Topological indices and QSPR/QSAR analysis of some antiviral drugs being investigated for the treatment of COVID-19 patients.

The spread of novel virus SARS-CoV-2, well known as COVID-19 has become a major health issue currently which has turned up to a pandemic worldwide. The treatment recommendations are variable. Lack of appropriate medication has worsened the disease. On the basis of prior research, scientists are testing drugs based on medical therapies for SARS and MERS. Many drugs which include lopinavir, ritonavir and thalidomide are listed in the new recommendations. A topological index is a type of molecular descriptor that simply defines numerical values associated with the molecular structure of a compound that is effectively used in modeling many physicochemical properties in numerous quantitative structure-property/activity relationship (QSPR/QSAR) studies. In this study, several degree-based and neighborhood degree sum-based topological indices for several antiviral drugs were investigated by using a M-polynomial and neighborhood M-polynomial methods. In addition, a QSPR was established between the various topological indices and various physicochemical properties of these antiviral drugs along with remdesivir, chloroquine, hydroxychloroquine and theaflavin was performed in order to assess the efficacy of the calculated topological indices. The obtained results reveal that topological indices under study have strong correlation with the physicochemical characteristics of the potential antiviral drugs. A biological activity (pIC50) of these compounds were also investigated by using multiple linear regressions (MLR) analysis.

INTRODUCTION

On 31st December 2019, 27 cases of pneumonia of unknown etiology were reported by the Wuhan health commission in the Hubei province of the Republic of China. The symptoms witnessed in these patients were fever, dyspnea, dry cough and bilateral lung opacities [1]. Dense population, busy marketplaces and lack of early containment advocated the spread of infection [2]. Within a short span of time, by 11 March 2020, WHO declared it as a global pandemic.

The causative agent of SARS‐ CoV‐2 belongs to betacoronavirus family. MERS and SARS virus belong to the same family. Phylogenetic analysis of the SARS‐CoV‐2 genome reveals that it is closely related to bat‐derived‐SARS like coronaviruses (bat‐SL‐CoVZC45 and bat‐SL‐CoVZXC21) and genetically distinct from SARS‐CoV and MERS‐CoV [3]. The reports on this disease are continuously evolving and are subject to selection bias. Like other members of the coronavirus family, betacoronavirus exhibits high species specificity, but it can greatly alter its tissue tropism, host selection and pathogenicity [4, 5]. It has been reported by Forbes on 31st July 2020, children under 5 years of age or younger who developed mild to moderate symptoms had 10 to 100 times more SARS‐CoV‐2 in nasopharynx as compared to that found in older children and adults (July 31, 2020 Forbes).

Currently, there is no available beneficial antiviral treatment against the disease, as in the case of MERS‐CoV and SARS‐CoV [6]. However, many drugs are being tested based on previous studies in order to pave way for an effective treatment against SARS and MERS [7]. Although there are certain antiviral drugs available in the market but there is no conformity about their efficiency for COVID‐19 infection. Lopinavir (LPV) and ritonavir (RTV) are protease inhibitors that target papain like protease and 3C‐like protease of corona viruses. A study demonstrated decline in deaths among infected patients receiving LPV/RTV in the last stage [8]. Clinical trials are being conducted in order to investigate safety and efficacy of lopinavir‐ritonavir and interferon‐α2b among infected patients [9]. Moreover, it has been reported that Russia and China are using arbidol hydrochloride, which is a broad‐spectrum inhibitor of Influenza A and B viruses, parainfluenza and Hepatitis C virus, but it has not yet gained approval in other countries for coronavirus infection [10]. In addition, the anti‐inflammatory and immunomodulatory properties of thalidomide makes this drug a potential candidate for the treatment of complications of infected patients [11]. A brief study shows that a combination therapy of lopinavir/ritonavir, arbidol and Shufeng Jiedu Capsule (SFJDC), a conventional Chinese medication can be a potent drug as it has benifited three out of four COVID‐19 patients clinically. Although the guidelines for treating infected patients are not yet established hydroxychloroquine, chloroquine phosphate, remedesivir, lopinavir/ritonavir and thalidomide are being used for treatment purposes [10, 12, 13].

