Docking of Polyethylenimines Derivatives on Cube Rhombellane Functionalized Homeomorphs

Nowadays, in the world of science, an important goal is to create new nanostructures that may act as potential drug carriers. Among different, real or hypothetical, polymeric networks, rhombellanes are very promising and, therefore, attempts were made to deposit polyethylenimines as possible nano-drug complexes on the cube rhombellane homeomorphs surface. For the search of ligand–fullerene interactions, was used AutoDockVina software. As a reference structure, the fullerene C60 was used. After the docking procedure, the ligands–fullerenes interactions were tested. The important factor determining the mutual affinity of the tested ligands and nanocarriers is the symmetry of the analyzed nanostructures. Here, this feature has the influence on the distribution of such groups like donors and acceptors of hydrogen bonds on the surface of nanoparticles. We calculated the best binding affinities of ligands, values of binding constants and differences relative to C60 molecules. The best binding efficiency was found for linear ligands. It was also found that the shorter the molecule, the better the binding performance, the more the particle grows and the lower the yield. Small structures of ligands react easily with small structures of nanoparticles. The highest positive percentage deviations were obtained for ligand–fullerene complexes showing the highest binding energy values. Detailed analysis of structural properties after docking showed that the values of affinity of the studied indolizine ligands to the rhombellanes surface are correlated with the strength/length of hydrogen bonds formed between them.


Introduction
The development of multifunctional nanoparticles has a huge impact on the future of personalized medicine.Nanoparticles can be used for therapeutic purposes in anti-cancer therapy at the molecular level, which has been difficult so far.In earlier work [1][2][3], an enzyme GOx (3QVR) that fulfills the role of a biosensor, had been used for immobilization on the gel, while the polymer was polyethylenimine (PEI).By assembling polymeric nano-gels (for example PEI) and antibodies on nano-molecules [1][2][3], it was possible to recognize receptors of certain integrins on lung cancer tissues and to identify new cancer vessels.
In the present article, polyethylenimines (PEI) were studied.Molecular docking analysis of fifteen PEI derivatives acting as ligands on some cube rhombellane homeomorphs was carried out for the first time.Fourteen types of cube rhombellanes and three groups of polyethylenimines (PEIs), namely, branched (B-PEI), linear (L-PEI) and dendrimer (D-PEI) were used.
Rhombellanes have certain specific traits which define these group of structures.At first, all strong rings are squares/rhombs.The second vertex classes consist of only non-connected vertices.Omega polynomial has a single term: 1X^|E| and they contain one K2.3 complete bipartite subgraph or the smallest rhombellane rbl.5.The end line graph of the parent graph has a Hamiltonian circuit.
To explore the internal molecular mobility that is important in the bioactivity study of these compounds we used the Molecular mechanics (MMFF94) [27] method.In this way we study the pharmaceutical important parameters of rhombellanes [27].
Rhombellans (Figure 2) seem to be structures suitable for medical chemistry, with a new class of structures which could be an used in personalized medicine as new carrier nanostructures.
Because rhombellane homeomorphs may be bound to a protein, an attempt was made to deposit PEI derivatives on rhombellanes, as possible nano-drug complexes.Detailed analysis of structural properties after docking showed many interesting features.Behavior of polyethylenimine (linear LPEI, branched BPEI and/or dendrimers DPEI) with respect to rhombellane homeomorphs, in terms of their (interacting) topology, geometry and energy, was studied.After the docking procedure, the best values of ligand-rhombellane affinity were found, which is an important result for homeomorphs.
Rhombellanes have certain specific traits which define these group of structures.At first, all strong rings are squares/rhombs.The second vertex classes consist of only non-connected vertices.Omega polynomial has a single term: 1Xˆ|E| and they contain one K2.3 complete bipartite subgraph or the smallest rhombellane rbl.5.The end line graph of the parent graph has a Hamiltonian circuit.
To explore the internal molecular mobility that is important in the bioactivity study of these compounds we used the Molecular mechanics (MMFF94) [27] method.In this way we study the pharmaceutical important parameters of rhombellanes [27].
Rhombellans (Figure 2) seem to be structures suitable for medical chemistry, with a new class of structures which could be an used in personalized medicine as new carrier nanostructures.
Because rhombellane homeomorphs may be bound to a protein, an attempt was made to deposit PEI derivatives on rhombellanes, as possible nano-drug complexes.Detailed analysis of structural properties after docking showed many interesting features.Behavior of polyethylenimine (linear LPEI, branched BPEI and/or dendrimers DPEI) with respect to rhombellane homeomorphs, in terms of their (interacting) topology, geometry and energy, was studied.After the docking procedure, the best values of ligand-rhombellane affinity were found, which is an important result for homeomorphs.
Rhombellanes have certain specific traits which define these group of structures.At first, all strong rings are squares/rhombs.The second vertex classes consist of only non-connected vertices.Omega polynomial has a single term: 1X^|E| and they contain one K2.3 complete bipartite subgraph or the smallest rhombellane rbl.5.The end line graph of the parent graph has a Hamiltonian circuit.
To explore the internal molecular mobility that is important in the bioactivity study of these compounds we used the Molecular mechanics (MMFF94) [27] method.In this way we study the pharmaceutical important parameters of rhombellanes [27].
Rhombellans (Figure 2) seem to be structures suitable for medical chemistry, with a new class of structures which could be an used in personalized medicine as new carrier nanostructures.
Because rhombellane homeomorphs may be bound to a protein, an attempt was made to deposit PEI derivatives on rhombellanes, as possible nano-drug complexes.Detailed analysis of structural properties after docking showed many interesting features.Behavior of polyethylenimine (linear LPEI, branched BPEI and/or dendrimers DPEI) with respect to rhombellane homeomorphs, in terms of their (interacting) topology, geometry and energy, was studied.After the docking procedure, the best values of ligand-rhombellane affinity were found, which is an important result for homeomorphs.
The article is a collection of new data in the new field of rhombellanes (Figure 2).
During the docking stage, there were structures of ligand and nano-systems containing only polar hydrogen atoms.In the case of all nanoparticles, the grid box dimensions were established equal to 26 × 26 × 26 Å, boxing coordinates amount to (0,0,0).All initial procedures related with preparation of ligand and nano-systems during the docking procedure were realized with the use of the AutoDock Tools package [35].Using the AutoDockVina software in the docking procedure, after assigning hydrogen bonds, the molecules were loaded and stored as pdb-files [36].The investigated ligands were loaded and their torsions along the rotatable bonds were assigned and next saved as "ligand.pdbqt".The grid menu after loading "pdbqt" was toggled [37].For the search of ligand-rbl fullerene interactions, the map files were selected directly with setting up the grid points separately for each structure.The Lamarckian genetic algorithm completed the docking parameter files [38].As a reference structure, the fullerene C60 was used, the most referred to structure in nanoscience.
All calculations during the docking stage were realized with the exhaustiveness parameter equal to 20, since such a value ensures an appropriate reproducibility of the results and a reasonable time of calculation.The structural analysis of considered systems and visualization of obtained complexes were realized with use of the VMD package [39].The value of binding constant was calculated based on the formula: where ∆G max represent maximal value of binding affinity obtained during docking stage, R represent value of gas constant and T temperature.

