# Köln-Timişoara Molecular Activity Combined Models toward Interspecies Toxicity Assessment

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background Models

#### 2.1. Köln ESIP Model for Biological Activity

_{50}) [Mol/L] in order to calculate toxicity values Clog(1/MRC

_{50}) [Mol/L] for untested derivatives [5]. The molecular structures have always saturated hydrocarbon or aromatic substructures, so the first ESIP-parameter corresponds to saturated-carbon ESIPc-sat, followed by the aromatic-carbon ESIPc-ar, and ESIP-organic function (alcohol, amino, etc.). In the case of saturated hydrocarbons the ESIPc-sat have an average value of 0.50 log units, calculated on the basis of measured values M and saturated carbon numbers C.

- the toxicity of a compound can be subdivided into that of components (ESIP’s) in such a way that the sum of these components results in the total toxicity value;
- these components (ESIP’s) are identical in different substances;
- the ESIP’s components have a dynamical value (they depend on the determined number or are derived from newly available data) for one organism and a test-system, while varying for different test-systems. However, if a deviation between the measured M and the calculated C values is observed, there is an indication of an overlooked interaction between different parts of the molecule, or may indicate an activity of a substance specific for a certain biochemical pathway.

#### 2.2. Timişoara Spectral-SAR Model

- ○ Any molecular structural state (dynamical, since undergoes interactions with organisms) may be represented by a | ket〉 state vector, in the abstract Hilbert space, following the 〈bra | ket〉 Dirac formalism [14]; such states are to be represented by any reliable molecular index, or, in particular in our study by hydrophobicity |LogP〉, polarizability |POL〉, and total optimized energy |E
_{tot}〉, just to be restrained only the so called Hansch parameters, usually employed for accounting the diffusion, electrostatic and steric effects for molecules acting on organisms’ cells, respectively. - ○ The (quantum) superposition principle assuring that the various linear combinations of molecular states map onto the resulting state, here interpreted as the bio-, eco- or toxico-logical activity, e.g., |Y〉 = |Y
_{0}〉 +C_{LogP}|LogP〉 +C_{POL}|POL〉 +..., with |Y_{0}〉 meaning the free or unperturbed activity (when all other influences are absent). - ○ The orthogonalization feature of quantum states, a crucial condition providing that the superimposed molecular states generates new molecular state (here quantified as the organism activity); analytically, the orthogonalization condition is represented by the 〈bra | ket〉 scalar product of two envisaged states (molecular indices); if it is evaluated to zero value, i.e., 〈bra | ket〉 = 0, then the convoluted states are said to be orthogonal (zero-overlapping) and the associate molecular descriptors are considered as independent, therefore suitable to be assumed as eigen-states (of a spectral decomposition) in the resulted activity state, while quantified by the degree their molecular indices enter the activity correlation. Further details on scalar product and related properties are given in Appendix A1, whereas in what follows the Spectral-based SAR correlation method (thereby called as Spectral-SAR) is resumed.

_{0}〉 = 1 1 ... 1

_{N}〉was added to account for the free activity term.

_{PRED}(with all its sub-intended states $|{X}_{i=\overline{0,M}}\rangle $) belong to disjoint (thus orthogonal) Hilbert (sub)spaces; or, even more, one can say that the Hilbert space of the observed activity |Y

_{OBS}〉 may be decomposed into a predicted and error independent Hilbert sub-spaces of states.

_{PRED}〉 and the given descriptors $|{X}_{i=\overline{0,M}}\rangle $.

_{PRED}〉 on left side and the rest of states/indicators on the right side the sought QSAR solution for the initial observed-predicted correlation problem of Equation (1a) is obtained under the Spectral-SAR vectorial expansion (from where the “spectral” name is justified) without the need to minimize the predicted error vector anymore, being this stage absorbed in its orthogonal behavior with respect to the predicted activity.

_{1}, until the models with maximum factors of correlation, say A

_{M}– i.e., containing M number of indicators, see Table 1) [7–10]. Since each of these models is now characterized by its predicted activity norm ‖|Y

_{PRED}〉‖ along the algebraic (RA) and/or statistical (R) correlation factors, the elementary paths of Equation (9) are constructed as the Euclidian measure between two consecutive models (endpoints) [7–10,22–24]:

## 3. Spectral-SAR Results

- if the overall minimum is reached by many equivalent paths (as is the case of Mlog-algebraic column for H.e. in Table 5, for instance) the minimum path will be considered that one connecting the starting endpoint with the closest endpoint in the sense of norms (as is for H.e./ Mlog the norm of |2> state the closest to the norm of |2,3> state, as compared with |1,2> and |1,3>, see Spectral-SAR norm column of Table 3, for example);
- the overall minimum path will set the dominant hierarchical path in assessing the mechanistically mode of action towards the given/measured activity; it is called as the alpha path (α);
- once the alpha path has been set the next minimum path will be looked for in such a way that the new starting endpoint is different from that one already involved in the alpha path (that is, if in the established alpha path for H.e./ Mlog the starting model correspond to the |2> state, the next path to be identified will originate either on models/states |1> or |3>);
- the remaining minimum paths are identified on the same rules as before and will be called like beta and gamma paths, β and γ, respectively;
- at the end of this procedure each mode of action is to be “touched” only one, excepting the final endpoint state {|1,2,3>} that can present degeneracy, i.e., may be found with the same influence at the end of various paths, herein called as degenerate paths (e.g., the states |1,2,3>, |2,1,3>, and |3,1,2> in the case of Hydractinia echinata and Tetrahymena pyriformis at their ending toxicity paths of Table 5); Yet, such behavior may leave with the important idea the degenerate paths, although different in the start and intermediate states, while ending with the same ordering influences, e.g., the state |2,1,3> of Table 5 (with “1” for LogP, “2” for POL, and “3” for E
_{tot}, see Tables 3 and 4), provides weaker contribution to the recorder activity since two paths have to produce the same (final) effect in order it to be activated; this is nevertheless one remarkable mechanistic consequence of the present combined (algebraic or statistical) correlations with minimization (optimization) principle applied for the spectral path lengths through Equations (6)–(10); - the alpha, beta and gamma paths can be easily identified for algebraic and statistical treatments in Tables 5 and 6 and there are accordingly marked; the degeneracy behavior is readily verified in Table 5 where the alpha path is found as the only (non-degenerate) path out of all possible ones. Of course, the same rationalization applies also for alpha path of Table 6, however displaying the trivial situation in which the absence of any degeneracy is recorded due to the restrained structural parameters considered for activity modeling since less available data for Pimephales promelas (P.p.) and Vibrio fisheri (V.f.) species in Table 2, according with the above specified Topliss-Costello rule.

- h) models with higher correlation/probability (either within statistic or algebraic approaches) will firstly enter molecular mechanism of toxicity through their considered structural parameter, i.e., LogP, POL and E
_{tot}for the |1>, |2> and |3> end-points, respectively.

_{P.p.}for the actual case.

_{Clog}of Figure 2 are clearly individuated as having no crossing toxicity with other species eventually submersed in the same ecological area, while the carried toxicity may be transmitted to V.f. species according with the statistical approach R

_{Clog}of Figure 4.

_{Mlog}picture of Figure 1. Yet, a different situation is noted for the statistical R

_{Mlog}analysis of Figure 3, according which H.e. species is highly mixed from a toxicological point of view with the species T.p. and P.p., but not with the V.f. one, either by means of first (alpha), second (beta) or third (gamma) toxicity paths.

_{Clog}and statistical R

_{Clog}frameworks of Figures 2 and 4 due to POL and LogP parameters specific influence - identified on the grid region of their path crossings, respectively. Such a situation is no longer valid when Mlog values are modeled, since the algebraic RA

_{Mlog}approach predicts moderate inter-toxicity influence (through alpha-beta crossing paths due E

_{tot}or steric influence) (see Figure 1), in contrast with no recorded interaction within the statistical R

_{Mlog}analysis (see Figure 3).

## 4. Discussion of ESIP

_{50}and calculated Clog/MRC

_{50}values are in concordance with the numerical structural parameter counterparts in the Table 2. This also indicates that the individual structural parameters or their combinations are specific and the organism H.e. can be successful employed as a suitable test-system for further toxicity determinations.

## 5. Conclusions and Outlook

_{tot}); the influence of the amino-group for aliphatic amines is comparatively predominant to the relative extent of the hydrocarbon chain [5]; the influence of the first methyl in the phenol case is negligible; the steric influence of isopropyl-radicals on the phenolic active centre for 2,6-derivative is stronger (by increased POL value) as in the case of the t-butyl substituent (while logP and POL values are diminished), etc.

