Z Adibipour

N Fani

A.-K Bordbar
email: bordbar@chem.ui.ac.ir

M Sahihi

Exploring the Interaction Mechanism of Coumarin with Bovine β-Casein: Spectrofluorometric and Molecular Modeling Studies

Keywords: β-casein (βCN), Coumarin, Molecular docking, Molecular dynamics simulation, Fluorescence spectroscopy

This paper is designed to examine the binding behavior of Coumarin with bovine -casein (βCN) through fluorescence spectroscopy and molecular modeling techniques. The data analysis on fluorescence titration experiments at various temperatures represents the enthalpy driven nature for the formation of Coumarin-βCN complex and the prevailed role of hydrogen bonds and van der Waals interactions in the binding process. The results also represent the static quenching of tryptophan and dynamics quenching of tyrosine and phenylalanine residues due to the binding of Coumarin. It can be concluded from molecular docking studies that Coumarin binds to several polar and nonpolar residues in the hydrophobic core of βCN with the binding energy of -6.96 kcal mol-1 . Finally, analysis of molecular dynamics (MD) simulation results suggested that the interactions between βCN and Coumarin are very stable and the binding of Coumarin restricted the flexibility of important residues in the binding site of this protein.

INTRODUCTION

Coumarin derivatives are found in cosmetic products, perfumes and food additives such as cinnamon and vanilla. These heterocyclic compounds are also found in many edible plants such as beans, lavender, licorice, strawberry, blackcurrant, apricot, sweet clover and cherry [START_REF] Marles | Coumarin in vanilla extracts: Its detection and significance[END_REF]. These derivatives as benzo-αpyrone compounds have several characteristics such as high fluorescence quantum yield [START_REF] Zahradník | The Production and Application of Fluorescent Brightening Agents[END_REF], large Stokes shift, excellent light stability and less toxicity [START_REF] Bullock | Coumarin-based luminescent ligand that forms helicates with dicationic metal ions[END_REF]. The use of these compounds as fluorescent probes of pH, for detection of nitric oxide, nitroxide and hydrogen peroxide in food products is well established. The importance of these compounds also refers to their brilliant biological properties such as anticoagulant [START_REF] Ghazaryan | Peculiarities of the anticoagulant properties for the newly synthesized preparations of coumarin-related compounds[END_REF], free radical scavenger [START_REF] Kancheva | Structure-activity relationships of new 4-hydroxy bis-coumarins as radical scavengers and chain-breaking antioxidants[END_REF], antibacterial [START_REF] Morabito | Antioxidant properties of 4-methylcoumarins in in vitro cell-free systems[END_REF], antifungal [START_REF] Grimm | Substituted coumarins as potent 5-lipoxygenase inhibitors[END_REF], cytotoxic [START_REF] Resch | 5Lipoxygenase and cyclooxygenase-1 inhibitory active compounds from atractylodes lancea[END_REF], anti-tumor, anti-viral, anti-inflammatory, antioxidant and antimicrobial activities [START_REF] Bansal | Design and synthesis of coumarin substituted oxathiadiazolone derivatives having anti-inflammatory activity possibly through p38 MAP kinase inhibition[END_REF]. The main problem with Coumarin, limiting its biological applications, is its tremendously low aqueous solubility and poor bioavailability. Several attempts have been made to encapsulate this compound in polymeric micelles and nanoparticles, liposomes, lipid-based nanoparticle and hydrogels in order to overcome its shortcomings [START_REF] Rassu | Encapsulation and modified-release of thymol from oral microparticles as adjuvant or substitute to current medications[END_REF]. Caseins are phosphoproteins that are produced from raw skimmed milk. The store and transport of bio-available metal ions such as calcium and phosphate through the formation of soluble assemblies are the main functions of caseins [START_REF] Bhattacharyya | Molecular chaperonelike properties of an unfolded protein, alpha(s)casein[END_REF]. Beta-casein (βCN) as the predominant constituent of bovine milk casein has 209 residues in a single chain with the molecular weight of 23.983 kDa. Bovine βCN is a highly amphiphilic protein made up of a hydrophilic Nterminal part with a cluster of five phosphoryl residues and a hydrophobic C-terminal part [START_REF] De Kruif | Colloids Surfaces A Physicochem[END_REF]. The most common forms of this protein in dairy cattle breeds are A1 and A2, while B is the less common form [START_REF] Eigel | Nomenclature of proteins of cow's milk: Fifth Revision[END_REF][START_REF] Kamiński | Polymorphism of bovine beta-casein and its potential effect on human health[END_REF][START_REF] Keating | A note on the evaluation of a beta-casein variant in bovine breeds by allele-specific PCR and relevance to β-casomorphin[END_REF]. It should be noted that bovine βCN is known as a dynamic protein with noncompact and flexible conformation [START_REF] O'connell | Association behavior of ??-casein[END_REF].

The utilizing of βCN micelles as natural nano-delivery vehicles for lipid-soluble drugs have been reported in several studies. The easy digestion of βCN guarantees the easily release of encapsulated chemotherapeutics in the stomach [START_REF] Semo | Casein micelle as a natural nano-capsular vehicle for nutraceuticals[END_REF]. Considering the ability of βCN for encapsulation of bioactive materials, it is of interest to gain further insight into the delivery mechanism of some drug candidates with bovine βCN nanoparticles. In this regard, some studies have investigated the binding properties of some compounds such as bisdemethoxycurcumin (BDMC) and Quercetin with βCN by using various experimental and computational techniques [START_REF] Mehranfar | Binding analysis for interaction of diacetylcurcumin with β-casein nanoparticles by using fluorescence spectroscopy and molecular docking calculations[END_REF][START_REF] Mehranfar | A combined spectroscopic, molecular docking and molecular dynamic simulation study on the interaction of quercetin with β-casein nanoparticles[END_REF]. In the present work, a comprehensive study has been done on the interaction mechanism of Coumarin as a wellknown bioactive compound with bovine βCN as a nanocarrier by spectrofluorimetric, molecular docking and molecular dynamics (MD) simulation techniques. The detail structural information regarding binding poses, binding affinities and the effect of Coumarin on the protein stability and secondary structure of βCN are reported here.

MATERIALS AND METHODS

Materials

Bovine βCN (>99%; Sigma-Aldrich, Germany) was dissolved in pH 7.0 phosphate buffer (PBS) containing 80 mM NaCl, 5.65 mM Na2HPO4 and 3.05 mM NaH2PO4 with an ionic strength of 0.1. All salts used for buffer preparation were of analytical grade and were dissolved in deionized water. To avoid large protein aggregation, the protein solution was filtered through a permeable membrane. The exact concentration of protein was determined spectrophotometrically using molecular absorption coefficient of 4.6 (1%) at 280 nm. Coumarin with analytical grade was purchased from Merck Company and used without further purification. According to the low solubility of Coumarin in water, its stock solution was prepared in ethanol and the exact concentration was measured by determining its light absorption at λmax (logε274 = 4.2).

Fluorescence Spectroscopy

Spectrofluorimeter RF-5000 (Shimadzu, Japan) was used for fluorescence measurements equipped with a xenon lamp source, a 1.0 cm quartz cell and a thermostat bath that kept temperature constant within ±0.1 ○C. For spectrofluorometric experiments, 2 ml of βCN solution (11.5 μM) was put in quartz cell and titrated with specified volumes of Coumarin stock solution, over and over again. Emission spectra were separately recorded from 300 to 450 nm at an excitation wavelength of 295 nm and from 290 to 430 nm at an excitation wavelength of 280 nm. Both excitation and emission slit widths were set at 5 nm. It should be noted that emission spectrum of the βCN solution was measured as the control experiment and during the fluorescence measurements, ethanol concentration did not exceed over 1% v/v. The fluorescence titration experiments were run at four different temperatures ranging from 293K to 308 K and the gained results were analyzed for estimation of binding and thermodynamic quantities.

Preparation of βCN and Coumarin Structures

Since, the crystal structure of βCN does not exist in Protein Data Bank (PDB), in this paper, its structure was modeled using I-Threading ASSEmbly Refinement (TASSER) server which is a protein structure modeling approach based on the secondary structure enhanced profileprofile threading alignment (PPA) and the iterative implementation of the TASSER program [START_REF] Sokkar | Multiple templates-based homology modeling enhances structure quality of AT1 receptor: Validation by molecular dynamics and antagonist docking[END_REF]. Structural templates are first identified from the PDB by multiple threading approach LOMETS; full-length atomic models are then constructed by iterative template fragment assembly simulations. For query proteins that have no structurally related protein in the PDB library, the structure must be built from scratch by ab initio modeling [START_REF] Liwo | Protein structure prediction by global optimization of a potential energy function[END_REF]. This is a challenging task and success is limited to small proteins with < 120 amino acids [START_REF] Jain | Synthesis and structure-activity relationship studies on tryprostatin A, an inhibitor of breast cancer resistance protein[END_REF]. Lastly, function in slights of the target is derived by threading the 3D models through protein function database BioLiP. I-TASSER server is in active development with the goal to provide the most accurate structural and functional predictions using state-ofthe-art algorithms. The structure of βCN which was made with I-TASSER server contains 5 stands and 1 helix and other positions were made up of random coils. To accomplish more stable and better structure of the model obtained from I-TASSER, a 50 ns molecular dynamic simulation was performed on the βCN structure using GROMACS 4.5.4 software package. The obtained structure from the molecular dynamic simulation of βCN was used as the beginning point of molecular docking calculations. The Coumarin structure was optimized using the quantum chemistry software Gaussian 03 and its calculation was approved at the 6-31G** level by employing the Becke three-parameters Lee-Yang-Parr (B3LYP) hybrid density functional theory.

