Abstract

We exploit recent advances in single-particle tracking to perform fluorescence correlation spectroscopy on individual fluorescent particles, in contrast to traditional methods that build up statistics over a sequence of many measurements. By rapidly scanning the focus of an excitation laser in a circular pattern, demodulating the measured fluorescence, and feeding these results back to a piezoelectric translation stage, we track the Brownian motion of fluorescent polymer microspheres in aqueous solution in the plane transverse to the laser axis. We discuss the estimation of particle diffusion statistics from closed-loop position measurements, and we present a generalized theory of fluorescence correlation spectroscopy for the case that the motion of a single fluorescent particle is actively tracked by a time-dependent laser intensity. We model the motion of a tracked particle using Ornstein-Uhlenbeck statistics, using a general theory that contains a number of existing results as specific cases. We find good agreement between our theory and experimental results, and discuss possible future applications of these techniques to passive, single-shot, single-molecule fluorescence measurements with many orders of magnitude in time resolution.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. D. Magde, E. L. Elson, and W. W. Webb, �??Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,�?? Phys. Rev. Lett. 29, 705-708 (1972).
    [CrossRef]
  2. E. L. Elson and D. Magde, �??Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,�?? Biopolymers 13, 1-27 (1974).
    [CrossRef]
  3. D. Magde, E. L. Elson, and W. W. Webb, �??Fluorescence correlation spectroscopy. 2. Experimental realization,�?? Biopolymers 13, 29-61 (1974).
    [CrossRef] [PubMed]
  4. O. Krichevsky and G. Bonnett, �??Fluorescence correlation spectroscopy: the technique and its applications,�?? Rep. Prog. Phys. 65, 251-297 (2002).
    [CrossRef]
  5. S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, �??Biological and Chemical Applications of Fluorescence Correlation Spectroscopy: A Review,�?? Biochemistry 41, 697-705 (2002).
    [CrossRef] [PubMed]
  6. A. J. Berglund, A. C. Doherty, and H. Mabuchi, �??Photon statistics and dynamics of Fluorescence Resonance Energy Transfer,�?? Phys. Rev. Lett. 89, 068101 (2002).
    [CrossRef] [PubMed]
  7. H. D. Kim, G. U. Nienhaus, T. Ha, J. W. Orr, J. R. Williamson, and S. Chu, �??Mg2+-dependent conformational changes of RNA studied by fluorescence correlation and FRET on immobilized single molecules,�?? Proc. Natl. Acad. Sci. U.S.A. 99, 4284-4289 (2002).
    [CrossRef] [PubMed]
  8. K. C. Neuman and S. M. Block, �??Optical Trapping,�?? Rev. Sci. Instrum. 75, 2787-2809 (2004).
    [CrossRef]
  9. M. J. Saxton and K. Jacobson, �??Single-particle tracking: applications to membrane dynamics,�?? Annu. Rev. Biophys. Biomolec. Struct. 26, 373-399 (1997).
    [CrossRef]
  10. A. E. Cohen and W. E. Moerner, �??Method for trapping and manipulating nanoscale objects in solution,�?? Appl. Phys. Lett. 86, 093109 (2005).
    [CrossRef]
  11. A. E. Cohen, �??Control of Nanoparticles with Arbitrary Two-Dimensional Force Fields,�?? Phys. Rev. Lett. 94, 118102 (2005).
    [CrossRef] [PubMed]
  12. T. Meyer and H. Schindler, �??Simultaneous Measurement of Aggregation and Diffusion of Molecules in Solutions and in Membranes,�?? Biophys. J. 54, 983-993 (1988).
