer the local confinement of individual tracked beads.
This work was supported by the German Research Foundation (DFG) in the framework of SFB 755 “Nanoscale Photonic Imaging”.
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