## Appendix

## A1. Vectorial Scalar Product, Norms, and Cauchy-Schwarz Inequality

_{1},u

_{2},u

_{3}〉:

## A2. Algebraic Correlation Factor

## Acknowledgments

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**Figure 1.**The Hydractinia echinata (H.e.), Tetrahymena pyriformis (T.p.), Pimephales promelas (P.p.), Vibrio fisheri (V.f.), and Daphnia magna (D.m.) interspecies Spectral-SAR map modeling the molecular mechanisms for Mlog-algebraic toxicity paths of Tables 5 and 6 connecting the algebraic correlations of Table 3 across the ordered models of Table 7; the difference between species is made by the assignments of distinct icons, while alpha, beta and gamma paths are differentiated by thickness decreasing of lines joining the same icons; the D.m. pseudo-path (interrupted line on map) is considered from the highest correlation model towards the lowest one in Table 3.

**Figure 2.**The same type of representation as of Figure 1, here at the Clog-algebraic level.

**Figure 3.**The same type of representation as of Figure 1, here at the Mlog-statistic level.

**Figure 4.**The same type of representation as of Figure 1, here at the Clog-statistic level.

**Table 1.**The vectorial (molecular) descriptors in a Spectral-SAR analysis represented as states, within the Hilbert N-dimensional space of investigated molecules.

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

|Y_{OBS} _{(}_{ERVED}_{)}〉 | |X_{0}〉 | |X_{1}〉 | … | |X_{k}〉 | … | |X_{M}〉 |

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

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

… | … | … | … | … | … | … |

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

**Table 2.**The measured Mlog(1/MRC50) and ESIP-computed Clog(1/MRC50) toxicities for Hydractinia echinata and other organisms: for compounds nos. 2–7 from Ref. [5], for compounds nos. 13–21 from Ref. [6], new data for the rest; the Hansch molecular parameters as hydrophobicity (LogP), polarizability (POL) and the steric optimized total energy (E

_{tot}) were computed by HyperChem environment [26].

No. | Compound | Species Toxicities |Y> | Structural parameters | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Hydractinia echinata | Tetrahymena pyriformis | Pimephales promelas | Vibrio fischeri | Daphnia magna | |X_{1}> =Log P | |X_{2}> =POL (A^{3}) | |X_{3}>=Etot (kcal/mol) | |||||||

Mlog | Clog | Mlog | Clog | Mlog | Clog | Mlog | Clog | Mlog | Clog | |||||

1 | Water | −1.23 | −0.91 | −0.51 | 1.41 | −8038.2 | ||||||||

2 | Methanol | −0.22 | −0.41 | 0.33 | 0.15 | 0.04 | 0.24 | −0.27 | 3.25 | −11622.9 | ||||

3 | Ethanol | 0.02 | 0.09 | 0.59 | 0.60 | 0.51 | 0.74 | 0.11 | 1.11 | 0.93 | 0.25 | 0.08 | 5.08 | −15215.4 |

4 | 1-Butanol | 0.99 | 1.08 | 1.48 | 1.50 | 1.63 | 1.73 | 1.34 | 2.18 | 1.57 | 1.68 | 0.94 | 8.75 | −22402.8 |

5 | 1,2,3-Propanetriol | 0.34 | 0.37 | −1.08 | 8.19 | −33600. | ||||||||

6 | Triphenylmethanol | 5.69 | 5.27 | 4.87 | 32.23 | −68532.5 | ||||||||

7 | 1,10-Diaminodecane | 3.26 | 2.91 | 1.48 | 21.83 | −46754.2 | ||||||||

8 | 2-Benzylpyridine | 3.75 | 3.46 | 3.41 | 4.85 | 3.53 | 21.22 | −43675.3 | ||||||

9 | 4-Benzylpyridine | 4.08 | 3.46 | 3.68 | 4.85 | 3.75 | 21.22 | −43676.8 | ||||||

10 | 4-Phenylpyridin | 4.13 | 3.46 | 3.66 | 3.46 | 3.98 | 3.81 | 4.91 | 4.84 | 3.35 | 19.38 | −40083.1 | ||

11 | 4-Toluidine | 2.85 | 2.02 | 2.98 | 2.81 | 3.43 | 3.26 | 1.73 | 13.62 | −28300.3 | ||||

12 | 1,2-Dichlorobenzene | 3.04 | 3.45 | 4.00 | 3.66 | 4.19 | 4.17 | 3.08 | 14.29 | −36217.2 | ||||

13 | Phenol(3,15/2,66/2,85)* | 2.89* | 2.87 | 2.79 | 2.58 | 3.41 | 3.21 | 3.42 | 3.68 | 3.32 | 3.32 | 1.76 | 11.07 | −27003.1 |

