Abstract

Single nanometric apertures in a metallic film are used to develop a simple and robust setup for dual-color fluorescence crosscorrelation spectroscopy (FCCS) at high concentrations. If the nanoaperture concept has already proven to be useful for single-species analysis, its extension to the dual-color case brings new interesting specificities. The alignment and overlap of the two excitation beams are greatly simplified. No confocal pinhole is used, relaxing the requirement for accurate correction of chromatic aberrations. Compared to two-photon excitation, nanoapertures have the advantage to work with standard fluorophore constructions having high absorption cross-section and well-known absorption/emission spectra. Thanks to the ultra-low volume analysed within one single aperture, fluorescence correlation analysis can be performed with single molecule resolution at micromolar concentrations, resulting in 3 orders of magnitude gain compared to conventional setups. As applications of this technique, we follow the kinetics of an enzymatic cleavage reaction at 2 µM DNA oligonucleotide concentration.We also demonstrate that FCCS in nanoapertures can be applied to the fast screening of a sample for dual-labeled species within 1 s acquisition time. This offers new possibilities for rapid screening applications in biotechnology at high concentrations.

©2006 Optical Society of America

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  1. M. Eigen and R. Rigler, “Sorting single molecules: Application to diagnostics and evolutionary biotechnology,” Proc. Natl. Acad. Sci. USA 91, 5740–5747 (1994).
    [Crossref] [PubMed]
  2. P. Schwille, F.-J. Meyer-Almes, and R. Rigler, “Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution,” Biophys. J. 72, 1878–1886 (1997).
    [Crossref] [PubMed]
  3. T Weidemann, M Wachsmuth, M. Tewes, K. Rippe, and J. Langowski, “Analysis of Ligand Binding by Two-Colour Fluorescence Cross-Correlation Spectroscopy,” Single Mol. 3, 49–61 (2002).
    [Crossref]
  4. K. Bacia, S. A. Kim, and P. Schwille, “Fluorescence cross-correlation spectroscopy in living cells,” Nat. Methods 3, 83–89 (2006).
    [Crossref] [PubMed]
  5. U. Kettling, A. Koltermann, P. Schwille, and M. Eigen, “Real-time enzyme kinetics monitored by dual-color fluorescence cross-correlation spectroscopy,” Proc. Natl. Acad. Sci. USA 95, 1416–1420 (1998).
    [Crossref] [PubMed]
  6. A. Koltermann, U. Kettling, J. Bieschke, T. Winkler, and M. Eigen, “Rapid assay processing by integration of dualcolor fluorescence cross-correlation spectroscopy: high throughput screening for enzyme activity,” Proc. Natl. Acad. Sci. USA 95, 1421–1426 (1998).
    [Crossref] [PubMed]
  7. T. Winkler, U. Kettling, A. Koltermann, and M. Eigen, “Confocal fluorescence coincidence analysis: an approach to ultra high throughput screening,” Proc. Natl. Acad. Sci. USA 96, 1375–1378 (1999).
    [Crossref] [PubMed]
  8. N. Baudendistel, G. Müller, W. Waldeck, P. Angel, and J. Langowski, “Two-Hybrid Fluorescence Cross-Correlation Spectroscopy Detects ProteinProtein Interactions In Vivo,” Chem. Phys. Chem. 6, 984–990 (2005).
    [Crossref] [PubMed]
  9. K.G. Heinze, A. Koltermann, and P. Schwille, “Simultaneous two-photon excitation of distinct labels for dual-color fluorescence cross-correlation analysis,” Proc. Natl. Acad. Sci. USA 97, 10377–10382 (2000).
    [Crossref] [PubMed]
  10. S. A. Kim, K. G. Heinze, M. N. Waxham, and P. Schwille, “Intracellular calmodulin availability accessed with two-photon crosscorrelation,” Proc. Natl. Acad. Sci. USA. 101, 105–110 (2004).
    [Crossref]
  11. D. R. Larson, J. A. Gosse, D. A. Holowka, B. A. Baird, and W.W Webb, “Temporally resolved interactions between antigen-stimulated IgE receptors and Lyn kinase on living cells,” J. Cell Biol. 171, 527–536 (2005).
    [Crossref] [PubMed]
  12. T. Kogure, S. Karasawa, T. Araki, K. Saito, M. Kinjo, and A. Miyawaki, “A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy,” Nat. Biotechnol. 24, 577–581 (2006).
    [Crossref] [PubMed]
  13. M. Burkhardt, K. G. Heinze, and P. Schwille, “Four-color fluorescence correlation spectroscopy realized in a gratingbased detection platform,” Opt. Lett. 30, 2266–2268 (2005).
    [Crossref] [PubMed]
  14. L. C. Hwang, M. Leutenegger, M. Gösch, T. Lasser, P. Rigler, W. Meier, and T. Wohland, “Prism-based multicolor fluorescence correlation spectrometer,” Opt. Lett. 31, 1310–1312 (2006).
    [Crossref] [PubMed]
  15. M. Leutenegger, H. Blom, J. Widengren, C. Eggeling, M. Gösch, R. A. Leitgeb, and T. Lasser, “Dual-color total internal reflection fluorescence cross-correlation spectroscopy,” J. Biomed. Opt. 11, 040502 (2006).
    [Crossref] [PubMed]
  16. M.J. Levene, J. Korlach, S.W. Turner, M. Foquet, H.G. Craighead, and W.W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
    [Crossref] [PubMed]
  17. K. T. Samiee, M. Foquet, L. Guo, E. C. Cox, and H. G. Craighead, “Lambda repressor oligomerization kinetics at high concentrations using fluorescence correlation spectroscopy in zero-mode waveguides,” Biophys. J. 88, 2145–2153 (2005).
    [Crossref]
  18. H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
    [Crossref] [PubMed]
  19. J. Wenger, P.-F. Lenne, E. Popov, H. Rigneault, J. Dintinger, and T.W. Ebbesen, “Single molecule fluorescence in rectangular nano-apertures,” Opt. Express 13, 7035–7044 (2005).
    [Crossref] [PubMed]
  20. M. Leutenegger, M. Gösch, A. Perentes, P. Hoffmann, O. J. F. Martin, and T. Lasser, “Confining the sampling volume for Fluorescence Correlation Spectroscopy using a sub-wavelength sized aperture,” Opt. Express 14, 956–969 (2006).
    [Crossref] [PubMed]
  21. J. B. Edel, M. Wu, B. Baird, and H. G. Craighead, “High spatial resolution observation of single molecule dynamics in living cell membranes,” Biophys. J. 88, L43–L45 (2005).
    [Crossref] [PubMed]
  22. J Wenger, H. Rigneault, J. Dintinger, D Marguet, and P. F. Lenne, “Single-fluorophore diffusion in a lipid membrane over a subwavelength aperture,” J. Biol. Phys. 32, SN1–SN4 (2006).
    [Crossref] [PubMed]
  23. K. T. Samiee, J. M. Moran-Mirabal, Y. K. Cheung, and H. G. Craighead, “Zero Mode Waveguides for Single-Molecule Spectroscopy on Lipid Membranes,” Biophys. J. 90, 3288–3299 (2006).
    [Crossref] [PubMed]
  24. E. Popov, M. Nevière, J. Wenger, P.-F. Lenne, H. Rigneault, P. Chaumet, N. Bonod, J. Dintinger, and T.W. Ebbesen, “Field enhancement in single subwavelength apertures,” J. Opt. Soc. Am. A 23, 2342–2348 (2006).
    [Crossref]