    [CrossRef] [PubMed]
  13. T. Ha, D. S. Chemla, T. Enderle, and S.Weiss, �??Single molecule spectroscopy with automated positioning,�?? Appl. Phys. Lett. 70, 782-784 (1997).
    [CrossRef]
  14. J. Enderlein, �??Tracking of fluorescent molecules diffusing within membranes,�?? Appl. Phys. B 71, 773-777 (2000).
    [CrossRef]
  15. J. Enderlein, �??Positional and Temporal Accuracy of Single Molecule Tracking,�?? Sing. Mol. 1, 225-230 (2000).
    [CrossRef]
  16. R. S. Decca, C.-W. Lee, and S. R. Wassall, �??Single molecule tracking scheme using a near-field scanning optical microscope,�?? Rev. Sci. Instr. 73, 2675-2679 (2002).
    [CrossRef]
  17. V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, �??Scanning FCS, an novel method for three-dimensional particle tracking,�?? Biochem. Soc. Trans. 31, 997-1000 (2003).
    [CrossRef] [PubMed]
  18. A. J. Berglund and H. Mabuchi, �??Feedback controller design for tracking a single fluorescent molecule,�?? Appl. Phys. B 78, 653-659 (2004).
    [CrossRef]
  19. K. Kis-Petikova and E. Gratton, �??Distance measurement by circular scanning of the excitation beam in a two-photon microscope,�?? Microsc. Res. Tech. 63, 34-49 (2004).
    [CrossRef]
  20. V. Levi, Q. Ruan, and E. Gratton, �??3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,�?? Biophys. J. 88, 2919-2928 (2005).
    [CrossRef] [PubMed]
  21. M. A. Digman, P. Sengupta, P. W. Wiseman, C. M. Brown, A. R. Horwitz, and E. Gratton, �??Fluctuation Correlation Spectroscopy with a Laser-Scanning Microscope: Exploiting the Hidden Time Structure,�?? Biophys. J. 88, L33-L36 (2005).
    [CrossRef] [PubMed]
  22. M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, �??Measuring fast dynamics in solutions and cells with a laser scanning microscope,�?? Biophys. J. 89, 1317-1327 (2005).
    [CrossRef] [PubMed]
  23. M. H. DeGroot, Probability and Statistics (Addison-Wesley, Reading, MA, 1986).
  24. G. Chirico, C. Fumagalli, and G. Baldini, �??Trapped Brownian Motion in Single- and Two-Photon Excitation Fluorescence Correlation Experiments,�?? J. Phys. Chem. B 106, 2508-2519 (2002).
    [CrossRef]
  25. C. W. Gardiner, Handook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, 1985).
  26. N. G. Van Kampen, Stochastic processes in physics and chemistry (Elsevier Science Pub. Co., North-Holland, Amsterdam, 2001).
  27. A. Gennerich and D. Schild, �??Fluorescence correlation spectroscopy in small cytosolic compartments depends critically on the diffusion model used,�?? Biophys. J. 79, 3294-3306 (2000).
    [CrossRef] [PubMed]
  28. A. J. Berglund and H. Mabuchi, �??Performance bounds on single-particle tracking by fluorescence modulation,�?? in preparation (2005).
  29. X. Zhuang, L. E. Bartley, H. P. Babcock, R. Russell, T. Ha, D. Hershlag, and S. Chu, �??A Single-Molecule Study of RNA Catalysis and Folding,�?? Science 288, 2048-2051 (2000).
    [CrossRef] [PubMed]
  30. B. Okumus, T. J. Wilson, D. M. J. Lilley, and T. Ha, �??Vesicle Encapsulation Studies Reveal that Single Molecule Ribozyme Heterogeneities Are Intrinsinc,�?? Biophys. J. 87, 2798-2806 (2004).
    [CrossRef] [PubMed]
  31. E. Rhoades, E. Gussakovsky, and G. Haran, �??Watching proteins fold one molecule at a time,�?? Proc. Natl. Acad. Sci. U.S.A. 100, 3197-3202 (2003).
    [CrossRef] [PubMed]

