# Introducing Catastrophe-QSAR. Application on Modeling Molecular Mechanisms of Pyridinone Derivative-Type HIV Non-Nucleoside Reverse Transcriptase Inhibitors

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background Theories

#### 2.1. QSAR Phenomenology

- Y stands for the computed activity, not the observed activity, from the statistical characteristics of the present approach; thus the validation of Equation (1) should be done for another (preferably external or testing) set of compounds with which the predictive power of Equation (1) is tested.
- Because the right side of Equation (1) unfolds as a linear summation of the structural characteristics, it corresponds in fact with the quantum superposition principle, which provides a global Eigen-solution for a quantum system from its particular realization in orthogonal or projective sub-space; from where the need arises for structural indices X
_{1}, ..., X_{M}to be either linearly independent or orthogonal in algebraic space built from their associate vectors presented in Table 1.

- QSAR 1: a defined endpoint
- QSAR-2: an unambiguous algorithm
- QSAR-3: a defined domain of applicability
- QSAR-4: appropriate measures of goodness-of–fit, robustness and predictivity
- QSAR-5: a mechanistic interpretation, if possible

- QSAR-1. why does one do modeling ?
- QSAR-2. how does one do modeling ?
- QSAR-3. with what tools do I model ?
- QSAR-4. how reliable is what I modeled ?
- QSAR-5. what knowledge did the model provide ?

#### 2.2. Thom’s Catastrophe Theory

^{k}), parameterized by C

_{k}’s, provides a sudden change in its behavior space (I

^{m}), described by x

_{m}variables through stable singularities of the smooth map [34,35]

_{k}, x

_{m}) called the generic potential of the system. Therefore, catastrophes are given by the set of critical points (c

_{k}, x

_{m}) for which the field gradient of the generic potential vanishes

^{k}

^{×}

^{m}→ C

^{k}.

^{k}(also called the co-dimension, k) and on the number of variables in space I

^{m}(also called the co-rank, m), Thom classified the generic potentials (or maps) given by Equation (2) as seven unfolding elementary (in the sense of universal) catastrophes, i.e., providing the multi-variable (with the co-rank up to two) and multi-parametrical (with the co-dimension up to four) polynomials listed in Table 2. Going to the higher derivatives of the generic potential (the fields), the control parameter c

_{k}* for which the Laplacian of the generic potential vanishes

^{#}for which the Laplacian of a critical point is non-zero defines the domain of stability of the critical point. It is clear now that small perturbations of η(c*, x) bring the system from one domain of stability to another; otherwise, the system is located within a domain of structural stability.

_{m}(c

_{k}, t). The connection with equilibrium is recovered through the stationary time regime imposed on the critical points. In this way, the set of points giving a critical point in the stationary t → +∞ regime (the so-called ω-limit) corresponds to an attractor, and it forms a basin, whereas the stationary regime t→ −∞ (the so-called α-limit) describes a repellor. In this way, the catastrophe polynomials may be regarded either as an asymptotic solution of a dynamical evolutionary system or as a steady state solution allowing the quasi-equilibrium of the ligand-receptor or inhibitor-organism interactions to be described. However, in complex binding systems with multiple evolutionary phases, e.g., the HIV-1 life cycle, the possibility of “linking” the various classes of catastrophes themselves may provide a striking analytical approach to the dynamics and mutational sensitivity of the studied interaction that starts with the actual catastrophe-QSAR method.

## 3. Catastrophe-QSAR Method

- Determine the norms for each model$$\Vert |Y\rangle \Vert =\sqrt{\langle Y|Y\rangle}=\sqrt{\sum _{i=1}^{N}{y}_{i}^{2}}$$
- Calculate the algebraic correlation factor for each model [31]$${R}_{ALG}=\frac{\Vert |Y\rangle \Vert}{\Vert |A\rangle \Vert}=\sqrt{\frac{{\displaystyle \sum _{i=1}^{N}}{y}_{i}^{2}}{{\displaystyle \sum _{i=1}^{N}}{A}_{i}^{2}}}$$
- Calculate the so-called “statistical relative power” index for each model with each set of descriptors$$\mathrm{\Pi}=\sqrt{{r}^{2}+{t}^{2}+{f}^{2}}$$where the components are defined as follows:
- relative index of correlation:$$r=\frac{{R}_{ALG}}{{R}_{Pearson}}$$
- relative index for Student’s t-test$$t=\frac{{t}_{Computed}}{{t}_{\begin{array}{l}Tabulated\\ (1-\alpha =0.99;\\ N-M-2)\end{array}}}$$
- relative index for Fisher’s test$$f=\frac{{F}_{Computed}}{{F}_{\begin{array}{l}Tabulated\\ (1-\alpha =0.99;\\ M,N-M-1)\end{array}}}$$

- Determine the generalized Euclidian distances between corresponding type-I and type-II models employing different descriptors$$\mathrm{\Delta}\mathrm{\Pi}=\sqrt{{(r-{r}^{\prime})}^{2}+{(t-{t}^{\prime})}^{2}+{(f-{f}^{\prime})}^{2}}$$and establish formal matrices for the models’ differences for single descriptors, respectively$${\mathrm{\Delta}}^{2}{\mathrm{\Pi}}_{I({X}_{1},{X}_{2})}=\left|\mathrm{\Delta}{\mathrm{\Pi}}_{I({X}_{1})}-\mathrm{\Delta}{\mathrm{\Pi}}_{I({X}_{2})}\right|$$where$$\begin{array}{l}\mathrm{\Delta}{\mathrm{\Pi}}_{I(X={X}_{1}\vee {X}_{2})}\\ =\left(\begin{array}{cccc}{\mathit{QSAR}}_{I(X)}-{F}_{(X)}& {\mathit{QSAR}}_{I(X)}-{C}_{(x)}& {\mathit{QSAR}}_{I(X)}-{ST}_{(X)}& {\mathit{QSAR}}_{I(X)}-{B}_{(X)}\\ \hspace{0.17em}& {F}_{(X)}-{C}_{(X)}& {F}_{(X)}-{ST}_{(X)}& {F}_{(X)}-{B}_{(X)}\\ \hspace{0.17em}& \hspace{0.17em}& {C}_{(X)}-{ST}_{(X)}& {C}_{(X)}-{B}_{(X)}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& {ST}_{(X)}-{B}_{(X)}\end{array}\right)\end{array}$$and for pair descriptors$$\begin{array}{l}\mathrm{\Delta}{\mathrm{\Pi}}_{II({X}_{1}\wedge {X}_{2})}\\ =\left(\begin{array}{ccc}{\mathit{QSAR}}_{II({X}_{1},{X}_{2})}-{HU}_{({X}_{1},{X}_{2})}& {\mathit{QSAR}}_{II({X}_{1},{X}_{2})}-{EU}_{({X}_{1},{X}_{2})}& {\mathit{QSAR}}_{II({X}_{1},{X}_{2})}-{PU}_{({X}_{1},{X}_{2})}\\ \hspace{0.17em}& {HU}_{({X}_{1},{X}_{2})}-{EU}_{({X}_{1},{X}_{2})}& {HU}_{({X}_{1},{X}_{2})}-{PU}_{({X}_{1},{X}_{2})}\\ \hspace{0.17em}& \hspace{0.17em}& {EU}_{({X}_{1},{X}_{2})}-{PU}_{({X}_{1},{X}_{2})}\end{array}\right)\end{array}$$
- Identify all minimum paths across all differences ΔΠ
_{I}_{(}_{X}_{1∨}_{X}_{2)}, Δ^{2}Π_{I}_{(}_{X}_{1,}_{X}_{2)}and ΔΠ_{II}_{(}_{X}_{1∧}_{X}_{2)}for a given set of descriptors (X_{1}, X_{2})$$\{\begin{array}{l}\delta \left\{\mathrm{\Delta}{\mathrm{\Pi}}_{I(X)}\right\}=0\hfill \\ \delta \left\{{\mathrm{\Delta}}^{2}{\mathrm{\Pi}}_{I({X}_{1}\vee {X}_{2})}\right\}=0\hfill \\ \delta \left\{\mathrm{\Delta}{\mathrm{\Pi}}_{II({X}_{1}\wedge {X}_{2})}\right\}=0\hfill \end{array}$$The combination of descriptors that fulfills this system provides the molecular mechanism of the interaction. The correlation models involved are ordered according to their relative statistical power within the same molecular mechanism, thereby providing the best models. Because pair-descriptors are primarily involved in the present analysis, one can consider the first two such “waves” and their best correlation models up to the second order minimum paths, as in Equation (16). - For selected correlation models, in either structure-driven or molecular mechanistic “waves,” one employs them to compute the associated predicted activities for test molecules and to provide the statistics regarding the observed activity. If the obtained relative statistical power is close to those characteristic for the trial set of molecules, then these models may be validated for the specific eco-, bio-, or pharmacological problem. Moreover, further insight will be provided by the analysis of the catastrophe shape of the models involved and discussed accordingly.

