Soft and hard multiway FRET-based investigation of

Chemometrics and Intelligent Laboratory Systems 139 (2014) 33–41

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Soft and hard multiway FRET-based investigation of interaction between drug and QD labeled DNA Mohsen Kompany-Zareh a,b,⁎, Somayeh Gholami a a b

Department of Chemistry, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran Department of Food Science, Faculty of Life Sciences, University of Copenhagen, Rolighedsvej 30, DK-1958 Frederiksberg C, Denmark

a r t i c l e

i n f o

Article history: Received 7 June 2014 Received in revised form 3 August 2014 Accepted 25 August 2014 Available online 6 September 2014 Keywords: Quantum dot FRET HTD Restricted Tucker3 Actinomycin D Drug–DNA interaction

a b s t r a c t Interaction of 7-aminoactinomycin D (7AAD) with duplex form of quantum dot (QD) conjugated DNA, is the subject of this report. Excitation emission fluorescence measurements based on FRET mechanism offer a powerful means of studying this process. During titration experiments of DNA with 7AAD, three-way data are obtained—fluorescence measurements as a function of excitation and emission wavelengths at different 7AAD concentrations. The PARAFAC algorithm is applied to resolve the recorded data array. Upon inspection of the results, however, deviation from trilinearity as an intrinsic property of FRET data is observed that results in chemically less meaningful profiles from PARAFAC. This difficulty is addressed by using the restricted Tucker3 algorithm. The restricted Tucker3 shows a better performance compared to PARAFAC in resolving the data sets. Whereas this approach is more flexible in the modeling of profiles, the non-unique resolution is the major disadvantage of the algorithm. Therefore, to assess complete analysis of the FRET data hard trilinear decomposition is tried. It is shown that HTD algorithm has succeeded in the unique calculation of concentration profiles and pure spectra of all species. Also, equilibrium constants for the hybridization (3.1 × 106 M−1) and the intercalation equilibrium (1.4 × 107 M−1) are estimated in a unique way using hard modeling. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Investigation of interactions between drugs and DNA is required not only to understand biological process and the study of some diseases but also to design more efficient anticancer drugs [1,2]. Among the various techniques that have been used to consider drug–DNA interactions, electrochemical methods and fluorescence spectroscopies using univariate data analysis showed great ability [2,3]. Fluorescence techniques are of great importance in this field due to their rapidness and high sensitivity to elucidate the drug–DNA interactions. However, some limitations still existed such as the low intrinsic fluorescence of natural oligonucleotides and rapid photobleaching of organic dyes [4]. Fluorescence resonance energy transfer (FRET) as an effective tool provides valuable information about the dynamics of biomacromolecules [5]. Recently, application of this method has dramatically raised to investigate the interactions between biological macromolecules such as DNA hybridization [6], protein–DNA binding [7] and conformational changes in nucleic acids [4]. Although many conventional organic dye labels have been intensively applied in FRET-based studies, narrow excitation, untunable emission ⁎ Corresponding author at: Department of Chemistry, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran. Tel.: +98 241 415 3123; fax: +98 241 415 3232. E-mail addresses: [email protected], [email protected] (M. Kompany-Zareh).

http://dx.doi.org/10.1016/j.chemolab.2014.08.009 0169-7439/© 2014 Elsevier B.V. All rights reserved.

spectra and photobleaching of these fluorophores are the major complications associated with them. To overcome these shortcomings of traditional organic dye-based chemosensors, luminescent semiconductor inorganic nanoparticles (quantum dots, QDs) were introduced in FRET-based studies [8–10]. There are several published reports on the use of QDs as an efficient fluorophore in FRET data of nanobiological process such as DNA hybridization [11–13]. Traditionally, the FRET based analytical applications involving QD resume to a zero- or first-order instrument. The major drawback for such data is the low ability to deal with high overlapped spectra. In such condition, it is almost impossible total resolution of all species by single wavelength measurements [14]. To overcome such problems along with univariate data and achieve maximum information from multicomponent complex systems, measuring the excitation− emission fluorescence (EEM) is offered. If the FRET based drug–DNA interaction experiments is performed over EEMs, a three-way data structure is obtained during titration process (excitation wavelength × emission wavelength × sample). To analyze three-way data, some well-established multiway chemometric methods such as PARAFAC and Tucker3 can be used. PARAFAC is the most frequently multiway technique used in analyzing the EEM data [15]. PARAFAC decomposition of trilinear multiway data arrays allows unique and robust estimations of the excitation and emission spectra and concentration profiles. Indeed, PARAFAC is particularly suitable for

