Topological Indices of Novel Drugs Used in Diabetes Treatment and Their QSPR Modeling

Parveen, Saima; Hassan Awan, Nadeem Ul; Mohammed, Mogtaba; Farooq, Fozia Bashir; Iqbal, Nisha

doi:https://doi.org/10.1155/2022/5209329

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Volume 2022 | Article ID 5209329 | https://doi.org/10.1155/2022/5209329

Topological Indices of Novel Drugs Used in Diabetes Treatment and Their QSPR Modeling

Saima Parveen,¹Nadeem Ul Hassan Awan,¹Mogtaba Mohammed,^2,3Fozia Bashir Farooq,⁴and Nisha Iqbal¹

Academic Editor: Gohar Ali

Received17 Feb 2022

Revised23 Mar 2022

Accepted26 Mar 2022

Published19 May 2022

Abstract

A topological index is a real number obtained from the chemical graph structure. It is helpful to calculate the physicochemical and biological properties of numerous drugs. This is done through degree-based topological indices. In this paper, acarbose, tolazamide, miglitol, prandin, metformin, and so on used to treat diabetes are discussed, and the purpose of the QSPR study is to determine the mathematical relation between the properties under investigation (e.g., boiling point and flash point) and different descriptors related to the molecular structure of the drugs. In this study, it is observed that topological indices (TIs) applied to said drugs have a good correlation with physicochemical properties in this course.

1. Introduction

Diabetes is a metabolic disease that develops when the pancreas cannot generate enough insulin, and its effective use is retarded. This chronic disease leads to high blood glucose and is named blood sugar. Glucose is obtained from food and becomes an energy source in our body. Pancreas generates a hormone named insulin. The hormone helps glucose to digest in cells and is used as energy. Glucagon is another hormone that associates with insulin and controls blood sugar. When our immune system is not functioning properly, it fights infections and kills insulin-producing agents. As a result, there is a chance that insulin does not work well and glucose becomes to stay in our blood and will not become part of our cells. With the passage of time, diabetes leads to severe damage to nerves and blood vessels, the amount of glucose becomes very high, and it will generate other health issues. Every year 422 million persons become sick with diabetes, and 1.5 million lead to death because of this disease. The risk of diabetes increases when you are 45 or above, and high blood pressure will also increase the chance of the said disease [1–4]. The disease has no proper treatment, but effective care can help you manage diabetes and live a healthier life. Drugs are used to cure this malignant disease, and many drugs tests are accompanied to fight the fatal disease. This needs timely diagnosis, screening, and medication that benefits patients to control the deadly disease in future. The ten essential drugs acarbose, tolazamide, miglitol, prandin, metformin, glimepiride, linagliptin, pioglitazone, bromocriptine, and alogliptin are safe and the most efficient medicines that are required for well-being community. Figure 1 depicts the chemical structure of the said drugs.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Topological indices (TIs) are termed numeric descriptors that are obtained through a molecular graph in order to completely mention the chemical system and widely used in the investigation of physicochemical properties of many drugs. Since there are several kinds of polynomials and topological indices which are extensively calculated, represent the chemical structure, and have a vital position in chemical graph theory, among such families, degree-based topological indices are of great significance and play a vital part in chemical graph theory. The use of graph invariants (TIs) in QSPR and QSAR studies has taken key interest in recent years. Topological indices have applications in various areas of mathematics, bioinformatics, mathematics, informatics, biology, and so on, but their utmost significant use to date is in the nonempirical Quantitative Structure-Property Relationships (QSPR) [5, 6].

ABC index, Wiener index, and Randic index are helpful to predict the bioactivity of drugs. The QSPR models assist in determining the optimal relationship between topological indices and psychochemical characteristics. These psychochemical qualities are being studied because they have a big impact on bioactivity and drug transit in the human body. In this paper, we have computed degree-based TIs related to diabetes drugs. Similarly, antidiabetes drugs represent a chemical compound on which the given topological indices are thoroughly defined and deliberate QSPR analysis. The corresponding characteristic estimated through this method is highly correlated with the characteristic of diabetes drugs with the help of linear regression. It is observed that a high correlation exists between the properties of drugs and TIs.

2. Material and Method

In drug structures, atoms denote vertices, and the corresponding bonds connecting the atoms are termed edges. Graph G (V, E) is considered as simple, finite, and connected, whereas V and E represented in the chemical graph are termed as vertex and edge set, respectively. The degree of a vertex in a graph G is the number of vertices adjacent to it and is denoted by d_u. Valence of a compound in chemistry and the degree of a vertex in a graph are meticulously related concepts [7–10]. Degree-based topological indices used are given as follows.

