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Analysis Methods and Models

Software performs the Global Analysis of auto(cross)correlation functions, photon counting distributions and fluorescence factorial cumulants. FCS, FCCS, TIRFCS, PCH, PCMH, FIDA, FIMDA, FCA, TIFCA, coincidence bursts counting and coincidence analysis are supported in FFS Data Processor. The complete protocol of the global analysis of auto(cross)correlation functions and photon counting distributions is published in (Skakun et al. 2014). 
1. Fluorescence Correlation Spectroscopy.
The model for analysis of correlation functions is defined by the following general equation:


where:

is the level of autocorrelation function when
;

denotes the average number of fluorescent particles;

denotes the background correction multiplier;

is kinetics term that denotes some kinetic process. The following kinetic terms are supported:

PureDiffusion:
;

TripletState:
;

Conformational:
;

Protonation:
;

Flow:
.

is motion term that describes the motion type of the particles. The following motion terms are supported:
 Free 2D Diffusion;
 Free 3D Diffusion;
 Anomalous 2D Diffusion;
 Anomalous 3D Diffusion;
 Confined 2D Diffusion;
 Confined 3D Diffusion.


2. Photon Counting Histogram
The
PCH model is used to analyze Photon Counting Distribution (PCD). PCD here refers
to the data to be analyzed, whereas PCH is a commonly used term to specify the
method of analysis. The total PCD from a number of molecules is calculated by
successive convolutions of a single molecule PCD (Chen et al., 1999, Perroud et al. 2003, Huang et al. 2004):
(1)
where Poi(n,
l) denotes the Poisson distribution with the mean value l, T
is the counting time interval, V_{ref} is the reference observation volume, B(r)
is the brightness profile function and Q is taken so that the
product QV_{ref} is large enough to completely include
the illuminated volume. The total distribution P(n)
is the weighted average of
convolved M times
(2)
An additional convolution to the background term can be taken in order to account
for the background photons.
Four different combinations of fitted parameters are available:

Absolute N and q values,

N ratio,

q ratio,

N and q
ratio.
The following brightness profile approximations are
available:

2D Gaussian,

3D Gaussian,

GaussianLorenzian, squared,

Polynomial (accordingly to Palo et al., 2000).
For the first three approximations the first and second order
correction for the outoffocus emission (and other types of profile nonideality)
accordingly to Huang et al., 2005 is supported.
Normalization of model parameters (scaling) can be either to the effective
volume, i.e. by setting V_{eff} = 1 or to the first and second PSF
moments, i.e. by setting

Photon Counting Histogram with correction for outoffocus emission
Outoffocus correction is performed by introduction of additional fitting parameters F_{k}
defined as relative difference between
integral of the kth power of the actual observation profile χ_{k}
and that of its approximation
,.(3)
In the most cases only the first order correction (all F_{k} equal zero except F_{1})
is sufficient to get the best fit to the experimental data. F_{1}
can be treated as an outoffocus emission ratio. The second order correction (F_{1}
and F_{2} are different from zero) can be also applied.

Photon
Counting Histogram with polynomial profile
B(r)
is approximated by an exponential function of a single argument with polynomial transformation of a unit of volume (Kask et al., 1999, Palo
et al., 2000)
,
(4)
where
a_{1}, a_{2} are instrumental
parameters and B_{0} is the value of B(r)
at r equal to 0. B_{0} and A_{0 }are
calculated from a system of normalization equations:
.
(5)
The
total PCD is calculated by Eq. 1 and 2
taking into account Eq. 4,5.
In this definition, results of PCH analysis with the polynomial profile are fully
equivalent to results of FIDA.
Correction for a number
of timedependent processes:

Triplet, Conformation, Protonation

Free and anomalous 2D/3D diffusion
can be applied.
Correction for timedependent processes is performed such that the first and
second factorial cumulants of PCD are exact. According to this theory one has to
calculate the socalled binning correction factor
where g(t)
is a time dependent term of the autocorrelation function in FCS and to correct the
brightness and the number of molecules in the following form:
where q(T) and N(T) are apparent parameters of the
model dependent on bin time T and q_{0} and N_{0}
are absolute values of brightness and concentration that are independent of T.
Correction for the
detector dead time and afterpulsing can be additionally performed (accordingly
to Palo et al., 2006).
Global analysis of a number of autocorrelation functions and
a number of photon counting distributions with linking of all related parameters
is possible (Skakun
et al., 2005, 2011, 2014).


