Docking of Polyethylenimines Derivatives on Cube Rhombellane Functionalized Homeomorphs

Abstract

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.

1. 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.

The choice of ligands and docked nanostructures was guided by our earlier studies [1,2,3,4,5,6,7,8].

PEIs (polyethylenimines) are polymeric molecules built of two aliphatic carbons and repeating units of amine groups. There are L-, B-, and DPEI (Figure 1). Linear PEI (LPEI) are built of secondary and primary amino groups (Figure 1, left); branched PEI (B-PEI) are built of all types amino groups such as primary, secondary and tertiary (Figure 1, middle), while dendrimers [9,10] are symmetric around the core (Figure 1, right). PEI despite the fact that are cytotoxic [11], have many applications, first of all, as transfection reagents [12].

Calculations at the B3LYP/6-31G (d, p) level of theory [16,17,18,19] confirmed the hypothesis that rhombellanes are energetically feasible in the hope of a real synthesis [20,21,22,23,24,25,26].

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).

2. Methods

Docking Procedure

The ligand molecule was obtained from others study [1,2,3,4,5,6,7,8,13,14,15,28,29,30,31] Rhombellane homeomorphs, were received from Topo Cluj Group [32], the C60 structure was downloaded from Brookhaven Protein Database PDB [33], while C60 functionalized derivatives were obtained from PubChem database [34].

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:

$${\mathrm{K}}_{\max }={\exp }^{\left(\frac{-\Delta {\mathrm{G}}_{\max }}{\mathrm{RT}}\right)}$$

where ΔG max represent maximal value of binding affinity obtained during docking stage, R represent value of gas constant and T temperature.

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

Gibbs Free Energy (kcal/mol)
NANO-STRUCTURES	B10230_PEI_C22N12	B10230_PEI_C44N23	B10230_PEI_C20N11	B10230_PEI_C40N21	B10230_PEI_C60N31
144_ex_ex	−2.4	−2.9	−2.4	−2.8	−4.4
144_in_ex	−2.3	x	−2.2	−2.7	−4.3
156_ex_ex	−2.4	−2.8	−2.5	−2.6	−4.5
156_in_ex	−2.6	−2.9	−2.6	−2.9	−4.8
308a4	−3.9	−4.8	−3.9	−4.6	−6.4
308b4	−4.1	−5.0	−4.2	−4.6	−6.9
360a	−4.0	−4.5	−3.9	−4.5	−6.5
360b	−3.9	−4.5	−3.7	−4.2	−6.1
372AB	−4.3	−4.8	−4.0	−4.4	−6.2
396	−3.9	−4.7	−3.9	−4.2	−5.8
420	−4.2	−4.7	−3.7	−4.1	−5.4
444	−3.7	−4.3	−3.6	−4.0	−5.6
456	−3.9	−4.5	−3.5	−4.0	−5.3
ADA_132	−5.0	−4.4	−4.4	−4.2	x
C60	−3.8	−4	−3.8	−3.8	−4.8

), 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 C₆₀ 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 C₆₀ (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,

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

Docked Energy (kcal/mol)
NANO-STRUCTURES	D2_PEI_C10N6	D3_PEI_C26N14	D4_PEI_C58N30
144_ex_ex	−2.4	−2.6	−2.6
144_in_e	−2.4	−2.8	−2.6
156_ex_ex	−3.1	−3.3	−3
156_in_ex	−3.1	−3.4	−3.1
308a4	−3.1	−3.6	−0.6
308b4	−3.3	−3.7	−2
360a	−3.1	−3.6	−2.9
360b	−2.9	−3.3	2.5
372AB	−3.2	−3.7	−1.1
396	−2.8	−3.6	4.9
420	−2.9	−3.5	5
444	−2.8	−3.3	10.4
456	−2.9	−3.3	8.2
ADA_132	−3.4	−2.7	−1.9
C60	−2.1	−2.3	2.4

and

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 A-L_PEI_C06N4; B-L_PEI_C08N5; C-L_PEI_C10N6; D-L_PEI_C26N14; E-PEI_C14N8_01_0Linear; F-PEI_C14N8_07_B22; G-PEI_C18N10_01_0Linear.

