‘White’ and ‘grey’ methods of data modeling have been employed to resolve the heterogeneous fluorescence from a fluorophore mixture of 9-cyanoanthracene (CNA), 10-chloro-9-cyanoanthracene (ClCNA) and 9,10-dicyanoanthracene (DCNA) into com- ponent individual fluorescence spectra. The three-component spectra of fluorescence quenching in methanol were recorded for increasing amounts of lithium bromide used as a quencher. The associated intensity decay profiles of differentially quenched fluorescence of single components were modeled on the basis of a linear Stern-Volmer plot. These profiles are necessary to initiate the fitting procedure in both ‘white’ and ‘grey’ modeling of the original data matrices. ‘White’ methods of data modeling, called also ‘hard’ methods, are based on chemical/physical laws expressed in terms of some well-known or generally accepted mathematical equations. The parameters of these models are not known and they are estimated by least squares curve fitting. ‘Grey’ approaches to data modeling, also known as hard-soft modeling techniques, make use of both hard-model and soft-model parts. In practice, the difference between ‘white’ and ‘grey’ methods lies in the way in which the ‘crude’ fluorescence intensity decays of the mixture components are estimated. In the former case they are given in a functional form while in the latter as digitized curves which, in general, can only be obtained by using dedicated techniques of factor analysis. In the paper, the initial values of the Stern-Volmer constants of pure components were evaluated by both ‘point-by-point’ and ‘matrix’ versions of the method making use of the concept of wavelength dependent intensity fractions as well as by the rank annihilation factor analysis applied to the data matrices of the difference fluorescence spectra constructed in two ways: from the spectra recorded for a few excitation lines at the same concentration of a fluorescence quencher or classically from a series of the spectra measured for one selected excitation line but for increasing concentration of the quencher. The results of multiple curve resolution obtained by all types of the applied methods have been scrutinized and compared. In addition, the effect of inadequacy of sample preparation and increasing instrumental noise on the shape of the resolved spectral profiles has been studied on several datasets mimicking the measured data matrices. . . . . Keywords Multiple curve resolution Stern-Volmer plot Difference fluorescence spectra Rank annihilation factor analysis Non-linear least squares optimization Introduction multi-component mixture of spectrally active components. The main objective of such approaches is to decompose the The rapidly developing methods of chemical analysis are measured data matrix into the product of two matrices: first nowadays those involving self-modeling curve resolution containing the spectra of pure components and another one (SMCR) of a spectral data matrix representing a representing their relative concentrations. Preliminary step in this analysis consists, however, of decomposition of the orig- inal data matrix into the product of the matrices containing the * Andrzej M. Turek so called abstract spectral and concentration profiles. firstname.lastname@example.org Typically, this is achieved by using the Jacobi algorithm of the principal component analysis (PCA) or its more elegant Faculty of Chemistry, Jagiellonian University, 2 Gronostajowa St, 30 version called the singular value decomposition (SVD) . 387 Cracow, Poland 616 J Fluoresc (2018) 28:615–632 Upon the use of a proper transformation matrix the abstract The regions of existence of unique contributions from sin- matrices could easily be converted into the predicted profiles gle components in some portions of the measured data matrix of both types of variability . (selective regions) as well as those signalizing the absence of a For the first time, the concept of SMCR was successfully contribution from a specific component (Bzero’ regions) are of elaborated and applied in the early 1970s by Lawton and uttermost importance for reducing the number of feasible so- Sylvestre . The analyzed data matrix was a spectrophoto- lutions and reliability of the resolved profiles. These regions metric dataset representing a mixture of only two chemical were intensively utilized in multivariate curve resolution of species. Since the proposed method was based on two rather overlapping chromatographic peaks in HPLC-DAD chro- obvious premises concerning non-negativity of the predicted matograms [20–23]. A simple tutorial on how to use this in- spectra of pure components as well as non-negativity of the formation obtained from evolutionary rank analysis of the data coefficients of a linear combination used to build up each matrix provided by Maeder’s evolving factor analysis (EFA) measured two-component spectrum, the obtained solutions [20, 21] and Kvalheim and Liang heuristic evolving latent were not unique and classified later on as belonging to the projections (HELP)  has been reliably crafted by Toft . category of soft data modeling. Soon, an attempt to extend A significant improvement or even unique curve resolution this approach to a three-component system was made by can be achieved if instead of one data matrix two or more Ohta . By keeping the same minimum set of constraints matrices with altered evolution of the concentration profiles and imposing a constant value on all three elements of one are factor-analyzed. These model-free techniques include gen- vector of the transformation matrix, the three-dimensional eralized rank annihilation method (GRAM) [24, 25] and/or problem was reduced to two dimensions. This allowed to de- Kubista’s approach  for a pair of two-way matrices as well termine an appropriate set of the elements of the remaining as parallel factor analysis (PARAFAC) [27–29]for a two other vectors of the transformation matrix and conse- three-way data array (a stack of matrices). In this context, an quently also to visualize the area of feasible solutions (AFS) instructive example of effective application of such trilinear for the pure component spectra. The selection of this so called decomposition technique to several excitation-emission matri- T-space representation of the three-component data was car- ces (EEMs) measured for different concentrations of a fluo- ried out by the Monte Carlo method producing feasible spec- rescence quenching agent has been provided by Wentzell et al. tral bands for all components of the three-component system . As highlighted by