Physiochemical Properties of Benzophenone and Curcumin-conjugated PAMAM Dendrimers Using Topological Indices

Abstract Curcumin longa (Turmeric) contains an extremely strong phytochemical called curcumin, which has various medicinal and industrial benefits, including anti-inflammatory and anti-cancer effects, textile dyeing, and food and beverage colorants. However, due to its poor bioavailability and low aqueous solubility, it is unstable. These physiochemical properties limit curcumin’s clinical application. In order to solve this problem, scientists recently made new PAMAM dendrimers of the zero, first, and second generations conjugated with curcumin and benzophenone, which is a water-soluble secondary metabolite. Here, we use topological indices to examine these novel PAMAM dendrimers. Adriatic indices, which are a subclass of topological indices, can be used to measure how the arrangement of atoms in a chemical structure affects its most important physical properties. In this study, the parameters of the aforementioned dendrimer, including its density, retention duration, enthalpy, and lipophilicity, are calculated and analyzed based on 16 distinct adriatic indices. Enhanced bioavailability of curcumin in the near future is likely to bring this promising natural product to the forefront of therapeutic agents for the treatment of human diseases.


Introduction
The study of phytochemistry is used in bioinformatics and computational chemistry to find beneficial chemicals.One of these phytochemicals is curcumin, also known as 1,7-bis(4-hydroxy-3methoxyphenyl)-1, 6-heptadiene-3, 5-dione, a low molecular weight yellow polyphenol obtained from the roots of turmeric, the plant Curcuma longa Linn. 1 This has numerous pharmacological as well as physiological qualities, namely antioxidant, 2 anti-inflammatory, 3 anti-microbial (antifungal, anti-bacterial, anti-viral, etc.), anti-tumoral, anti-HIV and anti-cancer effects, anti-amyloid, anti-neoplastic, immune-modulating, metabolism regulating, anti-depressant, neuroprotective, and tissue protective effects. 4It is used in food (colorant, preservative), cosmetic, and textile (dyeing) industries also.Despite these facts, curcumin could not be employed as a medicine since it has poor bio availability and low aqueous solubility, which prevent it from fully satisfying the ADMET (Absorption, Digestion, Metabolism, Excretion, Toxicity) criteria.The research revealed that curcumin was poorly bioavailable, insoluble in water at physiological pH, rapidly metabolized, and excreted.Curcumin was found to be poorly absorbed from the stomach and to reach tissues beyond the gut in pharmacologically insignificantly small amounts, according to measurements of blood plasma levels and biliary excretion. 5Curcumin compounds are currently linked to diverse macromolecular platforms, incorporating synthetic polymers, and addressing proteins in order to produce biocompatible polyphenols with potential improved biological activity. 6Drugs' oral bioavailability issues can be solved through nanoencapsulation employing nanostructures like dendrimers.It has been discovered that liposomes, lipid nanoparticles, microparticles made of serum albumin and chitosan, 7 complexed with phospholipids 8 and cyclodextrin, 1 are used to deal with the problems of inadequate solubility and minimal bioavailability. 9Nanoencapsulation using nanostructures like dendrimers can resolve the challenges with regard to enhancing the oral bioavailability of drugs.
Dendrimers are a subclass of polymeric synthetic macromolecules with high branching points, globular form in three dimensions, monodispersed nature, and size range in the nanometres. 10ue to appealing qualities such nanoscopic size, a low polydispersity index, superior molecular structure control, the availability of different functional groups at the periphery, and cavities in the interior, they stand out among the available polymers.They are suitable for use in biological and drug delivery applications due to their distinct physiochemical properties and unimolecular micelle nature.Dendrimers are used for a variety of medicinal purposes, including medication administration and diagnostics, gene delivery, bioavailability, and targeting. 11These dendrimerbased formulations also make it simpler to construct highly physically stable chemical preparations, opening the door to future commercialization.These dendrimers' primary characteristics include generation size, surface modification, and three-dimensional spherical structure.
Five classes of dendrimers have been recognized, which differ from each other by their mode of synthesis, structural behavior, base core, and terminal groups. 12PAMAM dendrimers are one among them in which 'PAMAM' stands for Poly(AMido)AMine, which refers to one of the original dendrimer types built by polyamide branches with tertiary amines as focal points.PAMAM dendrimers have the ability to encapsulate or conjugate medicinal payloads.They are anticipated to have potential uses in improving drug delivery systems' solubility. 13Cancer can be targeted by surface-modifying these dendrimers with molecular recognition ligands.They potentially react with nucleic acids to deliver substances into cells.Nucleic acids are released into the cytosol during endosomal evacuation. 14AMAM dendrimers incorporated with curcumin, can maintain its conjugated structure through interactions with hydrophobic molecules, hydrogen bonds, and van der Waals forces. 15AMAM G4 dendrimers, the novel nanocarrier for curcumin, are created which increases the curcumin's oral bioavailability.This in turn encourage sustained therapeutic drug delivery and enhanced absorption by means of gastrointestinal tract. 16Fluorescence, UV spectroscopy, and molecular modeling techniques were used to examine the interface behavior amongst curcumin and PAMAM dendrimers. 17Curcumin-loaded dendrimers specifically reduce viability of glioblastoma cell lines. 18Yet, the uses of curcumin-loaded PAMAM dendrimers are less explored since the study on their physiochemical properties need time and expensive laboratory experiments.To overcome these factors, molecular descriptors (study based on the molecular topology) having good correlation with physiochemical properties can be studied.
The structure of dendrimers has a direct correlation with physiochemical properties.For instance, the extent of hydrophobicity of the dendrimer core is related to its retention efficacy.The flexibility of the dendrimer is determined by the quantified entropy.The structure relationship with the activity and property of dendrimers and other structures are explored through topological indices (TIs) by many researchers,  which is briefly narrated in the forthcoming paragraphs.
Two groups of polyphenylene dendrimers, PAMAM, tetrathiafulvalene, polyphenylene dendrimer line graph, and POPAM dendrimers' TIs are represented by analytical closed formulas.The results are useful for bioactivity prediction, molecular data mining, and in particular when assessing the uniqueness of tested (hyper-branched) molecular graphs. 33The outcomes also illustrated dendrimer behavior and its physiochemical characteristics, including boiling point and solubility partition. 30dditionally, topological structural properties can be used to express molecular electronic characteristics.Hydrophobicity/hydrophilicity is thought to be essential for opiates to pass through the blood-brain barrier. 43Dendrimer-Curcumin conjugates performed well in the hydrophobicity/hydrophilicity test. 44he assessment of the pharmacokinetic properties and the therapeutic potential of innovative as well as existing and utilized drug compounds may depend on the prediction of physiochemical properties.As a result, numerous studies are being carried out in the areas of QSAR/QSPR/QSTR (Quantitative Structure-Activity/Property/Toxicity Relationships) analysis to assess the properties of biologically active compounds and to investigate its correlation with pharmacological activity.
Mathematical chemistry has a field called chemical graph theory that focuses on processing chemical graphs related to chemical systems.The relationship between a molecule's properties and its features is among the fundamental ideas in chemistry.Therefore, the study of all connectivity-related effects in a chemical system falls under the purview of chemical graph theory.In the other terms, the implementation of graph theory to the field of chemistry is the focus of chemical graph theory.

