1 Introduction

Acquired immunodeficiency syndrome (AIDS) is an end-stage disease that is manifested by the gradual deterioration of the immune competence of infected patients (Alberts et al., 2002). Human immunodeficiency virus type 1 (HIV-1) is the causative organism for AIDS (Volberding and Deeks, 2010); it belongs to the family lentiviridae of pathogenic retroviruses, which depends on RNA to encode the genetic message (Chakravarty, 2006). The protease (PR) of HIV-1 is a homodimeric aspartyl PR; it cleaves a 55-kDa polyprotein precursor: by that process, it produces smaller functional protein fragments (e.g., p17, p24, p9, p7), which are responsible for packing and infectivity for budding virions. The inhibition of PR inhibits the processing steps; it causes noninfectious and immature progeny virions. Some HIV-1 PR inhibitors (such as indinavir, ritonavir, and nelfinavir) are marketed as anti-HIV-1 drugs, none of which is devoid of adverse effects (Katzung, 2004). Phenotypic cross/resistance restricted the use of these drugs (Tripathi, 2003). The search for HIV-1 PR inhibitors with better therapeutic efficacy and lesser toxicity is in progress.

Ali et al. (2006) described the design, synthesis, and bioevaluation of HIV-1 PR inhibitors, incorporating N-phenyloxazolidinone-5-carboxamides into (hydroxyethylamino)sulphonamide scaffold as P2 ligands. Series of inhibitors were synthesized with changes at P2 phenyloxazolidinone and P2’ phenylsulphonamide moieties. The compounds with the (S)-enantiomer of substituted phenyloxazolidinones at P2 showed potent inhibitory activities versus HIV-1 PR. Inhibitors possessing 3/4-acetyl and 3-trifluoromethyl groups at oxazolidinone phenyl ring were the most potent, with K_i values in the low pM range. The electron-donating groups 4-methoxy and 1,3-dioxolane were preferred at P2’ phenyl, as compounds with other substitutions showed less binding affinity. Attempts to replace the isobutyl at P1’ with small cyclic moieties caused a loss of affinity. Crystal structure analysis of both most potent inhibitors, in a complex with HIV-1 PR, provided information on the inhibitor–PR interactions. In inhibitor and enzyme complexes, oxazolidinone carbonyl H-bonded with conserved PR Asp²⁹. Potent inhibitors were selected from each series by incorporating various phenyloxazolidinone-based P2 ligands, and their activities versus a panel of multidrug-resistant (MDR) PR variants were determined. The most potent PR inhibitor started with tight affinity for the wild-type enzyme (K_i = 0.8 pM) and, even versus MDR variants, it retained pM to low nM K_i, which is comparable to the best PR inhibitors approved by the US Food and Drug Administration (FDA). Halder and Jha (2010) applied quantitative structure–activity relationships (QSARs) to some N-aryloxazolidinone-5-carboxamides (NCAs, cf. Figure 5.1) to find structural requirements for more active anti-HIV-1 PR agents. Wang et al. (2011) reported mangiferin as anti-HIV-1 targeting PR and effective versus resistant strains. Zhang et al. (2012) investigated interactions for HIV drug cross-resistance among PR inhibitors. Liu et al. (2013) informed 4862 F, an inhibitor of HIV-1 PR, from a Streptomyces culture.

A simple code is proposed that could be useful for establishing a chemical structure–biosignificance relationship (Benzecri, 1984; Varmuza, 1980). The starting point is to use information entropy (IE) for pattern recognition. The IE is formulated based on the similarity matrix between two biochemical species. As IE is weakly discriminating for classification, the more powerful concepts of IE production and its equipartition conjecture (EC) are introduced (Tondeur and Kvaalen, 1987). In earlier publications, it was analyzed the PTs of local anesthetics (Castellano-Estornell and Torrens-Zaragozá, 2009; Torrens and Castellano, 2006, 2011a), HIV inhibitors (Torrens and Castellano, 2009a, 2010, 2011b, 2012a–d, 2014), anti-cancers (Torrens and Castellano, 2009b, 2013a, 2013b), phenolics (Castellano et al., 2012), flavomoids (Castellano et al., 2013), stilbenoids (Castellano et al., 2014) in Ganoderma (Castellano and Torrens, 2015). The main aim of this chapter is to develop the code-learning potentialities and, since molecules are more naturally described by structured representation of varying sizes, study general approaches to structured information processing. A second goal is to present NCA PT. A third objective is to validate PT with an external property, anti-HIV-1 PR activity, which is not used in the development of PT.

