Principles of Fluorescence Spectroscopy

148 TIME-DOMAIN LIFETIME MEASUREMENTS
Figure 4.62. Lifetime distribution of Annexin V Domain III in the
presence of phospholipid at various lipid-to-protein molar ratios
(L/P). The lipid was an 80/20 molar ratio of DOPC and DOPS, where
S indicates serine. Revised from [214]. The upper panel shows a
schematic of Annexin insertion into membranes. Reprinted with permission
from [218].
4.14. DATA ANALYSIS: MAXIMUM ENTROPY
METHOD
Intensity decays of biomolecules are usually multi-exponential
or non-exponential. The decays can be fitted using
the multi-exponential model. However, it is difficult to
obtain an intuitive understanding of the results by examining
table of α i and τ i values. Analysis of the decays in terms
of lifetime distribution (Section 4.11.2) is often useful for
visualizing the decay. However, when using NLLS the lifetime
distribution analysis is usually performed in terms of
assumed shape functions (eqs. 4.35 and 4.36). Analysis
using the maximum entropy method (MEM) allows recovery
of lifetime distributions without assumptions about the
shape of the components.
The MEM is mathematically complex 213–216 and the fitting
criteria somewhat subjective. Most of the published
analyses were performed using commercial algorithms
which are not completely explained. Nonetheless, the MEM
is now widely utilized and provides insight into complex
intensity decays.
The MEM is based on maximizing a function called
the Skilling-Jaynes entropy function:
∞
α(τ)
S α(τ)
m(τ) α(τ) log
m(τ)
0
dτ
(4.43)
In this expression α(τ) is the recovered distribution and
m(τ) is an assumed starting model that is flat in log τ space.
The MEM method is not used alone, but the fits are performed
while calculating χ R 2 according to eq. 4.22 to
ensure that the recovered distribution is consistent with the
data. In contrast to NLLS there does not appear to be a welldefined
stopping point for the MEM analysis. The analysis
is stopped when χ R 2 does not decrease more than 2% for 20
interactions. The MEM is advantageous because it provides
smooth α(τ) spectra that have enough detail to reveal the
shape of the distribution. The MEM method is claimed to
not introduce α(τ) components unless they are needed to fit
the data.
An example of an MEM analysis is shown in Figure
4.62 for domain III of Annexin V. Annexins are peripheral
membrane proteins that interact with negatively charged
phospholipids. Annexins can become inserted into membranes
(Figure 4.62), so the tryptophan intensity decays are
expected to be dependent on the presence of phospholipids.
This domain of Annexin V contains a single tryptophan
residue at position 187 (W187). The intensity decays of
W187 were measured by TCSPC. The excitation source
was synchrotron radiation that appeared as pulses at 13.6
MHz with a pulse width of 1.4 ns. 215 The maximum entropy
analysis shows a shift from a dominant component near 0.9
ns in the absence of membrane to a longer-lived component