Results and Discussion
The results are presented in the following tables and figures.Rhombellane structures are given by their atom number.
In relation to the ligands of the D and L groups, the largest affinity values of the ligand-fullerenes were found for all ligands from B group (Figure 3, Table 1), for which the affinity values range from −2 to −7 kcal/mol.With the B_1320_PEI_C60N31 ligand the values are the lowest, thus showing the best affinity for all proposed fullerenes, with affinity values from −4 to −7 kcal/mol.All values of the interactions were compared with the values for C 60 fullerene, which was, as always, used as the reference structure in nanostructures family.Therefore, in all cases of ligands from the B group, there are affinities with better and worse values of energy compared with affinity ligand-fulleren C 60 (Figure 3, Table 1).When comparing the values of affinity of D ligand-fulleren and L ligand-fulleren complexes, it is clear that these complexes show the lowest values with all study fullerenes, compared with fulleren C60 (Figure 3, Tables 2 and 3).When comparing the values of affinity of D ligand-fulleren and L ligand-fulleren complexes, it is clear that these complexes show the lowest values with all study fullerenes, compared with fulleren C 60 (Figure 3, Tables 2 and 3).

Docked Energy (kcal/mol)
Table 2.The best binding affinity of ligands, BPEI, with the active site of Rbl-nano-structures (first column) during nine conformations.For the other proposed ligands of type D, L and PEI_C14N8_01_Linear; PEI_C14N8_07_B22; PEI_C18N10_01_0, their affinities ranged from −2 to −4 kcal/mol (Tables 2 and 3).However, the ligand D4_PEI_C58N30 shows not only affinity in this range, i.e., (−2 to −4 kcal/mol), but also low affinities for fullerenes with high values of around 0 kcal/mol, and it does not even have the possibility of interacting with fullerenes as their affinity values are positive (Figure 3, Tables 2 and 3)