Drug discovery process encompassing a series of events starting from identification of a novel, fascinating chemical entity to regulatory approval, is a multifaceted, expensive, and onerous process. In traditional paradigms, synthesis of compounds and biological screening involved a significant time delay due to inefficient usage of data in the iterative molecular design practice, compelling scientific communities to find lead compounds promptly and efficiently [14]. In the postgenomic era, application of computer‐aided drug design (CADD) approaches offers an alternative to the traditional hit‐and‐trial methods of design, synthesis as well as biological screening of the compounds [15, 16]. In particular, quantitative structure–property/activity relationship (QSAR/QSPR) modeling endeavors to mathematically predict or relate a molecule's physicochemical, pharmacokinetic and pharmacodynamic properties to its molecular structural features such as functional groups, and moieties. The authenticity of a QSPR/QSAR models is based on their capability to envisage physicochemical or pharmacological properties for new chemical entities with a high degree of accuracy. A robust predictive ability of these models relies on several factors such as: i) selection of the molecules in the training set; ii) the set of chosen descriptors; and iii) the algorithm or methodology employed in model development.

A descriptor represents molecular characteristics in terms of a number that describes the molecule. There are innumerable kinds of descriptors such as constitutional, steric, geometric, electrostatic, quantum chemical, lipophilic, electronic and topological indices. Among these, topological indices represent graph invariants that encrypt the topology of compounds depicted as molecular graphs after suppressing hydrogen atoms. [17] Interestingly, topological descriptors are routinely exploited in various drug discovery programs because they are information‐rich, quickly calculated and possess robust predictive power [18, 19]. Recently, several topological indices were used to develop a QSAR model by Sarkar et al for predicting DNA‐binding constant and growth‐inhibiting concentrations of 23 anthracycline drugs [20]. In another report, a set of 17 anticancer drugs from amathaspiramide‐E to tambjamine‐K were studied in terms of QSPR by using 13 degree‐based topological indices [21]. Recently, inspired by Qamar et al. [22] Hosamani consider 12 phytochemicals screened against SARS‐CoV‐2 3CLpro and develop QSPR model to predict Docking score, Binding energy, Molecular weight and Topological polar surface through various topological indices [23].. Very recently Rafi et al. used combination of QSAR, molecular docking, molecular dynamic simulation to study analogues of lopinavir and favipiravir [24].

In order to compute topological indices, a chemical compound is transformed into a graph that considers atoms as vertices and bonds as edges, known as a molecular graph. Let be a molecular graph with vertex set and edge set . We denote the number of vertices and edges in a graph G by | and | respectively. The degree of vertex is denoted by and is the number of vertices that are adjacent to and denote the degree sum of all vertices of G that are adjacent to u known as neighborhood degree sum of u in G. The edge connecting the vertices and is denoted by , where [25].

Topological index (Wiener index) was first used by Wiener in QSPR who showed that his index was well aligned with the boiling points of the alkane [26]. Subsequently, Wiener index has been used to explain various chemical, physical properties of molecules and to correlate the structure of molecules with their biological activity [27]. Randić index, one of the oldest, most popular and most effectively used topological index in QSPR and QSAR [28, 29, 30, 31, 32]. Later Bollabas and Erdos introduced generalized Randić index [33]. Harmonic index is an another variant of the Randić index first introduced by Fajtlowicz [34]. First and Second Zagreb indices are among the oldest topological indices proposed by Gutman and Trinajstic in 1972 and were determined to be efficient in calculating the total π‐electron energy of the molecules [35]. Later is modified as the second modified Zagreb index [36]. Ranjini et al. [37] redefined the Zagreb indices as the redefined first, second and third Zagreb indices for a graph G. The augmented Zagreb index provides the best estimation of the heat of formation of alkanes [38]. The notion of F‐index or Forgotten index of a graph was introduced by Gutman et al. [35] in 1972 and is further reinvestigated by Furtula and Gutman [39]. It was shown that the predictive ability of this index is almost similar to that of first Zagreb index. For more Zagreb index refer to [40, 41, 42, 43]. The symmetric division index (SDD) is a good indicator of the aggregate surface area for polychlorobiphenyls (PCBs) and the inverse sum index, is also a significant indicator of the aggregate surface area of octane isomers for which the acquired extremal graphs have a particularly simple and rich structure [44, 45].

A topological indices based on neighborhood degree sum‐based topological indices can predict different physicochemical properties with powerful accuracy [46, 47]. Gorbani et al [48] define a new type of Zagreb index known as The Third Version of Zagreb index based on degrees of neighbors of vertices in a given graph. Later, motivated by augmented Zagreb index, Hosamani [49] defined the Sanskruti index S(G) that shows good correlation with entropy of an octane isomers. We encourage readers to consult [46, 50, 51, 52] for several other neighborhood degree sum‐based topological indices defined in parallel with their corresponding classical degree based topological indices.

