# Introducing Spectral Structure Activity Relationship (S-SAR) Analysis. Application to Ecotoxicology

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Spectral-SAR Method

#### 2.1. Background Concepts

_{i}, i=1, …, M through the fixed parameters b

_{j}, j=0, …, M, while e stands as the residual or error value between the assumed multi-linear model and measurements.

_{1}, …, e

_{N}are potentially different although the ideal case would demand that they be equal with zero.

^{T}X is taken into account.

#### 2.2. Spectral-SAR Algorithm

_{0}〉 = |1 1... 1〉 for accounting of the coefficients of the free term (b

_{0}) of system (2).

_{0}〉, |X

_{1}〉,..., |X

_{k}〉,..., |X

_{M}〉}:

_{0}〉, |X

_{1}〉,..., |X

_{k}〉,..., |X

_{M}〉} into an orthogonal one, say {|Ω

_{0}〉, |Ω

_{1}〉,..., |Ω

_{k}〉,..., |Ω

_{M}〉}, is now considered. In this respect the consecrated Gram-Schmidt procedure is employed. It is worth noting that this procedure is well known in quantum chemistry when searching for an orthogonal basis for an orthogonal basis set in atomic and molecular wave function spectral decomposition [62].

_{0}〉, |Ω

_{1}〉,..., |Ω

_{k}〉,..., |Ω

_{M}〉}, can be constructed from the set {|X

_{0}〉, |X

_{1}〉,..., |X

_{k}〉,..., |X

_{M}〉} according with the iterative recipe:

- Choose$$\mid {\mathrm{\Omega}}_{0}\rangle =\mid {X}_{0}\rangle ;$$
- Then, by picking |X
_{1}〉 as the next vector to be transformed, one can write that:$$\mid {\mathrm{\Omega}}_{1}\rangle =\mid {X}_{1}\rangle -{r}_{0}^{1}\mid {\mathrm{\Omega}}_{0}\rangle ,\hspace{0.17em}{r}_{0}^{1}=\frac{\langle {X}_{1}\mid {\mathrm{\Omega}}_{0}\rangle}{\langle {\mathrm{\Omega}}_{0}\mid {\mathrm{\Omega}}_{0}\rangle}$$so that 〈Ω_{0}|Ω_{1}〉 = 0 assuring so far that |Ω_{0}〉 and |Ω_{1}〉 are orthogonal. - Next, repeating steps i. and ii. above until the vectors |Ω
_{0}〉, |Ω_{1}〉, …, |Ω_{k}_{−1}〉 are orthogonally constructed, we can, for instance, further transform the vector |X_{k}〉 into :$$\mid {\mathrm{\Omega}}_{k}\rangle =\mid {X}_{k}\rangle -\sum _{i=0}^{k-1}{r}_{i}^{k}\mid {\mathrm{\Omega}}_{i}\rangle ,\hspace{0.17em}{r}_{i}^{k}-\frac{\langle {X}_{k}\mid {\mathrm{\Omega}}_{i}\rangle}{\langle {\mathrm{\Omega}}_{i}\mid {\mathrm{\Omega}}_{i}\rangle}$$so that the vector |Ω_{k}〉 is orthogonal on all previous ones. - Step (iii) is repeated and extended until the last orthogonal predictor vector |Ω
_{M}〉 is obtained.

_{0}〉, |X

_{1}〉,..., |X

_{k}〉,..., |X

_{M}〉}is replaced with the orthogonal one {|Ω

_{0}〉, |Ω

_{1}〉,..., |Ω

_{k}〉,..., |Ω

_{M}〉} by appropriately subtracting from the original vectors the non-wished non-orthogonal contributions. Note that the above procedure holds for any arbitrary order of original vectors to be orthogonalized.

_{0}, ω

_{1},...,ω

_{k},...,ω

_{M}) are determined. These new coefficients can be immediately deduced based on the orthogonal peculiarities of the spectral decomposition (13) grounded on the fact that:

_{k}, respectively.

^{T}X and Q

^{T}Q in equations (7) and (19), respectively, there is clear that the last case certainly furnishes a diagonal form which for sure is easier to handle (i.e. to take its inverse) when searching for the vector B of SAR coefficients.

## 3. Application to Ecotoxicology

#### 3.1 Basic Characteristics of QSAR in Ecotoxicology

_{50}, 50% effect concentration-EC

_{50}, 50% grow inhibition concentration-IGC

_{50}) [74–77].