Docked Energy (kcal/mol
The diagram (Figure 4) shows two populations of affinity values of ligand B-fullerenes.The first with values ranging from −2 to −3 kcal/mol and in the case of second population from −3.3 to −5 kcal/mol for B1320_PEI_C20N11; B10230_PEI_C22N12; B1320_PEI_C40N21; B10230_PEI_C44N23.For the other proposed ligands of type D, L and PEI_C14N8_01_Linear; PEI_C14N8_07_B22; PEI_C18N10_01_0, their affinities ranged from −2 to −4 kcal/mol (Tables 2 and 3).However, the ligand D4_PEI_C58N30 shows not only affinity in this range, i.e., (−2 to −4 kcal/mol), but also low affinities for fullerenes with high values of around 0 kcal/mol, and it does not even have the possibility of interacting with fullerenes as their affinity values are positive (Figure 3, Tables 2 and 3) The diagram (Figure 4) shows two populations of affinity values of ligand B-fullerenes.The first with values ranging from −2 to −3 kcal/mol and in the case of second population from −3.3 to −5 kcal/mol for B1320_PEI_C20N11; B10230_PEI_C22N12; B1320_PEI_C40N21; B10230_PEI_C44N23.Similarly, two populations are visible in the case of B1320_PEI_C60N31 with markedly reduced affinity values relative to the values of the affinity represented by the first four ligands of the B group, described above (Tables 3 and 4).The first population has interaction values ranging from −4 kcal/mol to −5 kcal/mol, the second shows much higher affinity values from −5 to −7 kcal/mol (Tables 3 and 4).Two populations are also visible for other ligands from the D and L groups forming interactions with tested fullerenes.Quantitatively, the largest number of ligand-fullerene interactions is expressed by affinity values in the range of −3 to −4 kcal/mol, and this is a representative population.The second population is expressed by a small representation of the number of affinities with values ranging from −2 to −3 kcal/mol.
The existence of these two populations is closely related to the interaction with two fullerene groups.Small structures of ligands easily react with small structures of nanoparticles and vice versa.
Unfortunately, the energy parameter itself is insufficient.By using energy per quantity of carbon atoms (kcal/mol) parameter it is clearly visible that the elongation of carbon chain does not affect the binding efficiency, but only increases affinity (Figure 5).
Two populations are also visible for other ligands from the D and L groups forming interactions with tested fullerenes.Quantitatively, the largest number of ligand-fullerene interactions is expressed by affinity values in the range of −3 to −4 kcal/mol, and this is a representative population.The second population is expressed by a small representation of the number of affinities with values ranging from −2 to −3 kcal/mol.
The existence of these two populations is closely related to the interaction with two fullerene groups.Small structures of ligands easily react with small structures of nanoparticles and vice versa.
Unfortunately, the energy parameter itself is insufficient.By using energy per quantity of carbon atoms (kcal/mol) parameter it is clearly visible that the elongation of carbon chain does not affect the binding efficiency, but only increases affinity (Figure 5).
The best binding efficiency is shown by linear ligands L, with highest values of this parameter, compared with values of ligands from B and D groups (Figure 5).The shorter the molecule, the better the binding performance, the more the particle grows and the lower the yield.For linear LPEI structures, as the carbon chain length increases, the binding efficiency decreases and the saturation around the length of the chain with twenty carbon atoms is clearly visible (Figure 6).The best binding efficiency is shown by linear ligands L, with highest values of this parameter, compared with values of ligands from B and D groups (Figure 5).The shorter the molecule, the better the binding performance, the more the particle grows and the lower the yield.For linear LPEI structures, as the carbon chain length increases, the binding efficiency decreases and the saturation around the length of the chain with twenty carbon atoms is clearly visible (Figure 6).
