# Quantum-SAR Extension of the Spectral-SAR Algorithm. Application to Polyphenolic Anticancer Bioactivity

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## Abstract

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## 1. Introduction

_{50}), associated with various types of bioaccumulation and toxicity [24].

## 2. QuaSAR Methodology

- endpoint spectral norm$$\left|\left||{Y}_{l}\rangle \right|\right|=\sqrt{\langle {Y}_{l}|{Y}_{l}\rangle}=\sqrt{\sum _{i=1}^{N}{y}_{il}^{2},}\hspace{1em}l=\overline{1,C}$$

- algebraic correlation factor$${R}_{ALG,l}=\frac{\Vert |{Y}_{l}\rangle \Vert}{\Vert |A\rangle \Vert}=\sqrt{\frac{\sum _{i=1}^{N}{y}_{il}^{2}}{\sum _{i=1}^{N}{A}_{i}^{2}},}l=\overline{1,C}$$

- spectral path, with the distance defined in the Euclidian sense as:$$\left[l,{l}^{\prime}\right]=\sqrt{{\left(\Vert |{Y}_{l}\rangle \Vert -\Vert |{Y}_{{l}^{\prime}}\rangle \Vert \right)}^{2}+{\left({R}_{l}-{R}_{{l}^{\prime}}\right)}^{2}},\forall \left(l,{l}^{\prime}\right)=\overline{1,C}$$

_{l}corresponds to the measured activity A

_{l}defined as logarithm of inverse of 50%-effect concentration (EC50), see bellow, both modulus of Y

_{l}vectors and R values have no units so assuring the consistency of the Equation (4).

- least spectral path principle, formally shaped as:$$\delta \left[{l}_{1},\dots {l}_{k}\dots ,{l}_{M}\right]=0;\hspace{1em}{l}_{1},\dots ,{l}_{k},\dots ,{l}_{M}\hspace{0.17em}:ENDPOINTS$$

_{1},..., α

_{M}) are enough to be considered for a comprehensive (and self-consistent) mechanistic analysis [34–40].

- ▪ inter-endpoint norm difference (IEND),$$\Delta {Y}_{l|{l}^{\prime}}=\Vert |{Y}_{{l}^{\prime}}\rangle \Vert -\Vert |{Y}_{l}\rangle \Vert ,\hspace{1em}(l,{l}^{\prime})\in \hspace{0.17em}\left\{{\alpha}_{1},\dots ,{\alpha}_{M}\right\}$$

- ▪ inter-endpoint molecular activity difference (IEMAD),$$\Delta {A}_{i|j}^{l|{l}^{\prime}}={A}_{j}^{l\text{'}}-{A}_{i}^{l}=\text{ln}\frac{1}{{\left(E{C}_{50}\right)}_{j}^{{l}^{\prime}}}-\text{ln}\frac{1}{{\left(E{C}_{50}\right)}_{i}^{l}}=\text{ln}\frac{{\left(E{C}_{50}\right)}_{i}^{l}}{{\left(E{C}_{50}\right)}_{j}^{{l}^{\prime}}}$$

_{1},..., α

_{M}for which the inter-endpoint norm difference is given by Equation (6).

_{50}inter-molecular transformation:

_{l|l′}→ iΔY

_{l|l′}outside the factor ${q}_{i|j}^{l|{l}^{\prime}}$. Remark that although the differences in Eqs. (6) and (7) were consider mathematically, along the “arrow” i-to-j, the “quantum transformation” from Equation (9) suggests that the bio-chemical-physical equivalence (metabolization) of the concentration effects evolves from-j-to-i, revealing a typical quantum behavior with the factor ${q}_{i|j}^{l|{l}^{\prime}}$ playing the propagator role as the quantum kernels in path integral formulation of quantum mechanics [48].

