Abstract

The phenomenon of the fluorescence polarization of solutions has found numerous applications in biophysics, biochemistry, immunology, and diagnostic and clinical medicine. The current theory to explain the phenomenon of fluorescence polarization in solutions was developed by F. Perrin in 1926. Perrin based his theory on the belief that fluorescence polarization is a manifestation of rotational Brownian motion. Fluorescence polarization, however, is an electromagnetic radiation phenomenon. Using Maxwell’s equations, suitably modified to account for the quantum behavior of fluorescence, E. Collett developed a theory of fluorescence polarization (the electrodynamic theory) based on a model of dipole–dipole interactions. The electrodynamic theory is used to investigate protein–protein assays to determine the minimum and maximum binding distances between the proteins for (1) an estrogen receptor DNA bound to a fluorescein labeled estrogen response element and (2) a green fluorescent protein chimera of S-peptide (S65T-His6) bound to a free S-protein.

© 2009 Optical Society of America

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References

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  1. G. G. Stokes, “On the change of refrangibility of light,” Proc. Royal Soc. London vi, 195-208 (1852).
  2. F. Weigert, “Über polarisiertes fluoreszenzlicht,” Verh. Dtsch. Phys. Ges. 23, 100-102 (1920).
  3. F. Perrin, “Polarisation de la lumière de fluorescence. Vie moyenne des molécules dans l'etat excité,” J. Phys. Radium 7, 390-401 (1926).
    [CrossRef]
  4. G. Weber, “Polarization of the fluorescence of macromolecules: 1. Theory and experimental method,” Biochemistry J. 51, 145-155 (1952).
  5. J. R. Lackowicz, Principles of Fluorescence Spectroscopy (Plenum, 1983).
  6. E. Collett, “An electrodynamic theory for the emission of fluorescence polarization by solutions,” Opt. Commun. 64, 516-522 (1987).
    [CrossRef]
  7. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).
  8. A. L. McClellan, Table of Experimental Dipole Moments (W. H. Freeman, 1963).
  9. A. J. Clark, “The Mode of Action of Drugs on Cells” (Edward Arnold, 1933).
  10. “Beacon Applications Guide,” PanVera Corporation, Madison, Wisconsin, USA.
  11. S.-H. Park and R. T. Raines, “Green fluorescent protein as a signal for protein-protein interactions,” Protein Sci. 6, 2344-2349 (1997).
    [CrossRef]
  12. “RCSB Protein Data Bank,” www.rcsb.org.
  13. “Server and database for dipole moments of proteins,” http://bip.weizmann.ac.il/dipol.
  14. T. Förster, “Intermolecular energy migration and fluorescence,” Ann. Phys. 2, 55-75 (1948).
    [CrossRef]
  15. J. Lippincott-Schwartz and G. H. Patterson, “Development and use of fluorescent protein markers in living cells,” Science 300, 87-91 (2003).
    [CrossRef] [PubMed]
  16. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
    [CrossRef] [PubMed]
  17. H. Lorenz, D. W. Hally, and J. Lippincott-Schwartz, “Fluorescence protease protection of GFP chimeras to reveal protein topology and subcellular localization,” Nature Meth. 3, 205-210 (2006).
    [CrossRef]
  18. G. Chirico, M. Collini, K. Tóth, N. Brun, and J. Langowski, “Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy,” Eur. Biophys. J. 29, 597-606 (2001).
    [CrossRef] [PubMed]
  19. S. Zorilla, G. Rivas, A. U. Acuna, and M. P. Lillo, “Protein self-association in crowded protein solutions: a time-resolved fluorescence polarization study,” Protein Sci. 13, 2960-2969 (2004).
    [CrossRef]
  20. D. W. Piston and M. A. Rizzo, “FRET by fluorescence polarization microscopy,” Methods Cell Biol. 85, 415-430 (2008).
    [CrossRef]
  21. T. L. Mann and U. J. Krull, “Fluorescence polarization spectroscopy in protein analysis,” Analyst (Amsterdam) 128, 313-317 (2003).

