Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 143
acceptor, which is given by eq. 4.31. When using the results
of a multi-exponential analysis, the transfer efficiency
should be calculated using values, since these are proportional
to the steady-state intensity.
4.11.2. Lifetime Distributions
There are many situations where one does not expect a limited
number of discrete decay times, but rather a distribution
of decay times. Such behavior may be expected for a
fluorophore in a mixture of solvents, so that a range of environments
exists. One can imagine a fluorophore being surrounded
by one, two, three, or more polar molecules, each
resulting in a different intensity decay. Another possibility
is a protein with many tryptophan residues, so that it is not
practical to consider individual decay times.
In such cases the intensity decays are typically analyzed
in terms of a lifetime distribution. In this case the α i
values are replaced by distribution functions α(τ). The component
with each individual τ value is given by
(4.33)
However, one cannot observe these individual components
with lifetime τ, but only the entire decay. The total decay
law is the sum of the individual decays weighted by the
amplitudes:
(4.34)
where Iα(τ)dτ = 1.0.
Lifetime distributions are usually used without a theoretical
basis for the α(τ) distribution. One typically uses
arbitrarily selected Gaussian (G) and Lorentzian (L) lifetime
distributions. For these functions the α(τ) values are
α G(τ)
α L(τ) 1
π
I(τ,t) α(τ)e t/τ
∞
I(t) α (τ)e
τ0
t/τdτ 1
σ√2π exp{1
τ τ ( ) 2 σ 2}
Γ/2
(τ τ ) 2 (Γ/2) 2
(4.35)
(4.36)
where (τ) is the central value of the distribution, σ the standard
deviation of the Gaussian, and Γ the full width at half
maximum (FWHM) for the Lorentzian. For a Gaussian the
full width at half maximum is given by 2.345σ. For ease of
interpretation we prefer to describe both distributions by the
full width at half maxima. An alternative approach would
be to use α(τ) distributions that are not described by any
particular function. This approach may be superior in that it
makes no assumptions about the shape of the distribution.
However, the use of functional forms for α(τ) minimizes
the number of floating parameters in the fitting algorithms.
Without an assumed function form it may be necessary to
place restraints on the adjacent values of α(τ).
By analogy with the multi-exponential model, it is possible
that α(τ) is multimodal. Then
α(τ) ∑ i
(4.37)
where i refers to the ith component of the distribution centered
at α i , and g i represents the amplitude of this component.
The g i values are amplitude factors and α i 0(τ) the
shape factors describing the distribution. If part of the distribution
exists below τ = 0, then the α i (τ) values need additional
normalization. For any distribution, including those
cut off at the origin, the amplitude associated with the ith
mode of the distribution is given by
α i
(4.38)
The fractional contribution of the ith component to the total
emission is given by
f i
g i α 0 i (τ) ∑ i
∞
0 α i(τ) dτ
∞
0 ∑ i
∞
0 αi(τ)τdτ ∞
0 ∑ i
α i(τ)dτ
α i(τ)τdτ
α i(τ)
(4.39)
In the use of lifetime distributions each decay time component
is associated with three variables, α i , f i and the half
width (σ or Γ). Consequently, one can fit a complex decay
with fewer exponential components. For instance, data that
can be fit to three discrete decay times can typically be fit
to a bimodal distribution model. In general, it is not possible
to distinguish between the discrete multi-exponential
model (eq. 4.27) or the lifetime distribution model (eq.
4.34), so the model selection must be based on one's knowledge
of the system. 197–199