Similar observations have been made for group B ligands.Twofold chain elongation results in a two-fold decrease in binding efficiency (Figure 5, Table 1).In the case of dendrimeric structures, as the complexity of the system increases, the value of binding efficiency decreases (Figure 5, Table 2).Among the fullerenes tested, the best effects were found for fullerenes ADA and 308a4/b4.Small ligands easily form complexes primarily with fullerene ADA, long-chain ligands interact with the 308a4/b4 nanostructure (Figure 5).Thus, for small ligands, the best binding efficiency is with small fullerenes, while large fullerenes require large ligands.Similar observations have been made for group B ligands.Twofold chain elongation results in a two-fold decrease in binding efficiency (Figure 5, Table 1).In the case of dendrimeric structures, as the complexity of the system increases, the value of binding efficiency decreases (Figure 5, Table 2).Among the fullerenes tested, the best effects were found for fullerenes ADA and 308a4/b4.Small ligands easily form complexes primarily with fullerene ADA, long-chain ligands interact with the 308a4/b4 nanostructure (Figure 5).Thus, for small ligands, the best binding efficiency is with small fullerenes, while large fullerenes require large ligands.The best binding affinity of ligands BPEI (Table 4) DPEI (Table 5) and LPEI (Tables 6 and 7), k max values of binding constant estimated with use of binding free energy obtained for the best complex of ligand with nanostructure and k max differences relative C 60 molecule, defining the difference in the quality of ligand binding with considered nanostructure in comparison to reference system, were estimated for the tested Rbl-structures in relation to the affinity value obtained for the fullerene C 60 .The highest positive percentage deviations from the affinity of ligands to fullerene C 60 were obtained for those Rbl-structures showing the highest binding values (Tables 4-7, in boldface).Two last columns show the equilibrium K value of the bonds.The higher the K value, the more the reaction proceeds towards the formation of the complex.Detailed analysis of structural properties after docking showed that the affinities of the ligands to the rhombellanes surface are correlated with the quality of hydrogen bonds formed between them.The distance between acceptor and hydrogen atoms is the criterion for classification of the strength of hydrogen bonds: weak interactions are characterized by distances <3 Å, strong interactions by a distance <1.6 Å and medium strength by values in the range from 1.6 Å to 2.0 Å.
Ligand B1320_PEI_C20N11 and nanostructure ADA_132 form two hydrogen bonds with medium strength between amino groups of ligand and oxygen atoms of fullerenes with binding lengths 2.79Å and 2.95Å (Figure 7).Also, in the case of ligand B1320_PEI_C60N31-fullerene 308b4 four medium hydrogen bonds were created with bond lengths 2.69 Å, 2.87 Å and 2.93 Å (Figure 7).case of fullerene ADA_132 and ligands D2_PEI_C10N6 there is only one week hydrogen bond, while for fullerene 308b4 with D3_PEI_C26N14 ligand there are several strong and medium hydrogen bonds, first of all between amino groups of ligand and oxygen atom of nanostructure with bond lengths 1.9 Å, 2.47 Å., 2.52 Å and 2.94 Å.In the case of fullerene 156_in_ex with ligand D4_PEI_C58N30, there are three hydrogen bonds of medium strength; with bond lengths 2.50 Å, 2.80 Å and 2.83 Å (Figure 8).After the docking of ligands from D group, different interactions could be found, namely in the case of fullerene ADA_132 and ligands D2_PEI_C10N6 there is only one week hydrogen bond, while for fullerene 308b4 with D3_PEI_C26N14 ligand there are several strong and medium hydrogen bonds, first of all between amino groups of ligand and oxygen atom of nanostructure with bond lengths 1.9 Å, 2.47 Å., 2.52 Å and 2.94 Å.In the case of fullerene 156_in_ex with ligand D4_PEI_C58N30, there are three hydrogen bonds of medium strength; with bond lengths 2.50 Å, 2.80 Å and 2.83 Å (Figure 8).After the docking of ligands from L group, different interactions were also found, namely in the case of fullerene ADA_132 and ligands L_PEI_C10N6 there is only one-week hydrogen bond, the same as in the case D2_PEI_C10N6-ADA_132 (Figure 9).Again, as in DPEI-308b4 case, there are many interactions between hydrogen atoms of nitrogen groups of ligand and oxygen atoms of nanostructure with values 1.99 Å, 2.47 Å, 2.52 Å, 2.94 Å and 3.05 Å (Figure 9).
In the case of fullerene 308b4 and ligand PEI_C18N10_01_0Linear, there are several strong and medium hydrogen bonds, while in the case of fullerene C 60 as references structure with B1320_PEI_C60N31 ligand there are no important interactions (Figure 10).
After the docking of ligands from L group, different interactions were also found, namely in the case of fullerene ADA_132 and ligands L_PEI_C10N6 there is only one-week hydrogen bond, the same as in the case D2_PEI_C10N6-ADA_132 (Figure 9).Again, as in DPEI-308b4 case, there are many interactions between hydrogen atoms of nitrogen groups of ligand and oxygen atoms of nanostructure with values 1.99 Å, 2.47 Å, 2.52 Å, 2.94 Å and 3.05 Å (Figure 9).In the case of fullerene 308b4 and ligand PEI_C18N10_01_0Linear, there are several strong and medium hydrogen bonds, while in the case of fullerene C60 as references structure with B1320_PEI_C60N31 ligand there are no important interactions (Figure 10).

