Calibration of the optical torque wrench

Francesco Pedaci; Zhuangxiong Huang; Maarten van Oene; Nynke H. Dekker

doi:10.1364/OE.20.003787

Calibration of the optical torque wrench

Francesco Pedaci, Zhuangxiong Huang, Maarten van Oene, and Nynke H. Dekker

Optics Express
Vol. 20,
Issue 4,
pp. 3787-3802
(2012)
•https://doi.org/10.1364/OE.20.003787

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Francesco Pedaci, Zhuangxiong Huang, Maarten van Oene, and Nynke H. Dekker, "Calibration of the optical torque wrench," Opt. Express 20, 3787-3802 (2012)

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Related Topics
Table of Contents Category
- Optical Trapping and Manipulation
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Optomechanics (200.4880)
- Optical tweezers or optical manipulation (350.4855)

About this Article
History
- Original Manuscript: September 30, 2011
- Revised Manuscript: December 15, 2011
- Manuscript Accepted: December 15, 2011
- Published: February 1, 2012
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 7, Iss. 4

Accessible

Open Access

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 k_BT (∼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.

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References

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Year
|
Author
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Publication

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5, 318–321 (2011).
[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]
K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
[Crossref] [PubMed]
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]
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]
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]
Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys. 41, 103–132 (2008).
[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]
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]
M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[Crossref]
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]
A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).
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]
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, 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]
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, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. 101, L05–L07 (2011).
[Crossref]
K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol. 298, 460–489 (1998).
[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]
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]
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]
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]
R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[Crossref]
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]
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]
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]
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]
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]
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]
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]
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]

1908 (1)

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

Abbondanzieri, E. A.

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.

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]

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.

Bowman, G. D.

Bowman, R.

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

Capitanio, M.

Celedon, A.

Daniels, B.

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.

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]

den Broeck, C. V.

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]

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.

Dunlap, D.

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.

Finzi, L.

Flyvbjerg, H.

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]

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.

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.

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]

Gucciardi, P. G.

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.

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.

Huang, Z.

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.

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]

Jagadish, C.

Jager, T.

Jones, P. H.

Jülicher, F.

Kerssemakers, J. W. J.

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.

Lipfert, J.

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.

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.

Oene, M. v.

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]

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Fig. 1 Experimental configuration. a) Schematic depicting of the torque generation inside a birefringent crystal that has a larger susceptibility along its extraordinary axis χ_e than along its ordinary axis χ_o. 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.

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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 > ω_c. 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| < ω_c ≃ 37(×2π) Hz the torque is constant in time (τ_m = β_τγω), while for |ω| > ω_c 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 ω_c than the one used in the subsequent figures and table.

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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 ω_c = 429 rad/s, δτ_m(0) = 5.7 mV, and

\sqrt{2} δ τ_{m} (\infty) = 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.

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Fig. 4 Calibration approaches involving separate measurements at two frequencies. a) Power spectrum analysis at ω = 0 followed by fast rotation at ω > ω_c. 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 f_c = 152 Hz, A_o = 3.1E-3 V²Hz, and V_o = 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 δτ_m = 5.5 mV, 〈T_s〉 = 3.8 ms, and V_o = 66 mV.

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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 (f_mod = 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 f_c = 150 Hz, A_o = 3.2E-3 V²Hz, and A_m = 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 > ω_c, and a histogram of the measured torque period T_s. From these data, we obtain 〈T_s〉 = 3.8 ms, δT_S = 0.16 ms, and

δ τ_{m} = V_{o} / \sqrt{2} = 45.3

mV.

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Tables (1)

Table 1 Experimental results of the different calibration methods obtained with the same trapped birefringent cylinder. For every method, the notation $m \pm_{b}^{a}$ 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.

View Table

Equations (35)

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τ = τ_{o} sin (2 x) .

- γ (\dot{x} + {\dot{θ}}_{pol}) - τ_{o} sin (2 x) + η (t) = 0

γ \dot{x} = - U^{'} (x) + η (t)

T_{o} = \frac{π}{\sqrt{ω^{2} - ω_{c}^{2}}} .

τ_{m} = β_{τ} τ .

δ τ_{m} = β_{τ} \sqrt{2 τ_{o} k_{B} T} {[1 - {(ω / ω_{c})}^{2}]}^{\frac{1}{4}} for ω < ω_{c}

= β_{τ} τ_{o} {[\frac{\sqrt{{(ω / ω_{c})}^{2} - 1}}{(ω / ω_{c}) + \sqrt{{(ω / ω_{c})}^{2} - 1}}]}^{\frac{1}{2}} for ω > ω_{c}

τ_{o} = 4 k_{B} T δ τ_{m}^{2} (\infty) / δ τ_{m}^{2} (0)

γ = τ_{o} / ω_{c}

β_{τ} = \sqrt{2} δ τ_{m} (\infty) / 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}]}^{- \frac{1}{2}}

β_{τ} = V_{o} / τ_{o}

P (τ_{m}, f) = 2 τ_{o}^{2} β_{τ}^{2} [\frac{2 k_{B} T}{γ π^{2} (f^{2} + f_{c}^{2})} + \frac{A^{2}}{(1 + f_{c}^{2} / f_{\mod}^{2})} δ (f - f_{\mod})]