14 | 2-Methylphenol(3,18/3,24)* | 3.21* | 2.82 | 2.72 | 2.58 | 3.77 | 3.21 | 3.75 | 3.68 | 3.64 | 3.32 | 2.23 | 12.91 | −30596.6 |

15 | 2,4,6-Trimethyphenol(3,19/4,00)* | 3.60* | 3.82 | 3.42 | 3.48 | 4.02 | 4.21 | 4.08 | 4.75 | 4.49 | 4.75 | 3.16 | 16.58 | −37783.7 |

16 | 1,2-Dihydroxibenzene | 5.11 | 5.11 | 3.75 | 3.47 | 4.08 | 4.08 | 3.54 | 3.54 | 4.68 | 4.24 | 1.48 | 11.71 | −34396.4 |

17 | 2-Methoxyphenol(2,89/2,77)* | 2.83* | 3.28 | 2.49 | 2.54 | 3.29 | 1.51 | 13.54 | −37974.4 | |||||

18 | 1,4-Dihydroxybenzene(6,14/6,06)* | 6.10* | 6.10 | 3.47 | 3.59 | 6.40 | 6.40 | 6.42 | 6.42 | 1.48 | 11.71 | −34395.8 | ||

19 | t-Butylhydroquinone(5,05/5,00)* | 5.30* | 7.60 | 4.94 | 7.78 | 8.03 | 3.11 | 19.05 | −48758.1 | |||||

20 | 1,2,3-Trihydroxibenzene | 5.15 | 5.15 | 3.85 | 3.65 | 1.19 | 12.35 | −41789.9 | ||||||

21 | 4(3',5'-dimethyl--3'-heptyl) phenol | 7.65 | 6.81 | 4.87 | 25.75 | −55742. | ||||||||

22 | 4-Chlorophenol | 3.25 | 3.04 | 3.55 | 3.56 | 4.18 | 3.66 | 4.19 | 3.88 | 4.13 | 3.95 | 2.28 | 13 | −35307.6 |

23 | 2,6-Diisopropylphenol | 3.73 | 5.31 | 4.82 | 5.21 | 6.36 | 6.90 | 4.15 | 22.08 | −48554.7 | ||||

24 | 2-Aminophenol | 3.15 | 3.04 | 3.94 | 2.93 | 0.98 | 12.42 | −32098.6 | ||||||

25 | 2,4,6-Trinitrophenol | 2.92 | 2.99 | 2.84 | 2.63 | 3.77 | −4.17 | 16.59 | −84472.1 | |||||

26 | Chloranil | 5.15 | − | 1.12 | 18.51 | −66928.2 | ||||||||

27 | Chloranilic acid | 3.40 | 2.99 | −0.48 | 15.93 | −65113.6 | ||||||||

28 | 4-Methoxyazobenzene | 5.20 | 3.70 | 4.10 | 24.63 | −59069.5 |

^{*}The M* values represents the experimental results accomplished by different time interval with different generations of H.e. These results clearly prove the reproducibility of the test-system.

**Table 3.**Mlog-Spectral Spectral-SAR results employing the molecular parameters and the Hydractinia echinata (H.e.), Tetrahymena pyriformis (T.p.), Pimephales promelas (P.p.), Vibrio fisheri (V.f.), and Daphnia magna (D.m.) toxicities of Table 2; the models are characterized either by algebraic norms and correlation factors (RA), computed upon the Equations (6) and (8), and by Pearson statistical correlation (R) of Equation (7), for all possible mono-, bi-, and all- end-points, respectively. The referential algebraic norms of the considered species were estimated with the aid of Equation (6) from the Mlog input toxicity data of Table 2 as: ║|Y

_{H.e.}>║ = 20.8547, ║|Y

_{T.p.}>║ = 13.2774, ║|Y

_{P.p.}>║ = 12.8515, ║|Y

_{V.f.}>║ = 12.1055, ║|Y

_{D.m.}>║ = 9.31242.