- ○ it involves the wave-type expression of molecular effect of concentration, however, for special selected molecules (the fittest out of the C-models) and for special selected paths (the least for the M-ergodic assembly), being M and C related by Equation (1a);
- ○ it provides the specific transition or specific transformation of the effect of a certain molecule into the effect of another special molecule out from the N-trained molecules, paralleling the phenomenology of consecrated quantum transitions;
- ○ it has the amplitude of transformation driven by the so called quantum-SAR factor of an exponential form$${q}_{i|j}^{l|{l}^{\prime}}=\text{exp}\left(\Delta {A}_{i|j}^{l|l\text{'}}-\Delta {Y}_{l|{l}^{\prime}}\right)$$
- ○ it allows the identity$${\left(E{C}_{50}\right)}_{i}^{l}={\left(E{C}_{50}\right)}_{i}^{l}$$$${\left(E{C}_{50}\right)}_{i}^{{l}^{\prime}}={\left(E{C}_{50}\right)}_{i}^{l}\frac{1}{{q}_{i|j}^{l|{l}^{\prime}}}\text{exp}\left(-i\Delta {Y}_{l|{l}^{\prime}}\right)$$
- ○ it has a “phase” with unity norm, in the same manner as ordinary quantum wave functions, allowing the inter-molecular “real” quantum-SAR transformation$$\left|{\left(E{C}_{50}\right)}_{i}^{l}\right|={q}_{i|j}^{l|{l}^{\prime}}\xb7\left|{\left(E{C}_{50}\right)}_{j}^{{l}^{\prime}}\right|$$
- ○ when multiple transformations take place across paths with multiple linked models, say (l, l’, l’ ’), the inter-molecular transformation i→j→t is characterized by the overall quantum-SAR factor (10) written as product of intermediary ones$${q}_{i|t}^{l|{l}^{\u2033}}={q}_{i|j}^{l|{l}^{\prime}}\xb7{q}_{j|t}^{{l}^{\prime}|{l}^{\u2033}}$$$$\begin{array}{l}\left|{\left(E{C}_{50}\right)}_{i}^{l}\right|={q}_{i|t}^{l|{l}^{\u2033}}\xb7\left|{\left(E{C}_{50}\right)}_{t}^{{l}^{\u2033}}\right|\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={q}_{i|j}^{l|{l}^{\prime}}\xb7\left|{\left(E{C}_{50}\right)}_{j}^{{l}^{\prime}}\right|={q}_{i|j}^{l|{l}^{\prime}}\xb7\left({q}_{j|t}^{{l}^{\prime}|{l}^{\u2033}}\xb7\left|{\left(E{C}_{50}\right)}_{t}^{{l}^{\u2033}}\right|\right)\end{array}$$$${q}_{{i}_{1}|{i}_{M}}^{{l}_{1}|{l}_{M}}=\prod _{w=2}^{M}{q}_{{i}_{w-1|{i}_{w}}}^{{l}_{w-1}|{l}_{w}}$$
- ○ Equation (9) supports the self-transformation as well, with the driven qua-SAR factor given by:$${q}_{i|j=i}^{l|{l}^{\prime}}=\text{exp}\left(-\Delta {Y}_{l|{l}^{\prime}}\right)$$

## 3. Application to Flavonoids’ Anticancer Bioactivity

_{50}) of the maximum increase in mitoxantrone (MX) inhibitor substrate accumulation (interaction) with breast cancer resistance protein (BCRP), helping in reversing the multidrug resistance (MDR) mechanism of overexpressing MCF-7 MX100 cancer cells [51–54].

_{TOT}) Hantsch correlation variables [56], see Table 1, to successively provide the QSAR, S-SAR and finally to unfold the Qua-SAR analysis.

_{TOT}unfolds some statistically sensitive role in ligand (MX)-receptor (BCRP) binding (model Ic). The last assertion may also be sustained by going to the two-correlated parameters endpoint models, when one can see the confirmation of the stericity role through E

_{TOT}correlation variable: while combination LogP∧POL does not improve the statistical correlation of model IIa significantly over single-parameter LogP∨POL correlations, the total energy presence provides better and better correlation behavior as it is combined with LogP (the model IIb) and with POL (the model IIc), respectively. Instead, when all the Hansch structural variables are taken into account the model III is generated with appreciable statistical correlation respecting the other computed combinations.

_{TOT}correlation of model Ic as well as mixed correlations of bi-variable models IIb and IIc. Therefore, the mechanistic “alchemy” of structural features on molecular activity seems complex enough when all hydrophobicity, electrostatic and stericity influences combine as they are reciprocally activating one each other with a superior resultant in modeling ligand-receptor binding.

