**Contents**


#### **Piotr Cysewski**


#### **About the Special Issue Editors**

**Lorentz J ¨antschi** was born in Fagaras, Romania in 1973. In 1991 he moved to Cluj-Napoca, Cluj, where he completed his studies. In 1995 he was awarded a B.Sc. and M.Sc. in Informatics, in 1997 a B.Sc. and M.Sc. in Physics and Chemistry, in 2000 a Ph.D. in Chemistry under the supervision of Prof. Mircea V. Diudea, in 2002 an M.Sc. in Agriculture, in 2010 a Ph.D. in Horticulture, and, finally, in 2013 a postdoctorate in Horticulture. That same year (2013), he became a Full Profesor of chemistry at the Technical University of Cluj-Napoca and an associate at Babes-Bolyai University, where he advises Ph.D. studies in chemistry. Both positions are to date. During his research and education in Cluj, he performed activities under the auspicies of other institutions as well: G. Baritiu (1995–1999) and Balcescu (1999–2001) at National Colleges, the I. Hatieganu University of Medicine and Pharmacy (2007–2012), Oradea University (2013–2015), and the institute of Agricultural Sciences and Veterinary Medicine at University of Cluj-Napoca (2011–2016). He serves as an editor for the journals Notulae Scientia Biologicae, Notulae Horti Agro Botanici Cluj-Napoca, Open Agriculture and Symmetry. He was Editor-in-Chief of the Leonardo Journal of Sciences and the Leonardo Electronic Journal of Practices and Technologies.

**Beata Szefler** as a Doctor of Chemistry, is a Professor hab. of Collegium Medicum in Bydgoszcz, Poland, a laboratory diagnostician, and a researcher in the Department of Physical Chemistry, Collegium Medicum, Nicolaus Copernicus University in Bydgoszcz, Poland. A graduate of the Medical Academy in Bydgoszcz, Faculty of Pharmacy, with a Ph.D. in Chemical Sciences from the University of Technology & Life Sciences in Bydgoszcz, Poland, 2010, and awarded habilitation in pharmaceutical sciences from Collegium Medicum, Nicolaus Copernicus University, Bydgoszcz, Poland, in 2019, her experience lies in microbiology, hematology, pharmacokinetics and in silico research (ab initio calculations, HF, DFT, MD, and fullerenes). Her research interests are molecular modeling of aromatic systems with particular emphasis on the importance of biochemical molecules, the study of properties of fullerenes using HF, DFT and MD methods, docking ligand proteins, and QSAR. She is a principal investigator of parameterization of force fields, molecular dynamic simulations of selected protein and ligand–protein systems, quantum-chemical studies of small ligands, the topological description of the electronic structure of ligands (the atoms-in-molecules methodology), and data analysis. She is a member of the European Society of Mathematical Chemistry (ESMC) and the National Chamber of Laboratory Diagnosticians (KIDL, Poland).

#### **Preface to "Applied Designs in Chemical Structures with High Symmetry"**

An important factor in determining chemical and biological activity is structural symmetry or asymmetry. Structural symmetry is one of the most basic properties of chemical compounds. The best example of a highly symmetrical compound is C60 fullerene, which appears as a pure synthetic form of carbon. This structure is the best starting compound for the formation of fullerene derivatives with high symmetry. The application of new fullerene derivatives is endless, including in materials science, chemistry, biology, pharmacy and medicine. Thousands of papers on this subject are published annually. Editors Professor Dr. Lorenz Jantschi and Professor Dr. Beata Szefler are the authors ¨ of several dozen scientific papers in which structural symmetry plays a dominant role. The book "Applied Designs in Chemical Structures with High Symmetry" will be a collection of new works describing new structures with these properties.