## 4. Application to Non-Nucleoside Reverse Transcriptase Pyridinone Inhibitors

#### 4.1. Input Data

^{®}, Delavirdine-Rescriptor

^{®}, Efavirenz-Sustiva

^{®}, Etravirine-Intelence

^{®}), all of which bind to the hydrophobic pocket of HIV-1 reverse transcriptase [38]. The pyridinone derivatives were divided into a training set of 23 compounds and a test set of 9 compounds according to the methods of normal/Gaussian (G) and non-normal/non-Gaussian (NG) fitted activity [39–41] (Figure 1).

#### 4.2. Results and Discussion

- - First, it is clear that consideration of the catastrophe (polynomial) correlations is an improvement over the old multi-linear QSAR statistics (see also Appendix-A2).
- - The hydrophobicity indicator gives generally low correlations with any polynomial (linear, multilinear or catastrophe) approach, being a quite irrelevant linear QSAR descriptor (Table 5) but improving up to twice its influence within the swallow tail and butterfly phenomenologies once its fifth and sixth power involvement are considered. Nevertheless, this provides a sign of the value of catastrophe-QSAR for achieving a deeper understanding of the molecular mechanics of specific interactions when the normal multi-linear QSAR does not assign transport descriptors with much predictive power.
- - The relative statistical power, as defined by Equation (8), does not always parallel the Pearson coefficient or the relative correlation factors, as is evident from Tables 5 and 6. However, because it includes more statistical information, we consider a model as relevant when it has greater individual output of this newly introduced statistical index. In particular, neither the linear nor the multilinear QSAR framework provides a good fit between the statistical correlation and the relative statistical power using the structural parameter combinations considered. Instead, parabolic catastrophe correlations, the cusp and butterfly models, are revealed to be quite relevant, in particular regarding the formation energy (H) for which they show the highest Pearson correlation and relative statistical power values in comparison with the other descriptors plugged into these models. Unfortunately, for the two-variable descriptor models of Table 6, no consistency was found between the highest Pearson value and the relative statistical power apart from a few degenerate cases of descriptors for the parabolic models where the highest relative statistical power value corresponds with the highest Pearson correlation. Note that for the degenerate cases of Table 6, when two mixed descriptors can be combined in two distinct ways, the working model is considered to have maximum relative statistical power.

- - Table 7: At the individual descriptor level, the cusp and butterfly models are very close to each other for Log P and the forming energy H, which is even more relevant for the hydrophobicity, because for the forming energy it transpires from Table 5 that the butterfly model practically reduces to the cusp model because the sixth contribution virtually vanishes. However, for the structural influence on polarizability (POL) the butterfly and swallow tail are the closest models. When one considers the hierarchy of the individual descriptors according to their QSAR-I models in Table 5 in terms of the reduction in relative statistical power$$Log\hspace{0.17em}P\to H\to POL$$

- - Table 8: When the second order distance difference is considered between the individual inter-modeling paths of Table 7, it can nevertheless be considered through the further variations of paths of Table 7. Also, the QSAR-I and the fold (F) catastrophe model intervene in changing the influence on specific interactions from POL to H. Therefore, by counting the minimum hierarchy of these paths, the distance ordering is obtained as follows:$$(LogP\xf7H)\to (H\xf7POL)\to (POL\xf7LogP)$$

- - Table 9: Interestingly, in terms of the two structural descriptors, the QSAR model is present even though its individual statistics are not the highest in Table 6; however, judging by the ordering of minimum paths recorded, the coupling descriptors hierarchy is established as:$$(H\&POL)\to (POL\&Log\mathrm{\hspace{0.17em}\u200a\u200a}P)\to (Log\mathrm{\hspace{0.17em}\u200a\u200a}P\&H)$$

- The HIV-1 inhibitory activity is triggered by a hydrophobic interaction followed by energetic stabilization of the ligand/substrate (pyrididone derivative/viral protein) interaction here modeled by the heat of molecular formation and eventually completed by the ionic field influence herein represented by the polarizability descriptor.
- Although the QSAR multi-linear model should not be excluded from the molecular modeling of complex bio-chemical interactions, it should be complemented with other polynomial correlational catastrophe-type models that produce significant results comparable to those of other 3D-modeling procedures such as docking-based comparative molecular field analysis (CoMFA) and comparative molecular similarity indices analysis (CoMSIA) [24].

- Log P: For positive values, the compound behaves hydrophobically and requires dissolution in an organic solvent; by contrast, for negative values the compound is hydrophilic and can be dissolved directly in an aqueous buffer. For Log P equal to 0, the compound partitions at a 1:1 organic-to-aqueous phase ratio, meaning that it is likely soluble in both organic and aqueous solvents and in cellular environments; thus, values of Log P equal to or greater than zero are selected to achieve hydrophobicity and suitability for the cellular environment [43,44], while characterizing the stacking bonding of aromatic rings [45];
- H: Because the formation of a compound from its elements usually is an exothermic process, most heats of formation are negative, and this is also a characteristic of the dynamic equilibrium of ligand-substrate interactions [46]; note that the advantage of using heat of formation as QSAR descriptor resides in the following: it thermodynamically relates with the free energy ΔG= −RTlnK
_{eq}by the equilibrium constant_{eq}K which parallels the recorded activity at thermodynamic level [24]; it nevertheless expands the Gibbs free energy from the hydrogen to covalent bonding strength [45]; - Activity Models: Represent the same chemical-biological process providing their differences with respect to structural domains are minimized to zero.

**27**(Log P ≈ 2.72, H ≈ −39.459 kcal/mol, POL ≈ 35.55Ǻ

^{3}),

**28**(Log P ≈ 1.06, H ≈ −34.478kcal/mol, POL ≈ 34.88Ǻ

^{3}), and

**29**(Log P ≈ 0.96, H ≈ −21.361 kcal/mol, POL ≈ 35.17Ǻ

^{3}). Most impressively, these molecules were also predicted by the much more sophisticated methods of CoMFA and CoMSIA as having increased binding affinity between the aromatic ring (or wing 2 of the pyridinone derivative) and amino acid Tyr181 of the first molecule and Tyr188 of the last two. These two amino acids are very important in the inhibition of RT by NNRTIs because the most common mutations are Tyr181Cys and Tyr188Cys, and they are responsible for the emergence of viruses resistant to pyridinone derivatives. Therefore, designing pyridinone compounds that allow aromatic ring stacking interactions with Tyr181 and Tyr 188 may prevent these mutations and increase the activity of these anti-HIV drugs.

## 5. Conclusions

_{1}, X

_{2}), with X

_{1}, X

_{2}being structural physicochemical parameters (usually hydrophobicity, polarizability and/or forming heat energy in accordance with the basic recommendation of Hansch) [54] under the seven polynomial forms inspired by Thom’s catastrophe theory [1] (see Table 3).