Annu. Rev. Biophys. Biomolec. Struct. (1)

M. J. Saxton and K. Jacobson, �??Single-particle tracking: applications to membrane dynamics,�?? Annu. Rev. Biophys. Biomolec. Struct. 26, 373-399 (1997).
[CrossRef]

Appl. Phys. B (2)

J. Enderlein, �??Tracking of fluorescent molecules diffusing within membranes,�?? Appl. Phys. B 71, 773-777 (2000).
[CrossRef]

A. J. Berglund and H. Mabuchi, �??Feedback controller design for tracking a single fluorescent molecule,�?? Appl. Phys. B 78, 653-659 (2004).
[CrossRef]

Appl. Phys. Lett. (2)

T. Ha, D. S. Chemla, T. Enderle, and S.Weiss, �??Single molecule spectroscopy with automated positioning,�?? Appl. Phys. Lett. 70, 782-784 (1997).
[CrossRef]

A. E. Cohen and W. E. Moerner, �??Method for trapping and manipulating nanoscale objects in solution,�?? Appl. Phys. Lett. 86, 093109 (2005).
[CrossRef]

Biochem. Soc. Trans. (1)

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, �??Scanning FCS, an novel method for three-dimensional particle tracking,�?? Biochem. Soc. Trans. 31, 997-1000 (2003).
[CrossRef] [PubMed]

Biochemistry (1)

S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, �??Biological and Chemical Applications of Fluorescence Correlation Spectroscopy: A Review,�?? Biochemistry 41, 697-705 (2002).
[CrossRef] [PubMed]

Biophys. J. (6)

T. Meyer and H. Schindler, �??Simultaneous Measurement of Aggregation and Diffusion of Molecules in Solutions and in Membranes,�?? Biophys. J. 54, 983-993 (1988).
[CrossRef] [PubMed]

B. Okumus, T. J. Wilson, D. M. J. Lilley, and T. Ha, �??Vesicle Encapsulation Studies Reveal that Single Molecule Ribozyme Heterogeneities Are Intrinsinc,�?? Biophys. J. 87, 2798-2806 (2004).
[CrossRef] [PubMed]

V. Levi, Q. Ruan, and E. Gratton, �??3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,�?? Biophys. J. 88, 2919-2928 (2005).
[CrossRef] [PubMed]

M. A. Digman, P. Sengupta, P. W. Wiseman, C. M. Brown, A. R. Horwitz, and E. Gratton, �??Fluctuation Correlation Spectroscopy with a Laser-Scanning Microscope: Exploiting the Hidden Time Structure,�?? Biophys. J. 88, L33-L36 (2005).
[CrossRef] [PubMed]

M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, �??Measuring fast dynamics in solutions and cells with a laser scanning microscope,�?? Biophys. J. 89, 1317-1327 (2005).
[CrossRef] [PubMed]

A. Gennerich and D. Schild, �??Fluorescence correlation spectroscopy in small cytosolic compartments depends critically on the diffusion model used,�?? Biophys. J. 79, 3294-3306 (2000).
[CrossRef] [PubMed]

Biopolymers (2)

E. L. Elson and D. Magde, �??Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,�?? Biopolymers 13, 1-27 (1974).
[CrossRef]

D. Magde, E. L. Elson, and W. W. Webb, �??Fluorescence correlation spectroscopy. 2. Experimental realization,�?? Biopolymers 13, 29-61 (1974).
[CrossRef] [PubMed]

J. Phys. Chem. B (1)

G. Chirico, C. Fumagalli, and G. Baldini, �??Trapped Brownian Motion in Single- and Two-Photon Excitation Fluorescence Correlation Experiments,�?? J. Phys. Chem. B 106, 2508-2519 (2002).
[CrossRef]

Microsc. Res. Tech. (1)

K. Kis-Petikova and E. Gratton, �??Distance measurement by circular scanning of the excitation beam in a two-photon microscope,�?? Microsc. Res. Tech. 63, 34-49 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

D. Magde, E. L. Elson, and W. W. Webb, �??Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,�?? Phys. Rev. Lett. 29, 705-708 (1972).
[CrossRef]

A. E. Cohen, �??Control of Nanoparticles with Arbitrary Two-Dimensional Force Fields,�?? Phys. Rev. Lett. 94, 118102 (2005).
[CrossRef] [PubMed]

A. J. Berglund, A. C. Doherty, and H. Mabuchi, �??Photon statistics and dynamics of Fluorescence Resonance Energy Transfer,�?? Phys. Rev. Lett. 89, 068101 (2002).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. (1)

H. D. Kim, G. U. Nienhaus, T. Ha, J. W. Orr, J. R. Williamson, and S. Chu, �??Mg2+-dependent conformational changes of RNA studied by fluorescence correlation and FRET on immobilized single molecules,�?? Proc. Natl. Acad. Sci. U.S.A. 99, 4284-4289 (2002).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

E. Rhoades, E. Gussakovsky, and G. Haran, �??Watching proteins fold one molecule at a time,�?? Proc. Natl. Acad. Sci. U.S.A. 100, 3197-3202 (2003).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

O. Krichevsky and G. Bonnett, �??Fluorescence correlation spectroscopy: the technique and its applications,�?? Rep. Prog. Phys. 65, 251-297 (2002).
[CrossRef]

