Abstract

The use of optical traps to measure or apply forces on the molecular level requires a precise knowledge of the trapping force field. Close to the trap center, this field is typically approximated as linear in the displacement of the trapped microsphere. However, applications demanding high forces at low laser intensities can probe the light-microsphere interaction beyond the linear regime. Here, we measured the full nonlinear force and displacement response of an optical trap in two dimensions using a dual-beam optical trap setup with back-focal-plane photodetection. We observed a substantial stiffening of the trap beyond the linear regime that depends on microsphere size, in agreement with Mie theory calculations. Surprisingly, we found that the linear detection range for forces exceeds the one for displacement by far. Our approach allows for a complete calibration of an optical trap.

© 2011 Optical Society of America

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References

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  1. J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
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2009 (2)

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Y.-F. Chen, G. A. Blab, and J.-C. Meiners, Biophys. J. 96, 4701 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A 9, S196 (2007).
[CrossRef]

2006 (2)

A. A. R. Neves, A. Fontes, L. de Y Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, Opt. Express 14, 13101 (2006).
[CrossRef] [PubMed]

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, Rev. Sci. Instrum. 77, 3101 (2006).

2005 (2)

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, Nature 438, 460 (2005).
[CrossRef] [PubMed]

2004 (1)

K. Berg-Sørensen and H. Flyvbjerg, Rev. Sci. Instrum. 75, 594 (2004).
[CrossRef]

2002 (1)

1998 (1)

Abbondanzieri, E. A.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, Nature 438, 460 (2005).
[CrossRef] [PubMed]

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

Ander, M.

Barbosa, L. C.

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, Rev. Sci. Instrum. 75, 594 (2004).
[CrossRef]

Blab, G. A.

Y.-F. Chen, G. A. Blab, and J.-C. Meiners, Biophys. J. 96, 4701 (2009).
[CrossRef] [PubMed]

Block, S. M.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, Nature 438, 460 (2005).
[CrossRef] [PubMed]

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

Bormuth, V.

Branczyk, A. M.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A 9, S196 (2007).
[CrossRef]

Bustamante, C.

C. Bustamante, Y. Chemla, and J. Moffitt, in Single-Molecule Techniques: a Laboratory Manual, P.R.Selvin and T.Ha, eds. (CSHL, 2008).

Cesar, C. L.

Chemla, Y.

C. Bustamante, Y. Chemla, and J. Moffitt, in Single-Molecule Techniques: a Laboratory Manual, P.R.Selvin and T.Ha, eds. (CSHL, 2008).

Chen, Y.-F.

Y.-F. Chen, G. A. Blab, and J.-C. Meiners, Biophys. J. 96, 4701 (2009).
[CrossRef] [PubMed]

Chillce, E.

de Thomaz, A. A.

de Y Pozzo, L.

Falkenberg, M.

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Farge, G.

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Flyvbjerg, H.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, Rev. Sci. Instrum. 77, 3101 (2006).

K. Berg-Sørensen and H. Flyvbjerg, Rev. Sci. Instrum. 75, 594 (2004).
[CrossRef]

Fontes, A.

Gibson, G. M.

Gittes, F.

Greenleaf, W. J.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, Nature 438, 460 (2005).
[CrossRef] [PubMed]

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

Gross, P.

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Heckenberg, N. R.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A 9, S196 (2007).
[CrossRef]

Hooijman, P.

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Howard, J.

V. Bormuth, A. Jannasch, M. Ander, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, Opt. Express 16, 13831 (2008).
[CrossRef] [PubMed]

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, Rev. Sci. Instrum. 77, 3101 (2006).

Jannasch, A.

Jülicher, F.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, Rev. Sci. Instrum. 77, 3101 (2006).

Keen, S.

Knöner, G.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A 9, S196 (2007).
[CrossRef]

Landick, R.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, Nature 438, 460 (2005).
[CrossRef] [PubMed]

Leach, J.

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A 9, S196 (2007).
[CrossRef]

Meiners, J.-C.

Y.-F. Chen, G. A. Blab, and J.-C. Meiners, Biophys. J. 96, 4701 (2009).
[CrossRef] [PubMed]

Modesti, M.

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Moffitt, J.

C. Bustamante, Y. Chemla, and J. Moffitt, in Single-Molecule Techniques: a Laboratory Manual, P.R.Selvin and T.Ha, eds. (CSHL, 2008).

Neves, A. A. R.