_{TOT}in model III they considerably enrich the single E

_{TOT}correlation power of model Ic. Such behavior shows that orthogonal, i.e. independent, descriptors may provide better results when are combined than when considered apart due to the increase of the (inter) correlation space.

_{TOT}). This is the case of the most fitted molecule on the most correlated endpoint (III) appeared to be no.3 (naringenin), with low activity on the observed range compared with the no. 25 (7,8-benzoflavone) in Table 1. Consequently, one can say that the first half of the observed activities in Table 1 may be attributed to certain physicochemical indicators with clear mechanistically roles, while the rest of observed activities may be due to other unidentified specific structural descriptors or even to non-specific ones (rooting in the sub-quantum nature of the particular observer-observed system). Nevertheless, this lower activity prescribed by the computational results is in accordance with the so called “homeopathic principle” prescribing cure by moderate-to-low active drugs while better monitoring their effects through controlled physico-chemical descriptors.

_{TOT}used in this study; it offers a visual way for assessing the almost no-correlation of LogP with other concerned variables, POL and E

_{TOT}, respectively. This lead with conclusion that LogP is almost orthogonal (independent) on (respecting) the other two Hantsch variables. Instead, when further performing the factor analysis, the Table 6 is obtained while clearly revealing the scarce correlation carried by considering LogP variable alone. This is in close agreement with the Spectral-SAR results, see above. In any case, the hydrophobicity description and its descriptor cannot be rejected only by factor analysis since it drives (firstly or latter) the inter-membrane interaction that is essential for drug-cell binding. Spectral- and Qua-SAR highly proved the important role hydrophobicity plays in combination with electrostatic (POL) and steric (E

_{TOT}) interactions. Moreover, while PCA shows the POL factor influence equals that of E

_{TOT}, whereas their role in correlation is sensible different in Spectral-SAR analysis (compare model Ib-last column of Table 3 with POL-last column of Table 6). However, again, this discrepancy is in the favor of S-SAR since the PCA results are due to the sensitive degree of POL-E

_{TOT}correlation (see Figure 2), from where the PCA yield that POL and E

_{TOT}display similar correlation power, while S-SAR includes also the orthogonalization of POL and E

_{TOT}variables prior correlation takes effect and better discriminates among their influence in bonding.

_{ToT}) to IIb (LogP∧E

_{TOT}) corresponds to quantum free motion so that the null IEMAD for molecule no. 12 (4′-5,7-trimethoxyflavanone) is carried; here, the quantum metabolization factor ${q}_{12|12}^{Ic|IIb;\alpha}$ is consumed only for strongly activating the membrane transporter feature (LogP) of the same molecule. Instead, on the last passage of the α path the factor ${q}_{12|13}^{IIb|III;\alpha}$ is responsible for converting the electrostatic (POL) influence of the flavonoids no. 12 towards no.3 (naringenin) activity as well as for reverse-O-methylation (methoxylation) of oxygens in positions 5 and 7 (on ring A) and 4′ (on ring B) respecting the molecular pattern no.0 in Figure 1, respectively. Such result is in fully accordance with the reverse quantum influence that is at the foreground of quantum-SAR factor conversion prescribed by Equation (9), i.e. quantifying the power of back transformation of molecular EC50s respecting the “arrows” of IEND and IEMAND in Equation (10). However, the fact that such transformation is the first one acting at molecular level is sustained also by optimized 3D configurations of involved molecules no. 12 and 3, being both with rings A and B spatially bent in Figure 3 respecting the ring C of the planar pattern no.0 of Figure 1.

