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## Journal of the Chilean Chemical Society

##
*versión On-line* ISSN 0717-9707

### J. Chil. Chem. Soc. vol.57 no.1 Concepción mar. 2012

#### http://dx.doi.org/10.4067/S0717-97072012000100001

J. Chil. Chem. Soc, 57, N^{o} 1 (2012); págs.: 955-963

**AN IMPROVED TOPOLOGICAL DESCRIPTOR ED_{m} AND ITS APPLICATION**

**CHANGMING NIE ^{1*}, YAXIN WU^{1}, RONGYAN WU^{2} and SONGNIAN**

**WEN**^{2}^{1}School of Chemistry and Chemical Engineering, University of South China, Hengyang, Hunan Province 421001, P. R. China. * e-mail:*niecm196132@163.com** ^{2} College of electrical Engineering, University of South China,, Hengyang, Hunan Province 421001, P. R China*

**ABSTRACT**

In this paper, on the basis of the topological index *ED _{m}* derived from ionicity index matrix, improved distance matrix and branching degree matrix, we proposed the new topological descriptor

*ED*by introducing the bond angle into hidden hydrogen graph of molecules and using the geometric distance instead of the sum of bond length between two vertexes. The

_{m}'*ED*describes the molecular structure more accurately, and realizes unique characterization to cis-trans isomers. The quantitative structure-property relationship (QSPR) models, with correlation coefficients (R) in the range of 0.984-1.000 for the boiling point

_{m}*(b.p.),*the standard enthalpy of formation ( D

_{f}H

_{m}

^{θ}), the molar refraction (R

_{m}) and the molar volume

*(V*of some cis-trans isomers for alkenes, are subsequently developed by index ED

_{m})_{m}'. Moreover, the good stability and predictive ability of the models were demonstrated by LOO (leave-one-out) method and RSP (random sampling prediction) method, which further manifests the index

*ED*has high potential of wide applications in QSPR study.

_{m}**Key words:** cis/trans isomers, ionicity index matrix, improved distance matrix, branching degree matrix

**1. INTRODUCTION**

Since the Wiener index *W* introduced in 1947^{1}, many chemists such as Balaban A.T.^{2-5}, Randic M.^{6-10}, Gutman I.^{11-14}, Trinajstic N.^{15-18}, and Estrada E.^{19-21} have taken lots of time and gotten many brilliant achievements in quantitative structure-property/activity relationships (QSPR/QSAR) study. These studies have played significant roles in various fields such as drug design, physicochemical property/ biological activity forecast, environmental pollutant judgment, etc^{22-30}. Recently, our group has proposed some new topological descriptor, such as *ED _{m}* index

^{31},

*PE*index

^{32}and

*Nt*index

^{33}, which have been applied to QSPR study for series of compounds.

Stereochemistry is one of the crucial parts in organic chemistry, and the cis-trans isomerism that belongs to diastereoisomeric category, is of practical importance in stereochemistry^{34}. The cis-trans isomers have diverse chemical and physical properties, such as biological activity and inhibiting ability. Many experimental results demonstrated that stereo structural character has a great influence on chemical and vital processes. For instance, the cis-trans isomerism for alkenes often determines whether a reaction could go on wheels or not. Chiral features of drug molecules usually have decisive effect on drug effect^{35}. So it is very necessary to explore the QSPR/QSAR models for cis/ trans-isomers.

The traditional topological indices, although denote the molecular structural features effectively, are not able to differentiate the stereo structural characteristics of molecules^{35}. More and more attention has been given to develop new topological descriptors adapted to cis/trans isomers^{35-39}. Chenzhong Cao and Hua Yuan presented the modified vertex degree-distance index (MVDI) and modified edge degree-distance index (MEDI), and used them together with the odd-even index (OEI) and the number of carbon atoms *(N)* to characterize the molecular structural information for the cis- and trans-isomers of alkenes^{35}. Golbraikh, A. et al introduced several series of novel ZE-isomerism descriptors using ZE-isomerism correction, which is added to the vertex degrees of atoms connected by double bonds in Z and E configurations^{39}.

Based on the previous work^{31-33,40-45}, we proposed the index *ED _{m}* by revising

*ED*index

_{m}^{31}. The new topological descriptor

*ED*possesses good distinguishing ability to the molecular structures for cis-trans isomers. Superior to reported work, the

_{m}*ED*break with the tradition methods that all depend on using correction factor.

_{m}**2. Theories and methods**

For alkene, the double bonded carbons and the four attached atoms lie in a plane. C=C is made up of a σ bond and a π bond. The π bond is formed by sideway overlap of *p* orbitals and without symmetry axis. Since double bond can not be rotated, two alkenes with the same substituents in different spatial location are different compounds, and known as cis-trans isomers.