Nieminen, T. A.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A 9, S196 (2007).
[CrossRef]

Oddershede, L. B.

Padgett, M. J.

Pavone, F. S.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, Rev. Sci. Instrum. 77, 3101 (2006).

Peterman, E. J. G.

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Reihani, S. N. S.

Richardson, A. C.

Rodriguez, E.

Rohrbach, A.

Rubinsztein-Dunlop, H.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A 9, S196 (2007).
[CrossRef]

Schäffer, E.

V. Bormuth, A. Jannasch, M. Ander, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, Opt. Express 16, 13831 (2008).
[CrossRef] [PubMed]

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, Rev. Sci. Instrum. 77, 3101 (2006).

Schmidt, C. F.

Shaevitz, J. W.

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, Nature 438, 460 (2005).
[CrossRef] [PubMed]

Stelzer, E. H.

Stilgoe, A. B.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A 9, S196 (2007).
[CrossRef]

Tolic-Nørrelykke, S. F.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, Rev. Sci. Instrum. 77, 3101 (2006).

van Blaaderen, A.

van Kats, C. M.

van Mameren, J.

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Woodside, M. T.

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

Wright, A. J.

Wuite, G. J. L.

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Appl. Opt. (1)

Biophys. J. (1)

Y.-F. Chen, G. A. Blab, and J.-C. Meiners, Biophys. J. 96, 4701 (2009).
[CrossRef] [PubMed]

J. Opt. A (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Opt. A 9, S196 (2007).
[CrossRef]

Nature (1)

E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, Nature 438, 460 (2005).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, Phys. Rev. Lett. 95, 208102 (2005).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

J. van Mameren, P. Gross, G. Farge, P. Hooijman, M. Modesti, M. Falkenberg, G. J. L. Wuite, and E. J. G. Peterman, Proc. Natl. Acad. Sci. USA 106, 18231 (2009).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (2)

K. Berg-Sørensen and H. Flyvbjerg, Rev. Sci. Instrum. 75, 594 (2004).
[CrossRef]

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, Rev. Sci. Instrum. 77, 3101 (2006).

Other (2)

C. Bustamante, Y. Chemla, and J. Moffitt, in Single-Molecule Techniques: a Laboratory Manual, P.R.Selvin and T.Ha, eds. (CSHL, 2008).

Optical alignment using back-reflections reduces the polarization cross talk between the two traps to below 1%.

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Figures (3)

Fig. 1
Fig. 1

Setup: beam path, starting at Trap- LASER (red), back reflection path, ending at CCDs (blue), imaging path, from LED to CCD (orange). FI, Faraday isolator; DM1–DM5, dichroic mirrors; HWP, half-wave plate, GLP, Glan–Laser polarizer; PBS, polarizing beam splitter; AOD, acousto- optical deflector; PM, piezo mirror; O1 and O2, water immersion objectives ( NA = 1.20 ); PSD C and PSD A, position-sensitive devices; CCD (BR), camera-to-monitor back-reflections [11] Inset: schematic of the experiment. d, intertrap distance; r, displacement of microsphere from center of T A .

Fig. 2
Fig. 2

2D maps of (a) optical forces on a 1.26 μm microsphere in the direction of polarization, (b) magnitude of radial force | F | = F x 2 + F y 2 . Force magnitudes are color-coded by corresponding heat maps. (c) Complete force response along the polarization axis [dashed line in (a)]. (d) Numerical differentiation, κ ( x ) = x F ( r ) , yielded trap stiffness with respect to microsphere displacement.

Fig. 3
Fig. 3

Data (points) and theory (lines) of (a) normalized force-extension curves and (b) thereof-derived trap stiffnesses for indicated microsphere sizes. For experimental purposes, different microspheres were investigated with different laser intensities; F max : ( 37 , 35 , 48 ) pN ; κ 0 : ( 72 , 22 , 30 ) pN / μm for microspheres of diameter ( 1.26 , 2.01 , 2.40 ) μm , respectively. (c) Comparing forces (dotted line, red) and displacements (dashed line, green) to the detector signal of T A for a 2.01 μm microsphere. Shaded areas indicate where residuals [see (d)] of linear fit are less than 5% (dark) or less than 10% (bright). (d) Force (dotted line, red) and displacement (dashed line, green) residuals of the fit in (c) normalized to F ( V max ) = 33 pN and x ( V max ) = 950 nm , respectively.

Equations (1)

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κ ^ C r d t a v = F A ( r ) t a v ,

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