_{ToT}seems to mainly drive the mechanistic molecular transformation in MX-BCRP binding phenomenology, while the molecule no. 3 (naringenin) appears as the best fitted molecules belonging to the most relevant endpoint, in clear disjunction with the roughly molecular selection upon initial input observed activity data. That is, naringenin (no. 3) is shown to be the best adapted molecule for the actual LogP∧POL∧E

_{TOT}structural (independent) factors being metabolized from molecules as 6,2′,3′-7-hydroxyflavanone (no. 8), 4′-5,7-trimethoxyflavanone (no. 12), and flavone (no.13) by specific molecular mechanistically paths. However, there appears that these molecules are not linked even through the paths with the most active compounds of Table 1; statistically, this can be explained by the so called “regression towards the mean” effects, in the sense that the best correlations translated to the compounds found in the middle of the mentioned sorted Table 1; from the structural point of view such behavior may attributed to the specific parameters used for correlations that best describe molecules with specific groups, most favorable for the descriptor’s nature.

## 4. Conclusions

## Acknowledgments

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**Figure 1.**The studied flavonoids (with basic structure of as no.0 while the others are in the Table 1 characterized by associate QSAR data), covering the flavones, isoflavones, chalcones, flavonols and flavanones, as they assist the increase of mitoxantrone (MX) accumulation in BCRP-overexpressing MCF-7 MX100 breast cancer cells [51].

**Figure 3.**Spectral representation of the endpoints employed in designing the bioactivity mechanism for the molecules of Table 1, according with the algebraic correlation factors of Equation (3) in Table 3, across the shortest (three) paths identified from Table 4, while marking the fittest molecules’ orbital 3D-distribution for each considered model, i.e. molecule no. 12 (4′–5,7-trimethoxyflavanone) for the models Ic and IIb, molecule no. 13 (Flavone) for models Ib, Ia, and IIa, molecule no. 8 (6,2′,3′–7-tydroxyflavanone) for model IIc, and molecule no. 3 (naringenin) for model III, respectively.

**Table 1.**The flavonoids of Figure 1 arranged by their ascending observed activities, defined as A= -log

_{10}(EC

_{50}[μM]) [51], along the associate computed structural parameters like the hydrophobicity (LogP), electronic cloud polarizability (POL) and the ground state configurationally optimized total energy (E

_{TOT}) [55].

No. | Molecular Name | Activity | Structural parameters | ||
---|---|---|---|---|---|

