Cube-Rhombellane Related Structures: A Drug Perspective

Rhombellanes represent a new class of structures, of which homeomorphs may be synthesized as real molecules. Cube-rhombellane is a double-shell structure, with vertices of degree 3 and 6, respectively. Several hypothetical structures/molecules were proposed and computed using molecular graph theory and coordination chemistry principles. Some geometries were optimized at the B3LYP/6-31G (d, p) level of theory, followed by harmonic vibrational frequency analysis at the same level of theory, single point data were collected in view of molecular stability evaluation. Some of the bioactive functionalized structures were also proposed and explored by molecular mechanics (MM)-based conformational analysis, to check their internal mobility. Drug-like properties of the proposed molecular structures were compared with some existing nano-molecules (fullerenes, nanotubes). ADME and other physico-chemical characteristics were computed using commercial software. Substructures of the proposed molecules, useful in a future synthesis, were provided by retro combinatorial synthesis (RECAP). Computational results obtained are promising regarding ADME properties, drug-likeness and nano-properties.


Introduction
Crystal engineering and self-assembly have recently promoted new classes of finite and periodic crystalline solids, with promising applications in material science and biosciences [1][2][3][4][5][6][7]. The self-assembly principles have been applied to synthesize nano-sized spheroid architectures, based on Platonic and Archimedean solids. A variety of appropriate (as angles) ligands have been used in such syntheses to provide homeomorphs (see below) of the above solids. As metals to join the ligands, the most used was Pd 2+ [3][4][5][6], then Zn 2+ [7] or Cu 2+ [8]. There are some polyhedra (like cuboctahedron, rhombicuboctahedron, etc.), eventually called "faceted polyhedra" [9], containing both open and closed faces; for the synthesis of such structures, self-assembly of triangles and squares is needed [7]. The procedure of structure self-assembly was termed "molecular meccano" [10]. The single-shell spherical structures, possessing large hollows, could be functionalized, both endo-and/or exohedrally. A self-assembled double-shell structure, a sphere-in-sphere, molecular "Russian doll", was also reported [11].
Rhombellanes are structures with all strong rings being rhombs/squares, some of them forming local propellane substructures; they have been proposed by Diudea in 2017 [12]. Propellane is an organic molecule, first synthesized in 1982 [13]; by IUPAC nomenclature, it is named tricyclo[1.1.1.01,3]pentane, a hydrocarbon with formula C 5 H 6 . The reduced form of propellane, C 5 H 8 , eventually named bicyclo[1.1.1]pentane, has only rhomb/square rings; it can be represented as K 2.3 -the complete bipartite graph. The two bridge carbon atoms can be functionalized, e.g., by bromine or COOH, or even by repeating the K 2.3 motif, as in the polymer called staffane [14].
A rhombellane was defined by Diudea [15][16][17] as the structure having: (a) All strong rings are squares/rhombs; (b) Vertex classes consist of all non-connected vertices; (c) Omega polynomial has a single term: 1X |E| ; (d) Line-graph of the parent graph has a Hamiltonian circuit; (e) It contains at least one K2.3 subgraph.
A homeomorph of a graph contains on each parent edge one (or more) point(s) of degree two, e.g., the cube homeomorph 2 in Figure 1 (middle). The structure (3) (Figure 1, right), which is the homeomorph of (1) has seventy points/atoms; the vertex connectivity in (3) is 6; 3 and 2, respectively.
Molecules 2018, 23, x FOR PEER REVIEW 2 of 14 A rhombellane was defined by Diudea [15][16][17] as the structure having: (a) All strong rings are squares/rhombs; (b) Vertex classes consist of all non-connected vertices; (c) Omega polynomial has a single term: 1X |E| ; (d) Line-graph of the parent graph has a Hamiltonian circuit; (e) It contains at least one K2.3 subgraph.
A homeomorph of a graph contains on each parent edge one (or more) point(s) of degree two, e.g., the cube homeomorph 2 in Figure 1 (middle). The structure (3) (Figure 1, right), which is the homeomorph of (1) has seventy points/atoms; the vertex connectivity in (3) is 6; 3 and 2, respectively.
Cube-rhombellane 1 comprises an adamantane motif (Figure 3, left) that, in homeomorphs, becomes a hyper-adamantane, e.g., an ADA-ether ( Figure 3, middle and right).An ADA-cube-rbl 5 (Figure 4, left) is a functionalized hyper-adamantane, consisting of a rbl-core bound by six units (one for each face of the cube) derived from benzene-1,2,4,5-tetracarboxylic acid. Then, by completing the second/external shell (by adding four or eight tri-connected units), double-shell Cube-rbl-amides 6/7 ( Figure 4, middle and right) are obtained [21]; the envelope of 7 (or the second, external shell) will be counted as 8. There is a trend in supramolecular chemistry in the synthesis of Platonic or Archimedean polyhedral clusters by chemical fragments, eventually with metallic ions complexing suitable ligands. In this respect, homeomorphs of the cube, of various chemical composition, have been already synthesized [6,8,22,23]. For cubane and related structures see refs. [24][25][26].

