Abstract
A topological index of graph is a numerical quantity which describes its topology. If it is applied to molecular structure of a chemical compounds, then it reflects the theoretical properties of the chemical compounds. In this paper, well-known degree-based topological indices are applied on chemical structures of antituberculosis drugs. Chemical structure is considered as graph, where elements are taken as vertices and bounds between them are taken as edges. Furthermore, QSPR analysis of the said topological indices are discussed, and it is shown that these topological indices are highly correlated with the physical properties of antituberculosis drugs. This theocratical analysis may help the chemist and people working in pharmaceutical industry to predict properties of antituberculosis drugs without experimenting.
1. Introduction
Before discovery/invention of antibiotics, lives of humans and animals were on great threat of being infected by some bacteria. In previous century, different bacterial infections were the most common causes of death until Alexander Flemings discovered Penicillin in 1929, the first antibiotic which was introduced to the world in 1940.
Tuberculosis (TB) is a contagious infection caused by bacteria “Mycobacterium tuberculosis” that usually attacks lungs. It can also spread to other parts of body, like brain and spine.
A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds between atoms. Cheminformatics is a new subject which is a combination of chemistry, mathematics, and information science. It studies quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of different chemical compounds. The molecular topological indices or simply the topological indices are used in chemistry. Wiener was the first who first showed that the Wiener index number is closely correlated with the boiling points of alkane molecules [18]. Later, work on quantitative structure-activity relationships showed that it is also correlated with other quantities including the parameters of its critical point [16], density, surface tension, viscosity of its liquid phase [14], and the van der Waals surface area of the molecule [10]. In the QSAR/QSPR study, physico-chemical properties and topological indices are used to predict bioactivity of the chemical compounds. In this work, it is shown that no single topological index exists that correlates with all the physical properties of chemical compounds.
A graph with vertex set and edge set is connected if there exists a connection between any pair of vertices in . The distance between two vertices and is denoted as and is defined as the length of shortest path between and in graph . The number of vertices of adjacent to a given vertex is the “degree” of this vertex and will be denoted by or if misunderstanding is not possible simply by . The concept of degree is somewhat closely related to the concept of valence in chemistry.
Some of the degree-based topological indices which we use in this work are defined as follows.
Definition 1. index is proposed by Estrada et al. in [7], as
Definition 2. The Randi index is proposed by Milan Randi in [13], as
Definition 3. The sum connectivity index is proposed by Zhou and Trinjstic in [19], as
Definition 4. The index is proposed by Vukicevic et al. in [17], as
Definition 5. The first and second Zagreb indices are proposed by Gutman and Trinajestic in [11], as
Definition 6. The Harmonic index is proposed by Fajtlowicz et al. in [9], as
Definition 7. The hyper-Zagreb index is proposed by Shirdel et al. in [15], as
Definition 8. The third Zagreb index is proposed by Fath-Tabar et al. in [1], as
Definition 9. The forgotten index is proposed by Furtula et al. in [8], as
Definition 10. The symmetric division index is proposed in [2], aswhere and. .
Some of the work in this area can be seen in [3–6, 12].
2. Results and Discussion
The above defined 11 topological indices are used for the modeling of six physical properties: boiling point (BP), enthalpy of vaporization (E), flash point (F), molar refractivity (MR), molar volume (MV), and polarizability (P) of 15 antituberculosis drugs: amikacin, bedaquiline, clofazimine, delamanid, ethambutol, ethionamide, imipenem-cilastatin, isoniazid, levofloxacin, linezolid, moxifloxacin, p-aminosalicylic acid, pyrazinamide, rifampin, and terizidone.
2.1. Regression Models
The following equation is used to correlate the various physical properties of various drugs used for the treatment of tuberculosis with some topological indices. We have used the following linear regression model:where is physical property of drug, is constant, is regression coefficient, and is topological index. Constant and regression coefficient is calculated from SPSS software for seven physical properties and eleven degree-based topological indices of molecular structure of fourteen drugs. Using equation (11), following are the linear regression model for the defined degree-based topological indices:(1)Regression models for atom bond connectivity index: (2)Regression models for Randi index: (3). Regression models for sum-connectivity index. (4)Regression models for geometric-arithmetic index () (5)Regression models for first Zagreb index: (6)Regression models for second Zagreb index: (7)Regression models for harmonic index: (8)Regression models for hyper-Zagreb index: (9)Regression models for third Zagreb index: . (10)Regression models for forgotten index: (11)Regression models for symmetric division index:
2.2. Computation of Topological Indices and Their Comparison with Correlation Coefficients of Some Physical Properties
Table 1 shows the abovementioned physical properties of 15 drugs used for the treatment of tuberculosis.
In Table 2, the 12 topological indices are computed of the graphs constructed from the molecular structures of the drugs.
In Table 3, the correlation coefficients of the 6 physical properties with respect to each topological index are computed. The graph of the correlation coefficient of all the physical properties such as boiling point, enthalpy, flash point, molar refraction, molar volume, and polarizability on different topological indices are shown in Figure 1. The graphical representation of topological indices of the different medicine is shown in Figure 2.