- A defined endpoint: The hydrophobic binding of the inhibitor in the pocket of the p66 subunit of reverse-transcriptase was confirmed herein through the identification of hydrophobicity as the major influence among all the mono-nonlinear catastrophes employed; see Equation (17).
- An unambiguous algorithm: The Spectral-SAR minimum path principle [31,55–57] is here generalized to include relevant combination of statistical information (e.g., the correlation factor R, Student’s t-test, Fischer’s F-test) to provide an equal footing multi-dimensional Euler distance [see Equations (8–16)], thus avoiding the previously identified discrepancy in judging the mid-range performance in terms of correlation or other statistical factors [56].
- A defined domain of applicability: By performing linear vs. non-linear QSARs, the present strategy allows for the identification of recommended applicable structural domains through setting their difference to zero via inter-model activity minimization, which is equivalent to assuring the “smoothness” of the inhibitor-protein binding evolution towards the final steric inhibition output.
- Appropriate measures of goodness-of-fit, robustness and predictivity: The trial results were evaluated by external validation employing a testing set, which was selected by means of Gaussian vs. non-Gaussian distributions of the compounds’ activities, an improvement over the earlier arbitrariness of sampling the compounds only within a certain activity range. For instance, for linear QSAR the predicted correlation was superior to the tested correlation, thus confirming the reliability of this validation technique.
- A mechanistic interpretation: The selected succession of catastrophe-QSARs indicates that the inhibitor-HIV protein binding mutations that are involved in “birth and death” processes are associated with “waves” of induced activity in certain structural domain variants (see Figure 2). Moreover, the flat QSAR hypersurface should be complemented with catastrophe analysis to determine the specific structural domains for optimum interactions (see Figure 3) and for the associated molecular structure design of NNRT inhibitors.

## Acknowledgements

## Appendix

#### A1. More on Catastrophe Theory Background

_{1},...,c

_{k},x

_{1},..., x

_{m}), say in its origin (c, x)= 0 under the form

_{1}

^{2}+ ... + x

_{m}

^{2}, eventually after re-parameterization) likewise any transverse path through any non-Morse function that can be found within a family of finite functions looks the same as all other transverse paths in the family (those of Table 2). Even more, the co-rank of those functions (as the co-rank of their Hessian on the critical/singular/turning points) fixes also the minimum of variables that function can be reduced to; for example, if a function of 2011 variables has a critical point of co-rank equal 1, the actual function to be studied is of only 1 variable! This makes the Catastrophe Theory extremely interesting for being applied on QSAR studies, where the available structural variables are listed on hundred pages [62], while in fact one searches for modeling functions that enter natural classes or family of functions with an universal character—as the Thom polynomials are—and therefore aiming to work with appropriate functions with considerable lower number of variables/structural descriptors, see Table 3.

#### A2. Catastrophe Theory Implication on Pearson Correlation

^{QSAR}, and of its transformation into the Catastrophe-QSAR one, say Y

^{Γ}

^{/}

^{QSAR}, through the Gaussian mapping

^{QSAR}, while the hole expression (A.8) having the Hessian co-rank of order 2 is in full consistence with the maximum co-rank universal unfolding for the polynomials of Table 2.

_{A}= σ

_{i}, $\forall i=\overline{1,N}$ likely to be valid when dealing with great number of structural descriptors; this way, one actually performs the asymptotic limit M → ∞ on (A.6) for all $i=\overline{1,N}$ and recognizes the Poison integral result