Gibbs Free Energy (kcal/mol)
NANO-STRUCTURES	A	B	C	D	E	F	G
144_ex_ex	−2.3	−2.5	−2.8	−2.8	−2.8	−2.3	−2.9
144_in_ex	−2.2	−2.5	−2.8	−2.7	−2.8	−2.4	−2.9
156_ex_ex	−2.8	−3.2	−3.4	−2.6	−3.6	−2.9	−3.7
156_in_ex	−2.9	−3.3	−3.5	−2.9	−3.7	−3.2	−3.6
308a4	−3.1	−3.1	−3.4	−4.6	−3.4	−3.1	−3.8
308b4	−3.2	−3.5	−3.4	−4.6	−3.7	−3.4	−4.1
360a	−2.6	−3.1	−3.2	−4.5	−3.6	−3.2	−3.5
360b	−2.8	−3.2	−3.4	−4.2	−3.5	−3.2	−3.5
372AB	−2.7	−3.2	−3.3	−4.4	−3.5	−3.2	−3.8
396	−2.7	−3.2	−3.4	−4.2	−3.4	−3.1	−3.6
420	−2.6	−3.1	−3.4	−4.1	−3.5	−2.9	−3.5
444	−2.8	−3.3	−3.4	−4.0	−3.6	−3.1	−3.4
456	−2.8	−3.3	−3.5	−4	−3.6	−3	−3.6
ADA_132	−3.3	−3.5	−3.5	−4.2	−3.4	−3.4	−3.3
C60	−1.6	−1.8	−2	−2.6	−2.2	−2	−2.1

).

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 (Table 2 and Table 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, Table 2 and Table 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 (Table 3 and

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.

NANO-STRUCTURES	B10230_PEI C22N12			B10230_PEI C44N23			B10230_PEI C20N11			B10230_PEI C40N21			B10230_PEI C60N31
	A	B	C	A	B	C	A	B	C	A	B	C	A	B	C
144_ex_ex	−2.4	57.4	−90.6	−2.9	133.6	−84.4	−2.4	57.4	−90.6	−2.8	112.8	−26.3	−4.4	1679.7	−8.3
144_in_ex	−2.3	48.5	−92.0		1.0	−99.9	−2.2	41.0	−93.3	−2.7	95.3	−28.9	−4.3	1418.9	−10.4
156_ex_ex	−2.4	57.4	−90.6	−2.8	112.8	−86.8	−2.5	68.0	−88.9	−2.6	80.5	−31.6	−4.5	1988.6	−6.3
156_in_ex	−2.6	80.5	−86.8	−2.9	133.6	−84.4	−2.6	80.5	−86.8	−2.9	133.6	−23.7	−4.8	3299.5	0.0
308a4	−4.0	855.1	40.2	−4.6	2354.2	175.3	−3.9	722.3	18.4	−4.6	2354.2	21.1	−6.4	49,120.4	33.3
308b4	−4.1	1012.4	65.9	−5.0	4624.3	440.8	−4.2	1198.5	96.4	−4.6	2354.2	21.1	−6.9	114,226.6	43.8
360a	−4.0	855.1	40.2	−4.5	1988.6	132.5	−3.9	722.3	18.4	−4.5	1988.6	18.4	−6.5	58,151.8	35.4
360b	−3.9	722.3	18.4	−4.5	1988.6	132.5	−3.7	515.4	−15.5	−4.2	1198.5	10.5	−6.1	29,604.6	27.1
372AB	−4.3	1418.9	132.5	−4.8	3299.5	285.8	−4.0	855.1	40.2	−4.4	1679.7	15.8	−6.2	35,047.8	14.6
396	−3.9	722.3	18.4	−4.7	2787.1	225.9	−3.9	722.3	18.4	−4.2	1198.5	10.5	−5.8	17,842.5	29.2
420	−4.2	1198.5	96.4	−4.7	2787.1	225.9	−3.7	515.4	−15.5	−4.1	1012.4	7.9	−5.4	9083.5	20.8
444	−3.7	515.4	−15.5	−4.3	1418.9	65.9	−3.6	435.3	−28.6	−4.0	855.1	2.6	−5.6	12,730.8	12.5
456	−3.9	722.3	18.4	−4.5	1988.6	132.5	−3.5	367.7	−39.7	−4.0	855.1	5.3	−5.3	7672.8	16.7
ADA_132	−5.0	4624.3	657.9	−4.4	1679.7	96.4	−4.4	1679.7	175.3	−4.2	1198.5	5.3		1.0	−6.3
C60	−3.8	610.2	0.0	−4.0	855.1	0.0	−3.8	610.2	0.0	−3.8	610.2	7.9	−4.8	3299.5	−100.0