2008

D. W. Piston and M. A. Rizzo, “FRET by fluorescence polarization microscopy,” Methods Cell Biol. 85, 415-430 (2008).
[CrossRef]

2006

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

H. Lorenz, D. W. Hally, and J. Lippincott-Schwartz, “Fluorescence protease protection of GFP chimeras to reveal protein topology and subcellular localization,” Nature Meth. 3, 205-210 (2006).
[CrossRef]

2004

S. Zorilla, G. Rivas, A. U. Acuna, and M. P. Lillo, “Protein self-association in crowded protein solutions: a time-resolved fluorescence polarization study,” Protein Sci. 13, 2960-2969 (2004).
[CrossRef]

2003

T. L. Mann and U. J. Krull, “Fluorescence polarization spectroscopy in protein analysis,” Analyst (Amsterdam) 128, 313-317 (2003).

J. Lippincott-Schwartz and G. H. Patterson, “Development and use of fluorescent protein markers in living cells,” Science 300, 87-91 (2003).
[CrossRef] [PubMed]

2001

G. Chirico, M. Collini, K. Tóth, N. Brun, and J. Langowski, “Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy,” Eur. Biophys. J. 29, 597-606 (2001).
[CrossRef] [PubMed]

1997

S.-H. Park and R. T. Raines, “Green fluorescent protein as a signal for protein-protein interactions,” Protein Sci. 6, 2344-2349 (1997).
[CrossRef]

1987

E. Collett, “An electrodynamic theory for the emission of fluorescence polarization by solutions,” Opt. Commun. 64, 516-522 (1987).
[CrossRef]

1952

G. Weber, “Polarization of the fluorescence of macromolecules: 1. Theory and experimental method,” Biochemistry J. 51, 145-155 (1952).

1948

T. Förster, “Intermolecular energy migration and fluorescence,” Ann. Phys. 2, 55-75 (1948).
[CrossRef]

1926

F. Perrin, “Polarisation de la lumière de fluorescence. Vie moyenne des molécules dans l'etat excité,” J. Phys. Radium 7, 390-401 (1926).
[CrossRef]

1920

F. Weigert, “Über polarisiertes fluoreszenzlicht,” Verh. Dtsch. Phys. Ges. 23, 100-102 (1920).

1852

G. G. Stokes, “On the change of refrangibility of light,” Proc. Royal Soc. London vi, 195-208 (1852).

Acuna, A. U.

S. Zorilla, G. Rivas, A. U. Acuna, and M. P. Lillo, “Protein self-association in crowded protein solutions: a time-resolved fluorescence polarization study,” Protein Sci. 13, 2960-2969 (2004).
[CrossRef]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Brun, N.

G. Chirico, M. Collini, K. Tóth, N. Brun, and J. Langowski, “Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy,” Eur. Biophys. J. 29, 597-606 (2001).
[CrossRef] [PubMed]

Chirico, G.

G. Chirico, M. Collini, K. Tóth, N. Brun, and J. Langowski, “Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy,” Eur. Biophys. J. 29, 597-606 (2001).
[CrossRef] [PubMed]

Clark, A. J.

A. J. Clark, “The Mode of Action of Drugs on Cells” (Edward Arnold, 1933).

Collett, E.

E. Collett, “An electrodynamic theory for the emission of fluorescence polarization by solutions,” Opt. Commun. 64, 516-522 (1987).
[CrossRef]

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).

Collini, M.

G. Chirico, M. Collini, K. Tóth, N. Brun, and J. Langowski, “Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy,” Eur. Biophys. J. 29, 597-606 (2001).
[CrossRef] [PubMed]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Förster, T.

T. Förster, “Intermolecular energy migration and fluorescence,” Ann. Phys. 2, 55-75 (1948).
[CrossRef]

Hally, D. W.

H. Lorenz, D. W. Hally, and J. Lippincott-Schwartz, “Fluorescence protease protection of GFP chimeras to reveal protein topology and subcellular localization,” Nature Meth. 3, 205-210 (2006).
[CrossRef]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Krull, U. J.

T. L. Mann and U. J. Krull, “Fluorescence polarization spectroscopy in protein analysis,” Analyst (Amsterdam) 128, 313-317 (2003).

Lackowicz, J. R.

