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Current
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Fluorescence Fluctuation
Spectroscopy and Binding Stoichiometry of PTB. (Artem Melnykov) Our stoichiometry
data describe only the size of the largest complex, a restriction of the
method. The stoichiometry of the intermediate complexes is also of great
interest, since these are likely to be highly populated in vivo. Fluorescence
fluctuation spectroscopy (FFS) is capable of quantitatively describing the
equilibrium composition of a complex, and the equilibrium concentrations of
bound vs unbound components (Rigler & Elson, 2001). For these PTB experiments, the protein is
fluorescently tagged for detection, and the RNA:protein complex is monitored
over time to describe the fluctuations in its composition through measurement
of the fluorescence intensity (brightness). Brightness analysis in
fluorescence fluctuation spectroscopy is a technique complementary to
fluorescence correlation spectroscopy (FCS). Rather than focusing on dynamic
character of fluctuations, photon counting histogram (PCH) considers their
magnitudes. Therefore, the two data analysis methods complement each other:
while FCS measures diffusion coefficients and rates of chemical reactions,
PCH determines brightness per molecule. While PCH analysis is a promising technique, very
few successful applications have been reported. One reason is that the
resolving power of the method is not sufficient in most cases. For example,
it is a challenge to resolve a mixture of monomers and dimers. The reason for
that can be understood in simple terms. When fitting a histogram, we are
optimizing the model parameters, brightness (q) and concentration (c),
so that the model histogram approximates the measured one. It is always
possible to find values for c and q such that we match the first two
moments of the histogram. While higher order moments are also important for
describing the shape of the distribution, they have a smaller effect and
therefore such differences are harder to pick up. One straightforward approach towards improving the
resolving power of PCH is to increase the bin time used to build histograms. Traditionally
histograms in brightness analysis are built using a time window that is much
shorter than the characteristic decay time for the process giving rise to
fluctuations. In the simplest case, this is the diffusion time as measured by
FCS. This constraint on the time window has to be imposed since strictly
speaking the PCH theory applies only when the molecule remains in the same
excitation intensity shell during the bin time. However, intuitively a longer
time window would be more appropriate for capturing properties of the sample.
Such a bin time could have the same order of magnitude as the characteristic
diffusion time. Performance of simple PCH model. As a model system for proof of principle for the
application of this new data analysis methodology, we have used DNA
oligonucleotides. For these experiments, we have used
RhodamineGreen-d(T)15 and Cy3-d(T)15 as model
compounds. These fluorophores differ in brightness by a factor of two, and so
serve as a model system for the mixtures of monomers (1X brightness) and
dimers (2X brightness) that we will need to resolve. Table 1 shows results of
fitting histograms calculated with different bin times for RhG-d(T)15
(similar results were obtained for Cy3-d(T)15). A simple one-species PCH model with
the correction factor for one-photon excitation (F) was used to fit the data.
As indicated by low values of χ2 for each fit, the PCH model is
able to describe the shape of the curve in all cases. Essentially, this fact
indicates that for any time window, we can use two parameters to describe the
properties of the sample: apparent brightness per molecule and apparent
number of molecules in the beam. Both these values increase with the
increasing bin time as could be expected.
Table 1. The results of fitting
histograms calculated from RhG-(dT)15 solution with different bin
times. tdiff is
74 μs as measured by FCS. c is the average number of molecules in the beam, q is brightness per molecule (that is,
the average number of counts per molecule per time unit at given power), and T denotes the time bin used in
calculating the histogram. While this model works well when a single species
is present, our goal is to characterize heterogeneous mixtures of PTB:RNA
complexes. Two properties of these complexes will complicate interpretation
of the data; one is their differences in brightness, and the other is their
differences in diffusion times. Strictly speaking, PCH theory is not
corrected for molecular motion, and using it to analyze histograms distorted
by diffusion may result in unpredictable systematic errors. In order to
eliminate this possibility, we designed a correction algorithm for PCH
analysis based on the cumulant expansion of the generating function for the
histogram. Correction of cumulant expansion for diffusion: binning functionsIt is possible to correct cumulant expansion of the
PCH generating function (Qian & Elson, 1990; Muller, 2004) for diffusion
and then use such correction to fit the histogram. The corrected cumulants
are
Example
graphs of the first six binning functions (Bn) are shown in Figure 2. The limit of short time
where all binning functions are equal to T
n corresponds to the case of stationary molecules. On the other
hand, in the limit of long time, binning functions decay to 0, and any
information about fluctuations is lost. It is also important to note that
higher order binning functions decay faster than the One
has to realize limitations of such an approach to generalizing the PCH model.