Various graph polynomials have been presented in the literature to attenuate the laborious approach of computing various types of graph indices. For instance, the Hosoya polynomial, PI Polynomial, Schultz polynomial, modified Schultz polynomial etc. are example of distance based polynomials [53, 54, 55, 56]. This prompts researchers to establish polynomials based on the valence of atoms of chemical compounds which would provide a closed formula for degree based topological indices that are routinely obtained. In this direction, the M‐polynomial was introduced by Deutsch and Klavzar in 2015 [57] to compute the various degree based topological indices. Due to its wide applicability it has been used in various papers to derive formulas of degree based topological indices [58, 59, 60, 61, 62, 63, 64]. The Duo [65] recently proposed a new approach to determine M‐polynomial of chemical graph in which each vertex has only degree 2 or 3. This approach is very convenient for many important families of chemical graphs such as benzeoid graphs, phenylenes, fluoranthenes and fullerenes.

The M‐polynomial of graph G is defined as follows

A polynomial that plays the same roles as of M‐polynomial for neighborhood degree sum‐based indices is known as Neighborhood M‐polynomial (NM‐Polynomial) [52, 66] and is defined as

Some neighborhood degree sum based topological indices on for bismuth tri‐iodide chain and sheet are derived using NM‐polynomial method [52].

The Table 1 relates various degree based and neighborhood degree based topological indices with M‐polynomial and NM‐polynomial.

TABLE 1

Description of topological indices and its derivation from M‐polynomial and NM‐polynomial

Topological index	Formula g(χ(u), χ(v))	Derivation from f(x, y) f(x, y) = M(G; x, y) or NM(G; x, y)
First Zagreb index: M 1(G) Third version Zagreb index: NM 1(G)	∑uv∈EGχu+χv	(D x  + D y )(f(x, y)) x = y = 1
Second Zagreb index: M 2(G) Neighborhood second Zagreb index: NM 2(G)	∑uv∈EGχuχv	(D x D y )(f(x, y)) x = y = 1
Second modified Zagreb index: mM 2(G) Neighborhood Second modified Zagreb index: NmM 2(G)	∑uv∈EG1χuχv	(S x S y )(f(x, y)) x = y = 1
Redefined third Zagreb index: ReZG 3(G) Third NDe index: ND 3(G)	∑uv∈EGχuχvχu+χv	D x D y (D x  + D y )(f(x, y)) x = y = 1
Forgotten topological index: F(G) Neighborhood Forgotten topological index: NF(G)	∑uv∈EGχ^2u+χ^2v	(D x 2 + D y 2)(f(x, y)) x = y = 1
Randić index: R k (G) Neighborhood Randić index: NR k (G)	∑uv∈EGχuχv^k	(D x k D y k )(f(x, y)) x = y = 1
Inverse Randić index: RR k (G) Neighborhood Inverse Randić index: NRR k (G)	∑uv∈EG1χuχv^k	(S x k S y k )(f(x, y)) x = y = 1
Symmetric Division index: SDD (G) Fifth NDe index: ND 5(G)	∑uv∈EGχ^2u+χ^2vχuχv	(D x S y  + S x D y )(f(x, y)) x = y = 1
Harmonic Index: H(G) Neighborhood Harmonic Index: NH(G)	∑uv∈EG2χu+χv	(2S x J)(f(x, y)) x = 1
Inverse sum indeg Index: I(G) Neighborhood Inverse sum Index: NI(G)	∑uv∈EGχuχvχu+χv	(S x JD x D y )(f(x, y)) x = 1
Augmented Zagreb Index: A(G) Sanskruti Index: S(G)	∑uv∈EGχuχvχu+χv−2^3	(S x 3 Q −2 JD x 3 D y 3)(f(x, y)) x = 1
Where D_x=x∂fx,y∂x,D_y=y∂fx,y∂y,S_x=∫0xft,ytdt,S_y=∫0yfx,ttdt,Jfx,y=fx,x,Q_kfx,y=x^kfx,y For degree based topological indices: χ(u) = d(u), χ(v) = d(v), f(x, y) = M(G; x, y) For neighborhood degree sum based topological indices: χ(u) = nd(u), χ(v) = nd(v), f(x, y) = NM(G; x, y)

First Zagreb index:

Third version Zagreb index:

Second Zagreb index:

Neighborhood second Zagreb index:

Second modified Zagreb index:

Neighborhood Second modified Zagreb index:

Redefined third Zagreb index:

Third NDe index:

Forgotten topological index:

Neighborhood Forgotten topological index:

Randić index:

Neighborhood Randić index:

Inverse Randić index:

Neighborhood Inverse Randić index:

Symmetric Division index: (G)

Fifth NDe index:

Harmonic Index:

Neighborhood Harmonic Index:

Inverse sum indeg Index:

Neighborhood Inverse sum Index:

Augmented Zagreb Index:

Sanskruti Index:

For degree based topological indices:

For neighborhood degree sum based topological indices:

Recently Mondal S. et al. [67] investigated four antiviral drugs namely Remedesivir, Clioroquine, Hydroxychloroqine and Theaflavin applied for the treatment of COVID‐19 patients using M‐polynomial and NM‐polynomial approach. This motivates us to investigate more on molecular structure of antiviral drugs for the treatment of COVID‐19 such as Lopinavir, Ritonavir, Arbidol and Thalidomide. Thus in this article, we obtain M‐polynomial and NM‐polynomials of Lopinavir, Ritonavir, Arbidol and Thalidomide and retrieve some degree‐based and neighborhood‐degree sum‐based indices from these polynomial. We also test predictive strength of these degree‐based and neighborhood‐degree sum‐based topological indices by using some physicochemical properties and biological activity of the drugs under investigation together with Chloroquine, Hydroxychloroqine, Theaflavin and Remedesivir. The chemical structure of these drugs are shown in Figure 1.

FIGURE 1

The molecular structure of Lopinavir, ritonavir, Arbidol, thalidomide

METHODOLOGY AND MAIN RESULTS

In this section we obtain the expressions for M‐polynomials and NM‐polynomials of molecular graphs of Lopinavir ( ), Ritonavir ( ), Arbidol ( ) and Thalidomide ( ) using combinatorial computation, edge partition technique, degree and neighborhood degree counting method. In addition, we make use of Chemsketch for plotting the molecular graphs and SPSS for correlation analysis. We also deduce many well‐known degrees based and neighborhood degree sum based topological indices for the above mentioned molecular graphs.

For the rest of the paper,

We start with establishing the expression of M‐polynomial and NM‐polynomial for Lopinavir.

Let be the molecular graph of Lopinavir, then

Proof: The molecular graph ( ) of Lopinavir is shown below in Figure 2.

FIGURE 2

Corresponding molecular graph of Lopinavi

From the Figure we have | and |

By means of structure analysis, the edge set of can be divided into four classes on the basis of degree of vertices as follows

and

By definition of M‐polynomial

Similarly on the basic of neighborhood degree sum of vertices, the edge set of can be divided into 13 classes as follows

Using definition of NM‐polynomial

Hence the result.

Surface representation of M‐polynomials and NM‐polynomials for Lopinavir is shown below in Figure 3.

FIGURE 3

Plot of M‐polynomial and NM‐polynomial for Lopinavir

Now we recover some degree‐based and neighborhood degree sum‐based topological indices of the molecular graph of Lopinavir in the following proposition using Theorem 1 and Table 1.

Let be the molecular graph of Lopinavir

1. .

2. 1435.

5.

11. 1495.2289

Proof:

Let, , then

(

Now, using Table 1 we have

1. .

2. .

6. .

Again, let

Now, applying the above operations and using Table 1, we obtain the neighborhood degree sum‐based indices. Hence the results.

In the next theorem, the M‐polynomial and NM‐polynomial for the molecular graph ( ) of Ritonavir is derived.

Let be the molecular graph of Ritonavir. Then

Proof: Let be the molecular graph of Ritonavir as shown below in Figure 4, we have | and |.

FIGURE 4

Corresponding molecular graph of ritonavir

On the basic of degree of vertices, the edge partition of the molecular graph is obtained as and |

Using definition of M‐polynomial

The edge set of can be divided into 11 classes on the basic of neighborhood degree sum of vertices as follows

From the of definition of NM‐polynomial we have

Hence the theorem.

Surface representation of M‐polynomials and NM‐polynomials for Ritonavir is shown below in Figure 5.

FIGURE 5

Plot of M‐polynomial and NM‐polynomial for ritonavir

Now, in the following proposition, we can retrieve different degree‐based and neighborhood degree sum‐based indices for molecular structure of Ritonavir just as we followed in Proposition 1

Let be the molecular graph of Ritonavir

5.

In the following theorem we establish the results for M‐polynomal and NM‐polynomial of molecular graph ( ) of Arbidol

Let be the molecular graph of Arbidol. Then

Proof: We have | and | from the molecular graph ( ) of Arbidol as shown in Figure 6.

FIGURE 6

Corresponding molecular graph of Arbidol

By means of structure analysis the edge set of , we have

The M‐polynomial can easily be evaluated like earlier

Likewise, on the basic of neighborhood degree sum of vertices, the edge set of can be divided into following classes

Using definition of NM‐polynomial and above edge partition we have

This gives the theorem.

Surface representation of M‐polynomials and NM‐polynomials for Arbidol is shown below in Figure 7.

FIGURE 7

Plot of M‐polynomial and NM‐polynomial of Arbidol

From the above theorem one can immediately compute the following preposition

Let be the molecular graph of Arbidol

1. .

2. 1112

4. =14 910.