J. R. Lackowicz, Principles of Fluorescence Spectroscopy (Plenum, 1983).

Langowski, J.

G. Chirico, M. Collini, K. Tóth, N. Brun, and J. Langowski, “Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy,” Eur. Biophys. J. 29, 597-606 (2001).
[CrossRef] [PubMed]

Lillo, M. P.

S. Zorilla, G. Rivas, A. U. Acuna, and M. P. Lillo, “Protein self-association in crowded protein solutions: a time-resolved fluorescence polarization study,” Protein Sci. 13, 2960-2969 (2004).
[CrossRef]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

H. Lorenz, D. W. Hally, and J. Lippincott-Schwartz, “Fluorescence protease protection of GFP chimeras to reveal protein topology and subcellular localization,” Nature Meth. 3, 205-210 (2006).
[CrossRef]

J. Lippincott-Schwartz and G. H. Patterson, “Development and use of fluorescent protein markers in living cells,” Science 300, 87-91 (2003).
[CrossRef] [PubMed]

Lorenz, H.

H. Lorenz, D. W. Hally, and J. Lippincott-Schwartz, “Fluorescence protease protection of GFP chimeras to reveal protein topology and subcellular localization,” Nature Meth. 3, 205-210 (2006).
[CrossRef]

Mann, T. L.

T. L. Mann and U. J. Krull, “Fluorescence polarization spectroscopy in protein analysis,” Analyst (Amsterdam) 128, 313-317 (2003).

McClellan, A. L.

A. L. McClellan, Table of Experimental Dipole Moments (W. H. Freeman, 1963).

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Park, S.-H.

S.-H. Park and R. T. Raines, “Green fluorescent protein as a signal for protein-protein interactions,” Protein Sci. 6, 2344-2349 (1997).
[CrossRef]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

J. Lippincott-Schwartz and G. H. Patterson, “Development and use of fluorescent protein markers in living cells,” Science 300, 87-91 (2003).
[CrossRef] [PubMed]

Perrin, F.

F. Perrin, “Polarisation de la lumière de fluorescence. Vie moyenne des molécules dans l'etat excité,” J. Phys. Radium 7, 390-401 (1926).
[CrossRef]

Piston, D. W.

D. W. Piston and M. A. Rizzo, “FRET by fluorescence polarization microscopy,” Methods Cell Biol. 85, 415-430 (2008).
[CrossRef]

Raines, R. T.

S.-H. Park and R. T. Raines, “Green fluorescent protein as a signal for protein-protein interactions,” Protein Sci. 6, 2344-2349 (1997).
[CrossRef]

Rivas, G.

S. Zorilla, G. Rivas, A. U. Acuna, and M. P. Lillo, “Protein self-association in crowded protein solutions: a time-resolved fluorescence polarization study,” Protein Sci. 13, 2960-2969 (2004).
[CrossRef]

Rizzo, M. A.

D. W. Piston and M. A. Rizzo, “FRET by fluorescence polarization microscopy,” Methods Cell Biol. 85, 415-430 (2008).
[CrossRef]

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Stokes, G. G.

G. G. Stokes, “On the change of refrangibility of light,” Proc. Royal Soc. London vi, 195-208 (1852).

Tóth, K.

G. Chirico, M. Collini, K. Tóth, N. Brun, and J. Langowski, “Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy,” Eur. Biophys. J. 29, 597-606 (2001).
[CrossRef] [PubMed]

Weber, G.

G. Weber, “Polarization of the fluorescence of macromolecules: 1. Theory and experimental method,” Biochemistry J. 51, 145-155 (1952).

Weigert, F.

F. Weigert, “Über polarisiertes fluoreszenzlicht,” Verh. Dtsch. Phys. Ges. 23, 100-102 (1920).

Zorilla, S.

S. Zorilla, G. Rivas, A. U. Acuna, and M. P. Lillo, “Protein self-association in crowded protein solutions: a time-resolved fluorescence polarization study,” Protein Sci. 13, 2960-2969 (2004).
[CrossRef]

Analyst (Amsterdam)

T. L. Mann and U. J. Krull, “Fluorescence polarization spectroscopy in protein analysis,” Analyst (Amsterdam) 128, 313-317 (2003).