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M. Kompany-Zareh, S. Gholami / Chemometrics and Intelligent Laboratory Systems 139 (2014) 33–41

the analysis of trilinear excitation–emission data sets. If any of the modes shows non-linearity, depending on the degree of deviation from the trilinearity, PARAFAC may become quite complex and fail to resolve the data properly. In the case of FRET between a donor–acceptor fluorophore couple, non-trilinearity is expected because of proportionality rank deficiency in both the excitation and emission profiles. Compared to PARAFAC, Tucker3 method as a more flexible model is able to deal with nontrilinear three-way data. However, rotational freedom is the main problem along with this method that causes nonunique results. On the other hand, interpretation of the obtained models arising from a high number of interaction terms between the factors is complicated. Consequently, a restricted Tucker3 model is proposed in the literature as a hybrid method of too restricted PARAFAC and too flexible Tucker3 models. In fact, this method involves some of the advantages of both methods to handle with small deviations from trilinearity [16]. Recently, a few papers were published on the application of multivariate methods to the study of biological process such as conformation transition of cyclic nucleic acids [17], duplex–quadruplex competition equilibria [14], interaction of DNA and AuNp [18], interaction of human BCL-2 gene with porphyrin [19] and folding process of DNA [4]. Also, there are limited reports on the application of multiway chemometric methods for the analysis of spectrofluorimetric data from QDs [20–22]. Multiway analysis study of the pH effect on excitation-emission fluorescence (EEM) of QDs is one of these reports [23]. Recently, our group has introduced multiway analysis of QD-mediated-FRET data as an attractive alternative to the traditional methods [1,2,24] of drug–DNA interaction detection [22]. Here, an almost similar chemical system (drug–DNA interaction) from another point of view is considered and authors have more attention on theoretical aspect of multiway analysis. Also, all possible forms for the photoluminescence signal of the drug–DNA complex were considered in this study, whereas just some of which were investigated in the previous one. Specifically, ability of hard trilinear decomposition (HTD) as a model based method was investigated in detail using simulated data. In summary, the performance of three powerful multi-way methods (PARAFAC, restricted Tucker3 [16,25,26] and HTD [27]) for analysis of FRET based data of drug–DNA interaction, was compared. The designed system was constructed by conjugation of a synthetic 22-base-long oligonucleotide to QD with peak emission at 705 nm (QD705). High informative excitation–emission measurements of the interaction of duplex DNA with actinomycin D as an attractive antitumor drug were carried out [28]. 2. Theory 2.1. PARAFAC In the PARAFAC model of a multi-way array, a tensor is decomposed into a sum of triple products of vectors. In fact, a PARAFAC model consists of loading matrices A, B and C with the same number of factors. For a three-way array D (I × J × K) the R component PARAFAC model can be represented in matrix notation as in Eq. (1): DIKP ¼ AIP CKP ⊙ B JP þ EIKP

ð1Þ

where A (I × P), B (J × P) and C (K × P) are the loading matrices in three modes; the symbol ⊙ denotes the Khatri-Rao product and E is the matricized residuals. 2.2. Restricted Tucker3 A (P, Q, R) component Tucker3 model decomposition for a given array D (I × J × K) can be expressed as in Eq. (2): T DIKP ¼ AIP GPQR CKR ⊗ B JQ þ EIKP

ð2Þ

where D (I × JK) is the matricized D; A (I × P), B (J × Q) and C (K × R) are the loading matrices in three modes; G (P × QR) represents the matricized core array in the first mode; the symbol ⊗ denotes the Kronecker product and E is the residual matrix. In restricted Tucker3 model, based on prior chemical/physical knowledge of the system some elements in the Tucker3 core array (G) are forced to zero. This means that certain meaningless interaction terms, from a chemical or physical point of view, can be removed. In this way, the constructed model with a smaller number of interaction terms is simpler and interpretable. 2.3. Hard trilinear decomposition For a given data matrix D, multivariate resolution analysis in the matrix equation can be demonstrated as:

T

D¼CS þE

ð3Þ

where C and ST contain the pure concentration and spectral profiles of the n mixture components associated with the row direction and the column direction of D, respectively, and E is the error-related matrix. Algorithm for model-based analysis of multivariate data will be discussed here briefly and more details are available in the literature. In hard-modeling approaches, multivariate data is fitted to previously defined models based on chemical equilibrium or kinetic laws [29]. The model allows the computation of concentration profiles C for each set of non-linear parameters. Then the accompanying set of molar absorptivities is estimated as a linear least-squares fit, ST = C+D, where C+ is the pseudo-inverse of the matrix C. Traditionally, hard trilinear decomposition (HTD) is applied for nonlinear fitting of the three-way data (D). To obtain the non-linear model parameters, one has to matricize the three-way array D and then fitting a proper model following Eq. (4). DKIJ ¼ CKP XPIJ þ EKIJ

