Abstract

We present a new method for calibrating an optical-tweezer setup that does not depend on input parameters and is less affected by systematic errors like drift of the setup. It is based on an inference approach that uses Bayesian probability to infer the diffusion coefficient and the potential felt by a bead trapped in an optical or magnetic trap. It exploits a much larger amount of the information stored in the recorded bead trajectory than standard calibration approaches. We demonstrate that this method outperforms the equipartition method and the power-spectrum method in input information required (bead radius and trajectory length) and in output accuracy.

© 2013 Optical Society of America

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2013 (3)

J. Mas, A. C. Richardson, S. N. Reihani, L. B. Oddershede, and K. Berg-Sørensen, “Quantitative determination of optical trapping strength and viscoelastic moduli inside living cells,” Phys. Biol. 10(4), 046006 (2013).
[Crossref] [PubMed]

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat Commun 4, 1768 (2013).
[Crossref] [PubMed]

S. Türkcan, M. U. Richly, A. Alexandrou, and J.-B. Masson, “Probing membrane protein interactions with their lipid raft environment using single-molecule tracking and Bayesian inference alysis,” PLoS ONE 8(1), e53073 (2013).
[Crossref] [PubMed]

2012 (4)

Y. von Hansen, A. Mehlich, B. Pelz, M. Rief, and R. R. Netz, “Auto- and cross-power spectral analysis of dual trap optical tweezer experiments using Bayesian inference,” Rev. Sci. Instrum. 83(9), 095116 (2012).
[Crossref] [PubMed]

S. Türkcan, A. Alexandrou, and J.-B. Masson, “A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories,” Biophys. J. 102(10), 2288–2298 (2012).
[Crossref] [PubMed]

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

I. De Vlaminck and C. Dekker, “Recent advances in magnetic tweezers,” Annu Rev Biophys 41(1), 453–472 (2012).
[Crossref] [PubMed]

2011 (3)

C. Veigel and C. F. Schmidt, “Moving into the cell: single-molecule studies of molecular motors in complex environments,” Nat. Rev. Mol. Cell Biol. 12(3), 163–176 (2011).
[Crossref] [PubMed]

P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite, “Quantifying how DNA stretches, melts and changes twist under tension,” Nat. Phys. 7(9), 731–736 (2011).
[Crossref]

Y. Li, V. Lubchenko, and P. G. Vekilov, “The use of dynamic light scattering and Brownian microscopy to characterize protein aggregation,” Rev. Sci. Instrum. 82(5), 053106 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

J.-B. Masson, D. Casanova, S. Türkcan, G. Voisinne, M. R. Popoff, M. Vergassola, and A. Alexandrou, “Inferring maps of forces inside cell membrane microdomains,” Phys. Rev. Lett. 102(4), 048103 (2009).
[Crossref] [PubMed]

2008 (3)

A. C. Richardson, S. N. S. Reihani, and L. B. Oddershede, “Non-harmonic potential of a single beam optical trap,” Opt. Express 16(20), 15709–15717 (2008).
[Crossref] [PubMed]

A. Borgia, P. M. Williams, and J. Clarke, “Single-molecule studies of protein folding,” Annu. Rev. Biochem. 77(1), 101–125 (2008).
[Crossref] [PubMed]

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

2007 (2)

Z. Gong, Z. Wang, Y. Li, L. Lou, and S. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273(1), 37–42 (2007).
[Crossref]

E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface Forces and Drag Coefficients of Microspheres near a Plane Surface Measured with Optical Tweezers,” Langmuir 23(7), 3654–3665 (2007).
[Crossref] [PubMed]

2006 (2)

K. C. Vermeulen, G. J. L. Wuite, G. J. M. Stienen, and C. F. Schmidt, “Optical trap stiffness in the presence and absence of spherical aberrations,” Appl. Opt. 45(8), 1812–1819 (2006).
[Crossref] [PubMed]

P. M. Hansen, I. M. Tolić-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, “tweezercalib 2.0: Faster version of MatLab package for precise calibration of optical tweezers,” Comput. Phys. Commun. 174(6), 518–520 (2006).
[Crossref]

2005 (1)

C. Cecconi, E. A. Shank, C. Bustamante, and S. Marqusee, “Direct observation of the three-state folding of a single protein molecule,” Science 309(5743), 2057–2060 (2005).
[Crossref] [PubMed]

2004 (3)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[Crossref] [PubMed]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
[Crossref]