## References

- Thom, R. Stabilitè Structurelle et Morphogènése; Benjamin-Addison-Wesley: New York, NY, USA, 1973. [Google Scholar]
- Viret, J. Reaction of the organism to stress: The survival attractor concept. Acta Biotheor
**1994**, 42, 99–109. [Google Scholar] - Lacorre, P. Predation and generation processes through a new representation of the cusp catastrophe. Acta Biotheor
**1997**, 45, 93–115. [Google Scholar] - Viret, J. Topological approach of Jungian psychology. Acta Biotheor
**2010**, 58, 233–245. [Google Scholar] - Cerf, R. Catastrophe theory enables moves to be detected towards and away from self-organization: The example of epileptic seizure onset. Biol. Cybern
**2006**, 94, 459–468. [Google Scholar] - Silvi, B.; Savin, A. Classification of chemical bonds based on topological analysis of electron localization functions. Nature
**1994**, 371, 683–686. [Google Scholar] - Putz, M.V. Markovian approach of the electron localization functions. Int. J. Quantum Chem
**2005**, 105, 1–11. [Google Scholar] - Aerts, D.; Czachor, M.; Gabora, L.; Kuna, M.; Posiewnik, A.; Pykacz, J.; Syty, M. Quantum morphogenesis: A variation on Thom’s catastrophe theory. Phys. Rev
**2003**, 67. [Google Scholar] [CrossRef] - De Clercq, E. Anti-HIV drugs: 25 compounds approved within 25 years after the discovery of HIV. Int. J. Antimicrob. Agents
**2009**, 33, 307–320. [Google Scholar] - De Clercq, E. The history of antiretrovirals: Key discoveries over the past 25 years. Rev. Med. Virol
**2009**, 19, 287–299. [Google Scholar] - El Safadi, Y.; Vivet-Boudou, V.; Marquet, R. HIV-1 reverse transcriptase inhibitors. Appl. Microbiol. Biotechnol
**2007**, 75, 723–737. [Google Scholar] - Ivetac, A; McCammon, J.A. Elucidating the inhibition mechanism of HIV-1 non-nucleoside reverse transcriptase inhibitors through multi-copy molecular dynamics simulations. J. Mol. Biol.
**2009**, 388, 644–658. [Google Scholar] - Gupta, S.P. Advances in QSAR studies of HIV-1 reverse transcriptase inhibitors. Prog. Drug Res
**2002**, 58, 223–264. [Google Scholar] - Prabhakar, Y.S.; Solomon, V.R.; Gupta, M.K.; Katti, S.B. QSAR studies on thiazolidines: A biologically privileged scaffold. Top. Heterocycl. Chem
**2006**, 4, 161–249. [Google Scholar] - Prajapati, D.G.; Ramajayam, R.; Yadav, M.R.; Giridhar, R. The search for potent, small molecule NNRTIs: A review. Bioorg Med. Chem
**2009**, 17, 5744–5762. [Google Scholar] - Zhan, P.; Chen, X.; Li, D.; Fang, Z.; de Clercq, E.; Liu, X. HIV-1 NNRTIs: Structural diversity, pharmacophore similarity, and implications for drug design. Med. Res. Rev
**2011**, in press. [Google Scholar] - Chen, X.; Zhan, P.; Li, D.; De Clercq, E.; Liu, X. Recent advances in DAPYs and related analogues as HIV-1 NNRTIs. Curr. Med. Chem
**2011**, 18, 359–376. [Google Scholar] - Rebehmed, J.; Barbault, F.; Teixeira, C.; Maurel, F. 2D and 3D QSAR studies of diarylpyrimidine HIV-1 reverse transcriptase inhibitors. J. Comput. Aided Mol. Des
**2008**, 22, 831–841. [Google Scholar] - Afantitis, A.; Melagraki, G.; Sarimveis, H.; Koutentis, P.A.; Markopoulos, J.; Igglessi-Markopoulou, O. A novel simple QSAR model for the prediction of anti-HIV activity using multiple linear regression analysis. Mol. Divers
**2006**, 10, 405–414. [Google Scholar] - Marino, D.J.G.; Castro, E.A.; Toropov, A. Improved QSAR modeling of anti-HIV-1 activities by means of the optimized correlation weights of local graphs invariants. Central Eur. J. Chem
**2006**, 4, 135–148. [Google Scholar] - Mandal, A.S.; Roy, K. Predictive QSAR modeling of HIV reverse transcriptase inhibitor TIBO derivatives. Eur. J. Med. Chem
**2009**, 44, 1509–1524. [Google Scholar] - Bak, A.; Polanski, J. A 4D-QSAR study on anti-HIV HEPT analogues. Bioorg. Med. Chem
**2006**, 14, 273–279. [Google Scholar] - Duda-Seiman, C.; Duda-Seiman, D.; Dragoş, D.; Medeleanu, M.; Careja, V.; Putz, M.V.; Lacrămă, A.-M.; Chiriac, A.; Nuţiu, R.; Ciubotariu, D. Design of anti-HIV ligands by means of minimal topological difference (MTD) Method. Int. J. Mol. Sci
**2006**, 7, 537–555. [Google Scholar] - Medina-Franco, J.L.; Rodríguez-Morales, S.; Juárez-Gordiano, C.; Hernández-Campos, A.; Castillo, R. Docking-based CoMFA and CoMSIA studies of non-nucleoside reverse transcriptase inhibitors of the pyridinone derivative type. J. Comput. Aided Mol. Des
**2004**, 18, 345–360. [Google Scholar] - Topliss, J. Quantitative Structure-Activity Relationships of Drugs; Academic Press: New York, NY, USA, 1983. [Google Scholar]
- Seyfel, J.K. QSAR and Strategies in the Design of Bioactive Compounds; VCH Weinheim: New York, NY, USA, 1985. [Google Scholar]
- Duchowicz, P.R.; Castro, E.A. The Order Theory in QSPR-QSAR Studies; Mathematical Chemistry Monographs, University of Kragujevac: Kragujevac, Serbia, 2008. [Google Scholar]
- Zhao, V.H.; Cronin, M.T.D.; Dearden, J.C. Quantitative structure-activity relationships of chemicals acting by non-polar narcosis—Theoretical considerations. Quant. Struct. Act. Relat
**1998**, 17, 131–138. [Google Scholar] - Pavan, M.; Netzeva, T.; Worth, A.P. Review of literature based quantitative structure-activity relationship models for bioconcentration. QSAR Comb. Sci
**2008**, 27, 21–31. [Google Scholar] - Pavan, M.; Worth, A.P. Review of estimation models for biodegradation. QSAR Comb. Sci
**2008**, 27, 32–40. [Google Scholar] - Putz, M.V.; Putz, A.M. Timisoara Spectral—Structure Activity Relationship (Spectral-SAR) Algorithm: From statistical and algebraic fundamentals to quantum consequences. In Quantum Frontiers of Atoms and Molecules; Putz, M.V., Ed.; NOVA Science Publishers Inc: New York, NY, USA, 2011; Volume Chapter 21, pp. 539–580. [Google Scholar]
- OECD Principles: Guidance Document on the Validation of (Q)SARModels; OECD Envioronment Diretorate: Paris, France, 2007.
- Putz, M.V.; Putz, A.M.; Barou, R. Spectral-SAR Realization of OECD-QSAR Principles. Int. J. Chem. Model
**2011**, 3. in press. [Google Scholar] - Krokidis, X.; Noury, S.; Silvi, B. Characterization of elementary chemical processes by catastrophe theory. J. Phys. Chem. A
**1997**, 101, 7277–7282. [Google Scholar] - Putz, M.V. Path integrals for electronic densities, reactivity indices, and localization functions in quantum systems. Int. J. Mol. Sci
**2009**, 10, 4816–4940. [Google Scholar] - Weisstein, E.W. Catastrophe. From MathWorld—A Wolfram Web Resource. Available online: http://mathworld.wolfram.com/Catastrophe.html accessed on 1 September 2011.
- Sanns, W. Catastrophe Theory with Mathematica: A Geometric Approach; DAV: Waghäusel, Germany, 2000. [Google Scholar]
- Lu, X.-F.; Chen, Z.-W. The development of anti-HIV-1 drugs. Acta Pharm. Sin
**2010**, 45, 165–176. [Google Scholar] - Putz, M.V. Residual-QSAR. Implications for genotoxic carcinogenesis. Chem. Central J
**2011**, 5. [Google Scholar] [CrossRef] - Putz, M.V.; Lazea, M.; Sandjo, L.P. Quantitative Structure Inter-Activity Relationship (QSInAR). Cytotoxicity study of some hemisynthetic and isolated natural steroids and precursors on human fibrosarcoma cells HT1080. Molecules
**2011**, 16, 6603–6620. [Google Scholar] - Putz, M.V.; Ionaşcu, C.; Putz, A.M.; Ostafe, V. Alert-QSAR. Implications for electrophilic theory of chemical carcinogenesis. Int. J. Mol. Sci
**2011**, 12, 5098–5134. [Google Scholar] - Hypercube, Inc. HyperChem 7.01 [Program Package]; Hypercube, Inc: Gainesville, FL, USA, 2002. [Google Scholar]
- Leo, A.; Hansch, C.; Elkins, D. Partition coefficients and their uses. Chem. Rev
**1971**, 71, 525–616. [Google Scholar] - Cronin, D.; Mark, T. The role of hydrophobicity in toxicity prediction. Curr. Comput. Aided Drug Design
**2006**, 2, 405–413. [Google Scholar] - Selassie, C.D. History of Quantitative Structure-Activity Relationships. In Burger’s Medicinal Chemistry and Drug Discovery, 6th ed; Abraham, D.J., Ed.; Wiley: New York, NY, USA, 2003; pp. 1–48. [Google Scholar]
- Masterton, W.L.; Slowinski, E.J.; Stanitski, C.L. Chemical Principles; CBS College Publishing: Philadelphia, PA, USA, 1983. [Google Scholar]
- Chattaraj, P.K.; Sengupta, S. Popular electronic structure principles in a dynamical context. J. Phys. Chem
**1996**, 100, 16126–16130. [Google Scholar] - Himmel, D.M.; Das, K.; Clark, A.D.; Hughes, S.H.; Benjahad, A.; Oumouch, S.; Guillemont, J.; Coupa, S.; Poncelet, A.; Csoka, I.; et al. Crystal structures for HIV-1 reverse transcriptase in complexes with three pyridinone derivatives: A new class of non-nucleoside inhibitors effective against a broad range of drug-resistant strains. J. Med. Chem
**2005**, 48, 7582–7591. [Google Scholar] - The European Bioinformatics Institute. Available online: http://www.ebi.ac.uk/pdbsum/2BE2 accessed on 11 September 2011.
- Duda-Seiman, C.; Duda-Seiman, D.; Putz, M.V.; Ciubotariu, D. QSAR modeling of anti-HIV activity with HEPT derivatives. Digest J. Nanomat. Biostruct
**2007**, 2, 207–219. [Google Scholar] - Croce, C.M. Oncogenes and cancer. N. Engl. J. Med
**2008**, 358, 502–511. [Google Scholar] - Dingli, D.; Nowak, M.A. Cancer biology: Infectious tumour cells. Nature
**2006**, 443, 35–36. [Google Scholar] - Benigni, R.; Bossa, C.; Jeliazkova, N.; Netzeva, T.; Worth, A. The Benigni/Bossa rules for mutagenicity and carcinogenicity—A module of Toxtree; European Commission report EUR 23241; Office for Official Publications of the European Communities: Luxembourg, 2008; pp. 1–69. [Google Scholar]
- Hansch, C.; Kurup, A.; Garg, R.; Gao, H. Chem-bioinformatics and QSAR: A review of QSAR lacking positive hydrophobic terms. Chem. Rev
**2001**, 101, 619–672. [Google Scholar] - 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. [Google Scholar] - Putz, M.V.; Putz, A.M.; Lazea, M.; Chiriac, A. Spectral vs. statistic approach of structure-activity relationship. Application on ecotoxicity of aliphatic amines. J Theor. Comput. Chem
**2009**, 8, 1235–1251. [Google Scholar] - QSAR & Spectral-SAR in Computational Ecotoxicology; Putz, M.V. (Ed.) Apple Academics: Ontario, Canada, 2012; in press.
- Zeeman, E.C. Catastrophe theory. Sci. Am
**1976**, 234, 65–83. [Google Scholar] - Morse, M. The critical points of a functional on n variables. Trans. Am. Math. Soc
**1931**, 33, 72–91. [Google Scholar] - Arnold, V.I. Local normal forms of functions. Invent. Math
**1976**, 35, 87–109. [Google Scholar] - Poston, T.; Stewart, I. Catastrophe Theory and Its Applications; Pitman Publishing: Boston, MA, USA, 1978. [Google Scholar]
- Todeschini, R.; Consonni, V. Handbook of Molecular Descriptors; Wiley-VCH: Weinheim, Germany, 2000. [Google Scholar]

**Figure 1.**Gaussian (G) and non-Gaussian (NG) screening of the observed activities of the working molecules in Table 4 grouped into trial and test congener series.

**Figure 3.**Determination of the structural domains of pyridinone-derivative type non-nucleoside reverse transcriptase inhibitors in the same range of structural descriptors by employing the principles of hydrophobicity, minimum polarizability, binding energy, and the minimum difference between the polynomial activity models of Figure 2; the hydrophobic pocket was identified in the p66 subunit of HIV-1-rt of specific transferase R221239 [48,49].

**Table 1.**The QSAR working table for Equation (1) in the presence of M-structural descriptors for N-compounds with known activities.

