Abstract

The detection of the precise movement of a vesicle during transport in a live cell provides key information for the intracellular delivery process. Here we report a novel numerical method for analyzing three-dimensional vesicle movement. Since the vesicle moves along a linear cytoskeleton during the active transport, our method first detects the orientation and position of the cytoskeleton as a linear section based on angle correlation and linear regression, after noise reduction. Then, the precise vesicle movement is calculated using vector analysis, in terms of rotation angle and translational displacement. Using this method, various vesicle trajectories obtained via high spatiotemporal resolution microscopy can be understood..

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref]
  29. J. Kurebayashi, T. Otsuki, C. Tang, M. Kurosumi, S. Yamamoto, K. Tanaka, M. Mochizuki, H. Nakamura, and H. Soono, “Isolation and characterization of a new human breast cancer cell line, KPL-4, expressing the Erb B family receptors and interleukin-6,” British Journal of Cancer 79, 707–717 (1999).
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    [Crossref] [PubMed]
  32. Y. Xiao, S. P. Forry, X. Gao, R. D. Holbrook, W. G. Telford, and A. Tona, “Dynamics and mechanisms of quantum dot nanoparticle cellular uptake,” J. Nanobiotechnol. 8:13 (2010).
    [Crossref]
  33. S. Lee, H. Kim, and H. Higuchi,“Focus stabilization by axial position feedback in biomedical imaging microscopy,” Proc. IEEE Sensors Applications Symposium (SAS). 309–314 (2018).
  34. I. Kulic, A. Brown, H. Kim, C. Kural, B. Blehm, P. Selvin, P. Nelson, and V. Gelfand, “The role of microtubule movement in bidirectional organelle transport,” Proceedings of the National Academy of Sciences 105, 10011–10016 (2008).
    [Crossref]

2017 (3)

M. Liu, Q. Li, L. Liang, J. Li, K. Wang, J. Li, M. Lv, N. Chen, H. Song, J. Lee, J. Shi, L. Wang, R. Lal, and C. Fan, “Real-time visualization of clustering and intracellular transport of gold nanoparticles by correlative imaging,” Nat. Commun. 8:15646 (2017).
[Crossref] [PubMed]

M. Nozumi, F. Nakatsu, K. Katoh, and M. Igarashi, “Coordinated movement of vesicles and actin bundles during nerve growth revealed by superresolution microscopy,” Cell Rep. 18, 2203–2216 (2017).
[Crossref] [PubMed]

R. Velmurugan, J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Intensity-based axial localization approaches for multifocal plane microscopy,” Opt. Express 25(4), 3394–3410 (2017).
[Crossref] [PubMed]

2016 (1)

2014 (2)

S. Can, M. A. Dewitt, and A. Yildiz, “Bidirectional helical motility of cytoplasmic dynein around microtubules,” eLife 3:e03205 (2014).
[Crossref] [PubMed]

M. Röding, M. Guo, D. A. Weitz, M. Rudemo, and A. Särkkä, “Identifying directional persistence in intracellular particle motion using Hidden Markov Models,” Math. Biosci. 248, 140–145 (2014).
[Crossref] [PubMed]

2013 (3)

Š. Bálint, I. V. Vilanova, Á. S. Álvarez, and M. Lakadamyali, “Correlative live-cell and superresolution microscopy reveals cargo transport dynamics at microtubule intersections,” Proc. Natl. Acad. Sci. U.S.A. 110(9), 3375–3380 (2013).
[Crossref] [PubMed]

A. Jayachandran and R. Dhanasekaran, “Automatic detection of brain tumor in magnetic resonance images using multi-texton histogram and support vector machine,” Int. J. Imaging Syst. Technol. 23(2), 1098 (2013).
[Crossref]

A. W. Harrison, D. A. Kenwright, T. A. Waigh, P. G. Woodman, and V. J. Allan, “Modes of correlated angular motion in live cells across three distinct time scales,” Phys. Biol. 10, 036002 (2013).
[Crossref] [PubMed]

2011 (4)

A. Ukil, V. H. Shah, and B. Deck, “Fast computation of arctangent functions for embedded applications: A comparative analysis,” Proc. IEEE Int. Symp. Ind. Electron. 20111206–1211 (2011).