I. M. Tolic-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

2002 (2)

C. Gosse and V. Croquette, “Magnetic Tweezers: Micromanipulation and Force Measurement at the Molecular Level,” Biophys. J. 82(6), 3314–3329 (2002).
[Crossref] [PubMed]

N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, “Measurements of trapping efficiency and stiffness in optical tweezers,” Opt. Commun. 214(1-6), 15–24 (2002).
[Crossref]

1999 (1)

A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich, M. S. Mooseker, and R. E. Cheney, “Myosin-V is a processive actin-based motor,” Nature 400(6744), 590–593 (1999).
[Crossref] [PubMed]

1997 (2)

M. S. Z. Kellermayer, S. B. Smith, H. L. Granzier, and C. Bustamante, “Folding-unfolding transitions in single titin molecules characterized with laser tweezers,” Science 276(5315), 1112–1116 (1997).
[Crossref] [PubMed]

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

1996 (2)

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: The elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271(5250), 795–799 (1996).
[Crossref] [PubMed]

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Quantum Electron. 2(4), 1066–1076 (1996).
[Crossref]

1994 (1)

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994).
[Crossref] [PubMed]

1991 (1)

1986 (1)

1923 (1)

H. Faxen, “Die Bewegung einer starren Kugel laengs der Achse eines mit zaeher Fluessigkeit gefuellten Rohres,” Ark. Mat. Aston. Fys. 17, 1–28 (1923).

Alexandrou, A.

S. Türkcan, M. U. Richly, A. Alexandrou, and J.-B. Masson, “Probing membrane protein interactions with their lipid raft environment using single-molecule tracking and Bayesian inference alysis,” PLoS ONE 8(1), e53073 (2013).
[Crossref] [PubMed]

S. Türkcan, A. Alexandrou, and J.-B. Masson, “A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories,” Biophys. J. 102(10), 2288–2298 (2012).
[Crossref] [PubMed]

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

J.-B. Masson, D. Casanova, S. Türkcan, G. Voisinne, M. R. Popoff, M. Vergassola, and A. Alexandrou, “Inferring maps of forces inside cell membrane microdomains,” Phys. Rev. Lett. 102(4), 048103 (2009).
[Crossref] [PubMed]

Arimondo, E.

N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, “Measurements of trapping efficiency and stiffness in optical tweezers,” Opt. Commun. 214(1-6), 15–24 (2002).
[Crossref]

Ashkin, A.

Berg-Sørensen, K.

J. Mas, A. C. Richardson, S. N. Reihani, L. B. Oddershede, and K. Berg-Sørensen, “Quantitative determination of optical trapping strength and viscoelastic moduli inside living cells,” Phys. Biol. 10(4), 046006 (2013).
[Crossref] [PubMed]

P. M. Hansen, I. M. Tolić-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, “tweezercalib 2.0: Faster version of MatLab package for precise calibration of optical tweezers,” Comput. Phys. Commun. 174(6), 518–520 (2006).
[Crossref]

I. M. Tolic-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
[Crossref]

Bjorkholm, J. E.

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[Crossref] [PubMed]

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Quantum Electron. 2(4), 1066–1076 (1996).
[Crossref]

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994).
[Crossref] [PubMed]

Bockelmann, U.

P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite, “Quantifying how DNA stretches, melts and changes twist under tension,” Nat. Phys. 7(9), 731–736 (2011).
[Crossref]

Boilot, J.-P.

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

Borgia, A.

A. Borgia, P. M. Williams, and J. Clarke, “Single-molecule studies of protein folding,” Annu. Rev. Biochem. 77(1), 101–125 (2008).
[Crossref] [PubMed]

Bouyer, P.

Bustamante, C.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

C. Cecconi, E. A. Shank, C. Bustamante, and S. Marqusee, “Direct observation of the three-state folding of a single protein molecule,” Science 309(5743), 2057–2060 (2005).
[Crossref] [PubMed]

M. S. Z. Kellermayer, S. B. Smith, H. L. Granzier, and C. Bustamante, “Folding-unfolding transitions in single titin molecules characterized with laser tweezers,” Science 276(5315), 1112–1116 (1997).
[Crossref] [PubMed]

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: The elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271(5250), 795–799 (1996).
[Crossref] [PubMed]

Casanova, D.