these authors, inevitable Rayleigh and . Almost 30 years later this approach was effectively im- Raman scattering caused by the solvent molecules and possi- proved by Leger and Wentzell and introduced to the literature ble primary absorption of the quencher lead, however, to ap- as the dynamic Monte Carlo SMCR . In the meantime, the parently distorted EEMs which hardly can be corrected with random AFS generation for three component systems was no left traces. neatly replaced by an approach taking advantage of the ideas In the case of a single experimental data matrix the same developed by computational geometricians. This was com- goal can be accomplished quite often by hard modelling that is menced by Borgen and Kovalski who developed the mathe- by taking into account the existing physical/chemical laws matical tools for confining the T-space convex hulls related to responsible for evolution of each individual concentration AFS . The so called Borgen plots, preserving the two in- profile. The evolving concentration profile can be directly trinsic assumptions of soft data modeling, were then succes- expressed as a function of time, pH or another non-random sively modified by adding some other constraints narrowing variable using the relevant mathematical formula (white meth- the bands of the AFS computed spectra and concentration od) or represented by its digitized form obtained by a partial profiles [7–11]. usage of the information concerning the existing law com- The classical soft modeling methods mentioned above bined with a complementary application of some soft-model [3–11] provide possibly the best estimated pure component approach [31–34]. The latter method is called a grey method. spectra but sometimes only the selection of the purest mea- In this paper a detailed analysis and comparison of the sured spectra is required and made. Such spectra can easily be results obtained using white (hard) and grey (hard+soft) sought by using the criterion of maximal spectral dissimilarity MCR methodologies applied to resolve the spectra of a as demonstrated by Cruciani et al.  or by applying any three-component system of quenched fluorescence has been other non-factor analysis method employing this concept such included. The presentation goes as follows: in Second section as simple-to-use-interactive-self-modeling-mixture-analysis with five subsections the essential theoretical foundations of (SIMPLISMA) , orthogonal projection approach (OPA) the employed methods are explicitly stated. Third section  or alternating least squares (ALS) . The same goal gives details of experimental conditions and sample prepara- is also achieved using iterative target transformation factor tion. Fourth section provides a discussion of the obtained re- analysis (ITTFA) [16, 17]. Some other less common rational sults and is organized around two subsections. In the first curve resolution methodologies are briefly characterized in subsection the results obtained for simulated dataset are ex- review papers by Jiang and Ozaki  or Jiang et al. . amined while the second subsection dwells on the analysis of J Fluoresc (2018) 28:615–632 617 the results referring to real experimental dataset. In closing Fifth section the outcome of this study is succinctly summa- rized in four subsections. Theoretical Background Fig. 1 Decomposition of multi-component data matrix of fluorescence quenching, Y, into a product of two matrices C and S containing re- Fluorescence Quenching solved intensity decays and spectra of pure fluorophores, respectively It is well known that in the case of collisional fluorescence −1 T T T þ quenching the ratio of the integrated intensity of the fluores- S ¼ C C C Y ¼ C Y ð3Þ cence spectra in the absence and in the presence of a specified amount of quencher, Q, can be replaced by the ratio of the Matrix C in Eq. (3) is called the left pseudoinverse of observed signal intensities at any emission wavelength, matrix C. Thus, the main task, as regards the decomposition λ λ of the spectra of multi-component mixture of fluorophores, F =F , if the shape of the emission spectrum is not modified consists in finding the Stern-Volmer constants, K , for all by quenching. If so, then the ratio of fluorescence intensities, SV λ λ involved components. F =F , increases linearly with the quencher concentration. This dependence shown below is called the linear Rank Annihilation Factor Analysis Stern-Volmer equation τ‐RAFA ¼ 1 þ K Q ð1Þ SV Recently, it has been demonstrated that the successful estima- tion of K s for a three-component mixture of fluorophores SV where K = k τ and k is the quenching rate constant, τ is SV q 0 q 0 can be easily carried out  by using an iterative version of the singlet state lifetime in the absence of quencher, and λ rank annihilation factor analysis (RAFA) as proposed by designates the selected emission wavelength. Deviations to Davidson et al. [37, 38]. In order to apply this method it is the simple Stern-Volmer plots defined above can be numerous necessary to measure a few series of quenched fluorescence as discussed in , however, at sufficiently low concentra- spectra with various excitation lines. Then the data matrices tions of the quencher (usually below 0.1 M) this linear rela- are constructed in such a way that each matrix, M ,contains tionship holds true for all components of the fluorophore in its rows the spectra recorded successively for all selected mixture. excitation lines but for a specified amount of the quencher. In general, an original data matrix Y containing, in its rows, Naturally, a reference matrix, M , for unquenched fluores- the multi-component spectra of the quenched fluorescence cence is generated alike. A three-component fluorescence recorded for successive quenching experiments can be repre- spectrum measured with a specified excitation wavelength, sented by a product of two matrices: the first with decays of λ, is thus a sum of the spectra of particular components refer- the emission intensities of individual components caused by ring to the same excitation wavelength, as given below quenching, C, and the second (transposed) matrix, S , having in its rows the fluorescence spectra of those pure components. λ A B C m ¼ m þ m þ m ð4Þ Q Q Q The dimension of these matrices are defined by the following numbers: n – number of chemical components, Q – number of On the right side of Eq. (4)someexpected λ symbols are added portions of quencher, and λ - number of the used emis- λ;A sion wavelengths (see Fig. 1). Hence, the matrix C is a matrix omitted for simplicity, e.g. it should be