2006 (8)

K. Bacia, S. A. Kim, and P. Schwille, “Fluorescence cross-correlation spectroscopy in living cells,” Nat. Methods 3, 83–89 (2006).
[Crossref] [PubMed]

T. Kogure, S. Karasawa, T. Araki, K. Saito, M. Kinjo, and A. Miyawaki, “A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy,” Nat. Biotechnol. 24, 577–581 (2006).
[Crossref] [PubMed]

M. Leutenegger, H. Blom, J. Widengren, C. Eggeling, M. Gösch, R. A. Leitgeb, and T. Lasser, “Dual-color total internal reflection fluorescence cross-correlation spectroscopy,” J. Biomed. Opt. 11, 040502 (2006).
[Crossref] [PubMed]

J Wenger, H. Rigneault, J. Dintinger, D Marguet, and P. F. Lenne, “Single-fluorophore diffusion in a lipid membrane over a subwavelength aperture,” J. Biol. Phys. 32, SN1–SN4 (2006).
[Crossref] [PubMed]

K. T. Samiee, J. M. Moran-Mirabal, Y. K. Cheung, and H. G. Craighead, “Zero Mode Waveguides for Single-Molecule Spectroscopy on Lipid Membranes,” Biophys. J. 90, 3288–3299 (2006).
[Crossref] [PubMed]

M. Leutenegger, M. Gösch, A. Perentes, P. Hoffmann, O. J. F. Martin, and T. Lasser, “Confining the sampling volume for Fluorescence Correlation Spectroscopy using a sub-wavelength sized aperture,” Opt. Express 14, 956–969 (2006).
[Crossref] [PubMed]

L. C. Hwang, M. Leutenegger, M. Gösch, T. Lasser, P. Rigler, W. Meier, and T. Wohland, “Prism-based multicolor fluorescence correlation spectrometer,” Opt. Lett. 31, 1310–1312 (2006).
[Crossref] [PubMed]

E. Popov, M. Nevière, J. Wenger, P.-F. Lenne, H. Rigneault, P. Chaumet, N. Bonod, J. Dintinger, and T.W. Ebbesen, “Field enhancement in single subwavelength apertures,” J. Opt. Soc. Am. A 23, 2342–2348 (2006).
[Crossref]

2005 (7)

K. T. Samiee, M. Foquet, L. Guo, E. C. Cox, and H. G. Craighead, “Lambda repressor oligomerization kinetics at high concentrations using fluorescence correlation spectroscopy in zero-mode waveguides,” Biophys. J. 88, 2145–2153 (2005).
[Crossref]

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[Crossref] [PubMed]

J. B. Edel, M. Wu, B. Baird, and H. G. Craighead, “High spatial resolution observation of single molecule dynamics in living cell membranes,” Biophys. J. 88, L43–L45 (2005).
[Crossref] [PubMed]

M. Burkhardt, K. G. Heinze, and P. Schwille, “Four-color fluorescence correlation spectroscopy realized in a gratingbased detection platform,” Opt. Lett. 30, 2266–2268 (2005).
[Crossref] [PubMed]

J. Wenger, P.-F. Lenne, E. Popov, H. Rigneault, J. Dintinger, and T.W. Ebbesen, “Single molecule fluorescence in rectangular nano-apertures,” Opt. Express 13, 7035–7044 (2005).
[Crossref] [PubMed]

D. R. Larson, J. A. Gosse, D. A. Holowka, B. A. Baird, and W.W Webb, “Temporally resolved interactions between antigen-stimulated IgE receptors and Lyn kinase on living cells,” J. Cell Biol. 171, 527–536 (2005).
[Crossref] [PubMed]

N. Baudendistel, G. Müller, W. Waldeck, P. Angel, and J. Langowski, “Two-Hybrid Fluorescence Cross-Correlation Spectroscopy Detects ProteinProtein Interactions In Vivo,” Chem. Phys. Chem. 6, 984–990 (2005).
[Crossref] [PubMed]

2004 (1)

