Fast Decomposition of Three-Component Spectra of Fluorescence Quenching by White and Grey Methods of Data Modeling
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Abstract
‘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 component 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 optimizationIntroduction
The rapidly developing methods of chemical analysis are nowadays those involving self-modeling curve resolution (SMCR) of a spectral data matrix representing a multi-component mixture of spectrally active components. The main objective of such approaches is to decompose the measured data matrix into the product of two matrices: first containing the spectra of pure components and another one representing their relative concentrations. Preliminary step in this analysis consists, however, of decomposition of the original data matrix into the product of the matrices containing the so called abstract spectral and concentration profiles. Typically, this is achieved by using the Jacobi algorithm of the principal component analysis (PCA) or its more elegant version called the singular value decomposition (SVD) [1]. Upon the use of a proper transformation matrix the abstract matrices could easily be converted into the predicted profiles of both types of variability [2].
For the first time, the concept of SMCR was successfully elaborated and applied in the early 1970s by Lawton and Sylvestre [3]. The analyzed data matrix was a spectrophotometric dataset representing a mixture of only two chemical species. Since the proposed method was based on two rather obvious premises concerning non-negativity of the predicted spectra of pure components as well as non-negativity of the coefficients of a linear combination used to build up each measured two-component spectrum, the obtained solutions were not unique and classified later on as belonging to the category of soft data modeling. Soon, an attempt to extend this approach to a three-component system was made by Ohta [4]. By keeping the same minimum set of constraints and imposing a constant value on all three elements of one vector of the transformation matrix, the three-dimensional problem was reduced to two dimensions. This allowed to determine an appropriate set of the elements of the remaining two other vectors of the transformation matrix and consequently also to visualize the area of feasible solutions (AFS) for the pure component spectra. The selection of this so called T-space representation of the three-component data was carried out by the Monte Carlo method producing feasible spectral bands for all components of the three-component system [4]. Almost 30 years later this approach was effectively improved by Leger and Wentzell and introduced to the literature as the dynamic Monte Carlo SMCR [5]. In the meantime, the random AFS generation for three component systems was neatly replaced by an approach taking advantage of the ideas developed by computational geometricians. This was commenced by Borgen and Kovalski who developed the mathematical tools for confining the T-space convex hulls related to AFS [6]. The so called Borgen plots, preserving the two intrinsic assumptions of soft data modeling, were then successively modified by adding some other constraints narrowing the bands of the AFS computed spectra and concentration profiles [7, 8, 9, 10, 11] .
The classical soft modeling methods mentioned above [3, 4, 5, 6, 7, 8, 9, 10, 11] provide possibly the best estimated pure component spectra but sometimes only the selection of the purest measured spectra is required and made. Such spectra can easily be sought by using the criterion of maximal spectral dissimilarity as demonstrated by Cruciani et al. [12] or by applying any other non-factor analysis method employing this concept such as simple-to-use-interactive-self-modeling-mixture-analysis (SIMPLISMA) [13], orthogonal projection approach (OPA) [14] or alternating least squares (ALS) [15]. The same goal is also achieved using iterative target transformation factor analysis (ITTFA) [16, 17]. Some other less common rational curve resolution methodologies are briefly characterized in review papers by Jiang and Ozaki [18] or Jiang et al. [19].
The regions of existence of unique contributions from single components in some portions of the measured data matrix (selective regions) as well as those signalizing the absence of a contribution from a specific component (“zero’ regions) are of uttermost importance for reducing the number of feasible solutions and reliability of the resolved profiles. These regions were intensively utilized in multivariate curve resolution of overlapping chromatographic peaks in HPLC-DAD chromatograms [20, 21, 22, 23]. A simple tutorial on how to use this information obtained from evolutionary rank analysis of the data matrix provided by Maeder’s evolving factor analysis (EFA) [20, 21] and Kvalheim and Liang heuristic evolving latent projections (HELP) [22] has been reliably crafted by Toft [23].
