## 1 Methods for the Analysis of Protein Fluorescence Spectra
### SIMS and PHREQ decomposition algorithms  
Several constraints were implemented into the SIMS and PHEQ decomposition algorithms, which reduced the complexity of the problem. Thus, stable solutions could be derived with an error not exceeding the experimental one [45]. The implemented constraints are:  
1. The spectrum of an elementary component on the frequency (wave number) scale is described by a bi-parametric (maximal amplitude and position of the maximum) log-normal function (Equation 2 and 3). 2. The shape and position of the tryptophan emission spectra remain unchanged by quenching of the fluorescence by small water-soluble quenchers [34-35]. Thus, a series of spectra measured at various quencher concentrations represents a sum of spectral components whose position and shape are constant at all quencher concentrations, while the relative spectral contributions (intensities) are changed. This allows the analysis of all spectra measured at different concentrations of quencher in a global mode. 3. The change in amplitudes of the individual components induced by quenching obeys the Stern-Volmer law (Equation 1). 4. The number of experimental points under analysis greatly exceeds the number of parameters sought. This approach attenuates the effect of occasional noise [49]. The input to the algorithms are the tryptophan fluorescence spectra measured at different concentrations _c(i)_ of the external fluorescence quenchers (acrylamide or \(\Gamma\), or Cs\({}^{+}\), or NO\({}_{3}\)). Since under different concentrations of the quenchers, the position and shape of the spectral componentsremain unchanged, while the relative contributions of the components change, the experimental spectra can be written as:  
\[F(i,j)=\sum_{k=1}^{L}I(k,i)\cdot\varphi(k,j) \tag{4}\]  
where \(i=1\),..., \(N\) is the number of spectra corresponding to the \(i\)-th quencher concentration, \(c(i)\); \(j=1\),..., \(M\) is the number of current frequency (wavenumber), \(\
u(j)\); \(k=1\),..., \(L\) is the number of component determined by the position of its spectral maximum, \(\
u_{m}(k)\); \(F(i,j)\) is the experimental intensity of fluorescence on the wavenumber scale in the \(i\)-th spectrum at the \(j\)-th frequency \(\
u(j)\); \(\varphi(k,j)\) is the value of the log-normal function (Equations 2 and 3) with a position of the maximum at \(\
u_{m}(k)\) at current frequency \(\
u(j)\) with unit maximal amplitude (at given \(k\) and \(j\), this value is the same for any of the \(N\) spectra); \(I(k,\,i)\) is the maximal amplitude of the \(k\)-th component in the \(i\)-th spectrum (Figure 4). **Figure 4.** Plot of three fluorescence spectra measured at three different concentration of quencher (\(i=1\), \(2\) and \(3\)) (black points). Each spectrum was decomposed into two components with position of maximum of spectral components at \(\
u_{m}(1)\) and \(\
u_{m}(2)\). The shape of the first and second spectral components is described by normalized log-normal functions \(\varphi(1,j)\) and \(\varphi(2,j)\), with intensities \(I(1,\,i)\) and \(I(2,\,i)\), respectively. In order to decompose the spectra into components, it is necessary to find the positions of maximum \(\
u_{m}(k)\) and the maximal intensities \(I(k,\,i)\) of the log-normal spectral components from the set of experimental spectra \(F(i,j)\). However, Equation 4 cannot be solved analytically because the log-normal function \(\varphi(k,j)\) is transcendental with respect to the unknown \(\
u_{m}(k)\) (see Equation 2). The position of maximum of spectral components can be found by fitting the \(\
u_{m}(k)\) values. The SIMS method employs the minimal least-square approach, according to which the canonical sets of linear equations are solved to find \(I(k,\,i)\). Thus, the variable parameters, \(\
u_{m}(k)\), are found by fitting, while \(I(k,\,i)\) values are calculated. The PHREQ method is based on the fact that any physical parameter could be used to characterize the transition from one physical state (\(A\)) to another (\(B\)).