> **Lorentz J¨antschi, Beata Szefler** *Special Issue Editors*

#### *Communication* **The Eigenproblem Translated for Alignment of Molecules**

#### **Lorentz Jäntschi 1,2**


Received: 12 July 2019; Accepted: 7 August 2019; Published: 9 August 2019

**Abstract:** Molecular conformation as a subproblem of the geometrical shaping of the molecules is essential for the expression of biological activity. It is well known that from the series of all possible sugars, those that are most naturally occurring and usable by living organisms as a source of energy—because they can be phosphorylated by hexokinase, the first enzyme in the glycolysis pathway—are D-sugars (from the Latin dextro). Furthermore, the most naturally occurring amino acids in living cells are L-sugars (from the Latin laevo). However, a problem arises in dealing with the comparison of their conformers. One alternative way to compare sugars is via their molecular alignment. Here, a solution to the eigenproblem of molecular alignment is communicated. The Cartesian system is rotated, and eventually translated and reflected until the molecule arrives in a position characterized by the highest absolute values of the eigenvalues observed on the Cartesian coordinates. The rotation alone can provide eight alternate positions relative to the reflexes of each coordinate.

**Keywords:** eigenproblem; eigenvalues; molecular alignment; orthogonal alignment

#### **1. Introduction**

The topological description of a molecule requires knowledge of the adjacencies (the bonds) between the atoms as well as their identities (the atoms). If this problem is simplified to the extreme, by disregarding the bond types and atom identities, then the adjacencies are simply expressed as 0 or 1 in the vertex adjacency matrix ([*Ad*]) and the identities are expressed as 0 or 1 in the identity matrix ([*Id*]). The characteristic polynomial (*ChP*) is the natural construction of a polynomial in which the eigenvalues of the [*Ad*] are the roots of the *ChP*, as follows:

λ is an eigenvalue of [*Ad*] ↔ it follows that [*v*] - 0 eigenvector such that λ·[*v*] = [*Ad*]·[*v*] (λ·[*Id*] − [*Ad*])·[*v*] = 0); since v - 0 → [λ·*Id* − *Ad*] is singular → det([λ·*Id* − *Ad*]) = 0. Therefore, the characteristic polynomial is defined by:

$$ChP \stackrel{\text{def}}{=} \left| \lambda \cdot [Id] - [Ad] \right|.$$

The characteristic polynomial is a polynomial in λ of the degree of the number of atoms. The eigenproblem (the determination of eigenvalues and eigenvectors) is applicable to any Hessian [1] matrix [*A*] ([*Ad*] → [*A*]). The mixed derivatives of a scalar-valued function *f* are the entries off the main diagonal in the Hessian. Assuming that the derivatives are continuous, the order of differentiation does not matter (a result known as Schwarz's, Clairaut's, or Young's theorem), and then the Hessian of *f* is a symmetric matrix.

Indeed, this is the case (a symmetric matrix) for the (vertex) adjacency matrix, and for the distance matrix—both topological (by bonds) and geometrical (by the atom coordinates).

Related to this problem is the issue of determining the best rotation to relate two sets of vectors. To this issue, a solution was proposed by calculating a symmetric matrix of Lagrange multipliers which is used to minimize the residuals of the linear association between the vectors [2]. Later, different approaches were proposed, such as geometric hashing [3], clique detection [4], the embedding problem [5], Gaussian molecular representation, Gaussian overlap optimization [6], and others covered in [7]. Some of the proposed solutions go a different way, involving physical means forcing the alignment [8,9], while the formulation of similarity metrics was one of the most recently proposed computational alternatives [10]. The alignment serves as a tool for other studies, including similarity analysis [11], docking [12], and structure–activity relationships [13].

The eigenproblem in relation to geometrical alignment was stated before in the context of surface analysis and control [14], and also can go another direction into the context of the molecule. In this context, the molecule is seen as more than a simple unweighted undirected molecular graph with undistinguishable atoms [15].

The eigenproblem of molecular alignment is analyzed in this paper.

#### **2. Materials and Methods**

The alignment of molecules can be stated in many ways, as listed in the introduction. For instance, one approach is to search for topological alignment, and another is to search for geometrical alignment. To anticipate the type of molecular alignment, it is necessary to employ the latter method—to search for geometrical alignment.