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

J.-B. Masson, D. Casanova, S. Türkcan, G. Voisinne, M. R. Popoff, M. Vergassola, and A. Alexandrou, “Inferring maps of forces inside cell membrane microdomains,” Phys. Rev. Lett. 102(4), 048103 (2009).
[Crossref] [PubMed]

Cecconi, C.

C. Cecconi, E. A. Shank, C. Bustamante, and S. Marqusee, “Direct observation of the three-state folding of a single protein molecule,” Science 309(5743), 2057–2060 (2005).
[Crossref] [PubMed]

Chemla, Y. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Cheney, R. E.

A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich, M. S. Mooseker, and R. E. Cheney, “Myosin-V is a processive actin-based motor,” Nature 400(6744), 590–593 (1999).
[Crossref] [PubMed]

Chu, S.

Clarke, J.

A. Borgia, P. M. Williams, and J. Clarke, “Single-molecule studies of protein folding,” Annu. Rev. Biochem. 77(1), 101–125 (2008).
[Crossref] [PubMed]

Croquette, V.

C. Gosse and V. Croquette, “Magnetic Tweezers: Micromanipulation and Force Measurement at the Molecular Level,” Biophys. J. 82(6), 3314–3329 (2002).
[Crossref] [PubMed]

Cui, Y.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: The elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271(5250), 795–799 (1996).
[Crossref] [PubMed]

De Vlaminck, I.

I. De Vlaminck and C. Dekker, “Recent advances in magnetic tweezers,” Annu Rev Biophys 41(1), 453–472 (2012).
[Crossref] [PubMed]

Dekker, C.

I. De Vlaminck and C. Dekker, “Recent advances in magnetic tweezers,” Annu Rev Biophys 41(1), 453–472 (2012).
[Crossref] [PubMed]

Dulin, D.

Dziedzic, J. M.

Faxen, H.

H. Faxen, “Die Bewegung einer starren Kugel laengs der Achse eines mit zaeher Fluessigkeit gefuellten Rohres,” Ark. Mat. Aston. Fys. 17, 1–28 (1923).

Flyvbjerg, H.

P. M. Hansen, I. M. Tolić-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, “tweezercalib 2.0: Faster version of MatLab package for precise calibration of optical tweezers,” Comput. Phys. Commun. 174(6), 518–520 (2006).
[Crossref]

I. M. Tolic-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
[Crossref]

Gacoin, T.

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

Gelles, J.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

Gong, Z.

Z. Gong, Z. Wang, Y. Li, L. Lou, and S. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273(1), 37–42 (2007).
[Crossref]

Gosse, C.

C. Gosse and V. Croquette, “Magnetic Tweezers: Micromanipulation and Force Measurement at the Molecular Level,” Biophys. J. 82(6), 3314–3329 (2002).
[Crossref] [PubMed]

Granzier, H. L.

M. S. Z. Kellermayer, S. B. Smith, H. L. Granzier, and C. Bustamante, “Folding-unfolding transitions in single titin molecules characterized with laser tweezers,” Science 276(5315), 1112–1116 (1997).
[Crossref] [PubMed]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

Gross, P.

P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite, “Quantifying how DNA stretches, melts and changes twist under tension,” Nat. Phys. 7(9), 731–736 (2011).
[Crossref]

Gross, S. P.

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Quantum Electron. 2(4), 1066–1076 (1996).
[Crossref]

Hansen, P. M.

P. M. Hansen, I. M. Tolić-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, “tweezercalib 2.0: Faster version of MatLab package for precise calibration of optical tweezers,” Comput. Phys. Commun. 174(6), 518–520 (2006).
[Crossref]

Howard, J.

E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface Forces and Drag Coefficients of Microspheres near a Plane Surface Measured with Optical Tweezers,” Langmuir 23(7), 3654–3665 (2007).
[Crossref] [PubMed]

Inaba, H.

Kellermayer, M. S. Z.

M. S. Z. Kellermayer, S. B. Smith, H. L. Granzier, and C. Bustamante, “Folding-unfolding transitions in single titin molecules characterized with laser tweezers,” Science 276(5315), 1112–1116 (1997).
[Crossref] [PubMed]

Landick, R.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

Laurens, N.

P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite, “Quantifying how DNA stretches, melts and changes twist under tension,” Nat. Phys. 7(9), 731–736 (2011).
[Crossref]

Le Gall, A.

Li, Y.

