Degree-based topological indices on anticancer drugs with QSPR analysis.

From last two to three decades, the world is facing the threat of finding treatment for Cancer. This disease is striking almost ten million people every year throughout the world. Anticancer drugs are those which are used to cure malignant disease i.e. Cancer. These anticancer drugs are available in different forms including alkalyting agents, hormones and anti metabolites. Various examinations reveals that, there will be a adjacent relationship between the characteristics of alkanes and the anticancer drugs viz. Boiling point, melting point, enthalpy etc. with their chemical structures. In this proposed work, various topological indices are defined on some anticancer drugs to help the researchers to know the physical characteristics and chemical reaction associated with them. We also discuss the QSPR analysis of thirteen degree based topological indices. Further, we showcase that the characteristics have good correlation with physico-chemical characteristics of anticancer drugs.

Introduction and terminologies

Cancer is the rapid growth of abnormal cells in the human body. Carcinogens are the substances that causes cancer. A carcinogen is a chemical substance with certain molecules in tobacco smoke. It has a potential to spread to other parts of the body. Some of the symptoms of this disease includes lump, abnormal bleeding, prolonger cough, weight loss etc. Main causes for this malignant disease are chewing tobacco, obesity, bad diet, laziness, more intake of alcohol. This dangerous disease can be cured by several treatments like surgery, radiotherapy, chemotherapy, hormone therapy, targeted therapy and more. Anticancer drugs are those which are used to cure the disease so called cancer, which includes alkylates and metabolites. The chemical graph theory is a discipline of mathematical chemistry that deals with the chemical graphs which shows chemical system. The chemical graph theory offers defining topological indices on anticancer drugs. In this work, several drugs are taken and using the degree based calculations, few topological indices are defined on various anticancer drugs to determine physical characteristics and chemical reactions associated with them [1,2,3].

Topological indices are the important attributes to analyse the physico-chemical characteristics of chemical compound structures. There are five different types of topological indices: Degree, distance, eigenvalue, matching and mixed. In this work degree based topological indices are stated on anticancer drugs. Generally, the chemical compound is represented as a graph where the elements denote vertices and the bonds connecting them denote edges. In a similar fashion, these anticancer drugs under this study are considered as chemical compounds and the said topological indices are defined. Graph theory offers some tools like QSAR, QSPR and QSTR where chemists or pharmacists use these data for further research work.

In this work, further we discuss QSPR analysis of said topological indices. We also show that the characteristics obtained are highly correlated with the characteristics of anticancer drugs using linear regression [4,5,6,7,8,9].

In theoretical chemistry drugs are represented as molecular graphs where vertex represents an atom and each edge represents link between the two atoms. Consider G (V, E) be a molecular graph with vertex and edge set respectively. The graphs considered are simple graphs with no cycle formation and multiple edges [4,10,11,12].

Definition 1.1. Estrada et al. in [13] proposed degree-based topological index ABC and defined as

Definition 1.2. Ghorbani et al. in [14] proposed ABC4 index and can be stated as,

Definition 1.3. The Randic index [15] proposed by Milan Randic and can be stated as

Definition 1.4. The sum-connectivity index is proposed by Zhou and Trinjstic [16], and is defined as

Definition 1.5. The GA index is proposed by Vukicevic et al. [17] as

Definition 1.6. The GA5 index proposed by Graovac et al. [18] and is stated as

Definition 1.7. The first and second Zagreb indices are proposed by Gutman and Trinajestic [19], as

Definition 1.8. Fajtlowicz proposed Harmonic index in [20] as,

Definition 1.9. Shirdel et al. in [21] proposed the hyper Zagreb index and is stated as,

Definition 1.10. Fath-Tabar et al. in [22] proposed the third Zagreb index as

Definition 1.11. Furtula et al. in [23] proposed the forgotten topological index and is stated as

Definition 1.12. In chemical graph theory, there are some new degree-based graph types, which plays an important role in finding total surface area and heat-formation of various chemical compounds. These graphs types are as follow Symmetric division index [24],where, P = min [dG(v),dG(w)] and Q = max [dG(v),dG(w)]