Rev. Sci. Instr. (1)

R. S. Decca, C.-W. Lee, and S. R. Wassall, �??Single molecule tracking scheme using a near-field scanning optical microscope,�?? Rev. Sci. Instr. 73, 2675-2679 (2002).
[CrossRef]

Rev. Sci. Instrum. (1)

K. C. Neuman and S. M. Block, �??Optical Trapping,�?? Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Science (1)

X. Zhuang, L. E. Bartley, H. P. Babcock, R. Russell, T. Ha, D. Hershlag, and S. Chu, �??A Single-Molecule Study of RNA Catalysis and Folding,�?? Science 288, 2048-2051 (2000).
[CrossRef] [PubMed]

Sing. Mol. (1)

J. Enderlein, �??Positional and Temporal Accuracy of Single Molecule Tracking,�?? Sing. Mol. 1, 225-230 (2000).
[CrossRef]

Other (4)

C. W. Gardiner, Handook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, 1985).

N. G. Van Kampen, Stochastic processes in physics and chemistry (Elsevier Science Pub. Co., North-Holland, Amsterdam, 2001).

A. J. Berglund and H. Mabuchi, �??Performance bounds on single-particle tracking by fluorescence modulation,�?? in preparation (2005).

M. H. DeGroot, Probability and Statistics (Addison-Wesley, Reading, MA, 1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

(Left) Schematic diagram of the optics and electronics for tracking-FCS. Key: TIA = time-interval analyzer, APD = avalanche photodiode, MC = microcontroller, PZT = piezoelectric translation stage, D550 = dichroic filter with 550 nm cutoff. (Right) Detail of the sample volume. 60-nm diameter fluorescent beads suspended in water diffuse freely in the xy plane, but are confined by glass coverslips in the z direction. The coverslips are mounted on a piezoelectric sample stage, so that the entire bulk fluid volume can be translated. A tracked particle diffuses freely (in the axially confined geometry) in the reference frame of the bulk fluid while the feedback control translates the entire sample volume in order to hold the particle on the laser axis defined by x = y = 0. Because the sense of the laser rotation, and therefore the sign of the feedback controller, reverses upon crossing the focal plane of the microscope optics, the sample mount is adjusted in the z direction so that the focal plane lies just outside of the sample volume.

Fig. 2.
Fig. 2.

Fluorescence data and motion of the sample stage during tracking of a 60 nm microsphere in water. The upper plot shows fluorescence data, and the lower plot shows the x (solid) and y (dotted) positions of the sample stage during the fluorescence trace. Just before 7.5 s, a particle diffused into the capture region and the controller correspondingly responded by moving the sample stage to track this particle. The irregular motion of the sample stage at 7.6 s resulted from an (expected) arithmetic overflow in the microcontroller. The residual fluorescence fluctuations during tracking arise from the competition between diffusion and feedback control and also from the uncontrolled motion of the particle in the z direction. The fluctuations are recorded in Fig. 4 and a detailed theory is given in the next section.

Fig. 3.
Fig. 3.

Time-converging estimate of the diffusion coefficient D for the microsphere tracking data in Fig. 2 with bin time Δt = 20 ms. The dotted lines are error estimates calculated for the estimator , assuming underlying Brownian statistics. Inset: Final estimate of D as a function of bin time Δt. For bin times larger than ~ 10 ms, the estimates are roughly constant with mean value D = 6.2 μm2/s. See the text for an explanation of the estimator convergence with Δt.

Fig. 4.
Fig. 4.

Fluorescence correlation functions recorded during the tracking period in Fig. 2, normalized to the mean fluorescence. The noisy curves were measured from the tracking data in Fig. 2, averaged over coarse-grained time bins of 100 (dotted) and 200 (solid) μs. At higher time resolution, the oscillations due to the deterministic laser rotation make it difficult to resolve the overall shapes of the autocorrelation curve. The smooth solid curve is a fit to Eq. (18). The fit parameters are γxy = 134 Hz, D = (6.2 s-1)wxy2, ρ 0 = 1.4wxy , γz = 11.3 Hz, wz = 4.5μm, and z 0 = 2.8wxy . γxy is the tracking controller bandwidth. All fit parameters are scaled by the true beam waist wxy , which is approximately 1μm. For this value, the diffusion coefficient D determined by the statistical estimate from Fig. 3 and the value from a fit to Eq. 18 are identical.

