Abstract

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.

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  1. S. Kamimura, “Direct measurement of nanometric displacement under an optical microscope,” Appl. Opt.26, 3425–3427 (1987).
    [CrossRef] [PubMed]
  2. W. Denk and W. W. Webb, “Optical measurement of picometer displacements of transparent microscopic objects,” Appl. Opt.29, 2382–2391 (1990).
    [CrossRef] [PubMed]
  3. R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap.” Biophys. J.70, 1813–1822 (1996).
    [CrossRef] [PubMed]
  4. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

2010 (1)

2009 (1)

2007 (1)

2006 (1)

2005 (1)

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, “Direct observation of base-pair stepping by rna polymerase,” Nature438, 460–465 (2005).
[CrossRef] [PubMed]

2003 (2)

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

G. Cappello, M. Badoual, A. Ott, J. Prost, and L. Busoni, “Kinesin motion in the absence of external forces characterized by interference total internal reflection microscopy,” Phys. Rev. E68, 021907 (2003).
[CrossRef]

2002 (1)

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J.82, 2775–2783 (2002).
[CrossRef] [PubMed]

2001 (1)

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).
[CrossRef] [PubMed]

1998 (1)

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J.74, 1074–1085 (1998).
[CrossRef] [PubMed]

1996 (1)

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap.” Biophys. J.70, 1813–1822 (1996).
[CrossRef] [PubMed]

1995 (1)

Z. Wang, S. Khan, and M. P. Sheetz, “Single cytoplasmic dynein molecule movements: Characterization and comparison with kinesin,” Biophys. J.69, 2011–2023 (1995).
[CrossRef] [PubMed]

1994 (2)

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct.23, 247–285 (1994).
[CrossRef]

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load.” Biophys. J.67, 766–781 (1994).
[CrossRef] [PubMed]

1993 (1)

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature365, 721–727 (1993).
[CrossRef] [PubMed]

1990 (1)

1988 (1)

J. Gelles, B. J. Schnapp, and M. P. Sheetz, “Tracking kinesin-driven movements with nanometre-scale precision,” Nature331, 450–453 (1988).
[CrossRef] [PubMed]

1987 (1)

Abbondanzieri, E. A.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, “Direct observation of base-pair stepping by rna polymerase,” Nature438, 460–465 (2005).
[CrossRef] [PubMed]

Alchenberger, D.

Allersma, M. W.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J.74, 1074–1085 (1998).
[CrossRef] [PubMed]

Badoual, M.

G. Cappello, M. Badoual, A. Ott, J. Prost, and L. Busoni, “Kinesin motion in the absence of external forces characterized by interference total internal reflection microscopy,” Phys. Rev. E68, 021907 (2003).
[CrossRef]

Block, S. M.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, “Direct observation of base-pair stepping by rna polymerase,” Nature438, 460–465 (2005).
[CrossRef] [PubMed]

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct.23, 247–285 (1994).
[CrossRef]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature365, 721–727 (1993).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon PressNew York, 1980).

Brunner, C.

Busoni, L.

G. Cappello, M. Badoual, A. Ott, J. Prost, and L. Busoni, “Kinesin motion in the absence of external forces characterized by interference total internal reflection microscopy,” Phys. Rev. E68, 021907 (2003).
[CrossRef]

Cappello, G.

G. Cappello, M. Badoual, A. Ott, J. Prost, and L. Busoni, “Kinesin motion in the absence of external forces characterized by interference total internal reflection microscopy,” Phys. Rev. E68, 021907 (2003).
[CrossRef]

Carter, A. R.

Chu, S.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap.” Biophys. J.70, 1813–1822 (1996).
[CrossRef] [PubMed]

Czerwinski, F.

deCastro, M. J.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J.74, 1074–1085 (1998).
[CrossRef] [PubMed]

Denk, W.

Finer, J. T.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap.” Biophys. J.70, 1813–1822 (1996).
[CrossRef] [PubMed]

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Gelles, J.

J. Gelles, B. J. Schnapp, and M. P. Sheetz, “Tracking kinesin-driven movements with nanometre-scale precision,” Nature331, 450–453 (1988).
[CrossRef] [PubMed]

Gittes, F.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J.74, 1074–1085 (1998).
[CrossRef] [PubMed]

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load.” Biophys. J.67, 766–781 (1994).
[CrossRef] [PubMed]

Goldman, Y. E.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Gornall, J. L.