S. A. Kim, K. G. Heinze, M. N. Waxham, and P. Schwille, “Intracellular calmodulin availability accessed with two-photon crosscorrelation,” Proc. Natl. Acad. Sci. USA. 101, 105–110 (2004).
[Crossref]

2003 (1)

M.J. Levene, J. Korlach, S.W. Turner, M. Foquet, H.G. Craighead, and W.W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[Crossref] [PubMed]

2002 (1)

T Weidemann, M Wachsmuth, M. Tewes, K. Rippe, and J. Langowski, “Analysis of Ligand Binding by Two-Colour Fluorescence Cross-Correlation Spectroscopy,” Single Mol. 3, 49–61 (2002).
[Crossref]

2000 (1)

K.G. Heinze, A. Koltermann, and P. Schwille, “Simultaneous two-photon excitation of distinct labels for dual-color fluorescence cross-correlation analysis,” Proc. Natl. Acad. Sci. USA 97, 10377–10382 (2000).
[Crossref] [PubMed]

1999 (1)

T. Winkler, U. Kettling, A. Koltermann, and M. Eigen, “Confocal fluorescence coincidence analysis: an approach to ultra high throughput screening,” Proc. Natl. Acad. Sci. USA 96, 1375–1378 (1999).
[Crossref] [PubMed]

1998 (2)

U. Kettling, A. Koltermann, P. Schwille, and M. Eigen, “Real-time enzyme kinetics monitored by dual-color fluorescence cross-correlation spectroscopy,” Proc. Natl. Acad. Sci. USA 95, 1416–1420 (1998).
[Crossref] [PubMed]

A. Koltermann, U. Kettling, J. Bieschke, T. Winkler, and M. Eigen, “Rapid assay processing by integration of dualcolor fluorescence cross-correlation spectroscopy: high throughput screening for enzyme activity,” Proc. Natl. Acad. Sci. USA 95, 1421–1426 (1998).
[Crossref] [PubMed]

1997 (1)

P. Schwille, F.-J. Meyer-Almes, and R. Rigler, “Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution,” Biophys. J. 72, 1878–1886 (1997).
[Crossref] [PubMed]

1994 (1)

M. Eigen and R. Rigler, “Sorting single molecules: Application to diagnostics and evolutionary biotechnology,” Proc. Natl. Acad. Sci. USA 91, 5740–5747 (1994).
[Crossref] [PubMed]

Angel, P.

N. Baudendistel, G. Müller, W. Waldeck, P. Angel, and J. Langowski, “Two-Hybrid Fluorescence Cross-Correlation Spectroscopy Detects ProteinProtein Interactions In Vivo,” Chem. Phys. Chem. 6, 984–990 (2005).
[Crossref] [PubMed]

Araki, T.

T. Kogure, S. Karasawa, T. Araki, K. Saito, M. Kinjo, and A. Miyawaki, “A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy,” Nat. Biotechnol. 24, 577–581 (2006).
[Crossref] [PubMed]

Bacia, K.

K. Bacia, S. A. Kim, and P. Schwille, “Fluorescence cross-correlation spectroscopy in living cells,” Nat. Methods 3, 83–89 (2006).
[Crossref] [PubMed]

Baird, B.

J. B. Edel, M. Wu, B. Baird, and H. G. Craighead, “High spatial resolution observation of single molecule dynamics in living cell membranes,” Biophys. J. 88, L43–L45 (2005).
[Crossref] [PubMed]

Baird, B. A.

D. R. Larson, J. A. Gosse, D. A. Holowka, B. A. Baird, and W.W Webb, “Temporally resolved interactions between antigen-stimulated IgE receptors and Lyn kinase on living cells,” J. Cell Biol. 171, 527–536 (2005).
[Crossref] [PubMed]

Baudendistel, N.

N. Baudendistel, G. Müller, W. Waldeck, P. Angel, and J. Langowski, “Two-Hybrid Fluorescence Cross-Correlation Spectroscopy Detects ProteinProtein Interactions In Vivo,” Chem. Phys. Chem. 6, 984–990 (2005).
[Crossref] [PubMed]

Bieschke, J.

A. Koltermann, U. Kettling, J. Bieschke, T. Winkler, and M. Eigen, “Rapid assay processing by integration of dualcolor fluorescence cross-correlation spectroscopy: high throughput screening for enzyme activity,” Proc. Natl. Acad. Sci. USA 95, 1421–1426 (1998).
[Crossref] [PubMed]

Blom, H.

M. Leutenegger, H. Blom, J. Widengren, C. Eggeling, M. Gösch, R. A. Leitgeb, and T. Lasser, “Dual-color total internal reflection fluorescence cross-correlation spectroscopy,” J. Biomed. Opt. 11, 040502 (2006).
[Crossref] [PubMed]

Bonod, N.

E. Popov, M. Nevière, J. Wenger, P.-F. Lenne, H. Rigneault, P. Chaumet, N. Bonod, J. Dintinger, and T.W. Ebbesen, “Field enhancement in single subwavelength apertures,” J. Opt. Soc. Am. A 23, 2342–2348 (2006).
[Crossref]

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[Crossref] [PubMed]

Burkhardt, M.

Capoulade, J.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[Crossref] [PubMed]

Chaumet, P.

Cheung, Y. K.

K. T. Samiee, J. M. Moran-Mirabal, Y. K. Cheung, and H. G. Craighead, “Zero Mode Waveguides for Single-Molecule Spectroscopy on Lipid Membranes,” Biophys. J. 90, 3288–3299 (2006).
[Crossref] [PubMed]

Cox, E. C.

K. T. Samiee, M. Foquet, L. Guo, E. C. Cox, and H. G. Craighead, “Lambda repressor oligomerization kinetics at high concentrations using fluorescence correlation spectroscopy in zero-mode waveguides,” Biophys. J. 88, 2145–2153 (2005).
[Crossref]

Craighead, H. G.