In modern quantum mechanics theory, the double bonded carbons in ethylene are *sp*^{2}-hybridized and have three equivalent orbitals, which lie in a plane at angles of 120° to one another. In this paper, we will use the C-C-C angle equal to 120° for sp^{2} hybridation and approximate it to 110° for sp^{3} hybridation, and consider all the saturated carbon chains as stable zigzag staggered conformations. We introduce the bond angle into hidden hydrogen graph of molecules, use the geometric distance instead of the sum of bond length between two atoms, and put forward the topological index *ED _{m}* based on branching degree matrix G, improved distance matrix ,S", and ionicity index matrix

*Q*.

Substituting Eq. (2)-(4) into Eq. (1), we obtain

**2.1 Branching degree matrix**

In Eq. (5), branching degree^{40} *g* is figured as

Where Σ*k* is the sum of the number of bonded non-hydrogen single bonds of atoms numbered *i.* Double bond is regarded as two single bonds and triple bond as three single bonds when calculated. For primary, secondary, tertiary, and quaternary carbons, obviously, *g _{i}* values are 1, 0.7071, 0.5774 and 0.5, respectively.

**2.2 Improved distance matrix**

For molecular graph of *n* vertexes, the distance matrix is a *n* x *n* symmetric matrix. We depict the improved distance matrix as

In Eq. (7), *(l _{m})_{ij}* is the ratio of the geometric distance between the two vertexes

*i*and

*j*to the bond-length of C—C, and defined as the relative distance. The actual bond lengths of C-C, C=C, C=C are 154, 134, 120 nm

^{46}. For example, Fig. 1 is the hidden hydrogen graph of cis-2-pentene.

*(l _{m})_{ij}* in Figure 1

The value of (*l*_{3})_{25} is calculated as follows (see Fig.1.b2):

The improved distance matrix *S* ' of cis-2-pentene

For trans-2-pentene, the calculations are as follows.

*(l _{m})_{ij}* in Figure 2

Values of S_{1}' and S_{2}' of trans-2-pentene are identical to that of cis-2-pentene, so we only list the *S _{3}'* of trans-2-pentene.

**2.3 Ionicity index matrix**

In Eq. (5), *q _{i}* is defined as

*X _{i,INI}* is the ionicity index

^{43}of the atom

*i*in molecules and is defined as

where *X _{iA}* is Pauling electronegativity of atom i, and

*X*is the equilibrium electronegativity

_{iE}^{42}of atom

*i*in a molecule. The equilibrium electronegativity

*X*is figured as

_{iE}In Eq. (10), Σ*l* is the sum of atoms or branching groups directly attached to the atom i, and Σ*X _{iG}* is the sum of electronegativities of atoms or groups directly attached to the atom i. X

^{44}is given by

As shown in Figure 3, the group can be divided into *k* layers, and the left atoms of the dotted line labeled 1, 2, 3, ...k are known as the ground atoms. The right suffix of atoms *l* is the numeration of the ground atom. *n _{1l}, n_{2l}…n_{kl}* are the sum of the ground atom l and other atoms or groups directly attached to the ground atoms in the grade 1, 2, 3,.

*k.*are the sum of electronegativities of the ground atom

*l*and other atoms or branching groups directly attached to the ground atoms in the grade 1, 2, 3,…

*k.*

For instance, Figure 4 is the structural graph of cis-2-pentene.

Where *X*_{c} and *X*_{H} are the Pauling electronegativities of carbon atom and hydrogen atom.

*X _{E}* of carbon atoms labeled 1, 2, 3, 4 and 5 in Fig. 4

*X _{INI}* of carbon atoms labeled 1, 2, 3, 4 and 5 in Fig. 4

*Q* of cis-2-pentene

Some ionicity index matrices *Q* are given in Table 2.

**2.4 Some ED_{m}' values**

We calculated the *ED _{1}', ED_{2}'* and

*ED*values of 44 cis-trans isomers for alkenes by Eq. (5)-(11), and listed them on Table 3.

_{3}'The descriptor *EDJ* possesses all the traits that the *ED _{m}* has: (1) The

*EDJ*indicates the size of the molecule, and the more atoms correspond to the bigger values of

*ED*(2) The index

_{m}';*EDJ*reveals the molecular branching character. The bigger degree of branching for the isomers, the bigger ΣED

_{m}'; (3) The

*EDJ*shows better structural selectivity than

*ED*in cis-trans isomers distinguishing, because the ED

_{m}_{m}can not distinguish from cis-trans isomers.

**3. RESULTS AND DISCUSSION**

**3.1 Model Development**

In this paper, we considered the interactions of α, β, γ-carbon atoms, and used ED_{1}', ED_{2}' and ED_{3}' to build a multiple linear regression (MLR) model:

Where a, b, c and d are constants, *P* represents the molecular properties. The correlation coefficient (*R*), the Fischer ratio value (*F*), and the standard deviation *(S)* can be used to assess the quality of the models.

**3.2 Relationships between the indices ED_{m}' and properties for alkenes**

We regress the boiling point *(b.p.* )^{36,47}, the standard enthalpy of formation (D_{i}*H*_{m}^{θ})^{46}, the molar refraction (*R*_{m})^{48} and the molar volume (*V*_{m})^{46} of 44 cis-trans isomers for alkenes with their values of *ED _{m}',* according to Eq. (12), and listed the statistical results of regression in Table 4.