A significant improvement or even unique curve resolution can be achieved if instead of one data matrix two or more matrices with altered evolution of the concentration profiles are factor-analyzed. These model-free techniques include generalized rank annihilation method (GRAM) [24, 25] and/or Kubista’s approach [26] for a pair of two-way matrices as well as parallel factor analysis (PARAFAC) [27, 28, 29] for a three-way data array (a stack of matrices). In this context, an instructive example of effective application of such trilinear decomposition technique to several excitation-emission matrices (EEMs) measured for different concentrations of a fluorescence quenching agent has been provided by Wentzell et al. [30]. As highlighted by these authors, inevitable Rayleigh and Raman scattering caused by the solvent molecules and possible primary absorption of the quencher lead, however, to apparently distorted EEMs which hardly can be corrected with no left traces.
In the case of a single experimental data matrix the same goal can be accomplished quite often by hard modelling that is by taking into account the existing physical/chemical laws responsible for evolution of each individual concentration profile. The evolving concentration profile can be directly expressed as a function of time, pH or another non-random variable using the relevant mathematical formula (white method) or represented by its digitized form obtained by a partial usage of the information concerning the existing law combined with a complementary application of some soft-model approach [31, 32, 33, 34]. The latter method is called a grey method.
In this paper a detailed analysis and comparison of the results obtained using white (hard) and grey (hard+soft) MCR methodologies applied to resolve the spectra of a three-component system of quenched fluorescence has been included. The presentation goes as follows: in Second section with five subsections the essential theoretical foundations of the employed methods are explicitly stated. Third section gives details of experimental conditions and sample preparation. Fourth section provides a discussion of the obtained results and is organized around two subsections. In the first subsection the results obtained for simulated dataset are examined while the second subsection dwells on the analysis of the results referring to real experimental dataset. In closing Fifth section the outcome of this study is succinctly summarized in four subsections.
Theoretical Background
Fluorescence Quenching
Matrix C^{ + } in Eq. (3) is called the left pseudoinverse of matrix C. Thus, the main task, as regards the decomposition of the spectra of multi-component mixture of fluorophores, consists in finding the Stern-Volmer constants, K_{SV}, for all involved components.
Rank Annihilation Factor Analysis
τ‐RAFA
κ‐RAFA
The iterative parameter, marked here as κ, becomes equivalent to Stern-Volmer constant. In other words, in the case of a three-component system, the expected value of a quenching constant should be equal to the κ value corresponding to the minimum value of the third eigenvalue of the covariance matrix formed from the difference matrix D.
In Authors’ opinion this simple κ-RAFA approach should be called a ‘direct’ method while the word ‘indirect’ would rather be reserved for the τ-RAFA methodology. In the present article the performance of both methods as well as the effect of the type of noise and its magnitude on the final results have been carefully investigated (for details see Results and Discussion).
‘Point-by-Point’ Optimization of Stern-Volmer Constants
A Brief Description of the Applied Algorithm
Matrix Representation of Stern-Volmer Profiles
The optimization methods discussed above allow for taking into account the fluorescence intensity at only one emission wavelength. Of course, it is possible to carry out a series of individual optimizations for all emission wavelengths, however, the obtained Stern-Volmer constants of a given fluorophore that theoretically should be equal, remain actually independent which may lead to a few hundreds of different values depending on a measuring point, λ. To solve this problem one has either to take an average or to get down to constructing some matrix versions of the optimization algorithm, which form the basis for the modern methods of the multivariate curve resolution.
By introducing corrections to matrix C which are brought about only by the change in the values of the Stern-Volmer quenching constants, a better conformity between the empirical data collected in matrix Y and the data contained in matrix Y_{opt} is achieved. Finally, as a result of the optimization process both the quenching constants and spectral profiles are obtained, on the basis of which the best description of the studied system can be proposed.
While in ‘classical’ approach to decomposition of multi-component spectra it is assumed that the recorded spectra of quenched fluorescence are inserted into a data matrix in their ‘unaltered’ form, yet in an approach that makes use of the idea of ‘spectral fractions’ this natural assumption is modified. The method takes advantage of a notably different form of the data matrix which is actually an extension of the original ‘point’ methods as proposed by Lehrer [39] and Acuña et al. [40]. The original data matrix Y is replaced by a matrix Y_{ f } in which the elements of each row are obtained by point by point division by the corresponding elements of the first raw of the original data matrix.
In Eq. (26) y_{0, i} represents the i-th element of vector y_{0}.