Results
The structures discussed in the following are listed in Table 1. The cube-rhombellane homeomorphs are named as: Cube-rbl-(a.b.c.d)-amide/ester, where the a.b.c.d bits represent: hexa-connected unit, tri-connected unit, connection to the second shell and the tri-connected units in the second/external shell, respectively. Structures were designed and computed by our original Nano Studio software [27]. Ether; Amide 7:(6(6).6(6).Mt(2).8BMt(3)) Env = envelope; B = benzene; Mt = methylene; (n) = fragment connectivity. All the molecular structures discussed herein are hypothetical molecules, not yet synthesized. The aim of this paper is to introduce the cube-rhombellanes, belonging to the newly introduced structural class, as possible biologically active compound candidates.

Results
The structures discussed in the following are listed in Table 1. The cube-rhombellane homeomorphs are named as: Cube-rbl-(a.b.c.d)-amide/ester, where the a.b.c.d bits represent: hexa-connected unit, tri-connected unit, connection to the second shell and the tri-connected units in the second/external shell, respectively. Structures were designed and computed by our original Nano Studio software [27].
There are two types of aromatic rings in the envelopes 8:(0.0.0.8BMt(3)).288 and 8:(0.0.Mt(2−OH).BMt(3)).336 (one three−connected and the other four connected). These fragments have been optimized as molecules (Tb2(10) and Tb2(11), respectively). Their symmetry corresponds to the symmetry axes along which they are aligned. Full geometry optimizations were performed using B3LYP/6−31G (d, p) on structures listed in Table 2. Computational results confirm that the investigated compounds have a good stability, attributed on the rather large HOMO−LUMO energy gaps. The binding energy, expressed as the difference between the total energy of the structure and the energies of the constituent atoms, shows that all the compounds are energetically feasible. Cluster 7:(6(6).B(3).Mt(2).8BMt(3)).420, (Tb2(1)), with eight three−connected units in the second shell, has the largest energy gap (4.390 eV) compared to the molecules with incomplete second shell (Tb2(7):4.020 eV; Tb2(8): 3.692 eV; Tb2(9): 4.276 eV) -for their complete name see Table 1. A similar trend can be seen for their binding energy E bind/N ( Table 2).