L. Y. T. Chou, K. Ming, and W. C. W. Chan, “Strategies for the intracellular delivery of nanoparticles,” Chem. Soc. Rev. 40, 233–245 (2011).
[Crossref]

J. Seo, K. Cho, S. Lee, and S. Joo, “Concentration-dependent fluorescence live-cell imaging and tracking of intracellular nanoparticles,” Nanotechnology 22, 235101 (2011).
[Crossref] [PubMed]

M. Smith, H. Li, T. Shen, X. Huang, E. Yusuf, and D. Vavylonis, “Segmentation and Tracking of Cytoskeletal Filaments Using Open Active Contours,” Biophysical Journal 100, 445a (2011).
[Crossref]

2010 (3)

A. Ilin and T. Raiko, “Practical approaches to principal component analysis in the presence of missing values,” J. Mach. Learn. Res 11, 1957–2000 (2010).

H. Abdi and L. J. Williams, “Principal component analysis,” Comput. Stat. 2(4), 433–459 (2010).
[Crossref]

Y. Xiao, S. P. Forry, X. Gao, R. D. Holbrook, W. G. Telford, and A. Tona, “Dynamics and mechanisms of quantum dot nanoparticle cellular uptake,” J. Nanobiotechnol. 8:13 (2010).
[Crossref]

2009 (1)

L. W. Zhang and N. A. Monteiro-Riviere, “Mechanisms of quantum dot nanoparticle cellular uptake,” Toxicol. Sci. 110(1), 138–155 (2009).
[Crossref] [PubMed]

2008 (3)

I. Kulic, A. Brown, H. Kim, C. Kural, B. Blehm, P. Selvin, P. Nelson, and V. Gelfand, “The role of microtubule movement in bidirectional organelle transport,” Proceedings of the National Academy of Sciences 105, 10011–10016 (2008).
[Crossref]

D. Arcizet, B. Meier, E. Sackmann, J. O. Rädler, and D. Heinrich, “Temporal analysis of active and passive transport in living cells,” Phys. Rev. Lett. 101(24), 248103 (2008).
[Crossref] [PubMed]

R. Dixit, J. L. Ross, Y. E. Goldman, and E. L. F. Holzbaur, “Differential regulation of dynein and kinesin motor proteins by tau,” Science 319, 1086–1089 (2008).
[Crossref] [PubMed]

2007 (2)

E. Toprak, H. Balci, B. Blehm, and P. Selvin, “Three-Dimensional Particle Tracking via Bifocal Imaging,” Nano Letters 7, 2043–2045 (2007).
[Crossref] [PubMed]

T. M. Watanabe, T. Sato, K. Gonda, and H. Higuchi, “Three-dimensional nanometry of vesicle transport in living cells using dual-focus imaging optics,” Biochem. Biophys. Res. Commun. 359(1), 1–7 (2007).
[Crossref] [PubMed]

2005 (1)

A. Yildiz and P. Selvin, “Fluorescence Imaging with One Nanometer Accuracy: Application to Molecular Motors,” Accounts of Chemical Research 38, 574–582 (2005).
[Crossref] [PubMed]

2004 (1)

H. S. Kruth, N. L. Jones, W. Huang, B. Zhao, I. Ishii, J. Chang, C. A. Combs, D. Malide, and W. Zhang, “Macropinocytosis is the endocytic pathway that mediates macrophage foam cell formation with native low density lipoprotein,” J. Biol. Chem. 280(3), 2352–2360 (2004).
[Crossref] [PubMed]

2003 (1)

