# Riemann-Symmetric-Space-Based Models in Screening for Gene Transfer Polymers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Discussion

^{oi}) = e

^{0}= 1; f(e

^{2πi}) = e

^{2πia}

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Mary, B.; Maurya, S.; Kumar, M.; Bammidi, S.; Kumar, V.; Jayandharan, G.R. Molecular engineering of Adeno-associated virus capsid improves its therapeutic gene transfer in murine models of hemophilia and retinal degeneration. Mol. Pharm.
**2019**, 16, 4738–4750. [Google Scholar] [CrossRef] [PubMed] - Aartsma-Rus, A.; van Putten, M. The use of genetically humanized animal models for personalized medicine approaches. Dis. Model. Mech.
**2019**, 13, 041673. [Google Scholar] [CrossRef] [PubMed] - Contin, M.; Garcia, C.; Dobrecky, C.; Lucangioli, S.; D’Accorso, N. Advances in drug delivery, gene delivery and therapeutic agents based on dendritic materials. Future Med. Chem.
**2019**, 11, 1791–1810. [Google Scholar] [CrossRef] [PubMed] - Richard, B.; Ho-Fung, C.; Jóhannes, R. Characteristics of known drug space. Natural products, their derivatives and synthetic drugs. Eur. J. Med. Chem.
**2010**, 45, 5646–5652. [Google Scholar] - Anderson, D.G.; Peng, W.; Akinc, A.; Naushad, H.; Kohn, A.; Pandera, R.; Langer, R.; Sawicli, A.J. A polymer library approach to suicide gene therapy for cancer. Proc. Natl. Acad. Sci. USA
**2004**, 9, 16028–16033. [Google Scholar] [CrossRef] [PubMed] - Beata, S.; Mircea, V.D.; Mihai, V.P.; Ireneusz, P.G. Docking linear ligands to glucose oxidase. Int. J. Mol. Sci.
**2016**, 17, 1796. [Google Scholar] - Nguyen, L.H.; Nguyen, T.H.; Truong, T.N. Quantum Mechanical-Based Quantitative Structure-Property Relationships for Electronic Properties of Two Large Classes of Organic Semiconductor Materials: Polycyclic Aromatic Hydrocarbons and Thienoacenes. ACS Omega
**2019**, 4, 7516–7523. [Google Scholar] [CrossRef] [PubMed] - Mathematica; Version 8.0; Wolfram Research, Inc.: Champaign, IL, USA, 2010.
- Schrödinger Release Prime; Schrödinger LLC: New York, NY, USA, 2009.
- Akhiezer, D.N.; Vinberg, E.B. Weakly symmetric spaces and spherical varieties. Transf. Groups
**1999**, 4, 3–24. [Google Scholar] [CrossRef] - Wolf Joseph, A. Spaces of Constant Curvature, 5th ed.; McGraw–Hill: New York, NY, USA, 1999. [Google Scholar]
- Conway John, B. Functions of One Complex Variable, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 1978. [Google Scholar]
- Arfken, G. Mathematical Methods for Physicists, 3rd ed.; Academic Press: Orlando, FL, USA, 1985; pp. 397–399. [Google Scholar]
- Morse, P.M.; Feshbach, H. Methods of Theoretical Physics, Part I; McGraw-Hill: New York, NY, USA, 1953; pp. 391–392, 399–401. [Google Scholar]
- Trott, M. The Mathematica Guidebook for Programming; Springer: Berlin/Heidelberg, Germany, 2004; pp. 188–191. [Google Scholar]
- Weisstein, E.W. “Branch Point” from MathWorld—A WolframWeb Resource. Available online: http://mathworld.wolfram.com/BranchPoint.html (accessed on 5 December 2018).
- Majumdar, S.; Basak, S.C.; Lungu, C.N.; Diudea, M.V.; Grunwald, G.D. Mathematical structural descriptors and mutagenicity assessment: A study with congeneric and diverse datasets. SAR QSAR Environ. Res.
**2018**, 29, 579–590. [Google Scholar] [CrossRef] [PubMed] - Petitjean, M. A definition of symmetry. Symmetry Cult. Sci.
**2007**, 18, 99–119. [Google Scholar] - Lungu, C.N. C-C chemokine receptor type 3 inhibitors: Bioactivity prediction using local vertex invariants based on thermal conductivity layer matrix. Stud. Univ. Babes-Bolyai Chem.
**2018**, 63, 177–188. [Google Scholar] [CrossRef] - Lungu, C.N.; Diudea, M.V.; Putz, M.V. Ligand shaping in induced fit docking of MraY inhibitors. Polynomial discriminant and Laplacian operator as biological activity descriptors. Int. J. Mol. Sci.
**2018**, 18, 1377. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 2.**Riemann surfaces for compounds 8 and 21 respectively with their respective branching point values.

**Figure 3.**Branching points for the concerned monomers; points of structures with promising gene transfer profile, in red (see Supplementary Material S1).

**Figure 4.**(

**a**) Pharmacophore hypothesis based on molecule #8; (

**b**) pharmacophore hypothesis based on molecule #21; (

**c**) merge pharmacophore hypothesis (a + b): Lyophilic (H-atom group, in green); hydrogen donor group (D, in blue); hydrogen acceptor group (A, in red).