Molecular Dynamics Simulation

MD simulation, as a very powerful method in modern molecular modeling, enables us to follow and understand structure and dynamics with extreme detail literally on scales where the motion of individual atoms can be tracked. This task will focus on the two most commonly used methods, namely energy minimization and molecular dynamics that respectively optimize the structure and simulate the natural motion of molecules. The Gromos43a1 was designated as force field [START_REF] Van Der Spoel | Gromacs User Manual version 4.0[END_REF] and 20680 SPC/E water molecules were employed according to the results of previous studies that revealed the SPC/E water model performs well in water box simulation [START_REF] Rivail | Large-scale molecular dynamics of a G protein-coupled receptor, the human 5-HT4 serotonin receptor, in a lipid bilayer[END_REF]. The classical way to minimize edge effects in a finite system is to apply periodic boundary conditions. The atoms of the system supposed to be simulated were put into a space-filling cubic box of approximate dimensions 8.727 × 8.727 × 8.727 Ǻ 3 surrounded by translated copies of itself. The adequate amount of sodium counter ions were exchanged with water molecules in order to neutralize the system for calculations. The added hydrogens and broken hydrogen bond network in water would lead to quite large forces and structure distortion if MD was started immediately. To eliminate these forces, a short energy minimization run is necessary. Hence, this run was done at first using the steepest descent method [START_REF] Berendsen | The missing term in effective pair potentials[END_REF]. Two-step equilibration procedures are needed to relax the solvent and ions while keeping the protein atom positions restrained. This was accomplished in two phases: 100 ps NVT followed by 100 ps NPT ensembles. The dynamics were performed at conditions of 300 K and 1.0 bar which is close to the circumstances of most laboratory environments. A standard cutoff of 1.0 nm for both of the neighbor list generation and the Coulomb and LennardJones interactions were selected. Energies and other statistical data were stored every 10 steps and the more expensive Particle-Mesh-Ewald summation was chosen for electrostatic interactions [START_REF] York | The effect of long? range electrostatic interactions in simulations of macromolecular crystals: A comparison of the Ewald and truncated list methods[END_REF].

Molecular Docking Calculation

The molecular docking calculations were carried out to visualize the binding site of Coumarin in βCN and to determine the involved residues in the interaction. Docking calculations were accomplished using AutoDock 4.2.2 package which is an automated procedure for predicting the interactions of ligands with biomacromolecular targets [START_REF] Morris | Automated docking using a Lamarckian genetic algorithm and an empirical binding free energy function[END_REF]. Protein model was first modified by adding all hydrogens using the builder module of Autodock. The macromolecule was kept rigid, while all the torsional bonds of ligand were set free to rotate. First, a blind docking with 126 lattice points along X, Y and Z axes was performed to find the binding site of ligand to βCN. After determination the binding site, the dimensions of the grid map were selected 60 points with a grid point spacing of 0.375 Å to allow the ligand to rotate freely. By using the Lamarckian genetic algorithm (LGA) method, 200 separated docking calculations consisted of the maximum 25,000,000 energy evaluations were performed. The clustering of docking results from each of the 200 calculations was done on the basis of root mean square deviations (RMSD) between the Cartesian coordinates of the ligand atoms also were ranked according to the free energy of binding. The structure with the lowest free energy of binding in the highest populated cluster was chosen as the optimal docking pose.

RESULTS AND DISCUSSION

Fluorescence Spectroscopy

In order to identify the binding mechanism, the measurement of intrinsic fluorescence of βCN in the presence of various amounts of Coumarin was performed. Trp residue of βCN which can provide important information about the formation of Coumarin-βCN complex was mainly placed in the hydrophobic region of this protein. The calculated Ksv at four temperatures ranging from 293-308 K is represented in Table 1. The decreasing of Ksv values with temperature designates static quenching mechanism [START_REF] Lakowicz | Principles of Fluorescence Spectroscopy[END_REF].

In order to calculate the number of binding sites of the ligand on the surface of the protein (n) and the associative binding constant (Ka), the following equation can be used [START_REF] Lakowicz | Principles of Fluorescence Spectroscopy[END_REF]:

Figure 1c shows the corresponding plots of above equation to estimate the values of Ka and n for binding of Coumarin to βCN. The calculated values of Ka and n at four temperatures are presented in Table 1. The estimated Ka values were used for calculation of the thermodynamic parameters of reaction at various temperatures.

The values of binding constants indicate the high affinity for binding of Coumarin to βCN. The energetics of protein-ligand equilibrium can be conveniently characterized by three thermodynamic parameters; the standard Gibbs free energy, ΔG°, the standard molar enthalpy, ΔH0, and the standard molar entropy, ΔS°. ΔG° can be calculated from the equilibrium constant, Ka, of the reaction using the relationship, ΔG° = -RT lnKa, where R and T are gas constants and the absolute temperature, respectively. Therefore, the temperature dependent thermodynamic parameters were calculated to elucidate the interaction mechanism. The standard enthalpy and entropy changes of binding could be estimated from the following equations, based on the binding constants at different temperatures:

The plot of ΔG°/T vs. 1/T is shown in Fig. S1 and their calculated thermodynamic parameters at different temperatures ranging from 293-308 K are reported in Table 2.

Hydrophobic, electrostatic, van der Waals interactions and hydrogen bonds are the main driving forces in the binding phenomena that could be designated from the thermodynamic parameters. The sign and magnitude of thermodynamic parameters corresponding to these driving forces in the protein association process were characterized by Ross and Subramanian [START_REF] Eftink | Fluorescence quenching studies with proteins[END_REF].

It can be concluded from Table 2 that the binding procedure between Coumarin and βCN is spontaneous and mainly enthalpy driven. In advance, the negative values obtained for ΔH° and ΔS° indicates the significance of hydrogen bonding and van der Waals interactions in the binding process. Therefore, it can be concluded that both hydrogen bonding and van der Waals interactions play an essential role in the binding of Coumarin to βCN.

Finally, to identify the quenching mechanism of tyrosine and phenylalanine residues of βCN, the intrinsic fluorescence of protein was measured in the presence of various amounts of Coumarin at four temperatures ranging from 293-308 K at the excitation wavelength of 280 nm. With respect to Fig. S1, a positive deviation from linearity of S0/S vs. [Coumarin] plot is observed. In fact, this usually happens when two different mechanisms of quenching (i.e. dynamic and static quenching) occur synchronously. Hence, the static quenching mechanism for tryptophan and dynamics for tyrosine and phenylalanine could be concluded.

Fluorescence Resonance Energy Transfer (FRET) from βCN to Coumarin

Fluorescence resonance energy transfer (FRET) is a mechanism describing energy transfer between two lightsensitive molecules (chromophores). A donor chromophore, initially in its electronic excited state, may transfer energy to an acceptor chromophore through the nonradiative dipole-dipole coupling. The efficiency of this phenomenon is inversely proportional to the sixth power of the distance between donor and acceptor, making FRET extremely sensitive to small changes in distance. Measurements of FRET efficiency could be determined if two fluorophores are within a certain distance of each other.

Fluorescence quenching of βCN by Coumarin indicated the possibility of energy transfer between the protein and the bound Coumarin. In the present study, we prepared a solution containing the equimolar concentration of βCN and Coumarin (5 μM), and its UV-Vis absorption and fluorescence spectra were recorded. There is a good overlap between the emission spectrum of βCN and the absorption spectrum of Coumarin (Fig. s1). The efficiency of energy transfer can be calculated by using Eq. ( 5), where F0 and F are the fluorescence intensity of β-casein in the absence and presence of Coumarin, respectively, r is the distance from the bound ligand to the tryptophan residue and R0 is the Forster critical distance at which 50% of the excitation energy is transferred to the acceptor and can be obtained from Eq. ( 6) using K2 = 2/3, N = 1.53, and φ = 1.49 [START_REF] Lakowicz | Principles of Fluorescence Spectroscopy[END_REF], where K2 is a factor describing the relative orientation of the transition dipoles of the protein donor and ligand acceptor, N is the average refractive index of the medium in the wavelength range where the spectral overlap is significant, and φ is the fluorescence quantum yield of the protein donor. Overlap integral (J) expresses the extent of overlap between the normalized fluorescence emission spectrum of the protein donor and the ligand acceptor absorption spectrum and is given by Eq. [START_REF] Grimm | Substituted coumarins as potent 5-lipoxygenase inhibitors[END_REF]. In this equation, F (λ) is the fluorescence intensity of the donor at wavelength λ and is dimensionless and ε (λ) is the molar absorption coefficient of the acceptor at wavelength λ. In this study, the calculated value for J was 7.389 × 10-15 (cm3 M-1). By using the obtained value for E from Eq. ( 5) (0.204) and R0 from Eq. ( 6) (63.250 nm), the r value was calculated to equal to 4.078 nm. It can be seen that the distance from the bound ligand to the tryptophan residue is less than 7 nm indicating a non-radiative energy transfer mechanism for quenching [START_REF] Vörös | The density and refractive index of adsorbing protein layers[END_REF]. Moreover, the value of r is higher than the respective critical distance (R0), hence, the static quenching is more likely responsible than the dynamic mechanism that is in agreement with our conclusion corresponding to the reducing of Ksv with temperature. Moreover, the short distance value between bound ligand and the tryptophan residue represents the significant interaction between Coumarin and βCN.