Degree based topological indices in QSPR studies

Here we defined 13° based topological indices, Atom-bond connectivity index ABC(G), Fourth atom-bond connectivity index ABC4(G), Randic index χ(G), Sum-connectivity index S(G), Geometric-arithmetic index GA(G), Fifth Geometric arithmetic index GA5(G), First Zagreb index M1(G), Second Zagreb index M2(G), Harmonic index H(G), Hyper Zagreb index HM(G), Third Zagreb index ZG3(G), Forgotten index F(G), Symmetric Division index SSD(G) for modelling Five representative physical properties [Boiling point (BP), Melting point (MP), Enthalpy (E), Flash point (FP), Molar refraction (MR)] of the 17 anticancer drugs from Amathaspiramide-E to Tambjamine-K. The values for these properties are taken from Chem Spider. The above mentioned degree based topological indices and the experimental values for the physical and chemical properties of 17 anticancer drugs (Figure 1) are represented in Tables 1, 2, and 3 respectively.

Figure 1

Molecular structures of anticancer drugs.

Table 1

Various Anticancer drugs with its physico-chemical properties.

S.No.	Drugs	BP	MP	E	FP	MR
1	Amathaspiramide E	572.7	209.72	90.3	300.2	89.4
2	Aminopterin	782.27	344.45			114
3	Aspidostomide E	798.8		116.2	436.9	116
4	Carmustine	309.6	120.99	63.8	141	46.6
5	Caulibugulone E	373	129.46	62	179.4	52.2
6	Convolutamide A	629.9		97.9	334.7	130.1
7	Convolutamine F	387.7	128.67	63.7	188.3	73.8
8	Convolutamydine A	504.9	199.2	81.6	259.2	68.2
9	Daunorubicin	770	208.5	117.6	419.5	130
10	Deguelin	560.1	213.39	84.3	244.8	105.1
11	Melatonin	512.8	182.51	78.4	264	67.6
12	Minocycline	803.3	326.3	122.5	439.6	116
13	Perfragilin A	431.5	187.62	68.7	214.8	63.6
14	Podophyllotoxin	597.9	235.86	93.6	210.2	104.3
15	Pterocellin B	521.6	199.88	79.5	269.2	87.4
16	Raloxifene	728.2	289.58	110.1	394.2	136.6
17	Tambjamine K	391.7		64.1	3190.7	76.6

Table 2

Various Anticancer drugs with Topological Indices values.

Drugs	ABC(G)	ABC4(G)	χ(G)	S(G)	GA(G)	GA5(G)	M1(G)
Amathaspiramide E	10.773	9.079	7.112	7.076	14.403	11.748	70
Aminopterin	24.65	18.96	15.23	15.68	32.700	33.630	162
Aspidostomide E	18.813	11.346	12.35	13.00	17.548	26.906	148
Carmustine	7.847	6.775	5.757	5.482	10.634	10.738	46
Caulibugulone E	10.664	8.342	6.736	6.946	14.574	18.966	72
Convolutamide A	24.463	19.369	17.93	17.74	35.702	34.208	167
Convolutamine F	10.773	8.616	7.113	7.077	14.403	14.599	70
Convolutamydine A	12.016	8.962	7.93	7.544	16.273	15.753	88
Daunorubicin	32.295	22.564	17.89	18.89	40.190	33.116	216
Deguelin	23.398	17.507	13.91	14.80	31.954	32.5264	168
Melatonin	12.865	9.676	8.203	8.419	17.493	17.809	84
Minocycline	26.081	19.093	15.54	16.12	34.271	35.014	184
Perfragilin A	12.992	9.836	7.968	8.171	17.172	17.491	90
Podophyllotoxin	22.02	16.42	12.95	13.86	30.090	30.53	158
Pterocellin B	19.027	11.250	11.69	12.93	26.452	20.788	132
Raloxifene	26.956	20.862	16.58	17.5	37.234	37.684	182
Tambjamine K	14.28	9.654	9.203	9.419	19.493	19.774	92

Table 3

Various Anticancer drugs with Topological Indices values.