A | LogP | POL(Å^{3}) | E_{TOT}(kcal/mol) | ||

(1) | Silybin | 3.74 | 2.03 | 45.68 | − 146625.1875 |

(2) | Daidzein | 4.24 | 1.78 | 26.63 | − 76984.7109 |

(3) | Naringenin | 4.49 | 1.99 | 27.46 | − 85032.9218 |

(4) | Flavanone | 4.6 | 2.84 | 25.55 | − 62849.3125 |

(5) | 7,8-Dihydroxyflavone | 4.7 | 1.75 | 26.63 | − 76982.1328 |

(6) | 7–Methoxyflavanone | 4.79 | 2.59 | 28.02 | − 73823.8046 |

(7) | Genistein | 4.83 | 1.50 | 27.27 | − 84380.7578 |

(8) | 6,2′,3′-7-Hydroxyflavanone | 4.85 | 1.70 | 28.10 | − 92422.6640 |

(9) | Hesperetin | 4.91 | 1.73 | 29.93 | − 96003.9921 |

(10) | Chalcone | 4.93 | 3.68 | 25.49 | − 55450.1093 |

(11) | Kaempferol | 5.22 | 0.56 | 27.90 | − 91770.5859 |

(12) | 4′-5,7-Trimethoxyflavanone | 5.25 | 2.08 | 32.96 | − 95768.9062 |

(13) | Flavone | 5.4 | 2.32 | 25.36 | − 62196.3437 |

(14) | Apigenin | 5.78 | 1.46 | 27.27 | − 84379.8593 |

(15) | Biochanin A | 5.79 | 1.53 | 29.10 | − 87961.2812 |

(16) | 5,7-Dimethoxyflavone | 5.85 | 1.81 | 30.30 | − 84139.4687 |

(17) | Galangin | 5.92 | 0.85 | 27.27 | − 84376.8359 |

(18) | 5,6,7–Trimethoxyflavone | 5.96 | 1.56 | 32.77 | − 94976.1875 |

(19) | Kaempferide | 5.99 | 0.60 | 29.74 | − 95351.3984 |

(20) | 8-Methylflavone | 6.21 | 2.79 | 27.19 | − 65789.9218 |

(21) | 6,4′–Dimethoxy-3-hydroxy-flavone | 6.35 | 0.41 | 31.13 | − 92162.7187 |

(22) | Chrysin | 6.41 | 1.75 | 26.63 | − 76986.1171 |

(23) | 2′-Hydroxy-α-naphtoflavone | 7.03 | 3.07 | 33.26 | − 82027.8359 |

(24) | 7,8 – Benzoflavone | 7.14 | 3.35 | 32.63 | − 74634.5234 |

**Table 2.**The anti-symmetric matrix of the inter-molecular activity differences for the working flavonoids of Table 1.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0.5 | 0.75 | 0.86 | 0.96 | 1.05 | 1.09 | 1.11 | 1.17 | 1.19 | 1.48 | 1.51 | 1.66 | 2.04 | 2.05 | 2.11 | 2.18 | 2.22 | 2.25 | 2.47 | 2.61 | 2.67 | 3.29 | 3.4 | 1 |

0 | 0.25 | 0.36 | 0.46 | 0.55 | 0.59 | 0.61 | 0.67 | 0.69 | 0.98 | 1.01 | 1.16 | 1.54 | 1.55 | 1.61 | 1.68 | 1.72 | 1.75 | 1.97 | 2.11 | 2.17 | 2.79 | 2.9 | 2 | |

0 | 0.11 | 0.21 | 0.3 | 0.34 | 0.36 | 0.42 | 0.44 | 0.73 | 0.76 | 0.91 | 1.29 | 1.3 | 1.36 | 1.43 | 1.47 | 1.5 | 1.72 | 1.86 | 1.92 | 2.54 | 2.65 | 3 | ||

0 | 0.1 | 0.19 | 0.23 | 0.25 | 0.31 | 0.33 | 0.62 | 0.65 | 0.8 | 1.18 | 1.19 | 1.25 | 1.32 | 1.36 | 1.39 | 1.61 | 1.75 | 1.81 | 2.43 | 2.54 | 4 | |||

0 | 0.09 | 0.13 | 0.15 | 0.21 | 0.23 | 0.52 | 0.55 | 0.7 | 1.08 | 1.09 | 1.15 | 1.22 | 1.26 | 1.29 | 1.51 | 1.65 | 1.71 | 2.33 | 2.44 | 5 | ||||

0 | 0.04 | 0.06 | 0.12 | 0.14 | 0.43 | 0.46 | 0.61 | 0.99 | 1 | 1.06 | 1.13 | 1.17 | 1.2 | 1.42 | 1.56 | 1.62 | 2.24 | 2.35 | 6 | |||||

0 | 0.02 | 0.08 | 0.1 | 0.39 | 0.42 | 0.57 | 0.95 | 0.96 | 1.02 | 1.09 | 1.13 | 1.16 | 1.38 | 1.52 | 1.58 | 2.2 | 2.31 | 7 | ||||||

0 | 0.06 | 0.08 | 0.37 | 0.4 | 0.55 | 0.93 | 0.94 | 1 | 1.07 | 1.11 | 1.14 | 1.36 | 1.5 | 1.56 | 2.18 | 2.29 | 8 | |||||||

0 | 0.02 | 0.31 | 0.34 | 0.49 | 0.87 | 0.88 | 0.94 | 1.01 | 1.05 | 1.08 | 1.3 | 1.44 | 1.5 | 2.12 | 2.23 | 9 | ||||||||

0 | 0.29 | 0.32 | 0.47 | 0.85 | 0.86 | 0.92 | 0.99 | 1.03 | 1.06 | 1.28 | 1.42 | 1.48 | 2.1 | 2.21 | 10 | |||||||||

0 | 0.03 | 0.18 | 0.56 | 0.57 | 0.63 | 0.7 | 0.74 | 0.77 | 0.99 | 1.13 | 1.19 | 1.81 | 1.92 | 11 | ||||||||||

0 | 0.15 | 0.53 | 0.54 | 0.6 | 0.67 | 0.71 | 0.74 | 0.96 | 1.1 | 1.16 | 1.78 | 1.89 | 12 | |||||||||||