Observed Activity | Structural | Predictor | Variables | ||
---|---|---|---|---|---|

A | X_{1} | … | X_{k} | … | X_{M} |

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

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

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

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

Name | Co-dimension | Co-rank | Universal unfolding | Parametric Representation |
---|---|---|---|---|

Fold | 1 | 1 | x^{3} + ux | |

Cusp | 2 | 1 | x^{4} + ux^{2} + vx | |

Swallow tail | 3 | 1 | x^{5} + ux^{3} + vx^{2} + wx | |

Butterfly | 4 | 1 | x^{6} + ux^{4} + vx^{3} + wx^{2} + tx | |

Hyperbolic umbilic | 3 | 2 | x^{3} + y^{3} + uxy + vx + wy | |

Elliptic umbilic | 3 | 2 | x^{3} − xy^{2} + u(x^{2} + y^{2} ) + vx + wy | |

Parabolic umbilic | 4 | 2 | x^{2} y + y^{4} + ux^{2} + vy^{2} + wx + ty |

**Table 3.**Algebraic realization of Thom’s elementary catastrophes as uni- and bi- nonlinear QSARs. The systematics of the sub-indices indicate consecutive coupled pairs, where each pair is interpreted as: the index of a structural factor followed by its power.

Model | QSAR Equation |
---|---|

GROUP I: with one descriptor only, |X_{1}〉 | |

QSAR-(I) | $|{Y}_{I}\rangle ={a}_{0}|1\rangle +{a}_{11}|{X}_{1}\rangle $ |

Fold | $|{Y}_{F}\rangle ={f}_{0}|1\rangle +{f}_{11}|{X}_{1}\rangle +{f}_{13}|{X}_{1}^{3}\rangle $ |

Cusp | $|{Y}_{C}\rangle ={c}_{0}|1\rangle +{c}_{11}|{X}_{1}\rangle +{c}_{12}|{X}_{1}^{2}\rangle +{c}_{14}|{X}_{1}^{4}\rangle $ |

Swallow tail | $|{Y}_{ST}\rangle ={s}_{0}|1\rangle +{s}_{11}|{X}_{1}\rangle +{s}_{12}|{X}_{1}^{2}\rangle +{s}_{13}|{X}_{1}^{3}\rangle +{s}_{15}|{X}_{1}^{5}\rangle $ |

Butterfly | $|{Y}_{B}\rangle ={b}_{0}|1\rangle +{b}_{11}|{X}_{1}\rangle +{b}_{12}|{X}_{1}^{2}\rangle +{b}_{13}|{X}_{1}^{3}\rangle +{b}_{14}|{X}_{1}^{4}\rangle +{b}_{16}|{X}_{1}^{6}\rangle $ |

GROUP II: with two descriptors, |X_{1}〉,|X_{2}〉 | |

QSAR- (II) | $|{Y}_{II}\rangle ={q}_{0}|1\rangle +{q}_{11}|{X}_{1}\rangle +{q}_{21}|{X}_{2}\rangle $ |

Hyperbolic umbilic | $|{Y}_{HU}\rangle ={h}_{0}|1\rangle +{h}_{11}|{X}_{1}\rangle +{h}_{21}|{X}_{2}\rangle +{h}_{1121}|{X}_{1}{X}_{2}\rangle +{h}_{13}|{X}_{1}^{3}\rangle +{h}_{23}|{X}_{2}^{3}\rangle $ |

Elliptic umbilic | $|{Y}_{EU}\rangle ={e}_{0}|1\rangle +{e}_{11}|{X}_{1}\rangle +{e}_{21}|{X}_{2}\rangle +{e}_{12}|{X}_{1}^{2}\rangle +{e}_{22}|{X}_{2}^{2}\rangle +{e}_{1122}|{X}_{1}{X}_{2}^{2}\rangle +{e}_{13}|{X}_{1}^{3}\rangle $ |

Parabolic umbilic | $|{Y}_{PU}\rangle ={p}_{0}|1\rangle +{p}_{11}|{X}_{1}\rangle +{p}_{21}|{X}_{2}\rangle +{p}_{12}|{X}_{1}^{2}\rangle +{p}_{22}|{X}_{2}^{2}\rangle +{p}_{1221}|{X}_{1}^{2}{X}_{2}\rangle +{p}_{24}|{X}_{2}^{4}\rangle $ |

**Table 4.**Actual working reverse transcriptase pyridinone inhibitors grouped in Gaussian (G) and non-Gaussian (NG) molecular congeneric sets with their structural information (hydrophobicity, Log P; molecular polarizability POL [Å

^{3}] and total optimized energy of formation H [kcal/mol]) computed upon the semi-empirical PM3 method [42], along with their observed activity A = Log (1/IC50) [24].

No. | Type | WORKING MOLECULES | A^{obs} | QSAR parameters | |||
---|---|---|---|---|---|---|---|

Structure | Name | Log (1/IC_{50}) | Log P | POL (Å^{3}) | H (kcal/mol) | ||

1. | G1 | 3-{[(6′-azabenzofuran-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 3.98 | −0.54 | 31.21 | −14.67 | |

2. | G2 | 3-{[(5′-azabenzofuran-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 4.49 | −0.54 | 31.21 | −16.195 | |

3. | G3 | 3-{[(pyridine-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 4.82 | 0.21 | 27.87 | -5.854 | |

4. | G4 | 3-benzylamino-5-ethyl-6-methylpyridin-2(1H)-one | 5.27 | 0.67 | 28.58 | −11.659 | |

5. | G5 | 3-{[(1′,3′-naftoxazol-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 5.57 | 1.20 | 38.48 | −1.878 | |

6. | G6 | 3-{[(1′-benzopyran-4′-one-3′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 5.96 | −0.71 | 33.84 | −61.455 | |

7. | G7 | 3-{[(benzopyridine-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 6.28 | 1.16 | 35.14 | 11.246 | |

8. | G8 | 3-{[(1′,3′-benzothiazole-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 6.46 | 0.54 | 33.57 | 17.808 | |

9. | G9 | 3-{[(4′-methylbenzoxazole-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 6.92 | 0.67 | 33.05 | −27.613 | |

10. | G10 | 3-{[(4′,7′-dichlorobenzofuran-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 7.24 | 0.88 | 35.78 | −33.749 | |

11. | G11 | 3-{[(4′,7′-dimethylbenzoxazol-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 7.7 | 1.13 | 34.88 | −38.048 | |

12. | G12 | 3-{[(4′,7′-dichlorobenzoxazol-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 7.72 | 1.24 | 35.07 | −30.071 | |

13. | G13 | 3-[(4′,7′-dimethylbenzoxazol-2′-yl) ethyl]-5-ethyl-6-methylpyridin-2(1H)-one | 7.55 | 2.62 | 35.37 | −47.701 | |

14. | G14 | 3-[(4′,5′,6′,7′-tetrahydrobenzoxazole-2′-yl) ethyl]-5-ethyl-6-methylpyridin-2(1H)-one | 7.24 | −0.02 | 32.08 | −63.299 | |

15. | G15 | 3-{[(4′-methoxybenzoxazole-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 6.74 | −0.05 | 33.68 | −54.452 | |

16. | G16 | 3-[(4′,5′,6′,7′-tetrahydrobenzoxazole-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 6.55 | −1.50 | 31.59 | −50.643 | |

17. | G17 | 3-{[(benzothiophene-2′-yl) methyl] amino}-5-ethyl-6-methylpyridin-2(1H)-one | 6.30 | 0.19 | 34.28 | 11.703 | |

18. | G18 | 3-{[(5′-methylbenzoxazole-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 5.90 | 0.67 | 33.05 | −27.741 | |

19. | G19 | 3-[(benzopyridine-2′-yl) ethyl]5-ethyl-6-methylpyridin-2(1H)-one | 5.61 | 2.71 | 35.62 | 3.331 | |

20. | G20 | 3-{[(indol-2′-yl) methyl] amino}-5-ethyl-6-methylpyridin-2(1H)-one | 5.36 | −0.34 | 32.63 | 4.727 | |

21. | G21 | 3-{[(quinazolin-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 5.12 | 0.02 | 31.92 | 8.171 | |

22. | G22 | 3-{[(indol-3′-yl)methyl] amino}-5-ethyl-6-methylpyridin-2(1H)-one | 4.65 | −0.43 | 32.63 | 2.957 | |