2 Computational method

The key problem in classification studies is to define similarity indices with several criteria. The first step in quantifying the similarity concept for NCAs is to list the most important moieties. A vector of properties $\overline{i}=<i_1,i_2,\dots i_k,\dots >$ should be associated with every NCA i, whose components correspond to characteristic groups in a hierarchical order according to the importance of their pharmacological potency. If the mth portion of a molecule is more significant for the inhibitory effect than the kth portion, then m < k. The components i_k are either 1 or 0, depending on whether an identical portion of rank k is either present or absent in NCA i, compared to a reference. The analysis includes seven regions of structural variation in NCAs: R_1–7 positions showing diverse substitution patterns. The structural elements of an NCA are ranked according to their contribution to inhibitory potency as R₃ > R₆ > R₇ > R₄ > R₅ > R₂ > R₁. Index i₁ = 1 denotes R₃ = H (0 otherwise), i₂ = 1, R₆ = H, i₃ = 1, R₇ = iPr, i₄ = 1, R₄ = H, i₅ = 1, R₅ = OCH₃, i₆ = 1, R₂ = H and i₇ = 1, and R₁ = Ac. In NCA 5, R₃ = R₆ = R₄ = R₂ = H, R₇ = iPr, R₅ = OCH₃, and R₁ = Ac; its vector is <1111111>, which was selected as a reference because of its greatest inhibitory activity versus HIV-1. Table 5.1 contains the vectors associated with 38 NCAs. Vector <1111110> is associated with NCA 1 since R₃ = R₆ = R₄ = R₂ = R₁ = H, R₇ = iPr and R₅ = OCH₃.

Denote by r_ij (0 ≤ r_ij ≤ 1) the similarity index of two NCAs associated with vectors $\overline{i}$ and $\overline{j}$, respectively. The similitude is characterized by the similarity matrix R = [r_ij]. The similarity index between two NCAs $\overline{i}=<i_1,i_2,\dots i_k\dots >$ and $\overline{j}=<j_1,j_2,\dots j_k\dots >$ is

$r_{ij}={\displaystyle \sum _k{t}_k{\left(a_k\right)}^k}\left(k=1,2,\dots \right),$

where 0 ≤ a_k ≤ 1, and t_k = 1 if i_k = j_k, but t_k = 0 if i_k ≠ j_k. The definition assigns a weight (a_k)^k to any property involved in the description of molecule i or j.

3 Classification algorithm

The grouping algorithm uses the stabilized similarity matrix obtained by applying the max–min composition rule o, defined by

${\left(\mathbf{R}\mathrm{o}\mathbf{S}\right)}_{ij}={max}_k\left[{min}_k\left(r_{ik},s_{kj}\right)\right],$

where R = [r_ij] and S = [s_ij] are matrices of the same type, and (RoS)_ij is the (i,j)th element of the RoS matrix (Cox, 1994; Kaufmann, 1975; Kundu, 1998; Lambert-Torres et al., 1999). When applying the composition rule max–min iteratively so that R(n + 1) = R(n) o R, an integer n exists such that R(n) = R(n + 1) = … Matrix R(n) is called the stabilized similarity matrix. The importance of stabilization lies in the fact that in classification, it generates a partition into disjoint classes. The stabilized matrix is designated by R(n) = [r_ij(n)]. The grouping rule follows, to wit: i and j are assigned to the same class if r_ij(n) ≥ b. The class of i noted that $\stackrel{⌢}{i}$ is the set of species j that satisfies the rule: r_ij(n) ≥ b. The matrix of classes is

$\stackrel{⌢}{\mathbf{R}}\left(n\right)=\left[{\stackrel{⌢}{r}}_{\stackrel{⌢}{i}\stackrel{⌢}{j}}\right]={max}_{s,t}\left(r_{st}\right)\left(s\in \stackrel{⌢}{i},t\in \stackrel{⌢}{j}\right),$

where s stands for any index of the species belonging to class $\stackrel{⌢}{i}$ (similarly for t and $\stackrel{⌢}{j}$). Eq. (5.3) means finding the largest similarity index between species of two different classes.