Bioactivity Evaluation
Rhombellanic structures discussed in this section are also listed in Table 1 Table 3). The corresponding MM−energies for the reference fullerenes C 60 , C 70 and nanotube (5,5), are: 2354.76; 1980.58; 2015.98, respectively. Only the structure Ada−C−rbl.276 (1) was also optimized at the B3LYP level of theory (Table 2-entry 9). The compounds Tb3(1 to 3) were further computed in order to assess their drug-like properties and consecutively their bioactive conformations.
have been optimized as molecules (Tb2(10) and Tb2(11), respectively). Their symmetry corresponds to the symmetry axes along which they are aligned.
Full geometry optimizations were performed using B3LYP/6−31G (d, p) on structures listed in Table 2. Computational results confirm that the investigated compounds have a good stability, attributed on the rather large HOMO−LUMO energy gaps. The binding energy, expressed as the difference between the total energy of the structure and the energies of the constituent atoms, shows that all the compounds are energetically feasible. Cluster 7:(6(6).B(3).Mt(2).8BMt(3)).420, (Tb2(1)), with eight three−connected units in the second shell, has the largest energy gap (4.390 eV) compared to the molecules with incomplete second shell (Tb2(7):4.020 eV; Tb2(8): 3.692 eV; Tb2(9): 4.276 eV)for their complete name see Table 1. A similar trend can be seen for their binding energy Ebind/N ( Table 2).

ADME Properties Evaluation
Values of ADME [31,32] properties are listed in Table 4. Disposition of a pharmaceutical compound in organism is regarded as a function of compound lipophilic/hydrophilic properties, binding to serum proteins and permeability among cellular membranes. Env_264 (Tb3 (3)) shows close to nanotube (5,5), the polarizability QPpolrz and decahexane gas partition coefficient QPlogPC16 values; the other investigated parameters are less optimistic in terms of drug likeness when compared to nanotubes and fullerenes C70 and C60.  Regarding of binding to albumin (QPlogKhsa parameter − Table 4), nanotube (3,3) and C70, C60 show optimal albumin binding; rhombellanes do not reflect such a property.

Molecule
When evaluating membrane permeability, rhombellanes show poor CNS permeability; BBB, i.e. blood brain barrier, (see QPPMDCK parameter − Table 5 or CNS values - Table 6)) is not crossed by rhombellanes in contrast with the nanotubes and fullerenes, which show high penetration coefficient values. An exception is remarked for rhombellanes, particularly for C_rbl_348(Tb3(2)), which computationally express good skin permeability (parameter QPlogKp − Table 5).
Log S, the aqueous solubility of a compound, affects its absorption and distribution. A low solubility is correlated with modes absorption. 80% of the drugs on the market have log values greater than (−)4. In comparison with nanotubules and fullerenes some rhombellanes have decent solubilities ((−)6.95;(−)9.82).

ADME Properties Evaluation
Values of ADME [31,32] properties are listed in Table 4. Disposition of a pharmaceutical compound in organism is regarded as a function of compound lipophilic/hydrophilic properties, binding to serum proteins and permeability among cellular membranes. Env_264 (Tb3 (3)) shows close to nanotube (5,5), the polarizability QPpolrz and decahexane gas partition coefficient QPlogPC16 values; the other investigated parameters are less optimistic in terms of drug likeness when compared to nanotubes and fullerenes C 70 and C 60 .
When evaluating membrane permeability, rhombellanes show poor CNS permeability; BBB, i.e., blood brain barrier, (see QPPMDCK parameter- Table 5 or CNS values- Table 6) is not crossed by rhombellanes in contrast with the nanotubes and fullerenes, which show high penetration coefficient values. An exception is remarked for rhombellanes, particularly for C_rbl_348(Tb3(2)), which computationally express good skin permeability (parameter QPlogKp- Table 5).