Equations (25)

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

Φ t ( ρ , θ ) = exp { 2 w xy 2 [ ( ρ cos θ r 0 cos ω 0 t ) 2 + ( ρ sin θ r 0 sin ω 0 t ) 2 ] } .
f c ( ρ , θ ) = h ( ρ ) cos θ , fs ( ρ , θ ) = h ( ρ ) sin θ
h ( ρ ) = 2 π Γ 0 ω 0 exp [ 2 w xy 2 ( ρ 2 + r 0 2 ) ] I 1 [ 4 r 0 ρ w xy 2 ] .
h ( ρ ) 4 π r 0 ρ w xy 2 exp [ 2 r 0 2 w xy 2 ] .
D ̂ = [ ( Δ x ) 2 + ( Δ y ) 2 ] ( 4 Δ t )
d X t = γ x X t dt + 2 D d W t
P 0 ( x ) = 1 2 π x ¯ 2 exp [ x 2 2 x ¯ 2 ]
P τ ( x 2 x 1 ) = 1 2 π b τ 2 exp [ ( x 2 a τ x 1 ) 2 2 b τ 2 ]
G ( τ ) = ∫∫ d x 1 d x 2 P τ ( x 2 x 1 ) P 0 ( x 1 ) Φ x ( x 2 w x χ t + τ ) Φ x ( x 1 w x χ t ) t
ζ x = γ x τ x 1 + γ x τ x = w x 2 w x 2 + 4 x ¯ 2
G x ( τ ; χ t ) = ζ x 1 λ τ , x 2 exp [ 2 ζ x ( χ t + τ 2 2 λ τ , x χ t χ t + τ + χ t 2 1 λ τ , x 2 ) ]
λ τ , x = ( 1 ζ x ) e γ x τ .
G ( τ ) = G x ( τ ; x ̂ · r t w x ) G y ( τ ; y ̂ · r t w y ) G z ( τ ; z ̂ · r t w z ) t
g ( τ ) = ( ζ x y 2 1 λ τ , xy 2 ) ( ζ z 1 λ τ , z 2 ) exp [ 4 ρ 0 2 ( ζ xy 1 λ τ , xy 2 ) 4 z 0 2 ( ζ z 1 + λ τ , z ) ] ,
G ( τ ) = g ( τ ) exp [ 4 ρ 0 2 ( ζ xy λ τ , xy 1 λ τ , xy 2 ) cos ω 0 τ ]
G ˜ ( τ ) = 1 T ˜ τ T ˜ 2 τ + T ˜ 2 G ( τ ) g ( τ ) T ˜ τ T ˜ 2 τ + T ˜ 2 exp [ 4 ρ 0 2 ( ζ xy λ τ , xy 1 λ τ , xy 2 ) cos ω 0 τ ]
1 0 exp [ z cos θ ] d θ = I 0 ( z )
G ˜ ( τ ) g ( τ ) I 0 [ 4 ρ 0 2 ( ζ xy λ τ , xy 1 λ τ , xy 2 ) ]
P 0 ( x ) = lim t P ( X t = x )
P τ ( x 2 x 1 ) = lim t P ( X t + τ = x 2 X t = x 1 )
G ( τ ) = σ t σ t + τ t = Φ t ( X t ) Φ t + τ ( X t + τ ) t .
G ( τ ) = ∫∫ d q 1 2 π d q 2 2 π e i q 1 X t i q 2 X t + τ Φ ˜ t ( q 1 ) Φ ˜ t + τ ( q 2 ) t
= ∫∫ d q 1 2 π d q 2 2 π e i q 1 X t i q 2 X t + τ t Φ ˜ t ( q 1 ) Φ ˜ t + τ ( q 2 ) t
e i q 1 X t i q 2 X t + τ t = ∫∫ d x 1 d x 2 e i q 1 x 1 e i q 2 x 2 P 0 ( x 1 ) P τ ( x 2 x 1 ) .
G ( τ ) = ∫∫ d x 1 d x 2 P 0 ( x 1 ) P τ ( x 2 x 1 ) Φ t ( x 1 ) Φ t + τ ( x 2 ) t .

Metrics