m but is m ,andso Q Q equivalent to the matrix of individual pure concentration pro- on. The next step involves construction of a difference matrix, files resolved from overlapping chromatographic structures [20–24]. All the above mentioned matrices appear in the following D ¼ M −τM ð5Þ Q 0 Q equation with successive λ-dependent rows defined as follows Y ¼ CS ð2Þ λ A A B B C C If matrix C is known then upon simple transformation the d ¼ m −τm þ m −τm þ m −τm ð6Þ Q 0 Q 0 Q 0 Q spectral profiles of all fluorescent components are given by 618 J Fluoresc (2018) 28:615–632 where τ is a floating parameter. By stepping τ in its predefined for different quencher concentrations and one excitation range it is possible to find such a value of τ that the line as demonstrated below for a specified quencher con- Stern-Volmer dependence for one of the components, say centration, Q, component A, can be satisfied A A B B d ¼ m −ðÞ 1 þ κQ m þ m −ðÞ 1 þ κQ m 0 Q 0 Q A A A A m ¼ 1 þ K Q m ¼ τm ð7Þ 0 SV Q Q C C þ m −ðÞ 1 þ κQ m ð10Þ 0 Q The spectral contribution from component A to the overall The iterative parameter, marked here as κ, becomes equiv- fluorescence intensity is then lost alent to Stern-Volmer constant. In other words, in the case of a three-component system, the expected value of a quenching λ B B C C constant should be equal to the κ value corresponding to the d ¼ 0 þ m −τm þ m −τm ð8Þ Q 0 Q 0 Q minimum value of the third eigenvalue of the covariance ma- trix formed from the difference matrix D. which is reflected in a substantial cutdown of the third eigen- In Authors’ opinion this simple κ-RAFA approach value of the covariance matrix D D due to efficient reduc- shouldbe calleda ‘direct’ method while the word ‘indi- tion of the number of significant components of the spectral rect’ would rather be reserved for the τ-RAFA methodol- mixture from three to two. The optimum value of the floating ogy. In the present article the performance of both parameter, τ, is found for each quencher concentration, Q,by methods as well as the effect of the type of noise and its tracing the changes in the third eigenvalue of the covariance magnitude on the final results have been carefully inves- matrix D D as a function of τ. Then a plot is made which in tigated (for details see Results and Discussion). the caseofcomponent A gives ‘Point-by-Point’ Optimization of Stern-Volmer Constants τðÞ Q ¼ 1 þ K Q ð9Þ SV Having at hand some initial estimates for the Stern-Volmer with the slope equal to the Stern-Volmer constant of compo- constants it is possible to refine these values and then to initi- nent A. The whole procedure is illustrated in Fig. 2 shown ate the process of resolution of the multi-component fluores- below. cence spectra. Historically, the first approach to this problem was made by Sherwin Lehrer . Originally applied to de- κ‐RAFA termine a fraction of unquenched fluorescence it was based on the ‘point’ Stern-Volmer dependence as defined by Eq. (1). A much more simple alternative to the τ-RAFA approach (For better readability of equations, in the next portions of this described above is also conceivable. The rows of a differ- article the symbols λ and SV will be omitted). The above cited ence matrix D can be formed not for different excitation author introduced a concept of the fraction of the emission lines and constant quencher concentration but conversely intensity of the i-th component, f , defined as the ratio of the Fig. 2 τ-RAFA analysis of the covariance matrix of an ideal three-component system of quenched fluorescence J Fluoresc (2018) 28:615–632 619 contribution of its fluorescence intensity to the overall fluo- Acuña et al. . Adopting the following form of the rescence intensity of the unquenched emission, at a fixed Stern-Volmer dependence emission wavelength. For a three-component system, with F f f f A B C components A, B and C, it reads like below ¼ þ þ ð17Þ F 1 þ K Q 1 þ K Q 1 þ K Q 0 A B C A A F F 0 0 leads to the sum of three curves, the parameters of which are f ¼ ¼ ð11Þ A B C F þ F þ F 0 0 0 the Stern-Volmer constants and intensity fractions of particular substances. The applied optimization allows then for deter- By modifying the classical expression for the mining the required values describing the studied system with- Stern-Volmer dependence through introduction of the out need of solving any additional equations (see Fig. 3). difference between the ‘point’ fluorescence intensities of the unquenched emission, F , and the quenched 0 A Brief Description of the Applied Algorithm emission, F, one gets The approaches described above are based on curve fitting F −F ΔF f K Q f K Q f K Q 0 A B C A B C ¼ ¼ þ þ ð12Þ with the use of the method of the least squares and therefore F F 1 þ K Q 1 þ K Q 1 þ K Q 0 0 A B C it seems quite appropriate to briefly quote what are the oper- ating principles of one commonly used optimization algo- Upon bringing the above expression to the common de- rithm, i.e. a Newton-Gauss algorithm with a nominator, a third degree rational function of Q is obtained Levenberg-Marquardt extension, as explained by Maeder 2 3 ΔF a þ a Q þ a Q þ a Q and Neuhold . The cited procedure is based on minimiza- 0 1 2 3 gQðÞ¼ ¼ ; ð13Þ 2 3 F tion of the difference, r, between the real data given in a form 0 b þ b Q þ b Q þ b Q 0 1 2 3 of a vector, y, and the data resulting from the optimal func- with the related parameters a and b calculated as shown below tional form, y opt a ¼ 0 rpðÞ ¼ y−y ðÞ p ð18Þ opt a ¼ f K þ f K þ f K 1 A B C A B C a ¼ f KðÞ K þ K þ f KðÞ K þ K þ f KðÞ K þ K 2 A B C B A C C A B A B C As it can be noticed the above difference depends on a ¼ K K KðÞ f þ f þ f 3 A B C A B C parameters p of the fitted function, thus by changing the b ¼ 1 vector of initial parameters by a certain value, δp,itis b ¼ K þ K þ K 1 A B C possible to obtain the error vector r(p + δp)withsmaller b ¼ K K þ K K þ K K 2 A B B C A C b ¼ K K K elements in the least-squares sense (the sum of squares 3 A B C ssq = r r should be minimal, or at least smaller), optimal- ð14Þ ly equal zero. The residuals r(p + δp) are approximated by After finding the optimal parameters of the rational func- a Taylor series expansion tion, for instance by curve fitting with the use of the method of ∂rpðÞ the least squares, it is possible to determine the Stern-Volmer rpðÞ þ δp¼ rpðÞþ ½ ðÞ p þ δp −pþ … ð19Þ ∂p quenching constants of particular species as well as the ‘point’ fluorescence intensity fractions assigned to these components. For this purpose one has to solve a system consisting of the following polynomial equations 