Molecule CNS Human Oral Absorption Rule of Five Violation Rule of Three Violation
Nanotube (5,5) A set of descriptors were computed (see Supplementary Materials-S2). The number of aromatic atoms is increased in rhombellanes in comparison with fullerenes and nanotubules and consecutively the atomic contribution to log P.
Flexibility of rhombellanes is about 30 times bigger than that of C 70 , according to Kier flexibility index [35]. Total hydrophobic van der Walls area and total positive van der Walls areas reflect the same properties. (see Supplementary Material S2).
Rhombellanes carry more chemical information than fullerenes and nanotubules, when considering connectivity−based descriptors. Energy profiles (Figure 8), calculated using a MM2 force field [29] (the energy descriptors are calculated using the potential energy (in kcal/mol) from the stored 3D conformations) vary according to the same dynamics and confirm their internal mobility, higher than of the reference nano−molecules (fullerenes and nanotubes-see Table 3).
Molecules 2018, 23, x FOR PEER REVIEW 8 of 14 stored 3D conformations) vary according to the same dynamics and confirm their internal mobility, higher than of the reference nano−molecules (fullerenes and nanotubes -see Table 3).  Table 3. Used abbreviations: Epotential energy; E_ang -angle between potential energy; E_ele -electrostatic component of potential energy; E_oop -out of plane potential energy; E_sol -solvation energy; E_str -bond stretch potential energy; E_strain -local strain energy; E_tor -torsional potential energy; E_vdw -van der Waals component of potential energy.

RECAP Analysis
For the identification of the molecular building blocks of rhombellane derivatives listed in Table  3, an analysis was performed using the Retrosynthetic Combinatorial Analysis Procedure (RECAP) method [36], the results being listed in Table 7.   Table 3. Used abbreviations: E-potential energy; E_ang-angle between potential energy; E_ele-electrostatic component of potential energy; E_oop-out of plane potential energy; E_sol-solvation energy; E_str-bond stretch potential energy; E_strain-local strain energy; E_tor-torsional potential energy; E_vdw-van der Waals component of potential energy.

RECAP Analysis
For the identification of the molecular building blocks of rhombellane derivatives listed in Table 3, an analysis was performed using the Retrosynthetic Combinatorial Analysis Procedure (RECAP) method [36], the results being listed in Table 7. Table 7. RECAP analysis result: molecules represented as 2D formulas, molecular mass (M), and the corresponding frequency of appearance (%) in the analysis.

Structure Frequency (%) Structure Frequency (%)
Molecules 2018, 23, x FOR PEER REVIEW 8 of 14 stored 3D conformations) vary according to the same dynamics and confirm their internal mobility, higher than of the reference nano−molecules (fullerenes and nanotubes -see Table 3).  Table 3. Used abbreviations: Epotential energy; E_ang -angle between potential energy; E_ele -electrostatic component of potential energy; E_oop -out of plane potential energy; E_sol -solvation energy; E_str -bond stretch potential energy; E_strain -local strain energy; E_tor -torsional potential energy; E_vdw -van der Waals component of potential energy.

RECAP Analysis
For the identification of the molecular building blocks of rhombellane derivatives listed in Table  3, an analysis was performed using the Retrosynthetic Combinatorial Analysis Procedure (RECAP) method [36], the results being listed in Table 7. stored 3D conformations) vary according to the same dynamics and confirm their internal mobility, higher than of the reference nano−molecules (fullerenes and nanotubes -see Table 3).  Table 3. Used abbreviations: Epotential energy; E_ang -angle between potential energy; E_ele -electrostatic component of potential energy; E_oop -out of plane potential energy; E_sol -solvation energy; E_str -bond stretch potential energy; E_strain -local strain energy; E_tor -torsional potential energy; E_vdw -van der Waals component of potential energy.

RECAP Analysis
For the identification of the molecular building blocks of rhombellane derivatives listed in Table  3, an analysis was performed using the Retrosynthetic Combinatorial Analysis Procedure (RECAP) method [36], the results being listed in Table 7. stored 3D conformations) vary according to the same dynamics and confirm their internal mobility, higher than of the reference nano−molecules (fullerenes and nanotubes -see Table 3).  Table 3. Used abbreviations: Epotential energy; E_ang -angle between potential energy; E_ele -electrostatic component of potential energy; E_oop -out of plane potential energy; E_sol -solvation energy; E_str -bond stretch potential energy; E_strain -local strain energy; E_tor -torsional potential energy; E_vdw -van der Waals component of potential energy.