A molecule is taken here as an example from PubChem CID 444173 ((2R,3S,4R,5R)-oxane-2,3,4,5-tetrol), as shown in Figure 1.

**Figure 1.** 3D representation of the model of PubChem CID 444173.

For convenience, hydrogen atoms are excluded from the data and the analysis. The next table (Table 1) contains the relevant information for the heavy atoms in the reference molecule.


**Table 1.** 3D structural data for CID 444173 (heavy atoms, geometric coordinates, and atom symbols).

The general way of constructing a characteristic polynomial is to provide an identity matrix [*Id*] and a Hessian matrix (herein labeled as [ *A*]). If considering the topology of the molecule, then it is necessary to have the information regarding the connections between the atoms (e.g., bonds). Since all bonds are single bonds for the selected molecule, listing the atoms pairs of the bonds is enough (Table 2).

**Table 2.** Topology data for CID 444173 (list of bonds between heavy atoms).


A deeper look into the eigenproblem (|λ·*I* − *A*| = 0) is performed in the next section, with a specific focus on changing of the mathematical properties of the eigenproblem when the adjacencies in [ *A*] change from symmetric to anti-symmetric.

#### **3. Results and Discussion**

From Table 2, the adjacency matrix [*Ad*] is immediate—zeros represent the entries without a bond between the labeled atoms, while ones appear otherwise. The adjacency matrix is Hessian. For convenience, its characteristic polynomial is:

$$\text{ChP}(\lambda; \text{CID}4444173, \text{"[Ad]"}) = 1 \cdot \lambda^{10} - 10 \cdot \lambda^8 + 31 \cdot \lambda^6 - 35 \cdot \lambda^4 + 11 \cdot \lambda^2 - 1 \cdot \lambda^0.$$

As can be seen, the degree of the polynomial is 10, which is equal to the number of the (connected) atoms in the molecule. The general rule is that a characteristic polynomial is always of a degree equal tothesizeofthesquarematrices[*I*]and[ *A*](seethebeforegivenequation),fromwhichitwasderived.

 The same strategy can be applied if the adjacency matrix [*Ad*] is replaced by the distance matrix [*Di*]. For convenience, Table 3 lists these two matrices.


**Table 3.** Adjacency and distance matrices for CID 444173 (heavy atoms).

*Symmetry* **2019**, *11*, 1027

It can be checked (but is also true for the general case) that the distance matrix is Hessian, and therefore a characteristic polynomial can be computed for it as well. The next equation lists the *ChP* computed for the distance matrix [*Di*]:

$$\text{CHP(A; CID444173, "[Dl]")}=1\cdot\text{l}^{10}-313\cdot\text{l}^{8}+3488\cdot\text{l}^{7}-15456\cdot\text{l}^{6}-34720\cdot\text{l}^{5}-40832\cdot\text{l}^{4}-23808\cdot\text{l}^{3}-5376\cdot\text{l}^{2}$$

The natural extension of this matrix is to employ 3D distances instead of topological distances. Of course, one consequence of this is that the characteristic polynomial would no longer have integer coe fficients.

The next table lists the 3D distance matrix (distances were cut to four significant digits) and the roots of the associated characteristic polynomial.

One interesting remark to the data listed in Table 4 is that all roots are real (this is the general behavior for the roots of a characteristic polynomial).


**Table 4.** *3D* distance matrix and its eigenvalues for CID 444173 (heavy atoms).

Up until this point, the ideas presented in this paper have been reported before. Herein follows the extension to the extant knowledge. What if the same formula is applied to define the *ChP* for Cartesian coordinate distance matrices instead of for the Euclidian distance matrix?

Next three tables (Tables 5–7) list those results (the number of digits is displayed according to the input data—see Table 1).