Molecular Docking Study

Molecular docking studies were implemented to understand the binding site location and the best conformation of Coumarin for binding to βCN using binding free energy assessment. Estimated energies by AutoDock are described by intermolecular energy, internal energy, and torsional free energy. Docking studies showed that Coumarin binds to the hydrophobic core of βCN. The representative build derived from the best pose of Coumarin with the minimal binding energy of -6.96 kcal mol-1 is shown in Fig. 2a. As this figure shows, βCN has 3 helixes and 8 strands in its structure which were held together with random coils. Coumarin is nearby to some hydrophobic residues in the core of βCN such as Leu103, Pro101, Pro100, Phe102 and Ile45. Moreover, some polar residues such as Glu51, Glu46, and Lys63 are in the neighborhood of Coumarin. The docking results showed that Coumarin is able to form a hydrogen bond interaction with Phe102 residue, and therefore, this residue plays a significant role in stabilizing the complex (Fig. 2b).

Molecular Dynamics Simulation Study

The lowest energy structure of Coumarin resulting from docking calculation was selected for the initial structure of 50 ns MD simulation on the ligand-protein complex. To this end, force field parameters of ligand were obtained from PRODRG web server [START_REF] Van Aalten | PRODRG, a program for generating molecular topologies and unique molecular descriptors from coordinates of small molecules[END_REF].

In order to investigate the stability of the βCNCoumarin complex and determination of the types of interactions involved in the binding reaction, some system properties including root mean square deviations (RMSD), radius of gyration (Rg), secondary structure, root mean square fluctuations (RMSF), accessible surface area of protein and hydrogen bonds were investigated.

Root mean square deviations. The standard way to measure the trajectory stability for both βCN and βCNCoumarin complex is to calculate the root-mean-square deviations (RMSD) of backbone atoms of protein from the initial structure, as a function of time. It can be seen in Fig. 3a that the RMSD of protein increases pretty rapidly in the first part of the simulation, but it reaches to equilibrium and oscillates around the average value after about 20000 ps simulation time. In another word, for most of the times (between 20 and 50 ns) the RMSD profile of protein backbone atoms shows similar trends for both systems indicating their stability and equilibration. The mean RMSD values of protein backbone atoms from last 20 ns trajectory were 0.435 ± 0.030 and 0.523 ± 0.021 nm for βCN and βCN-Coumarin complex, respectively.

Radius of gyration.

The radius of gyration (Rg) is a factor which designates the compactness of the system. The radius of gyration of the free and bound form of the protein was determined and plotted as a function of time. As shown in Fig. 3b, in both systems, the Rg values are stabilized at about 25000 ps, indicating the achievement of equilibrium after this time. As seen in this figure, for both systems, the radius of gyration of backbone atoms does not change significantly. This implies a small decrease in the compacting of the βCN structure during the simulation.

Secondary structure analysis. The secondary structure of βCN was calculated for each frame with DSSP program [START_REF] Kabsch | Dictionary of protein secondary structure: Pattern recognition of hydrogen bonded and geometrical features[END_REF]. The result provides the α-helix content and other secondary structures of the protein. It is easy to notice that the main secondary structure of the protein in the presence of Coumarin maintains rather stable during whole MD simulation time (Fig. 3c).

The surface area of protein. Gases command computes hydrophobic, hydrophilic and total solvent accessible surface area. The program will ask for a group for the surface calculation and a group for the output. The calculation group should always consist of all non-solvent atoms in the system. The output group can be the whole or part of the calculation group. The area of the protein in the presence and absence of Coumarin indicates that the surface of βCN due to interaction with Coumarin does not change significantly and this is in agreement with our conclusion corresponding to the radius of gyration (Fig. S2).

Hydrogen bonds. G-h bond command computes the number of hydrogen bonds. The maximum of three hydrogen bonds can be formed between Coumarin and βCN, however, most of the time, only one hydrogen bond is formed which is in good agreement with the docking results (Fig. S2).

Root mean square fluctuations. Vibrations around the equilibrium are not random, and depend on local structure flexibility. The time-averaged root mean square fluctuation (RMSF) values of free βCN and βCN-Coumarin complex were calculated, and the results were plotted against residue numbers (Fig. 4a). As seen in this figure, the binding of Coumarin restricted the conformational space explored by βCN as assessed by a general reduction of the protein RMSF. In the other word, the overall reduction in the RMSF values reveals that binding of Coumarin reduces βCN flexibility. The particular regions directly in contact with Coumarin, including Leu103, Pro101, Glu51, Pro100, Phe102, Glu46, Phe48, Lys47, Ile45, Lys63 and Pro168 residues (pointed in Fig. 4a) show the more significant reduction in RMSF that is related to intermolecular interactions of these residues with the ligand.

Moreover, the RMSF of the atomic positions of the Coumarin was calculated to examine its conformational variation (Fig. 4b). The results indicated that the Coumarin atoms show limited fluctuations (<0.03 nm). Hence, it can be concluded that the interactions of βCN and Coumarin were stable during the simulation time.

CONCLUSIONS

This paper describes the interaction mechanism of Coumarin with bovine β-Casein using fluorescence spectroscopy, molecular docking, and MD simulation studies. The fluorescence studies presented here revealed the static quenching of βCN fluorescence and the formation of 1:1 complex between this protein and Coumarin. The negative values of both enthalpy and entropy changes by running the experiments at various temperatures recommend that hydrogen bonding and van der Waals interactions play important roles in the binding process. Moreover, fluorescence quenching studies revealed the existence of single binding site for Coumarin on βCN with relatively high binding affinity. Förster energy transfer measurements suggested the distance between bound Coumarin and Trp143 residue is higher than the respective critical distance. Therefore, the static quenching is more likely responsible for fluorescence quenching of βCN. The results of computational docking studies indicated that Coumarin probably binds to the hydrophobic core of βCN and the interaction proceeds through van der Waals and hydrogen bond interactions. The RMSD and radius of gyration profiles obtained by MD calculations demonstrated the stability of βCN-Coumarin complex during the simulation time. Analysis of MD data represented the insignificant change of secondary and tertiary structures of βCN due to the binding of Coumarin. Finally, the RMSF profiles and hydrogen bond analysis revealed the insignificant movement of Coumarin from its pose during the whole MD simulation time. To sum up, all experimental and molecular modeling results reciprocally supported each other and clarified that βCN seems to be the promising Coumarin carrier.

  Figure 1a shows fluorescence emission spectra of βCN in various concentrations of Coumarin at 293 K. The quenching effect of Coumarin on fluorescence spectrum of βCN is observed in this figure. Fluorescence quenching refers to any process which decreases the fluorescence intensity of a given substance. The processes such as excited state reactions, energy transfer, complex-formation and collisional quenching can result in quenching. There are two main mechanisms for fluorescence quenching usually called as dynamic and static. Collisional (dynamic) quenching occurs when the excited fluorophore experiences contact with an atom or molecule that can facilitate non radiative transitions to the ground state. In some cases, the fluorophore can form a stable complex with another molecule. The fluorophore is statically quenched, if this ground-state complex is non-fluorescent. In the simplest case of dynamic and static quenching, the following relation, called the Stern-Volmer equation, holds (Juan et al., 2044): where S0 and S are the total area under the emission spectrum in the absence and presence of quencher, [Q] is the quencher concentration and KSV is the Stern-Volmer quenching constant. Plot of S0/S versus [Coumarin] at 293 K is shown in Fig. 1b. The high linearity of this plot represents a good agreement between the results and the Stern-Volmer equation within the investigated concentrations.

ACKNOWLEDGMENTS

The financial support of the Research Council of University of Isfahan and Iran National Science Foundation (grant number 93013717) are gratefully acknowledgment.

Conflict of interest statement

We have no conflicts of interest.

List of Abbreviations

βCN: β-Casein RMSD: Root Mean Square Deviations RMSF: Root Mean Square Fluctuations BDMC: BisDeMethoxy Curcumin MD: Molecular Dynamics TASSER: Threading Assembly Refinement


Monireh Dehkhodaei

Mehdi Sahihi

Amiri Hadi

Mahnaz Rudbari

Sajjad Ariaeefar

Reza Gharaghani

Salman Azadbakht

Abolghasem Abbasi Taheri

Kajani

Hadi Amiri Rudbari

Mahnaz Ariaeefar

Sajjad Gharaghani

Reza Azadbakht

Salman Taheri

Abbasi Kajani

Multi experimental and computational studies for DNA and HSA interaction of new nano-scale ultrasound-assisted synthesized Pd(II) complex as a potent anticancer drug