3 2 −K þ b K −b K þ b ¼ 0 1 2 A 3 A A ð15Þ −K þðÞ b −K K − ¼ 0 1 A C K ¼ b −K −K B 1 A C as well as the matrix equation 0 1 0 10 1 a K K K f 1 A B C @ A @ A@ A a ¼ KðÞ K þ K KðÞ K þ K KðÞ K þ K f 2 A B C B A C C A B a K K K K K K K K K f 3 A B C A B C A B C C ð16Þ A similar algorithm but operating directly on the apparent Fig. 3 Model curves used to determine the values of Stern-Volmer con- parameters of the fitted functional curve was proposed by stants by Lehrer (DIF) and Acuña (FRA) methods 620 J Fluoresc (2018) 28:615–632 Fig. 6 Measured absorption (dashed lines) and emission (solid lines) of pure components and of the mixture (MIX); vertical lines mark the se- lected excitation wavelengths not uniquely determined and should be suitably adjusted Fig. 4 Matrix illustration of the nglm algorithm in each optimization case. The detailed description of the construction of this algorithm goes far beyond the con- which upon retaining the first two terms and introducing a tents of this article, hence the Reader is suggested to refer Jacobi matrix of the first partial derivatives gives to other Literature dealing with this particular issue. 0≈rpðÞ þ δp¼ rpðÞþ Jδp ð20Þ Matrix Representation of Stern-Volmer Profiles which eventually upon simple transformation using the idea of pseudoinversion leads to the matrix equation that The optimization methods discussed above allow for tak- allows to determine the ‘best’ parameter shift vector δp ing into account the fluorescence intensity at only one emission wavelength. Of course, it is possible to carry δp ¼ −J rpðÞ ð21Þ out a series of individual optimizations for all emission wavelengths, however, the obtained Stern-Volmer con- stants of a given fluorophore that theoretically should be Upon performing a simple operation of addition of two equal, remain actually independent which may lead to a vectors few hundreds of different values depending on a measur- p’ ¼ p þ δp ð22Þ ing point, λ. To solve this problem one has either to take an average or to get down to constructing some matrix a better convergence, at least on the theory grounds, between the real and optimized functions is achieved. Sometimes, however, the input values of parameters p depart significantly from optimal values – in such case the Levenberg-Marquardt extension to the Gauss-Newton minimizer can be used to ‘protect’ the algorithm from taking a too big step or inappropriate direction. This cor- rection consists in ‘elongation’ of the error vector by an appropriate amount of zero rows and augmentation of the Jacobi matrix, J, by a diagonal matrix with all the ele- ments on the diagonal equal to a predefined value m (see Fig. 4). A numeric value of the Marquardt parameter m is Fig. 7 Quenched fluorescence spectra of pure substances: a CNA, b Fig. 5 Fluorophores used in this study DCNA, c CICNA and d the related Stern-Volmer plots J Fluoresc (2018) 28:615–632 621 Fig. 8 Quenched fluorescene spectra of fluorophore mixture after preprocessing; excitation line = 368 nm versions of the optimization algorithm, which form the concentrations of the quencher. The matrix, Y = CS ,is opt basis for the modern methods of the multivariate curve subsequently used in the nglm algorithm. resolution. rKðÞ¼ Y−Y ðÞ K ¼ Y−CS ð24Þ opt The ‘white’ algorithms are directly based on the matrix factorization illustrated in Fig. 1 and described by Eq. (2). By introducing corrections to matrix C which are The resulting S (see Eq. (3)) stems from the ‘concentra- brought about only by the change in the values of the tion’ matrix which is constructed on the basis of the used Stern-Volmer quenching constants, a better conformity be- concentrations of a quencher and some preliminary initial- tween the empirical data collected in matrix Y and the ized Stern-Volmer quenching constants, K. data contained in matrix Y is achieved. Finally, as a opt result of the optimization process both the quenching con- stants and spectral profiles are obtained, on the basis of CðÞ Q; n ¼ ð23Þ 1 þ K Q n which the best description of the studied system can be proposed. In the above formula if Q = 0 then C(0, n) = 1, so the num- While in ‘classical’ approach to decomposition of ber of rows in matrix C formed by n Stern-Volmer profiles is multi-component spectra it is assumed that the recorded actually by 1 greater than the number of different spectra of quenched fluorescence are inserted into a data Fig. 9 Simulated fluorescence spectra of single components A, B and C summed up into a mixture spectrum (MIX) 622 J Fluoresc (2018) 28:615–632 Fig. 10 Two types of data noise: a additive (independent of signal) and b multiplicative (proportional to signal) matrix in their ‘unaltered’ form, yet in an approach that (n = 3), the spectral fractions at each specified emission A B C makes use of the idea of ‘spectral fractions’ this natural wavelength when summed up, give one, i.e. f + f + f = assumption is modified. The method takes advantage of a 1. The size of S is alsoðÞ λ x n . Thus, the transformation notably different form of the data matrix which is actually from f to S is performed for each matrix entry using the an extension of the original ‘point’ methodsasproposed first fluorescence spectrum, y = Y(1, λ), measured in the by Lehrer and Acuñaetal. . The original data absence of the quencher. matrix Y is replaced by a matrix Y in which the elements s ¼ f yðÞ i ¼ 1; …; λ; j ¼ 1; …; n ð26Þ ij ij 0;i of each row are obtained by point by point division by the corresponding elements of the first raw of the original In Eq. (26) y represents the i-th element of vector y . 0, i 0 data matrix. The methods described above are classified as ‘white’ The matrix equation on which the mathematical operations methods because of the assumption concerning the fulfill- of the applied algorithm are performed remains unchanged, ment of the linear Stern-Volmer equation. Despite the use- however, as a result of the optimization procedures the frac- ful approximation provided by a chemical model, the hid- tions of the overall fluorescence intensity, f, instead of the den disadvantage carried by hard methods are, in the con- emission profiles are obtained. sidered case, the values of the quencher concentration T þ assumed to be absolutely constant. However, it is f ¼ C Y ð25Þ well-known that even the best measurement procedure is Matrix f is sizedðÞ λ x n where λ is the number of endowed with uncertainties and therefore, as regards the emission wavelengths, and n isthenumberofsignificant assumed values of Q, some almost imperceptible depar- components. In the case of a three-component