γ = \frac{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} = \sqrt{\frac{A_{m} Δ f}{2 τ_{o}^{2} A^{2}} (1 + f_{c}^{2} / f_{\mod}^{2})}

D_{eff} = \frac{π^{2} δ T_{s}^{2}}{2 {〈 T_{s} 〉}^{3}} = \frac{k_{B} T}{γ} f (r)

γ = \frac{2 k_{B} T {〈 T_{s} 〉}^{3}}{π^{2} δ T_{s}^{2}} f (r)

τ_{o} = γ ω_{c}

β_{τ} = \frac{δ τ_{m}}{τ_{o}} \sqrt{\frac{(ω / ω_{c}) + \sqrt{{(ω / ω_{c})}^{2} - 1}}{\sqrt{{(ω / ω_{c})}^{2} - 1}}}

\dot{x} = - \frac{τ_{o}}{γ} sin (2 x) - ω

〈 τ 〉 = - τ_{o} sin (2 x_{eq}) = γ ω

δ τ^{2} = {(\frac{\partial τ}{\partial x})}^{2} δ x^{2} = 2 k_{B} T τ_{o} cos (2 x_{eq})

= 2 k_{B} T τ_{o} \sqrt{1 - {(ω / ω_{c})}^{2}}

\begin{array}{l} 〈 τ 〉 = \frac{1}{T_{o}} \int_{0}^{T_{o}} τ d t = \frac{1}{T_{o}} \int_{0}^{T_{o}} γ (\dot{x} + ω) d t \\ = \frac{1}{T_{o}} γ [x (T_{o}) - x (0)] + γ ω = - \frac{1}{T_{o}} γ π + γ ω \\ = γ (ω - \sqrt{ω^{2} - ω_{c}^{2}}) \end{array}

δ τ^{2} = 〈 τ^{2} 〉 - {〈 τ 〉}^{2} = \frac{1}{T_{τ}} \int_{0}^{T_{τ}} τ γ (d x / d t + ω) d t - {〈 τ 〉}^{2} = γ ω 〈 τ 〉 - {〈 τ 〉}^{2}

= τ_{0}^{2} \frac{\sqrt{{(ω / ω_{c})}^{2} - 1}}{(ω / ω_{c}) + \sqrt{{(ω / ω_{c})}^{2} - 1}}

D_{eff} - k_{B} T μ_{eff} \approx \frac{k_{B} T}{γ} \frac{\sqrt{ω^{2} - ω_{c}^{2}}}{ω} [\frac{2 〈 τ^{2} 〉}{γ^{2} ω^{2}} + \frac{5 〈 τ^{3} 〉}{γ^{3} ω^{3}}]

μ_{eff} = - \frac{1}{γ} \frac{d}{d ω} 〈 v (ω) 〉 = \frac{1}{γ} \frac{d}{d ω} \frac{π}{〈 T_{s} 〉} \approx \frac{ω}{γ \sqrt{ω^{2} - ω_{c}^{2}}}

D_{eff} = \frac{k_{B} T}{γ} f (r)

f (r) \approx \frac{1}{\sqrt{1 - r^{2}}} + \frac{2 \sqrt{1 - r^{2}}}{1 + \sqrt{1 - r^{2}}} [r^{2} + \frac{5}{4} r^{4} (1 + \frac{1}{1 + \sqrt{1 - r^{2}}})] .

Method	Drag γ		Sensitivity β′_τ		Max. Torque τ_o
Method	(pN nms)	Eq.	(pN nm/mV)	Eq.	(pN nm)	Eq.
Fit of the standard deviation of the torque (N = 2, sec. 4.1, Fig. 3)	$2.4 \pm_{0.2}^{0.1}$	(9)	$15.9 \pm_{0.9}^{0.3}$	(10)	$1041 \pm_{63}^{7}$	(8)
Spectrum at ω = 0 and fast rotation at ω > ω_c (N = 6, sec. 4.2.1 Fig. 4a)	$2.1 \pm_{0.3}^{0.2}$	(11)	$15.7 \pm_{2.9}^{0.6}$	(13)	$1008 \pm_{189}^{37}$	(12)
Torque variance, period and amplitude (N = 6, sec. 4.2.2, Fig. 4b)	$2.1 \pm_{0.1}^{0.1}$	(15)	$16.7 \pm_{0.1}^{0.6}$	(16)	$1100 \pm_{25}^{41}$	(14)
Sinusoidal modulation of the polarization direction (N = 2, sec. 4.3.1, Fig. 5a)	$2.2 \pm_{0.3}^{0.1}$	(18)	$16.0 \pm_{2.9}^{0.3}$	(20)	$1044 \pm_{129}^{22}$	(19)
Diffusion in a tilted potential landscape (N = 6, sec. 4.3.2, Fig. 5b)	$2.4 \pm_{0.1}^{0.1}$	(22)	$15.9 \pm_{1.4}^{0.6}$	(24)	$1046 \pm_{47}^{40}$	(23)