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A 1
calculating the fluorescence lifetime t Fl ¼ t rad ¼
21 . This equivalency, however,
is only true if no fluorescence quenching mechanism takes place. This is another
meaning of absorption spectroscopy for fluorescence spectroscopy. Equation (1)
also shows that fluorescence, i.e., spontaneous emission, becomes more important
at higher optical frequencies n or shorter wavelengths l. We will get back to this
relation while comparing different proteins (Sect. 3 ).
One limitation of the early formulation by Einstein is that it accounts for only
one wavelength l or one optical frequency n, respectively. Absorption as well as
fluorescence spectra are characterized by band widths in the range of tens of
nanometres (Fig. 2 ). This was solved in a careful reformulation of (1), (2) [ 23 ].
n 0
ln 10
e abs ð n Þ= n
fl :
A 21 ¼
n 3
c 2
The first term in parentheses takes the refractive index n 0 of the surrounding
medium into account. The second term in parentheses, a constant, scales the molar
decade extinction coefficient down to a molecular value (see Sect. 1.3 ). The
brackets in the third term denote integrations over the frequency of the respective
spectrum. The upper integral uses as argument the absorption spectrum e abs (n),
divided by the frequency n. The lower integral mainly consists of the fluorescence
Fig. 2 Absorption spectra of eGFP ( black ) and eYFP ( grey ) and fluorescence spectrum of the
latter ( black dotted ), normalized to the spectral area of eYFP. Taking the approximation of
equal oscillator strengths f 12 for eGFP and eYFP, e max (eGFP) can only be 0.57 e max (eYFP) as
the latter exhibits a distinctly smaller absorption spectrum. On the basis of
e max (eYFP) ¼
83.400 M 1
cm 1 ,
f 12 ¼ 0.48 is calculated according to (4). Application of (2) leads to
A 1
t rad ¼
7ns [ 20 ]. The dependence of A 21 on the spectral changes between GFPs and
YFPs was observed to be lower than 7% [ 21 ], i.e., the radiative lifetime of eGFP can be as short
as 4.4 ns which is very close to the value of [ 22 ]
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