Abstract

Quantitative fluorescence correlation spectroscopy (FCS) and fluorescence photobleaching recovery (FPR) measurements in bulk solution require a well characterized confocal laser microscope optical system. The introduction of a characteristic function, the collection efficiency function (CEF), provides a quantitative theoretical analysis of this system, which yields an interpretation of the FCS and FPR measurements in three dimensions. We demonstrate that when the proper field diaphragm is introduced, the 3-D FCS measurements can be mimicked by a 2-D theory with only minor error. The FPR characteristic recovery time for diffusion is expected to be slightly longer than the corresponding time measured by FCS in the same conditions. This is because the profile of the laser beam used for photobleaching is not affected by the field diaphragm. The CEF is also important for quantitative analysis of standard scanning confocal microscopy when it is carried out using a finite detection pinhole.

© 1991 Optical Society of America

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  1. E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery,” Ann. Rev. Phys. Chem. 36, 379 (1985).
    [CrossRef]
  2. K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Fluorescence Photobleaching in Cell Biology,” Nature London 295, 283 (1982).
    [CrossRef] [PubMed]
  3. K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Application of Fluorescence Photobleaching Techniques to Problems in Cell Biology,” Fed. Proc. Fed. Am. Soc. Exp. Biol. 42, 72 (1983).
  4. R. D. Icenogle, E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery of Multiple Binding Reactions. I. Theory and FCS Measurements,” Biopolymers 22, 1919 (1983).
    [CrossRef] [PubMed]
  5. R. D. Icenogle, E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery of Multiple Binding Reactions. II. FPR and FCS Measurements at Low and High DNA Concentrations,” Biopolymers 22, 1949 (1983).
    [CrossRef] [PubMed]
  6. J. Schlessinger, E. L. Elson, “Fluorescence Methods for Studying Membrane Dynamics, Methods of Experimental Physics,” 20, 197 (1982).
  7. D. Magde, E. L. Elson, W. W. Webb, “Fluorescence Correlation Spectroscopy. II. An Experimental Realization,” Biopolymers 13, 29 (1974).
    [CrossRef] [PubMed]
  8. N. O. Petersen, E. L. Elson, “Measurements of Diffusion and Chemical Kinetics by Fluorescence Photobleaching Recovery and Fluorescence Correlation Spectroscopy,” Methods Enzymol. 130, 454 (1986).
    [CrossRef] [PubMed]
  9. H. Qian, E. L. Elson, “3-D Fluorescence Correlation Spectroscopy in Bulk Solution,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 352 (1988).
  10. D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, W. W. Webb, “Dynamics of Fluorescence Marker Concentration as a Probe of Mobility,” Biophys. J. 16, 1315 (1976).
    [CrossRef] [PubMed]
  11. G. J. Brakenhoff, H. T. M. van der Voort, Imaging, “Representation and Analysis of 3-Dimensional Biological Structures by Confocal Microscopy,” in Proceedings, Forty-Sixth Annual Meeting of EMSA (1988), p. 996.
  12. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).
  13. M. H. Weissman, H. Schindler, G. Feher, “Determination of Molecular Weights by Fluctuation Spectroscopy: Application to DNA,” Proc. Natl. Acad. Sci. USA 73, 2776 (1976).
    [CrossRef] [PubMed]
  14. N. O. Petersen, “Scanning Fluorescence Correlation Spectroscopy. I. Theory and Simulation of Aggregation Measurements,” Biophys. J. 49, 809 (1986).
    [CrossRef] [PubMed]
  15. N. O. Petersen, D. C. Johnson, M. J. Schlesinger, “Scanning Fluorescence Correlation Spectroscopy. II. Application to Virus Glycoprotein Aggregation,” Biophys. J. 49, 817 (1986).
    [CrossRef] [PubMed]
  16. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550–1567 (1966).
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  17. M. B. Schneider, W. W. Webb, “Measurement of Submicron Laser Beam Radii,” Appl. Opt. 20, 1382–1388 (1981).
    [CrossRef] [PubMed]
  18. The Gaussian or Airy profiles are relative, since any instrument has only a finite aperture. Therefore, there is always secondary diffraction. The approximation of a Gaussian beam is appropriate when secondary diffraction can be neglected.
  19. E. L. Elson, D. Magde, “Fluorescence Correlation Spectroscopy. I. Conceptual Basis and Theory,” Biopolymers 13, 1 (1974).
    [CrossRef]
  20. D. Axelrod, D. E. Koppel, J. Schlessinger, E. L. Elson, W. W. Webb, “Mobility Measurement by Analysis of Fluorescence Photobleaching Recovery Kinetics,” Biophys. J. 16, 1055 (1976).
    [CrossRef] [PubMed]
  21. T. Wilson, A. R. Carlini, “Three-Dimensional Imaging in Confocal Imaging System with Finite Sized Detectors,” J. Microsc. 149, 51 (1988).
    [CrossRef]
  22. H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 217, 91 (1955).
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  24. D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191 (1984).
    [CrossRef]
  25. H. Qian, E. L. Elson, “Characterization of Confocal Laser-Based Microscope by Digital Image Analysis: an Optical Sectioning Microscopy Approach,” in Optical Microscopy for Biology, B. Herman, K. Jacobson, Eds. (Alan R. Liss, New York, 1990).
  26. More accurately, image formation should take the lens maker’s law into account: r° = rf/(d − z), z° = zd°/(d − z). As discussed later, this will introduce asymmetry and nonlinearity into the system.
  27. Mathematically this corresponds to replacing a very narrow function by a Dirac delta-function. This introduces very little error if the delta-function multiplies a slowly varying function and the product is integrated.
  28. A. G. Palmer, N. L. Thompson, “Optical Spatial Intensity Profiles for High Order Autocorrelation in Fluorescence Spectroscopy,” Appl. Opt. 28, 1214–1220 (1989).
    [CrossRef]
  29. E. L. Elson, W. W. Webb, “Concentration Correlation Spectroscopy: A New Biophysical Probe Based on Occupation Number Fluctuations,” Ann. Rev. Biophys. Bioeng. 4, 311 (1975).
    [CrossRef]
  30. E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery: Measurement of Transport and Chemical Kinetics,” in Spectroscopic Membrane Probes, L. M. Loew, Ed. (CRC Press, Cleveland, 1988).
  31. Only a minor error results if we neglect the contribution from region z > zɛ. Then an analytical form will beωapp2=2ω02[1+(zε/z1)/[1+(zε/z1)2]/arctan(zε/z1)]−1.
  32. H. Qian, “Biophysical Characterization of Biopolymer Solutions and Gels by Fluorescence Fluctuation Studies,” Ph.D. Thesis, Washington U., St. Louis (1989).
  33. G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, N. Nanning, “Three-Dimensional Imaging by Confocal Fluorescence Microscopy,” Ann. N.Y. Acad. Sci. 483, 405 (1986).
    [CrossRef]
  34. W. A. Carrington, F. S. Fay, K. E. Fogarty, L. Lifshitz, “Analysis of 3D Molecular Distribution With Digital Image Microscopy,” in Proceedings, Forty-Sixth Annual Meeting of EMSA (1988), p. 40.
  35. P. A. Stokseth, “Properties of a Defocused Optical System,” J. Opt. Soc. Am. 59, 1314 (1969).
    [CrossRef]
  36. F. Lanni, L. D. Taylor, B. R. Ware, “Fluorescence Photobleaching Recovery in Solutions of Labeled Actin,” Biophys. J. 35, 351 (1981).
    [CrossRef] [PubMed]