Ann. Phys.

T. Förster, “Intermolecular energy migration and fluorescence,” Ann. Phys. 2, 55-75 (1948).
[CrossRef]

Biochemistry J.

G. Weber, “Polarization of the fluorescence of macromolecules: 1. Theory and experimental method,” Biochemistry J. 51, 145-155 (1952).

Eur. Biophys. J.

G. Chirico, M. Collini, K. Tóth, N. Brun, and J. Langowski, “Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy,” Eur. Biophys. J. 29, 597-606 (2001).
[CrossRef] [PubMed]

J. Phys. Radium

F. Perrin, “Polarisation de la lumière de fluorescence. Vie moyenne des molécules dans l'etat excité,” J. Phys. Radium 7, 390-401 (1926).
[CrossRef]

Methods Cell Biol.

D. W. Piston and M. A. Rizzo, “FRET by fluorescence polarization microscopy,” Methods Cell Biol. 85, 415-430 (2008).
[CrossRef]

Nature Meth.

H. Lorenz, D. W. Hally, and J. Lippincott-Schwartz, “Fluorescence protease protection of GFP chimeras to reveal protein topology and subcellular localization,” Nature Meth. 3, 205-210 (2006).
[CrossRef]

Opt. Commun.

E. Collett, “An electrodynamic theory for the emission of fluorescence polarization by solutions,” Opt. Commun. 64, 516-522 (1987).
[CrossRef]

Proc. Royal Soc. London

G. G. Stokes, “On the change of refrangibility of light,” Proc. Royal Soc. London vi, 195-208 (1852).

Protein Sci.

S. Zorilla, G. Rivas, A. U. Acuna, and M. P. Lillo, “Protein self-association in crowded protein solutions: a time-resolved fluorescence polarization study,” Protein Sci. 13, 2960-2969 (2004).
[CrossRef]

S.-H. Park and R. T. Raines, “Green fluorescent protein as a signal for protein-protein interactions,” Protein Sci. 6, 2344-2349 (1997).
[CrossRef]

Science

J. Lippincott-Schwartz and G. H. Patterson, “Development and use of fluorescent protein markers in living cells,” Science 300, 87-91 (2003).
[CrossRef] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642-1645 (2006).
[CrossRef] [PubMed]

Verh. Dtsch. Phys. Ges.

F. Weigert, “Über polarisiertes fluoreszenzlicht,” Verh. Dtsch. Phys. Ges. 23, 100-102 (1920).

Other

J. R. Lackowicz, Principles of Fluorescence Spectroscopy (Plenum, 1983).

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).

A. L. McClellan, Table of Experimental Dipole Moments (W. H. Freeman, 1963).

A. J. Clark, “The Mode of Action of Drugs on Cells” (Edward Arnold, 1933).

“Beacon Applications Guide,” PanVera Corporation, Madison, Wisconsin, USA.

“RCSB Protein Data Bank,” www.rcsb.org.

“Server and database for dipole moments of proteins,” http://bip.weizmann.ac.il/dipol.

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Figures (7)

Fig. 1
Fig. 1

Plots using Eq. (5) of the degree of polarization (in mP) versus the dipole separation (in angstroms). Figure 1a is a plot for fluorescent perylene dissolved in ethanol. The respective dipole moments are 2.10 × 10 - 18 esu cm and 2.89 × 10 - 18 esu cm ; the temperature T is 293 K . Figures 1b, 1c describe two proteins with dipole moments of 350 D and 800 D with the abscissa plotted in terms of decreasing dipole separation (corresponding to increasing concentration). Figure 1c is plotted logarithmically. In particular, the final plot shows the familiar S-curve obtained in protein titrations (see Fig. 4).

Fig. 2
Fig. 2

Plot of the assay data for the binding of estrogen receptor DNA binding domain to ERE50ds-F. The numerical values of the data points were obtained from the figure shown on pg. 4-3 of the Beacon Applications Guide [10].

Fig. 3
Fig. 3

Binding curve derived from Fig. 2 and using Eq. (18) is shown. The limits on f B are seen to be 0.0 and 1.0, as expected. We note that at f M B = 1 / 2 , the data point corresponds to the dissociation constant K d = 23.0 × 10 8 M listed in Table 1.