5. 392, 2328

11. A(D) = 250.76563, S(D) = 1478.720039

In the theorem below we calculate M‐polynomial and NM‐polynomial of the molecular graph ( ) of Thalidomide.

Let be the molecular graph of Thalidomide. Then

Proof: The molecular graph (T) of Thalidomide is described as below in Figure 8.

FIGURE 8

Corresponding molecular graph of thalidomide

From the molecular graph (T) of Thalidomide, we have |, | and |

The M‐polynomial can be evaluated as earlier similarly, on the basic of neighborhood degree sum of vertices, the edge set of can be divided into following classes

Using definition of NM‐polynomial and above edge partition we have

This completes the proof. Surface representation of M‐polynomials and NM‐polynomials for Thalidomide is shown below in Figure 9.

FIGURE 9

Plot of M‐polynomial and NM‐polynomial of thalidomide

Now in the light of the above theorem we yield the following results.

Let be the molecular graph of Thalidomide

1. .

4. ReZG 3(T) = 670 , ND 3(T) = 10628.

5. ,

DEGREE BASED AND NEIGHBORHOOD DEGREE SUM BASED TOPOLOGICAL INDICES AND QSPR/QSAR

The main objective of this section is to establish quantitative structure–property/activity relationship (QSPR/QSAR) between the various topological indices and some physicochemical properties/activity of the drugs understudy in order to determine the efficacy of the topological indices. Ten degree‐based and 10 neighborhood degree sum‐based topological indices were used for modeling antiviral activity and eight representative physicochemical properties such as boiling point (BP), enthalpy of vaporization (E), flash point (FP), molar refraction (MR), polar surface area (PSA), polarizability (P), surface tension (T), and molar volume (MV) of several drugs currently being investigated for the treatment of COVID‐19 which includes lopinavir, ritonavir, arbidol, thalidomide, chloroquine, hydroxychloroquine, theaflavin and remdesivir. However, thalidomide was excluded from QSAR study because of unavailability of the antiviral activity data.

The values for the various physicochemical properties, as presented in Table 2, were obtained from ChemSpider whereas IC50 of antiviral activity were collected from literature and converted to their logarithmic scale (pIC50) [68, 69, 70, 71]. In Table 3 we enlist the measured values of the degree‐based topological indices of the drugs obtained in the prepositions 1–4 (Lopinavir, Ritonavir, Arbidol, Thalidomide) and reported in Mondal S. et al. (Chloroquine, Hydroxychloroquine, Theaflavin and Remdesivir) [67].

TABLE 2

Various COVID‐19 drugs with its physicochemical properties and biological activity

Drugs	BP	E	FP	MR	PSA	P	T	MV	Antiviral activity a
									IC50	pIC50
Lopinavir	924.2	140.8	512.7	179.2	120	71.0	49.5	540.5	5.2568	5.28
Ritonavir	947.0	144.4	526.6	198.9	202	78.9	53.7	581.7	8.6369	5.06
Arbidol	591.8	91.5	311.7	121.9	80	48.3	45.3	347.3	3.5468	5.45
Thalidomide	487.8	79.4	248.8	65.2	87	25.9	71.6	161	—	—
Chloroquine	460.6	72.1	232.3	97.4	28	38.6	44.0	287.9	1.3868	5.86
Hydroxy‐chloroquine	516.7	83.0	266.3	99.0	48	39.2	49.8	285.4	0.7270	6.14
Theaflavin	1003.9	153.5	336.5	137.3	218	54.4	138.6	301.0	8.4471	5.07
Remdesivir	—	—	—	149.5	213	59.3	62.3	409	0.9968	6.01

IC50, half maximal inhibitory concentration; pIC50, negative log of the IC50.

TABLE 3

Various COVID‐19 drugs with degree based topological indices values

Drugs	M 1	M 2	mM 2	ReZG 3	F	SDD	H	I	A	R ‐1/2
Lopinavir	230	255	10.2778	1298	578	112	21.33	52.5	378.73	22.117
Ritonavir	248	255	11.14	1366	622	122.33	23.07	136.857	399.328	23.977
Arbidol,	150	179	6.5	930	392	76	13.27	35.4667	250.766	13.845
Thalidomide	104	127	5.286	670	276	48.333	8.733	24.7	173.234	9.092
Chloroquine	106	120	5.189	584	262	51.66	10.3	25.23	181.5	10.635
Hydroxy‐chloroquine	110	124	5.389	600	270	53.66	5.4	26.2	189.5	11.135
Theaflavin	234	288	8.556	1536	638	109	9.2	54.9	369.2	19.422
Remdesivir	216	257	9.08	1360	586	104.6	18.6	50.2	349.6	19.508