Y. Li, V. Lubchenko, and P. G. Vekilov, “The use of dynamic light scattering and Brownian microscopy to characterize protein aggregation,” Rev. Sci. Instrum. 82(5), 053106 (2011).
[Crossref] [PubMed]

Z. Gong, Z. Wang, Y. Li, L. Lou, and S. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273(1), 37–42 (2007).
[Crossref]

Li, Y.-M.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat Commun 4, 1768 (2013).
[Crossref] [PubMed]

Lou, L.

Z. Gong, Z. Wang, Y. Li, L. Lou, and S. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273(1), 37–42 (2007).
[Crossref]

Lubchenko, V.

Y. Li, V. Lubchenko, and P. G. Vekilov, “The use of dynamic light scattering and Brownian microscopy to characterize protein aggregation,” Rev. Sci. Instrum. 82(5), 053106 (2011).
[Crossref] [PubMed]

Malagnino, N.

N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, “Measurements of trapping efficiency and stiffness in optical tweezers,” Opt. Commun. 214(1-6), 15–24 (2002).
[Crossref]

Marqusee, S.

C. Cecconi, E. A. Shank, C. Bustamante, and S. Marqusee, “Direct observation of the three-state folding of a single protein molecule,” Science 309(5743), 2057–2060 (2005).
[Crossref] [PubMed]

Mas, J.

J. Mas, A. C. Richardson, S. N. Reihani, L. B. Oddershede, and K. Berg-Sørensen, “Quantitative determination of optical trapping strength and viscoelastic moduli inside living cells,” Phys. Biol. 10(4), 046006 (2013).
[Crossref] [PubMed]

Masson, J.-B.

S. Türkcan, M. U. Richly, A. Alexandrou, and J.-B. Masson, “Probing membrane protein interactions with their lipid raft environment using single-molecule tracking and Bayesian inference alysis,” PLoS ONE 8(1), e53073 (2013).
[Crossref] [PubMed]

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

S. Türkcan, A. Alexandrou, and J.-B. Masson, “A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories,” Biophys. J. 102(10), 2288–2298 (2012).
[Crossref] [PubMed]

J.-B. Masson, D. Casanova, S. Türkcan, G. Voisinne, M. R. Popoff, M. Vergassola, and A. Alexandrou, “Inferring maps of forces inside cell membrane microdomains,” Phys. Rev. Lett. 102(4), 048103 (2009).
[Crossref] [PubMed]

Mehlich, A.

Y. von Hansen, A. Mehlich, B. Pelz, M. Rief, and R. R. Netz, “Auto- and cross-power spectral analysis of dual trap optical tweezer experiments using Bayesian inference,” Rev. Sci. Instrum. 83(9), 095116 (2012).
[Crossref] [PubMed]

Mehta, A. D.

A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich, M. S. Mooseker, and R. E. Cheney, “Myosin-V is a processive actin-based motor,” Nature 400(6744), 590–593 (1999).
[Crossref] [PubMed]

Mialon, G.

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

Moffitt, J. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Mooseker, M. S.

A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich, M. S. Mooseker, and R. E. Cheney, “Myosin-V is a processive actin-based motor,” Nature 400(6744), 590–593 (1999).
[Crossref] [PubMed]

Netz, R. R.

Y. von Hansen, A. Mehlich, B. Pelz, M. Rief, and R. R. Netz, “Auto- and cross-power spectral analysis of dual trap optical tweezer experiments using Bayesian inference,” Rev. Sci. Instrum. 83(9), 095116 (2012).
[Crossref] [PubMed]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[Crossref] [PubMed]

Nørrelykke, S. F.

E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface Forces and Drag Coefficients of Microspheres near a Plane Surface Measured with Optical Tweezers,” Langmuir 23(7), 3654–3665 (2007).
[Crossref] [PubMed]

Oddershede, L. B.

J. Mas, A. C. Richardson, S. N. Reihani, L. B. Oddershede, and K. Berg-Sørensen, “Quantitative determination of optical trapping strength and viscoelastic moduli inside living cells,” Phys. Biol. 10(4), 046006 (2013).
[Crossref] [PubMed]

P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite, “Quantifying how DNA stretches, melts and changes twist under tension,” Nat. Phys. 7(9), 731–736 (2011).
[Crossref]

A. C. Richardson, S. N. S. Reihani, and L. B. Oddershede, “Non-harmonic potential of a single beam optical trap,” Opt. Express 16(20), 15709–15717 (2008).
[Crossref] [PubMed]

Ogawa, Y.

Ohyumi, M.

Pelz, B.