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Platinum-containing drugs such as cis-platin, carboplatin and oxaliplatin show their anti-cancer activity based on in-vivo interaction with the DNA of the cancerous cells. Studying on the interactions between DNA and the other transition metal complexes is key step to develop new metal based therapeutics [START_REF] Mahalingam | Dimethyl sulfoxide ruthenium (II) complexes of thiosemicarbazones and semicarbazone: synthesis, characterization and biological studies[END_REF]. Furthermore, some studies have investigated binding of the anticancer drugs to the Human Serum Albumin (HSA), as the most abundant plasma carrier protein in the blood stream [START_REF] Khosravi | Synthesis, characterization, crystal structure and HSA binding of two new N, O, O-donor Schiff-base ligands derived from dihydroxybenzaldehyde and tert-butylamine[END_REF][START_REF] Ray | Nickel (II)-Schiff base complex recognizing domain II of bovine and human serum albumin: spectroscopic and docking studies[END_REF]. The binding ability of a drug to the non-specific transporter protein HSA is an important research field in chemistry, biochemistry and clinical medicine [ [START_REF] Belatik | Binding sites of retinol and retinoic acid with serum albumins[END_REF], [START_REF] Dömötör | Antitumor pentamethylcyclopentadienyl rhodium complexes ofmaltol and allomaltol: synthesis, solution speciation and bioactivity[END_REF], [START_REF] Bakaeean | Binding effect of common ions to human serum albumin in the presence of norfloxacin: investigation with spectroscopic and zeta potential approaches[END_REF], [START_REF] Chamani | Cooperative α-helix formation of β-lactoglobulin induced by sodium n-alkyl sulfates[END_REF], [START_REF] Chamani | Calorimetric indication of the molten globule-like state of cytochrome c induced by nalkyl sulfates at low concentrations[END_REF], [START_REF] Hamed-Akbari Tousi | Comparing the interaction of cyclophosphamide monohydrate to human serum albumin as opposed to holotransferrin by spectroscopic andmolecularmodeling methods: evidence for allocating the binding site[END_REF], [START_REF] Marouzi | Study on effect of lomefloxacin on human holo-transferrin in the presence of essential and nonessential amino acids: spectroscopic and molecular modeling approaches[END_REF], [START_REF] Omidvar | Studies on the antagonistic behavior between cyclophosphamide hydrochloride and aspirin with human serum albumin: timeresolved fluorescence spectroscopy and isothermal titration calorimetry[END_REF], [START_REF] Rashidipour | Study of the interaction between DNP and DIDS with human hemoglobin as binary and ternary systems: spectroscopic and molecular modeling investigation[END_REF], [START_REF] Sharif-Barfeh | Multispectroscopic and HPLC studies of the interaction between estradiol and cyclophosphamide with human serum albumin: binary and ternary systems[END_REF]]. Hence, investigation of the interaction between HSA and small drug molecules is very important for understanding the mystery of life and the interaction mechanisms of drugs. The results of investigation show that more than 90% plasma protein can be combined with metal drugs in form of covalent bonds. Studying on the metal base drugs is one of the basic area in bioinorganic chemistry field. The researchers in this field try to discover compounds with more impressive properties and less side effects from cis-platin that is known as appropriate anti-cancer drug. One of the approaches is using the synergic effect between metals and biological ligands. Schiff bases are the class of inorganic ligands that their biological activities have been proven, previously [ [START_REF] Dehkhodaei | Spectroscopic and molecular docking studies on the interaction of Pd (II) & Co (II) Schiff base complexes with βlactoglobulin as a carrier protein[END_REF], [START_REF] Dehkhodaei | Studies of DNA-and HSA-binding properties of new nano-scale green synthesized Ni (II) complex as anticancer agent using spectroscopic methods, viscosity measurement, molecular docking, MD simulation and QM/MM[END_REF], [START_REF] Dehkhodaei | DNA and HSA interaction of Vanadium (IV), Copper (II), and Zinc (II) complexes derived from an asymmetric bidentate Schiff-base ligand: multi spectroscopic, viscosity measurements, molecular docking, and ONIOM studies[END_REF]]. On the other hand, the simplistic synthesis and wide range of applications of Schiff base ligands have attracted a lot of interests. These compounds play an important role in development of coordination chemistry related to catalysis, organic synthesis [START_REF] Antony | Synthesis, spectrochemical characterisation and catalytic activity of transition metal complexes derived fromSchiff basemodified chitosan[END_REF][START_REF] Rosu | Synthesis, structural and spectral studies of Cu (II) and V (IV) complexes of a novel Schiff base derived from pyridoxal. Antimicrobial activity[END_REF], antitumor, antimicrobial and cytotoxic activity [ [START_REF] El-Halim | Synthesis, structural, thermal studies and biological activity of a tridentate Schiff base ligand and their transition metal complexes[END_REF], [START_REF] Patil | Spectroscopic, in vitro antibacterial, and antifungal studies of Co (II), Ni (II), and Cu (II) complexes with 4-chloro-3-coumarinaldehyde Schiff bases[END_REF], [START_REF] Singh | Cobalt, nickel, copper and zinc complexes with 1, 3-diphenyl-1H-pyrazole-4-carboxaldehyde Schiff bases: antimicrobial, spectroscopic, thermal and fluorescence studies[END_REF], [START_REF] You | Schiff base complexes as novel inhibitors of xanthine oxidase[END_REF]]. In recent years, remarkable attention has been paid to biological applications of the metal complexes of Schiff bases due to their stability, biocompatibility and biological activity [START_REF] Arjmand | Synthesis of new chiral heterocyclic Schiff base modulated Cu (II)/Zn (II) complexes: their comparative binding studies with CTDNA, mononucleotides and cleavage activity[END_REF][START_REF] Pravin | Investigation of in vitro anticancer and DNA strap interactions in live cells using carboplatin type Cu (II) and Zn (II) metalloinsertors[END_REF]. Their biological activity is due to the existence of imine group in their chemical structure [ [START_REF] Antony | Synthesis, spectroscopic and catalytic studies of Cu (II), Co (II) and Ni (II) complexes immobilized on Schiff base modified chitosan[END_REF], [START_REF] Guo | Antifungal properties of Schiff bases of chitosan, N-substituted chitosan and quaternized chitosan[END_REF], [START_REF] Raman | Metallation of ethylenediamine based Schiff base with biologically active Cu (II), Ni (II) and Zn (II) ions: synthesis, spectroscopic characterization, electrochemical behaviour, DNA binding, photonuclease activity and in vitro antimicrobial efficacy[END_REF]]. Furthermore, complexation of Schiff base ligands with transition metal ions, enhances their biological activities [ [START_REF] Anacona | Synthesis, characterization and antibacterial activity of a Schiff base derived from cephalexin and sulphathiazole and its transition metal complexes[END_REF], [START_REF] Sundararajan | Synthesis of metal complexes involving Schiff base ligand with methylenedioxy moiety: spectral, thermal, XRD and antimicrobial studies[END_REF], [START_REF] Tarushi | Interaction of Zn (II) with quinolone drugs: structure and biological evaluation[END_REF]]. Some palladium complexes had shown promising results as antimalarial and anti-cancer agents in previous investigations [START_REF] Chellan | Synthesis, structure and in vitro biological screening of palladium (II) complexes of functionalised salicylaldimine thiosemicarbazones as antimalarial and anticancer agents[END_REF][START_REF] Prabhakaran | Structural and biological studies of mononuclear palladium (II) complexes containing N-substituted thiosemicarbazones[END_REF]. Also, due to remarkable similarity between Pd(II) and Pt(II) complexes [START_REF] Shoukry | Synthesis, DNA binding and complex formation reactions of 3-amino-5, 6-dimethyl-1, 2, 4-triazine with Pd (II) and some selected biorelevant ligands[END_REF]; and their less toxicity and higher activity in compared to platinum complexes; they are regarded as potential anticancer substances and have better anticancer activity than their Pt(II) analogues in some cases [START_REF] Kapdi | Anti-cancer palladium complexes: a focus on PdX 2 L 2, palladacycles and related complexes[END_REF]. Due to the importance and significant growth of nanotechnology in all sciences, its role in the field of biology and drug design cannot be ignored. Previous studies showed that the interaction between nanoparticles and biomolecules can change the biomolecule conformation and perturb the normal protein function which can be lead to unexpected biological reactions and toxicity. Therefore, investigation of these types of interactions could not be ignored [START_REF] Calzolai | Protein -nanoparticle interaction: identification of the ubiquitin -gold nanoparticle interaction site[END_REF]. Herein, a new Schiff base Pd(II) complex (PdL2) was synthesized and its molecular structure was determined by single crystal X-ray diffraction technique. The cell viability percent of HeLa cancer cells was first studied by MTT assay. In order to increase the colloidal stability and suitability for the biomedical applications, the nano-scale compound was also synthesized using ultrasoundassisted method and used for in vitro studies. Finally, binding ability of the nano-and bulk-scale PdL2 with calf thymus DNA (CT-DNA) and HSA was investigated using combination of experimental (fluorescence, circular dichroism (CD) and viscosity) and computational (molecular docking, molecular dynamics simulation and qm/mm) methods.

Experimental section

Materials sources

All of the used chemicals for synthesis of the complex including 2-amino-propane (Isopropylamine), 2-hydroxybenzaldehyde (salicylaldehyde), triethylamine and Palladium chloride (PdCl2) were purchased from Merck Co. and were used without further purification. All of the used salts for buffer preparation were analytical grade and were dissolved in double distilled water. All of the solutions were used freshly after preparation. Also, HSA, CT-DNA, RPMI-1640 medium, Fetal bovine serum (FBS), Dimethyl sulfoxide (DMSO), antibiotics (penicillinstreptomycin) solution, and 3-(4,5-Dimethylthiazol-2-yl)-2,5-Diphenyltetrazolium Bromide (MTT) were obtained from Sigma-Aldrich.

Synthesis of the bulk-and nano-scale Pd(II) Schiff base complex (PdL2)

A methanolic solution (30 ml) of Isopropylamine (IUPAC name: 2-amino-propane) (5 mmol) was added slowly to 30 ml of a methanolic stirred solution of salicylaldehyde (IUPAC name: 2hydroxybenzaldehyde) (5 mmol) in ambient temperature. The colour immediately changed to yellow and the mixture was then stirred for 2 h. Then a solution of triethylamine (7 mmol) in absolute methanol (10 ml) was added dropwise to ligand solution. The mixture was stirred for 15 min again. Then, a solution of appropriate PdCl2 (2.5 mmol) in methanol (30 ml) was added to the mixture, gently, for synthesis of the complex in bulk-scale. The solution was refluxed overnight to proceed completely. The resulting yellow powder was isolated by filtration and washed by dry methanol several times. Appropriate single crystals for X-ray crystallography were obtained directly from the reaction mixture. In order to prepare the complex in nano scale, a saturated solution was prepared by adding excess powder to 10 ml of DMF at room temperature. The solution was pumped from a small orifice into 100 ml of the antisolvent (water), which was placed in an ultrasonic bath at 180 W for 30 min and was continuously stirred. The suspension was centrifuged at 5000 g for 15 min, and the supernatant was withdrawn and filtered through a 0.2 mm pore-size syringe filter [START_REF] Azadbakht | A novel aluminum-sensitive fluorescent nanochemosensor based on naphthalene macrocyclic[END_REF].