The methods described above are classified as ‘white’ methods because of the assumption concerning the fulfillment of the linear Stern-Volmer equation. Despite the useful approximation provided by a chemical model, the hidden disadvantage carried by hard methods are, in the considered case, the values of the quencher concentration assumed to be absolutely constant. However, it is well-known that even the best measurement procedure is endowed with uncertainties and therefore, as regards the assumed values of Q, some almost imperceptible departures are unavoidable. The solution to this problem may 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.
The ‘hard-soft’ methods of data modeling incorporate advantages of both the methods obeying the restrictive criteria of ‘white’ methods and the ‘black’ procedures void of any constraints except for non-negativity. This approach seems to combine two things that are mutually exclusive but there is no contradiction as it has been proven on the example of the MCR ALS (Multivariate Curve Resolution Alternating Least Squares) algorithm elaborated by Tauler et al. [42, 43].
Likewise in the case of the discussed ‘hard’ methods, the ‘grey’ (hard-soft) algorithm is operating on three matrices: original data matrix containing the measured multi-component spectra, Y, and with regard to particular components, the matrix of the fluorescence intensity decays (‘concentration’ matrix), C, and the matrix of spectral profiles, S^{ T }. The first step is analogous: the matrix C is built on the basis of known concentrations of the quencher and tentatively determined Stern-Volmer constants (this stands for the ‘white’ element). Then the initial matrix S^{ T } and the trial matrix Y_{opt} are generated. In the next step, however, a significant difference emerges: the concentration matrix is no more optimized only on the basis of the quenching constants, but by itself as a whole constitutes a parameter which undergoes a permanent optimization and adaptation process (this represents the ‘black’ element). To avoid the values without physical meaning the non-negativity constraint (a ‘white’ element) becomes superimposed on the profiles in matrices C and S^{ T }. Eventually, a pair of vectors, c_{ n } and s_{ n }, representing the emission intensity decay and the spectral profile (a spectrum) of a given component n, is generated.
Experimental
c_{CNA} = 7.68 ⋅ 10^{−6} | c_{DCNA} = 1.84 ⋅ 10^{−5} |
c_{ClCNA} = 9.47 ⋅ 10^{−6} | c_{LiBr} = 0, 0.0103, 0.0206, …, 0.2056 |
Results and Discussion
Synthetic Data
Stern-Volmer constants of individual components determined by two versions of RAFA for different noise types and levels
‘Noise’ | τ-RAFA | κ-RAFA | ||||
---|---|---|---|---|---|---|
K_{A} | K_{C} | K_{B} | K_{A} | K_{C} | K_{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 |
Q5S05 | 4.94 | 18.3 | 89.2 | 3.50 | 18.7 | 117 |
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_{3} = b_{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 functions. A problem of ambiguity of solutions is eliminated and so are additional calculations required to obtain the Stern-Volmer constants and spectral fractions from the optimized parameters (see “Theoretical Background”).
According to the Authors’ assumptions, all initial spectral fractions should be equal, but the nglm algorithm in MATLAB requires at least slightly different starting values – if they are identical, the procedure simply does not work properly.
On the basis of the averaged values of the Stern-Volmer constants it is possible to restore the concentration matrix C, which then can be used to resolve the multi-component fluorescence spectra into the emission profiles of pure substances. Nevertheless, another step ahead can be made, since the ‘point-by-point’ optimization can readily be replaced by the optimization performed as a whole on full data matrices.
Stern-Volmer constants of single components before and after optimization on simulated data with assumed noise level Q5%S05‰
Initial | Optimized | ||||
---|---|---|---|---|---|
5.00 | 20.0 | 100 | 4.487 | 19.57 | 123.6 |
4.75 | 19.0 | 100 | 4.486 | 19.57 | 123.5 |
4.50 | 18.0 | 110 | 4.486 | 19.57 | 123.5 |
4.00 | 15.0 | 120 | 4.485 | 19.56 | 123.5 |
49.0 | 50.0 | 51.0 | 4.485 | 19.56 | 123.5 |
Empirical Data
Stern-Volmer constants obtained for real data using different methods
Method | K_{CNA} | K_{ClCNA} | K_{DCNA} |
---|---|---|---|
EMP | 2.044 | 13.85 | 122.7 |
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 |
The measured three-component spectra of quenched fluorescence upon preliminary SVD pretreatment were resolved as well. However, a difference between the results of both approaches (with and without the SVD data preprocessing) in the case of white methods was practically unnoticeable. Moreover, as regards the grey MRC-ALS algorithm, the emission profiles of single constituents resolved upon the preliminary use of the SVD procedure are even more divergent from the expected spectra (measured for individual components) than those obtained for raw experimental data of the fluorophore mixture.