Y. von Hansen, A. Mehlich, B. Pelz, M. Rief, and R. R. Netz, “Auto- and cross-power spectral analysis of dual trap optical tweezer experiments using Bayesian inference,” Rev. Sci. Instrum. 83(9), 095116 (2012).
[Crossref] [PubMed]

Perronet, K.

Pesce, G.

N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, “Measurements of trapping efficiency and stiffness in optical tweezers,” Opt. Commun. 214(1-6), 15–24 (2002).
[Crossref]

Peterman, E. J. G.

P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite, “Quantifying how DNA stretches, melts and changes twist under tension,” Nat. Phys. 7(9), 731–736 (2011).
[Crossref]

Popoff, M. R.

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

J.-B. Masson, D. Casanova, S. Türkcan, G. Voisinne, M. R. Popoff, M. Vergassola, and A. Alexandrou, “Inferring maps of forces inside cell membrane microdomains,” Phys. Rev. Lett. 102(4), 048103 (2009).
[Crossref] [PubMed]

Reihani, S. N.

J. Mas, A. C. Richardson, S. N. Reihani, L. B. Oddershede, and K. Berg-Sørensen, “Quantitative determination of optical trapping strength and viscoelastic moduli inside living cells,” Phys. Biol. 10(4), 046006 (2013).
[Crossref] [PubMed]

Reihani, S. N. S.

Richardson, A. C.

J. Mas, A. C. Richardson, S. N. Reihani, L. B. Oddershede, and K. Berg-Sørensen, “Quantitative determination of optical trapping strength and viscoelastic moduli inside living cells,” Phys. Biol. 10(4), 046006 (2013).
[Crossref] [PubMed]

A. C. Richardson, S. N. S. Reihani, and L. B. Oddershede, “Non-harmonic potential of a single beam optical trap,” Opt. Express 16(20), 15709–15717 (2008).
[Crossref] [PubMed]

Richly, M. U.

S. Türkcan, M. U. Richly, A. Alexandrou, and J.-B. Masson, “Probing membrane protein interactions with their lipid raft environment using single-molecule tracking and Bayesian inference alysis,” PLoS ONE 8(1), e53073 (2013).
[Crossref] [PubMed]

Rief, M.

Y. von Hansen, A. Mehlich, B. Pelz, M. Rief, and R. R. Netz, “Auto- and cross-power spectral analysis of dual trap optical tweezer experiments using Bayesian inference,” Rev. Sci. Instrum. 83(9), 095116 (2012).
[Crossref] [PubMed]

A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich, M. S. Mooseker, and R. E. Cheney, “Myosin-V is a processive actin-based motor,” Nature 400(6744), 590–593 (1999).
[Crossref] [PubMed]

Rock, R. S.

A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich, M. S. Mooseker, and R. E. Cheney, “Myosin-V is a processive actin-based motor,” Nature 400(6744), 590–593 (1999).
[Crossref] [PubMed]

Sasso, A.

N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, “Measurements of trapping efficiency and stiffness in optical tweezers,” Opt. Commun. 214(1-6), 15–24 (2002).
[Crossref]

Sato, S.

Schäffer, E.

E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface Forces and Drag Coefficients of Microspheres near a Plane Surface Measured with Optical Tweezers,” Langmuir 23(7), 3654–3665 (2007).
[Crossref] [PubMed]

Schmidt, C. F.

C. Veigel and C. F. Schmidt, “Moving into the cell: single-molecule studies of molecular motors in complex environments,” Nat. Rev. Mol. Cell Biol. 12(3), 163–176 (2011).
[Crossref] [PubMed]

K. C. Vermeulen, G. J. L. Wuite, G. J. M. Stienen, and C. F. Schmidt, “Optical trap stiffness in the presence and absence of spherical aberrations,” Appl. Opt. 45(8), 1812–1819 (2006).
[Crossref] [PubMed]

Shank, E. A.

C. Cecconi, E. A. Shank, C. Bustamante, and S. Marqusee, “Direct observation of the three-state folding of a single protein molecule,” Science 309(5743), 2057–2060 (2005).
[Crossref] [PubMed]

Shibata, H.

Smith, S. B.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

M. S. Z. Kellermayer, S. B. Smith, H. L. Granzier, and C. Bustamante, “Folding-unfolding transitions in single titin molecules characterized with laser tweezers,” Science 276(5315), 1112–1116 (1997).
[Crossref] [PubMed]

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: The elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271(5250), 795–799 (1996).
[Crossref] [PubMed]

Spudich, J. A.

