Journal of Chemistry
Volume 2016 (2016), Article ID 1791756, 8 pages
http://dx.doi.org/10.1155/2016/1791756

Research Article

Nanoquantitative Structure-Property Relationship Modeling on C₄₂ Fullerene Isomers

Sorana D. Bolboacă¹ and Lorentz Jäntschi2,3

¹Department of Medical Informatics and Biostatistics, Iuliu Haţieganu University of Medicine and Pharmacy, 6 Louis Pasteur Street, 400349 Cluj-Napoca, Romania
²Department of Physics and Chemistry, Technical University of Cluj-Napoca, 103-105 Muncii Bulevardul, 400641 Cluj-Napoca, Romania
³Doctoral Studies-Chemistry, Babeş-Bolyai University, 11 Arany Janos Street, 400028 Cluj-Napoca, Romania

Received 5 October 2015; Revised 1 January 2016; Accepted 4 January 2016

Academic Editor: Teik-Cheng Lim

Copyright © 2016 Sorana D. Bolboacă and Lorentz Jäntschi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The interest of scientists in nanostructures has been increased in the last years and proper methods for their assessment are needed. In silico methods found their usefulness in the replacement of experimental evaluation and are successfully used as efficient alternatives for estimation and prediction of compound’s properties or activities. In this paper, it is shown that a Quantitative Structure-Property Relationship method is proper to be applied also on nanostructures. Based on computational experiment, several models to describe the total strain energy of C₄₂ fullerene isomers were obtained and their characteristics are presented. Furthermore, the best performing model obtained on C₄₂ fullerene isomers was validated on C₄₀ fullerene isomers.

This paper is dedicated to Professor Mircea V. Diudea on the occasion of his 65th birthday.

1. Introduction

Since their discovery in 1985 [1], fullerenes attracted interest in different fields of science, including medical field (e.g., for potential use as antibiotics [2–4], as inhibitors of erythroid cells—fullerenol [5], as drug delivery system [6], or as inhibitors of inflammatory mediators [7]). Fullerene molecules are constructed from carbon atoms and take the shape of sphere (also known as buckyballs), ellipsoid, or tube [8]. First spherical fullerene, C₆₀, was discovered in 1985 [1]. Fullerenes have different properties and showed different number of associated isomers (Table 1) [9]. The smallest fullerene (C₂₈) was stabilized by metal encapsulation (with Ti, Zr, and U) by Dunk et al. [10]. Chen et al. showed that C₃₂ fullerene has stronger aromaticity compared with C₃₀ and C₃₄, respectively [11]. Fifteen distinct isomers with different energies were reported by Manna and Ghanty who encapsulate U into various C₃₆ cages [12]. Muhammad et al. showed that C₂₀ is a closed-shell fullerene and fullerenes C₂₆ and C₃₀ are pure open-shell compounds, whereas C₃₆, C₄₀, and C₄₂ are intermediate open-shell compounds [13].

Table 1: Several small fullerenes and their number of isomers.

Number
Fullerene
Number of isomers

1
C₂₈
2

2
C₃₀
3

3
C₃₂
6

4
C₃₄
6

5
C₃₆
15

6
C₃₈
17

7
C₄₀
40

8
C₄₂
45

9
C₄₄
89

10
C₄₆
116

Source: http://www.nanotube.msu.edu/fullerene/fullerene-isomers.html [accessed June 7, 2015].

The C₄₂ fullerenes are small, not necessary spherical cages. The C₄₂ cages enclosed high pentagon/hexagon ratio [14]. Fullerene C₄₂ along with C₆₀ showed highest values of the main peak on Matrix-Assisted Laser Desorption Ionization Time-of-Flight (MALDI-TOF) on mass spectrometric measurement [15].

Some activities of fullerenes have been modeled using quantitative structure-activity relationship (QSAR) approaches (such as anti-HIV protease inhibition activity [16], antiviral activity [17], and drug delivery system [18]). However, C₆₀ received the main attention while other fullerenes were neglected in regard of QSAR/QSPR (Quantitative Structure-Property Relationship) modeling. The aim of our research was to model the total strain energy of the isomers of C₄₂ fullerene using the structural information.

2. Materials and Methods

All C₄₂ fullerene isomers were included in the analysis. Data related to continuum elasticity expressed as total strain energy (TSE in eV) and the structures as files of C₄₂ fullerene isomers were taken from [19] (Table 2).