Mlog Model | Species | Spectral-SAR Activity Equation | Spectral-SAR Norm | RA (Algebraic) | R (Statistic) |
---|---|---|---|---|---|

|1> | H.e. | |Y_{H.e}^{|1>}〉 = 2.348 + 0.595 |LogP> | 19.0572 | 0.9138 | 0.5912 |

T.p. | |Y_{T.p}^{|1>}〉 = 2.526+0.267 |LogP> | 12.642 | 0.9521 | 0.4446 | |

P.p. | |Y_{P.p}^{|1>}〉 = 1.402 +1.071 |LogP> | 12.1481 | 0.9453 | 0.6972 | |

V.f. | |Y_{V.f}^{|1>}〉 = 2.981 + 0.364 |LogP> | 11.1235 | 0.9189 | 0.4396 | |

D.m. | |Y_{D.m}^{|1>}〉 = 1.192 + 1.208 |LogP> | 9.09749 | 0.9769 | 0.8300 | |

|2> | H.e. | |Y_{H.e}^{|2>}〉 = 0.022 + 0.221 |POL> | 19.7048 | 0.9449 | 0.7597 |

T.p. | |Y_{T.p}^{|2>}〉 = 0.72 + 0.168 |POL> | 12.9074 | 0.9721 | 0.7267 | |

P.p. | |Y_{P.p}^{|2>}〉 = −0.109 + 0.29 |POL> | 12.2254 | 0.9513 | 0.7357 | |

V.f. | |Y_{V.f}^{|2>}〉 = 0.121 + 0.262 |POL> | 11.3472 | 0.9374 | 0.6092 | |

D.m. | |Y_{D.m}^{|2>}〉 = −0.759 + 0.355 |POL> | 9.16099 | 0.9837 | 0.8832 | |

|3> | H.e. | |Y_{H.e.}^{|3>}〉 = 0.433 –0.00007 |E_{tot} > | 19.2139 | 0.9213 | 0.6355 |

T.p. | |Y_{T.p}^{|3>}〉 = 1.669 –3.6·10^{−5} |E_{tot} > | 12.6819 | 0.9551 | 0.4969 | |

P.p. | |Y_{P.p.}^{|3>}〉 = –1.767 – 1.7·10^{−4} |E_{tot} > | 12.5439 | 0.9761 | 0.8785 | |

V.f. | |Y_{V.f.}^{|3>}〉 = 2.755 – 1.89·10^{−5}|E_{tot} > | 10.926 | 0.9026 | 0.1982 | |

D.m. | |Y_{D.m}^{|3>}〉 = −1.826 –1.75·10^{−4} |E_{tot} > | 9.26686 | 0.9951 | 0.9662 | |

|1,2> | H.e. | |Y_{H.e}^{|1,2>}〉 = 0.206 + 0.163|LogP> +0.19 |POL> | 19.7462 | 0.9468 | 0.7694 |

T.p. | |Y_{T.p}^{|1,2>}〉 = 0.784 +0.093|LogP> +0.152 |POL> | 12.9228 | 0.9733 | 0.7398 | |

P.p. | |Y_{P.p}^{|1,2>}〉 = −0.324 –0.191|LogP> + 0.337 |POL> | 12.2271 | 0.9514 | 0.7365 | |

V.f. | |Y_{V.f.}^{|1,2>}〉 = −0.018 + 0.307|LogP> + 0.242 |POL> | 11.5146 | 0.9512 | 0.7116 | |

|1,3> | H.e. | |Y_{H.e}^{|1,3>}〉 = −0.296 + 0.541|LogP> – 0.00007 |E_{tot} > | 20.0182 | 0.9599 | 0.8307 |

T.p. | |Y_{T.p.}^{|1,3>}〉 = 0.433 + 0.413| LogP> – 5.27·10^{−5} |E_{tot} > | 13.018 | 0.9805 | 0.8171 | |

P.p. | |Y_{P.p}^{|1,3>}〉 = −3.541 −1.061|LogP> −2.96·10^{−4} |E_{tot} > | 12.646 | 0.9840 | 0.9203 | |

V.f. | |Y_{V.f.}^{|1,3>}〉 = −0.512 +0.82 |LogP> −8.1·10^{−5} |E_{tot} > | 11.6329 | 0.9610 | 0.7767 | |

|2,3> | H.e. | |Y_{H.e.}^{|2,3>}〉 = −0.134 + 0.193|POL> – 0.00001 |E_{tot} > | 19.7224 | 0.9457 | 0.7638 |

T.p. | |Y_{T.p.}^{|2,3>}〉 = 0.704 +0.163 |POL> – 2.18·10^{−6} |E_{tot} > | 12.9078 | 0.9722 | 0.7270 | |

P.p. | |Y_{P.p.}^{|2,3>}〉 = −2.269 −0.262|POL> −2.94·10^{−4} |E_{tot} > | 12.6208 | 0.9820 | 0.9101 | |

V.f. | |Y_{V.f.}^{|2,3>}〉 = 0.082 + 0.36 |POL> +3.35〉10^{−5} |E_{tot} > | 11.4347 | 0.9446 | 0.6645 | |

