Principles of Fluorescence Spectroscopy

910 ANSWERS TO PROBLEMS
dx rA A ( r0 dPA (13.52)
Hence the maximum and minimum values of κ 2 are
0.19 and 2.62 (eqs. 13.18 and 13.19). According to
eqs. 13.23 and 13.24, the D–A distance can range
from 0.81R 0 to 1.26R 0 , or from 20.3 to 31.5 Å.
A13.9. If f A = 1.0 then the transfer efficiency is given by eq.
13.13:
(13.53)
and the D–A distance is thus equal to R 0 . If f A = 0.5 the
transfer efficiency is given by eq. 13.17:
E 1
) 1/2
0.71
E 1 0.5
0.5
1.0
0.5 1.0(0.5)
1.0(0.5)
1.0
(13.54)
If f A = 0.5 then the transfer efficiency for the actual
D–A pair is 100%, and thus the D–A distance is less
than 0.5R 0. The presence of acceptor underlabeling
results in a higher intensity for the presumed D–A pair
and an overestimation of the true D–A distance.
A13.10. Equation 13.25 can be easily derived by writing
expression for the acceptor intensity. In the absence
(F A) and presence of donor (F AD) the intensities are
given by
F A(λ em
A ) ε A(λ ex
D )C A(λ em
A )
F AD(λ em
A ) ε A(λ ex
D ) Eε D(λ ex
D )C A(λ em
A )
(13.55)
(13.56)
where excitation is at λ D , intensities are measured at
λ A, and E is the transfer efficiency. C A(λ A em) is a constant
relating the intensity at λ A to the acceptor concentration.
Dividing 13.55 by 13.56, followed by
rearrangement, yields eq. 13.25.
If the extent of donor labeling is less than 1.0, then
the acceptor intensities are given by
F A(λ em
A ) ε A(λ ex
D )C A(λ em
A )
(13.57)
F A(λ em
A ) ε A(λ ex
D ) f D E ε D(λ ex
D )C A(λ em
A )
(13.58)
where f D is the fractional labeling with the donor.
These expressions can be understood by recognizing
that the directly excited acceptor intensity is independent
of f D , but the acceptor intensity due to energy
transfer depends on f D . Rearrangement of eqs. 13.57
and 13.58 yields 13.25.
A13.11. Let C A (λ A ) and C D (λ A ) be the constants relating the
intensities at λ A to the acceptor and donor concentrations,
respectively, when both are excited at λ D . Since
the donor is assumed to emit at λ A , eqs. 13.55 and
13.56 become
F A(λ A) ε A(λ D)C A(λ A)
F AD(λ A) ε A(λ D) E ε D(λ D)C A(λ A)
ε D(λ D)C DA(λ A)
(13.59)
(13.60)
In eq. 13.58 we considered the contribution of the
donor in the D–A pair to the intensity at λ A . In general
C DA (λ A ) will be smaller than C D (λ A ) due to FRET
quenching of the donor. However, C D (λ A ) can be
measured with the donor-alone sample. C DA (λ A ) can
be estimated using the shape of the donor emission to
estimate the donor contribution at λ A in the doubly
labeled sample. Eqs. 13.59 and 13.60 can be
rearranged to
FAD(λA) FD(λD) 1 E εD(λD) εA(λD) εD(λD)CDA(λA) εA(λD)CA(λA) (13.61)
The transfer efficiency as seen from the acceptor
emission is given by eq. 13.25, which assumes that the
donor does not emit at the acceptor wavelength.
Hence the acceptor emission increases the apparent
efficiency to
E app E C DA(λ A)
C A(λ A)
(13.62)
and would thus be larger than the actual efficiency. If
the donor does not contribute at λ A, then C DA(λ A) = 0