S. Conner and S. Schmid, “Regulated portals of entry into the cell,” Nature 422, 37–44 (2003).
[Crossref] [PubMed]

2002 (2)

M. Y. Ali, S. Uemura, K. Adachi, H. Itoh, K. Kinosita, and S. Ishiwata, “Myosin V is a left-handed spriral motor on the right-handed actin helix,” Nat. Str. Biol. 9(6), 464–467 (2002).
[Crossref]

A. Sorkin and M. von Zastrow, “Signal transduction and endocytosis: close encounters of many kinds,” Nat. Rev. Mol. Cell. Biol. 3(8), 600–614 (2002).
[Crossref] [PubMed]

2001 (2)

H. Hess and V. Vogel, “Molecular shuttles based on motor proteins: active transport in synthetic environments,” Rev. Mol. Biotechnol. 82, 67–85 (2001).
[Crossref]

S. Matveev, X. Li, W. Everson, and E. Smart, “The role of caveolae and caveolin in vesicle-dependent and vesicle-independent trafficking,” Advanced Drug Delivery Reviews 49, 237–250 (2001).
[Crossref] [PubMed]

1999 (1)

J. Kurebayashi, T. Otsuki, C. Tang, M. Kurosumi, S. Yamamoto, K. Tanaka, M. Mochizuki, H. Nakamura, and H. Soono, “Isolation and characterization of a new human breast cancer cell line, KPL-4, expressing the Erb B family receptors and interleukin-6,” British Journal of Cancer 79, 707–717 (1999).
[Crossref] [PubMed]

1987 (1)

F. Wood, K. Esbensen, and P. Geladi, “Principal component analysis,” Chemometr. Intel. Lab. Syst. 2, 37–52 (1987)
[Crossref]

Abdi, H.

H. Abdi and L. J. Williams, “Principal component analysis,” Comput. Stat. 2(4), 433–459 (2010).
[Crossref]

Adachi, K.

M. Y. Ali, S. Uemura, K. Adachi, H. Itoh, K. Kinosita, and S. Ishiwata, “Myosin V is a left-handed spriral motor on the right-handed actin helix,” Nat. Str. Biol. 9(6), 464–467 (2002).
[Crossref]

Alberts, Bruce

Bruce Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell (Garland Sciences, 2002).

Ali, M. Y.

M. Y. Ali, S. Uemura, K. Adachi, H. Itoh, K. Kinosita, and S. Ishiwata, “Myosin V is a left-handed spriral motor on the right-handed actin helix,” Nat. Str. Biol. 9(6), 464–467 (2002).
[Crossref]

Allan, V. J.

A. W. Harrison, D. A. Kenwright, T. A. Waigh, P. G. Woodman, and V. J. Allan, “Modes of correlated angular motion in live cells across three distinct time scales,” Phys. Biol. 10, 036002 (2013).
[Crossref] [PubMed]

Álvarez, Á. S.

Š. Bálint, I. V. Vilanova, Á. S. Álvarez, and M. Lakadamyali, “Correlative live-cell and superresolution microscopy reveals cargo transport dynamics at microtubule intersections,” Proc. Natl. Acad. Sci. U.S.A. 110(9), 3375–3380 (2013).
[Crossref] [PubMed]

Arcizet, D.

D. Arcizet, B. Meier, E. Sackmann, J. O. Rädler, and D. Heinrich, “Temporal analysis of active and passive transport in living cells,” Phys. Rev. Lett. 101(24), 248103 (2008).
[Crossref] [PubMed]

Balci, H.

E. Toprak, H. Balci, B. Blehm, and P. Selvin, “Three-Dimensional Particle Tracking via Bifocal Imaging,” Nano Letters 7, 2043–2045 (2007).
[Crossref] [PubMed]

Bálint, Š.

Š. Bálint, I. V. Vilanova, Á. S. Álvarez, and M. Lakadamyali, “Correlative live-cell and superresolution microscopy reveals cargo transport dynamics at microtubule intersections,” Proc. Natl. Acad. Sci. U.S.A. 110(9), 3375–3380 (2013).
[Crossref] [PubMed]

Blehm, B.