Drugs	M2(G)	H(G)	HM(G)	ZG3(G)	F(G)	SSD(G)
Amathaspiramide E	81	6.767	343	12	180	35.667
Aminopterin	185	14.53	786	32	416	80.33
Aspidostomide E	186	11.767	778	22	406	55
Carmustine	48	5.533	202	8	106	25.331
Caulibugulone E	86	6.5	358	10	186	29.5
Convolutamide A	167	17.205	793	21	419	86.583
Convolutamine F	81	6.767	522	12	432	29.167
Convolutamydine A	109	6.738	468	20	250	40.083
Daunorubicin	270	16.919	1146	38	606	101.666
Deguelin	208	13.4	878	28	462	76.166
Melatonin	96	7.933	402	14	210	40.666
Minocycline	229	14.567	970	30	512	89
Perfragilin A	110	7.5	466	16	246	44
Podophyllotoxin	198	12.47	824	22	428	70.66
Pterocellin B	161	11.4	664	16	342	58.999
Raloxifene	215	16.2	890	24	460	83
Tambjamine K	104	8.933	434	14	226	44.667

From the data of above Tables 2 and 3, it has been found that all the data values are normally distributed. Hence the regression model is suitable test to adopt and analyse the data.

Regression models

The above table data shows normally distributed values. Hence the study used regression analysis for the calculation purpose. Here we have checked the linear regression model as belowwhere P is the Physical property of anticancer drug, A is a constant and B is the regression coefficient and TI represents the topological index. These were calculated using SPSS software for the values of five physical properties and the thirteen topological indices of seventeen anticancer drugs.

Using (1), we can get the different linear models for the defined degree based topological indices, which are as follows.

1. Atom-bond Connectivity index ABC(G):

2. Fourth Atom-bond Connectivity index ABC4(G):

3. Randic index χ(G):

4. Sum-Connectivity index S(G):

5. Geometric-Arithmetic index GA(G):

6. Fifth Geometric-Arithmetic index GA5(G):

7. First Zagreb index M1(G):

8. Second Zagreb index M2(G):

9. Harmonic index H(G):

10. Hyper Zagreb index HM(G):

11. Third Zagreb index ZG3(G):

12. Forgotten index F(G):

13. Symmetric Division index SSD(G):

Conclusion, study implications, limitations and future study

Conclusion

The Table 4 and graphs (Figure 2) indicates the correlated values of Physico-chemical properties of anticancer drugs with the defined degree based topological indices. It can be observed that M1(G) = 0.849 index shows higher significant positive correlation with Boiling point (BP), when compared with other indices.

Table 4

Correlation coefficients.

Index	Boiling Point	Melting Point	Enthalpy	Flash Point	Molar Refraction
ABC(G)	0.826	0.726	0.810	0.733	0.913
ABC4(G)	0.789	0.762	0.777	0.679	0.903
χ(G)	0.819	0.767	0.804	0.740	0.941
S(G)	0.821	0.747	0.802	0.735	0.938
GA(G)	0.728	0.745	0.708	0.620	0.872
GA5(G)	0.794	0.779	0.761	0.672	0.889
M1(G)	0.849	0.727	0.836	0.754	0.919
M2(G)	0.844	0.698	0.837	0.749	0.877
H(G)	0.806	0.756	0.788	0.723	0.941
HM(G)	0.827	0.663	0.818	0.737	0.895
ZG3(G)	0.837	0.731	0.810	0.735	0.797
F(G)	0.744	0.559	0.730	0.664	0.841
SSD(G)	0.815	0.767	0.804	0.720	0.904

The significance of bold numbers denote highest correlation value.

Figure 2

Physico-chemical properties with topological indices.

Similarly GA5(G) = 0.779 index gives positive correlated value with melting point (MP). In case of enthalpy, M2(G) shows highest correlated value i.e. r = 0.837.

Flashing point (FP) offers highest correlated value of 0.754 from the physico-chemical properties.

Based on molar refraction (MR), χ(G) and M2(G) indices depicts highest positive correlation value i.e. r = 0.941.