**Table 5.** First Cartesian coordinate ("*x*") distances matrix for CID 444173 (heavy atoms).



**Table 6.** Second Cartesian coordinate ("*y*") distances matrix for CID 444173 (heavy atoms).

**Table 7.** Third Cartesian coordinate ("*z*") distances matrix for CID 444173 (heavy atoms).


It can be observed that the Cartesian coordinates distance matrices are no longer symmetric matrices, but are in fact anti-symmetric, meaning that *Mi,j* = <sup>−</sup>*Mj,i*.

The beauty of the result shown by taking a look at the eigenvalues. The next table (Table 8) lists the eigenvalues for all matrices.


**Table 8.** Eigenvalues for CID 444173 (heavy atoms).

It should be noted that the values listed in Table 7 reveal some computational errors. It is obvious (and it is so) that 3.6 × 10−<sup>15</sup> is actually a "0" and it is necessary to be aware of this type of error coming from "machine epsilon" [16] which is about 10−<sup>7</sup> for "single" precision, 10−<sup>16</sup> for "double" precision, and about 10−<sup>19</sup> for "extended" precision. Most floating-point implementations use "double" precision and thus the listed value (3 × <sup>10</sup>−15) "fits in range".

More important, as can be observed (see Table 7), the eigenvalues of [*Dx*], [*Dy*], [*Dz*] are all 0 excepting (always) two—which are (always) paired and (always) imaginary (i = √−1, see Table 8). This is the opposite of the traditional case of symmetric matrices, when the values are (always) real. This is the beauty of the result.

Moreover, it should be noted that the polynomial can be expressed with real-value coefficients as a product of a polynomial of degree 2 and a monomial of degree (*n* − 2), as listed in Table 9.


**Table 9.** The polynomials of [*Dx*], [*Dy*], and [*Dz*] for CID 444173 (heavy atoms).

A consequence is hidden behind this result—to obtain those two coefficients (which are actually the first and third coefficients, independent of how many atoms are in the molecule) it is necessary to obtain their roots. Therefore, it is not necessary to run an "eigenvalues" routine to obtain them; it is enough to run only two steps of a coefficient determination program (such as that described in [17]), which will produce a result much more quickly.

So, what if we conduct a rotation of the molecule?

For example, by rotating the molecule by 15◦ (15/180 radians; coordinates are given in Table 1), the values for the polynomials are changed—see Table 10. First it should be pointed out that the polynomial is no longer invariant due to the choice of the system of coordinates. If invariants are sought, this is not a good situation—but for the purpose of addressing the alignment problem, this setup is very useable.

**Table 10.** The polynomials of [*Dx*], [*Dy*], and [*Dz*] rotated (15◦, 15◦, 15◦) for CID 444173 (heavy atoms).


In the general case, with *a*1, *a*2, and *a*3 as rotation angles defining the rotation matrices (given below), it is necessary to maximize the variance along the axes of coordinates.


This results a two-step algorithm, described below:


Therefore, at this point we have the alignment of the molecule.

The problem of molecular 3D alignment involving the modified characteristic polynomial (eigenproblem) becomes a combinatorial problem since, after eigenvector minimization by each (two out of three) Cartesian coordinate, we obtain the molecules in their proper alignment or in the mirror of the proper alignment, when "*xi* ← −*xi*" and/or "*yi* ← −*yi*" and/or "*zi* ← −*zi*" transformation will align it.

Of course, a question may arise: what is the meaning of such alignment? This research is ongoing, but so far it has been found that this alignment corresponds to the minimization of the rotation inertia

of the coordinates. In other words, the thinnest part of the molecule aligns with one coordinate, and then the thinnest part of what remains (so the molecule can be rotated around that axis) aligns with the second coordinate.

