Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 141
Table 4.7. Multi-Exponential Analysis of the Three-Component Mixture of Indole, 2-Aminopurine, and Anthranilic Acid
Pre-exponential Fractional χ R 2, number
Observa- Lifetimes (ns) factors intensities a of exponents
tion wavelength
(nm) τ 1 τ 2 τ 3 α 1 α 2 α 3 f 1 f 2 f 3 3 2 1
360 4.79 7.51 11.43 0.314 0.004 0.682 0.161 0.003 0.836 1.10 1.10 17.67
(0.10) b (1.66) (0.02) (0.001) (0.001)
380 4.29 8.37 11.53 0.155 0.622 0.223 0.079 0.617 0.304 0.93 1.22 26.45
(0.33) (0.05) (0.02) (0.001) (0.001)
400 4.99 9.50 13.48 0.180 0.722 0.098 0.099 0.755 0.146 0.96 0.97 7.88
(0.16) (0.13) (0.25) (0.001) (0.001)
420 4.32 8.54 11.68 0.072 0.658 0.270 0.034 0.618 0.348 0.93 0.34 4.97
(0.47) (0.25) (0.09) (0.001) (0.001)
440 1.70 7.94 11.07 0.037 0.580 0.383 0.007 0.517 0.476 1.02 1.04 4.14
(0.61) (0.24) (0.06) (0.001) (0.001)
a fi = α i τ i /Σα j τ j .
b Asymptotic standard errors.
One may question why there are two tests for goodness
of fit: based on χ R 2 itself and based on the F statistic. The
values of χ R 2 are useful when the experimental errors can be
accurately estimated, which is usually the case with TCSPC
data. In this case the value of χ R 2 provides a test of both the
agreement of the measured and calculated N(t k ) values, and
whether the only source of noise is Poisson photon statistics.
In contrast to χ R 2, the F statistic can be used when the
experimental uncertainties (σ k values) are not precisely
known. This is usually the case with stroboscopic, gated
detection, and streak camera measurements, in which photon
counting is not used. This situation also occurs in frequency-domain
fluorometry, where the uncertainties in the
phase and modulation values can only be estimated. The
calculated values of χ R 2 can be very different from unity
even for a good fit, because the σ k 2 values may not be equal
to the values of [N(t k ) – N c (t k )] 2 . This is not a problem as
long as the relative values of χ R 2 are known. In these cases
one uses the F statistic, or relative decrease in χ R 2, to determine
the goodness of fit.
For closely spaced lifetimes, the ASEs will greatly
underestimate the uncertainties in the parameters. This
underestimation of errors is also illustrated in Table 4.7,
which lists the analysis of the three-component mixture
when measured at various emission wavelengths. It is clear
from these analyses that the recovered lifetimes differ by
amounts considerably larger than the asymptotic standard
errors. This is particularly true for the fractional intensities,
for which the asymptotic standard errors are "0.001. Simi-
lar results can be expected for any decay with closely
spaced lifetimes.
4.11. INTENSITY DECAY LAWS
So far we have considered methods to measure intensity
decays, but we have not considered the forms that are possible.
Many examples will be seen in the remainder of this
book. A few examples are given here to illustrate the range
of possibilities.
4.11.1. Multi-Exponential Decays
In the multi-exponential model the intensity is assumed to
decay as the sum of individual single exponential decays:
I(t) ∑ n
i1
α i exp(t/τ i)
(4.27)
In this expression τ i are the decay times, α i represent the
amplitudes of the components at t = 0, and n is the number
of decay times. This is the most commonly used model, but
the meaning of the parameters (α i and τ i ) depends on the
system being studied. The most obvious application is to a
mixture of fluorophores, each displaying one of the decay
times τ i . In a multi-tryptophan protein the decay times may
be assigned to each of the tryptophan residue, but this usually
requires examination of mutant protein with some of
the tryptophan residues deleted. Many samples that contain