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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)

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