Definition 1. The ABC index [10] of a molecular graph G is defined as

Definition 2. The first degree-based topological index is the Randic index introduced by Milan Randic in 1975 [11]. Randic index is defined as

Definition 3. The sum connectivity index [12] of a molecular graph G is defined as

Definition 4. The GA index [13] of a molecular graph G is defined as

Definition 5. The first and second Zagreb indices [14] of a molecular graph G are defined as

Definition 6. The harmonic index [15] of a molecular graph G is defined as

Definition 7. The hyper-Zagreb index [16] of a molecular graph G is defined as

Definition 8. The forgotten index [17] of a molecular graph G is defined asThe energy of molecule was found with the help of the first and second Zagreb indices in [18]. Alkane heat of formation is best predicted by applying the augmented Zagreb index in [19]. The values of physical properties are taken from ChemSpider. It is observed from data in Table 1 and is found that these data values are normally distributed. Therefore, the linear regression model is the most adequate to test and adopt for the said analysis. For more insight on degree-based topological indices, we offer the reader to visit the following articles [5, 7, 20–24].

3. Results and Discussions

In this section, degree-based TIs are executed on diabetes drugs. The relation between QSPR analysis and topological indices portrays that the properties are vastly correlated in terms of physicochemical properties for the said disease. The ten medicines acarbose, tolazamide, miglitol, prandin, metformin, glimepiride, linagliptin, pioglitazone, bromocriptine, and alogliptin are used in the analysis for diabetes. The drug structures are displayed in Figure 1. We consider the molecular structure as a graph, the drug elements denote vertices, and bonds among atoms are their edges. We use regression analysis calculation for drugs study.

3.1. Regression Model

In this paper, drug computable structure analysis about ten topological indices is done for QSPR modeling tenacity. The six physical properties, flash point (FP), polarity, boiling point (BP), molar volume (MV), and refractivity (R) for ten medicine arranged in Table 2 are used in diabetes treatment. We execute the regression analysis for the drugs, and the linear regression model is tested with the help of equation

Here, P represents the physicochemical property of given drugs. TI is the topological index, A is constant, and b represents the regression coefficient. The IBM SPSS Statistics version-24 software is helpful to find out the results. Nine TIs of diabetes drugs and physiochemical properties are analyzed with a linear QSPR model. Equation (9) is suitable for the said calculation purpose.

Theorem 1. Let be the graph metformin. Various topological indices of are given as follows:(i).(ii).(iii).(iv).(v).(vi).(vii).(viii).(ix).

Proof. Let be the graph of metformin with edge set E. Let represent the class of edges of joining vertices of degrees and . With , and , one has the following: (i)By using Definition 1 and the above given edge partitions , we get(ii)By using Definition 2 and the above given edge partitions , we get(iii)By using Definition 3 and the above given edge partitions , we get(iv)By using Definition 4 and the above given edge partitions , we get(v)By using Definition 5 and the above given edge partitions , we get(vi)By using Definition 5 and the above given edge partitions , we get(vii)By using Definition 6 and the above given edge partitions , we get(viii)By using Definition 7 and the above given edge partitions , we get(ix)By using Definition 8 and the above given edge partitions , we get

Theorem 2. Let be the graph of tolazamide. Various topological indices of are given as follows:(i).(ii).(iii).(iv).(v).(vi).(vii).(viii).(ix).