). 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 (Table 3 and Table 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).

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.

The best binding affinity of ligands BPEI (Table 4) DPEI (

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.

NANO-STRUCTURES	D2_PEI_C10_N6			D3_PEI_C26_N14			D4_PEI_C58_N30
	A	B	C	A	B	C	A	B	C
144_ex_ex	−2.4	57.44118	65.9216	−2.6	80.50545	65.9216	−2.6	80.50545	462,332.8
144_in_ex	−2.4	57.44118	65.9216	−2.8	112.8307	132.5439	−2.6	80.50545	462,332.8
156_ex_ex	−3.1	187.2105	440.7664	−3.3	262.3808	440.7664	−3	158.1354	908,248.5
156_in_ex	−3.1	187.2105	440.7664	−3.4	310.6226	540.1927	−3.1	187.2105	1,075,259
308a4	−3.1	187.2105	440.7664	−3.6	435.3464	797.2483	−0.6	2.752998	15,713.54
308b4	−3.3	262.3808	657.8996	−3.7	515.39	962.2179	−2	29.24283	167,874.3
360a	−3.1	187.2105	440.7664	−3.7	515.39	962.2179	−2.9	133.5759	767,175.8
360b	−2.9	133.5759	285.8405	−3.3	262.3808	440.7664	2.5	0.014705	−15.5307
372AB	−3.2	221.6313	540.1927	−3.7	515.39	962.2179	−1.1	6.401927	36,673.42
396	−2.8	112.8307	225.9168	−3.6	435.3464	797.2483	4.6	0.000425	−97.5601
420	−2.9	133.5759	285.8405	−3.5	367.7342	657.8996	5	0.000216	−98.7578
444	−2.8	112.8307	225.9168	−3.3	262.3808	440.7664	10.4	2.38E−08	−99.9999
456	−2.9	133.5759	285.8405	−3.3	262.3808	440.7664	8.2	9.76E−07	−99.9944
ADA_132	−3.4	310.6226	797.2483	−2.7	95.30732	96.42822	−1.9	24.70122	141,786.7
C60_2	−2.1	34.61947	0	−2.3	48.52017	0	2.4	0.017409	0

) and LPEI (

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 C₆₀ molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system.