1989 (1)

1988 (2)

H. Qian, E. L. Elson, “3-D Fluorescence Correlation Spectroscopy in Bulk Solution,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 352 (1988).

T. Wilson, A. R. Carlini, “Three-Dimensional Imaging in Confocal Imaging System with Finite Sized Detectors,” J. Microsc. 149, 51 (1988).
[CrossRef]

1986 (4)

N. O. Petersen, E. L. Elson, “Measurements of Diffusion and Chemical Kinetics by Fluorescence Photobleaching Recovery and Fluorescence Correlation Spectroscopy,” Methods Enzymol. 130, 454 (1986).
[CrossRef] [PubMed]

N. O. Petersen, “Scanning Fluorescence Correlation Spectroscopy. I. Theory and Simulation of Aggregation Measurements,” Biophys. J. 49, 809 (1986).
[CrossRef] [PubMed]

N. O. Petersen, D. C. Johnson, M. J. Schlesinger, “Scanning Fluorescence Correlation Spectroscopy. II. Application to Virus Glycoprotein Aggregation,” Biophys. J. 49, 817 (1986).
[CrossRef] [PubMed]

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, N. Nanning, “Three-Dimensional Imaging by Confocal Fluorescence Microscopy,” Ann. N.Y. Acad. Sci. 483, 405 (1986).
[CrossRef]

1985 (1)

E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery,” Ann. Rev. Phys. Chem. 36, 379 (1985).
[CrossRef]

1984 (1)

D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191 (1984).
[CrossRef]

1983 (3)

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Application of Fluorescence Photobleaching Techniques to Problems in Cell Biology,” Fed. Proc. Fed. Am. Soc. Exp. Biol. 42, 72 (1983).

R. D. Icenogle, E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery of Multiple Binding Reactions. I. Theory and FCS Measurements,” Biopolymers 22, 1919 (1983).
[CrossRef] [PubMed]

R. D. Icenogle, E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery of Multiple Binding Reactions. II. FPR and FCS Measurements at Low and High DNA Concentrations,” Biopolymers 22, 1949 (1983).
[CrossRef] [PubMed]

1982 (2)

J. Schlessinger, E. L. Elson, “Fluorescence Methods for Studying Membrane Dynamics, Methods of Experimental Physics,” 20, 197 (1982).