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2.3. Computation of Statistical Parameters
In this section, regression parameters have been computed. is the sample size, is constant or -intercept, is slope, is correlation coefficient, and is the percentage of the dependent variable variation that a linear model explains. The value for each term tests the null hypothesis that the coefficient is equal to zero (no effect), whereas a larger (insignificant) value suggests that changes in the predictor are not associated with changes in the response. Suppose we are doing a test in which the null hypothesis is that all of the regression coefficients are zero. The result of this kind of a test gives us a value called value. In this case, the model has no predictive capability. With the help of this test, one can compare their model with zero predictor variables and decides whether their added coefficients have improved the model. In Tables 4–13, these statistical parameters are computed for linear QSPR models for different topological indices. All of these analyses show that the value in each of the model is zero that indicates the significance of the results.
2.4. Standard Error of Estimate
The Standard error of estimate is the measure of variation of an observation made around the computed regression line. Table 14 shows the standard error of estimate for six physical properties corresponding to each topological index.
2.5. Correlation Determination
The correlation determination describes the percentage of relation which gives you more information about the relationship between variables. When you square the correlation coefficient, you end up with the correlation of determination .
2.6. Comparison
In this section, the comparison for known values and computed values from our regression models is drawn. In Tables 15–20, the comparison of each physical property is shown.
3. Conclusion and Future Study
3.1. Conclusion
Tuberculosis (TB) is a disease and its infection is caused by bacteria “Mycobacterium tuberculosis” that usually attacks lungs. It can also spread to other parts of body, like brain and spine and cause death.
A molecular descriptor is actually a mathematical formula that can be applied to any graph which models some molecular structure. More precisely, a single number represents a chemical structure. In graph theory, this number is called topological descriptor. When a topological descriptor correlates with a molecular property, it is called molecular index or topological index . Thus, a topological graph index is also called a molecular descriptor. Mathematically, a topological index is a numeric quantity associated with a graph which characterizes the topology of a graph. By computing these topological indices, it is possible to analyze mathematical values and further investigate some physio-chemical properties of a molecule. Actually topological indices are designed on the ground of transformation of molecular graph into a number which characterizes the topology of graph.
Molecular topological indices play a significant role in mathematical chemistry, especially in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) investigations. (11) is used to correlate the various physical properties of various drugs used for the treatment of tuberculosis with some topological indices. Therefore, it is an efficient way to avoid expensive and time-consuming laboratory experiments. The purpose of computing these 11 topological indices is that no single topological index is found yet that can be efficient (given in various tables) for all physical properties of these drugs.
Tables 1 and 2 gives the values of physical properties and topological indices of medicines (Figure 3), respectively. Table 3 and graphs (Figure 1) shows how degree-based topological indices and physical properties of medicine correlate. Upon examining correlation coefficients horizontally for physical properties under consideration, we see that index gives highest correlation coefficient for molar refraction , molar volume , and polarizability . index gives highest correlation coefficient for boiling point . has highest correlation coefficient for flash point . gives highest correlation coefficient for boiling point and enthalpy. When we look vertically, boiling point has also good correlation with and , i.e., for both. Enthalpy has also good correlation with and , i.e., . Flash point has also good correlation with , i.e., . Molar refraction has also good correlation with , i.e., . Molar volume has also good correlation with , i.e., . Polarizability has also good correlation with and , i.e., .
While examining correlation determination (Table 21) horizontally for physical properties under consideration, we see that index gives highest correlation determination for molar refraction , molar volume , and polarizability . index gives highest correlation determination for boiling point . has highest correlation determination for flash point . gives highest correlation determination for boiling point and enthalpy . When we look vertically, boiling point has also good correlation with and , i.e., for both. Enthalpy has also good correlation with and , i.e., . Flash point has also good correlation with and , i.e., . Molar refraction has also good correlation with , i.e., . Molar volume has also good correlation with , i.e., . Polarizability has also good correlation with and , i.e., . One can easily gather from the table of statistical parameters for the linear QSPR models for different degree-based topological indices that are as follows: Harmonic index has positive and highly significant correlation coefficient for molar refraction , molar volume , and polarizability . Sum-connectivity index and hyper-Zagreb index has highly significant correlation coefficient for boiling point, i.e., . Sum-connectivity index also has highly significant correlation coefficient for enthalpy. Forgotten topological index has highly significant correlation coefficient for flash point . From Table 21, we can also see high percentage correlation between degree-based topological index and physical properties of medicines. Harmonic index has positive and high percentage of correlation for molar refraction , molar volume , and polarizability . Hyper-Zagreb index has high percentage of correlation correlation for boiling point, i.e., . Sum-connectivity index also has high percentage of correlation for enthalpy ( ). Forgotten topological index has high percentage of correlation for flash point .
Tables 4–13 and 22 show different statistical parameters of correlation between values of eleven degree based topological indices and six physical properties of medicine. We could not find any correlation between degree-based topological index and melting point of antituberculosis drugs.
This work indicated that this theocratical analysis may help the chemist and people working in pharmaceutical industry to predict properties of antituberculosis drugs without experimenting. It is also possible that different composition of these drugs may be used for different diseases; of course it depends on the range of the topological indices which are computed in this work. In this work, we have found the correlation coefficient for different topological indices; this will help the chemist to design new drugs based on the combination of positively high correlated drugs.
3.2. Future Study
In a similar pattern, relation between physical properties for different drugs/medicine for the treatment/preventive measures of a particular disease and Topological indices can be established to estimate physical properties of newly discovered medicine or candidate drug(s) of particular disease.
It will be very important to find a topological index that can correlate with all the physical properties.
Data Availability
The physical property data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors have no conflicts of interest.