ð4Þ

where, XP × IJ is a combination matrix of loading matrices B and A. For each set of non-linear parameters, concentration profiles of Ck × P are calculated and its corresponding set of spectral information XP × IJ is calculated as a linear least-squares fit, XT = C+D. Thus, for a given model, the sum of squares can be defined as a function of only the non-linear parameters, and these non-linear parameters that define C, are the only ones that need to be fitted in an iterative process. 3. Material and methods 3.1. Reagents and instruments 7-aminoactinomycin D (7AAD), 3-Mercaptopropionic acid (MPA), 4 (dimethylaminopropyl) pyridine (DMAP) and N,N-dimethylformamide (DMF) were from Sigma-Aldrich and used without further purification. Dithiothreitol (DTT), methanol, isopropanol and ethyl acetate were purchased from Merck and used without further purification. TOPO/TOP capped CdSe/ZnS core/shell QD in decane was from Invitrogen and transferred into aqueous solution according to reported procedures [30]. The 22 mer 3′ propylthiol-terminated oligonucleotide 5′-AGGGTTAGGGTT AGGGTTAGGG-(CH2)3-SH-3′ used in this study belongs to the human telomeric DNA. The oligonucleotide strands were purchased from MWG Biotech and used as received. All solutions were prepared in 10 mM phosphate buffer saline (PBS) containing 0.05 M NaCl, pH = 7.3. The EEM fluorescence spectra were recorded using a Cary Eclipse Spectrofluorimeter equipped with an 80 Hz Xenon lamp.


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Scheme 1. Schematic of the QD-dsDNA:7AAD FRET system. (a) Conjugated QD with single strand DNA forming a duplex DNA (QD-dsDNA) in the presence of complementary strand. (b) Upon intercalation of 7AAD within duplex base pairs, energy is transferred from the excited 7AAD to QD through the FRET process [22].

3.2. QD-DNA conjugates The TOPO/TOP stabilized native QDs in decane must be made water soluble in order to conjugate the thiol-terminated DNA. For this purpose direct ligand exchange with 3-mercaptopropionic acid (MPA) was applied [30]. At first, 3.2 mL of the 75/25 anhydrous methanol/ isopropanol mixture was added to 0.8 mL of QD solution (1 μM). Then the solution was centrifuged for 8 min at 3000 rpm and the supernatant was discarded and the wet precipitate was reacted with an excess of 3-mercaptopropionic acid (0.1 mL, 1.15 mmol) in 1.0 mL of DMF. This solution was vortexed for about 1–2 min and sonicated for 30 min. Then it was stored for 1–4 days at room temperature and under argon. Finally, solubilization was completed by reaction of QD (150 μL) with DMAP (8.0 mg, 0.065 mmol) dissolved in 0.4 ml DMF and the solution was centrifuged for 3 min at 3000 rpm. Disulfide bonds of thiol-modified oligonucleotides were cleaved by dispersion of 2 ml DNA (1 μM) into 2 ml of a 4 mM DTT, 0.17 M phosphate buffer solution (pH 8) and stored for 16 h at 37 °C. Before conjugation of oligonucleotide with QD, the unreacted DTT was removed using extraction with ethyl acetate (4 ml) and this step was repeated two additional times. The precipitate MPA-QD was dissolved in 0.6 ml DNA solution (1–2 ODs/ml, 1 μM). After 12 h, 0.15 M NaCl was added to the QD-ssDNA solution and stored for an additional 12 h. Then, the concentration of NaCl was raised twice and the mixture was aged for a further 40 h before titration with 7AAD. To investigate the interaction of double strand DNA (dsDNA) with 7AAD, 100 μL solution of QD-ssDNA (OD b 1 μM) was diluted with 350 μL of PBS and fluorescence signal was recorded as EEM spectrum. At the 2nd step, an aliquot (50 μL) of complementary strand solution (cDNA, 0.3 μM) was added to form QD-dsDNA. In the next steps, small volumes of 7AAD stock solution (0.64 μM) were progressively added to the QD-dsDNA solution, and the EEM fluorescence spectra were measured. 4. Results and discussion