4 Information entropy

In information theory, the IE h measures the surprise that a source emitting sequences can give (Shannon, 1948a, 1948b). Consider the use of a qualitative spot test to determine the presence of Fe in a sample of water. Without any history of testing, an analyst must begin by assuming that the two outcomes 0/1 (Fe absent or present) are equiprobable with probabilities 1/2. When up to two metals may be present in the sample (e.g., Fe and Ni), four possible outcomes exist, ranging from neither being present (0,0) to both (1,1), with probabilities 1/2². Which of the four possibilities turns up is determined by using two tests, each with two observable states. With three elements, eight possibilities exist with probabilities 1/2³; three tests are needed. The following pattern relates the uncertainty and information needed to resolve it. The number of possibilities is expressed to a power of 2. The power to which 2 must be raised to give the number of possibilities N is defined as the logarithm to base 2 of that number. Information and uncertainty are defined by the logarithm to base 2 of the number of possible analytical outcomes: log₂ N.

The initial uncertainty is defined in terms of the probability of the occurrence of every outcome; e.g., for the above-mentioned probabilities, the following definition results: I = H = log₂ N = log₂ 1/p = –log₂ p, where I is the information contained in an answer given that there were N possibilities, with H as the initial uncertainty resulting from the need to consider N possibilities and p, the probability of every outcome if all N possibilities are equally likely to occur. The expression is generalized to a situation in which the probability of every outcome is not the same. If one knows that some elements are more likely to be present than others, the equation is adjusted so that the logarithms of individual probabilities suitably weighted are summed: H = –Σ p_i log₂ p_i, where Σ p_i = 1. Consider the original example, except that in this case, past experience showed that 90% of the samples contained no Fe. The degree of uncertainty is calculated by: H = –(0.9 log₂ 0.9 + 0.1 log₂ 0.1) = 0.469 bits. For a single event occurring with probability p, the degree of surprise is proportional to –ln p. By generalizing the result to a random variable X (which can take N possible values x₁, …, x_N with probabilities p₁, …, p_N), the average surprise received on learning the X value is –Σ p_i ln p_i. The IE associated with R similarity matrix is

$h\left(\mathbf{R}\right)=-{\displaystyle \sum _{i,j}{r}_{ij}ln{r}_{ij}}-{\displaystyle \sum _{i,j}\left(1-r_{ij}\right)ln\left(1-r_{ij}\right)}\text{.}$

Denote by C_b the set of classes and by ${\stackrel{⌢}{\mathbf{R}}}_b$, the similarity matrix at grouping level b. The IE satisfies several properties: (i) h(R) = 0 if either r_ij = 0 or r_ij = 1; (ii) h(R) is at its maximum if r_ij = 0.5 (i.e., when the imprecision is at its maximum); (iii) $h\left({\stackrel{⌢}{\mathbf{R}}}_b\right)\le h\left(\mathbf{R}\right)$ for any b (i.e., classification leads to IE loss); (iv) $h\left({\stackrel{⌢}{\mathbf{R}}}_{b_1}\right)\le h\left({\stackrel{⌢}{\mathbf{R}}}_{b_2}\right)$ if b₁ < b₂ (i.e., IE is a monotone function of the grouping level b).

5 The EC of entropy production

In the classification algorithm, every hierarchical tree corresponds to IE dependence on the grouping level, and an h–b diagram is obtained. Tondeur and Kvaalen (1987) proposed the EC of IE production is proposed as the selection criterion among different variants, resulting from classification among hierarchical trees. According to EC, for a given charge, the binary tree (BT) with the best configuration is one in which IE production is the most uniformly distributed. One proceeds by analogy by using IE instead of thermodynamic entropy. The EC implies linear dependence (i.e., constant IE production) along scale b so that the EC line is

$h_{\mathrm{e}\mathrm{q}\mathrm{p}}=h_{\text{max}}b\text{.}$

Since the classification is discrete, the way of expressing EC would be a regular staircase function. The best variant is chosen to be one that minimizes the sum of the squares of the deviations:

$SS={\displaystyle \sum _{b_i}{\left(h-h_{\mathrm{e}\mathrm{q}\mathrm{p}}\right)}^2}\text{.}$

6 Learning procedure

Learning procedures (LPs), similar to those encountered in stochastic methods, are implemented (White, 1989). Consider a partition into classes as good from observations, which corresponds to a reference similarity matrix S = [s_ij], obtained for equal weights a₁ = a₂ = … = a and for an arbitrary number of fictious properties. Consider also the same set of species as in the good classification and actual properties. The degree of similarity r_ij is computed by Eq. (5.1), giving matrix R. The number of properties for R and S differs. The LP consists in finding classification results for R, as close as possible to the good classification. The a₁ weight is taken to be constant and the following weights a₂, a₃,… are subjected to random variations. A new similarity matrix is obtained via Eq. (5.1) and new weights. The distance between the partitions into classes, characterized by R and S, is

$D=-{\displaystyle \sum _{ij}\left(1-r_{ij}\right)ln\frac{1-r_{ij}}{1-s_{ij}}}-{\displaystyle \sum _{ij}{r}_{ij}ln\frac{r_{ij}}{s_{ij}}}\forall 0\le r_{ij},s_{ij}\le 1\text{.}$

The definition was suggested by that introduced in information theory by Kullback (1959), in order to measure the distance between two probability distributions. It is a measure of the distance between the R and S matrices. Since a corresponding classification exists for every matrix, two classifications will be compared by the distance, which is a nonnegative quantity that approaches zero as the resemblance between R and S rises. The result of the algorithm is a set of weights allowing classification. The procedure was applied to the synthesis of complex BTs using IE (Baez and Dolan, 1995; Crans, 2000; Iordache, 2011, 2012; Iordache et al., 1993; Leinster, 2004).

Our code MolClas is a simple, reliable, efficient, and fast procedure for molecular classification, based on IE-production EC, according to Eqs. (5.1)–(5.7). With IE, but without EC, an excessive number of results appear compatible with the data and suffer a combinatorial explosion; however, after EC, the best configuration is that in which IE production is most uniformly distributed. The MolClas reads the number of properties and molecular properties; it allows the optimization of the coefficients; it optionally reads the starting coefficients and the number of iteration cycles. The correlation matrix can be either calculated by MolClas or read from the input file. The MolClas allows correlation-matrix transformation in the range [–1,1] to [0,1]; it calculates the similarity matrix of the property in symmetric storage mode; it applies the graphical correlation model to obtain the partial correlation diagram (PCD); it computes the classifications, tests if the groupings are different, calculates the distances between classifications, computes the similarity matrices of groupings, works out classifications of IE, optimizes the coefficients, performs single/complete-linkage hierarchical cluster analyses, and plots cluster diagrams; it was written not only to analyze IE-production EC, but also to explore the world of molecular classification.

7 Calculation results and discussion

Structural data of anti-HIV-1 NCAs reported by Halder and Jha (2010) were used as the model data set. The matrix of Pearson correlation coefficients (PCCs) was calculated between the pairs of vectors <i₁,i₂,i₃,i₄,i₅,i₆,i₇> of 38 NCAs. The PCCs are illustrated in a PCD, which could contain high (r ≥ 0.75), medium (0.50 ≤ r < 0.75), low (0.25 ≤ r < 0.50), and no (r < 0.25) partial correlation. Pairs of inhibitors with high partial correlations show similar vectors; however, the results should be taken with care because NCA with constant vector <1111111> (Entry 5) shows null standard deviation, causing the greatest partial correlations r = 1 with any NCA, which is an artifact. With EC, correlations are illustrated in PCD, which contains 598 high (cf. Figure 5.2, red lines) and 105 zero (black) partial correlation. Notice that 3 out of 37 high partial correlations of Entry 5 were corrected: its correlations with Entries 34–36 are zero partial correlations.

The grouping rule in the case of equal weights a_k = 0.5 for b₁ = 0.97 allows the following classes:

C–b₁ = (1–7)(8–13,24,25)(14–18,26,27)(19–23)(28)(29,30)(31–33,37,38)(34–36).