RECAP Analysis
For the identification of the molecular building blocks of rhombellane derivatives listed in Table  3, an analysis was performed using the Retrosynthetic Combinatorial Analysis Procedure (RECAP) method [36], the results being listed in Table 7. stored 3D conformations) vary according to the same dynamics and confirm their internal mobility, higher than of the reference nano−molecules (fullerenes and nanotubes -see Table 3).  Table 3. Used abbreviations: Epotential energy; E_ang -angle between potential energy; E_ele -electrostatic component of potential energy; E_oop -out of plane potential energy; E_sol -solvation energy; E_str -bond stretch potential energy; E_strain -local strain energy; E_tor -torsional potential energy; E_vdw -van der Waals component of potential energy.

RECAP Analysis
For the identification of the molecular building blocks of rhombellane derivatives listed in Table  3, an analysis was performed using the Retrosynthetic Combinatorial Analysis Procedure (RECAP) method [36], the results being listed in Table 7. stored 3D conformations) vary according to the same dynamics and confirm their internal mobility, higher than of the reference nano−molecules (fullerenes and nanotubes -see Table 3).   Table 3. Used abbreviations: Epotential energy; E_ang -angle between potential energy; E_ele -electrostatic component of potential energy; E_oop -out of plane potential energy; E_sol -solvation energy; E_str -bond stretch potential energy; E_strain -local strain energy; E_tor -torsional potential energy; E_vdw -van der Waals component of potential energy.

RECAP Analysis
For the identification of the molecular building blocks of rhombellane derivatives listed in Table  3, an analysis was performed using the Retrosynthetic Combinatorial Analysis Procedure (RECAP) method [36], the results being listed in Table 7. stored 3D conformations) vary according to the same dynamics and confirm their internal mobility, higher than of the reference nano−molecules (fullerenes and nanotubes -see Table 3).   Table 3. Used abbreviations: Epotential energy; E_ang -angle between potential energy; E_ele -electrostatic component of potential energy; E_oop -out of plane potential energy; E_sol -solvation energy; E_str -bond stretch potential energy; E_strain -local strain energy; E_tor -torsional potential energy; E_vdw -van der Waals component of potential energy.

RECAP Analysis
For the identification of the molecular building blocks of rhombellane derivatives listed in Table  3, an analysis was performed using the Retrosynthetic Combinatorial Analysis Procedure (RECAP) method [36], the results being listed in Table 7. Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3).  Tables 7 and 8, show the chemical synthetic space of rhombellanes. Frequency of computed appearance reveals the following: 6−connected hexagonal rings (frequency (%): 1.252-8.493, Table 7, entries 1 to 6); 3−connected hexagonal rings (frequency (%): 2.877-14.358, Table 7, entries 7 to 10), with more plausible substructures those having a higher frequency. Substitution of aromatic core seems to be an important variable in the synthesis; Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3).  Tables 7 and 8, show the chemical synthetic space of rhombellanes. Frequency of computed appearance reveals the following: 6−connected hexagonal rings (frequency (%): 1.252-8.493, Table 7, entries 1 to 6); 3−connected hexagonal rings (frequency (%): 2.877-14.358, Table 7, entries 7 to 10), with more plausible substructures those having a higher frequency. Substitution of aromatic core seems to be an important variable in the synthesis; Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3).  Tables 7 and 8, show the chemical synthetic space of rhombellanes. Frequency of computed appearance reveals the following: 6−connected hexagonal rings (frequency (%): 1.252-8.493, Table 7, entries 1 to 6); 3−connected hexagonal rings (frequency (%): 2.877-14.358, Table 7, entries 7 to 10), with more plausible substructures those having a higher frequency. Substitution of aromatic core seems to be an important variable in the synthesis; Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3). Some RECAP novel molecules suggested for the synthesis using templates arising from RECAP analysis are shown in Table 8. (full output is given in Supplementary Material S3).  Table 7, entries 7 to 10), with more plausible substructures those having a higher frequency. Substitution of aromatic core seems to be an important variable in the synthesis; stereochemistry of compounds has an important impact. Water molecules show a high frequency (94.106) suggesting its role as a reaction medium and possibly its formation as a by-product.
When analyzing the entire RECAP results, including all compounds (over 1000 results-see Supplementary Material S3), two major clusters were identified (Figure 9). Density analysis also demonstrates clusters identity. Among the novel molecules suggested synthesis using templates arising from RECAP analysis some structures, collected in Table 8, show ether constructions, possibly useful in a real synthesis.
Molecules 2018, 23, x FOR PEER REVIEW 10 of 14 stereochemistry of compounds has an important impact. Water molecules show a high frequency (94.106) suggesting its role as a reaction medium and possibly its formation as a by−product. When analyzing the entire RECAP results, including all compounds (over 1000 results − see Supplementary Material S3), two major clusters were identified (Figure 9). Density analysis also demonstrates clusters identity. Among the novel molecules suggested synthesis using templates arising from RECAP analysis some structures, collected in Table 8, show ether constructions, possibly useful in a real synthesis.

