Fluorescence Lifetime of Fluorescent Proteins - Fluorescent Proteins: From Understanding to Design - page 75

assume an equal force constant k for both states. The likelihood of the crossing from
the electronic S 1 state to the electronic ground state S 0 , which is then, instanta-
neously, highly vibrationally excited, depends on the overlap of the vibrational
wavefunctions of both states. This overlap is characterized by the Franck-Condon
factor f FC [(8), see also Sect. 1.3 ] and depends in the case of radiationless
transitions on several factors: (1) the difference of the vibrational quantum
numbers v between the populated vibrational states of S 1 and S 0 ,(2)theforce

constant k of the vibration, (3) the energy of the vibration hc

D

n and (4) on the

geometric displacement D q i of both potential curves [ 32 - 34 ].

2

v S 0

exp ðÞ g k qðÞ D

f FC 2 ¼ ; g ¼ : (8)

v S 0 !

2 hc n

In (8) and Fig. 3 , it is assumed that the radiationless transition occurs from the
vibrational ground state in the first electronically excited state S 1 , i.e., ( v S 1 ¼

0), to

some vibrationally excited state v S0 in the electronic ground state S 0 .
Any valence vibrations of hydrogen-atoms with n ~ 3,000 cm 1 exhibit the
strongest influence whenever they are coupled to the chromophore and undergo
geometric changes upon excitation. Only 5-6 quanta of suchlike vibrations, i.e.,
D

E between S 1 and S 0 in FPs.
The effect becomes even faster if there is a larger geometric change

v ¼ 5-6, are required to surpass the energy gap D

q i along
the considered coordinate q i between both electronic states. q i is called a normal
coordinate but could correspond e.g. to a bond length. It was experimentally found
and reproduced by theoretical calculations that

D

the rate constant k IC roughly

exponentially decreases with increasing E [ 34 ]. The contribution of this so-called

energy-gap law can be detected by a distinct change of

D

t Fl upon H/D-isotope

v is increased by a factor of 2 1/2 upon deuteration and, concomitantly,
k IC is even smaller.
Although the importance of the energy-gap law is known for molecules, which
exhibit fluorescence in the red region of the visible spectrum, little is found out
about its importance in FPs. Figure 4 shows the only example so far (to our
knowledge), where deuteration of the Gold Fluorescent Protein (GdFP) leads to a
1.5

exchange: D

longer t Fl [ 35 ]. However, the exact mechanism is still unknown. We propose
that GdFP can be used to map the water exchange of cells by fluorescence lifetime
microscopy once the incorporation of non-canonical amino acids into gene products
of higher organisms will be realized.
Other, well-described decay mechanisms for different FP chromophores and
their synthetic analogues are one-bond rotations [ 1 , 36 , 37 ]. Upon rotation, the
energy of the ground state is raised whereas the energy of S 1 becomes reduced thus
diminishing the energy gap D E . On the one hand, as the protein surrounding of the

protein matrix is rather rigid, only small displacements of the chromophore from
the adopted ground state conformation are possible and low energy-gap configura-
tions in S 1 are hardly accessible. On the other hand, most chromophore surround-
ings fit better to a twisted chromophore [ 38 ]. It appears on the basis of multiple