Eight classes are obtained with associated IE h–R–b₁ = 21.18. The BT matching to <i₁,i₂,i₃,i₄,i₅,i₆,i₇>/C–b₁ (cf. Figure 5.3) is calculated (IMSL, 1989; Jarvis and Patrick, 1973; Tryon, 1939); it provides a BT of Table 5.1 that separates the same classes: the data bifurcate into classes 5, 1–4, and 6–8 with 1, 7, 8, 7, 5, 2, 5, and 3 NCAs, respectively (Page, 2000). NCAs 1–7 with the greatest inhibitory activity are grouped into the same class. The NCAs in the same cluster appear highly correlated in PCD (Figure 5.2).

At level b₂ with 0.93 ≤ b₂ ≤ 0.94, the set of classes turns out to be:

C–b₂ = (1–13,24,25)(14–23,26,27)(28–30)(31–33,37,38)(34–36)

Five classes result, and IE decays to h–R–b₂ = 7.98. The BT matching to <i₁,i₂,i₃,i₄,i₅,i₆,i₇> and C–b₂ (cf. Figure 5.4) divides the same five classes: 1–5 with 15, 12, 3, 5, and 3 NCAs, respectively. Again, NCAs with the greatest inhibitory activity are grouped into the same class. The NCAs belonging to the same cluster appear highly correlated in a PCD, in qualitative agreement with BT (Figures 5.2 and 5.3).

Table 5.2 shows an analysis of the set containing 1–38 classes, agreeing with PCD and BTs (Figures 5.2–5.4).

In view of PCD and BTs (Figures 5.2–5.4), the data are split into the same classes above. Figure 5.5 displays the BT. Again, NCAs with the greatest inhibitory activity are grouped into the same class.

The illustration of the classification in a radial tree (RT; cf. Figure 5.6) shows the same groupings as already discussed, in qualitative agreement with the PCD and BTs (Figures 5.2–5.5). Again, NCAs with the greatest inhibitory activity (1–13, etc.) are grouped into the same cluster.

The SplitsTree program allows analyzing the cluster analysis (CA) data (Huson, 1998). Based on the method of split decomposition, it takes as input the distance matrix and produces a graph that represents the relationships between the taxa. For ideal data, the graph is an RT, whereas less ideal data will give rise to an RT-like net, which is interpreted as possible evidence for conflicting data. Furthermore, as split decomposition does not attempt to force the data onto an RT, it can provide a good indication of how RT-like are given data. The splits graph (SG) for 38 NCAs in Table 5.1 (cf. Figure 5.7) shows that 1–27 collapse, as well as 28–33-37-38 and 34–36. It reveals no conflicting relationship between classes. It is in qualitative agreement with PCD, BTs, and RT (Figures 2–6).

Usually, in structure–property relationships (SPRs), the data file contains less than 100 objects and more than1000 X-variables. So many X-variables exist that no one can discover by inspection the patterns in objects. The principal components analysis (PCA) is useful to summarize the information contained in an X-matrix and put it in an understandable form (Hotelling, 1933; Jolliffe, 2002; Kramer, 1998; Patra et al., 1999; Shaw, 2003; Xu and Hagler, 2002). The PCA works by decomposing X-matrix as the product of two smaller matrices P and T. The loading matrix (P) with information about variables contains a few vectors, the principal components (PCs), which are obtained as the linear combinations (LCs) of the original X-variables. The score matrix (T), with information about the objects, is such that every object is described in terms of the projections onto PCs instead of the original variables: X = TP’ + E, where ’ denotes the transpose matrix.

The information not contained in the matrices remains as the unexplained X-variance in the residual matrix (E). Every PC_i is a new coordinate expressed as LC of the old features x_j: PC_i = Σ_jb_ijx_j. The new coordinates PC_i are called scores or factors, while the coefficients b_ij, loadings. Scores are ordered according to their information content with regard to the total variance among all objects. Score–score plots (SPs) show the positions of compounds in the new coordinate system, while loading–loading plots (LPs) show the locations of features that represent compounds in the new coordinates. The PCs show two properties: (i) They are extracted in decaying order of importance. The first PC F₁ always contains more information than F₂, F₂ more than F₃, etc.; and (ii) every PC is orthogonal to one another. No correlation exists between information contained in the different PCs. A PCA was performed for NCAs. The importance of PCA factors F₁–F₇ for {i₁,i₂,i₃,i₄,i₅,i₆,i₇} is collected in Table 5.3. Factors are LCs of the vectors. Factor F₁ explains 36% of variance (64% error); F_1/2, 55% of variance (45% error); F_1–3, 72% of variance (28% error); etc.