I. Kulic, A. Brown, H. Kim, C. Kural, B. Blehm, P. Selvin, P. Nelson, and V. Gelfand, “The role of microtubule movement in bidirectional organelle transport,” Proceedings of the National Academy of Sciences 105, 10011–10016 (2008).
[Crossref]

E. Toprak, H. Balci, B. Blehm, and P. Selvin, “Three-Dimensional Particle Tracking via Bifocal Imaging,” Nano Letters 7, 2043–2045 (2007).
[Crossref] [PubMed]

Brown, A.

I. Kulic, A. Brown, H. Kim, C. Kural, B. Blehm, P. Selvin, P. Nelson, and V. Gelfand, “The role of microtubule movement in bidirectional organelle transport,” Proceedings of the National Academy of Sciences 105, 10011–10016 (2008).
[Crossref]

Can, S.

S. Can, M. A. Dewitt, and A. Yildiz, “Bidirectional helical motility of cytoplasmic dynein around microtubules,” eLife 3:e03205 (2014).
[Crossref] [PubMed]

Chan, W. C. W.

L. Y. T. Chou, K. Ming, and W. C. W. Chan, “Strategies for the intracellular delivery of nanoparticles,” Chem. Soc. Rev. 40, 233–245 (2011).
[Crossref]

Chang, J.

H. S. Kruth, N. L. Jones, W. Huang, B. Zhao, I. Ishii, J. Chang, C. A. Combs, D. Malide, and W. Zhang, “Macropinocytosis is the endocytic pathway that mediates macrophage foam cell formation with native low density lipoprotein,” J. Biol. Chem. 280(3), 2352–2360 (2004).
[Crossref] [PubMed]

Chao, J.

Chen, N.

M. Liu, Q. Li, L. Liang, J. Li, K. Wang, J. Li, M. Lv, N. Chen, H. Song, J. Lee, J. Shi, L. Wang, R. Lal, and C. Fan, “Real-time visualization of clustering and intracellular transport of gold nanoparticles by correlative imaging,” Nat. Commun. 8:15646 (2017).
[Crossref] [PubMed]

Cho, K.

J. Seo, K. Cho, S. Lee, and S. Joo, “Concentration-dependent fluorescence live-cell imaging and tracking of intracellular nanoparticles,” Nanotechnology 22, 235101 (2011).
[Crossref] [PubMed]

Chou, L. Y. T.

L. Y. T. Chou, K. Ming, and W. C. W. Chan, “Strategies for the intracellular delivery of nanoparticles,” Chem. Soc. Rev. 40, 233–245 (2011).
[Crossref]

Combs, C. A.

H. S. Kruth, N. L. Jones, W. Huang, B. Zhao, I. Ishii, J. Chang, C. A. Combs, D. Malide, and W. Zhang, “Macropinocytosis is the endocytic pathway that mediates macrophage foam cell formation with native low density lipoprotein,” J. Biol. Chem. 280(3), 2352–2360 (2004).
[Crossref] [PubMed]

Conner, S.

S. Conner and S. Schmid, “Regulated portals of entry into the cell,” Nature 422, 37–44 (2003).
[Crossref] [PubMed]

Deck, B.

A. Ukil, V. H. Shah, and B. Deck, “Fast computation of arctangent functions for embedded applications: A comparative analysis,” Proc. IEEE Int. Symp. Ind. Electron. 20111206–1211 (2011).

Dewitt, M. A.

S. Can, M. A. Dewitt, and A. Yildiz, “Bidirectional helical motility of cytoplasmic dynein around microtubules,” eLife 3:e03205 (2014).
[Crossref] [PubMed]

Dhanasekaran, R.