Table 4 indicates the values of correlation coefficient ( ) of physicochemical properties of eight COVID‐19 drugs with the defined degree based topological indices. It can be easily seen from the Table 4 that F‐index depicts strong positive correlation value (i.e. r = 0.992) with boiling point (BP) among all correlations. Figure 10 shows the corelation of F‐index with boiling point and enthalpy of vaporization. It can also be seen from Table 4 that the most convenient indices which are modeling the Boiling point (BP), Enthalpy(E), Flash point(FP), Molar refraction(MR), Polar Surface Area(PSA), Polarizability(P), Surface Tension(T), Molar Volume (MV)) are

TABLE 4

The correlation between degree based topological indices and physicochemical properties of various COVID‐19 drugs

Index	Boiling point (BP)	Enthalpy of vaporization: (E)	Flash point: (FP)	Molar refractivity (MR)	Polar surface area (PSA)	Polarizability (P)	Surface tension (T)	Molar volume (MV)
M 1	0.98683199	0.982405261	0.87621221	0.9121133	0.88000812	0.912249712	0.3598431	0.78463
M 2	0.98873009	0.985803347	0.77642452	0.82173094	0.90906601	0.821764922	0.4906564	0.661054
mM 2	0.92306794	0.917936354	0.96942137	0.96171783	0.79492295	0.962058072	0.1445281	0.891374
ReZG 3	0.98454636	0.982482786	0.76297569	0.8064583	0.92964965	0.806608262	0.5115195	0.639382
F	0.99203561	0.988798465	0.82466154	0.86305899	0.91657928	0.86322946	0.4360715	0.714623
SDD	0.97168741	0.965580433	0.93853029	0.93683624	0.85794379	0.936971325	0.2940312	0.823584
H	0.6296994	0.615242474	0.92584236	0.87721859	0.5370734	0.87792357	−0.276805	0.910836
I	0.70115506	0.697768574	0.80514313	0.81799007	0.65413774	0.818902027	0.0424164	0.774275
A	0.98093969	0.975693186	0.89456515	0.92765438	0.86028234	0.927750567	0.3196081	0.81025
R ‐1/2	0.946564	0.93967	0.9423247	0.9709078	0.802397	0.97102601	0.19241	0.886175

FIGURE 10

Correlation of F‐index with boiling point (BP) ad enthalpy of vaporization (E)

The F‐ index for Boiling point (BP) and Enthalpy of vaporization (E)

The Second modified Zagreb index for Flash point (FP).

Randić index for Molar refractivity(MR) and Polarizability (P)

The Redefined third Zagreb index for Polar surface area (PSA).

The Harmonic index for Molar volume (MV).

No topological index shows good correlation with Surface tension (T).

In similar fashion, using the prepositions 1–4 and results reported in Mondal S. et al. [67] we list the calculated values of the neighborhood degree sum based topological indices for the drugs in Table 5.

TABLE 5

Various COVID‐19 drugs with neighborhood degree sum based topological indices values

Drugs	NM 1	NM 2	NmM 2	Third NDe	NF	Fifth NDe	NH	NI	S	NR ‐1/2
Lopinavir	525	1435	1.8998	16 998	3009	126.7269	9.5357	127.9398	1495.2289	12.44419
Ritonavir	558	1485	2.1508	16 172	3078	109.7310	10.3748	136.8574	1827.7316	10.50131
Arbidol,	366	1112	1.3574	14 910	2328	70.4679	6.1450	89.0003	1478.7200	6.26086
Thalidomide	254	789	0.7139	10 628	1579	45.3588	3.6789	61.4830	1035.639	3.7575592
Chloroquine	240	665	1.052	7408	1342	48:704	4.662	58:626	802:238	4.745946
Hydroxy‐chloroquine	248	664	1.152	7600	1378	50:504	1.227	60.695	827.9	5.0373899
Theaflavin	576	1826	1.391	24 440	3810	98.409	3.819	140.301	1971:625	7.7958654
Remdesivir	514	1543	1.65	20 122	3266	95:236	8.01	124.363	1996.984	8.08357

Table 6 contains the correlated values of physicochemical properties of eight COVID‐19 drugs with the defined neighborhood degree sum based topological indices. It can be observed from the Table 6 that Neighborhood Second modified Zagreb index reflects a strong positive correlation value (i.e., r = 0.994) with Molar refractivity(MR) and Polarizability (P) among all correlations, depicted in Figure 11. It is easily visible from Table 6 that the most convenient indices which are modeling the Boiling point (BP), Enthalpy(E), Flash point(FP), Molar refraction(MR), Polar Surface Area(PSA), Polarizability(P), Surface Tension(T), Molar Volume(MV)) are