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Fluorescence Photobleaching in Cell Biology,” Nature London 295, 283 (1982).
[CrossRef] [PubMed]

1981 (2)

M. B. Schneider, W. W. Webb, “Measurement of Submicron Laser Beam Radii,” Appl. Opt. 20, 1382–1388 (1981).
[CrossRef] [PubMed]

F. Lanni, L. D. Taylor, B. R. Ware, “Fluorescence Photobleaching Recovery in Solutions of Labeled Actin,” Biophys. J. 35, 351 (1981).
[CrossRef] [PubMed]

1976 (3)

D. Axelrod, D. E. Koppel, J. Schlessinger, E. L. Elson, W. W. Webb, “Mobility Measurement by Analysis of Fluorescence Photobleaching Recovery Kinetics,” Biophys. J. 16, 1055 (1976).
[CrossRef] [PubMed]

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, W. W. Webb, “Dynamics of Fluorescence Marker Concentration as a Probe of Mobility,” Biophys. J. 16, 1315 (1976).
[CrossRef] [PubMed]

M. H. Weissman, H. Schindler, G. Feher, “Determination of Molecular Weights by Fluctuation Spectroscopy: Application to DNA,” Proc. Natl. Acad. Sci. USA 73, 2776 (1976).
[CrossRef] [PubMed]

1975 (1)

E. L. Elson, W. W. Webb, “Concentration Correlation Spectroscopy: A New Biophysical Probe Based on Occupation Number Fluctuations,” Ann. Rev. Biophys. Bioeng. 4, 311 (1975).
[CrossRef]

1974 (2)

D. Magde, E. L. Elson, W. W. Webb, “Fluorescence Correlation Spectroscopy. II. An Experimental Realization,” Biopolymers 13, 29 (1974).
[CrossRef] [PubMed]

E. L. Elson, D. Magde, “Fluorescence Correlation Spectroscopy. I. Conceptual Basis and Theory,” Biopolymers 13, 1 (1974).
[CrossRef]

1969 (1)

1966 (1)

1955 (1)

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 217, 91 (1955).

Agard, D. A.

D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191 (1984).
[CrossRef]

Axelrod, D.

D. Axelrod, D. E. Koppel, J. Schlessinger, E. L. Elson, W. W. Webb, “Mobility Measurement by Analysis of Fluorescence Photobleaching Recovery Kinetics,” Biophys. J. 16, 1055 (1976).
[CrossRef] [PubMed]

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, W. W. Webb, “Dynamics of Fluorescence Marker Concentration as a Probe of Mobility,” Biophys. J. 16, 1315 (1976).
[CrossRef] [PubMed]

Brakenhoff, G. J.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, N. Nanning, “Three-Dimensional Imaging by Confocal Fluorescence Microscopy,” Ann. N.Y. Acad. Sci. 483, 405 (1986).
[CrossRef]

G. J. Brakenhoff, H. T. M. van der Voort, Imaging, “Representation and Analysis of 3-Dimensional Biological Structures by Confocal Microscopy,” in Proceedings, Forty-Sixth Annual Meeting of EMSA (1988), p. 996.

Carlini, A. R.

T. Wilson, A. R. Carlini, “Three-Dimensional Imaging in Confocal Imaging System with Finite Sized Detectors,” J. Microsc. 149, 51 (1988).
[CrossRef]

Carrington, W. A.

W. A. Carrington, F. S. Fay, K. E. Fogarty, L. Lifshitz, “Analysis of 3D Molecular Distribution With Digital Image Microscopy,” in Proceedings, Forty-Sixth Annual Meeting of EMSA (1988), p. 40.

Elson, E. L.

H. Qian, E. L. Elson, “3-D Fluorescence Correlation Spectroscopy in Bulk Solution,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 352 (1988).

N. O. Petersen, E. L. Elson, “Measurements of Diffusion and Chemical Kinetics by Fluorescence Photobleaching Recovery and Fluorescence Correlation Spectroscopy,” Methods Enzymol. 130, 454 (1986).
[CrossRef] [PubMed]

E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery,” Ann. Rev. Phys. Chem. 36, 379 (1985).
[CrossRef]

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Application of Fluorescence Photobleaching Techniques to Problems in Cell Biology,” Fed. Proc. Fed. Am. Soc. Exp. Biol. 42, 72 (1983).

R. D. Icenogle, E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery of Multiple Binding Reactions. I. Theory and FCS Measurements,” Biopolymers 22, 1919 (1983).
[CrossRef] [PubMed]

R. D. Icenogle, E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery of Multiple Binding Reactions. II. FPR and FCS Measurements at Low and High DNA Concentrations,” Biopolymers 22, 1949 (1983).
[CrossRef] [PubMed]

J. Schlessinger, E. L. Elson, “Fluorescence Methods for Studying Membrane Dynamics, Methods of Experimental Physics,” 20, 197 (1982).

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Fluorescence Photobleaching in Cell Biology,” Nature London 295, 283 (1982).
[CrossRef] [PubMed]

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, W. W. Webb, “Dynamics of Fluorescence Marker Concentration as a Probe of Mobility,” Biophys. J. 16, 1315 (1976).
[CrossRef] [PubMed]

D. Axelrod, D. E. Koppel, J. Schlessinger, E. L. Elson, W. W. Webb, “Mobility Measurement by Analysis of Fluorescence Photobleaching Recovery Kinetics,” Biophys. J. 16, 1055 (1976).
[CrossRef] [PubMed]

E. L. Elson, W. W. Webb, “Concentration Correlation Spectroscopy: A New Biophysical Probe Based on Occupation Number Fluctuations,” Ann. Rev. Biophys. Bioeng. 4, 311 (1975).
[CrossRef]

E. L. Elson, D. Magde, “Fluorescence Correlation Spectroscopy. I. Conceptual Basis and Theory,” Biopolymers 13, 1 (1974).
[CrossRef]

D. Magde, E. L. Elson, W. W. Webb, “Fluorescence Correlation Spectroscopy. II. An Experimental Realization,” Biopolymers 13, 29 (1974).
[CrossRef] [PubMed]

H. Qian, E. L. Elson, “Characterization of Confocal Laser-Based Microscope by Digital Image Analysis: an Optical Sectioning Microscopy Approach,” in Optical Microscopy for Biology, B. Herman, K. Jacobson, Eds. (Alan R. Liss, New York, 1990).