Topological indices
TIs are numbers assigned to a graph that completely characterize the structure of the graph and are invariant under the automorphism of the graphs.QSARs are mathematical models that relate various types of chemical reactivity, pharmacological activities, equilibrium, physical, and physicochemical characteristics with electronic, lipophilic, adsorption, and other attributes of a chemical structure of a specified series of compounds, including such TIs transition constants, and also with solvent and many other physicochemical parameters. 45esearchers used TIs to predict the biological actions of particular medications by analyzing their characteristics.To anticipate the bioactivity of diverse chemicals, researchers used computational studies of degree-based TIs for various networks, edge versions of line graphs, and their numerical simulation.This is accomplished by obtaining edge versions of analytical closed formulae and polynomials.
The topological graph indices of the drugs decitabine (5-aza-20-deoxycytidine), guadecitabine, and azacitidine encourage research focused on drugs and also examine the causes of symmetry in actual networks.The aforementioned drugs are hypomethylating medications that are used to treat individuals with chronic myelomonocytic leukemia, acute myeloid leukemia, or higher risk myelodysplastic syndromes who were not suitable for induction chemotherapy. 29he following are a few additional uses of TIs in chemical compounds: The entropy of aluminum phosphate materials, which are used as humidity sensors and natural gas dehydrators, was investigated using the vertex form of distance-based TIs.The findings aid in the advancement of ongoing chemical and materials science research.35 Catalysis uses a unique chemical compound named diphenylene.Any two diphenylenes that cross by one edge result in a chain of diphenylenes.To determine the number of spanning trees for this chain, TIs produced two formulas of equal size.The neighborhood polynomial of some common graph networks, such as the oxide, hexagonal, and silicate networks, was determined.The omega polynomial and Cluj-Ilmenau index of an infinite class of titanium nanotubes, TiO2(m,n), have been calculated.The theoretical implications for numerous TIs are given by taking G into account as the line graph of subdivision of some convex polytopes and its complement.[35][36][37][38] Through the use of graph invariants, these studies offered practical functions that had accurate linear relationships with various characteristics of such materials.
Throughout this article, let G be a graph with vertex and edge set represented by VðGÞ and EðGÞ, respectively.The number of vertices and edges in these sets is indicated by their cardinality.Let e ¼ uv denote an edge in the graph of EðGÞ with the end vertices u and v: If there is an edge connecting two vertices, u and v, then they are said to be neighboring.
TIs are calculated using various analytical procedures from graph theory namely path numbers, cut methods, edge partition technique, vertex partition strategy, techniques of degree counts, and distance count.In this study, we examine a class of molecular descriptors called bond additive descriptors, which are incredibly predictive.They can be easily calculated and stored in any of these software packages.Since this method defines a lot of essential descriptors, we focus only on edge contributions among bond additive descriptors.Many descriptors can be stated as the sum of edge contributions due to their bond additive character.
Heat capacity, total surface area