system tures are unavoidable. The solution to this problem may Fig. 11 τ-RAFA applied to simulated data; data noise level Q5%, S05‰ J Fluoresc (2018) 28:615–632 623 Table 1 Stern-Volmer constants of individual components determined by two versions of RAFA for different noise types and levels ‘Noise’ τ-RAFA κ-RAFA K K K K K K A C B A C B Q0S0 5.00 20.0 100 5.00 20.0 100 Q3 5.01 19.9 100 4.24 20.4 100 Q5 5.13 20.5 103 3.54 19.2 120 Q10 5.19 20.7 104 2.44 18.7 138 S03 4.89 19.2 94.1 4.86 17.4 91.7 S05 4.81 18.4 88.3 4.76 16.0 86.9 S1 4.51 – 71.7 4.22 12.9 74.7 Q3S03 4.89 19.0 93.2 4.22 20.0 100 Fig. 12 Stern-Volmer plots based on results of τ-RAFA; data noise level Q5S05 4.94 18.3 89.2 3.50 18.7 117 Q5%, S05‰ Q in % and S in ‰ S-V constants in M be provided by so called ‘grey’ methods of data modeling that do not impose stiff constraints on the amount of the quenching substance contained in a sample. spectral profiles, S . The first step is analogous: the ma- The ‘hard-soft’ methods of data modeling incorporate trix C is built on the basis of known concentrations of the advantages of both the methods obeying the restrictive quencher and tentatively determined Stern-Volmer con- criteria of ‘white’ methods and the ‘black’ procedures stants(thisstandsforthe ‘white’ element). Then the ini- void of any constraints except for non-negativity. This tial matrix S and the trial matrix Y are generated. In opt approach seems to combine two things that are mutually the next step, however, a significant difference emerges: exclusive but there is no contradiction as it has been prov- the concentration matrix is no more optimized only on the en on the example of the MCR ALS (Multivariate Curve basis of the quenching constants, but by itself as a whole Resolution Alternating Least Squares) algorithm elaborat- constitutes a parameter which undergoes a permanent op- ed by Tauler et al. [42, 43]. timization and adaptation process (this represents the Likewise in the case of the discussed ‘hard’ methods, ‘black’ element). To avoid the values without physical the ‘grey’ (hard-soft) algorithm is operating on three ma- meaning the non-negativity constraint (a ‘white’ element) trices: original data matrix containing the measured becomes superimposed on the profiles in matrices C and multi-component spectra, Y, and with regard to particular S . Eventually, a pair of vectors, c and s ,representing n n components, the matrix of the fluorescence intensity de- the emission intensity decay and the spectral profile (a cays (‘concentration’ matrix), C,andthematrixof spectrum) of a given component n, is generated. Fig. 13 Stern-Volmer constants determined by application of ‘direct’ κ-RAFA to model data: a without noise, and b with noise level Q5%, S05‰ 624 J Fluoresc (2018) 28:615–632 Fig. 14 Characteristic spreads of Stern-Volmer constants determined by ‘point-by-point’ optimization using the methods of Acuña et al. - FRA,and Lehrer  – DIF Experimental data points and the related quenching constant was ex- tracted (see Fig. 7). As first the absorption and emission spectra of three pure Finally, a mixture of all three fluorophores was made. fluorophores: 9-cyanoanthracene (CNA), 9,10-dicyanoanthracene The concentrations were chosen in such a way that the (DCNA) and 10-chloro-9-cyanoanthracene (ClCNA) (see absorption maximum for each substance was below Fig. 5) were measured in methanol solutions using a Hitachi 0.08. 21 equal portions of the solution were taken and U-2900 spectrophotometer and a Hitachi F7000 fluorimeter, the increasing aliquots of the quencher solution (lithium respectively (see Fig. 6). Then, in order to minimize the inner bromide in methanol) were added. To prevent appearance filter effect the mixtures were diluted, so that the absorption of a possible interference of Rayleigh and Raman scatter- maximum of each spectrum was lower than 0.04. ing, inevitably associated with the excitation lines, these In order to determine the Stern-Volmer constants of were carefully selected far away from the range of the pure fluorophores, the quenched fluorescence spectra recorded fluorescence spectra at five fixed wavelengths: were measured for a series of samples using the previous- 365, 368, 371, 374 and 377 nm. Moreover, it goes with- ly prepared CNA, DCNA and ClCNA solutions with lin- out saying, as regards the mixture of fluorophores, that early increasing amounts of lithium bromide added as a varying of the excitation wavelength is responsible for quencher up to the highest concentration of 0.2 M. The (leads to) the change in the relative amounts of the fluo- values of the fluorescence intensities of subsequent sam- rescence emitting species. The typical measured spectra of ples at the wavelength for which the intensity of the the unquenched mixture as well as individual components unquenched emission was the highest were plotted versus are presented in Fig. 6. Additionally, the absorption spec- the concentration of a quencher. Next, for each substance tra were recorded for two outermost samples: without the the Stern-Volmer straight line was fitted to the calculated quencher and with the highest quencher concentration. Fig. 15 Spectral fractions for simulated component B: PURE – expected curve shape, FRA - Acuña et al. method , DIF – Lehrer method , and MFRA - Acuña et al. approach in matrix version J Fluoresc (2018) 28:615–632 625 Fig. 16 Two forms – classical and fractional of simulated data used in optimizations. Vertical lines confine the spectral region with reduced data noise The concentrations of the used fluorophores and lithium magnitude of a signal (in this case independent of the bromide expressed in M (mole/dm )wereasfollows: response of a fluorimeter). The second noise is a variant of the heteroscedastic noise , denoted as Q –and is linearly proportional to a variable - in this case to a −6 −5 quencher concentration; it simulates an imperfection in c = 7.68 ⋅ 10 c =1.84 ⋅ 10 CNA DCNA −6 the amounts of substances used to prepare a sample. c =9.47 ⋅ 10 c = 0, 0.0103, 0.0206, …,0.2056 ClCNA LiBr Because those two types of data noise are different, their noise level is defined distinctly as: The recorded raw spectra were preprocessed: the meth- fixed : nðÞ λ ¼ FðÞ λ þ y⋅r⋅maxðÞ F anol (solvent) baseline was subtracted, correction for proportional : nQðÞ¼ Q þ x⋅r⋅Q self-absorption (inner filter effect) was made and the data where: n –‘noisy’ data, r – random numbers from −1to reproduction using the SVD procedure, producing the 1, F(λ) – fluorescence intensity at wavelength λ,max(F) ‘smoothed’ data, was applied (if not, then it is marked – the highest value of fluorescence intensity in a series, Q in the text). A series of such spectra upon preprocessing – quencher concentration, y – spectral data noise level (in is showninFig. 