Interesting points were observed in the second step of the experiment where the complementary strand was added into the solution. A considerable change on the excitation spectrum without any substantial change on the emission spectrum was observed (Figure S1, Supporting Information). By examining the obtained results, it can be concluded that there is a significant difference between environments of QDs in two solutions. In the literature there are some reports on non-specific adsorption of oligonucleotides on MPA–QDs [31]. Therefore, in the first sample, it is likely for the single strand DNA (ssDNA) to be adsorbed on QD surface. According to the literature, adsorption of double strand DNA (dsDNA) on QD surface is less than that of ssDNA [31]. The 2nd step includes hybridization of complementary strand to form double strand DNA which results in the decrease in the adsorption of DNA on QD surface. In other words, in the 2nd sample oligonucleotide tendency to be adsorbed on QD was decreased. As a result, it is not surprising to observe a substantial change in spectral properties of QD-ssDNA during hybridization process. The absorption and photoluminescence emission spectra for QD-ssDNA and 7AAD are given in Figure S2. As it shows, the absorption spectrum of QD strongly overlaps with the emission spectrum of 7AAD. Therefore, within a distance on the order of the Förster radius of the 7AAD–QD FRET pair, the energy of excited 7AAD could be transferred to QD. As shown in Scheme 1, FRET sensitized emission of QD-dsDNA would be occurred as a result of 7AAD intercalation [22]. Taking a look at the recorded excitation–emission landscapes during titration of QD-dsDNA by 7-AAD, (titration step 2 to end), it is almost impossible to detect the FRET signal in constructed complex (QD-dsDNA:7AAD). Indeed, there is substantial overlap in peak emission of QD-dsDNA and 7AAD as acceptor and donor in FRET process, respectively. It was really difficult to explain about the observed increase variations in the fluorescence signal at peak emission nearly 705 nm (corresponding to λmax of QD-dsDNA). In other words, the FRET signal cannot be distinguished from donor signal by aid of common classic techniques. Consequently, appropriate multiway methods would be useful to deal with such complex data to detect FRET signal.

4.1. Initial studies 4.2. Three-way data analysis A QD705 labeled single strand DNA (QD-ssDNA) was hybridized to its complementary (cDNA) and interaction of 7AAD with the constructed double strand (QD-dsDNA) was investigated. The EEM fluorescence spectra monitored during the stepwise addition of aliquots of 7AAD solutions were stacked on each other in the third dimension to construct three-way data. To have an insight to the system under study, the recorded EEM data for each sample was investigated.

The three-way array was generated by stacking each of EEM data for a sample into an array D (I × J × K). Principal component analysis (PCA) of the unfolded data array in all of its three modes was used to estimate the number of components (Nc). The result of PCA indicated that there are at least three principal components in the three modes of array (Figure S3 in the Supporting Information).

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The considered three-way data were analyzed using PARAFAC, restricted Tucker3, and hard trilinear decomposition (HTD). The PARAFAC method was carried out with applying non-negativity constraint on all of three mode loadings. The applied constraints for the loadings of restricted Tucker3 method were non-negativity, selectivity on some profiles in the concentration mode, as well as restriction on core elements. The stop criterion for both approaches was a maximum number of 10,000 iterations or change (relative or absolute) in fit of 10− 12 (whichever fulfilled earlier). The nonlinear parameters together with analytical concentration of QD-ssDNA were optimized using HTD. The concentration (C) and spectral profiles of all species are reported as well. 4.2.1. Three-components PARAFAC analysis In the FRET case studies, some deviations from trilinearity are expected because of rank deficiency problem. Nonetheless, from a mathematical point of view, the three-component PARAFAC analysis is acceptable. Indeed, the low relative residual sum of squares of 0.04%, reasonable core consistency of 87.1%, and considerable explained variance of 99.95% imply that this model is proper. Resolved excitation, emission, and concentration profiles for the PARAFAC resolution approach are shown in Fig. 1a. Taking a look at the calculated profiles, it seems that the resolved profiles of the 1st (QD-ssDNA) and 3rd (7AAD) component indicate meaningful shapes. Their concentration profiles are in complete agreement with the expected experimental trends. For instance, the concentration profile of the QD-ssDNA species (profile in blue) is decreasing with addition of 7AAD, reflecting the consumption of this compound during titration steps. The estimated concentration profile of 7-AAD is increasing, as it is a titrant and added to solution. Also, the shapes of excitation and emission profiles of both components are highly similar to their pure profiles. In contrast,

