Abstract

The optical torque wrench is a laser trapping technique that expands the capability of standard optical tweezers to torque manipulation and measurement, using the laser linear polarization to orient tailored microscopic birefringent particles. The ability to measure torque of the order of kBT (∼4 pN nm) is especially important in the study of biophysical systems at the molecular and cellular level. Quantitative torque measurements rely on an accurate calibration of the instrument. Here we describe and implement a set of calibration approaches for the optical torque wrench, including methods that have direct analogs in linear optical tweezers as well as introducing others that are specifically developed for the angular variables. We compare the different methods, analyze their differences, and make recommendations regarding their implementations.

© 2012 OSA

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  1. F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5, 318–321 (2011).
    [CrossRef] [PubMed]
  2. K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
    [CrossRef] [PubMed]
  3. K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
    [CrossRef] [PubMed]
  4. B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem. 262, 10327–10334 (1987).
    [PubMed]
  5. L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A. 84, 7024–7027 (1987).
    [CrossRef] [PubMed]
  6. M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol. 2, 669–677 (2001).
    [CrossRef] [PubMed]
  7. S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol. 212, 1604–1610 (2009).
    [CrossRef] [PubMed]
  8. Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys. 41, 103–132 (2008).
    [CrossRef] [PubMed]
  9. J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods 7, 977–980 (2010).
    [CrossRef] [PubMed]
  10. M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
    [CrossRef]
  11. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
    [CrossRef]
  12. A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
    [CrossRef] [PubMed]
  13. A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).
  14. J. Inman, S. Forth, and M. Wang, “Passive torque wrench and angular position detection using a single-beam optical trap,” Opt. Lett. 35, 2949–2951 (2010).
    [CrossRef] [PubMed]
  15. F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011).
    [CrossRef]
  16. S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008).
    [CrossRef] [PubMed]
  17. B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E 80, 040901 (2009).
    [CrossRef]
  18. S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. 101, L05–L07 (2011).
    [CrossRef]
  19. K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol. 298, 460–489 (1998).
    [CrossRef]
  20. M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
    [CrossRef]
  21. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
    [CrossRef]
  22. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
    [CrossRef]
  23. C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90, 657–667 (2006).
    [CrossRef]
  24. B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010).
    [CrossRef]
  25. R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
    [CrossRef]
  26. C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
    [CrossRef] [PubMed]
  27. Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011).
    [CrossRef] [PubMed]
  28. W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett. 95, 208102 (2005).
    [CrossRef] [PubMed]
  29. O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
    [CrossRef] [PubMed]
  30. P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
    [CrossRef] [PubMed]
  31. S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
    [CrossRef]
  32. P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001).
    [CrossRef] [PubMed]
  33. K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. 74, 2226–2232 (2005).
    [CrossRef]
  34. M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys. 73, 198–1993 (1980).
    [CrossRef]

2011 (6)

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011).
[CrossRef]

S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. 101, L05–L07 (2011).
[CrossRef]

Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011).
[CrossRef] [PubMed]

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
[CrossRef] [PubMed]

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5, 318–321 (2011).
[CrossRef] [PubMed]

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[CrossRef]

2010 (3)

J. Inman, S. Forth, and M. Wang, “Passive torque wrench and angular position detection using a single-beam optical trap,” Opt. Lett. 35, 2949–2951 (2010).
[CrossRef] [PubMed]

J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods 7, 977–980 (2010).
[CrossRef] [PubMed]

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010).
[CrossRef]

2009 (3)

B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E 80, 040901 (2009).
[CrossRef]

S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol. 212, 1604–1610 (2009).
[CrossRef] [PubMed]

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

2008 (4)

Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys. 41, 103–132 (2008).
[CrossRef] [PubMed]

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
[CrossRef] [PubMed]

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
[CrossRef] [PubMed]

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008).
[CrossRef] [PubMed]

2007 (1)

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

2006 (2)

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90, 657–667 (2006).
[CrossRef]

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

2005 (2)

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. 74, 2226–2232 (2005).
[CrossRef]

2004 (2)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[CrossRef]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

2002 (1)

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
[CrossRef]

2001 (2)

P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001).
[CrossRef] [PubMed]

M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol. 2, 669–677 (2001).
[CrossRef] [PubMed]

1998 (2)

M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol. 298, 460–489 (1998).
[CrossRef]

1994 (1)

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

1987 (2)

B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem. 262, 10327–10334 (1987).
[PubMed]

L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A. 84, 7024–7027 (1987).
[CrossRef] [PubMed]

1980 (1)

M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys. 73, 198–1993 (1980).
[CrossRef]

1946 (1)

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[CrossRef]

Abbondanzieri, E. A.