8. ‰), x – concentration data noise level (in %). To con- clude: the notation Qx or Sy stands for: Q, S – type of noise; x, y – data noise level. Results and Discussion In order to faithfully reflect the nature of the applied ana- lytical procedure, the analysis of the simulated system was Synthetic Data commenced with estimation of the Stern-Volmer constants of all three substances A, B and C. To this purpose both ver- All required data processing, calculations and analyses for this sions of the RAFA technique (τ‐RAFA and κ‐RAFA, see study were performed with MATLAB R2012a (The BRank Annihilation Factor Analysis^ Subsection) were used. MathWorks, Inc., Natick, MA) software. For practical rea- Furthermore, at the same time, an influence of data type and sons, in some cases the Authors have taken the liberty of noise level on the final results was also investigated – the keeping the unchanged MATLAB notation. A model system findings are presented in Figs. 11, 12,and 13 and in Table 1. of quenched fluorescence of three components: A, B and C On the basis of the performed research, it can be said that the was simulated (see Fig. 9) by using a set of a few Gaussian faster κ-RAFA procedure remains more sensitive to noisy data envelopes, partially based on real spectra. Stern-Volmer con- and the calculated values are remarkably divergent from stants were defined as follows: −1 −1 −1 K ¼5:00 M K ¼100 M K ¼20:0 M A B C Table 2 Stern-Volmer constants of single components before and after or in MATLAB notation: K = [5.00; 100; 20.0]. optimization on simulated data with assumed noise level Q5%S05‰ Next, on the basis of the linear Stern-Volmer equation, Initial Optimized a series of 21 three-component spectra was formed, each one simulating fluorescence at particular quencher con- 5.00 20.0 100 4.487 19.57 123.6 centration – linearly increasing from 0 to a value of 4.75 19.0 100 4.486 19.57 123.5 0.2 M. In order to imitate real dataset, two types (see 4.50 18.0 110 4.486 19.57 123.5 Fig. 10) of data noise were added to these ideal spectra. 4.00 15.0 120 4.485 19.56 123.5 The first one, marked as S – fixed or additive noise 49.0 50.0 51.0 4.485 19.56 123.5 (known also as homoscedastic noise) – represents instru- ) S-V constants in M mental type of noise, which is independent of the 626 J Fluoresc (2018) 28:615–632 Fig. 17 Effect of reduction of data range on the shape of the resolved spectra: PURE – expected curve shape, FULL – complete data range, and RED – data range reduced to the region between vertical lines; data noise level Q5%, S05‰ expected ones. The accuracy of more complex τ-RAFA ap- One of the suggested solutions was to make a variable out proach is higher, however, reliable application of both of the constant in the denominator. methods is limited by the noise level – predominantly instru- 2 3 a Q þ a Q þ a Q 1 2 3 mental, and therefore the margin for an acceptable error gðÞ Q ¼ ð28Þ 2 3 p þ b Q þ b Q þ b Q should be kept below S1‰. 1 2 3 The next interesting issue addressed by the Authors Theoretically, it would allow to avoid extremely large or- was to assess the performance and outcome of the ‘point’ ders of magnitude by dividing all the coefficients a and b by a optimization methods as proposed by Lehrer and variable parameter p (p/p = 1). Unfortunately, that train of Acuña et al.  in the form of third-order rational func- thought is not compatible with the MATLAB nglm algorithm, tion and the sum of three rational functions of the first since the shift vector applied during the optimization process degree, respectively. In the first case, the use of the nglm is responsible merely for the increase of the parameters a and b optimization algorithm results in ambiguity when the re- but does not change their ratios. Another possible approach spective terms, containing Q as variable, in the formula was to transform the original mathematical formula, Eq. (27), shown below attain values substantially greater than 1. into the following: Then the constant term in the denominator – equal to 1 a a – begins to lose its significance and the parameters a and 1 2 2 3 Q þ Q þ Q b can be multiplied by any factor whatsoever, which may a a 3 3 gðÞ Q ¼ 1 b b lead to huge orders of magnitude and lack of the optimi- 1 2 2 3 þ Q þ Q þ Q zation progress. a a a 3 3 3 2 3 2 3 p Q þ p Q þ Q a Q þ a Q þ a Q 1 2 1 2 3 ¼ ð29Þ gðÞ Q ¼ ð27Þ 2 3 2 3 p þ p Q þ p Q þ Q 1 þ b Q þ b Q þ b Q 3 4 5 1 2 3 Fig. 18 Resolved fluorescence spectra of pure components obtained for the simulated data noise level Q5%, S05‰.PURE – expected curve shape, SPE – obtained by ‘classical’ approach, FRA – yielded using fractional data, and MCR – produced by grey MCR-ALS method of Tauler et al.  J Fluoresc (2018) 28:615–632 627 The equality of parameters a and b results from their 3 3 entanglement with the Stern-Volmer constants (see BTheoretical Background^) a ¼ K K KðÞ f þ f þ f¼ K K K ¼ b ð30Þ 3 A B C A B C 3 A B C Unfortunately, also that method did not provide a solution to the encountered optimization problems. Finally, a decision was arrived at to use the original third-order rational formula, but with the number of coefficients reduced to five (a = b ). 3 3 This approach turned out to be only partly successful. Apparently, it appears that the form most suitable for use by the optimizer is the sum of three first-degree rational func- Fig. 20 Optimized and normalized to maximum intensity decays of pure tions. A problem of ambiguity of solutions is eliminated and components. The applied methods as in Fig. 18 so are additional calculations required to obtain the Stern-Volmer constants and spectral fractions from the opti- if they are identical, the procedure simply does not work mized parameters (see BTheoretical Background^). properly. In order to compare the efficiency of the two ‘point’ Due to a divergence of the results (see Fig. 14), obtained methods of optimization mentioned above, all calculations independently for different emission lines, a mean of all the were related to the same simulated spectral dataset. Also, the optimized point-by-point parameters was calculated (FRA – same initial values of spectral fractions and Stern-Volmer con- sum of three hyperbolas, DIF – third degree rational): stants were used in both cases: K ¼½ 4:67; 24:8; 105 K ¼½ 4:60; 23:7; 115 FRA DIF f ¼½ 0:3000; 0:3333; 0:3337 K ¼½ 4:00; 15:0; 120 According