Conclusions
As a proposal for a new nanodrug, an attempt was made to implement PEI ligands on the cube rhombellane homeomorphic surface.Fourteen types of cube rhombellanes were used together with three groups of polyethylenimines (PEIs), namely, branched (B-PEI), linear (L-PEI) and dendrimer (D-PEI).Ligand-fullerenes interactions were described in terms of quality and quantity.Specifically, there were calculated the affinity values and affinity per quantity of carbon atoms after the docking procedure for ligand nanostructure.The best binding efficiency was shown by linear ligands L, with highest values of this parameter, compared with values of ligand from B and D groups.The shorter the molecule, the better the binding performance, the more the particle grows and the lower the yield.For linear structures LPEI, as the carbon chain length increases, the binding efficiency decreases and the saturation around the length of the chain with twenty carbon atoms is clearly visible.Similar observations have been made for group B ligands.Twofold chain elongation results in a two-fold decrease in binding efficiency.In the case of dendrimeric structures, as the complexity of the system increases, the value of binding efficiency decreases.Two populations of affinity values have been observed, which is closely related to the interaction with two fullerene groups.Small structures of ligands easily react with small structures of nanoparticles and vice versa.The best binding affinity of ligands and k max values of binding constant were estimated with the use of binding free energy obtained for the best complex of ligand with nanostructure.Also, k max differences relative C60 molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system, were calculated.The highest positive percentage deviations were obtained for ligand-fullerene complexes showing the highest binding energy values.Detailed analysis of structural properties after docking showed that the values of affinity of the studied indolizine ligands to the Rhombellanes surface are correlated with the strength/length of hydrogen bonds formed between them.

Conclusions
As a proposal for a new nanodrug, an attempt was made to implement PEI ligands on the cube rhombellane homeomorphic surface.Fourteen types of cube rhombellanes were used together with three groups of polyethylenimines (PEIs), namely, branched (B-PEI), linear (L-PEI) and dendrimer (D-PEI).Ligand-fullerenes interactions were described in terms of quality and quantity.Specifically, there were calculated the affinity values and affinity per quantity of carbon atoms after the docking procedure for ligand nanostructure.The best binding efficiency was shown by linear ligands L, with highest values of this parameter, compared with values of ligand from B and D groups.The shorter the molecule, the better the binding performance, the more the particle grows and the lower the yield.For linear structures LPEI, as the carbon chain length increases, the binding efficiency decreases and the saturation around the length of the chain with twenty carbon atoms is clearly visible.Similar observations have been made for group B ligands.Twofold chain elongation results in a two-fold decrease in binding efficiency.In the case of dendrimeric structures, as the complexity of the system increases, the value of binding efficiency decreases.Two populations of affinity values have been observed, which is closely related to the interaction with two fullerene groups.Small structures of ligands easily react with small structures of nanoparticles and vice versa.The best binding affinity of ligands and k max values of binding constant were estimated with the use of binding free energy obtained for the best complex of ligand with nanostructure.Also, k max differences relative C 60 molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system, were calculated.The highest positive percentage deviations were

Figure 4 .
Figure 4. Example of diagram showing two populations of affinity values of ligand B-fullerenes.Similarly, two populations are visible in the case of B1320_PEI_C60N31 with markedly reduced affinity values relative to the values of the affinity represented by the first four ligands of the B

Figure 4 .
Figure 4. Example of diagram showing two populations of affinity values of ligand B-fullerenes.

Figure 5 .
Figure 5. Affinity values per quantity of carbon atoms in kcal/mol for all ligands BPEI, DPEI and LPEI.Figure 5. Affinity values per quantity of carbon atoms in kcal/mol for all ligands BPEI, DPEI and LPEI.