Single crystal diffraction studies

X-ray data for PdL2 complex was collected at room temperature with a Bruker APEX II CCD area detector diffractometer using Mo Kα radiation (k = 0.71073 Å). Data collections, cell refinements, data reductions and absorption corrections were performed using multiscan methods with Bruker software [ [37], [38], [START_REF] Li | Studies of Bicalutamide-excipients interaction by combination of molecular docking and molecular dynamics simulation[END_REF]]. The structure was solved by direct methods using SIR2004 [START_REF] Burla | SIR2004: an improved tool for crystal structure determination and refinement[END_REF]. The non-hydrogen atoms were refined anisotropically by the full matrix least squares method on F 2 using SHELXL [START_REF] Sheldrick | SHELXL-97[END_REF]. All hydrogen atoms were added at ideal positions and constrained to ride on their parent atoms. Molecular graphics were prepared with the Olex 2 program [START_REF] Dolomanov | OLEX2: a complete structure solution, refinement and analysis program[END_REF]. Crystallographic data for complex are listed in Table S1. Selected bond distances and angles are summarized in Table S2.

Cell viability assay

MTT assay was used to investigate the anticancer potential of the bulk-and nano-scale PdL2 on human cervical cancer cell line (HeLa) according to the previously reported procedure [START_REF] Kajani | Gold nanoparticles as potent anticancer agent: green synthesis, characterization, and in vitro study[END_REF]. The cells were first cultured in RPMI-1640 medium supplemented with 10% FBS and 1% antibiotics solution and were maintained in a humidified 5% CO2 incubator at 37 °C. When the cell confluence was reached to 70%, the cells were harvested and seeded on 96-well plates at a density of 10 4 cells per well containing 200 μl medium and incubated overnight at the same conditions. After exposure to different concentrations of each compound for 48 h, the medium was removed and 100 μl MTT solution (0.5 mg•ml -1 in media) was added into each well and the plates were incubated again at 37 °C for 4 h. The medium was then completely removed and the formazan crystals were dissolved in 150 μl DMSO and the absorbance was measured at 570 nm. Each experiment was conducted in triplicate and the results were presented as the mean values obtained from three independent experiments. The cell viability was determined as ratio of absorbance values from each treatment and the control.

Preparation of DNA, HSA, bulk-and nano-scale PdL2 stock solutions for binding experiments

The stock solution of CT-DNA was prepared in 50 mM Tris buffer at pH = 7.5 using doubledistilled deionized water and was stored at 4 °C. The CT-DNA concentration per nucleotide was determined using absorption intensity at 260 nm after adequate dilution with the buffer and using the reported molar absorptivity of 6600 M -1 •cm -1 [START_REF] Reichmann | A further examination of the molecular weight and size of desoxypentose nucleic acid[END_REF]. Purity of CT-DNA solution was confirmed by ratio of UV absorbance at 260 and 280 nm (A260/A280 = 1.9), indicating that CT-DNA is free from protein impurity [START_REF] Zheng | Synthesis, structure and molecular docking studies of dicopper (II) complexes bridged by N-phenolato-N′-[2-(dimethylamino) ethyl] oxamide: the influence of terminal ligands on cytotoxicity and reactivity towards DNA and protein BSA[END_REF]. Also, a stock solution of HSA was prepared by dissolving the desired amount of HSA in 50 mM phosphate buffer (pH = 7). The HSA stock solution was stored at 4 °C in the dark and was used within 2 h. HSA concentration was determined by UV-Vis spectrophotometry using the molar absorption coefficient 35,700 M -1 •cm -1 at 278 nm [START_REF] Pace | How to measure and predict the molar absorption coefficient of a protein[END_REF]. For preparation of PdL2 solution in bulk-scale, appropriate amounts of obtained complex powder was dissolved in DMSO as co-solvent, and then diluted with corresponding buffer to the required concentration for all experiments. The volume of co-solvent never exceeded 5% (v/v), so the effect of DMSO is negligible. Also, the appropriate amount of the nano-scale PdL2 powder was dispersed in Tris and phosphate buffer for DNA and HSA binding experiments, respectively.

Fluorescence spectroscopy measurements

Fluorescence experiments were carried out using quartz cuvette with 1 cm optical path length and the excitation and emission slits were set at 5 and 10 nm, respectively. For investigation of HSA interaction with PdL2 by this technique, 2 ml of HSA solution was placed into the quartz cell and various amounts of PdL2 were added to the cell (for both of bulk-and nano-scale). The fluorescence intensities were measured with excitation wavelength at 295 nm and emission wavelength range of 300-450 nm. In each measurement, the mixture was allowed to incubate for 2 min after addition of the compound. Furthermore, all intensities were corrected for the dilution effect in fluorescence experiments. Also, for investigation of interaction between PdL2 compound and DNA, the CT-DNA solution was stirred with EB with molar ratio of CT-DNA: EB 10:1 and was incubated for 1 h at 4 °C for completion of interaction between CT-DNA and EB. Then, various amounts of PdL2 compound were added to the mixture of CT-DNA: EB. The fluorescence spectra were measured in the range of 500-700 nm with exciting wavelength at 520 nm. In each measurement after addition of PdL2, the mixture was incubated for 2 min. Moreover, the measured fluorescence intensities were corrected for the dilution effect (for both of bulk-and nano-scale). Also, the measured fluorescence intensities were corrected for the inner-filter effect. To eliminate the inner filter effects, absorption measurements were carried out at the fluorescence excitation and emission wavelengths. The extent of this effect can be roughly evaluated with the following relationship: [START_REF] Macdonald | Correction of fluorescence inner filter effects and the partitioning of pyrene to dissolved organic carbon[END_REF] where, Fcorr and Fobs are the corrected and observed fluorescence intensities, respectively, and Aex and Aem are the absorption of PdL2 at excitation and emission wavelengths, respectively. All the fluorescence intensities obtained in this study were corrected.

Circular dichroism spectroscopy

The circular dichroism (CD) spectra were recorded at room temperature using 0.1 cm quartz cell in the near-UV region (190-260 nm) for HSA. The CD spectrum of HSA solution was recorded before and after addition of the nano-scale PdL2 complex with molar ratio 1:1. The analysis CD of HSA spectra were carried out by CDNN software. The content of secondary conformation of HSA in each step was calculated from this technique.

Viscosity measurement

Viscosity experiments were carried out using a rotational viscometer and the measurements were performed at 200 rpm at room temperature. The viscosity of CT-DNA solution was measured in the presence of various amounts of the bulk and nano-scale PdL2. The obtained data are presented as (η/η0) 1/3 versus [PdL2]/[CT-DNA], where η and η0 are viscosity of CT-DNA in the absence and presence of PdL2, respectively.

Computational methods

Molecular docking calculations

Docking study was carried out to indicate the HSA and DNA-binding site for PdL2 complex. The 3D structure of PdL2 was obtained using the .cif file of its X-ray crystal structure. The .cif file was converted to the .pdb format using the Mercury software (http://www.ccdc.cam.ac.uk/). The crystal structures of HSA (PDB ID: 1AO6) and DNA (PDB ID: 423D) with sequence d(ACCGACGTCGGT)2 were taken from the Brookhaven Protein Data Bank (http://www.rcsb.org/pdb). The resolution of these files was 2.5 and 1.6 Å for HSA and DNA, respectively. Water molecules of the .pdb files were deleted, missing hydrogen atoms and Gasteiger charges were added. Flexible ligand docking was carried out by AutoDock 4.2.5.1 molecular docking program using the implemented empirical free energy function and the Lamarckian Genetic Algorithm [START_REF] Morris | Automated docking using a Lamarckian genetic algorithm and an empirical binding free energy function[END_REF]. The Gasteiger charges were added to prepare the macromolecule input file for docking and the AutoGrid was used to calculate the grids. For docking of the PdL2 complex to HSA and DNA, a blind docking with 126 lattice points along X, Y, and Z axes was performed to find the binding site of complex on the biomacromolecules. Grid point spacing was set to 0.375 Å, to allow the complex to rotate freely. In the next step, the centre of the grid box was located at the binding site and the second docking was performed using a cubic box with 60 × 60 × 60 dimensions. 250 docking runs with 25,000,000 energy evaluations for all runs were performed.

