Principles of Fluorescence Spectroscopy

420 ADVANCED ANISOTROPY CONCEPTS
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Figure 12.12. Prolate ellipsoid of revolution. A and E are the
directions of the absorption and emission transition moments, respectively.
Revised and reprinted with permission from [4]. Copyright ©
1985, Academic Press, Inc.
tions of the two rotational rates (D || and D ⊥ ). The amplitudes
of the anisotropy decay depend on the orientation of the
transition moments. If one of the transition moments is
directed along any of the symmetry axes of the ellipsoid,
then the anisotropy decay becomes a double exponential.
Hence, one can expect double-exponential anisotropy
decays for molecules like DPH or perylene, where the longwavelength
absorption and emission moments are directed
along the long axis of the molecules.
12.4. ELLIPSOIDS OF REVOLUTION
We now describe the theory for the anisotropy decays of
ellipsoids of revolution. It is important to remember that
this theory assumes the ellipsoid to be rigid. In the case of
a labeled protein this assumption implies no independent
motion of the fluorophore within the protein. For prolate or
oblate ellipsoids the anisotropy decay is expected to display
three correlation times:
r(t) r 1 exp(t/θ 1) r 2 exp(t/θ 2)
r 3 exp(t/θ 3 )
(12.11)
The amplitudes decaying with each correlation time depend
on the angles of the absorption (β A ) and emission (β E )
dipoles with the symmetry axis. Using the notation defined
in Figure 12.12 these anisotropy amplitudes are
r 1 0.3 sin 2β A sin 2β E cos ξ
r 2 0.3 sin 2 β A sin 2 β E cos 2ξ
r 3 0.1(3 cos 2 β A 1)(3 cos 2 β E 1)
(12.12)
(12.13)
(12.14)
Occasionally one finds alternative forms 37–38 for r 1 and r 2 :
r 1 1.2 sin β A cos β A sin β E cos β E cos ξ
r 2 0.3 sin 2 β A sin 2 β E cos 2ξ
The fundamental anisotropy is given as usual by
r 0 r 1 r 2 r 3 0.2(3 cos 2 β 1)
(12.15)
(12.16)
(12.17)
where β is the angle between the absorption and emission
transition moments. The three correlation times are determined
by the two different rotational rates:
θ 1 (D || 5D ) 1
θ 2 (4D || 2D ) 1
θ 3 (6D ) 1
(12.18)
(12.19)
(12.20)
Depending upon the shape of the ellipsoid of rotation, and
the orientation of the transition moments, a variety of complex
anisotropy decays can be predicted. 4,37
In the previous chapter we saw that one can calculate
these rotational correlation times for a spherical molecule
(θ S ) based on its volume and the solution viscosity. Similar