Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 907
Table 12.3. Associated Anisotropy Decay
t (ns) f 1 (t) f 2 (t) r 1 (t) r 2 (t) r(t)
0 0.5 0.5 0.3 0.3 0.30
1 0.45 0.55 0.0 0.29 0.16
5 0.25 0.75 0.0 0.27 0.20
anisotropy values for the non-associated decay can be
calculated using
r(t) r 00.5 exp(t/0.05) 0.5 exp(t/40)
(12.51)
For t = 0, 1, and 5 ns these values are 0.30, 0.146,
and 0.132, respectively.
The presence of an associated anisotropy decay can
be seen from the increase in anisotropy at 5 ns as compared
to 1 ns. For the non-associated decay the
anisotropy decreases monotonically with time.
A12.2. For a non-associative model the anisotropy decay is
given by
r(t) r 0g 1 exp(t/θ 1 ) g 2 exp(t/θ 2 )
(12.52)
where subscripts 1 and 2 refer to components in the
decay, not the location of the fluorophore. From Figure
12.5 the time-zero anisotropy appears to be about
0.32. Using the parameter values in this figure,
r(t) 0.320.7 exp(t/0.30) 0.3 exp(t/685)
(12.53)
A plot of r(t) would show a rapid decrease to 30% of
the time-zero value followed by a long tail where the
anisotropy does not decay during the lifetime of the
fluorophore.
A12.3. The anisotropy can be calculated using eqs. 10.6 and
10.22. The anisotropy from the three transitions can
be calculated using β = 0E and "120E. Hence, r =
0.33(0.40) + 0.33(–0.05) + 0.33(–0.05) = 0.10.
A12.4. The apparent r(0) values of melittin are near 0.16,
which is considerably less than r 0 = 0.26. This indicates
that the tryptophan residue in melittin displays
fast motions that are not resolved with the available
range of lifetimes (0.6 to 2.4 ns).
The apparent correlation times from melittin can be
calculated from the slopes in Figure 12.41. For example,
in the absence of NaCl the slope is near 5.8 x 10 9 ,
which is equal to (r(0)θ) –1 . Hence the apparent correlation
time is 1.08 ns.
CHAPTER 13
A13.1. The D–A distance can be calculated using eq. 13.12.
The transfer efficiency is 90%. Hence the D–A distance
is r = (0.11) 1/6 R 0 = 0.69R 0 = 17.9 Å. The donor
lifetime in the D–A pair can be calculated from eq.
13.14, which can be rearranged to τ DA = (1 – E)τ D =
0.68 ns.
A13.2. The equations relating the donor intensity to the transfer
efficiency can be derived by recalling the expressions
for relative quantum yields and lifetimes. The
relative intensities and lifetimes are given by
1
FDA
, τDA
ΓD knr kT ΓD knr kT (13.34)
(13.35)
where Γ D is the emission rate of the donor and k nr is
the non-radiative decay rate. The ratio of intensities is
given by
F DA
F D
Hence
1
FD , τD
ΓD knr ΓD knr Γ D
Γ D
ΓD knr
ΓD knr kT 1 F DA
F D
(13.36)
(13.37)
One can derive a similar expression for the transfer
efficiency E based on lifetime using the right-hand
side of eqs. 13.34 and 13.35.
It should be noted that k T was assumed to be a single
value, which is equivalent to assuming a single
distance. We also assumed that the donor population
was homogeneous, so that each donor had a nearby
acceptor, that is, labeling by acceptor is 100%.
A13.3. The excitation spectra reveal the efficiency of energy
transfer by showing the extent to which the excitation
of the naphthyl donor at 290 nm results in dansyl
k T
τ 1
D k T
τ1
D
τ 1
D k T
E