The PCA factor loadings are shown in Table 5.4.

The PCA F_1–7 profile for the vectors is listed in Table 5.5. For factors F₁ and F₇, variables {i₁,i₂} show the greatest weight in profile; however, F₁ cannot be reduced to both variables without a 29% error, although F₇ is reduced to both variables with a 0% error. For F₂, the variable i₆ presents the greatest weight; notwithstanding, F₂ cannot be reduced to two variables {i₆,i₇} without a 13% error. For F₃, the variable i₄ assigns greatest weight; nevertheless, F₃ cannot be reduced to two variables {i₄,i₅} without a 16% error. For F₄, the variable i₆ consigns the greatest weight; however, F₄ cannot be reduced to two variables {i₆,i₇} without a 25% error. For F₅, the variable i₅ represents the greatest weight; however, F₅ cannot be reduced to two variables {i₄,i₅} without a 24% error. For F₆, the variable i₃ shows the greatest weight; nevertheless, F₆ cannot be reduced to two variables {i₃,i₇} without a 31% error.

In PCA F₂–F₁ SP, NCAs with the same vector collapse. Five NCA classes are distinguished: (i) class 1, with 15 compounds (0 < F₁ < F₂, cf. Figure 5.8, top); (ii) grouping 2, with 12 substances (F₁ > F₂ ≈ 0, right); cluster 3, with 3 molecules (F₁ < F₂ ≈ 0, center); class 4, with 5 organics (0 > F₁ > F₂, bottom); and cluster 5 (3 units, F₁ < < F₂, left). Classification agrees with PCD, BTs, RT, and SG (Figures 5.2–5.7).

From the PCA factor loadings of NCAs (Table 5.4), F₂–F₁ LP (cf. Figure 5.9) depicts seven vectors, and properties R_3/6 collapse. As a complement to SP (Figure 5.8) for loadings (Figure 5.9), NCAs in class 1, located at the top, present a contribution of R₁ = Ac situated on the same position in Figure 5.8. The NCAs in grouping 2 on the right have greater contributions of R₇ = iPr. The NCAs in classes 3 and 4 in the middle-bottom present a contribution of R₃ = R₆ = H. The NCAs in class 5 on the left present a contribution of R₄ = H. Two classes of properties are distinguished in LP: class 1 {R₃,R₆,R₇,R₄,R₅,R₂} (F₁ > F₂, Figure 5.9 bottom); and grouping 2 {R₁} (F₁ < F₂, top).

Instead of 38 NCAs in the space ℜ⁷ of seven vectors, consider seven properties in the space ℜ³⁸ of 38 NCAs. The BT of the vectors (cf. Figure 5.10) separates the first property R₁ (class 2), then R₂, R₅, R₄, R₇, R₃, and R₆ (class 1), agreeing with PCA LP (Figure 5.9).

The RT for the vectors (cf. Figure 5.11) separates the same classes as given previously, agreeing with PCA LP and BT (Figures 5.9 and 5.10).

The SG for the vectors (cf. Figure 5.12) shows that R_3–7 collapse. No conflicting relationship appears between the classes. The SG agrees with PCA LP, BT, and RT (Figures 5.9–5.11).

Another PCA was performed for the vectors, which are described by their occurrence in molecules, and new factors are LCs of the compounds. The use of factor F₁ explains 41% of the variance (59% error), F_1/2, 58% of variance (42% error), F_1–3, 74% of variance (26% error), etc. In PCA F₂–F₁ SP, R_3 -6 collapse. Two classes of properties are distinguished: class 1 {R₃,R₆,R₇,R₄,R₅,R₂} (F₁ > F₂, cf. Figure 5.13, right); and grouping 2 {R₁} (F₁ < F₂, left), agreeing with PCA LP, BT, RT, and SG (Figures 5.9–5.12).