23. | G23 | 3-(β-phenilethyl)-5-ethyl-6-methylpyridin-2(1H)-one | 4.30 | 2.36 | 29.06 | −23.245 | |

24. | NG1 | 3-{[(4′-quinozolone-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 5.60 | −0.47 | 33.85 | −36.959 | |

25. | NG2 | 3-{[(3′,4′-diazobenzofuran-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 5.72 | 0.05 | 30.50 | −8.120 | |

26. | NG3 | 3-{[(7′-hydroxybenzoxazole-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 6.36 | −0.08 | 31.85 | −62.189 | |

27. | NG4 | 3-[(4′,7′-dichlorobenzoxazole-2′-yl) ethyl]-5-ethyl-6-methylpyridin-2(1H)-one | 7.85 | 2.72 | 35.55 | −39.459 | |

28. | NG5 | 3-{[(7′-ethylbenzoxazole-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 6.59 | 1.06 | 34.88 | −34.478 | |

29. | NG6 | 3-[(5′-phenyl-oxazole-2′-yl) ethyl]-5-ethyl-6-methylpyridin-2(1H)-one | 6.41 | 0.96 | 35.17 | −21.361 | |

30. | NG7 | 3-[(benzothiazole-2′-yl) ethyl]-5-ethyl-6-methylpyridin-2(1H)-one | 6.43 | 2.02 | 34.06 | 8.873 | |

31. | NG8 | 3-{[(2′naphtyl) methyl] amino}-5-ethyl-6-methylpyridin-2(1H)-one | 6.34 | 1.67 | 35.85 | 5.495 | |

32. | NG9 | 3-{[(5′-phenyl-oxazole-2′-yl) methyl]amino}-5-ethyl-6-methylpyridin-2(1H)-one | 5.63 | −0.53 | 34.69 | −10.850 |

Catastrophe | QSAR Model | R_{Pearson}(a) | R_{ALG}(b) | r(c) | t-Stud. | t(d) | Fisher | f(e) | Π(f) |
---|---|---|---|---|---|---|---|---|---|

QSAR (I) | $|{Y}_{I}^{LogP}\rangle =5.861|1\rangle +0.240|LogP\rangle $ | 0.228 | 0.984 | 4.317 | 22.344 | 7.854 | 1.150 | 0.143 | 8.963 |

$|{Y}_{I}^{POL}\rangle =-2.257|1\rangle +0.249|POL\rangle $ | 0.554 | 0.989 | 1.784 | −0.832 | −0.292 | 9.284 | 1.158 | 2.147 | |

$|{Y}_{I}^{H}\rangle =5.57|1\rangle -0.021|H\rangle $ | 0.476 | 0.987 | 2.074 | 20.597 | 7.24 | 6.156 | 0.768 | 7.57 | |

Fold (F) | $|{Y}_{F}^{LogP}\rangle =5.854|1\rangle +0.738|LogP\rangle -0.106|Log{P}^{3}\rangle $ | 0.382 | 0.986 | 2.581 | 22.936 | 8.062 | 1.705 | 0.213 | 8.468 |

$|{Y}_{F}^{POL}\rangle =-24.206|1\rangle +1.26|POL\rangle -3\xb7{10}^{-4}|PO{L}^{3}\rangle $ | 0.601 | 0.989 | 1.646 | −1.422 | −0.45 | 5.650 | 0.704 | 1.859 | |

$|{Y}_{F}^{H}\rangle =5.58|1\rangle -0.016|H\rangle -2\xb7{10}^{-6}|{H}^{3}\rangle $ | 0.481 | 0.987 | 2.053 | 20.095 | 7.063 | 3.01 | 0.375 | 7.365 | |

Cusp (C) | $|{Y}_{C}^{LogP}\rangle =5.707|1\rangle +0.426|LogP\rangle +0.372|Log{P}^{2}\rangle -0.071|Log{P}^{4}\rangle $ | 0.348 | 0.985 | 2.832 | 16.120 | 5.666 | 0.872 | 0.109 | 6.335 |

$|{Y}_{C}^{POL}\rangle =431.26|1\rangle -35.694|POL\rangle +0.833|PO{L}^{2}\rangle -{10}^{-4}|PO{L}^{4}\rangle $ | 0.713 | 0.992 | 1.391 | 2.240 | 0.787 | 6.558 | 0.818 | 1.796 | |

$|{Y}_{C}^{H}\rangle =5.006|1\rangle +0.042|H\rangle +0.003|{H}^{2}\rangle -{10}^{-6}|{H}^{4}\rangle $ | 0.764 | 0.993 | 1.300 | 19.802 | 6.960 | 8.864 | 1.105 | 7.166 | |

Swallow tail (ST) | $\begin{array}{l}|{Y}_{ST}^{LogP}\rangle =5.649|1\rangle +1.608|LogP\rangle +0.326|Log{P}^{2}\rangle \\ -0.978|Log{P}^{3}\rangle +0.0093|Log{P}^{5}\rangle \end{array}$ | 0.575 | 0.989 | 1.720 | 18.665 | 6.561 | 2.222 | 0.277 | 6.788 |

$\begin{array}{l}|{Y}_{ST}^{POL}\rangle =1476.244|1\rangle -156.079|POL\rangle \\ +5.791|PO{L}^{2}\rangle -0.079|PO{L}^{3}\rangle +5.5\xb7{10}^{-6}|PO{L}^{5}\rangle \end{array}$ | 0.715 | 0.992 | 1.387 | 0.45 | 0.158 | 4.708 | 0.587 | 1.515 | |

$\begin{array}{l}|{Y}_{ST}^{H}\rangle =4.884|1\rangle +0.031|H\rangle +0.004|{H}^{2}\rangle \\ +5.2\xb7{10}^{-5}|{H}^{3}\rangle +4\xb7{10}^{-10}|{H}^{5}\rangle \end{array}$ | 0.763 | 0.993 | 1.302 | 15.608 | 5.486 | 6.263 | 0.781 | 5.692 | |

Butterfly (B) | $\begin{array}{l}|{Y}_{B}^{LogP}\rangle =5.646|1\rangle +1.464|LogP\rangle +0.303|Log{P}^{2}\rangle \\ -0.688|Log{P}^{3}\rangle -0.041|Log{P}^{4}\rangle +0.027|Log{P}^{6}\rangle \end{array}$ | 0.578 | 0.989 | 1.711 | 15.169 | 5.332 | 1.704 | 0.212 | 5.604 |

$\begin{array}{l}|{Y}_{B}^{POL}\rangle =-16485.827|1\rangle +2491.049|POL\rangle -146.094|PO{L}^{2}\rangle \\ +4.037|PO{L}^{3}\rangle -0.047|PO{L}^{4}\rangle +2.9\xb7{10}^{-6}|PO{L}^{6}\rangle \end{array}$ | 0.718 | 0.992 | 1.382 | −0.355 | −0.125 | 3.619 | 0.451 | 1.459 | |

$\begin{array}{l}|{Y}_{B}^{H}\rangle =4.876|1\rangle +0.110|H\rangle +0.004|{H}^{2}\rangle \\ -2.3\xb7{10}^{-4}|{H}^{3}\rangle -7.67\xb7{10}^{-6}|{H}^{4}\rangle +6.3\xb7{10}^{-10}|{H}^{6}\rangle \end{array}$ | 0.856 | 0.996 | 1.163 | 19.088 | 6.709 | 9.349 | 1.166 | 6.908 |

^{(a)}the statistical Pearson correlation factor;

^{(b)}computed from Equation (7);

^{(c)}computed from Equation (9);

^{(d)}computed from Equation (10) with ${t}_{\begin{array}{l}Tabulated\\ (0.99;20)\end{array}}=2.845$;

^{(e)}computed from Equation (11) with ${F}_{\begin{array}{l}Tabulated\\ (0.99;1,21)\end{array}}=8.02$;

^{(f)}computed from Equation (8).