the resolved profiles of 2nd species (green) cannot be attributed to the QD-dsDNA:7AAD or QD-dsDNA individually. By considering the excitation and emission profiles of this component, it is clear that it is not explaining the FRET signal in QD-dsDNA:7AAD, QD signal in QD-dsDNA, or a combination of both. On the other hand, its concentration profile shows unreasonable variations during titration steps. Consequently, three-component PARAFAC was unable to resolve FRET signal as a one of the important sources of variation in the considered data. The results confirm that although the estimated dimensionality is [3 3 3], the array rank of data from PARAFAC is not three. 4.2.2. Four-components PARAFAC analysis This model with low relative residual sum of squares of 0.03%, explained nearly 99.97% of variation in the data set. However, the negative core consistency with four-component PARAFAC represents an unsuitable model. The negative core consistency can be attributed to the non-trilinear structure of data and also means that the number of applied components in the PARAFAC model is higher than the dimensionality in one, two or three modes. The obtained profiles are in Fig. 1b. The calculated profiles of the first three components in this model were the same as those explained for the three-component PARAFAC. The fourth component's excitation and emission profiles were the same as QD-ssDNA. Consequently, it did not also demonstrate spectral properties of FRET signal in QD-dsDNA:7AAD species. As the results denote, the entire information about the system is not available even with considering a higher number of factors in PARAFAC model. Due to the high spectral overlap of compounds, the PARAFAC model is not totally successful to resolve chemically meaningful profiles from FRET based data. Furthermore, the non-unique resolution of this method can be confirmed by examining Kruskal's condition. The Kruskal's condition is

Fig. 1. The calculated profiles from applying PARAFAC model on data with considering (a) 3 principal compounds and (b) 4 principal compounds. At both models just two out of three or four profiles corresponding to 7-AAD and QD-ssDNA species are meaningful.

M. Kompany-Zareh, S. Gholami / Chemometrics and Intelligent Laboratory Systems 139 (2014) 33–41 Table 1 Designed restricted cores for Tucker3 analysis using five different patterns. The core at each model contains 3 rows, 3 columns, and 4 layers, corresponding to excitation, emission and concentration profiles, respectively. Cells in black that are numbered with 1 are related to interaction terms.

Model

Restricted [3 3 4] Core 1

1 1

1

1

1

1 1

2

1

1

1

1

1

1

1

1

3

1

1 1

4

1

1 5

1

1 1 1

1

a sufficient condition for a PARAFAC model to achieve unique parameter estimation. This condition is kA þ kB þ kC ≥2R þ 2

ð5Þ

Where kA, kB and kC are the k-ranks of the loading matrices A, B and C, respectively and R is the number of components in the PARAFAC model. Due to the fact that the data is rank deficient in all three modes (rank overlap in modes A and B, closure rank deficiency in mode C), it gives a sum of k-ranks 1 + 1 + 3 = 5. Therefore, the Kruskal's condition is not met due to the sum of k-ranks is less than 2 × 4 + 2 = 10 (or 2 × 3 + 2 = 8). As a result, uniqueness is not guaranteed. 4.2.3. Restricted Tucker3 analysis Determining the correct core size to design an efficient restricted Tucker3 model would be achieved using physicochemical information of the system. Note the data involve four spectroscopically active chemical compounds namely QD-ssDNA, QD-dsDNA, 7AAD and QD-dsDNA:7AAD. In this way, the chemical rank of the system is four.

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As a consequence, there are 4 profiles with different shapes in the concentration mode and thus the core size of four was supposed in the concentration mode although dimensionality in this mode was estimated as three. Furthermore, in EEM array of FRET based data rank deficiency is unavoidable as an intrinsic property of FRET phenomena. The emission and excitation of FRET species are similar to acceptor and donor, respectively, hence such data is always rank deficient in both the excitation and emission modes. With regard to this information a core with size of 3(excitation) × 3(emission) × 4(concentration) was tried which is equal to the number of profiles with different shapes in each mode. In other words, this core contains 3 rows, 3 columns, and 4 layers which correspond to the excitation, emission, and concentration profiles, respectively. The 1st to 4th layers are corresponding to chemical compounds QD-ssDNA, QD-dsDNA, 7AAD, and QD-dsDNA:7AAD respectively. The considered core with the size of [3,3,4] contains 36 possible interaction terms of the components. Nonetheless, some of these interactions are remarkable and most of them are negligible based on the chemical information of the system. According to the rules, the number of remarkable interactions that is equal to the number of nonzero core elements is between 4 and 6. The difference between these cores was in their 4th layer which corresponds to the concentration profile of the drug–DNA complex (QD-dsDNA:7AAD). First to third layers contain only one fluorophore and thus one fluorescence signal is the source of variance in each layer. As a result, each of the first three layers of core includes only one nonzero element. The fourth layer (correspond to drug–DNA complex) involves two fluorophores, hence at least two fluorescence signals are the source of variance in this layer. On the other hand, FRET signal can also be another source of variation. It means that the drug–DNA complex signal may include any of the three fluorescence signals namely 7AAD, QD-dsDNA and/or FRET. If one of these fluorescence signals is the source of variance in the 4th layer, the number of remarkable interaction terms is four. The maximum number of 6 interaction terms is possible where all of three fluorescence signals are the source of variance in the 4th layer. Consequently, among the 36 possible interaction terms, the number of remarkable interactions is between 4 and 6. In this way, five possible patterns for the restricted core were considered (models #1 to #5). Overall, five different models of restricted core namely #1 to #5 were designed and examined to demonstrate which one(s) of the fluorescence signals is (are) the source(s) of photoluminescence in the complex species. In model #1 only FRET signal was examined as the source of fluorescence (Black cells in Table 1 are related to remarkable interactions). Therefore, the interaction of 7AAD excitation and QD-dsDNA emission was supposed as a remarkable interaction in this layer. Models # 2 to #4 investigate the models that in addition to the FRET, QD-dsDNA, 7AAD or both of them can produce signal in the complex product of intercalation, respectively. Model #5 examines the case