Methods
Rhombellanes are constructed by the "rhombellation" procedure; it starts with diagonalizing each face of an all−rhomb map Rh0 by a joint point (called "rbl−point"); then, add new vertices opposite to the parent vertices and join each of them with the rbl−vertices lying in the proximity of each parent vertex, thus local Rh−cells being formed. The process can continue, considering the envelope Rhn as "Rh0" for Rhn+1, in this way shell by shell being added to the precedent structure. Since the two diagonals of a rhomb may be topologically different, each generation may consist of two isomers.
Density functional theory (DFT) was used to investigate the stability of a series of rhombellanes (Table 2). Geometry optimizations was performed at the B3LYP/6−31G (d, p) level of theory using tight convergence criteria as implemented in the Gaussian 09 suite [28]. Harmonic vibrational frequencies obtained at the same level of theory confirmed that a true stationary point has been obtained. The obtained computational DFT results are collected in Table 2: the point group symmetry (PG), HOMO−LUMO energy gap (Egap), binding energy (Ebind), binding energy per number of heavy atoms (Ebind/N), and the total energy (Etot). DFT optimized Cartesian coordinates are given in Supplementary Material S4.
To obtain the ground state geometry of structure Tb2(11), multiple starting geometries were considered. The lowest energy configuration was the one where the hydroxyl groups are located on the same side of the benzene ring, while all carbonyl oxygen atoms are located on the opposite side. This arrangement is in agreement with the fragments observed in the double−shell cluster 7: (6(6).B(3).Mt(2).8BMt(3)).420 (Tb2(1)).
Molecular mechanics (MM) were utilized in order to explore the internal molecular mobility, accessed in the potential bioactivity study of these compounds. MMFF94 force field [29] was used to sample for 10 conformation poses of molecular structures Tb3(1 to 5). Steric energy of each pose (kcal/mol) was computed (Table 3). Each structure was minimized for maximum of 100 times, 10 conformations were sampled and conformational energy for each sample computed (kcal/mol).
Globularity (the state of being globular [30]) of rhombellanes was computed in comparison with that of fullerene C70. Globularity correlates with bioactivity of molecules or with manifestation of bioactivity (i.e., bioactive molecules are globular). Results were interpreted according to