Molecular dynamics (MD) simulation

MD simulation method was used to compare the structural changes of HSA and DNA in the absence and presence of PdL2 complex. The PdL2-HSA and PdL2-DNA complexes with the most negative free binding energy were considered as the initial conformations for the MD studies. GROMACS 5.1.2 package [START_REF] Berendsen | GROMACS: a message-passing parallel molecular dynamics implementation[END_REF] and Amber99 force field [START_REF] Pérez | Refinement of the AMBER force field for nucleic acids: improving the description of α/γ conformers[END_REF][START_REF] Wang | Development and testing of a general amber force field[END_REF] were used to carry out all MD studies. The partial atomic charges of PdL2 were calculated using Gaussian 03 [START_REF] Frisch | Gaussian 03. U. User's Reference: Manual Version 7.1[END_REF] at the level of B3LYP/LanL2dz [START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF], while the other intramolecular force-field parameters were generated with ACPYPE [START_REF] Da | ACPYPE -antechamber python parser interface[END_REF] and VFFDT [START_REF] Zheng | VFFDT: a new software for preparing AMBER force field parameters for metal-containing molecular systems[END_REF] for the organic moiety and the metal center, respectively. Free biomacromolecules and PdL2-biomacromolecule complexes were located in the cubic box with the periodic boundary conditions in the three directions. The solutes were placed in the center of box and the minimum distance between solute surface and the box was 1.0 nm. The box was filled with TIP3P water molecules [START_REF] Jorgensen | Comparison of simple potential functions for simulating liquid water[END_REF][START_REF] Mark | Structure and dynamics of the TIP3P, SPC, and SPC/E water models at 298 K[END_REF], and the solvated systems were neutralized by adding appropriate amounts of sodium ions (Na + ) and chloride ions (Cl -). After energy minimization using the steepest descent method, the systems were equilibrated for 200 ps at the temperature of 300 K. Finally, a 10 ns MD simulation was carried out at 1 bar and 300 K. Parrinello-Rahman barostate [START_REF] Parrinello | Polymorphic transitions in single crystals: a new molecular dynamics method[END_REF][START_REF] Rahman | Molecular dynamics study of liquid water[END_REF]] at 1 bar, Berendsen thermostat [START_REF] Lemak | On the Berendsen thermostat[END_REF] at 300 K. 9 Å cut off for Van der Walls (VdW) and Coulomb interactions and the particle mesh Ewald (PME) method [START_REF] Darden | Particle mesh Ewald: an N• log (N) method for Ewald sums in large systems[END_REF][START_REF] Essmann | A smooth particle mesh Ewald method[END_REF] for long range electrostatics were used. The leap-frog algorithm with the 2 fs time step was used to integrate the equation of motions. Finally, an all-bond constrain was used to keep the PdL2 from drifting and the atomic coordinates were recorded to the trajectory file every 0.5 ps for later analysis.

QM/MM study

Own N-layered Integrated Molecular Orbital and molecular Mechanics (ONIOM) methodology was employed to carry out QM/MM calculations. The ONIOM can be considered as a hybrid method of quantum mechanical method (QM) with a molecular mechanics (MM). This method is able to apply different ab initio or semi-empirical methods to different parts of a system and combine them to produce reliable geometry and energy at reduced computational time [START_REF] Farrokhpour | A rare dihydroxo copper (II) complex with ciprofloxacin; a combined experimental and ONIOM computational study of the interaction of the complex with DNA and BSA[END_REF]. In the two layers ONIOM method, the total energy (EONIOM) of the whole system is a sum of the three independent energy calculations: [START_REF] Vreven | Combining quantum mechanics methods with molecular mechanics methods in ONIOM[END_REF].

Real system contains full geometry of the molecule and is considered as MM layer while the model system contains the chemically most important (core) part of the system that is considered as QM layer. Furthermore, the stability of respective adducts was obtained by the evaluation the interaction energy (ΔE), which is calculated by the following equation:

where ΔEBiomacromolecule/PdL2, ΔEBiomacromolecule and ΔEPdL2 are the energy of the optimized adduct of Biomacromolecule-PdL2, Biomacromolecule and PdL2, respectively. Biomacromolecule is HSA or DNA here as receptors. A two layers QM/MM method was used for all calculations. HSA and DNA are considered as low layer while the PdL2 is considered as high layer in the current work. Molecular mechanics method (UFF) and semi empirical method (PM6) were selected for the low layer and the high layer, respectively.

Results and discussion

Results of experimental sections

PdL2 synthesis and single crystal X-ray diffraction study

The reaction of the Schiff base ligand with metal ion is shown in Scheme 1A. Also, An ORTEP view of PdL2 with common atom numbering is shown in Scheme 1B. In the solid state, PdL2 has a crystallographic center of symmetry that is right in the middle point of the palladium (II) ion in a planar-transoid conformation (Scheme 1B). The molecular unit is centrosymmetric and is made up of equivalent halves. The structure of PdL2 displays a four coordinate palladium atom coordinated to two bidentate N-isopropylsalicylaldimine anions in a slightly distorted square planer geometry. Each Schiff base ligand acts as a chelate using a phenolato oxygen and an imino nitrogen. The deprotonated isopropylsalicylaldimine ligands are arranged with respect to each other in a trans orientation such that one isopropyl substituent lies above the [PdO2N2] plane and the other below. The Pd O distance [1.9756( 15) Å] and the Pd N bond length [2.0368( 15) Å] are similar to those were seen in our previous work [START_REF] Kazemi | Synthesis, characterization and biological application of four novel metal-Schiff base complexes derived from allylamine and their interactions with human serum albumin: experimental, molecular docking and ONIOM computational study[END_REF][START_REF] Kazemi | Synthesis, characterization and separation of chiral and achiral diastereomers of Schiff base Pd (II) complex: a comparative study of their DNA-and HSA-binding[END_REF]. For instance, distances of 1.982 [START_REF] Khosravi | Synthesis, characterization, crystal structure and HSA binding of two new N, O, O-donor Schiff-base ligands derived from dihydroxybenzaldehyde and tert-butylamine[END_REF] Å for Pd O bond and 2.017(2) Å for Pd N bond are found in Pd(II) complex with R/S-(1phenylethylimino)methylnaphtalen-2-ol ligand [START_REF] Kazemi | Synthesis, characterization and separation of chiral and achiral diastereomers of Schiff base Pd (II) complex: a comparative study of their DNA-and HSA-binding[END_REF]. 6)°] also confirm the distorted square planer geometry for this complex. The crystallographic data and selected bond lengths and angles are collected in Tables S1-2.

Scanning electron microscopy (SEM) measurements

Scheme 1C and D show the morphology of synthesized particles that were investigated under SEM for bulk-and nano-scale PdL2, respectively. In Scheme 1C the particles exhibit an approximately filament shape and their particles size distribution is about 500-1000 nm. While, in Scheme 1D the particles have sphere shape and their size distribution is under 200 nm. As it can be seen in Scheme 1C and D, the synthesized nano particles have more uniform distribution than the particles in bulk-scale.

MTT assay

The results of MTT assay after 48 h exposure of HeLa cancer cells to different concentrations of bulk and nano-scale PdL2 were presented in Fig. 1. Interestingly, the nano-scale PdL2 displayed potent anticancer activity as more than 35% cell mortality was obtained after exposure to 2.4 μM while increasing the concentration to 10 μM led to more than 53% of cell mortality. In contrast, significant less anticancer activity was observed in the presence of the bulk-scale PdL2 as only 29.56% mortality was obtained in the presence of 10 μM bulk-scale compound and increasing the concentration to 100 μM was necessary in order to reach the IC50 level. The results clearly indicated the higher anticancer activity of nano-scale PdL2 than bulk-scale one. It seems that decreasing the size of the compound significantly affect its biological activity probably due to its higher bioavailability.

Fluorescence study for interaction of bulk and nano-scale PdL2 with CT-DNA and HSA

The fluorescence spectroscopy has been performed as a reliable and sensitive technique to obtain information about interaction between bulk-and nano-complex PdL2 with CT-DNA and HSA. Fig. 2A andC represent the fluorescence quenching of HSA in the presence of various amount of bulk-and nano-scale PdL2, respectively. Actually, the observed fluorescence emission of HSA is mainly due to the tryptophan residue. Fig. 2A and C represent that adding the complex in both of scales, leads to changes of microenvironment around the Trp-214 residue in HSA and causes the fluorescence quenching of HSA. This implies the appropriate interaction of the compound with HSA as a carrier protein. The appearance of isosbestic point signifies that equilibrium is established between HSA and the compound and reflects the formation of the ground state complex between them. Scachard equation was used to obtain the binding affinity: [START_REF] Lakowicz | Plasmonics in biology and plasmon-controlled fluorescence[END_REF]. In this equation, F0 and F are the fluorescence intensity of biomacromolecule in the absence and presence of the compound, respectively. [Q] is the concentration of quencher that quencher is bulkand nano-scale PdL2, here. "Kb" is binding constant and is obtained from the plot of Ln((F0-F)/F) versus Ln [Q] as y-intercept. These plots are shown in Fig. 2B andD. "n" is the number of binding site per biomacromolecule and is determined as slope of the plot. The values of n are nearly 1 for PdL2 in both of scales indicating the compound binds to HSA with molar ratio of 1:1. The calculated results are presented in Table 1. To determine the interaction mode of the bulk and nano-scale PdL2 dispersion with HSA, the Stern-Volmer quenching plots were obtained by monitoring the fluorescence quenching of biomacromolecules with increasing the concentration of the compound according to the Stern-Volmer equation [START_REF] Boaz | The quenching of fluorescence. Deviations from the Stern-Volmer law[END_REF]:

where, F0 and F are the fluorescence intensity of biomacromolecules in the absence and presence of the compound, respectively. Ksv is the Stern-Volmer quenching constant, kq is the quenching rate constant of biomacromolecules and τ is the average lifetime of biomacromolecules without quencher which is typically equal to 10 -8 s for biomacromolecules. Ksv is determined from the plot of F0/F vs. [Q] (Fig. S1). Fluorescence quenching is classified as two mechanisms: static quenching and dynamic quenching. In the static mechanism, the fluorophore and the quencher collide together in the ground state while fluorophore and quencher collide together in the excited state in dynamic mechanism. Linearity of the Stern-Volmer plot indicates that fluorescence quenching has only one mechanism: dynamic or static. In this case, if the calculated value of kq was greater than limiting diffusion rate constant of the diffusional quenching for biopolymers (2 × 10 10 M -1 •S -1 ) support that quenching fluorescence occurs by static mechanism and against that will be dynamic mode. Here, based on the linearity of Stern-Volmer plot for nano scale (Fig. S1B) and the obtained value of kq (about 10 12 M -1 •S -1 ) fluorescence quenching occurs by static mechanism. While, nonlinearity of Stern-Volmer plot for bulk scale Pd-HSA complex indicates that fluorescence quenching is occurred by combination of static and dynamic modes (see Fig. S1A).