Table 2: C₄₂ fullerene isomers: identification number (IsoID) and total strain energy (TSE).

IsoID
TSE (eV)

#01
31.060

#02
30.537

#03
29.791

#04
29.805

#05
30.618

#06
29.850

#07
30.608

#08
29.782

#09
28.527

#10
29.393

#11
29.475

#12
28.340

#13
28.157

#14
27.147

#15
29.955

#16
28.175

#17
28.276

#18
29.474

#19
27.408

#20
28.175

#21
27.283

#22
29.140

#23
28.765

#24
27.743

#25
27.487

#26
28.353

#27
28.014

#28
29.051

#29
27.489

#30
28.972

#31
27.484

#32
26.657

#33
26.639

#34
27.371

#35
26.554

#36
27.973

#37
29.764

#38
31.101

#39
26.639

#40
27.501

#41
26.672

#42
28.665

#43
28.284

#44
26.737

#45
25.661

The analysis was conducted on the downloaded file of the C₄₂ isomers without any modification on the available geometry. According to [19], the fullerene geometries were based on the geometry of the structures in Yoshida’s Fullerene Library (UNIX files) and reoptimized using Dreiding-like force-field [20]. Here the obtained geometry is used.

The steps applied in the analysis are depicted in Scheme 1.

Scheme 1: Flowchart of the applied methods. The pool of filtered SMPI (Szeged Matrix Property Indices) descriptors contains those descriptors with absolute values between 10⁻⁷ and 10⁷.

In the first step of the analysis the downloaded files were translated into file with Spartan software (https://www.wavefun.com/products/spartan.html). In the second step the file is transformed as file using Babel software (http://openbabel.org). The partial charges were calculated in the third step using HyperChem software (http://www.hyper.com/) by applying PM3 (Parameterized Model number 3 [21]) single point (energy) semiempirical calculations. The structural features of the investigated nanoclass of compounds were extracted using unsymmetrical Szeged set, an extension of corresponding Szeged Matrix [22] (forth step). The calculated values of the structural descriptors and the collected values of total strain energy were included in nano-QSPR modeling in the fifth step of the analysis and models with the highest goodness-of-fit (defined as highest correlation coefficients) were analyzed and validated in leave-one-out and leave-many-out analyses [23, 24].

Leave-one-out analysis retrieves valid models if determination coefficient () takes values higher than 0.5. Leave-many-out analysis was conducted for the models with highest abilities in estimation expressed as the highest value of the correlation coefficient. The set was split using a simple random technique [25] in training and test with 2/3 of compounds in training set. The models obtained in training sets were used to predict the TSE in the test sets. The leave-many-out analysis was run five times for equations identified as being with highest estimation and internal prediction abilities in order to assess their prediction abilities.

The assessment of the prediction ability was done on an external dataset represented by C₄₀ isomers considering the same property. The TSE values and the structures for external validation were taken from the same source as C₄₂ isomers: http://nanotube.msu.edu/fullerene/fullerene.php?C=40 (accessed December 20, 2015). Several metrics were used to assess the prediction ability of the model [23, 24]: determination coefficient on the external set (), predictive square correlation coefficient on external set (, [26]), external prediction ability (), root mean square error of prediction (RMSEP), mean absolute error of prediction (MAEP), percentage predictive error (%PredErr), and concordance correlation coefficient (CCC [27]).

3. Results and Discussion

Structural information of the investigated C₄₂ isomers was obtained by calculation of the pool of descriptors given by Szeged Matrix Property Indices (SMPI) method [28]. Performing models in regard of goodness-of-fit (highest correlation coefficient) with 1, 2, 3, and 4 SMPI descriptors was obtained and is given in (1)–(4):

where is total strain energy estimated by the model; IJUGE, IIUGF, IFEGE, IFETB, and IFUGB are SMPI descriptors. Two descriptors (IFETB and IFUGB) account for the atomic number as atomic property; the other two descriptors account for electronegativity (IJUGE and IFEGE), while one accounts for the first ionization energy (IIUGF). The investigated property is related to the geometry of compounds (fourth letter “G” in the name of descriptors) with one exception when it is related to topology (fourth letter “T” in the IFETB descriptor). The other letters reflect the linearization operator (first letter), matrix operation (second letter), and interaction descriptor (third letter).

As expected, the determination coefficient increases as the number of descriptors in the models increases, while the standard error of the estimate decreases (Table 3).