A. Jayachandran and R. Dhanasekaran, “Automatic detection of brain tumor in magnetic resonance images using multi-texton histogram and support vector machine,” Int. J. Imaging Syst. Technol. 23(2), 1098 (2013).
[Crossref]

Diebel, J.

J. Diebel, “Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors,” Matrix (2006).

Dixit, R.

R. Dixit, J. L. Ross, Y. E. Goldman, and E. L. F. Holzbaur, “Differential regulation of dynein and kinesin motor proteins by tau,” Science 319, 1086–1089 (2008).
[Crossref] [PubMed]

Esbensen, K.

F. Wood, K. Esbensen, and P. Geladi, “Principal component analysis,” Chemometr. Intel. Lab. Syst. 2, 37–52 (1987)
[Crossref]

Everson, W.

S. Matveev, X. Li, W. Everson, and E. Smart, “The role of caveolae and caveolin in vesicle-dependent and vesicle-independent trafficking,” Advanced Drug Delivery Reviews 49, 237–250 (2001).
[Crossref] [PubMed]

Fan, C.

M. Liu, Q. Li, L. Liang, J. Li, K. Wang, J. Li, M. Lv, N. Chen, H. Song, J. Lee, J. Shi, L. Wang, R. Lal, and C. Fan, “Real-time visualization of clustering and intracellular transport of gold nanoparticles by correlative imaging,” Nat. Commun. 8:15646 (2017).
[Crossref] [PubMed]

Forry, S. P.

Y. Xiao, S. P. Forry, X. Gao, R. D. Holbrook, W. G. Telford, and A. Tona, “Dynamics and mechanisms of quantum dot nanoparticle cellular uptake,” J. Nanobiotechnol. 8:13 (2010).
[Crossref]

Fujita, H.

Gao, X.

Y. Xiao, S. P. Forry, X. Gao, R. D. Holbrook, W. G. Telford, and A. Tona, “Dynamics and mechanisms of quantum dot nanoparticle cellular uptake,” J. Nanobiotechnol. 8:13 (2010).
[Crossref]

Geladi, P.

F. Wood, K. Esbensen, and P. Geladi, “Principal component analysis,” Chemometr. Intel. Lab. Syst. 2, 37–52 (1987)
[Crossref]

Gelfand, V.

I. Kulic, A. Brown, H. Kim, C. Kural, B. Blehm, P. Selvin, P. Nelson, and V. Gelfand, “The role of microtubule movement in bidirectional organelle transport,” Proceedings of the National Academy of Sciences 105, 10011–10016 (2008).
[Crossref]

Goldman, Y. E.

R. Dixit, J. L. Ross, Y. E. Goldman, and E. L. F. Holzbaur, “Differential regulation of dynein and kinesin motor proteins by tau,” Science 319, 1086–1089 (2008).
[Crossref] [PubMed]

Gonda, K.

T. M. Watanabe, T. Sato, K. Gonda, and H. Higuchi, “Three-dimensional nanometry of vesicle transport in living cells using dual-focus imaging optics,” Biochem. Biophys. Res. Commun. 359(1), 1–7 (2007).
[Crossref] [PubMed]

Guo, M.

M. Röding, M. Guo, D. A. Weitz, M. Rudemo, and A. Särkkä, “Identifying directional persistence in intracellular particle motion using Hidden Markov Models,” Math. Biosci. 248, 140–145 (2014).
[Crossref] [PubMed]

Harrison, A. W.

A. W. Harrison, D. A. Kenwright, T. A. Waigh, P. G. Woodman, and V. J. Allan, “Modes of correlated angular motion in live cells across three distinct time scales,” Phys. Biol. 10, 036002 (2013).
[Crossref] [PubMed]

Heinrich, D.

D. Arcizet, B. Meier, E. Sackmann, J. O. Rädler, and D. Heinrich, “Temporal analysis of active and passive transport in living cells,” Phys. Rev. Lett. 101(24), 248103 (2008).
[Crossref] [PubMed]

Hess, H.