E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery: Measurement of Transport and Chemical Kinetics,” in Spectroscopic Membrane Probes, L. M. Loew, Ed. (CRC Press, Cleveland, 1988).

Fay, F. S.

W. A. Carrington, F. S. Fay, K. E. Fogarty, L. Lifshitz, “Analysis of 3D Molecular Distribution With Digital Image Microscopy,” in Proceedings, Forty-Sixth Annual Meeting of EMSA (1988), p. 40.

Feher, G.

M. H. Weissman, H. Schindler, G. Feher, “Determination of Molecular Weights by Fluctuation Spectroscopy: Application to DNA,” Proc. Natl. Acad. Sci. USA 73, 2776 (1976).
[CrossRef] [PubMed]

Fogarty, K. E.

W. A. Carrington, F. S. Fay, K. E. Fogarty, L. Lifshitz, “Analysis of 3D Molecular Distribution With Digital Image Microscopy,” in Proceedings, Forty-Sixth Annual Meeting of EMSA (1988), p. 40.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hopkins, H. H.

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 217, 91 (1955).

Icenogle, R. D.

R. D. Icenogle, E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery of Multiple Binding Reactions. II. FPR and FCS Measurements at Low and High DNA Concentrations,” Biopolymers 22, 1949 (1983).
[CrossRef] [PubMed]

R. D. Icenogle, E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery of Multiple Binding Reactions. I. Theory and FCS Measurements,” Biopolymers 22, 1919 (1983).
[CrossRef] [PubMed]

Jacobson, K.

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Application of Fluorescence Photobleaching Techniques to Problems in Cell Biology,” Fed. Proc. Fed. Am. Soc. Exp. Biol. 42, 72 (1983).

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Fluorescence Photobleaching in Cell Biology,” Nature London 295, 283 (1982).
[CrossRef] [PubMed]

Johnson, D. C.

N. O. Petersen, D. C. Johnson, M. J. Schlesinger, “Scanning Fluorescence Correlation Spectroscopy. II. Application to Virus Glycoprotein Aggregation,” Biophys. J. 49, 817 (1986).
[CrossRef] [PubMed]

Kogelnik, H.

Koppel, D. E.

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Application of Fluorescence Photobleaching Techniques to Problems in Cell Biology,” Fed. Proc. Fed. Am. Soc. Exp. Biol. 42, 72 (1983).

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Fluorescence Photobleaching in Cell Biology,” Nature London 295, 283 (1982).
[CrossRef] [PubMed]

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, W. W. Webb, “Dynamics of Fluorescence Marker Concentration as a Probe of Mobility,” Biophys. J. 16, 1315 (1976).
[CrossRef] [PubMed]

D. Axelrod, D. E. Koppel, J. Schlessinger, E. L. Elson, W. W. Webb, “Mobility Measurement by Analysis of Fluorescence Photobleaching Recovery Kinetics,” Biophys. J. 16, 1055 (1976).
[CrossRef] [PubMed]

Lanni, F.

F. Lanni, L. D. Taylor, B. R. Ware, “Fluorescence Photobleaching Recovery in Solutions of Labeled Actin,” Biophys. J. 35, 351 (1981).
[CrossRef] [PubMed]

Li, T.

Lifshitz, L.

W. A. Carrington, F. S. Fay, K. E. Fogarty, L. Lifshitz, “Analysis of 3D Molecular Distribution With Digital Image Microscopy,” in Proceedings, Forty-Sixth Annual Meeting of EMSA (1988), p. 40.

Magde, D.

E. L. Elson, D. Magde, “Fluorescence Correlation Spectroscopy. I. Conceptual Basis and Theory,” Biopolymers 13, 1 (1974).
[CrossRef]

D. Magde, E. L. Elson, W. W. Webb, “Fluorescence Correlation Spectroscopy. II. An Experimental Realization,” Biopolymers 13, 29 (1974).
[CrossRef] [PubMed]

Nanning, N.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, N. Nanning, “Three-Dimensional Imaging by Confocal Fluorescence Microscopy,” Ann. N.Y. Acad. Sci. 483, 405 (1986).
[CrossRef]

Palmer, A. G.

Petersen, N. O.

N. O. Petersen, “Scanning Fluorescence Correlation Spectroscopy. I. Theory and Simulation of Aggregation Measurements,” Biophys. J. 49, 809 (1986).
[CrossRef] [PubMed]

N. O. Petersen, D. C. Johnson, M. J. Schlesinger, “Scanning Fluorescence Correlation Spectroscopy. II. Application to Virus Glycoprotein Aggregation,” Biophys. J. 49, 817 (1986).
[CrossRef] [PubMed]

N. O. Petersen, E. L. Elson, “Measurements of Diffusion and Chemical Kinetics by Fluorescence Photobleaching Recovery and Fluorescence Correlation Spectroscopy,” Methods Enzymol. 130, 454 (1986).
[CrossRef] [PubMed]

Qian, H.