{|1,2,3>} | H.e. | |Y_{H.e}^{{|1,2,3>}}〉 = −0.259 + 0.979|LogP> −0.214|POL> −0.00013|E_{tot} > | 20.1085 | 0.9642 | 0.8502 |

T.p. | |Y_{T.p}^{{|1,2,3>}}〉 = 0.456 + 0.773 |LogP> −0.185|POL> −0.00011|E_{tot} > | 13.0541 | 0.9832 | 0.8447 |

**Table 4.**The same type of Spectral-SAR models as those of Table 3, here for Clog data of Table 2. The referential algebraic norms of the considered species were estimated with the Equation (6) from the Clog input toxicity data of Table 2 as: ║|Y

_{H.e.}>║ = 20.1051, ║|Y

_{T.p.}>║ = 14.8984, ║|Y

_{P.p.}>║ = 15.5929, ║|Y

_{V.f.}>║ = 16.6682, ║|Y

_{D.m.}>║ = 11.3438.

Clog Model | Species | Spectral-SAR Activity Equation | Spectral-SAR Norm | RA (Algebraic) | R (Statistic) |
---|---|---|---|---|---|

|1> | H.e. | |Y_{H.e}^{|1>}〉 = 2.242 + 0.583 |LogP> | 18.1498 | 0.9027 | 0.5744 |

T.p. | |Y_{T.p}^{|1>}〉 = 1.248+ 0.919 |LogP> | 14.6075 | 0.9805 | 0.8572 | |

P.p. | |Y_{P.p}^{|1>}〉 = 1.436 + 1.107 |LogP> | 14.7182 | 0.9439 | 0.7011 | |

V.f. | |Y_{V.f}^{|1>}〉 = 3.598 + 0.41 |LogP> | 15.6785 | 0.9406 | 0.4605 | |

D.m. | |Y_{D.m}^{|1>}〉 = 0.57 + 1.483 |LogP> | 11.2079 | 0.9880 | 0.9432 | |

|2> | H.e. | |Y_{H.e}^{|2>}〉 = 0.242 + 0.201 |POL> | 18.5655 | 0.9234 | 0.6831 |

T.p. | |Y_{T.p}^{|2>}〉 = –0.118 + 0.237 |POL> | 14.7122 | 0.9875 | 0.9108 | |

P.p. | |Y_{P.p}^{|2>}〉 = –0.092 + 0.29 |POL> | 14.871 | 0.9537 | 0.7604 | |

V.f. | |Y_{V.f}^{|2>}〉 = 0.111 + 0.298 |POL> | 16.1385 | 0.9682 | 0.7565 | |

D.m. | |Y_{D.m}^{|2>}〉 = –1.241 + 0.379 |POL> | 11.2605 | 0.9927 | 0.9655 | |

|3> | H.e. | |Y_{H.e}^{|3>}〉 = 0.518 – 0.00007 |E_{tot} > | 18.16 | 0.9033 | 0.5773 |

T.p. | |Y_{T.p}^{|3>}〉 = –1.176 –1.27·10^{−4} |E_{tot} > | 14.8013 | 0.9935 | 0.9544 | |

P.p. | |Y_{P.p}^{|3>}〉 = –1.597 – 1.64·10^{−4} |E_{tot} > | 15.2359 | 0.9771 | 0.8882 | |

V.f. | |Y_{V.f.}^{|3>}〉 = 2.546 – 4.51·10^{−5} |E_{tot} > | 15.6221 | 0.9372 | 0.4106 | |

D.m. | |Y_{D.m}^{|3>}〉 = –2.546 –1.94·10^{−4} |E_{tot} > | 11.3184 | 0.9978 | 0.9896 | |

|1,2> | H.e. | |Y_{H.e}^{|1,2>}〉 = 0.488 + 0.217 |LogP> + 0.16 |POL> | 18.6415 | 0.9272 | 0.7014 |

T.p. | |Y_{T.p}^{|1,2>}〉 = –0.208 – 0.081|LogP> + 0.255 |POL> | 14.7128 | 0.9875 | 0.9112 | |

P.p. | |Y_{P.p.}^{|1,2>}〉 = –1.038 – 0.931|LogP> + 0.509 |POL> | 14.908 | 0.9561 | 0.7742 | |

V.f. | |Y_{V.f.}^{|1,2>}〉 = 0.228 + 0.188 |LogP> + 0.268 |POL> | 16.187 | 0.9711 | 0.7816 | |

|1,3> | H.e. | |Y_{H.e}^{|1,3>}〉 = –0.134 + 0.522 |LogP> – 0.00006 |E_{tot} > | 18.9449 | 0.9423 | 0.7708 |