H. Hess and V. Vogel, “Molecular shuttles based on motor proteins: active transport in synthetic environments,” Rev. Mol. Biotechnol. 82, 67–85 (2001).
[Crossref]

Higuchi, H.

T. M. Watanabe, T. Sato, K. Gonda, and H. Higuchi, “Three-dimensional nanometry of vesicle transport in living cells using dual-focus imaging optics,” Biochem. Biophys. Res. Commun. 359(1), 1–7 (2007).
[Crossref] [PubMed]

S. Lee, H. Kim, and H. Higuchi,“Focus stabilization by axial position feedback in biomedical imaging microscopy,” Proc. IEEE Sensors Applications Symposium (SAS). 309–314 (2018).

Holbrook, R. D.

Y. Xiao, S. P. Forry, X. Gao, R. D. Holbrook, W. G. Telford, and A. Tona, “Dynamics and mechanisms of quantum dot nanoparticle cellular uptake,” J. Nanobiotechnol. 8:13 (2010).
[Crossref]

Holzbaur, E. L. F.

R. Dixit, J. L. Ross, Y. E. Goldman, and E. L. F. Holzbaur, “Differential regulation of dynein and kinesin motor proteins by tau,” Science 319, 1086–1089 (2008).
[Crossref] [PubMed]

Huang, W.

H. S. Kruth, N. L. Jones, W. Huang, B. Zhao, I. Ishii, J. Chang, C. A. Combs, D. Malide, and W. Zhang, “Macropinocytosis is the endocytic pathway that mediates macrophage foam cell formation with native low density lipoprotein,” J. Biol. Chem. 280(3), 2352–2360 (2004).
[Crossref] [PubMed]

Huang, X.

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Figures (7)