Fig. 2. The resolved emission, excitation and concentration profiles using restricted Tucker3 decomposition of core #4. All resolved profiles are completely consistent with experimental trends and knowledge.

38


profiles and their agreement with experimental information can be the criterion to select the best core. Models #1, #3, and #5 were not completely successful to describe the chemically meaningful concentration profile for double strand DNA (QD-dsDNA). However, the obtained profiles for all compounds by models #2 and #4 were totally similar and acceptable as shown in Fig. 2. The resolved emission and excitation profiles for QD-ssDNA (green profile) and QD-dsDNA (red profile in Fig. 2) illustrate a significant difference between the excitation profiles without any substantial changes on emission profiles. The EEM data for individual sample of QD-ssDNA and QD-dsDNA confirm this observation (Figure S1). Both #2 and #4 models confirm that along with the FRET signal the signal of QD-dsDNA can be observed in the complex species. As the results clearly demonstrate, restricted Tucker3 analysis not only proves the FRET signal in drug–DNA complex but also distinguishes it from the signal of QD-dsDNA and 7AAD. The estimated FRET profiles by the proposed approach were meaningful and interpretable whereas such robust information could not be obtained from classical analysis. The restricted Tucker3 is able to handle perfectly the data with deviations from trilinearity. This method is completely successful to resolve chemically meaningful profiles from non-trilinear FRET data. It deals with non-trilinear data by applying chemical/physical knowledge of the considered system to remove some chemically meaningless interaction terms. In this way, new models will be generated which are chemically more meaningful and also interpretation of models is simpler than the unrestricted one. However, it does not succeed in a unique resolution. As a consequence, to attain unique and complete analysis of the FRET based intercalation of anti-cancer drug, hardmodeling method was tried. 4.2.4. Hard modeling analysis In completely unknown systems, it is rather difficult and time consuming to postulate a correct chemical model. Any additional information or chemical knowledge contained in the data would be strongly helpful to construct the proper model. As it was mentioned above, titration of DNA solution with 7AAD was carried out and EEM fluorescence spectra were recorded throughout the titration. There are several reports on the literature that confirm the formation of a spectroscopically active complex between actinomycin D and synthesized oligonucleotides [32]. Moreover, hybridization of two complementary single strands DNA is unavoidable. With regard to this information, a two-step equilibrium model was suggested to form a ternary complex QD-dsDNA:7AAD. At first, two single strands DNA (QD-ssDNA and cDNA) were hybridized and then the 7AAD binds on the QD-dsDNA to form the drug–DNA complex (QD-dsDNA:7AAD). The proposed model can be represented by the following equilibria: β110

QD−ssDNA þ cDNA ⇔ QD−dsDNA

Fig. 3. Sum of square of residual surface as a function of two nonlinear parameters β110 and β111 (a) for real data, (b) simulated data, and (c) simulated data after augmentation with ML formation data.

that there is no FRET signal and just the two fluorophores (QD-dsDNA, 7AAD) are the source of variance in the 4th layer. Consequently, all possible forms for the photoluminescence signal of drug–DNA complex would be considered in these five models. It is interesting to note that all the five models reach the same error and explained variance values that confirm that there is no significant difference between the models. On the other hand, all models result in the same excitation and emission profiles. Thus the model selection according to the fit and spectral profiles shape is impossible. There is just one difference between these models which is chemically meaningless or meaningful shape of concentration profiles. Therefore, the shape of obtained concentration