TABLE 6

The correlation between neighborhood degree sum based topological indices and physicochemical properties of various COVID‐19 drugs

Index	Boiling point (BP)	Enthalpy of vaporization: (E)	Flash point: (FP)	Molar refractivity (MR)	Polar surface area (PSA)	Polarizability (P)	Surface tension (T)	Molar volume (MV)
NM 1	0.98998544	0.986429746	0.82299836	0.86523802	0.90530378	0.865357016	0.4392171	0.716849
NM 2	0.96241305	0.960727236	0.68945929	0.74688044	0.92592471	0.746976459	0.5775063	0.563534
NmM 2	0.79972934	0.787580724	0.95173756	0.99427464	0.62774418	0.994296858	−0.098138	0.97877
Third NDe	0.88304124	0.883429492	0.52756156	0.59396178	0.89595364	0.594001606	0.6682426	0.388835
NF	0.96292531	0.960934598	0.69280217	0.7506981	0.92309504	0.750763856	0.5654859	0.569431
Fifth NDe	0.8981797	0.888619462	0.90132539	0.92729886	0.55967275	0.926760237	−0.0253388	0.870262
NH	0.60376318	0.58734941	0.88738981	0.84990145	0.51235015	0.850632969	−0.27003	0.882473
NI	0.990147297	0.986528991	0.82582862	0.86957692	0.90304555	0.869691721	0.43647729	0.7220741
S	0.908134765	0.905760929	0.67836824	0.71727292	0.94980194	0.717757979	0.47782155	0.54603609
NR ‐1/2	0.853372773	0.844318773	0.96290613	0.94621374	0.58412497	0.945956574	0.00567605	0.92285901

FIGURE 11

Correlation of with molar refractivity (MR) and Polarizability (P)

The Neighborhood Inverse sum Index and Third version Zagreb index for Boiling point (BP) and Enthalpy of vaporization (E)

The Neighborhood Randić index for Flash point (FP).

The Neighborhood Second modified Zagreb index for Molar refractivity(MR), Polarizability (P) and Molar volume (MV).

The Sanskruti index for Polar surface area (PSA).

No index shows good correlation with Surface tension (T). However, Third NDe index ( ) shows positive correlation (0.668) with Surface tension (T).

Observations 1 to 11 exhibit the applicability of topological indices for the physicochemical properties of mentioned drugs in QSPR study. Overall, second modified Zagreb index and its neighborhood degree sum version give strong correlation for Flash point (FP), Molar refractivity(MR) and Polarizability (P). For the above reasons, it can be argued that the topological indices considered above are potential tools for QSPR analysis of antiviral drugs.

Next we attempt to develop QSAR models for predicting the biological activity (pIC50) of aforementioned drugs using multiple linear regression (MLR). QSAR model development are restricted to a maximum of four topological indices. To select the topological indices, we obtain the intercorrelation matrix of topological indices to investigate the occurrence of multicolinearity (Table 7). We decide to avoid the combinations between the highly inter‐correlated topological indices with | where r is the simple linear coefficient.

TABLE 9

Some statistical parameters of generated model

Model no.	R	RA 2	PE	F
1	0.637479791	0.287656581	0.149578058	3.42290
2	0.750999971	0.346001434	0.109861433	2.587166
3	0.754092	0.13731	0.108688665	1.318331
4	0.801329525	−0.07361298	0.090175014	0.897151516

We start building simple linear regression models with topological indices having a least inter‐correlation (i.e 0.406 between Nde3 and I) giving two mono parameter models. Both models exhibit weak correlation with pIC50, However the simple linear regression analysis yields that the following model shows better statistical results

n = 7, r = 0.637479791, R2 = 0.406380484, = 0.287656581, Se = 0.378885417, F = 3.42290, PE = 0.149578058

Here and thereafter:

n: number of compounds, Se: standard error of estimation, r (R):simple (multiple) correlation coefficient, : adjustable R 2, F: Fisher's statistics, PE: Probability error

Step wise regression analysis using different combinations of two topological indices, the following bi –parametric model is found best among all having significantly better statistics than mono‐parametric model (Model 1)

n = 7, R = 0.750999971, R2 = 0.564000956, = 0.346001434, Se = 0.363037577, F = 2.587166, PE = 0.109861433

Trials are made to correlate three combined topological indices with the biological activity (pIC50) to improve the statistical parameters of the obtained models. The following model showed little improved statistics.

n = 7, R = 0.754092, R2 = 0.568655, = 0.13731, Se = 0.416956, F = 1.318331, PE = 0.108688665

Finally, successive stepwise regression yielded the following tetra‐parametric model with improvement in statistical parameters

n = 7, R = 0.801329525, R2 = 0.642129007, = −0.073612978,Se = 0.465143609, F = 0.897151516, PE = 0.090175014

The observed activity (Obs. pIC50) together with the predicted activity (Pred. pIC50) for the tested drugs calculated using MLR model 4 are listed in Table 8.