On the other hand, investigation of interaction of the compounds with DNA as the first intracellular target for cancer cells is very important. Plots of Ln((F-F0)/F) versus Ln[PdL2] for obtaining the binding affinity (Kb) of the compound to CT-DNA and the number of binding sites on CT-DNA are present in Fig. 3B andD. The results obtained are collected in Table 1. The values of n for PdL2 on CT-DNA is similar to HSA and is nearly 1. It means that compound binds to CT-DNA with molar ratio of 1:1 (see Table 1). The obtained Kb values indicate that PdL2 in its nano-scale interacts stronger than its bulk-scale to CT-DNA. However, the effect of its size on interaction with HSA is negligible.

To determine the interaction mode of the bulk and nano-scale PdL2 dispersion with DNA, the F/F0 vs. [E] plots were obtained by monitoring the fluorescence enhancement of biomacromolecules with increasing the concentration of the compound according to the following equation [START_REF] Anger | Enhancement and quenching of single-molecule fluorescence[END_REF]:

where, F0 and F are the fluorescence intensity of biomacromolecules in the absence and presence of the compound, respectively. KE is the enhancement rate constant of biomacromolecules. KE is determined from the plot of F/F0 vs. [E] (Fig. S1).

Circular dichroism study

The CD spectra of HSA in free and bonded to nano-scale PdL2 complex states are shown in Fig. 4. The CD spectrum of HSA indicates two negative peaks at 208 and 222 nm which are characteristic absorption peaks of α-helical structure of protein attributed to the n-π* transfer for the bonds of the α-helix [START_REF] Frisch | Gaussian 03. U. User's Reference: Manual Version 7.1[END_REF][START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF]. According to Fig. 4, the intensity of HSA peak has been altered slightly due to the binding of nano-complex. This observation was confirmed by CDNN software analysis for protein secondary structure contents (Fig. 4).

Viscosity measurement

To further verify the interaction mode of the bulk-and nano-scale PdL2 with CT-DNA, viscosity measurements of CT-DNA upon addition of the bulk-and nano-scale PdL2 were carried out. A classical intercalative mode causes an increase in the DNA viscosity due to increasing of overall length of DNA [START_REF] Kazemi | Synthesis, characterization and separation of chiral and achiral diastereomers of Schiff base Pd (II) complex: a comparative study of their DNA-and HSA-binding[END_REF]. On the other hand, groove binding and electrostatic interactions could bend the DNA helix, reduce its length and cause the reduction or no change in the DNA viscosity [START_REF] Kazemi | Synthesis, characterization and separation of chiral and achiral diastereomers of Schiff base Pd (II) complex: a comparative study of their DNA-and HSA-binding[END_REF].

According to Fig. S2, the viscosity of CT-DNA decreased until molar ratio of [PdL2]/[DNA] = 0.3 and 1.2 in the presence of bulk-and nano-scale PdL2, respectively and remained constant afterward. The observed trends confirmed that the binding mode of PdL2 in its both of scales, may be groove binding.

Results of computational sections

Molecular docking studies

Molecular docking is a virtual powerful tool for drug design. This computational technique helps researchers to predict and obtain first insight about stable structure of receptor-ligand complex. In order to find out the location of the compound on DNA and HSA, molecular docking studies were carried out. The results of molecular docking for interaction of PdL2 and HSA indicate that the compound is situated in the IB subdomain that is known as third drug binding site of HSA. HSA acts as a carrier for drugs such as Bilirubin, Lidocaine, Indomethacin, Hemin and so on. These drugs bind to IB subdomain of HSA and transfer in body [START_REF] Varshney | Ligand binding strategies of human serum albumin: how can the cargo be utilized?[END_REF]. Fig. 5A represents the location of PdL2 and its related cluster during its interaction with HSA. The standard binding free energy (ΔG°) of this interaction is about -8.08 kcal•mol -1 . This score is related to the docking cluster with maximum population and minimum binding energy (see Fig. 5A). Furthermore, intermolecular energy which covers H-bonding interactions, Van der Walls, desolvation and electrostatic energies [START_REF] Li | Studies of Bicalutamide-excipients interaction by combination of molecular docking and molecular dynamics simulation[END_REF], was estimated as -8.63 kcal•mol -1 . The amino acids residues with major role in the stability of PdL2-HSA system are exhibited in Fig. 5A. Also, the docked model in Fig. 5B for PdL2-DNA system suggests that PdL2 binds in the minor groove of DNA with ΔG° and intermolecular energy of -5.48 and - 6.03 kcal•mol -1 , respectively. These energies are related to docking conformation (1-1) which has maximum population and minimum binding energy. This result is in good agreement with obtained results of viscosity measurements.

Molecular dynamics simulation

One of the exact computational approaches for investigation of interaction details between biomacromolecules and small molecules is molecular dynamics (MD) simulation. MD simulation studies for PdL2 complex with DNA and HSA were carried out and the beginning structures for the MD analyses were selected from the conformations with lowest docking energies. The stability of the system (biomacromolecule and compound) properties was examined by means of RMS deviations (RMSD) of unbound and ligand conjugated biomacromolecule with respect to the initial structure, RMS fluctuations (RMSF), the solvent accessible surface area (SASA) of biomacromolecule, radius of gyration (Rg) number of H-bonds. Preliminary simulations over 10 ns time were performed on DNA and HSA with PdL2. Table 2 provides the average values of RMSD, SASA, Rg and number of H-bonds of the HSA, HSA-PdL2, DNA and DNA-PdL2 systems. Fig. 6 represents time dependence of RMSD, Rg, H-bond and RMSF values of HSA and HSA-PdL2 systems. Fig. 6A indicates that the trajectories of both systems are stable and their RMSD reached equilibrium and fluctuated around its mean value after about 5 ns simulation time. Figs. 6B andS3 show that complexation with PdL2 changed the structure of HSA with compared to free protein. It can be clearly seen that the Rg was smaller upon binding of the PdL2 suggesting a more compact structure of HSA in the presence of PdL2. Decreasing the number of intramolecular H-bonds is another evidence that confirms a modified structure of HSA in the presence of PdL2 (Fig. 6C). Also, local protein mobility was analyzed by calculating the time averaged RMSF values of free HSA and HSA-PdL2 complex and was plotted against residue numbers based on the last 5000 ps trajectory (Fig. 6D). The profiles of atomic fluctuations were found to be very similar to those of free HSA and HSA-PdL2 complex. RMSF highlights the conformational adjustments of the protein structure, in conjunction with PdL2 conformational adaptation to its binding sites. The results indicate that the residues that were in contact with the PdL2 are stable and have low RMSF values suggesting that the structure of drug binding site remains rigid during simulation. Hence, it can be concluded that the interactions of protein and PdL2 were stable during the simulation time. Fig. 7 represents time dependence of RMSD and RMSF values of DNA and DNA-PdL2 systems. Fig. 7A indicates that the trajectories of both systems are stable and their RMSDs reach equilibrium and fluctuate around their mean values after about 1 ns simulation time. Also, local DNA mobility was analyzed by calculating the time averaged RMSF values of free DNA and DNA-PdL2 complex and was plotted against atom numbers based on the last 8000 ps trajectory (Fig. 7B). The results indicate that the atoms of DNA bound PdL2 are stable and have lower RMSF values than the free DNA. Hence, it can be concluded that the interactions of DNA and PdL2 were stable during the simulation time.

QM/MM studies

In this work a two-layer ONIOM calculations including (PM6: UFF) were employed to perform QM/MM studies. The molecular mechanics (MM) was described using the UFF force field for HSA and DNA (low layer) while semi-empirical quantum mechanics (QM) method (PM6) was chosen for the PdL2 (high layer). The most stable geometries obtained from the molecular docking calculations were opted as starting points of HSA-and DNA-PdL2 for the two layer ONIOM studies. All calculations were performed using Gaussian 03 quantum chemistry package. The partial atomic charges on the atoms of the PdL2, DNA and HSA were considered to reoptimize the optimized geometries. The Van der Waals and electrostatic interaction were considered in structure optimization. Moreover, the polarization of the wave function of the PdL2 was considered to obtain more truthful geometries. The results of ONIOM calculations represented that structure of the PdL2 deviates from its initial geometry due to binding to HSA and DNA. Along with the interaction of PdL2 with biomacromolecules some bond lengths and bond angles are changed. Such the changes can be indicative of the strength of the interaction between biomacromolecules and PdL2. Fig. 8, Fig. 9 show the changes take place in some geometrical parameters including bond angles of PdL2 along with binding to HSA and DNA. Also, Fig. 9 shows that the distances between the nucleotides were changed due to DNA interaction with PdL2 complex. The calculated binding energies (ΔEONIOM) of PdL2 are 4853.58 and - 2883.28 kcal•mol -1 for HSA and DNA, respectively. In view of considering the polarization of the PdL2 in electrostatic field of HSA and DNA in QM/MM calculation, this energy is more accurate than calculated energy by molecular docking method.