K. T. Samiee, J. M. Moran-Mirabal, Y. K. Cheung, and H. G. Craighead, “Zero Mode Waveguides for Single-Molecule Spectroscopy on Lipid Membranes,” Biophys. J. 90, 3288–3299 (2006).
[Crossref] [PubMed]

K. T. Samiee, M. Foquet, L. Guo, E. C. Cox, and H. G. Craighead, “Lambda repressor oligomerization kinetics at high concentrations using fluorescence correlation spectroscopy in zero-mode waveguides,” Biophys. J. 88, 2145–2153 (2005).
[Crossref]

J. B. Edel, M. Wu, B. Baird, and H. G. Craighead, “High spatial resolution observation of single molecule dynamics in living cell membranes,” Biophys. J. 88, L43–L45 (2005).
[Crossref] [PubMed]

Craighead, H.G.

M.J. Levene, J. Korlach, S.W. Turner, M. Foquet, H.G. Craighead, and W.W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[Crossref] [PubMed]

Dintinger, J.

E. Popov, M. Nevière, J. Wenger, P.-F. Lenne, H. Rigneault, P. Chaumet, N. Bonod, J. Dintinger, and T.W. Ebbesen, “Field enhancement in single subwavelength apertures,” J. Opt. Soc. Am. A 23, 2342–2348 (2006).
[Crossref]

J Wenger, H. Rigneault, J. Dintinger, D Marguet, and P. F. Lenne, “Single-fluorophore diffusion in a lipid membrane over a subwavelength aperture,” J. Biol. Phys. 32, SN1–SN4 (2006).
[Crossref] [PubMed]

J. Wenger, P.-F. Lenne, E. Popov, H. Rigneault, J. Dintinger, and T.W. Ebbesen, “Single molecule fluorescence in rectangular nano-apertures,” Opt. Express 13, 7035–7044 (2005).
[Crossref] [PubMed]

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[Crossref] [PubMed]

Ebbesen, T. W.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[Crossref] [PubMed]

Ebbesen, T.W.

Edel, J. B.

J. B. Edel, M. Wu, B. Baird, and H. G. Craighead, “High spatial resolution observation of single molecule dynamics in living cell membranes,” Biophys. J. 88, L43–L45 (2005).
[Crossref] [PubMed]

Eggeling, C.

M. Leutenegger, H. Blom, J. Widengren, C. Eggeling, M. Gösch, R. A. Leitgeb, and T. Lasser, “Dual-color total internal reflection fluorescence cross-correlation spectroscopy,” J. Biomed. Opt. 11, 040502 (2006).
[Crossref] [PubMed]

Eigen, M.

T. Winkler, U. Kettling, A. Koltermann, and M. Eigen, “Confocal fluorescence coincidence analysis: an approach to ultra high throughput screening,” Proc. Natl. Acad. Sci. USA 96, 1375–1378 (1999).
[Crossref] [PubMed]

A. Koltermann, U. Kettling, J. Bieschke, T. Winkler, and M. Eigen, “Rapid assay processing by integration of dualcolor fluorescence cross-correlation spectroscopy: high throughput screening for enzyme activity,” Proc. Natl. Acad. Sci. USA 95, 1421–1426 (1998).
[Crossref] [PubMed]

U. Kettling, A. Koltermann, P. Schwille, and M. Eigen, “Real-time enzyme kinetics monitored by dual-color fluorescence cross-correlation spectroscopy,” Proc. Natl. Acad. Sci. USA 95, 1416–1420 (1998).
[Crossref] [PubMed]

M. Eigen and R. Rigler, “Sorting single molecules: Application to diagnostics and evolutionary biotechnology,” Proc. Natl. Acad. Sci. USA 91, 5740–5747 (1994).
[Crossref] [PubMed]

Foquet, M.

K. T. Samiee, M. Foquet, L. Guo, E. C. Cox, and H. G. Craighead, “Lambda repressor oligomerization kinetics at high concentrations using fluorescence correlation spectroscopy in zero-mode waveguides,” Biophys. J. 88, 2145–2153 (2005).
[Crossref]

M.J. Levene, J. Korlach, S.W. Turner, M. Foquet, H.G. Craighead, and W.W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[Crossref] [PubMed]

Gösch, M.

Gosse, J. A.

D. R. Larson, J. A. Gosse, D. A. Holowka, B. A. Baird, and W.W Webb, “Temporally resolved interactions between antigen-stimulated IgE receptors and Lyn kinase on living cells,” J. Cell Biol. 171, 527–536 (2005).
[Crossref] [PubMed]

Guo, L.

K. T. Samiee, M. Foquet, L. Guo, E. C. Cox, and H. G. Craighead, “Lambda repressor oligomerization kinetics at high concentrations using fluorescence correlation spectroscopy in zero-mode waveguides,” Biophys. J. 88, 2145–2153 (2005).
[Crossref]

Heinze, K. G.

M. Burkhardt, K. G. Heinze, and P. Schwille, “Four-color fluorescence correlation spectroscopy realized in a gratingbased detection platform,” Opt. Lett. 30, 2266–2268 (2005).
[Crossref] [PubMed]

S. A. Kim, K. G. Heinze, M. N. Waxham, and P. Schwille, “Intracellular calmodulin availability accessed with two-photon crosscorrelation,” Proc. Natl. Acad. Sci. USA. 101, 105–110 (2004).
[Crossref]

Heinze, K.G.

K.G. Heinze, A. Koltermann, and P. Schwille, “Simultaneous two-photon excitation of distinct labels for dual-color fluorescence cross-correlation analysis,” Proc. Natl. Acad. Sci. USA 97, 10377–10382 (2000).
[Crossref] [PubMed]

Hoffmann, P.

Holowka, D. A.