TABLE 7

Intercorrelation matrix of the topological indices

	M1	M2	mM2	ReZ3	F	SDD	H	I	A	NM1	NM2	NmM2	Third NDe	NF	Fifth NDe	NH	NI	S(G)
M1	1.000
M2	0.979	1.000
mM2	0.965	0.898	1.000
ReZ3	0.975	0.998	0.892	1.000
F	0.993	0.994	0.934	0.994	1.000
SDD	0.997	0.966	0.975	0.961	0.985	1.000
H	0.754	0.651	0.874	0.643	0.704	0.790	1.000
I	0.741	0.616	0.815	0.633	0.691	0.762	0.727	1.000
A	0.999	0.973	0.973	0.966	0.988	0.999	0.775	0.741	1.000
NM1	0.993	0.995	0.932	0.994	0.999	0.985	0.701	0.690	0.988	1.000
NM2	0.948	0.990	0.839	0.994	0.977	0.929	0.575	0.572	0.935	0.978	1.000
NmM2	0.877	0.778	0.942	0.762	0.824	0.908	0.884	0.816	0.896	0.825	0.697	1.000
Third NDe	0.856	0.940	0.708	0.948	0.908	0.830	0.432	0.406	0.838	0.909	0.975	0.540	1.000
NF	0.948	0.991	0.840	0.994	0.977	0.931	0.578	0.564	0.936	0.978	1.000	0.703	0.976	1.000
Fifth NDe	0.957	0.920	0.963	0.902	0.932	0.963	0.816	0.655	0.968	0.936	0.863	0.900	0.757	0.865	1.000
NH	0.732	0.634	0.846	0.628	0.685	0.768	0.993	0.708	0.752	0.684	0.566	0.851	0.431	0.567	0.793	1.000
NI	0.994	0.993	0.935	0.992	0.999	0.987	0.702	0.698	0.990	1.000	0.976	0.829	0.905	0.975	0.937	0.685	1.000
S(G)	0.903	0.942	0.807	0.957	0.939	0.895	0.610	0.601	0.890	0.935	0.962	0.687	0.947	0.964	0.785	0.602	0.932	1.000

In order to judge the quality of these models some statistical parameters such as , , F are summarized in Table 9.

TABLE 8

The observed and predicted biological activity of the drugs with residuals

Drugs	Observed pIC50	Predicted pIC50	Residuals
1. Lopinavir	5.28	5.355130613	−0.075130613
2. Ritonavir	5.06	5.076097539	−0.016097539
3. Arbidol	5.45	5.796583917	−0.346583917
4. Chloroquine	5.86	6.03247222	−0.17247222
5. Hydroxy‐chloroquine	6.14	5.834557251	0.305442749
6. Theaflavin	5.07	5.179482724	−0.109482724
7. Remdesivir	6.01	5.595675735	0.414324265

The QSAR studies performed on seven COVID‐19 drugs produced four models using MLR. The studied model shows that there is insufficient evidence to indicate that these drugs' antiviral activity (pIC50) is well correlated with the topological indices being used at a 5% significance level.

CONCLUSION

In pharmaceutical sciences, the properties of molecular structure are essential for the achievement of a new product and can be identified from the study of topological indices Therefore, in this paper we study several promising drugs for the treatment of COVID‐19 like Lopinavir, Ritonavir, Arbidol and Thalidomide and establish expression of M‐polynomial and NM‐polynomial for these drugs by analyzing their molecular structure and edge partition technique. Several degree based and neighborhood degree based topological indices are also derived from these polynomials.

Further, QSPR study shows that various topological indices computed in this article possess strong predictive ability for physicochemical properties of antiviral drugs. However, the QSAR analysis indicates that the studied biological activity model (pIC50) is not well associated with the topological indices being used, which requires further studies to determine a link between the topological indices studied and some other biological activities of these drugs.

In particular, F‐index depicts strong positive correlation value (i.e. r = 0.992) with Boiling point (BP) when compared with other indices. Similarly, it can be noticed that neighborhood second modified Zagreb index correlate Molar refractivity (MR) and Polarizability (P) strongly with correlation coefficient r = 0.994. The theoretical results obtained in this article have promising aspects in designing new drug and vaccine for the treatment of COVID‐19.

AUTHOR CONTRIBUTIONS

Syed Kirmani: Conceptualization; formal analysis; investigation; methodology; writing‐original draft. Parvez Ali: Conceptualization; validation. Faizul Azam: Resources.