Hence it can be remarked that all the physical and chemical properties of anticancer drugs are positively correlate with the defined degree based topological indices.

Tables 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17 shows the regression model of various physico-chemical properties. It can be observed that the regression model value r is more than 0.6 and p value shows less than 0.05. Hence it can be concluded that all the physico-chemical properties are highly significant.

Table 5

Statistical parameters for the linear QSPR model For ABC(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	232.702	18.457	0.826	32.119	0.000	significant
Melting Point	15	97.481	6.385	0.726	13.381	0.003	significant
Enthalpy	16	46.017	2.307	0.810	26.772	0.000	significant
Flash Point	16	105.864	9.791	0.733	16.221	0.001	significant
Molar Refraction	17	27.349	3.590	0.913	75.573	0.000	significant

The significance of bold numbers denote highest correlation value.

Table 6

Statistical parameters for the linear QSPR model For ABC4(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	242.956	24.290	0.789	24.649	0.000	significant
Melting Point	15	84.818	9.517	0.762	16.580	0.002	significant
Enthalpy	16	46.834	3.081	0.777	21.267	0.000	significant
Flash Point	16	114.955	12.646	0.679	11.977	0.004	significant
Molar Refraction	17	27.134	4.889	0.903	66.079	0.000	significant

The significance of bold numbers denote highest correlation value.

Table 7

Statistical parameters for the linear QSPR model For χ(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	207.524	31.676	0.819	30.638	0.000	significant
Melting Point	15	75.233	12.437	0.767	17.167	0.001	significant
Enthalpy	16	42.806	3.966	0.804	25.674	0.000	significant
Flash Point	16	89.052	17.119	0.740	16.907	0.001	significant
Molar Refraction	17	19.758	6.347	0.941	116.4160.000	0.000	significant

The significance of bold numbers denote highest correlation value.

Table 8

Statistical parameters for the linear QSPR model For S(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	219.568	29.619	0.821	31.014	0.000	significant
Melting Point	15	86.175	11.026	0.747	15.172	0.002	significant
Enthalpy	16	45.580	3.682	0.802	25.326	0.000	significant
Flash Point	16	97.414	15.829	0.735	16.448	0.001	significant
Molar Refraction	17	22.568	5.950	0.938	109.935	0.000	significant

The significance of bold numbers denote highest correlation value.

Table 9

Statistical parameters for the linear QSPR model For GA(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	276.572	12.115	0.728	16.912	0.001	significant
Melting Point	15	90.640	5.053	0.745	14.93	0.002	significant
Enthalpy	16	51.708	1.5	0.708	14.098	0.002	significant
Flash Point	16	134.852	6.163	0.620	8.735	0.010	Significant
Molar Refraction	17	31.169	2.552	0.872	47.644	0.000	Significant

The significance of bold numbers denote highest correlation value.

Table 10

Statistical parameters for the linear QSPR model For GA5(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	226.669	14.157	0.794	25.640	0.000	Significant
Melting Point	15	81.229	5.563	0.779	18.506	0.001	Significant
Enthalpy	16	45.953	1.745	0.761	19.266	0.001	Significant
Flash Point	16	109.675	7.234	0.672	11.526	0.004	Significant
Molar Refraction	17	25.435	2.784	0.889	56.261	0.000	Significant

The significance of bold numbers denote highest correlation value.

Table 11

Statistical parameters for the linear QSPR model For M1(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	232.771	2.686	0.849	38.616	0.000	Significant
Melting Point	15	100.234	0.913	0.727	13.472	0.003	Significant
Enthalpy	16	46.094	0.334	0.836	32.541	0.000	Significant
Flash Point	16	106.537	1.414	0.754	18.503	0.001	Significant
Molar Refraction	17	28.756	0.511	0.919	81.435	0.000	Significant

The significance of bold numbers denote highest correlation value.

Table 12

Statistical parameters for the linear QSPR model For M2(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	250.418	2.138	0.844	37.243	0.000	Significant
Melting Point	15	109.887	0.692	0.698	11.405	0.005	Significant
Enthalpy	16	48.129	0.266	0.837	32.841	0.000	Significant
Flash Point	16	116.671	1.115	0.749	17.849	0.001	Significant
Molar Refraction	17	34.562	0.391	0.877	50.156	0.000	Significant

The significance of bold numbers denote highest correlation value.