to the Authors’ assumptions, all initial spectral The computed Stern-Volmer constants are somewhat more fractions should be equal, but the nglm algorithm in convergent with the expected ones than the constants obtained MATLAB requires at least slightly different starting values – by the use of the τ− and κ-RAFAmethods. Moreover, the application of both Lehrer and Acuña approaches allowed to determine spectral fractions of all three substances for almost each selected emission line, though the fractional profiles re- constructed on their basis could be described as ‘rugged’ and discontinuous (see Fig. 15). In conclusion, it can be noted that the ‘point’ methods were historically justified. Fitting one ra- tional function was easier than fitting a sum of three hyperbol- ic curves, but nowadays a complexity of a fitted combinations of formulas has become less problematic. Fig. 19 Effect of spectral (S ‰) and quencher concentration (Q %) noise on the shape of resolved spectrum of component C. The applied methods Fig. 21 Stern-Volmer plots based on the results of τ-RAFA applied to as in Fig. 18 empirical dataset without use of the SVD procedure 628 J Fluoresc (2018) 28:615–632 Fig. 22 Stern-Volmer constants obtained using a ‘direct’ κ-RAFA applied to real dataset, not reproduced (F) and reproduced (F ) with use of SVD REP procedure; excitation line at λ =374 nm On the basis of the averaged values of the Stern-Volmer parameters: 21 versus 5 in case of the applied ‘point-by-point’ constants it is possible to restore the concentration matrix C, method and more than 21 × 1000 versus 3 in case of the which then can be used to resolve the multi-component fluo- employed matrix method. rescence spectra into the emission profiles of pure substances. The difference between ‘classical’ and originated by Acuña Nevertheless, another step ahead can be made, since the et al. ‘fractional’ approach to hard resolution of ‘point-by-point’ optimization can readily be replaced by the multi-component spectra lies in a form of data optimization performed as a whole on full data matrices. to-be-optimized. In the former case, a dataset is created ex- All white methods of data modeling that have been plicitly from the measured quenched fluorescence spectra, exploited in this article make use of the same form of the while in the latter case all spectral intensities of a given spec- concentration matrix C containing the individual intensity de- trum are divided ‘point-by-point’ by corresponding intensities cays recovered on the basis of the preliminary estimated of the fluorescence spectrum recorded for the sample without values of the Stern-Volmer constants. Therefore it seemed the quencher. An advantage of such normalized data is that the appropriate to investigate the influence of initial parameters areas of high and low noise level are now easily recognizable on final results obtained through application of the nglm algo- (see Fig. 16). If the range of the analyzed spectral data is rithm. A series of different initial vectors have been optimized narrowed to its less noisy portion, the accuracy of the whole and the available findings are collected in Table 2. The carried procedure is appreciably improved (Fig. 17). Moreover, it is out analysis has proven that the white (or hard) method based possible to use the concentration matrix optimized this way to on a full data matrix, unlike ‘point’ methods, is practically reproduce completely resolved spectra of pure components. independent of the user-entered starting values. A concluding As a result of application of the ‘fractional’ method the spec- statement can be made that the mentioned above indepen- tral fractions instead the spectra are obtained – emission pro- dence is likely a result of the increased ratio of two quantities: files could then be determined in at least two equivalent ways: the number of the data entries to the number of the modified by dividing the spectral data matrix ‘point-by-point’ by the resulting concentration matrix or by multiplying the measured spectrum of unquenched fluorescence of the fluorophore mix- ture ‘point-by-point’ by the fractional profiles. Table 3 Stern-Volmer Method K K K constants obtained for CNA ClCNA DCNA real data using different EMP 2.044 13.85 122.7 methods τ-RAFA 2.765 13.70 116.4 κ-RAFA 0.580 7.50 90.4 FRA 0.901 9.03 113.8 DIF 0.438 15.98 127.8 S-V constants in M EMP – determined for pure components, Fig. 23 Curve fitting by ‘point’ type optimizations of real dataset (x) RAFA – obtained using two versions of representing a fluorescence intensity decay at emission wavelength λ = RAFA, FRA/DIF – estimated with use of 445 nm; FRA - Acuña method, and DIF – Lehrer method white ‘point’ methods J Fluoresc (2018) 28:615–632 629 Fig. 24 Averaged fluorescence spectra of all three components of a real spectrum obtained by means of ‘classical’ white method, FRA – system resolved from the mixture spectra with no use of the SVD yielded using fractional data, and MCR – produced by grey MCR-ALS procedure; EMP – measured spectrum of a pure component; SPE – method of Tauler et al.  Finally, a few series of simulated three-component quenched Empirical Data fluorescence spectra with different type and noise level were resolved using ‘classical’ hard method of data modeling, ‘frac- As a final test, an attempt was made to resolve the empir- tional’ white method with reduced spectral range as well as ical spectra (See BExperimental^). Thewholeprocess MCR-ALS approach i.e. a grey method of Tauler et al. [42, started with estimation of the Stern-Volmer quenching 43]. The initial concentration matrix was the same in all ap- constants. First, the τ-RAFA method was applied, iterative proaches and it was constructed assuming the following values parameter versus quencher concentration plots were of the Stern-Volmer constants: drawn and the straight lines were fitted (Fig. 21). To allow a comparison, the results of the κ-RAFA approach applied K ¼½ 4:00; 120; 15:0 to the data with and without reduced dimensionality were also taken into account (Fig. 22). Because the results of The spectra of pure components resolved by the applied ‘indirect’ τ-RAFA method seemed to be more reliable (see methods are depicted in Fig. 18. previous section), only these were considered as appropri- The effect of the noise type and level on the shape of the ate for further processing. resolved spectrum is demonstrated here on the example of the In the next part of this study, which may also be called an worst resolved spectrum of substance C (see Fig. 19). overview of historical