Figure 5 .
Figure 5. Affinity values per quantity of carbon atoms in kcal/mol for all ligands BPEI, DPEI and LPEI.Figure 5. Affinity values per quantity of carbon atoms in kcal/mol for all ligands BPEI, DPEI and LPEI.

Figure 6 .
Figure 6.The affinity values per quantity of carbon atoms in function of quantity of carbon atoms for ligands PEI.

Figure 6 .
Figure 6.The affinity values per quantity of carbon atoms in function of quantity of carbon atoms for ligands PEI.

Figure 7 .
Figure 7. Interactions found in the complexes of fullerene ADA_132 and ligands B1320_PEI_C20N11 (left) and fullerene 308b4 with ligand B1320_PEI_C60N31 (right) after the docking procedure.

Figure 7 .
Figure 7. Interactions found in the complexes of fullerene ADA_132 and ligands B1320_PEI_C20N11 (left) and fullerene 308b4 with ligand B1320_PEI_C60N31 (right) after the docking procedure.

Symmetry 2019 , 14 Figure 8 .
Figure 8. Interactions found in the complexes of fullerene ADA_132 and ligands D2_PEI_C10N6 (left) and fullerene 308b4 with D3_PEI_C26N14 ligand (middle) and 156_in_ex with ligand D4_PEI_C58N30 (right) after the docking procedure.

Figure 8 .
Figure 8. Interactions found in the complexes of fullerene ADA_132 and ligands D2_PEI_C10N6 (left) and fullerene 308b4 with D3_PEI_C26N14 ligand (middle) and 156_in_ex with ligand D4_PEI_C58N30 (right) after the docking procedure.

Figure 9 .
Figure 9. Interactions found in the complexes of fullerene ADA_132 and ligands L_PEI_C10N6 (left) and fullerene 308b4 with ligand L_PEI_C26N14 (right) after the docking procedure.

Figure 9 . 14 Figure 10 .
Figure 9. Interactions found in the complexes of fullerene ADA_132 and ligands L_PEI_C10N6 (left) and fullerene 308b4 with ligand L_PEI_C26N14 (right) after the docking procedure.Symmetry 2019, 11, x FOR PEER REVIEW 12 of 14

Figure 10 .
Figure 10.Interactions found in the complexes of fullerene 308b4 and ligands PEI_C18N10_01_0Linear (left) and fullerene c60 as references structure with ligand B1320_PEI_C60N31 (right) after the docking procedure.

Table 1 .
The best binding affinity of ligands, BPEI, with the active site of Rbl-nano-structures (first column) during nine conformations.

Table 2 .
The best binding affinity of ligands, BPEI, with the active site of Rbl-nano-structures (first column) during nine conformations.

Table 3 .
The best binding energy of ligands, BPEI, with the active site of Rbl-nano-structures (first column) during nine conformations.In letters ABCD, etc., have been marked further nanostructures

Table 3 .
The best binding energy of ligands, BPEI, with the active site of Rbl-nano-structures (first column) during nine conformations.In letters ABCD, etc., have been marked further nanostructures

Table 4 .
The best binding affinity of ligands, BPEI, -columns A. Columns B represent k max values of binding constant estimated with use of binding free energy obtained for the best complex of ligand with nanostructure, while columns C represent k max differences relative C 60 molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system.

Table 4 .
The best binding affinity of ligands, BPEI, -columns A. Columns B represent k max values of binding constant estimated with use of binding free energy obtained for the best complex of ligand with nanostructure, while columns C represent k max differences relative C60 molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system.

Table 5 .
The best binding affinity of ligands, BPEI, -columns A. Columns B represent k max values of binding constant estimated with use of binding free energy obtained for the best complex of ligand with nanostructure, while columns C represent k max differences relative C 60 molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system.

Table 6 .
The best binding affinity of ligands, BPEI, -columns A. Columns B represent k max values of binding constant estimated with use of binding free energy obtained for the best complex of ligand with nanostructure, while columns C represent k max differences relative C 60 molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system.

Table 7 .
The best binding affinity of ligands, BPEI, -columns A. Columns B represent k max values of binding constant estimated with use of binding free energy obtained for the best complex of ligand with nanostructure, while columns C represent k max differences relative C 60 molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system.

Table 5 .
The best binding affinity of ligands, BPEI, -columns A. Columns B represent k max values of binding constant estimated with use of binding free energy obtained for the best complex of ligand with nanostructure, while columns C represent k max differences relative C60 molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system.