Bond additive TIs
A bond additive descriptor Des can be written as where EðGÞ is the set of edges and f is some mapping that assigns a real value to an ordered pair consisting of a graph and its edge.It is evident that this definition is fairly open-ended given the multitude of ways that f can be calculated.Adriatic indices fall into three categories: extended, discrete, and variable.One of the closest groups of these descriptors, discrete adriatic descriptors, contains 148 descriptors.Based on the features they describe a subset of 16 indices are chosen and discussed in this article.These indices are generalized as follows: where d u , d v are respectively the degrees of vertices u and v: The value of u i, a determines the nomenclature, which is represented by the second function as follows: 1,1/2 is equivalent to lor, 1,1 is equivalent to lo, 1,2 is equivalent to los, 2,1 is equivalent to in, 2,À1/2 is equivalent to ir, 2,1/2 is equivalent to ro, and 2,2 is equivalent to s. 46 The 16 discrete adriatic indices considered for discussion in this research is listed in Table 1.
The photostability of the PAMAM dendrimer conjugated to curcumin and benzophenone is studied for absorption and fluorescence spectroscopy.It has been discovered that adding curcumin to the dendrimer molecule increases its photostability, and that adding 2,4-dihydroxybenzophenone to the dendrimer molecule as an ultraviolet absorber results in an additional stabilization.Additionally, it has been shown that the generation of dendrimers has an impact on this parameter since the photo stabilizing effect is stronger as dendrimer generation increases. 47
The aforementioned PAMAM dendrimer of zeroth generation in the first growth stage is the core of the molecular structure and is denoted by G 0 [1].The corresponding molecular structure is represented in Figure 3.We consider the first and second generation of the same to calculate and compare their properties.The number of vertices in any graph is the size and the order is calculated using the number of edges.

First generation (CBD 1 )
The chemical graph of first generation in the second growth stage is represented by G 1 [2].The corresponding molecular structure is represented in Figure 4. Throughout the study, the properties of G 1 [2] are studied with the name CBD 1.The size of CBD 1 is ð64Þ 2 nþ1 þ 3, that is partitioned into three categories as ð8Þ2 nþ1 þ 4, ð32Þ2 nþ1 þ 6, ð18Þ2 nþ1 þ 4, which are the vertices of degree of 1, 2, and 3 respectively.The order of CBD 1 is ð59Þ 2 nþ1 þ 4: The edge sets of CBD 1 are divided into four parts by the edge partition technique as the following: The number of edges in each partition of CBD 1 denoted by jE i j is mentioned in Table 2.
Theorem 2.1.1.For the chemical structures of curcumin and 2,4-dihydroxybenzophenone conjugated to PAMAM dendrimer of first generation (CBD 1 ) n ! 1, with edge partitions as E 1 the adriatic indices are: The chemical graph of second generation in the third growth stage is represented by G 2 [3].The corresponding molecular structure is represented in Figure 5. Throughout the study, the properties of G 2 [3] are studied with the name CBD 2. The size of CBD 2 is 2 nþ1 61 ð Þ þ 4, that is partitioned into three categories as ð8Þ2 nþ1 þ 12, ð33Þ2 nþ1 þ 14, ð17Þ 2 nþ1 þ 10, which are the vertices of degree of 1, 2, and 3, respectively.The order of CBD 1 is ð66Þ 2 nþ1 þ 3: The edge sets of CBD 2 are divided into four parts by the edge partition technique as the following: The number of edges in each partition of CBD 2 denoted by jE i j is mentioned in Table 3.
. PAMAM dendrimers of second-generation covalently bonded with curcumin and benzophenone residues. Proof:

Biological experiments and results
Curcumin oral supplements have been used in several research studies to examine their absorption in humans and lab animals.9][50][51][52][53][54][55][56][57][58][59][60][61] A succinct description of these laboratory trials is provided in the Supplementary Information.These results establish that nanoparticle conjugates of curcumin gave fluorescence effects, improved bioavailability, better control over drug release, longer half-life, antioxidant effects, longer blood circulation, faster wound healing, anti-cancer, anti-inflammatory, brain delivery, and increased chemical stability and solubility. 62able 4. Graphical abstract of comparative analysis of the physio chemical properties of CBD 1 and CBD 2 . (continued) Regardless of the source-conjugated material employed in investigations, a variety of limited, uncertain, and contradictory outcomes are achieved even if these conjugates might be regarded as good transporters for curcumin.These incongruent results are due to few factors mentioned here: (1) The maximum tolerable dose varies depending on the investigation.(2) The drug release has no time-dependant impact because of the loaded medication's low rate of drug release.(3) A few of the negative effects are also recorded in a trial.When monitored for 72 h, seven participants in a dose-response trial receiving 500-12,000 mg reported having diarrhea, headaches, rashes, and yellow stools.(4) Another trial revealed nausea, diarrhea, and a rise in serum alkaline phosphate and dehydrogenate content at doses of 0.45-3.6g/day for 1-4 months.][63][64][65][66][67][68] Hence, curcumin is currently more commonly employed as a dietary supplement than a lead chemical because the desired goal in the research on biocompatibility has not yet been attained because of the contradictory outcomes seen in biological experimentation.Additional in-depth clinical research is required to transform it into an effective lead compound and pinpoint its impacts on human health. 66n this regard, curcumin conjugated with benzophenone PAMAM dendrimers synthesized in 2020 is studied.The study focuses on improving biocompatibility, which is dependent on several elements namely preservation from oxidation, a lengthy blood circulation time, and accumulation of curcumin in tumor tissues.The main determining factor of all these variables is physiochemical traits like log P, relative retention time, enthalpy, etc.
In this article, the physiochemical properties such as log P, relative retention time, standard enthalpy of vaporization, and enthalpy of vaporization are calculated using TIs, which are useful in identifying a hit molecule.
To achieve the goal, in this research article CBD 1 and CBD 2 are considered whose fluorescence effect alone is studied so far. 69The data of the experimental results of physiochemical properties are not yet made available by chemists in globally recognized databases.The results in this article predict CBD 1 and CBD 2 as hit molecules that elicit a desired activity in a screening assay.The outcomes facilitate chemists and pharmacologists to use chemical tests and subsequently biological investigations to turn this prediction into a lead compound.The following paragraph provides an explanation of several processes that make up the procedure for future work.

Future prospective
The molecular structure of CBD 1 and CBD 2 can be optimized as a lead compound drug using the following phases: (a) Numerous other physiochemical properties needed for drug designing can be predicted using various TIs/molecular descriptors.(b) Chemists can establish the experimental results of CBD 1 and CBD 2 in globally recognized databases like PubChem, PubMed, chEMBL, etc. (c) Regression models can validate the data obtained in (a) and (b) phases with QSAR/QSPR/QSTR analysis.(d) For additional outcome configuration, virtual screening using various software packages like DRAGON, CODESSA, and PaDEL can be employed.(e) The efficiency can be analyzed by cell culture experiments and animal studies (biological experiments).(f) Further optimization of the lead molecule delivers the drug candidate.

Conclusion
In this article, by edge partition method in terms of drug structure analysis we determine the adriatic indices of PAMAM dendrimers conjugated with curcumin and benzophenone, which exemplify its properties without any experiment.The outcomes found in this article demonstrate POLYCYCLIC AROMATIC COMPOUNDS encouraging claims for pharmacy engineering.Since our globe always faces newly discovered diseases, we are always in need of new drug discovery that can be made easier and successful by testing various chemicals by using TIs and evaluating them in a noticeably short period of time.

Table 1 .
List of discrete adriatic indices with formulation and correlated physiochemical properties.

Table 2 .
Edge partition of first-generation PAMAM dendrimers conjugated with curcumin and benzophenone.