Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 103
rophore. The rotational correlation times are determined by
the size, shape, and flexibility of the macromolecules. Both
the decay times and the rotational correlation times are
often on the nanosecond timescale. These conditions result
in anisotropies that are sensitive to the size of the protein
and its interactions with other macromolecules.
4.2.1. Resolution of Multi-Exponential Decays
Is Difficult
Why is so much attention given to data analysis and obtaining
high signal-to-noise in the time-resolved data? The need
for high signal-to-noise is due to the inherent difficulty in
recovering the amplitudes and lifetimes for a multi-exponential
process. This difficulty was well known to mathematicians,
and was pointed out to fluorescence spectroscopists
when time-resolved measurements were first being
applied to biochemical systems. 3 This paper defined a
method for analyzing time-resolved fluorescence data that
is still in use today. This paper illustrated how apparently
different multi-exponential decays can yield similar I(t) values.
Consider the following two double exponential decays:
I 1(t) 7500 exp(t/5.5) 2500 exp( t/8.0)
I 2(t) 2500 exp(t/4.5) 7500 exp(t/6.7)
(4.12)
(4.13)
The pre-exponential factor sum of 10,000 corresponds
to 10,000 photons in the highest intensity channel, which is
typical of data for time-correlated single-photon counting
(TCSPC). From examination of these equations one would
think that the intensity decays would be distinct. However,
a plot of the intensity decays on a linear scale shows that
they are indistinguishable at all times (Figure 4.6). On a
logarithmic scale one notices some minor differences at
30–50 ns. However, at 50 ns there are only about 3 photons
per channel with a 1-ns width. The difference between the
two decays at long times is just 1–2 photons. If one adds the
Poisson noise, which is unavoidable in photon-counting
data, the differences between the curves is seven-fold less
than the uncertainties due to the Poisson noise. 4 This illustrates
that it is difficult to distinguish between some multiexponential
functions, and that it is difficult to recover the
actual values of α i and τ i for a multi-exponential decay. A
similar result can be obtained from simulations of the frequency-domain
data. The simulated frequency responses
are visually indistinguishable for these two decay laws.
Figure 4.6. Comparison of two intensity decays: on a linear (left) and
logarithmic scale (right). The error bars represent Poisson noise on the
photon counts. The decay functions were described in [3].
Why is it difficult to resolve multi-exponential decays?
In I 1 (t) and I 2 (t) the lifetimes and amplitudes are different
for each decay law. In fact, this is the problem. For a multiexponential
decay one can vary the lifetime to compensate
for the amplitude, or vice versa, and obtain similar intensity
decays with different values of α i and τ i . In mathematical
terms the values of α i and τ i are said to be correlated. The
problem of correlated parameters is well known within the
framework of general least-squares fitting. 5–7 The unfortunate
result is that the ability to determine the precise values
of α i and τ i is greatly hindered by parameter correlation.
There is no way to avoid this problem, except by careful
experimentation and conservative interpretation of data.
4.3. TIME-CORRELATED SINGLE-PHOTON
COUNTING
At present most of the time-domain measurements are performed
using time-correlated single-photon counting, but
other methods can be used when rapid measurements are
needed. Many publications on TCSPC have appeared. 4,8–13
One book is completely devoted to TCSPC and provides
numerous valuable details. 8 Rather than present a history of
the method, we will start by describing current state-of-theart
instrumentation. These instruments use high repetition
rate mode-locked picosecond (ps) or femtosecond (fs) laser
light sources, and high-speed microchannel plate (MCP)