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

Adler, R.

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[CrossRef]

Andreasson, J. O. L.

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010).
[CrossRef]

Asakia, K. S.

K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. 74, 2226–2232 (2005).
[CrossRef]

Baker, T. A.

B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem. 262, 10327–10334 (1987).
[PubMed]

Ballerini, R.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
[CrossRef]

Barland, S.

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011).
[CrossRef]

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[CrossRef]

Berry, R. M.

Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys. 41, 103–132 (2008).
[CrossRef] [PubMed]

Block, S. M.

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5, 318–321 (2011).
[CrossRef] [PubMed]

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010).
[CrossRef]

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol. 298, 460–489 (1998).
[CrossRef]

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

Bonaccorso, F.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
[CrossRef] [PubMed]

Bowman, G. D.

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[CrossRef]

Capitanio, M.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
[CrossRef]

Celedon, A.

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

Daniels, B.

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008).
[CrossRef] [PubMed]

Daniels, B. C.

B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E 80, 040901 (2009).
[CrossRef]

Dejgosha, S.

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

Dekker, N. H.

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011).
[CrossRef]

Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011).
[CrossRef] [PubMed]

J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods 7, 977–980 (2010).
[CrossRef] [PubMed]

den Broeck, C. V.

P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001).
[CrossRef] [PubMed]

Deufel, C.

S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. 101, L05–L07 (2011).
[CrossRef]

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008).
[CrossRef] [PubMed]

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90, 657–667 (2006).
[CrossRef]

Dewan, R.

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

Dunlap, D.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
[CrossRef]

Fazal, F. M.

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5, 318–321 (2011).
[CrossRef] [PubMed]

Ferrari, A. C.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
[CrossRef] [PubMed]

Finzi, L.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
[CrossRef]

Flyvbjerg, H.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[CrossRef]

Forth, S.

S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. 101, L05–L07 (2011).
[CrossRef]

J. Inman, S. Forth, and M. Wang, “Passive torque wrench and angular position detection using a single-beam optical trap,” Opt. Lett. 35, 2949–2951 (2010).
[CrossRef] [PubMed]

B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E 80, 040901 (2009).
[CrossRef]

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008).
[CrossRef] [PubMed]

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Funnell, B. E.

B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem. 262, 10327–10334 (1987).
[PubMed]

Gao, Q.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
[CrossRef] [PubMed]

Garciadelatorre, J.

M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys. 73, 198–1993 (1980).
[CrossRef]

Giuntini, M.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
[CrossRef]

Greenleaf, W. J.

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010).
[CrossRef]

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

Gucciardi, P. G.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
[CrossRef] [PubMed]

Gutierrez-Medina, B.

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010).
[CrossRef]

Hanggi, P.

P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001).
[CrossRef] [PubMed]

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Hisabori, T.

M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol. 2, 669–677 (2001).
[CrossRef] [PubMed]

Howard, J.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

Huang, Z.

Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011).
[CrossRef] [PubMed]

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011).
[CrossRef]

Inman, J.

Jagadish, C.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
[CrossRef] [PubMed]

Jager, T.

J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods 7, 977–980 (2010).
[CrossRef] [PubMed]

Jones, P. H.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
[CrossRef] [PubMed]

Jülicher, F.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

Kerssemakers, J. W. J.

J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods 7, 977–980 (2010).
[CrossRef] [PubMed]

Kornberg, A.

B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem. 262, 10327–10334 (1987).
[PubMed]

LaPorta, A.

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010).
[CrossRef]

A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).

Linke, H.

P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001).
[CrossRef] [PubMed]

Lipfert, J.

J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods 7, 977–980 (2010).
[CrossRef] [PubMed]

Liu, L. F.

L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A. 84, 7024–7027 (1987).
[CrossRef] [PubMed]

Maragò, O. M.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
[CrossRef] [PubMed]

Mari, S. A.