T.p. | |Y_{T.p.}^{|1,3>}〉 = –0.859 + 0.219 |LogP> – 1.04·10^{−4} |E_{tot} > | 14.8152 | 0.9944 | 0.9611 | |

P.p. | |Y_{P.p}^{|1,3>}〉 = –3.524 – 1.327|LogP>– 3.1·10^{−4} |E_{tot} > | 15.4088 | 0.9882 | 0.9437 | |

V.f. | |Y_{V.f.}^{|1,3>}〉 = –0.12 + 0.713 |LogP> – 8.42·10^{−5} |E_{tot} > | 16.2777 | 0.9766 | 0.8267 | |

|2,3> | H.e. | |Y_{H.e}^{|2,3>}〉 = 0.093 + 0.175 |POL> – 0.00001 |E_{tot} > | 18.5804 | 0.9242 | 0.6868 |

T.p. | |Y_{T.p.}^{|2,3>}〉 = –1.045 + 0.062 |POL> – 9.77·10^{−5}|E_{tot} > | 14.8118 | 0.9942 | 0.9594 | |

P.p. | |Y_{P.p}^{|2,3>}〉 = –2.243 –0.355 |POL> –3.28·10^{−4} |E_{tot} > | 15.3717 | 0.9858 | 0.9320 | |

V.f. | |Y_{V.f}^{|2,3>}〉 = 0.2 + 0.337 |POL> + 1.65·10^{−5} |E_{tot} > | 16.1548 | 0.9692 | 0.7650 | |

{|1,2,3>} | H.e. | |Y_{H.e}^{{|1,2,3>}}〉 = –0.166 + 1.229|LogP> – 0.351|POL> –0.00017|E_{tot} > | 19.1684 | 0.9534 | 0.8188 |

T.p. | |Y_{T.p}^{{|1,2,3>}}〉 = –0.871 + 0.199|LogP> + 0.008|POL> –0.0001|E_{tot} > | 14.8153 | 0.9944 | 0.9611 |

**Table 5.**Synopsis of the statistic and algebraic values of paths connecting the Spectral-SAR models for Hydractinia echinata (H.e.) and Tetrahymena pyriformis (T.p.) in the Mlog/Clog and algebraic/statistical computational frames of Tables 3 and 4. The primary, secondary and tertiary - the so called alpha (α), beta (β) and gamma (γ) - paths are indicated according to the least path principle in spectral norm-correlation space, respectively.

Species Method Paths | H.e. | T.p. | ||||||
---|---|---|---|---|---|---|---|---|

Mlog | CLog | Mlog | Clog | |||||

Algebraic | Statistic | Algebraic | Statistic | Algebraic | Statistic | Algebraic | Statistic | |

|1>→|1,2>→|1,2,3> | 1.05246 | 1.08283^{γ} | 1.01981 | 1.0476^{γ} | 0.41325 | 0.575439^{γ} | 0.208278 | 0.232353 |

|1>→|1,3>→|1,3,2> | 1.05246^{γ} | 1.08273 | 1.01981^{γ} | 1.04754 | 0.41325^{γ} | 0.574673 | 0.208278^{γ} | 0.232342^{γ} |

|1>→|2,3>→|1,2,3> | 1.05246 | 1.08284 | 1.01981 | 1.0476 | 0.41325 | 0.575534 | 0.208278 | 0.232342 |

|2>→|1,2>→|2,1,3> | 0.404191 | 0.413755 | 0.603637 | 0.61798 | 0.147067 | 0.188257 | 0.103313^{β} | 0.114674^{β} |

|2>→|1,3>→|2,1,3> | 0.404191 | 0.413756 | 0.603637 | 0.617987 | 0.147067 | 0.188265 | 0.103313 | 0.114674 |

|2>→|2,3>→|2,3,1> | 0.404191^{α} | 0.413754^{α} | 0.603637^{α} | 0.617972^{α} | 0.147067^{α} | 0.188246^{α} | 0.103313 | 0.114674 |

|3>→|1,2>→|3,1,2> | 0.89559^{β} | 0.920041 | 1.00961^{β} | 1.03703 | 0.373261^{β} | 0.510175 | 0.191347 | 0.212443 |

|3>→|1,3>→|3,1,2> | 0.89559 | 0.919987^{β} | 1.00961 | 1.03697^{β} | 0.373261 | 0.509664^{β} | 0.0140336 | 0.0155182 |

|3>→|2,3>→|3,2,1> | 0.89559 | 0.920044 | 1.00961 | 1.03703 | 0.373261 | 0.510232 | 0.0140336^{α} | 0.0155182^{α} |

**Table 6.**The same type of information and analysis as in Table 5, here for Pimephales promelas (P.p.) and Vibrio fisheri (V.f.) species.