Revising the results communicated here, it should be noted that the classical eigenproblem is addressed to symmetric matrices—such as are the topological adjacency and topological distance matrices (shown in Table 3) and the geometrical distance matrix (Table 4). The peculiarity of the Cartesian distance matrices (shown in Tables 5–7) is the fact that they are anti-symmetric, sometimes called skew-symmetric matrices. This is, in mathematical terms, a strong property—as strong as the property of symmetry (please note that here the symmetry describes the matrices—namely, matrix *A* is symmetric if *A* = *A*<sup>T</sup> and it is anti-symmetric if *A* = −*<sup>A</sup>*T). On the other hand, the elements of the Cartesian coordinate matrices are mirrored relative to the main diagonal—this property is called reflection symmetry, line symmetry, or mirror symmetry—which makes these matrices very suitable for the same set of operations that are typically employed for symmetric matrices. Further, among the known properties of skew-symmetric matrices is the fact illustrated in Table 8—If *A* is a real skew-symmetric matrix and λ is a real eigenvalue, then λ = 0, i.e., the nonzero eigenvalues of a skew-symmetric matrix are purely imaginary". Since a skew-symmetric matrix is similar to its own transposition, they must have the same eigenvalues. It follows that the eigenvalues (λ) of a skew-symmetric matrix always come in pairs (±λ), a property which is also illustrated in Table 8.

It should be noted that the generation of Cartesian coordinates from the diagonalization of adjacency or distance-related matrices is quite standard in mathematical chemistry. For instance, the methods to generate fullerene cages from Schlegel diagrams are normally embedded in fullerene sw packages (see for example [18]). Thus, the results communicated here may have useful applications in this regard.

#### **4. Conclusions**

The change from symmetry to anti-symmetry in the adjacency matrix of the eigenproblem moves the eigenvalues from real space into imaginary space. When the eigenequation is applied to the Cartesian space of the molecule instead of the topological or Euclidean spaces, the resultant roots (corresponding to the eigenvalues) are all 0 (multiple roots) excepting two, which are always imaginary (and complementary). The rotation of a molecule induces into the Cartesian space a way of aligning the molecule by maximizing the magnitude of the roots in a preselected order of the Cartesian axes. This property can be further exploited for the alignment of multiple molecules, when for highly symmetric molecules the alignment problem is turned into the (S2) 3 conformational problem.

Though the programs provided in the Supplementary Materials can be used to align any molecule, they are not communicated as a novel tool. Aligning a molecule by its Cartesian coordinates via the simultaneous alignment of many molecules—such as for molecular docking purposes—will require further study.

**Supplementary Materials:** The datafile for CID 444173 and the MATLAB program implementing the Cartesian alignment of a molecule are available online at http://www.mdpi.com/2073-8994/11/8/1027/s1.

**Author Contributions:** The author designed and made the study and also wrote the paper.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

## **References**


©2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

#### *Article* **Docking Linear Ligands to Glucose Oxidase**

#### **Beata Szefler**

Department of Physical Chemistry, Faculty of Pharmacy, Collegium Medicum, Nicolaus Copernicus University, Kurpi ´nskiego 5, 85-096 Bydgoszcz, Poland; beatas@cm.umk.pl

Received: 12 June 2019; Accepted: 8 July 2019; Published: 10 July 2019

**Abstract:** GOX (3QVR), glucose oxidase, is an oxidoreductase enzyme, which has found many applications in biotechnology and modern diagnostics with typical assays including biosensors useful in the determination of free glucose in body fluids. PEI (polyethylenimines) are polymer molecules made up of amine groups and two aliphatic carbons, which are cyclically repeated. PEI are transfection reagents which, using positively charged units, bind well to anionic DNA residues. During the studies on GOX, PEI were used both in their linear and branched structures. Rhombellanes, RBL, are structures decorated with rhombs/squares. The aim of the paper is to study the interactions of two kinds of linear ligands: PEIs (Polyethylenimines) and CHRs (ethers of Hexahydroxy-cyclohexane) with the glucose oxidase enzyme, GOX (3QVR). To understand the structure-activity relationship between the GOX enzyme and the linear ligands PEI and CHR, two steps of docking simulation were performed; mapping the whole area of the 3QVR enzyme and docking on the first and second surface of the enzyme, separately. The studied ligands interacted with amino acids of GOX inside the protein and on its surface, with stronger and shorter bonds inside of the protein. However, long chain ligands can only interact with amino acids on the external protein surface. After the study, two domains of the enzyme were clearly evidenced; the external surface domain more easily creates interactions with ligands, particularly with CHR ligands.