NANO-STRUCTURES	L_PEI_C06N4			L_PEI_C08N5			L_PEI_C10N6			L_PEI_C26N14
	A	B	C	A	B	C	A	B	C	A	B	C
144_ex_ex	−2.3	48.5	225.9	−2.5	68.0	225.9	−2.8	112.8	285.	−2.8	112.8	40.2
144_in_ex	−2.2	41.0	175.3	−2.5	68.0	225.9	−2.8	112.8	285.8	−2.7	95.3	18.4
156_ex_ex	−2.8	112.8	657.9	−3.2	221.6	962.2	−3.4	310.6	962.2	−2.6	80.5	0.0
156_in_ex	−2.9	133.6	797.2	−3.3	262.4	1157.5	−3.5	367.7	1157.5	−2.9	133.6	65.9
308a4	−3.0	158.1	962.2	−3.3	262.4	1157.5	−3.4	310.6	962.2	−4.6	2354.2	2824.3
308b4	−3.2	221.6	1388.7	−3.5	367.7	1662.4	−3.4	310.6	962.2	−4.6	2354.2	2824.3
360a	−2.7	95.3	540.2	−3.1	187.2	797.2	−3.2	221.6	657.9	−4.5	1988.6	2370.1
360b	−2.8	112.8	657.9	−3.2	221.6	962.2	−3.4	310.6	962.2	−4.2	1198.5	1388.7
372AB	−2.7	95.3	540.2	−3.2	221.6	962.2	−3.4	310.6	962.2	−4.4	1679.7	1986.5
396	−2.7	95.3	540.2	−3.2	221.6	962.2	−3.4	310.6	962.2	−4.2	1198.5	1388.7
420	−2.7	95.3	540.2	−3.1	187.2	797.2	−3.6	435.3	1388.7	−4.1	1012.4	1157.5
444	−2.8	112.8	657.9	−3.3	262.4	1157.5	−3.4	310.6	962.2	−4.0	855.1	962.2
456	−2.8	112.8	657.9	−3.3	262.4	1157.5	−3.5	367.7	1157.5	−4.0	855.1	962.2
ADA_132	−3.3	262.4	1662.4	−3.5	367.7	1662.4	−3.5	367.7	1157.5	−4.2	1198.5	1388.7
C60	−1.6	14.9	0.0	−1.8	20.9	0.0	−2.0	29.2	0.0	−2.6	80.5	0.0

and

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 C₆₀ molecule, defining the difference in quality of ligand binding with considered nanostructure in comparison to reference system.

NANO-STRUCTURES	PEI_C14N8_01_0Linear			PEI_C14N8_07_B22			PEI_C18N10_01_0Linear
	A	B	C	A	B	C	A	B	C
144_ex_ex	−2.8	112.8307	40.15285	−2.3	48.52017	175.2998	−2.9	133.5759	37.5
144_in_ex	−2.8	112.8307	65.9216	−2.4	57.44118	225.9168	−2.9	133.5759	37.5
156_ex_ex	−3.6	435.3464	225.9168	−2.9	133.5759	657.8996	−3.7	515.39	62.5
156_in_ex	−3.7	515.39	657.8996	−3.2	221.6313	1157.519	−3.7	515.39	75
308a4	−3.2	221.6313	440.7664	−3.1	187.2105	1388.729	−3.7	515.39	87.5
308b4	−3.7	515.39	540.1927	−3.4	310.6226	2824.283	−4.1	1012.371	93.75
360a	−3.6	435.3464	440.7664	−3.2	221.6313	962.2179	−3.5	367.7342	62.5
360b	−3.5	367.7342	356.7818	−3.2	221.6313	797.2483	−3.5	367.7342	68.75
372AB	−3.5	367.7342	657.8996	−3.2	221.6313	1662.449	−3.8	610.1504	68.75
396	−3.4	310.6226	440.7664	−3.1	187.2105	1157.519	−3.6	435.3464	68.75
420	−3.5	367.7342	440.7664	−3	158.1354	962.2179	−3.5	367.7342	68.75
444	−3.6	435.3464	356.7818	−3.1	187.2105	797.2483	−3.4	310.6226	75
456	−3.6	435.3464	440.7664	−3	158.1354	1157.519	−3.6	435.3464	68.75
ADA_132	−3.4	310.6226	797.2483	−3.4	310.6226	440.7664	−3.3	262.3808	100
C60	−2.2	40.98466	0	−2	29.24283	0	−2.1	34.61947	0

), 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 C₆₀ 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 C₆₀. The highest positive percentage deviations from the affinity of ligands to fullerene C₆₀ were obtained for those Rbl-structures showing the highest binding values (Table 4, Table 5, Table 6 and Table 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).

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 C₆₀ as references structure with B1320_PEI_C60N31 ligand there are no important interactions (Figure 10).

4. 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.