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

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Discussion

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^{0}= 1; f(e

^{2πi}) = e

^{2πia}

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**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 |

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**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