#### 3.2 Bio-ecological Issues of Unicellular Organisms

#### 3.3 Spectral-SAR Ecotoxicity of Tetrahymena pyriformis

_{LUMO}) phenomena plays a particular place in explaining the ecotoxicology of the species.

_{TOT}) in its ground state, for the reason that it is calculated at the optimum molecular geometry where the stereospecificity is included.

_{50}) [86–90], from a series of xenobiotics of which majority are of phenol type is in Table 3 considered.

^{MEASURED}|| = 6.83243 the algebraic S-SAR correlation factors for the actual predicted models are given in Table 6 along the individual spectral norm of activity and the standard statistical correlation factor values.

_{50}) endpoint of Table 3 though considering all path combinations that contain a single model for each class, with one and two descriptors, towards the closest model, i.e. (III), with respect to the ideal one. It follows that the paths are grouped according to the intermediary passing model while extreme models (initial and final) are kept fixed. Such ordered paths can be rationalized since a selection criterion is further introduced. Since paths are involved, one may learn from the well-established principle of nature according to which the events are linked by closest paths (in all classical and quantum spaces).

## 4. Conclusions

**Figure 1.**Generic world of the quantitative structure-activity/property relationships - QSA(P)R - through classical, 3D, decisional and orthogonal methods of multivariate analysis of the chemical-biological interactions. In scheme MSD-MTD, CoMFA, and PCA stand for the “minimal steric difference-minimal topological difference”, “comparative molecular field analysis” and “principal component analysis”, respectively.

**Figure 2.**Generic mapping of data space containing the vectorial sets {|X〉, |O〉} into orthogonal basis {|Ω(X)〉, |Ω(O)〉}.

**Figure 3.**Illustration of the oral region of Tetrahymena pyriformis during ingestion as taken by electron micrograph technique [83].

**Figure 4.**Norm correlation spectral space of the statistical and algebraic correlation factors against the spectral norm of the predicted S-SAR models of Table 6, respectively.

**Figure 5.**Spectral-structural models, designed through the rules of minimal spectral-SAR paths of Table 7, emphasizing the primary, secondary and tertiary hierarchies forward the endpoints of the Tetrahymena pyriformis eco-toxicological activity according with data of Table 3, S-SAR equations of Table 5, and of the associated spectral norms computed upon Eq. (30).

Activity | Structural predictor variables | ||||
---|---|---|---|---|---|

y_{1} | x_{11} | … | x_{1}_{k} | … | x_{1}_{M} |

y_{2} | x_{21} | … | x_{2}_{k} | … | x_{2}_{M} |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

y_{N} | x_{N}_{1} | … | x_{Nk} | … | x_{NM} |

**Table 2.**The spectral (vectorial) version of SAR descriptors of Table 1.

Activity | Structural predictor variables | |||||
---|---|---|---|---|---|---|

|Y〉 | |X_{0}〉 | |X_{1}〉 | … | |X_{k}〉 | … | |X_{M}〉 |

y_{1} | 1 | x_{11} | … | x_{1}_{k} | … | x_{1}_{M} |

y_{2} | 1 | x_{21} | … | x_{2}_{k} | … | x_{2}_{M} |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

y_{N} | 1 | x_{N}_{1} | … | x_{Nk} | … | x_{NM} |

**Table 3.**The series of the xenobiotics of those toxic activities

**A**= Log(1/IGC

_{50}) were considered [86] along structural parameters LogP, POL (Å

^{3}), and E

_{TOT}(kcal/mol) as accounting for the hydrophobicity, electronic (polarizability) and steric (total energy at optimized 3D geometry) effects, respectively, derived with the help of HyperChem program [91].

No. | Compound | A | |1〉 | Log P | POL | E_{TOT} | |
---|---|---|---|---|---|---|---|

Name | Formulae | |Y〉 | |X_{0}〉 | |X_{1}〉 | |X_{2}〉 | |X_{3}〉 | |

1 | methanol | CH_{3}OH | −2.67 | 1 | −0.27 | 3.25 | −11622.9 |

2 | ethanol | C_{2}H_{5}OH | −1.99 | 1 | 0.08 | 5.08 | −15215.4 |

3 | butan-1-ol | C_{4}H_{9}OH | −1.43 | 1 | 0.94 | 8.75 | −22402.8 |

4 | butanone | C_{4}H_{8}O | −1.75 | 1 | 1.01 | 8.2 | −21751.8 |

5 | pentan-3-one | C_{5}H_{10}O | −1.46 | 1 | 1.64 | 10.04 | −25344.6 |

6 | phenol | C_{6}H_{5}OH | −0.21 | 1 | 1.76 | 11.07 | −27003.1 |

7 | aniline | C_{6}H_{5}NH_{2} | −0.23 | 1 | 1.26 | 11.79 | −24705.9 |

8 | 3-cresol | CH_{3}-C_{6}H_{4}-OH | −0.06 | 1 | 2.23 | 12.91 | −30597.6 |

9 | 4-methoxiphenol | OH-C_{6}H_{4}-O-CH_{3} | −0.14 | 1 | 1.51 | 13.54 | −37976.3 |