D. R. Larson, J. A. Gosse, D. A. Holowka, B. A. Baird, and W.W Webb, “Temporally resolved interactions between antigen-stimulated IgE receptors and Lyn kinase on living cells,” J. Cell Biol. 171, 527–536 (2005).
[Crossref] [PubMed]

Hwang, L. C.

Karasawa, S.

T. Kogure, S. Karasawa, T. Araki, K. Saito, M. Kinjo, and A. Miyawaki, “A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy,” Nat. Biotechnol. 24, 577–581 (2006).
[Crossref] [PubMed]

Kettling, U.

T. Winkler, U. Kettling, A. Koltermann, and M. Eigen, “Confocal fluorescence coincidence analysis: an approach to ultra high throughput screening,” Proc. Natl. Acad. Sci. USA 96, 1375–1378 (1999).
[Crossref] [PubMed]

A. Koltermann, U. Kettling, J. Bieschke, T. Winkler, and M. Eigen, “Rapid assay processing by integration of dualcolor fluorescence cross-correlation spectroscopy: high throughput screening for enzyme activity,” Proc. Natl. Acad. Sci. USA 95, 1421–1426 (1998).
[Crossref] [PubMed]

U. Kettling, A. Koltermann, P. Schwille, and M. Eigen, “Real-time enzyme kinetics monitored by dual-color fluorescence cross-correlation spectroscopy,” Proc. Natl. Acad. Sci. USA 95, 1416–1420 (1998).
[Crossref] [PubMed]

Kim, S. A.

K. Bacia, S. A. Kim, and P. Schwille, “Fluorescence cross-correlation spectroscopy in living cells,” Nat. Methods 3, 83–89 (2006).
[Crossref] [PubMed]

S. A. Kim, K. G. Heinze, M. N. Waxham, and P. Schwille, “Intracellular calmodulin availability accessed with two-photon crosscorrelation,” Proc. Natl. Acad. Sci. USA. 101, 105–110 (2004).
[Crossref]

Kinjo, M.

T. Kogure, S. Karasawa, T. Araki, K. Saito, M. Kinjo, and A. Miyawaki, “A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy,” Nat. Biotechnol. 24, 577–581 (2006).
[Crossref] [PubMed]

Kogure, T.

T. Kogure, S. Karasawa, T. Araki, K. Saito, M. Kinjo, and A. Miyawaki, “A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy,” Nat. Biotechnol. 24, 577–581 (2006).
[Crossref] [PubMed]

Koltermann, A.

K.G. Heinze, A. Koltermann, and P. Schwille, “Simultaneous two-photon excitation of distinct labels for dual-color fluorescence cross-correlation analysis,” Proc. Natl. Acad. Sci. USA 97, 10377–10382 (2000).
[Crossref] [PubMed]

T. Winkler, U. Kettling, A. Koltermann, and M. Eigen, “Confocal fluorescence coincidence analysis: an approach to ultra high throughput screening,” Proc. Natl. Acad. Sci. USA 96, 1375–1378 (1999).
[Crossref] [PubMed]

A. Koltermann, U. Kettling, J. Bieschke, T. Winkler, and M. Eigen, “Rapid assay processing by integration of dualcolor fluorescence cross-correlation spectroscopy: high throughput screening for enzyme activity,” Proc. Natl. Acad. Sci. USA 95, 1421–1426 (1998).
[Crossref] [PubMed]

U. Kettling, A. Koltermann, P. Schwille, and M. Eigen, “Real-time enzyme kinetics monitored by dual-color fluorescence cross-correlation spectroscopy,” Proc. Natl. Acad. Sci. USA 95, 1416–1420 (1998).
[Crossref] [PubMed]

Korlach, J.

M.J. Levene, J. Korlach, S.W. Turner, M. Foquet, H.G. Craighead, and W.W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[Crossref] [PubMed]

Langowski, J.

N. Baudendistel, G. Müller, W. Waldeck, P. Angel, and J. Langowski, “Two-Hybrid Fluorescence Cross-Correlation Spectroscopy Detects ProteinProtein Interactions In Vivo,” Chem. Phys. Chem. 6, 984–990 (2005).
[Crossref] [PubMed]

T Weidemann, M Wachsmuth, M. Tewes, K. Rippe, and J. Langowski, “Analysis of Ligand Binding by Two-Colour Fluorescence Cross-Correlation Spectroscopy,” Single Mol. 3, 49–61 (2002).
[Crossref]

Larson, D. R.

D. R. Larson, J. A. Gosse, D. A. Holowka, B. A. Baird, and W.W Webb, “Temporally resolved interactions between antigen-stimulated IgE receptors and Lyn kinase on living cells,” J. Cell Biol. 171, 527–536 (2005).
[Crossref] [PubMed]

Lasser, T.

Leitgeb, R. A.

M. Leutenegger, H. Blom, J. Widengren, C. Eggeling, M. Gösch, R. A. Leitgeb, and T. Lasser, “Dual-color total internal reflection fluorescence cross-correlation spectroscopy,” J. Biomed. Opt. 11, 040502 (2006).
[Crossref] [PubMed]

Lenne, P. F.

J Wenger, H. Rigneault, J. Dintinger, D Marguet, and P. F. Lenne, “Single-fluorophore diffusion in a lipid membrane over a subwavelength aperture,” J. Biol. Phys. 32, SN1–SN4 (2006).
[Crossref] [PubMed]

Lenne, P.-F.

Leutenegger, M.

Levene, M.J.

M.J. Levene, J. Korlach, S.W. Turner, M. Foquet, H.G. Craighead, and W.W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[Crossref] [PubMed]

Marguet, D

J Wenger, H. Rigneault, J. Dintinger, D Marguet, and P. F. Lenne, “Single-fluorophore diffusion in a lipid membrane over a subwavelength aperture,” J. Biol. Phys. 32, SN1–SN4 (2006).
[Crossref] [PubMed]

Martin, O. J. F.