Greenleaf, W. J.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, “Direct observation of base-pair stepping by rna polymerase,” Nature438, 460–465 (2005).
[CrossRef] [PubMed]

Ha, T.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Halsey, W.

Higuchi, H.

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).
[CrossRef] [PubMed]

Howard, J.

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load.” Biophys. J.67, 766–781 (1994).
[CrossRef] [PubMed]

Hunt, A. J.

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load.” Biophys. J.67, 766–781 (1994).
[CrossRef] [PubMed]

Inoue, Y.

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).
[CrossRef] [PubMed]

Jacobsen, V.

Kamimura, S.

Keyser, U. F.

Khan, S.

Z. Wang, S. Khan, and M. P. Sheetz, “Single cytoplasmic dynein molecule movements: Characterization and comparison with kinesin,” Biophys. J.69, 2011–2023 (1995).
[CrossRef] [PubMed]

King, G. M.

Landick, R.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, “Direct observation of base-pair stepping by rna polymerase,” Nature438, 460–465 (2005).
[CrossRef] [PubMed]

Larson, D. R.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J.82, 2775–2783 (2002).
[CrossRef] [PubMed]

Mahamdeh, M.

McKinney, S. A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Muto, E.

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).
[CrossRef] [PubMed]

Nishiyama, M.

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).
[CrossRef] [PubMed]

Oddershede, L. B.

Ott, A.

G. Cappello, M. Badoual, A. Ott, J. Prost, and L. Busoni, “Kinesin motion in the absence of external forces characterized by interference total internal reflection microscopy,” Phys. Rev. E68, 021907 (2003).
[CrossRef]

Otto, O.

Perkins, T. T.

Prost, J.

G. Cappello, M. Badoual, A. Ott, J. Prost, and L. Busoni, “Kinesin motion in the absence of external forces characterized by interference total internal reflection microscopy,” Phys. Rev. E68, 021907 (2003).
[CrossRef]

Sandoghdar, V.

Schäffer, E.

Schmidt, C. F.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J.74, 1074–1085 (1998).
[CrossRef] [PubMed]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature365, 721–727 (1993).
[CrossRef] [PubMed]

Schnapp, B. J.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature365, 721–727 (1993).
[CrossRef] [PubMed]

J. Gelles, B. J. Schnapp, and M. P. Sheetz, “Tracking kinesin-driven movements with nanometre-scale precision,” Nature331, 450–453 (1988).
[CrossRef] [PubMed]

Seidel, R.

Selvin, P. R.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Shaevitz, J. W.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, “Direct observation of base-pair stepping by rna polymerase,” Nature438, 460–465 (2005).
[CrossRef] [PubMed]

Sheetz, M. P.

Z. Wang, S. Khan, and M. P. Sheetz, “Single cytoplasmic dynein molecule movements: Characterization and comparison with kinesin,” Biophys. J.69, 2011–2023 (1995).
[CrossRef] [PubMed]

J. Gelles, B. J. Schnapp, and M. P. Sheetz, “Tracking kinesin-driven movements with nanometre-scale precision,” Nature331, 450–453 (1988).
[CrossRef] [PubMed]

Simmons, R. M.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap.” Biophys. J.70, 1813–1822 (1996).
[CrossRef] [PubMed]

Spudich, J. A.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap.” Biophys. J.70, 1813–1822 (1996).
[CrossRef] [PubMed]

Stewart, R. J.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J.74, 1074–1085 (1998).
[CrossRef] [PubMed]

Stober, G.

Stoller, P.

Svoboda, K.

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct.23, 247–285 (1994).
[CrossRef]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature365, 721–727 (1993).
[CrossRef] [PubMed]

Thompson, R. E.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J.82, 2775–2783 (2002).
[CrossRef] [PubMed]

Ulrich, T. A.

Vogel, V.

Wang, Z.

Z. Wang, S. Khan, and M. P. Sheetz, “Single cytoplasmic dynein molecule movements: Characterization and comparison with kinesin,” Biophys. J.69, 2011–2023 (1995).
[CrossRef] [PubMed]

Webb, W. W.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J.82, 2775–2783 (2002).
[CrossRef] [PubMed]

W. Denk and W. W. Webb, “Optical measurement of picometer displacements of transparent microscopic objects,” Appl. Opt.29, 2382–2391 (1990).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon PressNew York, 1980).

Yanagida, T.