Species Method Paths | P.p. | V.f. | ||||||
---|---|---|---|---|---|---|---|---|

Mlog | CLog | Mlog | Clog | |||||

Algebraic | Statistic | Algebraic | Statistic | Algebraic | Statistic | Algebraic | Statistic | |

|1>→|1,2> | 0.0792073 | 0.0881801 | 0.190201 | 0.2034 | 0.392451^{β} | 0.476398^{β} | 0.509418^{β} | 0.601392^{β} |

|1>→|1,3> | 0.499343^{γ} | 0.545515^{γ} | 0.692013^{γ} | 0.731962^{γ} | 0.511148 | 0.61083 | 0.600329 | 0.702271 |

|1>→|2,3> | 0.474093 | 0.518389 | 0.654817 | 0.69307 | 0.312213 | 0.383883 | 0.477152 | 0.565317 |

|2>→|1,2> | 0.00166893^{α} | 0.0018496^{α} | 0.0371086^{α} | 0.0395137^{α} | 0.167993 | 0.196303 | 0.0485269 | 0.0545366 |

|2>→|1,3> | 0.421805 | 0.459267 | 0.538921 | 0.568184 | 0.28669 | 0.331225 | 0.139438 | 0.155864 |

|2>→|2,3> | 0.396554 | 0.432134 | 0.501725 | 0.529282 | 0.0877552^{α} | 0.103489^{α} | 0.0162612^{α} | 0.0183134^{α} |

|3>→|1,2> | 0.3177 | 0.347114 | 0.328541 | 0.347126 | 0.590616 | 0.781086 | 0.565916 | 0.675813 |

|3>→|1,3> | 0.102435 | 0.11035 | 0.173271 | 0.181596 | 0.709313^{γ} | 0.913454^{γ} | 0.656827^{γ} | 0.776511^{γ} |

|3>→|2,3> | 0.0771849^{β} | 0.0832042^{β} | 0.136075^{β} | 0.142683^{β} | 0.510379 | 0.690032 | 0.53365 | 0.639803 |

Computational Modes | Minimum Interspecies Paths | Ordered Endpoints | |||
---|---|---|---|---|---|

alpha | beta | gamma | |||

Algebraic | Mlog | α_{P.p.} | β_{P.p.} | γ_{T.p.} | |

|2>→|1,2> | |3>→|2,3> | |1>→|1,3> | |2>→|3>→|1>→|1,2>→|2,3>→|1,3>→{|1,2,3>} | ||

Clog | α_{T.p.} | β_{T.p.} | γ_{T.p.} | ||

|3>→|2,3> | |2>→|1,2> | |1>→|1,3> | |3>→|2>→|1>→|2,3>→|1,2>→|1,3>→{|1,2,3>} | ||

Statistic | Mlog | α_{P.p.} | β_{P.p.} | γ_{P.p.} | |

|2>→|1,2> | |3>→|2,3> | |1>→|1,3> | |2>→|3>→|1>→|1,2>→|2,3>→|1,3>→{|1,2,3>} | ||

Clog | α_{T.p.} | β_{T.p.} | γ_{T.p.} | ||

|3>→|2,3> | |2>→|1,2> | |1>→|1,3> | |3>→|2>→|1>→|2,3>→|1,2>→|1,3>→{|1,2,3>} |

© 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/).

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**MDPI and ACS Style**

Chicu, S.A.; Putz, M.V.
Köln-Timişoara Molecular Activity Combined Models toward Interspecies Toxicity Assessment. *Int. J. Mol. Sci.* **2009**, *10*, 4474-4497.
https://doi.org/10.3390/ijms10104474

**AMA Style**

Chicu SA, Putz MV.
Köln-Timişoara Molecular Activity Combined Models toward Interspecies Toxicity Assessment. *International Journal of Molecular Sciences*. 2009; 10(10):4474-4497.
https://doi.org/10.3390/ijms10104474

**Chicago/Turabian Style**

Chicu, Sergiu A., and Mihai V. Putz.
2009. "Köln-Timişoara Molecular Activity Combined Models toward Interspecies Toxicity Assessment" *International Journal of Molecular Sciences* 10, no. 10: 4474-4497.
https://doi.org/10.3390/ijms10104474