Meier, W.

Meyer-Almes, F.-J.

P. Schwille, F.-J. Meyer-Almes, and R. Rigler, “Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution,” Biophys. J. 72, 1878–1886 (1997).
[Crossref] [PubMed]

Miyawaki, A.

T. Kogure, S. Karasawa, T. Araki, K. Saito, M. Kinjo, and A. Miyawaki, “A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy,” Nat. Biotechnol. 24, 577–581 (2006).
[Crossref] [PubMed]

Moran-Mirabal, J. M.

K. T. Samiee, J. M. Moran-Mirabal, Y. K. Cheung, and H. G. Craighead, “Zero Mode Waveguides for Single-Molecule Spectroscopy on Lipid Membranes,” Biophys. J. 90, 3288–3299 (2006).
[Crossref] [PubMed]

Müller, G.

N. Baudendistel, G. Müller, W. Waldeck, P. Angel, and J. Langowski, “Two-Hybrid Fluorescence Cross-Correlation Spectroscopy Detects ProteinProtein Interactions In Vivo,” Chem. Phys. Chem. 6, 984–990 (2005).
[Crossref] [PubMed]

Nevière, M.

Perentes, A.

Popov, E.

Rigler, P.

Rigler, R.

P. Schwille, F.-J. Meyer-Almes, and R. Rigler, “Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution,” Biophys. J. 72, 1878–1886 (1997).
[Crossref] [PubMed]

M. Eigen and R. Rigler, “Sorting single molecules: Application to diagnostics and evolutionary biotechnology,” Proc. Natl. Acad. Sci. USA 91, 5740–5747 (1994).
[Crossref] [PubMed]

Rigneault, H.

J Wenger, H. Rigneault, J. Dintinger, D Marguet, and P. F. Lenne, “Single-fluorophore diffusion in a lipid membrane over a subwavelength aperture,” J. Biol. Phys. 32, SN1–SN4 (2006).
[Crossref] [PubMed]

E. Popov, M. Nevière, J. Wenger, P.-F. Lenne, H. Rigneault, P. Chaumet, N. Bonod, J. Dintinger, and T.W. Ebbesen, “Field enhancement in single subwavelength apertures,” J. Opt. Soc. Am. A 23, 2342–2348 (2006).
[Crossref]

J. Wenger, P.-F. Lenne, E. Popov, H. Rigneault, J. Dintinger, and T.W. Ebbesen, “Single molecule fluorescence in rectangular nano-apertures,” Opt. Express 13, 7035–7044 (2005).
[Crossref] [PubMed]

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[Crossref] [PubMed]

Rippe, K.

T Weidemann, M Wachsmuth, M. Tewes, K. Rippe, and J. Langowski, “Analysis of Ligand Binding by Two-Colour Fluorescence Cross-Correlation Spectroscopy,” Single Mol. 3, 49–61 (2002).
[Crossref]

Saito, K.

T. Kogure, S. Karasawa, T. Araki, K. Saito, M. Kinjo, and A. Miyawaki, “A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy,” Nat. Biotechnol. 24, 577–581 (2006).
[Crossref] [PubMed]

Samiee, K. T.

K. T. Samiee, J. M. Moran-Mirabal, Y. K. Cheung, and H. G. Craighead, “Zero Mode Waveguides for Single-Molecule Spectroscopy on Lipid Membranes,” Biophys. J. 90, 3288–3299 (2006).
[Crossref] [PubMed]

K. T. Samiee, M. Foquet, L. Guo, E. C. Cox, and H. G. Craighead, “Lambda repressor oligomerization kinetics at high concentrations using fluorescence correlation spectroscopy in zero-mode waveguides,” Biophys. J. 88, 2145–2153 (2005).
[Crossref]

Schwille, P.

K. Bacia, S. A. Kim, and P. Schwille, “Fluorescence cross-correlation spectroscopy in living cells,” Nat. Methods 3, 83–89 (2006).
[Crossref] [PubMed]

M. Burkhardt, K. G. Heinze, and P. Schwille, “Four-color fluorescence correlation spectroscopy realized in a gratingbased detection platform,” Opt. Lett. 30, 2266–2268 (2005).
[Crossref] [PubMed]

S. A. Kim, K. G. Heinze, M. N. Waxham, and P. Schwille, “Intracellular calmodulin availability accessed with two-photon crosscorrelation,” Proc. Natl. Acad. Sci. USA. 101, 105–110 (2004).
[Crossref]

K.G. Heinze, A. Koltermann, and P. Schwille, “Simultaneous two-photon excitation of distinct labels for dual-color fluorescence cross-correlation analysis,” Proc. Natl. Acad. Sci. USA 97, 10377–10382 (2000).
[Crossref] [PubMed]

U. Kettling, A. Koltermann, P. Schwille, and M. Eigen, “Real-time enzyme kinetics monitored by dual-color fluorescence cross-correlation spectroscopy,” Proc. Natl. Acad. Sci. USA 95, 1416–1420 (1998).
[Crossref] [PubMed]

P. Schwille, F.-J. Meyer-Almes, and R. Rigler, “Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution,” Biophys. J. 72, 1878–1886 (1997).
[Crossref] [PubMed]

Tewes, M.

T Weidemann, M Wachsmuth, M. Tewes, K. Rippe, and J. Langowski, “Analysis of Ligand Binding by Two-Colour Fluorescence Cross-Correlation Spectroscopy,” Single Mol. 3, 49–61 (2002).
[Crossref]

Turner, S.W.

M.J. Levene, J. Korlach, S.W. Turner, M. Foquet, H.G. Craighead, and W.W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[Crossref] [PubMed]

Wachsmuth, M

T Weidemann, M Wachsmuth, M. Tewes, K. Rippe, and J. Langowski, “Analysis of Ligand Binding by Two-Colour Fluorescence Cross-Correlation Spectroscopy,” Single Mol. 3, 49–61 (2002).
[Crossref]

Waldeck, W.