Fig. 4
Fig. 4

Plots of f B for the GFP chimera; f B corresponds to f M B . The S-curve described by the open circles (∘) corresponds to the aqueous solution and has a dissociation constant of 1.1 × 10 - 8 M . The S-curve described by the solid circles (•) corresponds to the saline solution and has a dissociation constant of 4.2 × 10 8 M (courtesy of Protein Science).

Fig. 5
Fig. 5

Dipole–dipole interaction between a fluorescent molecule, a, and a host molecule, b, separated by a distance r a b showing the orientation polar angles θ a and θ b of the respective molecules. The azimuthal angles ϕ a and ϕ b are not shown.

Fig. 6
Fig. 6

Plot of the binding curve given by Eq. (65). When f B = 1 / 2 , we have L F = 1 and K d = 1 .

Fig. 7
Fig. 7

Change from a linear to a logarithmic abscissa not only allows the plotting over a large range of the free ligand concentration but also gives rise to a sigmoidal or S-curve. The logarithmic plot is called a Klotz plot. Klotz plots make the determination of the dissociation constant K d far easier to read than that obtained with the linear abscissa values shown in Fig. 6.

Tables (2)

Tables Icon

Table 1 Binding Parameters of Human Recombinant Estrogen Receptor (hER) to a Fluorescein-Labeled Estrogen Response Element a

Tables Icon

Table 2 Values of the Protein Separation for Different Concentrations Using Eq. (29) a

Equations (67)