Item | Monomers with Proven Gene Transfer Capabilities |
---|---|

1 | |

2 | |

3 | |

4 | |

5 | |

6 | |

7 | |

8 | |

9 | |

10 | |

11 | |

11 | |

12 | |

13 | |

14 | |

15 | |

16 | |

17 | |

18 | |

19 | |

20 | |

21 | |

22 | |

23 | |

24 | |

25 | |

26 | |

27 | |

28 | |

29 |

Monomer Equations | |
---|---|

# | Coordinate Based Equations |

1 | ∂_{H}(−13.4 + 19H) |

2 | ∂_{H}(−2.2 + 18.5H) |

3 | ∂_{H}(−13.0 + 28H) |

4 | ∂_{H}(−13.6 + 14H) |

5 | ∂_{H}(−2.4 + 15H) |

6 | ∂_{H}(−2.6 + 15H) |

7 | ∂_{H}(−4.9 + 17H) |

8 | ∂_{H}(−15.5 + 20H) |

9 | ∂_{H}(−4.5 + 23H) |

10 | ∂_{H}(−4.3 + 18H) |

11 | ∂_{H}(−19.1 + 24H) |

12 | ∂_{H}(−3.4 + 19H) |

13 | ∂_{H}(−3.7 + 24H) |

14 | ∂_{H}(−2.4 + 21H) |

15 | ∂_{H}(−4.6 + 26H) |

16 | ∂_{H}(−4.0 + 2.8H) |

17 | ∂_{H}(6.8H) |

18 | ∂_{H}(−1.8 + 19H) |

19 | ∂_{H}(−1.9 + 795.2H) |

20 | ∂_{H}(−13.4 + 28H) |

21 | ∂_{H}(−28.7 + 28H) |

22 | ∂_{H}(−16.7 + 59H) |

23 | ∂_{H}(83.1H) |

24 | ∂_{H}(−10.8 + 31H) |

25 | ∂_{H}(25H) |

26 | ∂_{H}(31H) |

27 | ∂_{H}(−7.3 + 323.5H) |

28 | ∂_{H}(398.6H) |

29 | ∂_{H}(−5.8 + 49H) |

# | Equations for Computing Riemann Surfaces |
---|---|

1 | z (−13.4 + 19 z)^(1/(z (−13.4 + 19 z))) |

2 | z (−2.2 + 18.5z)^(1/(z (−2.2 + 18.5z))) |

3 | z (−13.0 + 28z)^(1/(z (−13.0 + 28z))) |

4 | z (−13.6 + 14 z)^(1/(z (−13.6 + 14z))) |

5 | z (−2.4 + 15 z)^(1/(z (−2.4 + 15 z))) |

6 | z (−2.6 + 15z)^(1/(z (−2.6 + 15z))) |

7 | z (−4.9 + 17z)^(1/(z (−4.9 + 17z))) |

8 | z (−15.5 + 20z)^(1/(z (−15.5 + 20z))) |

9 | z (−4.5 + 23z)^(1/(z (−4.5 + 23 z))) |

10 | z (−4.3 + 18z)^(1/(z (−4.3 + 18z))) |

11 | z (−19.1 + 24z)^(1/(z (−19.1 + 24z))) |

12 | z (−3.4 + 19 z)^(1/(z (−3.4 + 19z))) |

13 | z (−3.7 + 24 z)^(1/(z (−3.7 + 24z))) |

14 | z (−2.4 + 21z)^(1/(−2.4 + 21 z))) |

15 | z (−4.6 + 26z)^(1/(z (−4.6 + 26 z))) |

16 | z (−4.0 + 2.8 z)^(1/(z (−4.0 + 28 z))) |

17 | z (6.8 z)^(1/(z (6.8 z))) |

18 | z (−1.8 + 19z)^(1/(z (−1.8 + 19 z))) |

19 | z (−1.9 + 795.2 z)^(1/(z (−1.9 + 795.2 z))) |

20 | z (−13.4 + 28 z)^(1/(z (−13.4 + 28 z))) |

21 | z (−28.7 + 28 z)^(1/(z (−28.7 + 28 z))) |

22 | z (−16.7 + 59 z)^(1/(z (−16.7 + 59 z))) |

23 | z ( 83.1 z)^(1/(z (83.1 z))) |

24 | z (−10.8 + 31 z)^(1/(z (−10.8 + 31 z))) |

25 | z (25 z)^(1/(z ( 25 z))) |

26 | z (31 z)^(1/(z ( 31z))) |

27 | z (−7.3 + 323.5 z)^(1/(z (−7.3 + 323.5 z))) |

28 | z ( 398.6 z)^(1/(z (398.6 z))) |

29 | z (−5.8 + 49 z)^(1/(z (−5.8 + 49 z))) |

**Table 4.**Riemann surfaces: 1—function plots, 2—complex space plots, 3—real part 3D plot, 4—imaginary part 3D plot, 5—branching number, and 6—polymers (#).

Riemann Surfaces | ||
---|---|---|

1 | ||

2 | ||

3 | ||

4 | ||

5 | 0.7754 | 1.027 |

6 | # 8 | # 21 |

© 2019 by the authors. 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/).

## Share and Cite

**MDPI and ACS Style**

Lungu, C.N.; Grudzinski, I.P.
Riemann-Symmetric-Space-Based Models in Screening for Gene Transfer Polymers. *Symmetry* **2019**, *11*, 1466.
https://doi.org/10.3390/sym11121466

**AMA Style**

Lungu CN, Grudzinski IP.
Riemann-Symmetric-Space-Based Models in Screening for Gene Transfer Polymers. *Symmetry*. 2019; 11(12):1466.
https://doi.org/10.3390/sym11121466

**Chicago/Turabian Style**

Lungu, Claudiu N., and Ireneusz P. Grudzinski.
2019. "Riemann-Symmetric-Space-Based Models in Screening for Gene Transfer Polymers" *Symmetry* 11, no. 12: 1466.
https://doi.org/10.3390/sym11121466