Calculation of binning functions is a time consuming effort; therefore these
functions have to be pre-calculated before fitting can be done. Also, the
point spread function of the instrument influences this calculation. Finally,
the effects of photophysical phenomena such as photobleaching, saturation and
inter-system crossing on the apparent beam profile have not been studied in
the case of PCH and may result in artifacts. In practice, a three-dimensional
Gaussian beam seems to be a very good approximation for the point spread
function as far as PCH is concerned, and the factor F introduced for
one-photon excitation takes care of the beam non-idealities manifested
through γn factors.
Photophysical problems can be minimized by a judicious choice of
fluorophores. Performance of PCH model corrected
for diffusion with different bin times. We next examined the effect of diffusion correction on the extracted
parameters and compared results to those presented from the simple model.
Apparently, the correction is successful in describing how the shape of the
histogram depends on the binning time. It then follows that we can fit
re-binned histograms globally and extract the characteristic diffusion time
of the fluorophore. As shown in Figure 3, that is true and the diffusion time
can be extracted together with concentration and brightness per molecule.
More results of global fits of 40-100 μs histograms are shown in Table
2, for varying order of correction. Two observations are important. One is
that even lowest (second) order of correction is very successful in extracting
tdiff
and removing bias in the values of c
and q. The other observation is
that the values of tdiff
are not equal to those determined by FCS analysis, which are 74 μs for
RhGreen-dT15 and 72 μs for Cy3-dT15.
Figure 3. Global fit of multiple
histograms with the PCH model corrected for diffusion to the sixth order
(Cy3-(dT)15). C is average number of molecules in the beam; q is
brightness; Td is diffusion constant. Table 2.
The results of global fitting of histograms calculated from RhGreen-(dT)15
solutions.
Characterization of a mixture of species: the effect of diffusion correctionFinally, we revisit our results for histograms collected
with binary mixtures with the diffusion-corrected model. From our experience,
fitting with the second-order corrected model (FIMDA/PCMH) results in
unpredictable bias in the extracted model parameters. This result may seem
surprising considering the success of second order correction in the case of
homogeneous solutions. The failure of the FIMDA model is most likely due to
the fact that all model parameters are highly correlated. Therefore,
relatively small systematic errors drive the minimization algorithm towards a
combination of parameters that is very far from the true set expected for the
sample. Table 3.
Comparison of model parameters extracted by PCH models corrected to various
order.
The results of global analysis of 40-100 μs
histograms for the case of a binary mixture of Cy3-(dT)15:
RhGreen-(dT)15 are given in Table 3. The first line of the table
serves as an example of poor performance of PCMH2 model. PCMH4 and PCMH6
models are able to extract parameter values that are much closer to the
expectation in this case (the last digit in the model name denotes the
correction order). In summary, PCH analysis can benefit from the
theory that explicitly takes diffusion into account. With such modification,
not only all major sample properties (concentration, brightness, and
diffusion coefficient) can be determined in a single experiment, but also the
resolving power of the method is improved. The correction for diffusion can
be cast in terms of the cumulant expansion of the PCH generating function
where the notion of correction order arises naturally. The corrected model
allows reliable determination of sample parameters for simple solutions of
fluorescent molecules and for their mixtures. It is the latter case where
higher order (above two) corrections are necessary in order to obtain
meaningful results. It should be emphasized that while the approach outlined above is
quite general, it requires prior time consuming calculations of the binning
functions. Therefore, in the cases where some prior knowledge about the
sample is available, PCMH2 can be used instead. For example, given two
brightness values for the binary mixture, PCMH2 can reliably determine
concentrations at which molecules are present in the mixture. Chen,Y., Muller, J.D., So, P.T.C.,
Gratton, E. 1999. The photon counting histogram in fluorescence fluctuation
spectroscopy. Biophys J. 77, 553-567 Muller, J.D., 2004. Cumulant
analysis in fluorescence fluctuation spectroscopy. Biophys J. 86(6):3981-92. Qian, H., Elson, E.L., 1990. Distribution of molecular aggregation
by analysis of fluctuation moments. Proc Natl Acad Sci U S A. 87, 5479-83. Rigler, R., Elson,E.L. 2001. Eds,
“Fluorescence correlation spectroscopy” Springer,
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Return to Washington University Biochemistry and Molecular Biophysics Homepage
Dr. Kathleen B. Hall (hall(at)biochem.wustl.edu)
Department of Biochemistry and
660
office: 314-362-4196
lab: 314-362-4197
or 314-747-8079
FAX: 314-362-7183
send email to kathleenhal(at)gmail.com
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