β111

QD−ssDNA þ cDNA þ 7AAD ⇔ QD−dsDNA : 7AAD

ð6Þ

ð7Þ

where, β110 and β111 are the overall equilibrium constants of hybridization and complexation, respectively. Nonlinear fit of the unfolded EEM fluorescence data (DMLX) to the described hard chemical model was performed utilizing hard trilinear decomposition (HTD) to estimate the optimum nonlinear parameters (β110 and β111), and as a result, the concentration profiles of components. Excitation and emission spectral profiles of components were estimated from the application of HTD on the three-way data, as well. Note that if the concentration profiles matrix is of full rank, the HTD algorithm leads to a unique estimation of nonlinear parameters and spectral profiles (excitation and emission spectra). However, if the C matrix is rank-deficient, all nonlinear parameters are well estimated but proper estimation of spectral profiles is impossible. It should be


mentioned that if the C matrix of concentration profiles is of full rank, the rank deficiency in two other modes of data (here, excitation and emission modes) is not problematic for HTD performance. In order to gain robust information about the performance of HTD under different conditions, it is imperative to simulate the different three way sets. For this, a non-rank-deficient set CML and a rankdeficient set CMLX of concentration profiles are created based on the mass-action law of the models ‘M + L → ML’ and ‘M + L → ML in the presence of interference X’, respectively. In both data sets, DML and D ML − X, all species are spectroscopically active while emission (Em) and excitation (Ex) spectra in both sets are rank-deficient. Figure S4 displays the result of the hard-modeling fit. In the case of data set DML the resolved spectral profiles are completely well defined (Figure S4a). Also, the nonlinear parameter (β110) and analytical concentration of M are estimated uniquely, as Figure S5a demonstrates. It is noteworthy that model-based analysis of DML data in the conditions that formation constant of ML species is relatively high, will not be unique. In fact, in this situation no observable changes happen in concentration profiles as the high value of formation constant changes. However, rank deficiency in CML − X does not allow unique resolution of spectral profiles (Figure S4b-S4C). Comparing Figure S4b with S4C demonstrates that the spectral profiles of X, which cause the linear dependency in the concentration mode of DML − X, cannot be successfully resolved while the other species such as M and L, despite the rank overlap problem in spectral modes, were resolved completely in all modes. Note that the nonlinear parameter is also calculated uniquely in this condition as shown in Figure S5b. It can be concluded that complete and robust HTD analysis of the three-way set is possible if and only if the C matrix is full rank. In other words, rank deficiency in concentration mode is the main problem for HTD performance and rank deficiency in other modes does not make any problem. The linear dependency of the concentration profiles must be circumvented before applying HTD. For that purpose, several methods were offered in the literature [33–35]. One of the most popular methods is the use of the independent experiments to gain suitable external spectral information of dependent species [35]. In that way it would be possible to completely resolve data using known spectra augmentation and analysis of all data. For the described experimental titration data of QD-dsDNA by 7-AAD the proposed model results in linearly dependent concentration profiles. In fact, the chemical rank that is equivalent to the number of spectroscopically active chemical compounds is four (cDNA is spectroscopically inactive). However, with regard to PCA results on concentration mode (Figure S3) the rank of concentration matrix C (pseudo rank) is reduced to three. The pseudo rank of this matrix is less than the chemical rank, whereby rank deficiency in this mode is confirmed. It is noteworthy that for the previously quoted equilibria model, closure rank deficiency seems unavoidable. The rank-deficiency problem is clearly the case in the two other modes of data, as well (Figure S3). Thus the nonlinear fit analysis would result in correct formation constants but wrong spectral profiles of some species. The task is to eliminate the rank-deficiency in the concentration matrix. To overcome this problem the available additional information with respect to the spectra of 7AAD is applied. Augmentation of the pure EEM fluorescence data of drug with three-way titration array, allowed us to successfully eliminate the dependences in concentration matrix. The interesting observation was the non-unique assignment of formation constants (β110 and β111) while the rank-deficiency was successfully eliminated. Fig. 3a displays the lack of fit (%lof) as a function of β110 and β111 parameters. A minimum line for %lof was observed whereas a minimum point was predicted for a full rank concentration mode, and in the presence of a hard model (unique solution). In order to solve this problem, the considered model and applied methodology was investigated precisely using simulated data. For this, a rank-deficient set CMLX of concentration profiles was simulated for titration of ML by ligand X based on the mass-action law of the model

39

Fig. 4. Variations of the lake of fit (lof) versus nonlinear parameters β110 and β111 with applying HTD method on the experimental data from titration of QD-dsDNA by 7-AAD, augmented with QD-dsDNA formation data.