3. Fluorescence Cumulants Analysis
The fluorescence factorial cumulants (FFC) are analyzed by the following model (Muller, 2004, Wu et al., 2005)
(7)
where q_{i}
is the mean number of photons (expressed in counts per second per
molecule) detected in a time interval T, N_{i}
is the mean number of molecules of the ith component in the
open observation volume and Bg is
the mean background count rate of the detector. γfactors are calculated as
,
where
B(r) is the normalized to unity brightness profile function and
is the diffusion
correction term.
is the normalized correlation function of kth order (Wu at
al., 2005). For the limit of short bin times
. We use normalization to the effective volume
in order to relate N_{i} obtained by both FCA and FCS
(additional factor γ_{2} in Eq. 7 is due to this type
of normalization).
To correct for deviations of the
actual observation profile from its ideal approximation three different
approaches can be used:
1) γfactors ()
can be fitted during the analysis (applicable only in the global analysis with
linking all fitted γfactors);
2) FIDAlike polynomial profile (Kask et al. 1999, Palo et al., 2000) can be
used for the approximation of the actual PSF;
3) outof focus correction of 2D/3D Gaussian or squared GaussLorenzian profiles
(Perroud et al. 2003, Huang et al. 2004) can be used.

The following brightness profiles are supported:
2D/3D
Gaussian. γfactors are calculated by
, where d is the dimensionality.
3D Gaussian
fitted. γfactors for two first
comulants are calculated as
respectively. All other γfactors
(actually products γ2γ3,
γ2γ4,
…) are fit parameters.
Squared
GaussLorenzian. γfactors are calculated by_{
}.
2D/3D Gaussian
or squared GaussLorenzian with outoffocus correction.
For the
outoffocus correction Eq. 7 becomes
_{
}
where F_{k}
is a correction factor defined as relative difference between
integral of the kth power of the actual observation profile χ_{k}
and that of its approximation_{}.
In many cases just correction of the first integral of 3D
Gaussian approximation of actual brightness profile is needed (all F_{k}
= 0 except F_{1} – first order correction).
Polynomial
profile. For the polynomial profile Eq. 7 becomes
where
,
and
,
(a_{1} and a_{2} are the fit
parameters). A_{0} and B_{0} are chosen
so that the normalization conditions assumed in FIDA (_{})
are satisfied.


4. Coincidence analysis
The coincidence histograms built from raw data can be fit by Gaussian model and separation quality parameter of two analyzed histograms can be calculated.
Coincidence analysis is performed accordingly to
Heinze et al. 2002.


5. Coincidence bursts counting
The relative number of coincidence bursts to the number of busts in a separate channel can be calculated.


Custom model with builtin script programming language provides creating userdefined models. Modular objectoriented architecture allows for easy extension of the model library

Raw data processing

 Calculating statistical characteristics (autocorrelation and crosscorrelation functions, photon counting, first event, single event and interevent time distributions, fluorescence cumulants and coincidence histograms) with correct weight factors from raw data (Carl Zeiss ConfoCor2, Carl Zeiss ConfoCor3, ISS ALBA, PicoQuant PicoHarp 300, PicoQuant Time Harp 200, PicoQuant Symphotime 64, Becker & Hickl TCSPC modules, binary and text files containing photon arrival times) (updated)
 Calculation of both bluetored and redtoblue crosscorrelation functions from the raw data. Calculation of averaged (bluetored + redtoblue)/2 crosscorrelation function. (new)
 Import, processing and analysis of data from PicoQuant PicoHarp 300
without the header
 Direct (without merging) import, processing and analysis of data from two SPC 130 cards (SPC 132)
(new)
 Sliding analysis of characteristics (PCD, AC(C)F, etc), calculated from the raw data either by the constant
time shift or the constant amount of photons
 Rejection of unwanted bursts of intensity while calculation of characteristics (PCD, AC(C)F, etc) from the raw data (updated)
 Bursts analysis. Selection of bursts in the intensity trace and calculation of characteristics (PCD, AC(C)F, etc) from each burst (new)
 Calculation of multiple characteristics (PCD, AC(C)F, etc) with binning time changed in a user defined range
 Simulation of raw data (Shingaryov et al., 2014)

Methods

 Global fit of correlation functions, photon counting distributions, fluorescence cumulants and coincidence histograms with the linked parameters (updated)
 Sequential fit: independent analysis of datasets one by one with
building dependencies of obtained fit parameters (updated)
 Automated generating initial guesses for fit parameters
(updated)
 Parameter fixing, constraints and linkage
 Confidence intervals by exhaustive search and standard errors (updated)
 Quality of fit is judged by chisquare criterion and visual inspection of residuals
 Builtin simulator of correlation functions

Interface

 Multidocument interface
 Advanced parameters management (sorting, quick linkage and easy navigation)
 Advanced managing of parameters of a large number of fitted data, e.g. finding parameters with the name that begins with a specific symbol and fix them to some value
 Automated creation of a global ACF + PCD analysis (FCS+PCH global analysis)
 Templates for quick saving and restoring analysis configurations
 Saving and loading experimental data and analysis results from databases
 Database support for saving and restoring userdefined models
 Import of external data and export of analysis results
 Creation of a Fit Parameter Dependence (FPD) curve from a number of
opened Experiments
 Calculation of histograms of fit parameters (new)
 Graphical representation of fit parameters depending on external parameters
 2D and 3D graphical data representation (updated)
 Visualization of standard deviations and relative standard deviations of each point of calculated characteristic
 Support for automatic calculation of a number of statistical characteristics
with different bin time from the selected region of raw data (updated)
 Support for automatic calculation of a number
of statistical characteristics from the raw data either by the
constant time shift or the constant amount of photons (sliding sheme).
Multiple characteristics can now be calculated not only with
different Time Step but also with different Start and End times
accordingly to the chosen sliding scheme. (updated)
 Rejection of unwanted bursts of intensity (updated)
 Selection of bursts (new)
 Selection and counting of coincidence bursts (updated)
 Saving and restoring interface settings