N. Baudendistel, G. Müller, W. Waldeck, P. Angel, and J. Langowski, “Two-Hybrid Fluorescence Cross-Correlation Spectroscopy Detects ProteinProtein Interactions In Vivo,” Chem. Phys. Chem. 6, 984–990 (2005).
[Crossref] [PubMed]

Waxham, M. N.

S. A. Kim, K. G. Heinze, M. N. Waxham, and P. Schwille, “Intracellular calmodulin availability accessed with two-photon crosscorrelation,” Proc. Natl. Acad. Sci. USA. 101, 105–110 (2004).
[Crossref]

Webb, W.W

D. R. Larson, J. A. Gosse, D. A. Holowka, B. A. Baird, and W.W Webb, “Temporally resolved interactions between antigen-stimulated IgE receptors and Lyn kinase on living cells,” J. Cell Biol. 171, 527–536 (2005).
[Crossref] [PubMed]

Webb, W.W.

M.J. Levene, J. Korlach, S.W. Turner, M. Foquet, H.G. Craighead, and W.W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[Crossref] [PubMed]

Weidemann, T

T Weidemann, M Wachsmuth, M. Tewes, K. Rippe, and J. Langowski, “Analysis of Ligand Binding by Two-Colour Fluorescence Cross-Correlation Spectroscopy,” Single Mol. 3, 49–61 (2002).
[Crossref]

Wenger, J

J Wenger, H. Rigneault, J. Dintinger, D Marguet, and P. F. Lenne, “Single-fluorophore diffusion in a lipid membrane over a subwavelength aperture,” J. Biol. Phys. 32, SN1–SN4 (2006).
[Crossref] [PubMed]

Wenger, J.

Widengren, J.

M. Leutenegger, H. Blom, J. Widengren, C. Eggeling, M. Gösch, R. A. Leitgeb, and T. Lasser, “Dual-color total internal reflection fluorescence cross-correlation spectroscopy,” J. Biomed. Opt. 11, 040502 (2006).
[Crossref] [PubMed]

Winkler, T.

T. Winkler, U. Kettling, A. Koltermann, and M. Eigen, “Confocal fluorescence coincidence analysis: an approach to ultra high throughput screening,” Proc. Natl. Acad. Sci. USA 96, 1375–1378 (1999).
[Crossref] [PubMed]

A. Koltermann, U. Kettling, J. Bieschke, T. Winkler, and M. Eigen, “Rapid assay processing by integration of dualcolor fluorescence cross-correlation spectroscopy: high throughput screening for enzyme activity,” Proc. Natl. Acad. Sci. USA 95, 1421–1426 (1998).
[Crossref] [PubMed]

Wohland, T.

Wu, M.

J. B. Edel, M. Wu, B. Baird, and H. G. Craighead, “High spatial resolution observation of single molecule dynamics in living cell membranes,” Biophys. J. 88, L43–L45 (2005).
[Crossref] [PubMed]

Biophys. J. (4)

P. Schwille, F.-J. Meyer-Almes, and R. Rigler, “Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution,” Biophys. J. 72, 1878–1886 (1997).
[Crossref] [PubMed]

K. T. Samiee, M. Foquet, L. Guo, E. C. Cox, and H. G. Craighead, “Lambda repressor oligomerization kinetics at high concentrations using fluorescence correlation spectroscopy in zero-mode waveguides,” Biophys. J. 88, 2145–2153 (2005).
[Crossref]

J. B. Edel, M. Wu, B. Baird, and H. G. Craighead, “High spatial resolution observation of single molecule dynamics in living cell membranes,” Biophys. J. 88, L43–L45 (2005).
[Crossref] [PubMed]

K. T. Samiee, J. M. Moran-Mirabal, Y. K. Cheung, and H. G. Craighead, “Zero Mode Waveguides for Single-Molecule Spectroscopy on Lipid Membranes,” Biophys. J. 90, 3288–3299 (2006).
[Crossref] [PubMed]

Chem. Phys. Chem. (1)

N. Baudendistel, G. Müller, W. Waldeck, P. Angel, and J. Langowski, “Two-Hybrid Fluorescence Cross-Correlation Spectroscopy Detects ProteinProtein Interactions In Vivo,” Chem. Phys. Chem. 6, 984–990 (2005).
[Crossref] [PubMed]

J. Biol. Phys. (1)

J Wenger, H. Rigneault, J. Dintinger, D Marguet, and P. F. Lenne, “Single-fluorophore diffusion in a lipid membrane over a subwavelength aperture,” J. Biol. Phys. 32, SN1–SN4 (2006).
[Crossref] [PubMed]

J. Biomed. Opt. (1)

M. Leutenegger, H. Blom, J. Widengren, C. Eggeling, M. Gösch, R. A. Leitgeb, and T. Lasser, “Dual-color total internal reflection fluorescence cross-correlation spectroscopy,” J. Biomed. Opt. 11, 040502 (2006).
[Crossref] [PubMed]

J. Cell Biol. (1)

D. R. Larson, J. A. Gosse, D. A. Holowka, B. A. Baird, and W.W Webb, “Temporally resolved interactions between antigen-stimulated IgE receptors and Lyn kinase on living cells,” J. Cell Biol. 171, 527–536 (2005).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

Nat. Biotechnol. (1)

T. Kogure, S. Karasawa, T. Araki, K. Saito, M. Kinjo, and A. Miyawaki, “A fluorescent variant of a protein from the stony coral Montipora facilitates dual-color single-laser fluorescence cross-correlation spectroscopy,” Nat. Biotechnol. 24, 577–581 (2006).
[Crossref] [PubMed]