Conclusion

Here, synthesis of novel Pd(II) complex in bulk and nano scales using ultrasonic waves was described. The structure of complex was established by single crystal X-ray diffraction. The MMT assay results indicated that the anticancer activity of the compound is affected by its size and notable point is that nano-scale complex reached to IC50 in 10 μM of concentration. Investigation of binding affinity, type of interactions and probable changing structure due to interaction of PdL2 with HSA and DNA, for both of bulk and nano scales complex were carried out by versatile experimental (Fluorescence spectroscopy, viscosity measurement, CD spectroscopy) and computational techniques (molecular docking, ONIOM and MD simulation). According to the obtained results of binding experiments, it can be found that the size decreasing has positive effect on the binding affinity. Circular dichroism spectroscopy was used to investigate the changes on the secondary structure of HSA upon its interaction with the complex. The existence of slight changing in the HSA peak intensity indicated that the secondary structure of protein has been altered slightly due to the binding of complex. The molecular docking studies exhibited that hydrophobic interaction have a dominant role in the stability of complex-HSA and DNA systems and PdL2 bind to the minor groove of DNA and IB-binding site of protein. MD simulation and ONIOM studies revealed that conformation of the complex and the biomacromolecules changed due to the strong interaction between the compound and these biomacromolecules.

Scheme 1 .

 1 Scheme 1. A) Synthetic routes for preparation of PdL2. B) ORTEP representation of PdL2. Displacement ellipsoids are drawn at the 30% probability level and H atoms are shown as small spheres of arbitrary radii. C) SEM image of the bulk-scale PdL2 complex. D) SEM image of the nano-scale PdL2 complex.

Fig. 1 .

 1 Fig. 1. The viability percentage of HeLa cancer cells after 48 h exposure to different concentrations of bulk-and nano-scale PdL2.

Fig. 2 .

 2 Fig. 2. A) Changes in the fluorescence spectra of HSA in the presence of increasing amounts of bulk-scale PdL2 in mol ratio of ([HSA]/[PdL2] = 0-5 ([HSA] = 5 μM, [PdL2] = 100 μM). B) Plot of Ln((F0-F)/F) versus Ln[PdL2]. C) Changes in the fluorescence spectra of HSA in the presence of increasing amounts of nano-scale PdL2 in mol ratio of ([HSA]/[PdL2] = 0-5 ([HSA] = 5 μM, [PdL2] = 25 μM). D) Plot of Ln((F0-F)/F) versus Ln[PdL2].

  Fig. 3. A) Fluorescence spectra of the EB-DNA in the absence and presence of increasing amounts of bulk-scale PdL2 in mol ratio of [DNA]/[PdL2] = 0-5 ([DNA] = 25 μM, [PdL2] = 250 μM). B) Plot of Ln((F-F0)/F) versus Ln[PdL2]. C) Fluorescence spectra of the EB-DNA in the absence and presence of increasing amounts of nano-scale PdL2 in mol ratio of [DNA]/[PdL2] = 0-5 ([DNA] = 5 μM, [PdL2] = 25 μM). D) Plot of Ln((F-F0)/F) versus Ln[PdL2].

Fig. 4 .

 4 Fig. 4. The CD spectra of HSA in the absence and presence of nano-scale PdL2 complex in the molar ratio of [Complex]/[HSA] = 1 and the results of CDNN software analysis contents of protein secondary structure.

Fig. 5 .

 5 Fig. 5. (A) Location of PdL2 in interaction with HSA, the amino acids residues of HSA in binding site and its docking clusters. (B) Location of PdL2 in its interaction with DNA, the base pairs of DNA in binding site and its docking clusters.

Fig. 5B represents

  Fig.5Brepresents the base pairs of DNA around of PdL2 complex. Also, molecular docking results shows that H-bond or π stacking interactions have not key role in the stability of DNA-or HSA-PdL2 systems and the nature of these interactions are mainly hydrophobic.

Fig. 6 .

 6 Fig. 6. (A) Time dependence of RMSD. RMSD values for HSA and HSA-PdL2 complex during 10 ns MD simulation. (B) Time-evolution of radius of gyration (Rg) for HSA and HSA-PdL2 complex during MD simulation. (C) The number of Intramolecular H-bonds for HSA and HSA-PdL2 complex during 10 ns MD simulation. (D) RMSF values of HSA and HSA-PdL2 complex plotted against residue numbers.

Fig. 8 .

 8 Fig. 8. The structure of the PdL2 complex and HSA optimized considering charged embedding of MM part in QM part in the gas phase (The blue part of system considers as QM part). The changes of structural parameters of PdL2 during binding to HSA (Arcs and double arrow show angle and length of bond, respectively).

Fig. 9 .

 9 Fig. 9. The structure of the PdL2 complex and DNA optimized considering charged embedding of MM part in QM part in the gas phase. The changes of structural parameters of PdL2 during binding to DNA (Arcs and double arrow show angle and length of bond, respectively).

Table 1 .

 1 The DNA and HSA-binding constants (Kb), the number of binding sites (n) of the PdL2.

	System	HSA		DNA
		Kb (M -1 )	n	Kb (M -1 )	n
	Nano-scale PdL2	1.68 × 10 4	0.93	5.59 × 10 4	0.95
	Bulk-scale PdL2	3.12 × 10 4	0.72	2.51 × 10 3	1.18

Table 2 .

 2 The average values of RMS deviation (RMSD), number of H-bonds, the radius of gyration (Rg) and the solvent accessible surface area (SASA) of the HSA, HSA-PdL2, DNA and DNA-PdL2 systems.

	System	RMSD (nm)	H-bond	Rg (nm)	SASA (nm 2 )
	HSA	0.26 ± 0.05	481.5 ± 10.9	2.70 ± 0.02	298.22 ± 3.71
	HSA + PdL2	0.25 ± 0.03	470.0 ± 10.9	2.68 ± 0.01	299.00 ± 5.61
	DNA	0.25 ± 0.10	33.14 ± 1.48	-	-
	DNA + PdL2	0.25 ± 0.12	33.17 ± 1.72	-	-

Acknowledgements

The financial supports of the research council of University of Isfahan are gratefully acknowledged.

Supplementary data CCDC 1554815 contain the supplementary crystallographic data for PdL2. These data can be obtained free of charge via http://www.ccdc. cam.ac.uk/conts/retrieving.html, or from the Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK. Fax: +44 1223 336 033; or e-mail:deposit@ccdc.cam.ac.uk. Supplementary data to this article can be found online at doi: https://doi.org/10. 1016/j.molliq.2018.05.077


domain: math spi

text:
{ |[h]| ∈ 2 R 2 } ⊃ △pqr from Theorem 1 & ∃gcat(E) : E = min {{ |[h]| ∈ 2 R 2 }} ⊃ {△pqr} ∩ r from Corollary 1.
This paper introduces results for several forms of the geometric Lusternik-Schirel'mann categories (LS gcat), namely, geometric realization of one or more homotopy classes containing multiple sinusoidal paths that minimally cover planar triangles and clusters of Delaunay triangles with a common vertex that provide an application of gcat in a minimal covering of a foreground object in a video frame.

Geometric Lusternik-Schnirel'mann Category

Let h : I → K be a continuous map called homotopy (briefly, path in a space K. A homotopic class for different maps h (denoted by

[h] = { h 0 , . . . , h i , . . . , h n-1[n] } with [n] = mod n ∈ Z + ) is a collection of h i[n]
homotopic maps that have the same endpoints, namely, h i (0) and h i (1).

The geometric realization of [n] (denoted by |[h]|

) is a collection of sinusoidal curves, each being the geometric realization of a path h.

1 Definition 1. [Geometric LS Category] [2].
For a topological space X, the geometric category of X is the minimal covering of X with contractible open subsets of X.

Lemma 1.

Let h be a homotopic sinusoidal path. The geometric realization |h| is a planar sinusoidal curve.

Lemma 2. Let [h] be a collection of homotopic paths with common endpoints. The geometric realization |[h]| is a collection space filling planar curves.

Lemma 3. Let △pqr be a planar filled triangle in a space K, geometric realization |[h]| such that each path h has endpoints h(0), h(1) ∈ K \ △pqr. Then limit

i→∞ h i ∈ [h] ⊇ △pqr.

Lemma 4. Let h be a homotopic path in space K. 1 o Every path h is contractible. 2 o There exists a minimal |[h]| in space K with h i ∈ [h] with the same boundary endpoints h(0), h(1) ∈ ∂△pqr such that |[h]| covers triangle △pqr ⊂ K. 3 o Every planar triangle in K has a minimal covering |[h]|.

Theorem 1. There exists gcat(|[h]|

∈ 2 K ) such that min |[h]| ⊇ △pqr ∈ 2 K .
Example 1. From Theorem 1, the triangle T in Fig. 1

.1 has a |[h]| minimal covering, which is a gcat(T ).

A cluster of triangles {△pqr} in a Euclidean space K is a collection of triangles attached to a common vertex. From Theorem 1, we have

Corollary 1. Let |{[h]}| ⊂ K such that each h ∈ {[h]} ⊂ K \ {△pqr} has the same endpoints. There exists gcat(|{[h]}|) such that min |{[h]}| ⊇ {△pqr} ∈ 2 K .

Example 2. From Corollary 1, there is a gcat(|{[h]}| such that gcat(|{[h]}| is a minimal covering of the triangle cluster {△pqr} ∩ p, which is a bounded region in

Figure 1 .

 1 Figure 1. ∃gcat(X) : X = min

Fig. 1 Figure 2 . 2 .Theorem 2 .Example 3 .

 12223 Figure 2. Delaunay Triangle cluster minimally covering bus in video frame.