The recommended format for the NCA PT (cf. Table 5.6) shows that they are classified first by i₁, then by i₂, i₃, i₄, i₅, i₆, and, finally, by i₇. Periods of eight units are assumed; e.g., group g00010 stands for <i₁,i₂,i₃,i₄,i₅> = <00010 >: <0001000> (–F –F –2-TP –H –F –Ac –H), etc. The NCAs in the same column appear close in PCD, BTs, RT, SG, and PCA SP (Figures 5.2–5.8).

The variation of property P (HIV-1 PR inhibitory activity) of vector < i₁,i₂,i₃,i₄,i₅,i₆,i₇ > (cf. Figure 5.14) is expressed in the decimal system P = 10⁶i₁ + 10⁵i₂ + 10⁴i₃ + 10³i₄ + 10²i₅ + 10i₆ + i₇ versus structural parameters {i₁,i₂,i₃,i₄,i₅,i₆,i₇} for NCAs. Most data collapse. Parameter i₁ shows greater variation i₂, i₃, i₄, i₅ and i₆. The property P was not used in the development of the PT and serves to validate it. The results agree with PT of properties with vertical groups defined by {i₁,i₂,i₃,i₄,i₅} and horizontal periods, by {i₆,i₇}.

The change of property P of vector < i₁,i₂,i₃,i₄,i₅,i₆,i₇ > in base 10 versus the number of the group in the NCA PT (cf. Figure 5.15) reveals minima and maxima corresponding to compounds with <i₁,i₂,i₃,i₄,i₅> ca. <00010> (group g00010) and <11111> (g11111), respectively. Most points collapse, especially in groups 5–8. Periods p00, p10, and p11 represent rows 1–3 in Table 5.6. The function P(i₁,i₂,i₃,i₄,i₅,i₆,i₇) denotes series of periodic waves (PWs) limited by minima and maxima, which suggest a PW behavior that recalls the form of a trigonometric function. For <i₁,i₂,i₃,i₄,i₅,i₆,i₇>, a minimum is shown. The distance in <i₁,i₂,i₃,i₄,i₅,i₆,i₇> units between each pair of consecutive minima is eight, which coincides with NCA sets in successive PWs. The minima occupy analogous positions in the curve and are in phase. The representative points in phase should correspond to elements in the same group in PT. For <i₁,i₂,i₃,i₄,i₅,i₆,i₇> minima, coherence exists between the two representations; however, the consistency is not general. Wave comparison shows two differences: (i) all PWs are incomplete, and (ii) PWs p00 and p10 are staircaselike. Most characteristic points of the plot are minima that lie about group g00010. Values of <i₁,i₂,i₃,i₄,i₅,i₆,i₇> are repeated, as the periodic law (PL) states.

An empirical function P(p) reproduces different <i₁,i₂,i₃,i₄,i₅,i₆,i₇> values. A minimum of P(p) has meaning only if it is compared with the former P(p–1) and later P(p + 1) points, needing to fulfill:

$P_{\text{min}}\left(p\right)<P\left(p-1\right)$

$P_{\text{min}}\left(p\right)<P\left(p+1\right)\text{.}$

Order relations [Eq. (5.8)] should repeat at determined intervals equal to PW size and are equivalent to

$P_{\text{min}}\left(p\right)-P\left(p-1\right)<0$

$P\left(p+1\right)-P_{\text{min}}\left(p\right)>0\text{.}$

As Eq. (5.9) is valid only for minima, other more general expressions are desired for all values of p. The differences D(p) = P(p + 1)–P(p) are calculated by assigning every value to NCA p:

$D\left(p\right)=P\left(p+1\right)-P\left(p\right)\text{.}$

Instead of D(p), R(p) = P(p + 1)/P(p) is taken by assigning them to NCA p. If PL were general, elements in the same group in analogous positions in different PWs would satisfy

$\mathrm{either}D\left(p\right)>0\mathrm{or}D\left(p\right)<0$

$\mathrm{either}R\left(p\right)>1\mathrm{or}R\left(p\right)<1$

However, the results show that this is not the case, so that PL is not general, existing some anomalies. Change of D(p) versus group number (cf. Figure 5.16) presents a lack of coherence between <i₁,i₂,i₃,i₄,i₅,i₆,i₇> Cartesian and PT representations. Most results collapse, mainly in groups 5–7. The datum for group 8, PW p00, should be taken with care because it was calculated with the first point in the following PW. If consistency were rigorous, all points in every PW would have the same sign. Trend exists in points to give D(p) > 0 for lower groups, but not for greater ones. However, irregularities exist in which NCAs for successive PWs are not always in phase.