Table 3: Characteristics of nano-QSPR models obtained on C₄₂ isomers.

Equation

se

%PredErr

(1)
0.8883
0.8857
0.4577
342
(4.38 × 10⁻²²)
18.05
(1.09 × 10⁻²¹)
46.77
0.8656
0.5039
275
(1.56 × 10⁻²⁰)

(2)
0.9612
0.9593
0.2729
520
(2.32 × 10⁻³⁰)
6.69
(4.10 × 10⁻⁸)
31.92
0.9545
0.2960
439
(8.48 × 10⁻³⁰)

(3)
0.9836
0.9824
0.1796
820
(1.30 × 10⁻³⁶)
4.37
(8.40 × 10⁻⁵)
19.76
0.9809
0.1939
701
(3.80 × 10⁻³⁷)

(4)
0.9898
0.9888
0.1431
974
(2.87 × 10⁻³⁹)
3.55
(1.01 × 10⁻³)
15.95
0.9768
0.2171
418
(2.28 × 10⁻³⁴)

: determination coefficient; : adjusted determination coefficient; se: standard error of estimate; : Fisher’s statistic (-value); : the minimum of absolute -statistic associated with the intercept and coefficients of the model; %PredErr: percentage prediction error; : determination coefficient in leave-one-out analysis; loo: leave-one-out analysis.

The distance between determination coefficient of the model and determination coefficient obtained in leave-one-out analysis varied from 0.0027 to 0.0227, the smallest distance being obtained by (3) (Table 3). On the other hand, the smallest difference between standard errors (estimation model and leave-one-out model) is obtained by the same model (3).

The analysis of the results presented in Table 3 showed that the model with four descriptors is the one with smallest percentage of prediction error. Furthermore, the data on the scatter closest to the straight line is observed for the model given by (4) (Figure 1). Figure 1 shows the absence of the differences between models from (3) and (4), with the dispersion of the point in the scatter closest to the line for model given by (4).

(a)

(b)

(c)

(d)

Figure 1: Observed versus estimated TSE by (1)–(4).

The main characteristics of the models given by (3) and (4) obtained in leave-many-out analysis (training versus test analysis; 2/3 of compounds in training set run 5 times) are presented in Table 4.

Table 4: Characteristics of nano-QSPR models in leave-many-out analysis: C₄₂ isomers.

Id
Model
Training
Test

Equation (3)
Intercept
IFEGE
IIUGF × 10⁻³
IJUGE

-stat

-stat

1
744.37
−1.27
−3.57
2.09
0.9797
360
0.9877
264

2
682.4
−1.21
−3.46
2.07
0.9788
369
0.9935
361

3
902.68
−1.48
−3.79
2.20
0.9794
376
0.9894
358

4
678.39
−1.22
−3.53
2.12
0.9853
534
0.9851
171

5
854.05
−1.49
−3.51
2.17
0.9828
458
0.9835
219

Equation (4)
Intercept
IFETB
IFUGB
IIUGF × 10⁻³
IJUGE

-stat

-stat

6
−120.59
−19.88
37.19
−2.87
1.55
0.9901
568
0.9637
72

7
−277.84
−20.87
40.14
−2.30
1.54
0.9819
310
0.9814
154

8
−91.46
−20.86
39.01
−3.00
1.56
0.9878
459
0.9636
64

9
−225.97
−21.69
42.02
−2.35
1.48
0.9830
331
0.9794
139

10
−223.28
−18.71
36.37
−2.35
1.47
0.9887
497
0.9701
85

The results presented in Table 4 showed the stability of the models, with internal prediction power (defined as determination coefficient in test sets) closed to the estimation power (determination coefficient in training set) from both investigated models. Therefore, the results obtained in training sets closely follow the results on the whole sample for (3) with in the same range when two decimals are of interest. The obtained in test set in all five runs of the leave-many-out analysis was equal to 0.99, so slightly higher than the obtained in training sets (0.98). In three cases out of five, the in training sets for (4) was in the same range for two decimals with the value given in Table 3. However, without any exception, the in test sets was smaller than the in training sets for (4), with values that varied from 0.0005 (id 7 in Table 4) to 0.0264 (id 6 in Table 4). These results showed that (3) performs slightly better in terms of determination coefficients in leave-many-out analysis.

The plots of the models obtained in the fourth run for (3) and fifth run for (4), as examples, are given in Figure 2.

(a)

(b)

Figure 2: Internal prediction versus estimation power in training and test analysis for (3) and (4).

