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Global Analysis of Fluorescence and Anisotropy decays [1, 2, 4] can be done by either MLE with Poissonian statistics of the noise or Least-Squares methods .

Deconvolution via Instrumental response function: Deconvolution via one-exponential Reference compound: where:
I(t)
is impulse response function of the sample;
g(t)
is measured Instrumental response function (IRF);
fref(t) is measured one-exponential reference compound;
B(t) is measured background;
D
denotes the  time shift between sample decay and IRF;
b
is time-uncorrelated background in the IRF or Reference compound;
g is background multiplication factor;
c
is time-uncorrelated background in the sample decay;
n is scattered light coefficient;
tref  is decay time of reference compound;
are fit parameters and

d(t) is Dirac delta-function.
Complete deconvolution for analysis of multi-excitation decays is also supported.
The detailed protocol of the global analysis, programmed in TRFA Data Processor, is published in .

• Multi-exponential model for fluorescence decays • Multi-exponential model for anisotropy decays The form of the multi-exponential model depends on the model property Fluor Parameters Type, which can be either “Amplitudes and decay times” or “Contributions and decay times” or “Ratio and decay times”. Switching between amplitudes and contributions is controlled by Normalization property of the multi-exponential model and can be done independently for positive and negative pre-exponential factors.

• Associative and non-associative models for anisotropy
Two approaches for performing of anisotropy analysis are supported:
1. Two stage anisotropy analysis based on a sequential fit of the measured sample total fluorescence decay and the measured sample parallel and perpendicular polarization components to the
multi-exponential model.
2. The anisotropy analysis based on a direct global fit of the sample parallel and perpendicular polarization components. This approach also supports an associative anisotropy analysis. The fluorescence decay is calculated accordingly to the following equation: where A is the normalization parameter; aj and tj are, respectively, contributions and decay times of the corresponding fluorescence exponents; Q is the polarization angle, bk and jk are, respectively, the amplitudes and rotational correlation times of the corresponding anisotropy exponents and Tjk controls the associations.

• Two compartmental models Fit parameters:
D  - the normalization parameter;
ci  - the emission weighting factor of species i* (i=1,2);
bi  - the concentration of species i* (i=1,2) at time zero;
k0i - the composite rate constant of species i* (i=1,2);
k12 - first-order rate constant for dissociation of 2* into 1* and co-reactant;
k21 - second-order rate constant for the association of 1* and co-reactant to 2*.

• Three compartmental models Fit parameters:
D  - the normalization parameter;
ci  - the emission weighting factor of species i* (i=1,…,3);
bi  - the concentration of species i* (i=1,…,3) at time zero;
k0i - the composite rate constant of species i* (i=1,…,3);
k1i - first-order rate constant for dissociation of i* (i=2,3) into 1* and co-reactant;
ki1 - second-order rate constant for the association of 1* and co-reactant to i* (i=2,3).
k
23
- interconversion rate constant from state 3* to 2*;
k32 - interconversion rate constant from state 2* to 3*.

• Poisson distribution of decay rates model Fit parameters:
A  - the normalization parameter;
l1 - the inverted decay time of first fluorescence component in absence of energy transfer or other quenching;
b1 - quenching efficiency of first fluorescence component;
m  - average number of quenchers;
e  - rate of quenching for a probe interacting with one quencher;
a  - the relative contribution of second fluorescence component;
l2 - the inverted decay time of second fluorescence component in absence of energy transfer or other quenching.

• Gaussian distribution of decay rates model Fit parameters:
A  - the normalization parameter;
s - standard deviation on decay rate;
m  - average decay rate;
k0  - minimum decay rate;
a  - the relative contribution of second fluorescence component;
l2 - the inverted decay time of second fluorescence component.

Modular object-oriented architecture allows for easy extension of the model library

Instrumental models

• Instrumental response function via scattering solution
• Instrumental response function via single exponential reference compound
• Fitting of reference lifetime, time shift, level of dark noise in the instrumental response function, level of time-uncorrelated background and scattered light coefficient
Methods

• MLE nonlinear fitting with Marquardt-Levenberg optimization assuming Poissonian statistics of the noise
• Least-squares nonlinear fitting with Marquardt-Levenberg optimization (c2 criterion)
• Global fit: several fluorescence decays are combined and simultaneously fitted
• Anisotropy analysis via simultaneous fit of parallel and perpendicular components of fluorescence
• Automatically generated initial guesses for parameters
• Parameter fixing, constraints and linkage
• Constraints by a functional relationships between model parameters
• Confidence intervals by exhaustive search and standard errors
• The “Complete convolution” method for processing multi-excitation decays
• Quality of fit is judged by c2 or MLE criterion, visual inspection of residuals and auto-correlation function of residuals, Zc2, Durbin Watson and Runs tests, Heterosedasticity and Normal probability function of the residuals
• Build-in simulation of fluorescence decays and anisotropy curves
• The Phasor analysis of fluorescence decays. The standard Phasor plot and complete Phasor analysis of bi-exponential data with estimation of all prameters including pre-exponential factors with final calculation of Chi2 criterion

Interface

• Multi-document interface
• Advanced managing of parameters of a large number of fitted data, e.g. finding parameters with the name that begins with F and fix them to some value
• Templates for quick saving and restoring analysis configurations
• Import of external data and export of analysis results
• Graphical representation of fit parameters depending on external parameters
• 2D and 3D graphical data representation
• Saving and restoring interface settings
Databases

• Importing PicoQuant PHD files
• Importing of decays from PicoQuant and B&H SPC raw data files
• Import column organized text file (datasheet like)
• Import of text files with simple dataformat (like one, two column list of double precision values with or without the header)
• Storing and managing experimental data, analysis results and often-used 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

1. Beechem J. M., Gratton E., Ameloot M. et al. (2002) The Global Analysis of Fluorescence Intensity and Anisotropy Decay Data: Second-Generation Theory and Programs. In: Lakowicz J. R. (ed) Topics in Fluorescence Spectroscopy Volume 2, Springer US, pp 241-305.

2. Verveer P. J., Squire A., Bastiaens P. I. (2000) Global analysis of fluorescence lifetime imaging microscopy data. Biophys. J. 78(4), 2127-2137

3. Maus M., Cotlet M., Hofkens J. et al. (2001) An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules. Anal. Chem. 73(9), 2078-2086

4. Digris, A.V.,E.G. Novikov, V.V. Skakun, and V.V. Apanasovich. Global Analysis of Time-Resolved Fluorescence Data // 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. 257-277. DOI 10.1007/978-1-62703-649-8_10
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