A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich, M. S. Mooseker, and R. E. Cheney, “Myosin-V is a processive actin-based motor,” Nature 400(6744), 590–593 (1999).
[Crossref] [PubMed]

Stienen, G. J. M.

Svoboda, K.

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994).
[Crossref] [PubMed]

Tolic-Nørrelykke, I. M.

P. M. Hansen, I. M. Tolić-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, “tweezercalib 2.0: Faster version of MatLab package for precise calibration of optical tweezers,” Comput. Phys. Commun. 174(6), 518–520 (2006).
[Crossref]

I. M. Tolic-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

Türkcan, S.

S. Türkcan, M. U. Richly, A. Alexandrou, and J.-B. Masson, “Probing membrane protein interactions with their lipid raft environment using single-molecule tracking and Bayesian inference alysis,” PLoS ONE 8(1), e53073 (2013).
[Crossref] [PubMed]

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

S. Türkcan, A. Alexandrou, and J.-B. Masson, “A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories,” Biophys. J. 102(10), 2288–2298 (2012).
[Crossref] [PubMed]

J.-B. Masson, D. Casanova, S. Türkcan, G. Voisinne, M. R. Popoff, M. Vergassola, and A. Alexandrou, “Inferring maps of forces inside cell membrane microdomains,” Phys. Rev. Lett. 102(4), 048103 (2009).
[Crossref] [PubMed]

Veigel, C.

C. Veigel and C. F. Schmidt, “Moving into the cell: single-molecule studies of molecular motors in complex environments,” Nat. Rev. Mol. Cell Biol. 12(3), 163–176 (2011).
[Crossref] [PubMed]

Vekilov, P. G.

Y. Li, V. Lubchenko, and P. G. Vekilov, “The use of dynamic light scattering and Brownian microscopy to characterize protein aggregation,” Rev. Sci. Instrum. 82(5), 053106 (2011).
[Crossref] [PubMed]

Vergassola, M.

J.-B. Masson, D. Casanova, S. Türkcan, G. Voisinne, M. R. Popoff, M. Vergassola, and A. Alexandrou, “Inferring maps of forces inside cell membrane microdomains,” Phys. Rev. Lett. 102(4), 048103 (2009).
[Crossref] [PubMed]

Vermeulen, K. C.

Villing, A.

Visscher, K.

A. Le Gall, K. Perronet, D. Dulin, A. Villing, P. Bouyer, K. Visscher, and N. Westbrook, “Simultaneous calibration of optical tweezers spring constant and position detector response,” Opt. Express 18(25), 26469–26474 (2010).
[Crossref] [PubMed]

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Quantum Electron. 2(4), 1066–1076 (1996).
[Crossref]

Voisinne, G.

J.-B. Masson, D. Casanova, S. Türkcan, G. Voisinne, M. R. Popoff, M. Vergassola, and A. Alexandrou, “Inferring maps of forces inside cell membrane microdomains,” Phys. Rev. Lett. 102(4), 048103 (2009).
[Crossref] [PubMed]

von Hansen, Y.

Y. von Hansen, A. Mehlich, B. Pelz, M. Rief, and R. R. Netz, “Auto- and cross-power spectral analysis of dual trap optical tweezer experiments using Bayesian inference,” Rev. Sci. Instrum. 83(9), 095116 (2012).
[Crossref] [PubMed]

Wang, M. D.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

Wang, Z.

Z. Gong, Z. Wang, Y. Li, L. Lou, and S. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273(1), 37–42 (2007).
[Crossref]

Wang, Z.-Q.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat Commun 4, 1768 (2013).
[Crossref] [PubMed]

Wei, X.-B.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat Commun 4, 1768 (2013).
[Crossref] [PubMed]

Westbrook, N.

Williams, P. M.

A. Borgia, P. M. Williams, and J. Clarke, “Single-molecule studies of protein folding,” Annu. Rev. Biochem. 77(1), 101–125 (2008).
[Crossref] [PubMed]

Wuite, G. J. L.

P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite, “Quantifying how DNA stretches, melts and changes twist under tension,” Nat. Phys. 7(9), 731–736 (2011).
[Crossref]

K. C. Vermeulen, G. J. L. Wuite, G. J. M. Stienen, and C. F. Schmidt, “Optical trap stiffness in the presence and absence of spherical aberrations,” Appl. Opt. 45(8), 1812–1819 (2006).
[Crossref] [PubMed]

Xu, S.