**Keywords:** PEI; CHR; 3QVR; Glucose oxidase; docking

#### **1. Introduction**

GOX, glucose oxidase, is an oxidoreductase that catalyzes the oxidation of β-<sup>d</sup>-glucose to d-glucono-β-lactone, which is non-enzymatically hydrolyzed to gluconic acid [1]. In these reactions, FADH2 is formed as the reduction product of the FAD ring of GOX. FADH2 is then re-oxidized by using molecular oxygen to obtain hydrogen peroxide (H2O2). GOX has found many applications in biotechnology and modern diagnostics. Typical assays utilize it as a biosensor useful in determination of free glucose in body fluids.

There are several types of PEI [2]: The branched PEI (BPEI), linear PEI (LPEI), and dendrymer PEI (DPEI). Branched PEI have all types of amino groups, while linear PEI contain primary and secondary amino groups.

BPEI is liquid at room temperature, while LPEI is solid because its melting point is about 73–75 ◦C. It is well soluble in water (hot with low pH), methanol, ethanol, and chloroform.

PEI binds to anionic residues of DNA by its positively charged units [3]. Despite high toxicity, PEI has many applications [4] due to its polycationic character. PEI has been used in studies on GOX, both in its linear and branched structures [5–8].

Rhombellanes, RBL, are structures built of rings in the form of rhombs or squares (Figure 1, left); they have recently been proposed by Diudea [9,10].

**Figure 1.** Rhombellane basic structures [9,10].

Rombellanes are organic molecules, C5H6, structurally related to [1,1,1] propellane, which contains only triangles [11]. Its reduced form, C5H8, is named bicyclo [1.1.1]pentane and is made of rhomb rings or square rings (Figure 1, right). [1,1,1] Propellane undergoes spontaneous polymerization to [*n*] staffanes (Figure 2, left) [12,13], which are rigid, linear structures. The number suffixing a molecule name stands for the number of its (heavy) atoms.

**Figure 2.** Linear oligomers related to rhombellanes [9,10].

Rhombellanes have five common features [14–16], but foremost they are structures made of rhomb rings or square rings, where vertex classes consist of all non-connected vertices. The cube-rhombellane is presented in Figure 1. Rhombellanes are, in general, designed by the "rhombellation" operation [9,10].

By analogy to staffanes [*n*]STF [5], (Figure 2, left), a linear rod-like polymer (a poly-ether of 1,2,3,4,5,6-hexahydroxy-cyclohexane, [*n*] CHR, (Figure 2, right) was proposed by Diudea [17]. This idea came from cube-rhombellane (Figure 1, right) [9,10], with vertex/atom connectivity 6 and 3, respectively. For its realization as a molecule, it was proposed [10] to use hexahydroxy-cyclohexane for connectivity 6 (connectivity 3 being more accessible).

#### **2. Materials and Methods**

In the first step of the docking simulation, the whole area of the 3QVR enzyme was mapped step by step (Figure 3). The protein was downloaded from Brookhaven Protein Database PDB [18,19].

To understand the structure–activity relationship of the ligand–protein complex (Supplementary Tables S1 and S2), molecular docking was carried out.

Two kinds of linear structures were used as ligands: PEI.C2<sup>+</sup>nN2+m; *n* = 2, 4, 6 ... ; *m* = 1, 2, 3 ... (Figure 4) and [*n*]CHR.24 + 15*n*; *n* = 0, 1, 2 ... , respectively (Figure 4).

**Figure 3.** *Cont.*

**Figure 3.** The protein 3QVR/GOX, with the inside pocket (**top**) and external binding domain (**bottom**) docking surface.