K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. 74, 2226–2232 (2005).
[CrossRef]

Muneyuki, E.

M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol. 2, 669–677 (2001).
[CrossRef] [PubMed]

Nagy, A.

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
[CrossRef] [PubMed]

Nelson, N.

S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol. 212, 1604–1610 (2009).
[CrossRef] [PubMed]

Neuman, K. C.

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
[CrossRef] [PubMed]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

Nieminem, T. A.

M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Nodelman, I. M.

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

Oene, M. v.

Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011).
[CrossRef] [PubMed]

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011).
[CrossRef]

Padgett, M.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[CrossRef]

Paiman, S.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
[CrossRef] [PubMed]

Patel, S. S.

S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. 101, L05–L07 (2011).
[CrossRef]

Pavone, F. S.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
[CrossRef]

Pedaci, F.

Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011).
[CrossRef] [PubMed]

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011).
[CrossRef]

Pérez-Madrid, A.

P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001).
[CrossRef] [PubMed]

Reece, P. J.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
[CrossRef] [PubMed]

Reimann, P.

P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001).
[CrossRef] [PubMed]

Romano, G.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
[CrossRef]

Rozhin, A. G.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
[CrossRef] [PubMed]

Rubi, J. M.

P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Saroussi, S.

S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol. 212, 1604–1610 (2009).
[CrossRef] [PubMed]

Scardaci, V.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
[CrossRef] [PubMed]

Schäffer, E.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

Searson, P.

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

Sethna, J. P.

B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E 80, 040901 (2009).
[CrossRef]

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008).
[CrossRef] [PubMed]

Sheinin, M. Y.

B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E 80, 040901 (2009).
[CrossRef]

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008).
[CrossRef] [PubMed]

Simmons, C. R.

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

Sowa, Y.

Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys. 41, 103–132 (2008).
[CrossRef] [PubMed]

Sun, S. X.

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

Svoboda, K.

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

Tan, H. H.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
[CrossRef] [PubMed]

Tirado, M. M.

M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys. 73, 198–1993 (1980).
[CrossRef]

Toe, W. J.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
[CrossRef] [PubMed]

Tolic-Nørrelykke, S. F.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

Visscher, K.

K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol. 298, 460–489 (1998).
[CrossRef]

Wang, F.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
[CrossRef] [PubMed]

Wang, J. C.

L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A. 84, 7024–7027 (1987).
[CrossRef] [PubMed]

Wang, M.

Wang, M. D.

S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. 101, L05–L07 (2011).
[CrossRef]

B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E 80, 040901 (2009).
[CrossRef]

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008).
[CrossRef] [PubMed]

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90, 657–667 (2006).
[CrossRef]

A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).

Wiggin, M.

Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011).
[CrossRef] [PubMed]

Wildt, B.

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

Wirtz, D.

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

Woodside, M. T.

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

Yoshida, M.

M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol. 2, 669–677 (2001).
[CrossRef] [PubMed]

ACS Nano (1)

Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011).
[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 (1994).
[CrossRef] [PubMed]

Biophys. J. (2)

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90, 657–667 (2006).
[CrossRef]

S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. 101, L05–L07 (2011).
[CrossRef]

J. Biol. Chem. (1)

B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem. 262, 10327–10334 (1987).
[PubMed]

J. Chem. Phys. (1)

M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys. 73, 198–1993 (1980).
[CrossRef]

J. Exp. Biol. (1)

S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol. 212, 1604–1610 (2009).
[CrossRef] [PubMed]

J. Phys. Soc. Jpn. (1)

K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. 74, 2226–2232 (2005).
[CrossRef]

Method Enzymol. (2)

K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol. 298, 460–489 (1998).
[CrossRef]

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010).
[CrossRef]

Nano Lett. (3)

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008).
[CrossRef] [PubMed]

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011).
[CrossRef] [PubMed]

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009).
[CrossRef] [PubMed]

Nat. Methods (3)

J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods 7, 977–980 (2010).
[CrossRef] [PubMed]

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
[CrossRef] [PubMed]

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

Nat. Photonics (2)

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5, 318–321 (2011).
[CrossRef] [PubMed]

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[CrossRef]

Nat. Phys. (1)

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011).
[CrossRef]

Nat. Rev. Mol. Cell Biol. (1)

M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol. 2, 669–677 (2001).
[CrossRef] [PubMed]

Nature (1)

M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (1)

B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E 80, 040901 (2009).
[CrossRef]

Phys. Rev. Lett. (4)

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008).
[CrossRef] [PubMed]

A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).