Nat. Methods (1)

K. Bacia, S. A. Kim, and P. Schwille, “Fluorescence cross-correlation spectroscopy in living cells,” Nat. Methods 3, 83–89 (2006).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. USA (5)

U. Kettling, A. Koltermann, P. Schwille, and M. Eigen, “Real-time enzyme kinetics monitored by dual-color fluorescence cross-correlation spectroscopy,” Proc. Natl. Acad. Sci. USA 95, 1416–1420 (1998).
[Crossref] [PubMed]

A. Koltermann, U. Kettling, J. Bieschke, T. Winkler, and M. Eigen, “Rapid assay processing by integration of dualcolor fluorescence cross-correlation spectroscopy: high throughput screening for enzyme activity,” Proc. Natl. Acad. Sci. USA 95, 1421–1426 (1998).
[Crossref] [PubMed]

T. Winkler, U. Kettling, A. Koltermann, and M. Eigen, “Confocal fluorescence coincidence analysis: an approach to ultra high throughput screening,” Proc. Natl. Acad. Sci. USA 96, 1375–1378 (1999).
[Crossref] [PubMed]

K.G. Heinze, A. Koltermann, and P. Schwille, “Simultaneous two-photon excitation of distinct labels for dual-color fluorescence cross-correlation analysis,” Proc. Natl. Acad. Sci. USA 97, 10377–10382 (2000).
[Crossref] [PubMed]

M. Eigen and R. Rigler, “Sorting single molecules: Application to diagnostics and evolutionary biotechnology,” Proc. Natl. Acad. Sci. USA 91, 5740–5747 (1994).
[Crossref] [PubMed]

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

S. A. Kim, K. G. Heinze, M. N. Waxham, and P. Schwille, “Intracellular calmodulin availability accessed with two-photon crosscorrelation,” Proc. Natl. Acad. Sci. USA. 101, 105–110 (2004).
[Crossref]

Science (1)

M.J. Levene, J. Korlach, S.W. Turner, M. Foquet, H.G. Craighead, and W.W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[Crossref] [PubMed]

Single Mol. (1)

T Weidemann, M Wachsmuth, M. Tewes, K. Rippe, and J. Langowski, “Analysis of Ligand Binding by Two-Colour Fluorescence Cross-Correlation Spectroscopy,” Single Mol. 3, 49–61 (2002).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Scheme of the experimental setup for dual-color FCCS with a nanoaperture (APD: avalanche photodiode, DPSSL : diode-pumped solid state laser).

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Fig. 2.
Fig. 2. Autocorrelation functions of Alexa-647 dye excited at 633 nm (red) and 488 nm (blue) in a 340 nm diameter nanoaperture. Thin solid lines correspond to raw experimental data, while thick lines refer to numerical fits assuming Eq.(5). The dashed curve is the 488 nm autocorrelation after correction for the background noise. Amplitudes and characteristic times of the 633 nm and corrected 488 nm autocorrelations are almost equal, showing that the effective observation volumes for each laser line accurately overlap inside the nanoaperture, even without minute alignment.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. Auto- and cross-correlation curves for labeled DNA oligonucleotides at 2 µM concentration in a 340 nm diameter nanoaperture. Thin lines correspond to raw experimental data, thick lines are numerical fits.

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Fig. 4.
Fig. 4. Cross-correlation curves during an endonucleotic cleavage reaction carried on a droplet containing 20 µL DNA substrate at 2 µM, 2 µL EcoRI of activity 10 u/µL and 2 µL Tris-HCl/MgCl2/NaCl buffer (10×) in a 340 nm diameter nanoaperture. Thin lines correspond to raw experimental data, thick lines are numerical fits.

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Fig. 5.
Fig. 5. Typical cross-correlation curves with (green) and without (gray) double-labeled molecules in a 340 nm nanoaperture. For rapid assay processing, the integration time was set to 1 s (dotted line). The cross-correlation obtained after 100 s integration time is displayed in thick line for reference.

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Fig. 6.
Fig. 6. (Left) Histograms of plots showing the distribution of cross-correlation g (2)(0) with (green) and without (gray) double-labeled molecules in a 340 nm nanoaperture. These histograms are obtained from a set of 250 FCCS experiments of 1 s integration time, as displayed on Fig. 5. Line curves represent fitted Gaussian distributions. (Right) Error and detection probability versus the threshold value T.

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Equations (8)

Equations on this page are rendered with MathJax. Learn more.

g ij ( 2 ) ( τ ) = 〈 F i ( t ) F j ( t + τ ) 〉 〈 F i ( t ) 〉 〈 F j ( t ) 〉 ,
g GG ( 2 ) ( τ ) = 1 + C G Diff G ( τ ) + C GR Diff GR ( τ ) V eff C G , tot 2 ,
g RR ( 2 ) ( τ ) = 1 + C R Diff R ( τ ) + C GR Diff GR ( τ ) V eff C R , tot 2 ,
g GR ( 2 ) ( τ ) = 1 + C GR Diff GR ( τ ) V eff C G , tot C R , tot ,
Diff i = G , R , GR ( τ ) = ( 1 + n T , i exp ( − τ τ T , i ) ) ( 1 − 〈 B i 〉 〈 I i 〉 ) 2 1 ( 1 + τ τ d , i ) 1 + s i 2 τ τ d , i .
C GR = g GR ( 2 ) ( 0 ) − 1 1 + n T , GR 1 + n T , R g RR ( 2 ) ( 0 ) − 1 C G , tot = g GR ( 2 ) ( 0 ) − 1 1 + n T , GR 1 + n T , G g GG ( 2 ) ( 0 ) − 1 C R , tot .
P r det = 1 𝒩 + ∫ T + ∞ H + ( g ) dg
P r err = 1 𝒩 − ∫ T + ∞ H − ( g ) dg , ,

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