/LQHDUOLJDQG3(,DXQLWOHIWDQGDQROLJRPHUB&Q1PULJKW

**Figure 4.** Linear ligands PEI

/LQHDUOLJDQG&+5+H[DK\GUR[\F\FORKH[DQH+&+OHIWDQGDQROLJRPHUULJKW (**top** 

**row**) and CHR (**bottom row**).

All structures used during the molecular docking stage, including the ligands and proteins, were created with the use of the AutoDockTools package and non-polar hydrogen atoms were removed from each molecule. The dimensions of the grid box were fitted to the size of the protein; while during modeling, all possible interactions of the ligand with the whole protein surface were evaluated. The docking procedure was realized with the use of AutoDockVina [20,21], using the Lamarckian genetic algorithm [22]. The calculations were realized with exhaustiveness parameter equal 20, with the chosen value being a compromise between good reproducibility of calculations and computation time cost. Finally, the validity of the used algorithm was verified [23,24].

#### **3. Results**

#### *3.1. The First Step of the Study*

By performing the docking procedure, it is clear that ligands can interact with amino acids of GOX both inside of protein (Figure 5, top) and on its surface (Figure 5, bottom). Selected values of the free energy of binding to the proteins active site are given in Supplementary Tables S1 and S2 for ligands PEI and CHR, respectively.

**Figure 5.** Hydrogen bond lengths (in Å) in the PEI–GOX complexes: Inside the protein (3QVR–PEI.C18N10.01.0Lin2, **top**) and on its surface (3QVR–PEI.C10N6.01.0Lin3, **bottom**).

However, the bonds created inside are stronger and shorter, compared with the bonds formed on the outside surface of enzyme, which is what could be expected (see Supplementary Tables S1 and S2; Supplementary Figures S1 and S2. The structural analysis of the formed complexes clearly shows that the ligand affinity values on the external surface are lower compared to the values of affinity inside of enzyme (see also Supplementary Figures S1–S3; Supplementary Tables S1 and S2.

With the increasing chain length of PEI and CHR ligands, the values of ligand–protein affinity also increase (Supplementary Figure S4; Supplementary Tables S1 and S2. This is caused by the fact that increasing the structural elongation of the ligand increases the number of atoms participating in the ligand–enzyme hydrogen bond formation (Figure 6).

Hydrogen bonds of ligand and protein are detailed in Supplementary Figures S1 and S2 (amino acids are hidden for the sake of image clarity).

The ligand affinity depends on the number of hydrogen bonds and the strength of the interactions between ligand and protein (Figure 7; Supplementary Table S3). It also depends on whether these interactions occur on the external surface of the enzyme or inside it.

**Figure 6.** *Cont.*

**Figure 6.** Binding affinity (kcal/mol) in PEI (**top**) and CHR (**bottom**) ligands, after the first step of the docking procedure.

**Figure 7.** Binding affinity (kcal/mol) vs. the number of H-bonds created between ligand and enzyme.

Jeffrey [25] categorizes H bonds by donor–acceptor distances. Based on such categorization, H-bonds created with the protein by PEI and CHR ligands are strong and moderate (Supplementary Table S3). The number and strength of the hydrogen bonds between the ligand and protein will, finally, determine the values of binding affinity, and consequently, the ligand–protein overall distance. Thus, a high value of ligand–protein binding affinity is directly related to the formation of many strong hydrogen bonds.

In the case of PEI ligands, all the tested molecules, except for PEI.C10N6.01.0Lin3, form H-bonds inside of the protein. In the case of CHR molecules, only those with a short chain (namely [0]CHR.24 and [1]CHR.39) find room to interact with amino acids inside of the enzyme. Ligands with a longer chain ([*n*]CHR.24 + 15*n*; *n* = 3 to 7) can only embed on the external binding domain of the enzyme surface (Figure 3).

That is why, in the second step of this study, the docking was made on the surface of the enzyme, where the studied ligands can create H-bonds with the protein in an easy and quick way.

#### *3.2. The Second Step of the Study*

Again, with the growing chain length of PEI and CHR molecules, the values of ligand–protein affinity also increase (Figure 8), by increasing the number of hydrogen bonds created between the enzyme and ligand.