0 | 0.38 | 0.39 | 0.45 | 0.52 | 0.56 | 0.59 | 0.81 | 0.95 | 1.01 | 1.63 | 1.74 | 13 | ||||||||||||

0 | 0.01 | 0.07 | 0.14 | 0.18 | 0.21 | 0.43 | 0.57 | 0.63 | 1.25 | 1.36 | 14 | |||||||||||||

0 | 0.06 | 0.13 | 0.17 | 0.2 | 0.42 | 0.56 | 0.62 | 1.24 | 1.35 | 15 | ||||||||||||||

0 | 0.07 | 0.11 | 0.14 | 0.36 | 0.5 | 0.56 | 1.18 | 1.29 | 16 | |||||||||||||||

0 | 0.04 | 0.07 | 0.29 | 0.43 | 0.49 | 1.11 | 1.22 | 17 | ||||||||||||||||

0 | 0.03 | 0.25 | 0.39 | 0.45 | 1.07 | 1.18 | 18 | |||||||||||||||||

0 | 0.22 | 0.36 | 0.42 | 1.04 | 1.15 | 19 | ||||||||||||||||||

0 | 0.14 | 0.2 | 0.82 | 0.93 | 20 | |||||||||||||||||||

0 | 0.06 | 0.68 | 0.79 | 21 | ||||||||||||||||||||

0 | 0.62 | 0.73 | 22 | |||||||||||||||||||||

0 | 0.11 | 23 | ||||||||||||||||||||||

0 | 24 |

**Table 3.**QSAR equations through Spectral-SAR multi-linear procedure [32–34] for all possible correlation models considered from data of Table 1; here |X

_{0}〉 is the unitary vector|11...1

_{24}〉, while the structural variables are set as |X

_{1}〉 = LogP, |X

_{2}〉 = POL, and |X

_{3}〉 = E

_{TOT}; the predicted activities’ norms where calculated with Equation (2), while the algebraic correlation factor of Equation (3) uses the measured activity of ‖| A〉‖ = 26.9357 computed upon Equation (2) with data of Table 1; R

_{Statistic}is the traditional Pearson correlation factor [1–8].

Model | Variables | (Q/S-)SAR Equation | ‖|Y〉 ^{PREDICTED}‖ | R_{Algebraic} | R_{Statistic} |
---|---|---|---|---|---|

Ia | |X_{0}>, |X_{1}> | |Y>^{Ia} = 5.39837|X_{0}>+0.0179106|X_{1}> | 26.6138 | 0.988049 | 0.0175601 |

Ib | |X_{0}>, |X_{2}> | |Y>^{Ib} = 5.67735 X_{0}>–0.00834411|X_{2}> | 26.61425 | 0.988065 | 0.0409922 |

Ic | |X_{0}>, |X_{3}> | |Y>^{Ic} = 6.48303|X_{0}>+0.0000124625|X_{3}> | 26.6344 | 0.988812 | 0.252513 |

IIa | |X_{0}>, |X_{1}〉,|X_{2}> | |Y>^{IIa} = 5.64318|X_{0}> +0.0178242 |X_{1}〉–0.00833676|X_{2}> | 26.614349 | 0.988069 | 0.0445618 |

IIb | |X_{0}>, |X_{1}>,|X_{3}> | |Y>^{IIb} = 6.93331|X_{0}> − 0.120924|X_{1}>+0.0000150708|X_{3}> | 26.638 | 0.988947 | 0.273909 |

IIc | |X_{0}>, |X_{2}>,|X_{3}> | |Y>^{IIc} = 4.99884|X_{0}> +0.122989|X_{2}>+0.0000376701 |X_{3}> | 26.6681 | 0.990063 | 0.409837 |

III | |X_{0}>, |X_{1}>,|X_{2}>, |X_{3}> | |Y>^{III} = 5.59424|X_{0}>–1.05993|X_{1}>+0.400704|X_{2}>+0.000117452|X_{3}> | 26.7758 | 0.994064 | 0.708509 |