Databases

 Importing experimental data of following dataformats:
Carl Zeiss ConfoCor^{ tm}, Carl Zeiss ConfoCor2^{ tm},
Carl Zeiss ConfoCor3^{ tm}, PicoQuant PicoHarp 300, PicoQuant Time Harp 200,
PicoQuant Symphotime 64, ISS ALBA and B&H SPC (SPC 4x1, 4x2, 6x0 256 ch, 6x0 4096 ch, 130/134, 830) (updated)
 Import and processing of raw data of PicoQuant PicoHarp 300
without the header (new)
 Direct (without merging) import, processing and analysis of data from two SPC 130 cards (SPC 132)
(new)
 Import column organized text file (datasheet like,
can be used for import of ALV data) (updated)
 Import of binary and text files with simple dataformat (like one
column list of double precision values with or without the header)
(new)
 Direct (noniterative) fast estimation of model parameters of measured (or calculated from raw data files) auto(cross)correlations, photon counting distributions and fluorescence cumulants.
 Import of SIN files
 Storing and managing experimental data, analysis results and
oftenused model and parameter linkage configurations (Templates)
 Searching, sorting and filtering data
 Support of data integrity and validity
 Avoiding of data redundancy
 2D graphical data previewing
 Report generation

References

Chen Y., J.D.
Muller, P.T. So, and E. Gratton. 1999. The Photon Counting Histogram
in Fluorescence Fluctuation Spectroscopy. Biophis. J. 77:553567.
Perroud T.D., B.
Huang, M.I. Wallace, and R.N. Zare. 2003. Photon Counting Histogram
for OnePhoton Excitation. ChemPhysChem. 4:11211123.
Huang B., T.D.
Perroud, and R.N. Zare. 2004. Photon Counting Histogram: OnePhoton
Excitation. ChemPhysChem. 5:15231531.
Kask, P., K.
Palo, D. Ullmann, and K. Gall. 1999. Fluorescenceintensity
distribution analysis and its application in biomolecular detection
technology. Proc. Natl. Acad. Sci. USA. 96:1375613761.
Palo K., U. Mets,
S. Jager, P. Kask, and K. Gall. 2000. Fluorescence intensity multiple
distribution analysis: concurrent determination of diffusion times
and molecular brightness. Boiphys. J. 79:28582866.
Muller J.D. 2004.
Cumulant analysis in fluctuation spectroscopy. Biophys. J.
86:39813992.
Wu B., and J.D.
Muller. 2005, TimeIntegrated Fluorescence Cumulant Analysis in
Fluorescence Fluctuation Spectroscopy. Biophys. J. 89: 27212735.
Heinze K.G., Markus Rarbach, Michael Jahnz, and Petra Schwille. 2002.
TwoPhoton Fluorescence Coincidence Analysis: Rapid Measurements of
Enzyme Kinetics. Biophys. J. 83. 1671–1681.
Skakun V.V., M.A. Hink, A.V. Digris, R. Engel, E.G. Novikov, V.V.
Apanasovich, A.J.W.G. Visser. 2005. Global Analysis of Fluorescence Fluctuation
Data. Eur. Biophys. J. 34: 323–334.
Skakun V.V., R. Engel, A.V. Digris, J.W. Borst, A.J.W. Visser. 2011. Global
analysis of autocorrelation functions and photon counting distributions.
Frontiers in Bioscience (Elite Ed) 3: 489505.
Skakun, V.V. Global Analysis of Autocorrelation Functions and Photon Counting Distributions in Fluorescence Fluctuation Spectroscopy / V.V. Skakun, A.V. Digris, and V.V. Apanasovich // In book Fluorescence Spectroscopy and Microscopy: Methods and Protocols: Methods in Molecular Biology, Springer Protocols, Yves Engelborghs and Antonie J.W.G. Visser (eds.). Springer Science+Business Media, LLC. vol. 1076. – 2014. P. 719741. DOI 10.1007/9781627036498_33
Shingaryov, I.P. Simulation of Autocorrelation Function and Photon Counting Distribution in Fluorescence Fluctuation Spectroscopy / I.P. Shingaryov, V.V. Skakun, and V.V. Apanasovich // In book Fluorescence Spectroscopy and Microscopy: Methods and Protocols: Methods in Molecular Biology, Springer Protocols, Yves Engelborghs and Antonie J.W.G. Visser (eds.). Springer Science+Business Media, LLC. vol. 1076. – 2014. P. 743755. DOI 10.1007/9781627036498_34