**Figure 8.** Binding affinity (kcal/mol) in PEI (**top**) and CHR molecules (**bottom**) after docking of ligands on the external surface of protein.

One can see that the values of ligand–enzyme affinity on the external domain (Supplementary Tables S4 and S5) are much smaller than in the inside domain (Supplementary Tables S6 and S7), which suggests the creation of longer and weaker hydrogen bonds.

#### **4. Discussion**

When comparing the energy of hydrogen bonds created between ligands and GOX enzyme, it can be found that differences were around 1% in favor of the external domain of docking (Figures 4 and 9, top) for PEI ligands; while in the case of CHR ligands, the differences ranged from 40% to 70%, also in favor of the external domain (Figure 9, bottom).

All of the above means that the external surface domain of enzyme (Figure 3, bottom) interacts more easily with ligands, particularly with CHR molecules. Here, the larger differences are in the case of longer chain molecules (Figure 9, bottom).

**Figure 9.** *Cont.*

**Figure 9.** Binding a ffinities (in kcal/mol) of enzyme (GOX protein)–ligands (PEI (**top**) and CHR (**bottom**)), bound inside (first protein surface) and outside/externally (second protein surface), respectively.

Detailed structural analysis of the ligand-protein complexes, formed after the docking procedure, confirmed the observation regarding the existing di fferences in the strength of hydrogen bonds, on the inside and outside domains of GOX enzyme, in docking of the ligands PEI and CHR, respectively (Figure 9). The number and strength of hydrogen bonds created between PEI and amino acids were comparable on the two docking domains of the protein. Contrarily, in the case of CHR ligands, the external surface domain provided more room to form a large number of interactions with the protein, in comparison to the internal pocket of enzyme (Figure 3).

#### **5. Conclusions**

To understand the structure–activity relationship between the GOX (3QVR) enzyme and linear ligands PEI and CHR, in the first step of this docking study, the whole area of the enzyme was mapped. It was found that ligands can interact with amino acids of GOX in two domains, namely, the inside of the protein and on its external surface. The bonds created in the inside domain are stronger and shorter, compared with the bonds formed on the external surface of enzyme. Also, in the latter, the bonding energy values were lower compared with the values recorded for the complexes formed with the inside docking domain of enzyme. Despite this, with the increasing chain length of PEI and CHR ligands, the values of ligand–protein a ffinity also increased, which is what is connected to the increasing ability to create more ligand–protein hydrogen bonds. The binding a ffinity depends on the number and strength of interactions between ligand and protein, and also depends on whether these interactions occur on the external surface of the enzyme or inside it. The longer ligand chains can only find room for docking with the amino acids lying on the external surface of the protein.

Comparison between the two docking sites of GOX revealed di fferences in binding a ffinity in favor of the external domain of docking; around 1% in the case of PEI ligands and from 40% to 70% in the case of CHR molecules.

The number and strength of hydrogen bonds created between PEI and GOX amino acids are comparable on the two sites of protein docking; while in case of CHR ligands, the docking is favored on the external domain of the protein surface. This means that the CHR ligands act selectively on the external binding site of the GOX enzyme.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2073-8994/11/7/901/s1, Figure S1: The length of hydrogen bonds in Å formed between ligands PEI and GOX enzyme. Table S1: The final lamarckian genetic algorithm docked state—Binding energy of PEI ligand to the active sites of type of 3QVR during the nine explored conformations. The number 1,2,3 in the ligand name means the use of another random variable in the docking procedure.

**Funding:** This research was supported by PL-Grid Infrastructure http://www.plgrid.pl/en.

**Acknowledgments:** This work was supported by GEMNS project granted in the European Union's Seventh Framework Programme under the frame of the ERA-NET EuroNanoMed II (European Innovative Research and Technological Development Projects in Nanomedicine).

**Conflicts of Interest:** The author declares no conflict of interest.
