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Fig. 4 Experimental isotope effect on the basis of the energy-gap law. (a) The longer component

of the fluorescence decay in Gold fluorescent Protein (GdFP) [
35
] prolongs from t
Fl

¼
2.7 ns to

t
Fl

¼
4.2 ns upon H
2
O/D
2
O exchange. The decay faster component is, probably, a result of the

structural heterogeneity of GdFP. (b) The prolongation can be explained by the energy-gap law:

D

n
N
H
~ 3,300 cm
1
) but

E
of GdFP roughly corresponds to

D

v

¼

5 quanta of an N-H vibration (

n
N
D
~ 2,300 cm
1
). Calculations on the model of Fig.
3a,

b
and (8) (
inset
; wavefunctions are only half depicted) yield a reduction of
k
IC
by a factor 400 upon

isotope exchange which is then too slow to compete with
k
rad

to

D

v

¼

7 quanta of an N-D vibration (

structural analyses that the more planar a chromophore is in the crystallographic

structure the longer is t
Fl
.

One-dimensional potential curves as depicted in Fig.
3
are only a rough approxi-

mation of the real circumstances of complex potential energy surfaces (PES). Even

the chromophore itself exhibits a conformational space with more than 40 degrees

of freedom. Displacements along some combinations of the underlying normal

coordinates
q
i
lead to close approaches of the
S
0
and
S
1
PES, i.e.,

E
is small and

efficient IC can occur. Even crossings of the different PES can exist in so-called

conical intersections (CIs). However, isotope effects are unexpected for

D

E
~0,

and only quantum-mechanical treatments can give evidence for the corresponding

radiationless transition from
S
1
to
S
0
[
37
]. It should be emphasized that only the

knowledge of the coordinates
q
i
, i.e., geometric changes of the chromophore after

excitation, gives a basis for the suppression of these movements and, thus, the

fundament for a rational design of FPs with prolonged t
Fl
.

D

1.5 Other Mechanisms Which Compete with Fluorescence

There are other mechanisms which can lead to a reduction of

t
Fl
. They are

accounted for in (9) as additional rate constants.

d
S
½

d
t
¼

ð

A
21
þ

k
IC
þ

k
Iso
þ

k
PET
þ

k
FRET
þ

Þ

S½

1

t
Fl
¼

!

A
21
þ

k
IC
þ

k
Iso
þ

k
PET
þ

k
FRET
þ:

(9)

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