The equations identified with estimation power and internal prediction abilities, namely, (3) and (4), were further applied on C₄₀ isomers to test the external prediction abilities. The prediction power of (4) proved to be better compared with prediction power of (3) (see Figure 3 and Table 5).

Table 5: Prediction power of nano-QSPR given by (3) and (4) on C₄₀ isomers.

Equation

RMSEP
MAEP

%PredErr

(3)
0.6183
0.9501
NR
1.60
51.28
324 (8.37 × 10⁻⁶⁹)
63.19

(4)
0.8462
0.5144
NR
1.60
5.27
52 (4.96 × 10⁻³⁸)
6.49

: determination coefficient on the external set; : predictive square correlation coefficient on external set; : external prediction ability; RMSEP: root mean square error of prediction; MAEP: mean absolute error of prediction; %PredErr: percentage predictive error; NR: not reliable value.

(a)

(b)

Figure 3: Analysis of (3) and (4) on external dataset represented by C40 isomers.

Despite the fact that the predictive square correlation coefficient on external set is higher for (3) compared with the value obtained with (4), all other calculated metrics sustain that the model given by (4) has better prediction abilities (highest determination coefficient on external set, lowest mean absolute error of prediction, and lowest percentage of predictive error; see Table 5). Furthermore, the analysis of the overall spread of the points in the scatter-plot leads to the conclusion that (4) had better prediction abilities compared with (3). Nevertheless, the mean of residuals proved to be significantly different than the expected value (zero). It could be concluded that the model given by (4) better fit the data on which it was constructed compared with all other models. Nevertheless, are the structural features extracted by SMPI descriptors on C ₄₂ isomers able to predict the TSE on C ₄₀ isomers?

SMPI descriptors used by (3) and, respectively, (4) were used to predict the TSE on C₄₀ isomers. One out the three descriptors from (3) proved to have the slope not significantly different by zero and was not included in further analysis. The identified models obtained on C₄₀ isomers are given in

where is total strain energy estimated by the model; IJUGE, IIUGF, IFETB, and IFUGB are SMPI descriptors. Two descriptors (IFETB and IFUGB) account for the atomic number as atomic property, one descriptor accounts for electronegativity (IJUGE), and one accounts for the first ionization energy (IIUGF). The investigated property is related to the geometry of compounds (fourth letter “G” in the name of descriptors) with one exception that is related with compounds topology (IFETB descriptor). The other letters reflect the linearization operator (first letter), matrix operation (second letter), and interaction descriptor (third letter). Note that both models have the mean of residual not significantly different by zero ().

The analysis of the metrics associated with (5) and (6) leads to the conclusion that model given by (6) perform better than the model given by (5). The same conclusion is obtained by analyzing the plots of observed versus predicted TSE (Figure 4).

(a)

(b)

Figure 4: Analysis of (5) and (6) on external dataset represented by C40 isomers.

The results of our study showed that the identified nano-QSPR models fit the data based on which the model was identified (C₄₂ isomers) but could be used for selection of those structural descriptors with fair abilities in prediction on external dataset (C₄₀ isomers). To sum up, equations relating electronegativities, ionization potential, and energy have been identified on C₄₂ isomers and proved to work also on C₄₀ isomers. Note that electronegativities and ionization potential are atomic properties and since the investigated set contains just C and H atoms, the identified relation between the three properties could be assigned also to the topology and geometry of the investigated compounds.

To the best of our knowledge, structure-property relationship approaches were not applied on C₄₂ or C₄₀ fullerene isomers. The small-diameter fullerenes (C₂₀, C₃₄, C₄₂, and C₆₀) were mainly investigated in regard of properties (such as adsorption [29], distribution of CC distance [14], and Schlegel diagrams of molecular structures [30]). Therefore, this is the first report of a quantitative relationship between structure and property of C₄₂ fullerene. Undoubtedly, the advancement from theoretical to experimental studies is desired.

4. Conclusions

The C₄₂ fullerene isomers were successfully modeled and the total strain energy was characterized as function of information extracted from structure of the compounds. The models with goodness-of-fit in leave-one-out () and leave-many-out analyses proved also that prediction power is the one with four descriptors. The total strain reaction proved a function of electronegativity and first ionization energy, in relation to geometry of compounds. The structural descriptors able to fairly explain the total strain energy on C₄₂ isomers proved also able to explain the same property on C₄₀ fullerene isomers.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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