P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001).
[CrossRef] [PubMed]

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

Proc. IRE (1)

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A. 84, 7024–7027 (1987).
[CrossRef] [PubMed]

Q. Rev. Biophys. (1)

Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys. 41, 103–132 (2008).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (4)

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687 (2002).
[CrossRef]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[CrossRef]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Experimental configuration. a) Schematic depicting of the torque generation inside a birefringent crystal that has a larger susceptibility along its extraordinary axis χe than along its ordinary axis χo. b) SEM image of a nano-fabricated birefringent quartz cylinder used in the OTW. c) Diagram of the optical setup. OI: optical isolator, AOM: acousto-optic modulator, EOM: electro-optic modulator, NPBS: 50% non-polarizing beam splitter, λ/4: quarter wave-plate, PBS: polarizing beam splitter, PD: photo-detector, PSD: position sensitive detector, OBJ: 1.2NA microscope objective. The optical trap is surrounded by red dashed lines, the torque reference system is labeled and surrounded by grey dots, and the polarization state controlled by the EOM is indicated by red arrows inside the black dashed squares.

Fig. 2
Fig. 2

Torque on a birefringent cylinder. a) Calibrated optical torque as a function of the angle x between the polarization and the extraordinary axis for two laser intensities (blue points: 100 mW, red points: 50 mW, power measured at the objective input). The traces are reconstructed from torque traces recorded at ω > ωc. Calibration was performed following the method described in sec. 4.2.1. The black lines are sinusoidal fits (Eq. (1)) of the experimental points. b) Mean value of the measured torque as a function of the polarization rotation frequency. Within the region |ω| < ωc ≃ 37(×2π) Hz the torque is constant in time (τm = βτγω), while for |ω| > ωc the torque becomes periodic, with mean period given by Eq. (4). Negative frequencies indicate opposite rotation direction. Note that the cylinder used here has a larger volume and lower ωc than the one used in the subsequent figures and table.

Fig. 3
Fig. 3

Calibration approach involving measurement over the full range of frequencies. We plot the standard deviation of the measured torque (in Volts) as a function of the polarization rotation frequency. The quantities needed for calibration are indicated by arrows. The red line is a fit of the data to Eqs. (6) and (7), yielding ωc = 429 rad/s, δτm(0) = 5.7 mV, and 2 δ τ m ( ) = V o = 64.9 mV (see text). The top diagram uses blue circles to schematically indicate the polarization frequencies generated by the EOM in this method.

Fig. 4
Fig. 4

Calibration approaches involving separate measurements at two frequencies. a) Power spectrum analysis at ω = 0 followed by fast rotation at ω > ωc. From top to bottom: a schematic of the EOM frequencies used (blue circles), the power spectrum at ω = 0 (where the red points result from binning the experimental points (blue) into bins of variable size and the green line is a fit of the red points to a Lorentzian), and a segment of the torque trace acquired at ω/2π = 300Hz. For this dataset, the measured variables (indicated by arrows) are fc = 152 Hz, Ao = 3.1E-3 V2Hz, and Vo = 67 mV. b) Measurement of the torque variance, period, and amplitude. From top to bottom: a schematic of the EOM frequencies used (blue circles), the probability distribution of the torque readout at ω = 0, and a segment of the torque trace acquired at ω/2π = 300Hz. For this data set, the measured variables (indicated by arrows) are δτm = 5.5 mV, 〈Ts〉 = 3.8 ms, and Vo = 66 mV.

