## 1 Methods for the Analysis of Protein Fluorescence Spectra
### Log-normal function for describing spectral curves  
The decomposition methods described above do not employ any analytical descriptions of the spectral curves. There were many attempts to describe spectral components by using various mathematical functions [38-41]. The quadric-parametric (maximal amplitude, \(I_{m}\) position of the maximum, \(\
u_{m}\) and positions of half-maximal amplitudes, \(\
u_{-}\) and \(\
u_{+}\); Figure 3) log-normal function originally proposed by [PERSON] and [PERSON] [38], was found to be one of the best functions for describing absorption spectra of complex molecules with a minimal number of variable parameters [39-43]. The log-normal function used in its mirror-symmetric form has been shown to accurately describe fluorescence spectra as well [44-47] and can be written as:  
\[\begin{cases}I(\
u)=I_{m}\cdot\exp\Biggl{\{}-\frac{\ln 2}{\ln^{2}\rho}\cdot \ln^{2}\Biggl{(}\frac{a-\
u}{a-\
u_{m}}\Biggr{)}\Biggr{\}}&\quad\mbox{(at $\
u<a$)}\\ I(\
u)=0&\quad\mbox{(at $\
u\geq a$)}\end{cases} \tag{2}\]  
where \(I_{m}\) is the maximal intensity; \(\
u\) is the current wavenumber; \(\rho\) is the band asymmetry parameter and \(a\) is the function limiting point position (Figure 3):  
\[\rho=\frac{\
u_{m}-\
u_{-}}{\
u_{+}-\
u_{m}} \tag{3}\]  
\[a=\
u_{m}+\frac{\rho\cdot(\
u_{+}-\
u_{-})}{\rho^{2}-1}\]  
Despite the fact that the log-normal function describes the shape of fluorescence spectra very well, it is not of any practical use in decomposition algorithms, since it has too many variable parameters (seven or eleven unknown parameters for two- or three- component solutions, respectively).  
Figure 3: The log-normal function.  
Estimating so many parameters would lead to unstable solutions. Thus, an extremely important result was obtained by [PERSON] and [PERSON] [44], who experimentally established the existence of linear relationships between the positions of maximal (\(\
u_{m}\)) and two half-maximal amplitudes (\(\
u_{-}\) and \(\
u_{+}\)) for a large series of monocomponent spectra of small tryptophan derivatives in various solvents:  
\[\
u_{+} = 0.831\cdot\
u_{m}+7070\] \[\
u_{-} = 1.177\cdot\
u_{m}-7780 \tag{3}\]  
These relations allow a reduction in the number of unknown parameters for individual spectral component from four (\(I_{m}\), \(\
u_{m}\), \(\
u_{-}\), \(\
u_{+}\)) to two (\(I_{m}\), \(\
u_{m}\)) (three or five unknown parameters for two- or three- component solutions, respectively). Such a significant reduction in the number of parameters makes decomposition analysis much less ambiguous [48]. By implementing this additional information (constraints) and using the log-normal function to describe the spectral components, two mathematically different algorithms for the decomposition of fluorescence spectra were developed: SIMS - SImple fitting procedure using the root-Mean-Square criterion, which is based on the minimal least-square approach, and PHREQ - PHase-plot-based RESolution using Quenchers that uses an analytical pseudo-graphic solving technique [45]. Later, the linear relations between the position of maximum and the width of spectra were established for environment sensitive fluorophores Prodan and Acrylodan [47].