**Table 4.**Synopsis of paths connecting the endpoints of Table 3 in the norm-correlation spectral-space.

PC1 | PC2 | PC3 | Multiple | |

Eigenvalue: | 1.958158 | 0.892127 | 0.149715 | PC1–PC3 |

% total variance: | 65.27195 | 29.73757 | 4.99049 | factors’ |

Variable | Factors’ coefficients | R^{2} | ||

LogP | 0.232179 | −0.997780 | 0.20467 | 0.083712 |

POL | −0.472177 | −0.302902 | −1.79349 | 0.716820 |

E_{TOT} | 0.483556 | 0.183309 | −1.84956 | 0.728872 |

**Table 7.**Determination of the quantum-SAR, see Equation (10) with Eqs. (6) and (7), associate with certain couple of molecules involved in activating specific structural quantum indices (or their combinations) driving spectral paths of Table 4, by employing minimum residue recipe throughout Table 5 for each considered endpoint, as well as the associate recorded bioactivity differences of Table 2, respectively.

Path | $\Delta {Y}_{l|{l}^{\prime}}^{PATH}$ (IEND)# | $\Delta {A}_{i|j}^{l|{l}^{\prime}}$ (IEMAD)^{♣} | ${q}_{i|j}^{l|{l}^{\prime};PATH*}$* | q^{PATH} |
---|---|---|---|---|

α | $\Delta {Y}_{Ic|IIb}^{\alpha}=0.00364573$ | ${A}_{12|12}^{Ic|IIb}=0$ | ${q}_{12|12}^{Ic|IIb;\alpha}=0.991641$ | q^{α} =0.125464 |

$\Delta {Y}_{IIb|III}^{\alpha}=0.137836$ | ${A}_{12|3}^{IIb|III}=-0.76$ | ${q}_{12|3}^{IIb|III;\alpha}=0.126521$ | ||

β | $\Delta {Y}_{Ib|IIa}^{\beta}=0.0000989324$ | ${A}_{13|13}^{Ib|IIa}=0$ | ${q}_{13|13}^{Ib|IIa;\beta}=0.999772$ | q^{β}=0.0848036 |

$\Delta {Y}_{IIa|III}^{\beta}=0.161487$ | ${A}_{13|3}^{IIa|III}=-0.91$ | ${q}_{13|3}^{IIa|III;\beta}=0.0848229$ | ||

γ | $\Delta {Y}_{Ia|IIc}^{\gamma}=0.0542592$ | ${A}_{13|8}^{Ia|IIc}=-0.55$ | ${q}_{13|8}^{Ia|IIc;\gamma}=0.248737$ | q^{γ} =0.0847168 |

$\Delta {Y}_{IIc|III}^{\gamma}=0.107771$ | ${A}_{8|3}^{IIc|III}=-0.36$ | ${q}_{8|3}^{IIc|III;\gamma}=0.340588$ |

^{#}Inter-Endpoint Norm Difference, Equation (6);

^{♣}Inter-Endpoint Molecular Activity Difference, Equation (7);

^{*}Note that here the basic relation of Equation (10) was considered in decimal base since originally, the associated activities in Table 1 were as such defined.

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/3.0/). This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Putz, M.V.; Putz, A.-M.; Lazea, M.; Ienciu, L.; Chiriac, A.
Quantum-SAR Extension of the Spectral-SAR Algorithm. Application to Polyphenolic Anticancer Bioactivity. *Int. J. Mol. Sci.* **2009**, *10*, 1193-1214.
https://doi.org/10.3390/ijms10031193

**AMA Style**

Putz MV, Putz A-M, Lazea M, Ienciu L, Chiriac A.
Quantum-SAR Extension of the Spectral-SAR Algorithm. Application to Polyphenolic Anticancer Bioactivity. *International Journal of Molecular Sciences*. 2009; 10(3):1193-1214.
https://doi.org/10.3390/ijms10031193

**Chicago/Turabian Style**

Putz, Mihai V., Ana-Maria Putz, Marius Lazea, Luciana Ienciu, and Adrian Chiriac.
2009. "Quantum-SAR Extension of the Spectral-SAR Algorithm. Application to Polyphenolic Anticancer Bioactivity" *International Journal of Molecular Sciences* 10, no. 3: 1193-1214.
https://doi.org/10.3390/ijms10031193