Z. Gong, Z. Wang, Y. Li, L. Lou, and S. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273(1), 37–42 (2007).
[Crossref]

Yin, H.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

Zhong, M.-C.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat Commun 4, 1768 (2013).
[Crossref] [PubMed]

Zhou, J.-H.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat Commun 4, 1768 (2013).
[Crossref] [PubMed]

Annu Rev Biophys (1)

I. De Vlaminck and C. Dekker, “Recent advances in magnetic tweezers,” Annu Rev Biophys 41(1), 453–472 (2012).
[Crossref] [PubMed]

Annu. Rev. Biochem. (2)

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

A. Borgia, P. M. Williams, and J. Clarke, “Single-molecule studies of protein folding,” Annu. Rev. Biochem. 77(1), 101–125 (2008).
[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(1), 247–285 (1994).
[Crossref] [PubMed]

Appl. Opt. (1)

Ark. Mat. Aston. Fys. (1)

H. Faxen, “Die Bewegung einer starren Kugel laengs der Achse eines mit zaeher Fluessigkeit gefuellten Rohres,” Ark. Mat. Aston. Fys. 17, 1–28 (1923).

Biophys. J. (4)

S. Türkcan, A. Alexandrou, and J.-B. Masson, “A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories,” Biophys. J. 102(10), 2288–2298 (2012).
[Crossref] [PubMed]

S. Türkcan, J.-B. Masson, D. Casanova, G. Mialon, T. Gacoin, J.-P. Boilot, M. R. Popoff, and A. Alexandrou, “Observing the confinement potential of bacterial pore-forming toxin receptors inside rafts with nonblinking Eu(3+)-doped oxide nanoparticles,” Biophys. J. 102(10), 2299–2308 (2012).
[Crossref] [PubMed]

C. Gosse and V. Croquette, “Magnetic Tweezers: Micromanipulation and Force Measurement at the Molecular Level,” Biophys. J. 82(6), 3314–3329 (2002).
[Crossref] [PubMed]

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997).
[Crossref] [PubMed]

Comput. Phys. Commun. (2)

I. M. Tolic-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[Crossref]

P. M. Hansen, I. M. Tolić-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, “tweezercalib 2.0: Faster version of MatLab package for precise calibration of optical tweezers,” Comput. Phys. Commun. 174(6), 518–520 (2006).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Quantum Electron. 2(4), 1066–1076 (1996).
[Crossref]

Langmuir (1)

E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface Forces and Drag Coefficients of Microspheres near a Plane Surface Measured with Optical Tweezers,” Langmuir 23(7), 3654–3665 (2007).
[Crossref] [PubMed]

Nat Commun (1)

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat Commun 4, 1768 (2013).
[Crossref] [PubMed]

Nat. Phys. (1)

P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite, “Quantifying how DNA stretches, melts and changes twist under tension,” Nat. Phys. 7(9), 731–736 (2011).
[Crossref]

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

C. Veigel and C. F. Schmidt, “Moving into the cell: single-molecule studies of molecular motors in complex environments,” Nat. Rev. Mol. Cell Biol. 12(3), 163–176 (2011).
[Crossref] [PubMed]

Nature (2)

A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich, M. S. Mooseker, and R. E. Cheney, “Myosin-V is a processive actin-based motor,” Nature 400(6744), 590–593 (1999).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

Opt. Commun. (2)