Table 13

Statistical parameters for the linear QSPR model For H(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	218.618	32.191	0.806	27.735	0.000	Significant
Melting Point	15	79.495	12.656	0.756	16.024	0.002	Significant
Enthalpy	16	44.375	4.011	0.788	22.892	0.000	Significant
Flash Point	16	96.112	17.286	0.723	15.343	0.002	Significant
Molar Refraction	17	20.814	6.610	0.941	115.783	0.000	Significant

The significance of bold numbers denote highest correlation value.

Table 14

Statistical parameters for the linear QSPR model For HM(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	242.453	0.508	0.827	32.364	0.000	Significant
Melting Point	15	109.795	0.161	0.663	9.413	0.010	Significant
Enthalpy	16	47.199	0.063	0.818	28.407	0.000	Significant
Flash Point	16	111.462	0.267	0.737	16.689	0.001	Significant
Molar Refraction	17	30.608	0.097	0.895	60.345	0.000	Significant

The significance of bold numbers denote highest correlation value.

Table 15

Statistical parameters for the linear QSPR model For ZG3(G).

Physical Properties	N	A	b	r	F	p	Indicator
Boiling Point	17	247.098	16.151	0.837	35.057	0.000	Significant
Melting Point	15	102.564	5.462	0.731	13.733	0.003	Significant
Enthalpy	16	47.108	2.087	0.810	26.767	0.000	Significant
Flash Point	16	109.841	8.890	0.735	16.490	0.001	Significant
Molar Refraction	17	38.887	2.703	0.797	26.072	0.000	Significant

The significance of bold numbers denote highest correlation value.

Table 16

Statistical parameters for the linear QSPR model For F(G).

Physical Properties	N	a	b	r	F	p	Indicator
Boiling Point	17	269.670	0.865	0.744	18.0619	0.001	Significant
Melting Point	15	123.82	0.257	0.559	5.462	0.038	Significant
Enthalpy	16	50.889	0.106	0.730	15.942	0.001	Significant
Flash Point	16	125.652	0.453	0.664	11.012	0.005	Significant
Molar Refraction	17	33.304	0.172	0.841	36.186	0.000	Significant

The significance of bold numbers denote highest correlation value.

Table 17

Statistical parameters for the linear QSPR model For SSD(G).

Physical Properties	N	a	b	r	F	p	Indicator
Boiling Point	17	253.080	5.425	0.815	29.571	0.000	Significant
Melting Point	15	94.601	2.054	0.767	17.119	0.001	Significant
Enthalpy	16	48.291	0.683	0.804	25.633	0.000	Significant
Flash Point	16	117.165	2.87	0720	15.055	0.002	Significant
Molar Refraction	17	31.147	1.058	0.904	66.734	0.000	Significant

The significance of bold numbers denote highest correlation value.

Study implications

The work implies that these anti-cancer drugs may be considered for further study by pharmacists and chemists in designing the drugs using these topological indices values. May be the composition of these drugs, like the combinations may be tried for different ailments based on the range of the topological indices that are determined in the study. As the correlation coefficient has been found for the topological indices, the positively high correlated drugs may be considered for the combination of design of novel drugs.

Limitations

As the range of topological indices are not published by chemists anywhere in web/internet, the mathematicians may not be able to decide upon the values they obtain for different chemical compounds whether the compounds the researchers chose have future study or not. The best solution for this would be a joint venture of the study in future may be carried out by both mathematicians/statisticians and chemists/pharmacists.

Future study

In a similar fashion, a study may be carried out for different chemical structures and a conclusion may be given based on their topological indices range. May it be benzene structure or polymers or any chemical compounds can be taken for future study. A multidisciplinary project may be taken up by various disciplines researchers for a better result.

Declarations

Author contribution statement

Shanmukha M C, Basavarajappa N S, Shilpa K C, Usha A: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Competing interest statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.