Fig. 5
Fig. 5

Calibration approaches using measurements at a single frequency. a) Sinusoidal modulation of the laser polarization direction. Top: schematic of the EOM frequency used (the blue circle indicates the frequency of the sinusoidal modulation). Bottom: power spectrum of the measured torque signal including the contribution from the imposed modulation of the direction of the laser polarization (fmod = 300Hz, A = 0.018 rad, Δf = 2 Hz); red points result from binning the experimental points (blue) into bins of variable size and the green line fits the red points to a Lorentzian. From the data shown, we obtain fc = 150 Hz, Ao = 3.2E-3 V2Hz, and Am = 2.18E-6 V2/Hz. b) Analysis of the diffusion in a tilted potential landscape. From top to bottom: schematic of the EOM frequency used (blue circle), a segment of the torque trace recorded at ω > ωc, and a histogram of the measured torque period Ts. From these data, we obtain 〈Ts〉 = 3.8 ms, δTS = 0.16 ms, and δ τ m = V o / 2 = 45.3 mV.

Tables (1)

Tables Icon

Table 1 Experimental results of the different calibration methods obtained with the same trapped birefringent cylinder. For every method, the notation m ± b a indicates the mean value m, obtained in N successive measurements, the standard deviation a of the N measurements, and the error propagated from the uncertainties of the parameters measured in the method and calculated from the analytical expression of m. Here β τ ' = β τ 1. References to sections, figures and equations of this work for each method and parameter are provided.

Equations (35)

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τ = τ o sin ( 2 x ) .
γ ( x ˙ + θ ˙ pol ) τ o sin ( 2 x ) + η ( t ) = 0
γ x ˙ = U ( x ) + η ( t )
T o = π ω 2 ω c 2 .
τ m = β τ τ .
δ τ m = β τ 2 τ o k B T [ 1 ( ω / ω c ) 2 ] 1 4 for ω < ω c
= β τ τ o [ ( ω / ω c ) 2 1 ( ω / ω c ) + ( ω / ω c ) 2 1 ] 1 2 for ω > ω c
τ o = 4 k B T δ τ m 2 ( ) / δ τ m 2 ( 0 )
γ = τ o / ω c
β τ = 2 δ τ m ( ) / τ o
γ = 4 k B T V o 2 / ( π 2 A o )
τ o = π γ f c
β τ = V o / τ o
τ o = 2 k B T [ V o / δ τ m ( 0 ) ] 2
γ = τ o [ ω 2 ( π / T s ) 2 ] 1 2
β τ = V o / τ o
P ( τ m , f ) = 2 τ o 2 β τ 2 [ 2 k B T γ π 2 ( f 2 + f c 2 ) + A 2 ( 1 + f c 2 / f mod 2 ) δ ( f f mod ) ]
γ = 2 k B T A m π 2 A 2 A o ( 1 + f c 2 / f mod 2 ) Δ f
τ o = π γ f c
β τ = A m Δ f 2 τ o 2 A 2 ( 1 + f c 2 / f mod 2 )
D eff = π 2 δ T s 2 2 T s 3 = k B T γ f ( r )
γ = 2 k B T T s 3 π 2 δ T s 2 f ( r )
τ o = γ ω c
β τ = δ τ m τ o ( ω / ω c ) + ( ω / ω c ) 2 1 ( ω / ω c ) 2 1
x ˙ = τ o γ sin ( 2 x ) ω
τ = τ o sin ( 2 x eq ) = γ ω
δ τ 2 = ( τ x ) 2 δ x 2 = 2 k B T τ o cos ( 2 x eq )
= 2 k B T τ o 1 ( ω / ω c ) 2
τ = 1 T o 0 T o τ d t = 1 T o 0 T o γ ( x ˙ + ω ) d t = 1 T o γ [ x ( T o ) x ( 0 ) ] + γ ω = 1 T o γ π + γ ω = γ ( ω ω 2 ω c 2 )
δ τ 2 = τ 2 τ 2 = 1 T τ 0 T τ τ γ ( d x / d t + ω ) d t τ 2 = γ ω τ τ 2
= τ 0 2 ( ω / ω c ) 2 1 ( ω / ω c ) + ( ω / ω c ) 2 1
D eff k B T μ eff k B T γ ω 2 ω c 2 ω [ 2 τ 2 γ 2 ω 2 + 5 τ 3 γ 3 ω 3 ]
μ eff = 1 γ d d ω v ( ω ) = 1 γ d d ω π T s ω γ ω 2 ω c 2
D eff = k B T γ f ( r )
f ( r ) 1 1 r 2 + 2 1 r 2 1 + 1 r 2 [ r 2 + 5 4 r 4 ( 1 + 1 1 + 1 r 2 ) ] .

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