Z. Gong, Z. Wang, Y. Li, L. Lou, and S. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273(1), 37–42 (2007).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 A sample trajectory recorded for a bead in an optical trap created by a focused laser intensity of 377 mW (A) and the confinement potential inferred with the Bayesian inference algorithm (B). The trap stiffness in the x and y direction, kx and ky, is 0.05 and 0.08 pN/nm, respectively.
Fig. 2
Fig. 2 Bias curves obtained from simulated trajectories for Δt = 91.8 µs and D = 0.3µm2/s. The trap stiffness kx = ky is varied from 10−6 to 0.2 pN/nm. The inferred values Qinferred are normalized to the input values. Solid lines are fits according to Eq. (11) and (12). The data points are averages of 30 numerical trajectories for each u value, each trajectory has 3000 points.
Fig. 3
Fig. 3 Potentials inferred from experimental trajectories using the Bayesian approach for increasing laser power (60, 138, 251, 377, 466, 500 mW; green to red curves) (A) and for decreasing distance from the coverslip surface (21.5, 16, 11, 6, 4, 2 µm; red to green curves) (B).
Fig. 4
Fig. 4 Spring constants extracted from experimental bead trajectories with the Bayesian inference (black data points), the power-spectrum (red) and the equipartition method (blue) as a function of laser power (A) and distance from the coverslip (B). The solid lines are a guide to the eye. The dashed black line in (A) is a linear fit to the spring constants extracted with Bayesian inference (BI). Error bars for the power spectrum data are derived from the error in the Lorentzian fit; for the Bayesian inference data, the error bars indicate the width of the posteriori distribution and, for the equipartition data, the error bars are the error on the mean of <x2>. (C) Diffusion coefficient as a function of the distance from the coverslip inferred from the data set in (B) (orange), effective viscosity η normalized to the bulk water viscosity η0 (purple) and diffusion coefficient after correcting for the effective viscosity effect (green). The green dashed line represents the resulting diffusion coefficient: 0.63 ± 0.02 µm2/s (the error is that of the fit with a constant value).
Fig. 5
Fig. 5 Biases found for the stiffness values kx determined with the inference method (black curve) and with the equipartition method (blue curve) from trajectories with drift. Each data point is the average value for 20 simulated trajectories where the stated drift occurs linearly over the entire trajectory. The time interval between trajectory points, the diffusion coefficient and spring constant used were 15.3 µs (x6 for the Bayesian Inference), 0.3 µm2/s, and 0.06 pN/nm, respectively. The simulated trajectory length was 45 000 points for the equipartition method and 7500 points for the Bayesian inference approach. The error bars are the errors on the mean of the values determined.
Fig. 6
Fig. 6 Simulated trajectories for a varying number of points. (A) Spring constants determined with the Bayesian inference (black curve), equipartition (blue curve) and the power spectrum (red curve) approach normalized with the input spring constant. For each trajectory length, the average value determined for 20 trajectories generated with Δt = 15.3 µs, D = 0.3 µm2/s, and kx = 0.06 pN/nm is shown. The error bars are the errors on the mean of the values determined. (B) Posteriori probability distributions of kx for two numerical trajectories generated with D = 0.3 µm2/s and kx = 0.06 pN/nm with N = 600 and 3000 points. For N = 600, if only every 6th point is taken into account, the posterior distribution is given by the black curve which is broad and more biased. If all 600 points are taken into account (see text), the blue posteriori distribution is obtained which is narrow and peaks at the same value as that corresponding to the longer N = 3000 trajectory (red). The small bias observed with respect to the input value of kx = 0.06 pN/nm (vertical dotted line) is corrected by using the bias curves of Fig. 2.
Fig. 7
Fig. 7 2nd (red), 4th (dark blue) and 6th (light blue) order potentials inferred from an experimental trajectory obtained for a laser power of 251 mW (same trajectory as in Fig. 4(a)).

Equations (15)

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1 2 k B T= 1 2 k x < x 2 >
P(f)= k B T 2γ π 2 ( f 2 + f c 2 ) .
f c = k x 2πγ .
dr(t) dt =- V(r) γ + 2D ξ(t).
t P(r,t| r 0 , t 0 )=-[ V(r) γ P(r,t| r 0 , t 0 )(DP(r,t| r 0 , t 0 )) ]
P(r,t| r 0 , t 0 )| F ij ,D)= exp( (r r 0 F ij (t t 0 )/γ) 2 4(D+ σ 2 t t 0 )(t t 0 ) ) 4π(D+ σ 2 t t 0 )(t t 0 )
P(T|F,D)= i,j=1 i max , j max μ: r μ S ij P(( r μ+1 , t μ+1 | r μ , t μ )| F ij ,D)
P( Q 1,2,...,n |T)= P(T| Q 1,2,...,n ) P 0 ( Q 1,2,...,n ) P 0 (T)
V 2 nd order =C+ C x x+ C y y+ C xx x 2 + C xy xy+ C yy y 2
k x =2 C xx and k y =2 C yy .
D inf D =0.32 e u 0.024 +1.29
k xinf k x =0.023 e u 0.0085 +1.01
g(t)= α (diode) δ(t)+(1 α (diode) ) 1 τ e t τ .
S (det) (t)= t dt' g(tt')S(t'),
γ Faxen (R/h)= γ 0 1(9R/19h)+( R 3 /8 h 3 )(45 R 4 /256 h 4 )( R 5 /16 h 5 )+

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