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).
[CrossRef] [PubMed]

Yildiz, A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Annu. Rev. Biophys. Biomol. Struct. (1)

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct.23, 247–285 (1994).
[CrossRef]

Appl. Opt. (3)

Biophys. J. (5)

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J.82, 2775–2783 (2002).
[CrossRef] [PubMed]

Z. Wang, S. Khan, and M. P. Sheetz, “Single cytoplasmic dynein molecule movements: Characterization and comparison with kinesin,” Biophys. J.69, 2011–2023 (1995).
[CrossRef] [PubMed]

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J.74, 1074–1085 (1998).
[CrossRef] [PubMed]

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap.” Biophys. J.70, 1813–1822 (1996).
[CrossRef] [PubMed]

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load.” Biophys. J.67, 766–781 (1994).
[CrossRef] [PubMed]

Nat. Cell Biol. (1)

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).
[CrossRef] [PubMed]

Nature (3)

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature365, 721–727 (1993).
[CrossRef] [PubMed]

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, “Direct observation of base-pair stepping by rna polymerase,” Nature438, 460–465 (2005).
[CrossRef] [PubMed]

J. Gelles, B. J. Schnapp, and M. P. Sheetz, “Tracking kinesin-driven movements with nanometre-scale precision,” Nature331, 450–453 (1988).
[CrossRef] [PubMed]

Opt. Express (3)

Phys. Rev. E (1)

G. Cappello, M. Badoual, A. Ott, J. Prost, and L. Busoni, “Kinesin motion in the absence of external forces characterized by interference total internal reflection microscopy,” Phys. Rev. E68, 021907 (2003).
[CrossRef]

Science (1)

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science300, 2061–2065 (2003).
[CrossRef] [PubMed]

Other (1)

M. Born and E. Wolf, Principles of Optics (Pergamon PressNew York, 1980).

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Figures (7)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Phase fronts of the illuminating laser beam in the object plane. The different refractive indices of air, glass, and water do not affect the distance λx = λ0/ sin(α) between two consecutive wavefronts in x-direction. The angle between the laser beam and the normal to the object plane is α = 80°. From geometrical considerations a relation can be obtained between the displacement ds of the probe particle relative to the reference particle and a shift of the phase difference dΦ of the illuminating laser beam at both particles at a particular time.

Fig. 4
Fig. 4

Experimental verification of the phase-distance relation in Eq. (3). Focussed (A) and slightly defocussed (B) image of an immobilized pair of gold nanospheres with diameter 200 nm in a flow chamber. (C) Gaussian functions (blue curve) are fit to the intensity profile (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 Φ between the waves of scattered light from both particles by fitting a cosine function to the central part. (E) s plotted against Φ for several particle pairs (circles) compared with the computed relation from Eq. (3) (solid straight line).

Fig. 5
Fig. 5

Demonstration of the instrumentation error and drift elimination: Reference and probe particle are both immobilized on a dry glass surface. Measurement over 60 s at a bandwidth of 5 kHz. The reconstructed particle separation ss0 (orange curve, s0 is the mean value) has a standard deviation of 0.9 nm. The black curve represents the filtered data which has been smoothed with a moving average with window width 1s. The filtered data covers a range of 0.4 nm.

Fig. 6
Fig. 6

(A) Starting of a kinesin-driven bead movement (at t = 25 s). ATP has been flushed into 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 bead movement (blue) and of a bead stuck to the glass surface of the flow chamber (black). In the red curve three 8-nm steps can be clearly recognized (marked by arrows). The blue and the black curve are shifted in s-direction. Horizontal lines are spaced at 8-nm intervals.

Fig. 7
Fig. 7

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

Equations (6)

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I 1 ( X , Y ) = A 1 2 ( 1 + cos ( K X + Φ ( s ) ) ) + B 1 ,
I 2 ( X , Y ) = A 2 2 ( 1 cos ( K X + Φ ( s ) ) ) + B 2 .
d Φ = 2 π λ x d s = 2 π sin ( α ) λ 0 d s .
S 1 ( Φ ) = aperture I 1 ( X , Y , Φ ) d X d Y = χ ( t ) ( a 1 2 ( 1 + cos ( Φ ) ) + b 1 ) ,
S 2 ( Φ ) = aperture I 2 ( X , Y , Φ ) d X d Y = χ ( t ) ( a 2 2 ( 1 cos ( Φ ) ) + b 2 ) .
Φ ^ = arccos ( 2 S 1 ( b 2 + a 2 / 2 ) S 2 ( b 1 + a 1 / 2 ) S 1 a 2 + S 2 a 1 ) .

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