Catastrophe | QSAR Model | R_{Pearson}(a) | R_{ALG}(b) | r(c) | t-Stud. | t(d) | Fisher | f(e) | Π(f) |
---|---|---|---|---|---|---|---|---|---|

QSAR (II) | $|{Y}_{II}^{LogP,POL}\rangle =-2.044|1\rangle +0.051|LogP\rangle +0.242|POL\rangle $ | 0.556 | 0.989 | 1.778 | −0.702 | −0.245 | 4.464 | 0.763 | 1.9504 |

$|{Y}_{II}^{LogP,H}\rangle =5.379|1\rangle +0.304|LogP\rangle -0.023|H\rangle $ | 0.556 | 0.989 | 1.778 | 18.564 | 6.489 | 4.468 | 0.764 | 6.771 | |

$|{Y}_{II}^{POL,H}\rangle =-2.637|1\rangle +0.248|POL\rangle -0.021|H\rangle $ | 0.728 | 0.992 | 1.363 | −1.151 | −0.402 | 11.302 | 1.932 | 2.398 | |

Hyperbolic umbilic (HU) | $\begin{array}{l}|{Y}_{HU}^{LogP,POL}\rangle =-39.499|1\rangle -2.463|LogP\rangle +2.043|POL\rangle \\ +0.104|(LogP)(POL)\rangle -0.145|Log{P}^{3}\rangle -6\xb7{10}^{-4}|PO{L}^{3}\rangle \end{array}$ | 0.715 | 0.992 | 1.387 | −2.215 | −0.774 | 3.561 | 0.609 | 1.701 |

$\begin{array}{l}|{Y}_{HU}^{LogP,H}\rangle =5.319|1\rangle +1.083|LogP\rangle -0.002|H\rangle \\ -0.003|(LogP)(H)\rangle -0.161|Log{P}^{3}\rangle -9\xb7{10}^{-6}|{H}^{3}\rangle \end{array}$ | 0.736 | 0.992 | 1.3485 | 19.328 | 6.756 | 4.019 | 0.687 | 6.923 | |

$\begin{array}{l}|{Y}_{HU}^{POL,H}\rangle =-13.192|1\rangle +0.766|POL\rangle +0.122|H\rangle \\ -0.004|(POL)(H)\rangle -2\xb7{10}^{-4}|PO{L}^{3}\rangle -5.1\xb7{10}^{-7}|{H}^{3}\rangle \end{array}$ | 0.755 | 0.993 | 1.315 | −0.79 | −0.276 | 4.503 | 0.770 | 1.549 | |

Elliptic umbilic (EU) | $\begin{array}{l}{|{Y}_{EU}^{LogP,POL}\rangle}_{A}=-69.262|1\rangle -0.556|LogP\rangle +4.531|POL\rangle \\ +0.443|Log{P}^{2}\rangle -0.068|PO{L}^{2}\rangle \\ +0.002|(LogP)(PO{L}^{2})\rangle -0.322|Log{P}^{3}\rangle \end{array}$ | 0.757 | 0.993 | 1.312 | −2.548 | −0.891 | 3.582 | 0.612 | 1.670 |

$\begin{array}{l}{|{Y}_{EU}^{LogP,POL}\rangle}_{B}=644.623|1\rangle +0.022|LogP\rangle -59.934|POL\rangle \\ +0.467|Log{P}^{2}\rangle +1.855|PO{L}^{2}\rangle \\ -0.015|(POL)(Log{P}^{2})\rangle -0.019|PO{L}^{3}\rangle \end{array}$ | 0.722 | 0.992 | 1.374 | 1.866 | 0.652 | 2.908 | 0.497 | 1.600 | |

$\begin{array}{l}{|{Y}_{EU}^{LogP,H}\rangle}_{A}=5.022|1\rangle +0.974|LogP\rangle +0.025|H\rangle \\ +0.530|Log{P}^{2}\rangle +0.001|{H}^{2}\rangle \\ +2.87\xb7{10}^{-4}|(LogP)({H}^{2})\rangle -0.359|Log{P}^{3}\rangle \end{array}$ | 0.843 | 0.995 | 1.181 | 20.638 | 7.214 | 6.542 | 1.118 | 7.395 | |

Elliptic umbilic (EU) | $\begin{array}{l}{|{Y}_{EU}^{LogP,H}\rangle}_{B}=4.779|1\rangle +0.643|LogP\rangle +0.029|H\rangle \\ -0.211|Log{P}^{2}\rangle +0.004|{H}^{2}\rangle \\ +0.001|(H)(Log{P}^{2})\rangle +5\xb7{10}^{-5}|{H}^{3}\rangle \end{array}$ | 0.851 | 0.995 | 1.170 | 17.047 | 5.958 | 7.015 | 1.199 | 6.189 |

$\begin{array}{l}{|{Y}_{EU}^{POL,H}\rangle}_{A}=802.877|1\rangle -74.631|POL\rangle -0.02|H\rangle \\ +2.291|PO{L}^{2}\rangle +0.005|{H}^{2}\rangle \\ -2\xb7{10}^{-4}|(POL)({H}^{2})\rangle -0.023|PO{L}^{3}\rangle \end{array}$ | 0.857 | 0.996 | 1.162 | 3.124 | 1.092 | 7.346 | 1.256 | 2.029 | |

$\begin{array}{l}{|{Y}_{EU}^{POL,H}\rangle}_{B}=11.888|1\rangle -0.562|POL\rangle +0.068|H\rangle \\ +0.011|PO{L}^{2}\rangle +0.004|{H}^{2}\rangle \\ -4\xb7{10}^{-5}|(H)(PO{L}^{2})\rangle +4\xb7{10}^{-5}|{H}^{3}\rangle \end{array}$ | 0.853 | 0.996 | 1.167 | 0.532 | 0.186 | 7.120 | 1.217 | 1.696 | |

Parabolic umbilic (PU) | $\begin{array}{l}{|{Y}_{PU}^{LogP,POL}\rangle}_{A}=474.915|1\rangle +0.021|LogP\rangle -39.256|POL\rangle \\ +0.454|Log{P}^{2}\rangle +0.914|PO{L}^{2}\rangle \\ -0.015|(Log{P}^{2})\left(POL\right)\rangle -{10}^{-4}|PO{L}^{4}\rangle \end{array}$ | 0.722 | 0.992 | 1.374 | 1.817 | 0.635 | 2.905 | 0.497 | 1.593 |

$\begin{array}{l}{|{Y}_{PU}^{LogP,POL}\rangle}_{B}=-67.522|1\rangle -1.539|LogP\rangle +4.444|POL\rangle \\ +0.573|Log{P}^{2}\rangle -0.067|PO{L}^{2}\rangle \\ +0.002|(PO{L}^{2})\left(LogP\right)\rangle -0.115|Log{P}^{4}\rangle \end{array}$ | 0.703 | 0.992 | 1.411 | −2.219 | −0.776 | 2.611 | 0.446 | 1.671 | |

Parabolic umbilic (PU) | $\begin{array}{l}{|{Y}_{PU}^{LogP,H}\rangle}_{A}=4.852|1\rangle +0.700|LogP\rangle +0.041|H\rangle \\ -0.240|Log{P}^{2}\rangle +0.004|{H}^{2}\rangle \\ +0.002|\left(Log{P}^{2}\right)(H)\rangle -{10}^{-6}|{H}^{4}\rangle \end{array}$ | 0.874 | 0.996 | 1.140 | 20.243 | 7.075 | 8.645 | 1.478 | 7.317 |

$\begin{array}{l}{|{Y}_{PU}^{LogP,H}\rangle}_{B}=5.10|1\rangle +0.552|LogP\rangle +0.020|H\rangle \\ +0.460|Log{P}^{2}\rangle +9.57\xb7{10}^{-4}|{H}^{2}\rangle \\ +1.93\xb7{10}^{-4}|({H}^{2})\left(Log{P}^{2}\right)\rangle -0.099|Log{P}^{4}\rangle \end{array}$ | 0.767 | 0.993 | 1.295 | 16.828 | 5.882 | 3.815 | 0.652 | 6.058 | |

$\begin{array}{l}{|{Y}_{PU}^{POL,H}\rangle}_{A}=8.876|1\rangle -0.366|POL\rangle +0.069|H\rangle \\ +0.008|PO{L}^{2}\rangle +0.003|{H}^{2}\rangle \\ -3.7\xb7{10}^{-5}|(PO{L}^{2})(H)\rangle -4.5\xb7{10}^{-7}|{H}^{4}\rangle \end{array}$ | 0.841 | 0.995 | 1.183 | 0.386 | 0.135 | 6.447 | 1.102 | 1.623 | |

$\begin{array}{l}{|{Y}_{PU}^{POL,H}\rangle}_{B}=595.212|1\rangle -48.906|POL\rangle -0.019|H\rangle \\ +1.129|PO{L}^{2}\rangle +5\xb7{10}^{-3}|{H}^{2}\rangle \\ -1.49\xb7{10}^{-4}|({H}^{2})(POL)\rangle -1.73\xb7{10}^{-4}|PO{L}^{4}\rangle \end{array}$ | 0.856 | 0.996 | 1.163 | 3.074 | 1.074 | 7.292 | 1.246 | 2.015 |

^{(a)}the statistical Pearson correlation factor;

^{(b)}computed from Equation (7);

^{(c)}computed from Equation (9);

^{(d)}computed from Equation (10) with ${t}_{\begin{array}{l}Tabulated\\ (0.99;19)\end{array}}=2.861$;

^{(e)}computed from Equation (11) with ${F}_{\begin{array}{l}Tabulated\\ (0.99;2,20)\end{array}}=5.85$;

^{(f)}computed from Equation (8).