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

1 P 1 3 = ( 1 P 0 1 3 ) [ 1 + ( R T η V ) τ ] ,
2 E ( r , t ) μ ε c 2 2 E ( r , t ) t 2 = ( 4 π μ c 2 ) j ( r , t ) t ,
S = S 0 S 1 S 2 S 3 = 2 15 ( ω n m 4 c 4 R 2 ) ( α 3 α 1 ) 2 I 0 1 + 13 42 μ a 2 μ b 2 k 2 T 2 r a b 6 1 14 μ a 2 μ b 2 k 2 T 2 r a b 6 0 0 .
S = S 0 S 1 S 2 S 3 = 2 15 ( ω n m 4 c 4 R 2 ) ( α 3 α 1 ) 2 I 0 1 0 0 0 .
P = | S 1 S 0 | = 1 14 μ a 2 μ b 2 k 2 T 2 r a b 6 1 + 13 42 μ a 2 μ b 2 k 2 T 2 r a b 6 ,
r a b = [ μ a 2 μ b 2 ( 3 13 P ) 42 P k 2 T 2 ] 1 6 .
μ a = 2.10 × 10 18 esu · cm , μ b = 2.89 × 10 18 esu · cm .
r a b = [ μ a 2 μ b 2 ( 3 13 P ) 42 P k 2 T 2 ] 1 6 = 4.35 × 10 8     cm = 4.35 Å .
L F + R F L F · R F = B ,
[ B ] [ R T ] = f B = [ L F ] [ K d + L F ] , 0 f B 1.
f B = P P m i n P m a x P m i n .
f B = P P m i n P m a x P m i n = [ L F ] [ K d + L F ] .
P m i n P P m a x .
0 P 3 / 13.
0 f B 1.
P = P m i n K d + P m a x L F K d + L F .
r a b = [ μ a 2 μ b 2 ( 3 13 X ) 42 k 2 T 2 X ] 1 6 ,
f M B = P P m i n P m a x P m i n .
f T B = 13 3 1 14 μ a 2 μ b 2 k 2 T 2 r a b 6 1 + 13 42 μ a 2 μ b 2 k 2 T 2 r a b 6 .
f B = P P m i n P m a x P m i n = ( 1 14 μ a 2 μ b 2 k 2 T 2 r a b 6 1 + 13 42 μ a 2 μ b 2 k 2 T 2 r a b 6 ) P m i n P m a x P m i n ,
r a b ( f B ) = ( μ a 2 μ b 2 42 k 2 T 2 ) 1 6 [ 3 13 P m i n 13 f B ( P m a x P m i n ) P m i n + f B ( P m a x P m i n ) ] 1 6 .
r a b ( f B = 1 ) = r a b ( m i n ) = ( μ a 2 μ b 2 42 k 2 T 2 ) 1 6 ( 3 13 P m a x P m a x ) 1 6 ,
r a b ( f B = 1 2 ) = r a b ( m i d ) = ( μ a 2 μ b 2 42 k 2 T 2 ) 1 6 [ 3 13 ( P m a x + P m i n 2 ) P m a x + P m i n 2 ] 1 6 ,
r a b ( f B = 0 ) = r a b ( m a x ) = ( μ a 2 μ b 2 42 k 2 T 2 ) 1 6 ( 3 13 P m i n P m i n ) 1 6 .
r a b ( f T B ) = ( 13 μ a 2 μ b 2 42 k 2 T 2 ) 1 6 ( 1 f T B f T B ) 1 6 .
r a b ( f T B = 1 2 ) = ( 13 μ a 2 μ b 2 42 k 2 T 2 ) 1 6 .
r a b ( f T B = 1 2 ) = ( 13 μ a 2 μ b 2 42 k 2 T 2 ) 1 6 = 134.0 Å .
r a b ( 0.99 P m a x ) = 0.382 ( μ a 2 μ b 2 k 2 T 2 ) 1 6 .
Labeled DNA (unbound) + DNA Binding Protein Labeled DNA (bound) · DNA Binding Protein
r a b ( L F ) = { μ a 2 μ b 2 [ 3 13 ( 1.46 × 10 9 + 0.156 L F ) 23.0 × 10 9 + L F ] 42 [ ( 1.46 × 10 9 + 0.156 L F ) 23.0 × 10 9 + L F ] k 2 T 2 } 1 6 ,
r a b ( f B = 1 ) = r a b ( m i n ) = ( μ a 2 μ b 2 42 k 2 T 2 ) 1 6 ( 3 13 P m a x P m a x ) 1 6 = 132.4 ± 0.4 Å ,
r a b ( f B = 1 2 ) = r a b ( m i d ) = ( μ a 2 μ b 2 42 k 2 T 2 ) 1 6 [ 3 13 ( P m a x + P m i n 2 ) P m a x + P m i n 2 ] 1 6 = 152.2 ± 0.5 Å ,
r a b ( f B = 0 ) = r a b ( m a x ) = ( μ a 2 μ b 2 42 k 2 T 2 ) 1 6 ( 3 13 P m i n P m i n ) 1 6 = 176.1 ± 0.6 Å .
r a b ( 0.99 P m a x ) = 0.382 ( μ a 2 μ b 2 k 2 T 2 ) 1 6 = 69.6 Å .
r a b ( L F , K d ) = ( 13 μ a 2 μ b 2 K d 42 L F k 2 T 2 ) 1 6 .