10 | 2-hydroxyaniline | OH-C_{6}H_{4}-NH_{2} | 0.94 | 1 | 0.98 | 12.42 | −32095.4 |

11 | Benzaldehyde | C_{6}H_{5}-CHO | −0.2 | 1 | 1.72 | 12.36 | −29946.9 |

12 | 2-cresol | CH_{3}-C_{6}H_{4}-OH | −0.27 | 1 | 2.23 | 12.91 | −30597.2 |

13 | 3,4-dimeyhylphenol | C_{6}H_{3}(CH_{3})_{2}OH | 0.12 | 1 | 2.7 | 14.74 | −34190.8 |

14 | 3-nitrotoluene | CH_{3}-C_{6}H_{4}-NO_{2} | 0.05 | 1 | 0.94 | 13.98 | −42365.1 |

15 | 4-chlorophenol | C_{6}H_{5}-O-Cl | 0.55 | 1 | 2.28 | 13 | −35307.6 |

16 | 2,4-dinitroaniline | C_{6}H_{3}(NO_{2})NH_{2} | 0.53 | 1 | −1.75 | 15.22 | −63030.2 |

17 | 2-methyl-1-4-naphtoquinone | C_{11}H_{8}O_{2} | 1.54 | 1 | 2.39 | 20.99 | −49768.3 |

18 | 1,2-dichlorobenzene | C_{6}H_{4}Cl_{2} | 0.53 | 1 | 3.08 | 14.29 | −36217.2 |

19 | 2,4-dinitrophenol | C_{6}H_{3}(NO_{2})OH | 1.08 | 1 | 1.67 | 14.5 | −65318 |

20 | 1,4-dinitrobenzene | C_{6}H_{4}N_{2}O_{4} | 1.3 | 1 | 1.95 | 13.86 | −57926.7 |

21 | 2,4-dinitrotoluene | C_{7}H_{6}(NO_{2})_{2} | 0.87 | 1 | 2.42 | 15.7 | −61520.7 |

22 | 2,6-ditertbutil 4-methyl phenol | C_{15}H_{23}OH | 1.8 | 1 | 5.48 | 27.59 | −59316.5 |

23 | 2,3,5,6-tetrachloroaniline | C_{6}H_{3}NCl_{4} | 1.76 | 1 | 3.34 | 19.5 | −57920.2 |

24 | penthaclorophenol | C_{6}Cl_{5}OH | 2.05 | 1 | −0.54 | 20.71 | −68512.4 |

25 | phenylazophenol | C_{12}H_{10}N_{2}O | 1.66 | 1 | 4.06 | 22.79 | −55488.9 |

26 | pentabromophenol | C_{6}Br_{5}OH | 2.66 | 1 | 5.72 | 24.2 | −66151.5 |

Model | Variables | QSAR Equation | r | s | F |
---|---|---|---|---|---|

Ia | logP | ${A}^{Ia}=-0.547836+0.435669logP$ | 0.539 | 1.15 | 9.834 |

Ib | POL | ${A}^{Ib}=-2.84021+0.2166POL$ | 0.908 | 0.574 | 112.15 |

Ic | E_{TOT} | ${A}^{Ic}=-2.50233-0.00007\hspace{0.17em}{E}_{TOT}$ | 0.882 | 0.644 | 84.015 |

IIa | logP, POL | ${A}^{IIa}=-2.91377-0.08109logP+0.23233POL$ | 0.911 | 0.58 | 55.930 |

IIb | logP, E_{TOT} | ${A}^{IIb}=-2.64602+0.22991logP-0.00006\hspace{0.17em}{E}_{TOT}$ | 0.922 | 0.54 | 65.339 |

IIc | POL, E_{TOT} | ${A}^{IIc}=-2.98407+0.13427POL-0.00003\hspace{0.17em}{E}_{TOT}$ | 0.939 | 0.478 | 86.503 |

III | logP, POL, E_{TOT} | ${A}^{III}=-2.94395+0.06335logP+0.11206POL-0.00004{E}_{TOT}$ | 0.941 | 0.48 | 56.598 |

**Table 5.**Spectral structure activity relationships (S-SAR) through determinants of Equations (27)–(29) for all possible correlation models considered from the data in Table 3.