Proof. Let be the graph of tolazamide with edge set . Let represent the class of edges of joining vertices of degrees and . With , , , , and , one has the following:(i)By using Definition 1 and edge partitions , we get(ii)By using Definition 2 and edge partition , we get(iii)Definition 3 and edge partition give(iv)By using Definition 4 and edge partition , we get(v)By using Definition 5 and edge partition , we get(vi)By using Definition 5 and edge partition , we get(vii)By using Definition 6 and edge partition , we get(viii)By using Definition 7 and edge partition , we get(ix)By using Definition 8 and edge partition , we getOne can calculate topological indices of the remaining drugs by adopting a similar procedure applied in Theorem 1 and Theorem 2 and using Definitions 1 to 8. Moreover, the calculated values of all drugs are listed in Table 1.
Using (1), we have calculated the following diverse linear models for all degree-based topological indices, which are given as follows:(1)Regression models for atom bond connectivity index ABC(G): Boiling point = 103.153 + 10.44 [ABC(G)]. Refractivity = 18.569 + 2.093 [ABC(G)]. Flash point = 20.160 + 7.125 [ABC(G)]. Polarity = 0.172 + 0.980 [ABC(G)]. Molar volume = 31.785 + 6.065 [ABC(G)]. Complexity = −182.330 + 19.139 [ABC(G)].(2)Regression models for atom bond connectivity index RA(G): Boiling point = 100.512 + 18.552 [RA(G)]. Refractivity = 18.020 + 3.731 [RA(G)]. Flash point = 10.148 + 13.033 [RA(G)]. Polarity = −0.380 + 1.759 [RA(G)]. Molar volume = 29.935 + 10.820 [RA(G)]. Complexity = −186.139 + 34.060 [RA(G)].(3)Regression models for atom bond connectivity index S(G): Boiling point = 105.293 + 18.056 [S(G)]. Refractive index = 18.429 + 3.638 [S(G)]. Flash point = 15.429 + 12.603 [S(G)]. Polarity = 0.122 + 1.710 [S(G)]. Molar volume = 32.905 + 10.520 [S(G)]. Complexity = −180.010 + 33.247 [S(G)].(4)Regression models for atom bond connectivity index GA(G): Boiling point = 113.440 + 8.420 [GA(G)]. Refractivity = 20.596 + 1.694 [GA(G)]. Flash point = 22.547 + 5.853 [GA(G)]. Polarity = 1.044 + 0.795 [GA(G)]. Molar volume = 38.721 + 4.888 [GA(G)]. Complexity = −465.288 + 15.519 [GA(G)].(5)Regression models for atom bond connectivity index M1(G): Boiling point = 126.868 + 1.316 [M1(G)]. Refractivity = 24.500 + 0.261 [M1(G)]. Flash point = 34.273 + .908 [M1(G)]. Polarity = 2.745 + 0.123 [M1(G)]. Molar volume = 49.701 + 0.755 [M1(G)]. Complexity = −135.869 + 2.413 [M1(G)].(6)Regression models for atom bond connectivity index HM(G): Boiling point = 104.181 + 20.696 [HM(G)]. Refractivity = 18.567 + 4.176 [HM(G)]. Flash point = 10.181 + 14.677 [HM(G)]. Polarity = −0.136 + 1.970 [HM(G)]. Molar volume = 32.218 + 12.080 [HM(G)]. Complexity = −181.037 + 38.124 [HM(G)].(7)Regression models for atom bond connectivity index M2(G): Boiling point = 151.218 + .952 [M2(G)]. Refractivity = 30.535 + 0.187 [M2(G)]. Flash point = 48.647 + 0.665 [M2(G)]. Polarity = 5.362 + 0.088 [M2(G)]. Molar volume = 67.501 + 0.539 [M2(G)]. Complexity = −86.964 + 1.741 [M2(G)].(8)Regression models for atom bond connectivity index F(G): Boiling point = 144.280 + 0.380 [F(G)]. Refractivity = 29.397 + 0.074 [F(G)]. Flash point = 44.851 + 0.264 [F(G)]. Polarity = 4.737 + 0.035 [F(G)]. Molar volume = 63.476 + 0.215 [F(G)]. Complexity = −97.844 + 0.692 [F(G)].(9)Regression models for atom bond connectivity index H(G): Boiling point = 147.316 + 0.211 [H(G)]. Refractivity = 29.878 + 0.041 [H(G)]. Flash point = 46.427 + 0.147 [H(G)]. Polarity = 5.004 + 0.020 [H(G)]. Molar volume = 65.21.0 + 0.120 [H(G)]. Complexity = −93.085 + 0.386 [H(G)].Tables 3–11 represent the statistical parameters used in the QSPR models of TIs.

3.2. Quantitative Structure Analysis and Comparison between Topological Indices and Correlation

3.2.1. Coefficient of Physicochemical Properties

Physical properties for ten diabetes drugs are listed in Table 12, and their TIs computed through the molecular structure are recorded in Table 1. The correlation coefficient between six physical properties and TIs is itemized in Table 2. The graph of TIs and physical properties is shown in Figure 2.

3.3. Calculation of Statistical Parameters

In this section, we find the relation between degree base TIs and physical properties of diabetes drugs such as medicines acarbose, tolazamide, miglitol, prandin, metformin, glimepiride, linagliptin, pioglitazone, bromocriptine, and alogliptin, and this is done with the help of QSPR modeling, whereas TIs, N, b, and r represent regression model constant, correlation coefficient, and sample size, respectively. This will be helpful to compare and improve the model. It is noted that the value of r is greater than 0.6, and value is less than 0.05. Hence, the calculation verifies that all properties are significant.

3.4. Standard Error (SE) of Estimate and Comparison

Measure of variation for an observation calculated around the computed regression line is said to be the standard error estimate. It measures the amount of accuracy of predictions done around the computed regression line in Table 13. We also compare the physicochemical properties of the experimental and theoretical calculated values of the models as presented in Tables 14–19.

4. Conclusions

It is obvious from statistical parameters used in linear QSPR models and topological indices that ABC(G) index provides a maximum high correlated value for polarity r = 0.993, refractive index r = 0.961, and molar volume r = 0.960. M1 index provides a high correlated value of boiling point; that is, r = 0.912 and complexity r = 0.947. M2 index depicts the utmost correlation coefficient of flash point r = 0.950 and boiling point r = 0.912.

In this paper, we have computed topological indices and related them to the linear QSPR model for the drugs used to cure diabetes. The results gained in the following means will be supportive for designing various new drugs to attain defensive measures for the said disease in the pharmaceutical industry. The correlation coefficient plays a substantial impact on the range of topological indices of the drugs and may be considered for the combination to designing novel drugs. The outcomes are helpful to research workers on drugs science in the pharmaceutical industry and offer a pathway to approximate physical properties for novice discoveries of diabetes medicines to cure other specific diseases [25].

Data Availability

The data used to support the findings of this study are included within the paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Saima Parveen et al. 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.

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