Fluorescence Lifetime of Fluorescent Proteins - Fluorescent Proteins: From Understanding to Design - page 73

as in (2), provided by the second term in (4). Thus, this formula can be applied to
almost any dye, but caution is advised when used for FPs.
A speciality of FPs is that e abs (n) is related to 1 mole proteins but not to 1 mole of
chromophoric units. The value neglects, on the one hand, that not every protein has a
fully completed chromophore, i.e., incomplete chromophore maturation [ 31 ]. On the
other hand, equilibria between different chromophore forms, e.g. the neutral and the
anionic chromophore forms in wt-GFP, are also neglected. Mutations in FPs which do
not alter the chromophore mainly influence these two parameters. Hence, a correction
factor x might be introduced: x should denote the fraction of proteins of a certain
mutant, which effectively contribute to the S 0 !

S 1 transition under investigation (5).

ð

4 m e e 0 c
e 2

ln 10
x

f 12 ¼ e abs ðÞ dn : (5)

N A

The meaning of (4) is that, as f 12 and both terms in the parentheses are constant,
the integral over the frequency is also constant. In other words, the area under the
absorption spectrum is constant for a specific transition of a certain chromophore
form. A higher e max in an absorption spectrum results in a reduced width of the
absorption band. In terms of quantum mechanics, the shape of the excitation as well
as the emission spectrum is modulated by the Franck-Condon factors f FC (see also
Sect. 1.4 ). However, the natural lifetime t rad of the S 1 state is virtually constant if
the spectral changes of the same transition upon mutation are minor. This relation
was experimentally verified in a series of GFP and YFP variants; it turned out that
the whole variation of A 21 is less than 10% [ 21 ]. It is worth to emphasize that an
accurate calculation of the extinction coefficients or the oscillator strength is
absolutely necessary for a correct and unbiased determination of t rad (2).

1.4 Relevant Radiationless Processes

The decay of the upper state of a TLS like in Fig. 1 obeys first-order kinetics (6)
with A 21 as the decay rate constant. Sometimes, k rad is used instead of A 21 .

d S ½
d t ¼

A 21 S½! S½ð t Þ¼ S½ exp ð A 21 t Þ: (6)

The purely radiative lifetime, which could be measured if no fluorescence
quenching took place, is t rad ¼

A 1

21 . Fluorescence lifetime measurements, however,
always lead to t fl , which is shorter than t rad . A faster decay of the excited state S 1 is
the result of additional decay channels from the excited state S 1 to the electronic
ground state S 0 . In the kinetic description, these exclusively non-radiative channels
are accounted for by introducing additional rate constants. They are subsumed by
k IC , the rate constant for internal conversion (7).

d S ½
d t ¼

1
t Fl ¼

ð A 21 þ k IC Þ S½! A 21 þ k IC : (7)