Methods
Rhombellanes are constructed by the "rhombellation" procedure; it starts with diagonalizing each face of an all−rhomb map Rh 0 by a joint point (called "rbl−point"); then, add new vertices opposite to the parent vertices and join each of them with the rbl−vertices lying in the proximity of each parent vertex, thus local Rh−cells being formed. The process can continue, considering the envelope Rh n as "Rh 0 " for Rh n+1 , in this way shell by shell being added to the precedent structure. Since the two diagonals of a rhomb may be topologically different, each generation may consist of two isomers.
Density functional theory (DFT) was used to investigate the stability of a series of rhombellanes (Table 2). Geometry optimizations was performed at the B3LYP/6−31G (d, p) level of theory using tight convergence criteria as implemented in the Gaussian 09 suite [28]. Harmonic vibrational frequencies obtained at the same level of theory confirmed that a true stationary point has been obtained. The obtained computational DFT results are collected in Table 2: the point group symmetry (PG), HOMO−LUMO energy gap (E gap ), binding energy (E bind ), binding energy per number of heavy atoms (E bind/N ), and the total energy (E tot ). DFT optimized Cartesian coordinates are given in Supplementary Material S4.
To obtain the ground state geometry of structure Tb2(11), multiple starting geometries were considered. The lowest energy configuration was the one where the hydroxyl groups are located on the same side of the benzene ring, while all carbonyl oxygen atoms are located on the opposite side. This arrangement is in agreement with the fragments observed in the double−shell cluster 7: (6(6).B(3).Mt(2).8BMt(3)).420 (Tb2(1)).
Molecular mechanics (MM) were utilized in order to explore the internal molecular mobility, accessed in the potential bioactivity study of these compounds. MMFF94 force field [29] was used to sample for 10 conformation poses of molecular structures Tb3(1 to 5). Steric energy of each pose (kcal/mol) was computed (Table 3). Each structure was minimized for maximum of 100 times, 10 conformations were sampled and conformational energy for each sample computed (kcal/mol).
Globularity (the state of being globular [30]) of rhombellanes was computed in comparison with that of fullerene C 70 . Globularity correlates with bioactivity of molecules or with manifestation of bioactivity (i.e., bioactive molecules are globular). Results were interpreted according to globularity descriptor definition (i.e. a sphere shape molecule has globularity equal to 1). MD was used to explore globularity. The descriptor was computed for 200 conformations corresponding to 200 MD steps computed for a target temperature of 310.15K, uth a step interval of 1fs, frame interval of 1fs, and a heating/cooling rate of 1 kcal/atom/ps. For each structure the computation was terminated after 200 steps. ADME properties evaluation was performed using QikProp [37] module included in Schrodinger software. A series of descriptors related to bioactivity and drug-ability of these compounds were computed (see supplemental material S2 and refs. [38][39][40]). Also, a comparative energy profile was calculated using the same module.
RECAP computes a molecule by breaking certain bonds, estimated to be those that can be reformed by common reliable chemistry. Each resulting fragment is assigned a unique extended SMILES name that retains the chemical context of the broken bond. By applying this fragmentation methodology to a collection of compounds, statistics on individual fragments can be gathered and the fragments subsequently recombined in the "synthesis" procedure to generate new molecules.
RECAP resulted molecules were computed as clusters in order to identify their dispersion and correlation to chemical space. To make a cluster representation, the density analysis of resulted clusters was performed. RECAP analysis was used to generate novel molecules that can be, eventually, utilized in rhombellane synthesis.