Fig. 1
Fig. 1 An example of complex trajectory of vesicle movement acquired by three-dimensional microscopy. (A) Cell image taken in phase contrast microscopy. (B) Vesicle labeled with quantum dot imaged via dual focus optics [13]. The x, y coordinates and intensity information are exploited for acquiring three-dimensional position data. (C) Three-dimensional trajectory of the target vesicle tracked for 10 s at 100 Hz frame rate. The complexity in trajectory reflects the fluctuating vesicle dynamics in cytoplasm, transferring among multiple intracellular structures.
Fig. 2
Fig. 2 Process overview of the presented numerical analysis method. (A) Raw position data distribution P i * in three dimensions. (B) Noise reduction using Weierstrass transform with the filter size of 16 data points to acquire the filtered position data Pi. (C) Recognition of the linear section (colored in pink) by determining the local curvature. Nonlinear sections are colored in mint green. The orientation and position of cytoskeleton in linear section (red line) are estimated via linear regression based on PCA. (D) Analysis of vesicle movement on the estimated cytoskeleton is conducted by projecting the data points using vector analysis.
Fig. 3
Fig. 3 Determination of the local curvature in order to recognize a linear section. The angle θi is defined as an angle between two vectors, consisting of the data points separated by the filter size m. Note that the closer value of θi to π implies the more linear and persistent movement of a vesicle.
Fig. 4
Fig. 4 Simulation for determining proper noise filter size m in order to extract the linear section. (A) Three consecutive linear segments in space as mimicking the crossed cytoskeletons. (B) Vesicle movement trajectory generated along the segments. (C) Angle θi changes when the filter size m varies from 1 to 40 data points. Ground truth represents the angle calculated from the base segments. Note that there exists a trade-off between the noise reduction and excessive smoothing according to the parameter m. (D) The efficiency of the filter size is defined based on the occupation ratio with respect to the ground truth above the critical angle, θc. The graph on the right indicates the occupation ratio calculated when θ c = 3 π 4. The optimal value of m is about 16 data points in this simulation.
Fig. 5
Fig. 5 Estimation of linear section and corresponding cytoskeleton location. (A) The detected linear section (colored in pink) and the corner of the crossing segments (mint green) by simulation. (B) Estimated orientation of cytoskeleton ê and the scheme for finding perpendicular projection onto the linear axis. Q1 and Qn are the starting and ending point of estimated cytoskeleton, respectively. Qc is determined by the mean coordinates of Pl. Note that the notation Q is used for the points on the linear axis. (C) Comparison between the ground truth (black line) and the estimation result by PCA (red line). The discrepancy in localization is defined using δ, the angle between the estimation and ground truth. The angle ϕ represents the angle between estimated cytoskeletons. (D) Influence of system noise level and the diameter of vesicle upon δ.
Fig. 6
Fig. 6 Vesicle movement analysis on the cytoskeleton. The movement between consecutive two data points can be calculated as a two-dimensional rotation transformation with coherent direction vector , which is the orientation of the estimated cytoskeleton. (A) Qi and Qi+1 are respectively the projected points of the data points Pi and Pi+1 onto the estimated linear axis, which can be interpreted as the positions where a vesicle directly interacts with the cytoskeleton. (B) Relative angular position Pi+1 to Pi around the axis of cytoskeleton can be expressed by the rotation angle ρi. The angle ψi is defined as the cumulative angle of Σi with respect to Σ1.
Fig. 7
Fig. 7 Actual vesicle movement data analyzed by the proposed numerical method. (A) Tracking precision of the imaging system in the experiment. (B) Dual view images of vesicles labeled by quantum dot (red) and corresponding phase contrast image of a live cell (gray scale). One vesicle marked as Pc near the cell edge and the other vesicle Pd near nucleus were selected and tracked. (Scale bar = 10 μm). (C) In the trajectory of Pc, three linear sections (c1, c2, and c3) were detected, and the angles between the linear sections were calculated as 158° and 19°, respectively. Blue arrow indicates the initial point while the yellow arrow refers to the final point of the trajectory. (D) In the trajectory of Pd, three linear sections (d1, d2, d3, and d4) were detected, and the angles between the linear sections were calculated as 156°, 159°, and 98°, respectively. (E) Cumulative angle ψi around the axis of cytoskeleton and corresponding translational distance in each linear section of Pc trajectory. Note that the initial position of vesicle in each linear section is respectively set at 0° and 0 nm, for simplicity. (F) Cumulative angle ψi around the axis of cytoskeleton and corresponding translational distance in each linear section of Pd trajectory.

Equations (15)

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x i = ( x * G ) ( i ) = j = m m x i * G ( j i )
G ( x ) = 1 2 π σ e x 2 2 σ 2 .
θ i = cos 1 [ ( r i + m r i ) ( r i m r i ) r i + m r i r i m r i ] , 0 θ i π .
{ nonlinear movement : 0 θ i < 3 π 4 linear and persistent movement : 3 π 4 θ i π .
C = 1 n 1 M T M
C = A T Λ A
Q c Q c P i Q i = 0 .
( e x , e y , e z ) [ κ e x + ( x c x i ) , κ e y + ( y c y i ) , κ e z + ( z c z i ) ] = 0
κ = e x ( x i x c ) + e y ( y i y c ) + e z ( z i z c ) .
{ Q 1 = [ e x ( x 1 x c ) + e y ( y 1 y c ) + e z ( z 1 z c ) ] ( e x , e y , e z ) + ( x c , y c , z c ) Q n = [ e x ( x n x c ) + e y ( y n y c ) + e z ( z n z c ) ] ( e x , e y , e z ) + ( x c , y c , z c ) .
ϕ = atan 2 ( Q 1 Q n × Q 1 Q n , Q 1 Q n Q 1 Q n ) .
Σ i + 1 = R i Σ i + t i
R i = [ cos ρ i sin ρ i 0 sin ρ i cos ρ i 0 0 0 1 ] , t i = [ 0 0 Q i Q i + 1 . ]
ψ i = j = 1 i ρ j .
v i = Q i Q i + 1 t i + 1 t i .

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