curiosities related to the estimation of In addition, in Fig. 20 the obtained concentration profiles the Stern-Volmer constants by means of ‘point’ methods, the (or properly Stern-Volmer intensity decays) of pure best possible sets of coefficients for the Lehrer and Acuña fluorophores are shown revealing that a major difference can schemes were determined using the nglm algorithm for two be noticed between the functional curves provided by the hard hundred different wavelengths at which the decays of the ‘fractional’ method and those digitalized curves retrieved quenched fluorescence intensity were observed (Fig. 23). using the hard-soft method of MCR-ALS in which only a The averaged values of the Stern-Volmer constants obtained non-negativity constant was imposed. by the ‘point’ techniques are collected in Table 3 and Fig. 25 Comparison of measured (solid lines) and resolved (dashed lines) spectra of pure components. The latter are resolved with the use of Stern-Volmer constants determined from the quenched fluorescence measurements per- formed for single fluorophores 630 J Fluoresc (2018) 28:615–632 compared with those determined for pure components and expected ones. The κ-RAFA algorithm should only be used as extracted using both versions of RAFA. a tool in pre-analysis of the collected data or to confirm find- The last section of this study includes the resolution of ings already unveiled by other methods - the Authors suggest three-component system of fluorescence quenching by means using both RAFA approaches concomitantly, after initial de- of white and grey methods of data modeling. The initial con- termination of the number of principal components. centration matrix was constructed on the basis of the τ-RAFA factor analysis. At first, the raw data, not ‘smoothed’ by the ‘Point’ Optimization Methods SVD procedure, were analyzed. Prior to using the ‘fractional’ hard algorithm, the data range was reduced to 420–500 nm. The ‘point’ methods accounted for as rather historical epi- The resolved emission profiles of pure components obtained sodes are yet still used even today, especially to pinpoint some for five different excitation lines by all the applied methods specific fragments of proteins in biochemical systems as mo- were then averaged since the shape of the fluorescence profile lecular markers of cancer or virus spread [45, 46]. Application of each fluorophore, unlike the intensity, should remain un- of these methods allows for minimization of computational changed regardless the applied excitation line. The resulting resources since it does not require a continuous recording of spectra of pure components are portrayed in Fig. 24. the fluorescence spectra. A basic knowledge of the measured The measured three-component spectra of quenched fluo- system is, however, essential due to a high sensitivity of the rescence upon preliminary SVD pretreatment were resolved optimization algorithm to initial values of the optimized pa- as well. However, a difference between the results of both rameters. The experimenter should at least be aware of what approaches (with and without the SVD data preprocessing) are the possible values of the Stern-Volmer quenching con- in the case of white methods was practically unnoticeable. stants of individual fluorophores and which emission lines are, Moreover, as regards the grey MRC-ALS algorithm, the emis- in a broad sense, the most suitable for the analysis –‘point’ sion profiles of single constituents resolved upon the prelim- approaches are highly sensitive to the data noise level. inary use of the SVD procedure are even more divergent from the expected spectra (measured for individual components) White and Grey Methods of Data Modeling than those obtained for raw experimental data of the fluorophore mixture. It turns out that white algorithms of data modeling allow to In order to assess, for assumed quencher concentrations, resolve the composite fluorescence spectra with acceptable how inaccurate the whole procedure of sample preparation resemblance to the original pure component spectra. Shape might be the recorded spectra of the fluorophore mixture have of the calculated emission profile depends mainly on the spec- also been resolved on the basis of the concentration matrix tral data noise level – calculations above 1‰ may be treated as generated for the Stern-Volmer constants determined by the uncertain. Concentration inadequacy, by contrast, results in analysis of fluorescence quenching of one component sys- shifting of the whole spectrum to lower or higher wave- tems. The graphical outcome is depicted in Fig. 25. lengths. Fortunately, the initial values of the variable parame- ters entered to the optimization nglm algorithm do not have apparent influence on final outcome. Conclusions In order to evade, at least some of the mentioned above limitations, the range of the scrutinized noisy spectra should Application of RAFA be reduced – this may easily be done through data transfor- mation which is a part of the Acuña et al. matrix approach. The analysis of the obtained results reveals that the method of Due to low costs of calculations and significant advantages, it ‘indirect’ rank annihilation factor analysis, τ-RAFA, can be is recommended to estimate an effective spectral range with successfully employed to determine the number of compo- the use of the ‘fractional’ data type technique prior to applica- nents in the multi-component system of quenched fluores- tion of any hard optimization algorithms. cence and to estimate their Stern-Volmer quenching constants. The grey MCR ALS approach operating on digitalized Imperfection in determining the quencher concentration curves, i.e. curves with unknown functional forms, becomes seems to have rather negligible effect on the final outcome, independent of the assumed values of the quencher concen- while a spectral noise influence cannot be ignored and appears tration which is an undeniable advantage of the method. to be a main limitation of the method – the results of RAFA for Furthermore, the data noise at least up to the noise level of the data with spectral noise level higher than 1‰, should not 1‰ seems to have a negligible effect on the final results, but be considered as reliable. the algorithm due to a decreased number of applied constraints The ‘direct’ κ-RAFA method is very ‘noise-prone’– both is sensitive to initial entries of the concentration matrix – a concentration inaccuracy and spectral noise insert influence pre-factor analysis should be performed. 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