The change of R(p) versus group number (cf. Figure 5.17) confirms the lack of constancy between Cartesian and PT charts. Most data collapse, especially in groups 5–7. If steadiness were exact, all points in every PW would show R(p) as either lesser or greater than 1. A trend in the points exists to give R(p) > 1 for lower groups, but not for greater ones. Notwithstanding, confirmed incongruities exist in which NCAs for the successive PWs are not always in phase.

The IE approach identifies cliffs. The method was applied to 74 flavonoids and is being used for 177 phenolic compounds. The technique is not sensitive to the number of compounds; otherwise, it could result in one-case classes that could be outliers. The impact of noisy experimental data on the method performance is expected to be small. Matched molecular pair analysis (MMPA) focuses on the effects of specific structural changes on properties of interest; its assumption is that differences in a property are predicted more accurately than the property itself; it is applied to structural diverse data sets, which rises confidence that observed effects of structural changes are globally relevant; it is related to bioisosterism in its focus on specific substructural transformations (Birch et al., 2009; Griffen et al., 2011; Kenny and Sadowski, 2005; Leach et al., 2006; Papadatos et al., 2010; Schultes et al., 2012; Warner et al., 2012). However, consider the following: (i) it goes further in providing quantitative estimates of the changes that result from the application of particular transformations and provides an inverse QSAR; (ii) it models not only bioactivity, but also any chemical, physicochemical, or pharmacokinetic property.

It is an inverse QSAR, which is gaining popularity in the retrospective analysis of large experimental data sets. While much focus was on the differences in properties between structurally related groups of existing compounds, attempts to extend it to the de novo design of structures were limited. Using IE, one looks for trends and central tendencies for novel drugs, mixtures, or properties; however, when one uses MMPA to look at activities that involve some kind of interaction with a binding site, one is looking for exceptions.

8 Conclusions

From these results and discussion, the following conclusions can be drawn:

1. Several criteria, selected to reduce the analysis to a manageable quantity of N-aryloxazolidinone-5-carboxamides, refer to substitutions at positions R_1/2, R_3–6 on different phenyls and R₇. Many classification algorithms are based on IE. For sets of moderate size, an excessive number of results appear compatible with the data and suffer a combinatorial explosion. However, after the EC, the best configuration is that in which the entropy production is most uniformly distributed. Molecular structural elements are ranked according to inhibitory activity: R₃ > R₆ > R₇ > R₄ > R₅ > R₂ > R₁. In compound 5, R₃ = R₆ = R₄ = R₂ = H, R₇ = iPr, R₅ = OCH₃, and R₁ = Ac <1111111>, which was selected as reference. Substances are grouped into five classes. This method avoids the problem of others of continuum variables because for <1111111>, null standard deviation causes a Pearson correlation coefficient of 1. The results agree with the principal component analyses.

2. The periodic law does not satisfy the laws of physics: (i) the inhibitory activity of N-aryloxazolidinone-5-carboxamides is not repeated, as the periodic law states, which is perhaps due to their chemical character; (ii) the order relationships are repeated with exceptions. The analysis forces the statement: The relationships that any compound p has with its neighbor p + 1 are approximately repeated for every period. The periodicity is not general; however, if a natural order of substances is accepted, the periodic law must be phenomenological. The inhibitory activity was not used in the generation of the PT and serves to validate it. Work to be done is the periodic analysis of any other molecular properties (e.g., cytotoxicity) that are not used in the construction of the PT. It would give insight into the possible generality of the PL. The authors are happy with the PT, although it should be tested for all properties: adverse effects, etc. The method is thought for medicinal, nutritional and phylogenetic chemists. However, the obtained PT deserves a broader spectra (e.g., for doctors to mix carboxamides of different classes with dissimilar types of properties: high anti-viral activity, low cytotoxicity, etc.)

3. Code MolClas is a simple, reliable, efficient and fast procedure for molecular classification, based on EC of entropy production. It was written not only to analyze EC, but also to explore the world of molecular classification.