r a b ( L F , K d ) = ( 13 μ a 2 μ b 2 K d 42 L F k 2 T 2 ) 1 6 = 109.7 Å .
K d ( aqueous ) = 1.1 × 10 8 M r a b = 75.9 Å ,
K d ( saline ) = 4.2 × 10 8 M r a b = 95.0 Å .
2 E ( r , t ) μ ε c 2 2 E ( r , t ) t 2 = ( 4 π μ c 2 ) j ( r , t ) t ,
E = e c [ n × ( n × β ˙ ) R ] ,
E θ = ( e c 2 R ) [ x ¨ ( t ) cos θ z ¨ ( t ) sin θ ] , E ϕ = ( e c 2 R ) [ y ¨ ( t ) ] ,
x ( t ) x n m = x n m ( 0 ) e i ω n m t , y ( t ) y n m = y n m ( 0 ) e i ω n m t , z ( t ) z n m = z n m ( 0 ) e i ω n m t ,
E θ = ( ω n m 2 c 2 R ) ( p x n m cos θ p z n m sin θ ) , E ϕ = ( ω n m 2 c 2 R ) ( p y n m ) ,
E θ = ( ω n m 2 c 2 R ) ( p z ) , E ϕ = ( ω n m 2 c 2 R ) ( p y ) .
p = α ¯ ¯ · E ,
p x p y p x = α 1 0 0 0 α 2 0 0 0 α 3 E z E y E Z .
p x p y p x = A 1 α 1 0 0 0 α 2 0 0 0 α 3 A E z E y E Z ,
A = ( cos ψ     cos ϕ cos θ sin ϕ sin ψ cos ψ sin ϕ + cos θ cos ϕ sin ψ sin ψ sin θ sin ψ cos ϕ cos θ sin ϕ cos ψ sin ψ sin ϕ + cos θ cos ϕ cos ψ cos ψ sin θ sin θ sin ϕ sin θ     cos ϕ cos θ ) .
E θ = ( ω n m 2 c 2 R ) ( α 3 α 1 ) ( sin 2 θ     sin ϕ cos ϕ ) E x , E ϕ = ( ω n m 2 c 2 R ) ( α 3 α 1 ) ( sin θ cos θ sin ϕ ) E x .
S 0 = E x E x * + E y E y * , S 1 = E x E x * E y E y * , S 2 = E x E y * + E y E x * , S 3 = i ( E x E y * E y E x * ) ,
S 0 = E θ E θ * + E ϕ E ϕ * , S 1 = E θ E θ * E ϕ E ϕ * , S 2 = E θ E ϕ * + E ϕ E θ * , S 3 = i ( E θ E ϕ * E ϕ E θ * ) ,
E θ E θ * = ( ω n m 4 c 4 R 2 ) ( α 3 α 1 ) 2 sin 4 θ sin 2 ϕ cos 2 ϕ E x E x * , E ϕ E ϕ * = ( ω n m 4 c 4 R 2 ) ( α 3 α 1 ) 2 sin 2 θ cos 2 θ sin 2 ϕ E x E x * , E θ E ϕ * = ( ω n m 4 c 4 R 2 ) ( α 3 α 1 ) 2 sin 3 θ cos θ sin 2 ϕ cos ϕ E x E x * , E ϕ E θ * = ( ω n m 4 c 4 R 2 ) ( α 3 α 1 ) 2 sin 3 θ cos θ sin 2 ϕ cos ϕ E x E x * .
= a b exp ( U ( a , b ) k T ) d τ a d τ b a b exp ( U ( a , b ) k T ) d τ a d τ b .
U ( θ a , ϕ a , θ b , ϕ b ) = μ a μ b r a b 3 [ 2 cos θ a cos θ b sin θ a sin θ b cos ( ϕ a ϕ b ) ] ,
exp [ U ( a , b ) k T ] 1 U ( a , b ) k T + 1 2 U ( a , b ) 2 k 2 T 2 .
S 0 = ( ω n m 4 c 4 R 2 ) ( α 3 α 1 ) 2 I 0 ( 2 15 + 13 315 μ a 2 μ b 2 k 2 T 2 r a b 6 ) , S 1 = ( ω n m 4 c 4 R 2 ) ( α 3 α 1 ) 2 I 0 ( 1 105 μ a 2 μ b 2 k 2 T 2 r a b 6 ) , S 2 = 0 , S 3 = 0 ,
S = S 0 S 1 S 2 S 3 = 2 15 ( ω n m 4 c 4 R 2 ) ( α 3 α 1 ) 2 I 0 1 + 13 42 μ a 2 μ b 2 k 2 T 2 r a b 6 1 14 μ a 2 μ b 2 k 2 T 2 r a b 6 0 0 .
P = S 1 2 + S 2 2 + S 3 2 S 0 , 0 P 1.
S 0 2 S 1 2 + S 2 2 + S 3 2 .
P = | S 1 S 0 | = 1 14 μ a 2 μ b 2 k 2 T 2 r a b 6 1 + 13 42 μ a 2 μ b 2 k 2 T 2 r a b 6 .
L F + R F L F · R F = B .
[ L F ] [ R F ] [ B ] = K d ,
R F = R T B .
[ L F ] [ R T B ] [ B ] = K d .
[ B ] = [ R T ] [ L F ] [ K d + L F ] ,
[ B ] [ R T ] = f B = [ L F ] [ K d + L F ] .
f B ( 10 2 ) = 0.99 % , f B ( 10 2 ) = 99.1 % .

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