Models | Vectors | S-SAR Equation |
---|---|---|

Ia | |X_{0}〉, |X_{1}〉 | ${\mid Y\rangle}^{Ia}=-0.547836\mid {X}_{0}\rangle +0.435669\mid {X}_{1}\rangle $ |

Ib | |X_{0}〉, |X_{2}〉 | ${\mid Y\rangle}^{Ib}=-2.84021\mid {X}_{0}\rangle +0.216598\mid {X}_{2}\rangle $ |

Ic | |X_{0}〉, |X_{3}〉 | ${\mid Y\rangle}^{Ic}=-2.50233\mid {X}_{0}\rangle -0.000067863\mid {X}_{3}\rangle $ |

IIa | |X_{0}〉, |X_{1}〉, |X_{2}〉 | ${\mid Y\rangle}^{IIa}=-2.91377\mid {X}_{0}\rangle -0.0810929\mid {X}_{1}\rangle +0.232325\mid {X}_{2}\rangle $ |

IIb | |X_{0}〉, |X_{1}〉, |X_{3}〉 | ${\mid Y\rangle}^{IIb}=-2.64602\mid {X}_{0}\rangle +0.229913\mid {X}_{1}\rangle -0.0000608117\mid {X}_{3}\rangle $ |

IIc | |X_{0}〉, |X_{2}〉, |X_{3}〉 | ${\mid Y\rangle}^{IIc}=-2.98407\mid {X}_{0}\rangle +0.134274\mid {X}_{2}\rangle -0.0000324573\mid {X}_{3}\rangle $ |

III | |X_{0}〉, |X_{1}〉, |X_{2}〉, |X_{3}〉 | ${\mid Y\rangle}^{III}=-2.94395\mid {X}_{0}\rangle +0.0633549\mid {X}_{1}\rangle -0.112056\mid {X}_{2}\rangle -0.0000363728\mid {X}_{3}\rangle $ |

Ia | Ib | Ic | IIa | IIb | IIc | III | |
---|---|---|---|---|---|---|---|

|||Y〉^{PREDICTED}|| | 3.86176 | 6.22803 | 6.0607 | 6.24858 | 6.32297 | 6.43641 | 6.44557 |

${r}_{S-SAR}^{STATISTIC}$ | 0.53905 | 0.90759 | 0.88193 | 0.91074 | 0.92214 | 0.9395 | 0.9409 |

${r}_{S-SAR}^{ALGEBRAIC}$ | 0.56521 | 0.91154 | 0.88705 | 0.91455 | 0.92543 | 0.94204 | 0.94338 |

Path | Value | |
---|---|---|

Statistic | Algebraic | |

Ia-IIa-III | 2.61485 | 2.61132 |

Ia-IIb-III | 2.61485 | 2.61132 |

Ia-IIc-III | 2.61485 | 2.61132 |

Ib-IIa-III | 0.220072 | 0.219855 |

Ib-IIb-III | 0.220072 | 0.219855 |

Ib-IIc-III | 0.220072 | 0.219855 |

Ic-IIa-III | 0.389359 | 0.388969 |

Ic-IIb-III | 0.389359 | 0.388969 |

Ic-IIc-III | 0.389359 | 0.388969 |

## Acknowledgements

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**MDPI and ACS Style**

Putz, M.V.; Lacrămă, A.-M. Introducing Spectral Structure Activity Relationship (S-SAR) Analysis. Application to Ecotoxicology. *Int. J. Mol. Sci.* **2007**, *8*, 363-391.
https://doi.org/10.3390/i8050363

**AMA Style**

Putz MV, Lacrămă A-M. Introducing Spectral Structure Activity Relationship (S-SAR) Analysis. Application to Ecotoxicology. *International Journal of Molecular Sciences*. 2007; 8(5):363-391.
https://doi.org/10.3390/i8050363

**Chicago/Turabian Style**

Putz, Mihai V., and Ana-Maria Lacrămă. 2007. "Introducing Spectral Structure Activity Relationship (S-SAR) Analysis. Application to Ecotoxicology" *International Journal of Molecular Sciences* 8, no. 5: 363-391.
https://doi.org/10.3390/i8050363