‘M + L → ML and ML + nX → MLXn’. In the created data set DMLX, four species M, ML, X, and MLX were spectroscopically active exactly similar to the experimental conditions. Overall, the simulation was carried out at equal conditions and using the same procedure as the experimental case. The linear dependence between the columns of CMLX matrix was totally removed by the augmentation with known spectral data of X species (DX). The observed results of HTD for real data were also confirmed using the simulated data for the considered model (Fig. 3b). A point is that in the considered methodology the 1st sample involves M and L to form ML complex. At the following titration steps the X species was added to the solution. In fact, there is not enough information about the formation of ML species and thus its properties are not well defined. It is not surprising that titration of ML solution with ligand X, to form the complex MLX, does not provide sufficient information about the formation constant of ML. This reasonably complex system renders impossible to find a unique estimation for the formation constant of ML complex. To confirm this suggestion, an individual three-way data of ML formation was created and augmented with the MLX formation data. In this way, the entire information of two equilibria was considered together and it would be possible to uniquely estimate the nonlinear parameters. As it was expected, in this case a minimum point for the sum of squares of residual surface is obtained as Fig. 3c clearly shows. Generally, the procedure mentioned above was applied for the considered experimental data. Complete titration of QD-ssDNA with cDNA was carried out and augmented with the QD-dsDNA:7-AAD formation data. In the next step, by applying HTD on the augmented data, the formation constant of hybridization (β110) and the intercalation constant (β111) were optimized (Fig. 4). Analytical concentration of QD-ssDNA was determined by nonlinear fit analysis, as well (Table 2). The concentration of QD-ssDNA is a Table 2 Calculated overall and stepwise equilibrium constants for DNA hybridization (β110, K1,10) and 7AAD intercalation for stoichiometry 1:2 of DNA:7-AAD (β112, K11,1), and analytical concentration of QD-ssDNA using nonlinear fit analysis. Parameters

a b c

β110

β112

a

3.1 × 106

6.3 × 1020

3.1 × 106

K1,10

2

a

b

Canal. of QDssDNA (μM)

c

1.4 × 107

0.24

5.3

K11,1

lof%

β110 = K1,10 and β112 = K1,10 × (K11,1) . Initial concentration of QD-ssDNA. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ∑ e2 Lack of fit (lof), lof ð%Þ ¼ 100 u t i;j ij ; where eij are the elements of the residuals 2

∑i;j dij

matrix E and dij are the elements of the data set, D.

40


quantum-dot-mediated FRET has been investigated. The resolved spectral and concentration profiles by restricted Tucker3 were chemically more meaningful than those obtained from PARAFAC. The restricted Tucker3 method could effectively resolve the sources of variation in the drug–DNA complex, although it does not succeed in a unique resolution. The unique and complete resolution of data was performed by HTD as a hard method with overcoming the rank deficiency problem in the concentration mode. The current study shows the usefulness of restricted Tucker3 and HTD approaches for dealing with a rank deficient system of drug–DNA interactions. This study is a successful application of multi-way chemometric methods in modeling of nanobiotechnological systems with highly overlapped spectra. Conflict of interest There is no conflict of interest among authors. Acknowledgments The authors wish to thank the Institute for Advanced Studies in Basic Sciences (IASBS) for supporting this study. Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.chemolab.2014.08.009. References

Fig. 5. (a) Resolved emission and excitation and (b) model based concentration profiles at the fitted (optimum) parameters using hard trilinear decomposition method.

measure of the surface modification yield of QD with DNA. It should be noted that the yield was calculated without applying any external standard and any multivariate calibration modeling. In accordance with lowest lack of fit (lof), the best fit was achieved with the model that described the 1:2 complex formation (QD-dsDNA: 7AAD). Details are given in Table 2. Fig. 5 displays the resolved excitation, emission and model based concentration profiles at the fitted parameters. Summarizing, all the spectra obtained by this method are well-resolved and completely in accord with the experimental data for individual sample of some independent experiments. For instance, the emission spectrum for QD-ssDNA remains globally conserved upon hybridization with its complementary single strand DNA. However, its excitation spectrum has remarkably changed during the process of QD-dsDNA formation. In other words, the excitation spectra of QD-dsDNA and QD-ssDNA are different and their emission spectra are almost equal. The resolved excitation and emission profiles of 7AAD are totally in agreement with its pure spectrum too. Also, hard-modeling methods are able to resolve the concentration profile of all species whether they are spectroscopically active or not. Therefore, the concentration profile of cDNA which did not contribute in the rank of the data matrix was resolved successfully (Fig. 5b, green profile). 5. Conclusion The performance of three common multi-way techniques PARAFAC, restricted Tucker3, and HTD in resolving a non-trilinear data set of

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