**Table 7.**Single-structure matrices of the Euclidean distances ΔΠ

_{I}of the QSAR and catastrophe models’ relative statistics of Table 5 employing Equation (12).

Log P | F | C | ST | B |
---|---|---|---|---|

QSAR | 1.750 | 2.645 | 2.905 | 3.627 |

F | 2.411 | 1.732 | 2.865 | |

C | 1.437 | 1.174 | ||

ST | 1.231 |

POL | F | C | ST | B |
---|---|---|---|---|

QSAR | 0.517 | 1.198 | 0.828 | 0.830 |

F | 1.317 | 0.717 | 0.524 | |

C | 0.670 | 0.983 | ||

ST | 0.314 |

H | F | C | ST | B |
---|---|---|---|---|

QSAR | 0.431 | 0.89 | 1.916 | 1.127 |

F | 1.054 | 1.793 | 1.242 | |

C | 1.509 | 0.292 | ||

ST | 1.29 |

**Table 8.**Differences Δ

^{2}Π

_{I}between the single-structure matrices of the Euclidean distances in Table 7.

|Log P ÷ POL| | F | C | ST | B |
---|---|---|---|---|

QSAR | 1.233 | 1.446 | 2.076 | 2.797 |

F | 1.094 | 1.015 | 2.341 | |

C | 0.767 | 0.191 | ||

ST | 0.917 |

|Log P ÷ H| | F | C | ST | B |
---|---|---|---|---|

QSAR | 1.32 | 1.755 | 0.988 | 2.501 |

F | 1.358 | 0.062 | 1.624 | |

C | 0.072 | 0.882 | ||

ST | 0.059 |

|POL ÷ H| | F | C | ST | B |
---|---|---|---|---|

QSAR | 0.086 | 0.309 | 1.088 | 0.297 |

F | 0.264 | 1.076 | 0.717 | |

C | 0.839 | 0.691 | ||

ST | 0.976 |

**Table 9.**Single-structure matrices of the Euclidean distances ΔΠ

_{II}of the QSAR and catastrophe models’ relative statistics of Table 6 employing Equation (12); note that for the degenerate models of Table 6 that one is employed that displays higher relative statistical power ( Π).

Log P^POL | HU | EU | PU |
---|---|---|---|

QSAR | 0.675 | 0.810 | 1.005 |

HU | 0.139 | 1.414 | |

EU | 1.531 |

Log P^H | HU | EU | PU |
---|---|---|---|

QSAR | 0.512 | 0.917 | 1.123 |

HU | 0.964 | 0.878 | |

EU | 1.152 |

POL^H | HU | EU | PU |
---|---|---|---|

QSAR | 1.170 | 1.652 | 1.640 |

HU | 1.46 | 1.440 | |

EU | 0.02 |

Model | $|{Y}_{C}^{LogP}\rangle $ | $|{Y}_{C}^{H}\rangle $ | $|{Y}_{ST}^{POL}\rangle $ | $|{Y}_{B}^{LogP}\rangle $ | $|{Y}_{B}^{POL}\rangle $ | $|{Y}_{B}^{H}\rangle $ |
---|---|---|---|---|---|---|

Molecule | ||||||

NG1 | 5.586 | 6.179 | 5.294 | 5.094 | −20.595 | 5.687 |

NG2 | 5.729 | 4.885 | 4.294 | 5.719 | −9.764 | 4.360 |

NG3 | 5.676 | 0.415 | 4.708 | 5.531 | −13.457 | −7.932 |

NG4 | 5.729 | 6.156 | 5.149 | 6.657 | −29.709 | 5.259 |

NG5 | 6.487 | 6.141 | 5.309 | 6.705 | −25.700 | 5.923 |

NG6 | 6.399 | 5.438 | 5.258 | 6.708 | −27.365 | 5.219 |

NG7 | 6.903 | 5.631 | 5.319 | 5.311 | −21.540 | 5.984 |

NG8 | 6.904 | 5.334 | 5.027 | 5.995 | −31.693 | 5.566 |

NG9 | 5.580 | 4.9357 | 5.328 | 5.054 | −24.666 | 4.383 |

R-Pearson | 0.195 | 0.129 | 0.174 | 0.701 | 0.488 | 0.026 |

Model | $|{Y}_{II}^{LogP,H}\rangle $ | $|{Y}_{HU}^{LogP,POL}\rangle $ | $|{Y}_{HU}^{LogP,H}\rangle $ | ${|{Y}_{EU}^{LogP,POL}\rangle}_{A}$ | ${|{Y}_{EU}^{POL,H}\rangle}_{A}$ | ${|{Y}_{PU}^{POL,H}\rangle}_{B}$ |
---|---|---|---|---|---|---|

Molecule | ||||||

NG1 | 6.0865 | 5.918 | 5.308 | 5.387 | 5.351 | 7.210 |

NG2 | 5.581 | 5.839 | 5.399 | 5.448 | 4.816 | 4.578 |

NG3 | 6.785 | 6.132 | 7.526 | 5.686 | 1.423 | 7.234 |

NG4 | 7.115 | 6.642 | 6.037 | 6.289 | 5.480 | 7.765 |

NG5 | 6.495 | 7.382 | 6.853 | 7.277 | 6.033 | 7.629 |

NG6 | 6.163 | 7.291 | 6.426 | 7.104 | 7.338 | 7.647 |

NG7 | 5.790 | 7.388 | 6.087 | 7.615 | 6.879 | 6.547 |

NG8 | 5.761 | 7.560 | 6.330 | 7.640 | 7.895 | 7.447 |

NG9 | 5.467 | 5.755 | 4.786 | 5.177 | 7.586 | 7.303 |

R-Pearson | 0.778 | 0.468 | 0.454 | 0.431 | 0.057 | 0.451 |

© 2011 by the authors; licensee MDPI, 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/).

## Share and Cite

**MDPI and ACS Style**

Putz, M.V.; Lazea, M.; Putz, A.-M.; Duda-Seiman, C.
Introducing Catastrophe-QSAR. Application on Modeling Molecular Mechanisms of Pyridinone Derivative-Type HIV Non-Nucleoside Reverse Transcriptase Inhibitors. *Int. J. Mol. Sci.* **2011**, *12*, 9533-9569.
https://doi.org/10.3390/ijms12129533

**AMA Style**

Putz MV, Lazea M, Putz A-M, Duda-Seiman C.
Introducing Catastrophe-QSAR. Application on Modeling Molecular Mechanisms of Pyridinone Derivative-Type HIV Non-Nucleoside Reverse Transcriptase Inhibitors. *International Journal of Molecular Sciences*. 2011; 12(12):9533-9569.
https://doi.org/10.3390/ijms12129533

**Chicago/Turabian Style**

Putz, Mihai V., Marius Lazea, Ana-Maria Putz, and Corina Duda-Seiman.
2011. "Introducing Catastrophe-QSAR. Application on Modeling Molecular Mechanisms of Pyridinone Derivative-Type HIV Non-Nucleoside Reverse Transcriptase Inhibitors" *International Journal of Molecular Sciences* 12, no. 12: 9533-9569.
https://doi.org/10.3390/ijms12129533