Conclusions
Application of RAFA
The analysis of the obtained results reveals that the method of ‘indirect’ rank annihilation factor analysis, τ-RAFA, can be successfully employed to determine the number of components in the multi-component system of quenched fluorescence and to estimate their Stern-Volmer quenching constants. Imperfection in determining the quencher concentration seems to have rather negligible effect on the final outcome, while a spectral noise influence cannot be ignored and appears to be a main limitation of the method – the results of RAFA for the data with spectral noise level higher than 1‰, should not be considered as reliable.
The ‘direct’ κ-RAFA method is very ‘noise-prone’ – both concentration inaccuracy and spectral noise insert influence on the final results, which are, moreover, diverging from the expected ones. The κ-RAFA algorithm should only be used as a tool in pre-analysis of the collected data or to confirm findings already unveiled by other methods - the Authors suggest using both RAFA approaches concomitantly, after initial determination of the number of principal components.
‘Point’ Optimization Methods
The ‘point’ methods accounted for as rather historical episodes are yet still used even today, especially to pinpoint some specific fragments of proteins in biochemical systems as molecular markers of cancer or virus spread [45, 46]. Application of these methods allows for minimization of computational resources since it does not require a continuous recording of the fluorescence spectra. A basic knowledge of the measured system is, however, essential due to a high sensitivity of the optimization algorithm to initial values of the optimized parameters. The experimenter should at least be aware of what are the possible values of the Stern-Volmer quenching constants of individual fluorophores and which emission lines are, in a broad sense, the most suitable for the analysis – ‘point’ approaches are highly sensitive to the data noise level.
White and Grey Methods of Data Modeling
It turns out that white algorithms of data modeling allow to resolve the composite fluorescence spectra with acceptable resemblance to the original pure component spectra. Shape of the calculated emission profile depends mainly on the spectral data noise level – calculations above 1‰ may be treated as uncertain. Concentration inadequacy, by contrast, results in shifting of the whole spectrum to lower or higher wavelengths. Fortunately, the initial values of the variable parameters entered to the optimization nglm algorithm do not have apparent influence on final outcome.
In order to evade, at least some of the mentioned above limitations, the range of the scrutinized noisy spectra should be reduced – this may easily be done through data transformation which is a part of the Acuña et al. matrix approach. Due to low costs of calculations and significant advantages, it is recommended to estimate an effective spectral range with the use of the ‘fractional’ data type technique prior to application of any hard optimization algorithms.
The grey MCR ALS approach operating on digitalized curves, i.e. curves with unknown functional forms, becomes independent of the assumed values of the quencher concentration which is an undeniable advantage of the method. Furthermore, the data noise at least up to the noise level of 1‰ seems to have a negligible effect on the final results, but the algorithm due to a decreased number of applied constraints is sensitive to initial entries of the concentration matrix – a pre-factor analysis should be performed. Despite the fact that the fluorescence spectra of single components resolved in this study by the MCR ALS approach are not so accurate as those obtained by hard modeling, the method remains undoubtedly a very useful tool for immediate evaluation of feasible solutions.
General Conclusions
Both white and grey methods of data modeling combined with two RAFA approaches made it possible to resolve a system of three-component spectra of fluorescence quenching. The retrieved emission profile appeared to be the most accurate for the substance with the highest Stern-Volmer quenching constant while the worst reproduced spectrum is that referring to the fluorophore with the Stern-Volmer constant in between the two extreme values. This probably might be justified by imperfection of sample preparation but in the case of simulated system such explanation fails – presumably neither hard nor hard-soft algorithms are good enough to correctly resolve the complex fluorescence spectra in the region of strongly overlapping emission bands. Another, more credible, explanation is that the values of both low Stern-Volmer constants are too close to each other so this substantially hinders the process of yielding the properly resolved spectra through optimization. The obtained spectra are, however, in Authors’ opinion, sufficiently consistent with the reference synthetic or experimental spectra of pure components.
Notes
Acknowledgments
The Authors wish to thank Mr. Ł. J. Witek for all his help, guidance and support provided in the lab.
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