Discussions
An active compound is a collection of molecular fragments according to fragment−based drug design principles. These fragments possess a specific three−dimensional arrangement that defines the whole properties of the compound: geometric, steric, conformational, topological, electronic and physicochemical properties define key aspects of bioactivity. By the medicinal chemistry point of view these properties define the therapeutic, metabolic and toxic properties.
Physicochemical properties are crucial to the pharmaceutical and pharmacokinetic phases of drug action; the pharmacodynamics are less discussed here and will be approached in another paper concerning these compounds.
Electronic properties reflect electron distribution within the drug molecule and determine the nature of the precise binding interaction between the drug and its receptor through hydrogen bonding and other interactions. On this subject computed conformational energies of rhombellanes take positive values, whereas rhombellanes envelopes possess negative energies (Table 3). All 10 distinct conformational samples for each structure present alternate conformational energies suggesting an encouraging conformational mobility specific for drug carriers and bio−nano−devices. Their energy profile, when compared to each other, is balanced, presenting the same proportional energy variation ( Figure 8).
Geometric, steric, and topological (globularity included) properties describe the structural arrangement of the atoms within the drug molecule and influence the interaction with the target.
Physicochemical properties reflect the solubility and absorption characteristics (in aqueous and lipid environments) of the drug and its ability to cross barriers, such as the blood-brain barrier, on its way toward the receptor. Thus, a highly significant property of drug molecules is their solubility (both in aqueous and non−aqueous environments), because only in solution can they interact with the cellular and subcellular structures that carry drug receptors, thus triggering pharmacological reactions. Partition coefficients are extremely important when understanding the properties of drug molecules. Most successful drugs exhibit solubility to some extent both in water and lipid environments; however, there are a few examples in which solubility in only one of these phases correlates with pharmacological activity.
Solubility is a function of many molecular parameters. Ionization, molecular steric structure, size, and electronic structure (by heteroatom presence), all influence the basic interactions between a solvent and the solute. Physicochemical properties are important in determining the ability of a drug molecule to survive the pharmacokinetic phase and to reach the region of the receptor−important both in bioactive molecules and in drug transporters. In this regard, rhombellanes present absorption properties comparable with carbon nanostructures. CNS penetration is less favorable in comparison to nanotubes and fullerenes.
The correct geometry, conformation, and stereochemistry of a molecule enable it to gain access to the receptor microenvironment [42,43]. However, it is the electronic molecular structure that finally decides the electrostatic, hydrogen bonding, and other drug-receptor binding interactions to actually occur.
The effect of electron distribution in organic molecules is manifested directly (short range) or indirectly (long range). Inductive forces, as van der Waals bonds or dipole−dipole interactions, are the result of polarization or polarizability-the permanent or induced distortions of the electron distribution within a molecule. These forces are especially important in studies of quantitative structure-activity relationships (QSAR) because the electronic effect of a substituent can, by resonance or an inductive or field effect, change the stereo−electronic properties of a molecule and thus influence its biological activity.
Statistical analysis is used frequently in bioinformatics. Density−based clustering defines clusters as areas of higher density than the remainder of the data set. In this study, OPTICS was used for clustering the RECAP results; OPTICS [44] is a generalization of DBSCAN which avoids the need to choose an appropriate value for the range parameter epsilon and produces a hierarchical result related to that of linkage clustering.
Based on this findings future developments of these compounds will be carried on. Using reverse synthesis techniques a chemical synthesis will be tested. Furthermore optimized compounds refined regarding geometry and outer surface and inner core-shell and envelope-are intended to be tested against biological targets.

Conclusions
Rhombellanes represent a new class of hypothetical structures, of which homeomorphs may be synthesized as real molecules. Cube−rhombellane, in the first step of rhombellation operation, is a double−shell structure showing vertices of degree 3 and 6 respectively, which may be realized by means of cyclohexane (and/or benzene) derivatives. Some suggestions of molecular realization of cube−rhombellane as functionalized structures (ether core, ester or amide envelope) were given. Topology of the discussed structures was detailed. Quantum calculations supported the herein hypothesis: both substructures (at every level of complexity) and the whole double−shell molecules show negative binding energy and a rather high energy gap, proving that all the compounds are energetically feasible in the hope of a real synthesis.
Rhombellanes seem to be appropriate and worthwhile molecules for medicinal chemistry. This new class of compounds may be an alternative to classical nano−structures regarding carrier, drug like properties and use for creating bio−nano devices. In contrast to fullerenes and nanotubules, where most of functionalization (and derivatives) occur symmetrically, rhombellanes present a variety of distinct derivatization sites thus making them easier to be chemically manipulated. As drug carriers, rhombellanes show decent membrane-passing properties. In close relation with fullerenes and nanotubules, our compounds present "per se" drug-like properties which could be further developed to access a novel chemical space.