H. Qian, E. L. Elson, “3-D Fluorescence Correlation Spectroscopy in Bulk Solution,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 352 (1988).

H. Qian, “Biophysical Characterization of Biopolymer Solutions and Gels by Fluorescence Fluctuation Studies,” Ph.D. Thesis, Washington U., St. Louis (1989).

H. Qian, E. L. Elson, “Characterization of Confocal Laser-Based Microscope by Digital Image Analysis: an Optical Sectioning Microscopy Approach,” in Optical Microscopy for Biology, B. Herman, K. Jacobson, Eds. (Alan R. Liss, New York, 1990).

Schindler, H.

M. H. Weissman, H. Schindler, G. Feher, “Determination of Molecular Weights by Fluctuation Spectroscopy: Application to DNA,” Proc. Natl. Acad. Sci. USA 73, 2776 (1976).
[CrossRef] [PubMed]

Schlesinger, M. J.

N. O. Petersen, D. C. Johnson, M. J. Schlesinger, “Scanning Fluorescence Correlation Spectroscopy. II. Application to Virus Glycoprotein Aggregation,” Biophys. J. 49, 817 (1986).
[CrossRef] [PubMed]

Schlessinger, J.

J. Schlessinger, E. L. Elson, “Fluorescence Methods for Studying Membrane Dynamics, Methods of Experimental Physics,” 20, 197 (1982).

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, W. W. Webb, “Dynamics of Fluorescence Marker Concentration as a Probe of Mobility,” Biophys. J. 16, 1315 (1976).
[CrossRef] [PubMed]

D. Axelrod, D. E. Koppel, J. Schlessinger, E. L. Elson, W. W. Webb, “Mobility Measurement by Analysis of Fluorescence Photobleaching Recovery Kinetics,” Biophys. J. 16, 1055 (1976).
[CrossRef] [PubMed]

Schneider, M. B.

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Stokseth, P. A.

Taylor, L. D.

F. Lanni, L. D. Taylor, B. R. Ware, “Fluorescence Photobleaching Recovery in Solutions of Labeled Actin,” Biophys. J. 35, 351 (1981).
[CrossRef] [PubMed]

Thompson, N. L.

van der Voort, H. T. M.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, N. Nanning, “Three-Dimensional Imaging by Confocal Fluorescence Microscopy,” Ann. N.Y. Acad. Sci. 483, 405 (1986).
[CrossRef]

G. J. Brakenhoff, H. T. M. van der Voort, Imaging, “Representation and Analysis of 3-Dimensional Biological Structures by Confocal Microscopy,” in Proceedings, Forty-Sixth Annual Meeting of EMSA (1988), p. 996.

van Spronsen, E. A.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, N. Nanning, “Three-Dimensional Imaging by Confocal Fluorescence Microscopy,” Ann. N.Y. Acad. Sci. 483, 405 (1986).
[CrossRef]

Ware, B. R.

F. Lanni, L. D. Taylor, B. R. Ware, “Fluorescence Photobleaching Recovery in Solutions of Labeled Actin,” Biophys. J. 35, 351 (1981).
[CrossRef] [PubMed]

Webb, W. W.

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Application of Fluorescence Photobleaching Techniques to Problems in Cell Biology,” Fed. Proc. Fed. Am. Soc. Exp. Biol. 42, 72 (1983).

K. Jacobson, E. L. Elson, D. E. Koppel, W. W. Webb, “Fluorescence Photobleaching in Cell Biology,” Nature London 295, 283 (1982).
[CrossRef] [PubMed]

M. B. Schneider, W. W. Webb, “Measurement of Submicron Laser Beam Radii,” Appl. Opt. 20, 1382–1388 (1981).
[CrossRef] [PubMed]

D. Axelrod, D. E. Koppel, J. Schlessinger, E. L. Elson, W. W. Webb, “Mobility Measurement by Analysis of Fluorescence Photobleaching Recovery Kinetics,” Biophys. J. 16, 1055 (1976).
[CrossRef] [PubMed]

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, W. W. Webb, “Dynamics of Fluorescence Marker Concentration as a Probe of Mobility,” Biophys. J. 16, 1315 (1976).
[CrossRef] [PubMed]

E. L. Elson, W. W. Webb, “Concentration Correlation Spectroscopy: A New Biophysical Probe Based on Occupation Number Fluctuations,” Ann. Rev. Biophys. Bioeng. 4, 311 (1975).
[CrossRef]

D. Magde, E. L. Elson, W. W. Webb, “Fluorescence Correlation Spectroscopy. II. An Experimental Realization,” Biopolymers 13, 29 (1974).
[CrossRef] [PubMed]

Weissman, M. H.

M. H. Weissman, H. Schindler, G. Feher, “Determination of Molecular Weights by Fluctuation Spectroscopy: Application to DNA,” Proc. Natl. Acad. Sci. USA 73, 2776 (1976).
[CrossRef] [PubMed]

Wilson, T.

