Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 393
ble to measure the maximum values of ∆ ω due to the faster
motion. In these cases the observed value of ∆ ω increases up
to the highest measured frequency. The presence of a rapid
correlation time also results in complex behavior for the
modulated anisotropy (r ω ). Depending on the upper frequency
limit of the measurement the values of r ω may not
reach the value of r 0 .
11.4.4. Correlation Time Distributions
Anisotropy decays can also be analyzed in terms of distributions
of correlation times. 37–39 One approach is to
describe the correlation time spread in terms of a Gaussian,
Lorentzian, or other distribution. The Gaussian (G) and
Lorentzian (L) distributions are given by
p G(θ) 1
σ√2π
p L(θ) 1
π
(11.42)
(11.43)
In these expressions θ are the central values, σ the standard
deviation of the Gaussian, and Γ the full width at half maximum
of the Lorentzian.
Suppose the anisotropy decay is described by a
single modal distribution, with a single mean value (θ).
That part of the anisotropy that displays a correlation time
θ is given by
(11.44)
where p(θ) is the probability of a particular correlation time
θ. It is not possible to selectively observe the fraction of the
anisotropy that decays with θ. Hence, the observed
anisotropy decay is given by the integral equation
(11.45)
It is also possible to describe the anisotropy decay by a multimodal
correlation time distribution. In this case the amplitude
that decays with a correlation time θ is given by
r(t,θ) ∑ j
θ θ
exp [ 1 ( ) 2 σ 2 ]
Γ/2
(θ θ ) 2 (Γ/2) 2
r(t,θ) r 0 p(θ) exp(t/θ)
∞
r(t) r0 p(θ) exp(t/θ)dθ
0
r 0j p j(θ) exp(t/θ)
(11.46)
and the observed anisotropy decay is given by
r(t) ∑ j
∞
r0j (11.47)
In this formulation the distribution shape factors are normalized
so that the integrated probability of each mode of
the distribution is equal to unity. Equations 11.45 and 11.47
are properly normalized only if none of the probability
occurs below zero. 39 Depending on the values of σ, or
Γ, part of the probability for the Gaussian or Lorentzian distributions
(eqs. 11.42 and 11.43) can occur below zero,
even if is larger than zero. This component should be
normalized by the integrated area of the distribution function
above θ = 0. The correlation time distributions can also
be obtained using maximum entropy methods, typically
without using assumed shapes for the distribution functions.
37–38
θ,
θ
11.4.5. Associated Anisotropy Decays
0
p j(θ) exp(t/θ)dθ
Multi-exponential anisotropy decay can also occur for a
mixture of independently rotating fluorophores. Such
anisotropy decay can occur for a fluorophore when some of
the fluorophores are bound to protein and some are free in
solution. The anisotropy from the mixture is an intensity
weighting average of the contribution from the probe in
each environment:
r(t) r 1(t)f 1(t) r 2(t)f 2(t)
(11.48)
where r 1 (t) and r 2 (t) are the anisotropy decays in each environment.
The fractional time-dependent intensities for each
fluorophore are determined by the decay times in each environment.
For single exponential decays these fractional
contributions are given by
αi exp(t/τi) fi(t)
α1 exp(t/τ1) α2 exp(t/τ2) (11.49)
Such systems can yield unusual anisotropy decays that
show minima at short times and increase at long times. 40–43
Associated anisotropy decays are described in more detail
in Chapter 12.