All the values of *R* in Table 4 are larger than 0.984, and the adjusted R square (*R*^{2}) are above 96.12%. The average absolute relative errors between calculated values and experimental ones for *b.p.,* D_{f}*H*_{m}^{θ}, *R*_{m}, and *V*_{m} are 1.84%, 2.55%, 1.39% and 0.57%, respectively. One can observe that the calculations agree well with the experimental values (Figure 5), and most factors describing the property change of cis-trans isomers are opened out. The calculated values and experimental ones for *b.p.,* D_{f}*H*_{m}^{θ}, *R*_{m} and *V*_{m} are listed in Table 5 and Table 6.

**3.3 Model validation**

**3.3.1 LOO method**

In this method^{40}, we take one sample from *N* samples, do the regression with the experimental values of the remaining *N-*1 samples, and obtain a regression equation for *N-*1 samples, then predict the value of the sample taken-out by the regression equation. Using the same method, we get *N* prediction values. Finally, *N* prediction values are regressed with their experimental values, and the regression results, such as the correlation coefficient *R _{cv},* the Fischer ratio value

*F*the standard error

_{cv},*S*are used to test the stability and validity of the model. The results by LOO method are shown in Table 7.

_{CV},The values of *R _{cv}*by LOO method are close to the

*R*in Table 4 (See Figure 6). The multiple correlation coefficients of models about

*b.p.*, D

_{f}

*H*

_{m}

^{θ},

*R*

_{m}and

*V*

_{m}by LOO method are among 0.9949-0.9958, 0.9796-0.9893, 0.99997-0.99998 and 0.9993-0.9997, respectively. Most of the multiple correlation coefficients fluctuate in a narrow range (See Fig. 7), which demonstrated the high stability and good reliability of the index.

Figure 7. The distribution plot of multi-correlation coefficient R in LOO method versus frequency of R for four properties of alkenes. |

In this method^{31}, the *M* samples are randomly taken from all the *N* samples as the training set, and the *N-M* remained samples as the test set. First, build a model with the training set, then predict the *N-M* values of the test set using the model of the test set, and finally regress *N-M* prediction values with their experimental ones. The correlation coefficient (*R _{P}*), the Fischer ratio value (

*F*), and the standard error

_{p}*(S*are used to test the predictive ability of the model. The results are shown in Table 8.

_{P})**3.3.2 RSP method**

All the values of *R* for *b.p.*, D_{f}*H*_{m}^{θ}, *R*_{m}, and V_{m} in Table 8 are larger than 0.98. Moreover, the plots (Fig. 8) of the predicted values and the experimental ones for the test set and the training set can be an evidence of the good quality for the models. All the experimental values, the calculated values and the predicted values mentioned in this paper are listed in Table 5 and Table 6.

Figure 8. The experimental values vs. the predicted values based on the RSP method. |

**3.4 The contributions of the different ED_{m}**

We studied the different effects that different *ED _{m}'* have on calculated properties for compounds. The relative and fraction contribution of each index were estimated. The relative contribution (Ψ

_{r}) and fraction contribution Ψ

_{f}) of the relevant indexes are defined as follows

^{49}:

where Ψ_{r}*, *Ψ_{t}, and* * are, respectively, the relative contribution, fraction contribution, and the average value of the ith topological index. Square of the correlation coefficient, R^{2}, is the coefficient of the determination. The sum is over all the indexes in the model. Results of the contribution analysis are summarized in Table 9.

From Table 9, one can observe that the contributions of each *ED _{m}'* to the properties cover a wide range of values.

*ED*

_{1}' plays the most important role in these models. On the contrary, the contributions of

*ED*

_{2}' and

*ED*

_{3}' are less significant. From Fig. 9, it is clear that Ψ

*values of ED*

_{f}_{1}' for alkenes range between 70% and 90%, which reflects the interaction of α-carbons to be the strongest. Meantime, we can see that the Ψ

*values of*

_{f}*ED*

_{2}' and

*ED*

_{3}' are in the range of 2%~30% and 1%~9%, respectively.

**4 CONCLUSIONS**

In this paper, we introduce the bond angle into hidden hydrogen graph of molecules, use the linear distance instead of the sum of bond length between the two atoms, and proposed the new topological descriptor *ED _{m} .* High-quality models of the cis-trans isomers for QSPR study were put forward on the ground of the index

*EDJ.*By these models, properties of

*b.p.,*D

_{f}

*H*

_{m}

^{θ},

*R*

_{m}and

*V*

_{m}were calculated and predicted, and the results agreed well with the experimental values. Moreover, the LOO method and the RSP method demonstrate that the models have statistical significance and good stability.

**ACKNOWLEDGEMENTS**

The project was supported by Science and Technology Projects Foundation of Hunan province (No. 06FJ4104).

(Received: June 30, 2010 - Accepted: January 26, 2011)

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