T. Wilson, A. R. Carlini, “Three-Dimensional Imaging in Confocal Imaging System with Finite Sized Detectors,” J. Microsc. 149, 51 (1988).
[CrossRef]

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Ann. N.Y. Acad. Sci. (1)

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, N. Nanning, “Three-Dimensional Imaging by Confocal Fluorescence Microscopy,” Ann. N.Y. Acad. Sci. 483, 405 (1986).
[CrossRef]

Ann. Rev. Biophys. Bioeng. (2)

E. L. Elson, W. W. Webb, “Concentration Correlation Spectroscopy: A New Biophysical Probe Based on Occupation Number Fluctuations,” Ann. Rev. Biophys. Bioeng. 4, 311 (1975).
[CrossRef]

D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191 (1984).
[CrossRef]

Ann. Rev. Phys. Chem. (1)

E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery,” Ann. Rev. Phys. Chem. 36, 379 (1985).
[CrossRef]

Appl. Opt. (3)

Biophys. J. (5)

F. Lanni, L. D. Taylor, B. R. Ware, “Fluorescence Photobleaching Recovery in Solutions of Labeled Actin,” Biophys. J. 35, 351 (1981).
[CrossRef] [PubMed]

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, W. W. Webb, “Dynamics of Fluorescence Marker Concentration as a Probe of Mobility,” Biophys. J. 16, 1315 (1976).
[CrossRef] [PubMed]

N. O. Petersen, “Scanning Fluorescence Correlation Spectroscopy. I. Theory and Simulation of Aggregation Measurements,” Biophys. J. 49, 809 (1986).
[CrossRef] [PubMed]

N. O. Petersen, D. C. Johnson, M. J. Schlesinger, “Scanning Fluorescence Correlation Spectroscopy. II. Application to Virus Glycoprotein Aggregation,” Biophys. J. 49, 817 (1986).
[CrossRef] [PubMed]

D. Axelrod, D. E. Koppel, J. Schlessinger, E. L. Elson, W. W. Webb, “Mobility Measurement by Analysis of Fluorescence Photobleaching Recovery Kinetics,” Biophys. J. 16, 1055 (1976).
[CrossRef] [PubMed]

Biopolymers (4)

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

R. D. Icenogle, E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery of Multiple Binding Reactions. II. FPR and FCS Measurements at Low and High DNA Concentrations,” Biopolymers 22, 1949 (1983).
[CrossRef] [PubMed]

E. L. Elson, D. Magde, “Fluorescence Correlation Spectroscopy. I. Conceptual Basis and Theory,” Biopolymers 13, 1 (1974).
[CrossRef]

Fed. Proc. Fed. Am. Soc. Exp. Biol. (1)

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Fluorescence Methods for Studying Membrane Dynamics, Methods of Experimental Physics (1)

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Proc. Soc. Photo-Opt. Instrum. Eng. (1)

H. Qian, E. L. Elson, “3-D Fluorescence Correlation Spectroscopy in Bulk Solution,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 352 (1988).

Other (11)

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W. A. Carrington, F. S. Fay, K. E. Fogarty, L. Lifshitz, “Analysis of 3D Molecular Distribution With Digital Image Microscopy,” in Proceedings, Forty-Sixth Annual Meeting of EMSA (1988), p. 40.

E. L. Elson, “Fluorescence Correlation Spectroscopy and Photobleaching Recovery: Measurement of Transport and Chemical Kinetics,” in Spectroscopic Membrane Probes, L. M. Loew, Ed. (CRC Press, Cleveland, 1988).

Only a minor error results if we neglect the contribution from region z > zɛ. Then an analytical form will beωapp2=2ω02[1+(zε/z1)/[1+(zε/z1)2]/arctan(zε/z1)]−1.

H. Qian, “Biophysical Characterization of Biopolymer Solutions and Gels by Fluorescence Fluctuation Studies,” Ph.D. Thesis, Washington U., St. Louis (1989).

The Gaussian or Airy profiles are relative, since any instrument has only a finite aperture. Therefore, there is always secondary diffraction. The approximation of a Gaussian beam is appropriate when secondary diffraction can be neglected.

H. Qian, E. L. Elson, “Characterization of Confocal Laser-Based Microscope by Digital Image Analysis: an Optical Sectioning Microscopy Approach,” in Optical Microscopy for Biology, B. Herman, K. Jacobson, Eds. (Alan R. Liss, New York, 1990).

More accurately, image formation should take the lens maker’s law into account: r° = rf/(d − z), z° = zd°/(d − z). As discussed later, this will introduce asymmetry and nonlinearity into the system.

Mathematically this corresponds to replacing a very narrow function by a Dirac delta-function. This introduces very little error if the delta-function multiplies a slowly varying function and the product is integrated.

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Figures (5)

Fig. 1
Fig. 1

Schematic of a confocal laser based microscope system. Planes A, B, and C are conjugate to each other. Since the laser does not fill the aperture of the objective lens, the focused laser beam is not limited by the numerical aperture of the objective but rather tanθ = λ/nπω0. (Note that in the calculations in the text θ appears implicitly based on the relationship tanθ = ω0/z1.) When a fluorescent emitter is off the objective plane, its image on the image plane is a defocused spot, the size of the spot can be derived by knowing the true image position (which is out of the image plane), and the light cone angle α′, tanα′ = tanα/M. Therefore, if M = 1, the light cone angle at the image plane is identical to the one at the objective plane. Also n sinα is the N.A. of the objective lens. (B) The excitation laser intensity in the object space is characterized by a Gaussian-Lorentzian function. That is, the distribution of intensity is Gaussian in planes normal to the optical (z-) axis, becoming broader the further off the focal plane. The distribution is Lorentzian along the z-axis. (C) At a first-order approximation, the CEF can be derived by a geometric optical approach. In this schematic in which the field aperture has been projected onto the object plane with magnification M = 1, the shaded area, designated CEF = 1, depicts the region of space in which all the light emitted from a point source and imaged by the objective lens passes the field aperture. The unshaded cone depicts the cone of light rays emitted from a point source on the optic axis in the object space focal plane (i.e., at r = 0, z = 0). No light from a point source in the regions designated CEF = 0 and a fraction of the light emitted from the intervening regions pass the field aperture. The geometrical optics approximation to the CEF used in the text is based on this conception.

Fig. 2
Fig. 2

Comparison of several different CEF(r,z). (A) CEF, calculated using the PSF as described in the text, for s 0 = s 0 tanα/λ = 3.7 (dotted curve), 7.5 (continuous curve), and 18.6 (dashed curve). (B) For s 0 = 7 . 5 comparison between CEF calculated from PSF (pCEF) (continuous curve) and by a geometrical optics approximation (gCEF) (dotted curve). The Gaussian representation is also plotted (dashed curve). Adjacent to each group of curves in panels A and B are the values of z / s 0. (C) The function CEF(0,Z) for s 0 = 0 . 37, 3.7, and 18.6. The geometric approximation (dotted curve) and Gaussian representation (dashed curve), both for s 0 = 7 . 5, are also plotted. Note that only when s 0 is small is the Gaussian representation close to the CEF and only for a small range of z.

Fig. 3
Fig. 3

Apparent excitation profile for s 0 = 7 . 0 and ω 0 = 3 . 15: (A) ——,-----, I(r,z) and I(r,z) CEF(r,z), respectively, for various z. The numbers adjacent to the curves provide z / s 0 for I(r,z) CEF(r,z); the values of this ratio are of the same order for I(r,z); (B) —— ·····, ○ are I(0,z), CEF(0,z), and I(0,z) CEF(0,z), respectively (also see Table II).

Fig. 4
Fig. 4

Integrated collection efficiency E(z). The a and b are N.A. = 0.75 and 0.35, respectively, calculated by using the geometric optics approximation to the CEF: a, ——, ·····, ----- are s0 = 4.0, 10.0, and from Eq. (12); b, ——, ·····, ----- are s0 = 6.26, 15.75, and Eq. (12).

Fig. 5
Fig. 5

Histogram of intensity for different ω. The apparent value of ω, ωapp can be thought of as the mean of this histogram. The sharper the distribution about ω = ω0, the closer the behavior to 2-D theory.

Tables (4)

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Table I Dimensionless Field Diaphragm Radii

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Table II Apparent Beam Profile

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Table III Focused Laser Intensity Profile and Apparent Beam Radii

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Table IV Halfwldth of E(z)

Equations (38)

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CEF ( r , z ) = 1 Δ circ ( r ° / s 0 ) PSF ( r ° , r , z ) d r ° .
circ ( r ) = { 1 | r | 1 , 0 | r | > 1 ,
Δ = circ ( r ° / s 0 ) PSF ( r ° , 0 , 0 ) d r °
CEF ( r / s 0 , z / s 0 ) ,
CEF ( r , δ z ) d r = Δ 1 circ ( r ° / s 0 ) PSF ( r ° r , δ z ) d r ° d r = Δ 1 circ ( r ° / s 0 ) d r ° PSF ( r ° r , δ z ) d r = Δ 1 circ ( r ° / s 0 ) d r ° = π s 0 2 / Δ .
g CEF ( R , Z ) = { circ ( R 1 ) circ [ ( R 1 R ) / Z ] d R 1 / π Z 2 Z < 1 , [ ( 1 Z / tan 2 α + Z 2 ) / ( 1 cos α ) ] circ ( R 1 ) × circ [ ( R 1 R ) / Z ] d R 1 / π Z > 1 ,
g CEF ( 0 , Z ) = 1 Z / Z 2 + tan 2 α / ( 1 cos α ) ,
1 Z ε / Z ε 2 + tan 2 α = ( 1 cos α ) / 2 = sin 2 ( α / 2 ) ,
Z ε = tan α cos 2 ( α / 2 ) / 1 cos 4 ( α / 2 ) = tan α / tan ( α / 2 ) / 1 + sec 2 ( α / 2 ) ,
G ( t ) = q 2 + I ( r , z ) CEF ( r , z ) δ c ( r , z , 0 ) δ c ( r ° , z ° , t ) × I ( r ° , z ° ) CEF ( r ° , z ° ) d r d z d r ° d z ° ,
I ( r , z ) = I 0 ( z ) exp [ 2 r 2 ω 2 ( z ) ] ( Gaussian ) ,
ω 2 ( z ) = ω 0 2 + ( λ n π ω 0 ) 2 z 2 ( Lorentzian ) ,
I 0 ( z ) = ω 0 2 I 0 ω 2 ( z ) ,
I c ( r , z ) = I c ( 0 , z ) exp [ 2 r 2 / ω c 2 ( z ) ] .
I c ( 0 , z ) / I 0 { 1 / ( 1 + z 2 / z 1 2 ) z < s 0 / tan α , ( 1 z / s 0 2 + z 2 ) / ( 1 cos α ) / ( 1 + z 2 / z 1 2 ) z > s 0 / tan α ,
ω c ( z ) { ω 0 z < s 0 / tan α , ω 0 ( 1 + z 2 / z 1 2 ) z > s 0 / tan α .
G ( 0 ) = D I c ( r ) I c ( r ° ) δ c ( r , 0 ) 2 δ c ( r ° , 0 ) d r d r ° ,
G ( 0 ) = D I c ( r ) 2 I c ( r ° ) δ c ( r , 0 ) δ c ( r ° , 0 ) d r d r ° = D c ¯ I c ( r ) 2 I c ( r ° ) δ ( r r ° ) d r d r ° = D c ¯ I c ( r ) 2 I c ( r ) d r = D c ¯ [ I c ( r ) ] 2 d r .
G ( 0 ) = D c ¯ [ I c ( r ) ] 2 d r = D c ¯ [ ( I c / x ) 2 + ( I c / y ) 2 + ( I c / z ) 2 ] d x d y d z = D c ¯ { π I c 2 ( 0 , z ) d z + 2 [ I c ( r , z ) / z ] 2 d r d z } ,
G ( 0 ) = c ¯ π / 4 I c 2 ( 0 , z ) ω c 2 ( z ) d z ,
g ( 0 ) = 4 D I c 2 ( 0 , z ) d z / ω c 2 ( z ) I c 2 ( 0 , z ) d z 8 D [ I c ( r , z ) / z ] 2 d r d z / ω c 2 ( z ) I c 2 ( 0 , z ) d z .
ω app 2 = ω c 2 ( z ) I c 2 ( 0 , z ) d z / I c 2 ( 0 , z ) d z ,
τ x y = ω app 2 / 4 D .
τ z = z app 2 / 2 D ,
z app 2 = 4 [ I c ( r , z ) / z ] 2 d r d z / ω c 2 ( z ) I c 2 ( 0 , z ) d z = π { ω c 2 ( z ) [ I c ( 0 , z ) z ] 2 + 2 ω c ( z ) ω c ( z ) z I c ( 0 , z ) I c ( 0 , z ) z + 2 I c 2 ( 0 , z ) [ ω c ( z ) z ] 2 } d z ω c 2 ( z ) I c 2 ( 0 , z ) d z
τ 3 D 1 = τ x y 1 + τ z 1 .
ω app 2 = ω 0 2 .
I c ( r , z ) = I 0 exp ( z 2 / z app 2 ) exp ( 2 r 2 / ω 0 2 ) .
ω app 2 = ω c 2 ( z ) I 2 ( 0 , z ) d z / I 2 ( 0 , z ) d z = ω 0 2 I 0 I ( 0 , z ) d z / I 2 ( 0 , z ) d z = 2 ω 0 2 , z app 2 = 4 [ I c ( r , z ) / z ] 2 d r d z / ω c 2 ( z ) I c 2 ( 0 , z ) d z = 1 / z 1 2 ,
E ( z ) = CEF ( r , z ) I ( r , z ) d r / CEF ( r , 0 ) I ( r , 0 ) d r = I c ( r , z ) d r / I c ( r , 0 ) d r .
E ( z ) = ω c 2 ( z ) I c ( 0 , z ) ω 0 2 I 0 { 1 / ( 1 + z 2 / z 1 2 ) z < s 0 / tan α , ( 1 z / s 0 2 + z 2 ) / ( 1 cos α ) z > s 0 / tan α .
F ( t ) = c ¯ [ 1 ( κ / I 0 ) Δ I ( 0 ) Δ I c ( t ) ] ,
ω FPR 2 = ω c 2 ( z ) / [ ω c 2 ( z ) + ω 2 ( z ) ] I c ( 0 , z ) d z 2 ω c 2 ( z ) / [ ω c 2 ( z ) + ω 2 ( z ) ] 2 I c ( 0 , z ) d z .
i ( r ° , z ° ) = o ( r r ° , z z ° ) T ( r , z ) d r d z = T ( r ° + r , z ° + z ) o ( r , z ) d r d z ,
T ( r , z ) = PSF ( r r ° , z ) Π e ( r ° ) d r · PSF ( r ° r , z ) Π d ( r ° ) d r ° ,
T ( r 1 / 2 , 0 ) = T ( 0 , z 1 / 2 ) = T ( 0 , 0 ) / 2 .
PSF ( r , r ° , z ) = PSF ( r r ° , z ) , PSF ( r r ° , z ) = PSF ( r r ° , z ) ,
PSF ( r , r ° , z ) = PSF [ r r ° / ( 1 + z / d ) , z ] .

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