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**Jiapu Zhang ^{1,2*}**

^{1}Molecular Model Discovery Laboratory, Department of Chemistry and Biotechnology, Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn Campus, Hawthorn, Victoria 3122, Australia

^{2}Graduate School of Sciences, Information Technology and Engineering and Centre of Informatics and Applied Optimisation, Faculty of Science and Technology, The Federation University Australia, Mount Helen Campus, Mount Helen, Ballarat, Victoria 3353, Australia

- *Corresponding Author:
- Jiapu Zhang

Molecular Model Discovery Laboratory

Department of Chemistry and Biotechnology

Faculty of Science Engineering and Technology

Swinburne University of Technology Hawthorn Campus

Hawthorn, Victoria 3122, Australia+61-3-92145596/+61-3-53276335

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**E-mail:**[email protected]/[email protected]

**Received date:** February 20, 2016; **Accepted date:** March 29, 2016; **Published date:** March 31, 2016

**Citation:** Zhang J (2016) Mathematical Formulas for Prion All Cross-Structures Listed in the Protein Data Bank. Med chem (Los Angeles) 6:179-188. doi:10.4172/2161-0444.1000343

**Copyright:** © 2016 Zhang J. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Prion protein (PrP) has two regions: unstructured region PrP(1-120) and structured region PrP(119-231). In the structured region, there are many segments which have the property of **amyloid fibril** formation. By theoretical calculations, PrP(126-133), PrP(137-143), PrP(170-175), PrP(177-182), PrP(211-216) have the amyloid fibril forming property. PrP(142-166) has a X-ray **crystallography **experimental β-hairpin structure, instead of a pure cross-β amyloid fibril structure; thus we cannot clearly find it by our theoretical calculations. However, we can predict that there must be a laboratory X-ray crystal structure in PrP(184-192) segment that will be produced in the near future. The experiments of X-ray crystallography laboratories are agreeing with our theoretical calculations. This article summarized mathematical formulas of prion amyloid fibril cross-β structures of all the above PrP segments currently listed in the **Protein Data Bank**.

PrP structured region; Amyloid fibril formation peptides; Theoretical calculations; Experimental laboratories; Mathematical formulas

**Table 1 **lists the cross-*β* structures of all PrP segments that were listed in the **Protein Data Bank** (PDB, www.rcsb.org), produced by X-ray crystallography experiments. In the below, 1- 24 give some mathematical formulas to describe all these cross-*β* structures.

PrP segment |
Species |
PDB ID |
Class of the cross-β |

1 PrP(126–131) | human | 4TUT | Class 7[1,4] |

2 | human | 4UBY | Class 8[1,4] |

3 | human | 4UBZ | Class 8[1,4] |

4 PrP(126–132) | human | 4W5L | Class 8[1,4] |

5 | human | 4W5M | Class 8[1,4] |

6 | human | 4W5P | Class 8[1,4] |

7 PrP(127–133) | human | 4W5Y | Class 6[1,4] |

8 | human | 4W67 | Class 6[1,4] |

9 | human | 4W71 | Class 6[1,4] |

10 PrP(127–132) | human | 4WBU | Class 8[1,4] |

11 | human | 4WBV | Class 8[1,4] |

12 | human | 3MD4 | antiparallel (P 21 21 21 ) |

13 | human | 3MD5 | parallel (P 1 21 1) |

14 | human-M129 | 3NHC | Class 8[1] |

15 | human-V129 | 3NHD | Class 8[1] |

16 PrP(137–142) | mouse | 3NVG | Class 1[1] |

17 PrP(137–143) | mouse | 3NVH | Class 1[1] |

18 PrP(138–143) | Syrian hamster | 3NVE | Class 6[1] |

19 | human | 3NVF | Class 1[1] |

20 PrP(142–166) | sheep | 1G04 | β-hairpin[3] |

21 PrP(170–175) | human | 2OL9 | Class 2[1] |

22 | elk | 3FVA | Class 1[1] |

23 PrP(177–182, 211–216) | human | 4E1I | β-prism I fold[2] (P 21 21 21 ) |

24 | human | 4E1H | β-prism I fold[2](P 21 21 21 ) |

**Table 1: **The cross-*β* structures known in the PDB Bank of PrP segments.

1. **Figure 1**, the mathematical formula for B Chain got from A Chain is

(1)

2. **Figure 2**, mathematical formulas for EFGH, IJKL, MNOP Chains obtained from ABCD Chains respectively are

(2)

(3)

(4)

3. **Figure 3**, mathematical formulas for CD, EF, GH, IJ, KL Chains obtained from AB Chains respectively are

(5)

(6)

(7)

4. **Figure 4**, mathematical formulas for CD, EF, GH, IJ, KL Chains obtained from the basic AB Chains respectively are

(8)

(9)

(10)

(11)

(12)

(13)

5. **Figure 5**, mathematical formulas for CD, EF, GH, IJ Chains obtained from the basic AB Chains respectively are

(14)

(15)

(16)

6. **Figure 6**, mathematical formulas for CD, EF, GH Chains obtained from the basic AB Chains respectively are

(17)

(18)

(19)

7. **Figure 7**, mathematical formulas for CD, EF, G H Chains obtained from the basic AB Chains respectively are

(20)

(21)

(22)

(23)

8. **Figure 8**, mathematical formulas for CD, EF, GH Chains obtained from the basic AB Chains respectively are

(24)

(25)

9. **Figure 9**, mathematical formulas for CD, EF, GH Chains obtained from the basic AB Chains respectively are

(26)

(27)

10. **Figure 10**, mathematical formulas for CD, EF, GH Chains obtained from the basic AB Chains respectively are

(28)

(29)

(30)

11. **Figure 11**, mathematical formulas for CD, EF, GH Chains obtained from the basic AB Chains respectively are

(31)

(32)

(33)

12. **Figure 12**, basing on AB Chains, other chains CD, EF, GH in the asym-metric one unit cell can be obtained by the mathematical formulas

(34)

(35)

(36)

13. **Figure 13**, basing on AB Chains, other Chains CD in the asymmetric unit cell can be obtained by mathematical formula

(37)

14. **Figure 14** we see that G(H) chains (i.e., *β*-sheet 2) of 3NHC.pdb can be obtained from A(B) chains (i.e., *β*-sheet 1) by

(38)

and other chains can be got by

(39)

(40)

15. By observations of the 3rd column of coordinates of 3NHD.pdb and **Figure 15**, G(H) chains (i.e., *β*-sheet 2) of 3NHD.pdb can be calculated from A(B) chains (i.e., *β*-sheet 1) by Equation (41) and other chains can be calculated by Equations (42)~(43):

(41)

(42)

(43)

16. In **Figure 16** we see that H Chain (i.e., *β*-sheet 2) of 3NVG.pdb can be obtained from A Chain (i.e., *β*-sheet 1) by

(44)

and other chains can be got by

(45)

(46)

17. In **Figure 17** we see that H chain (i.e., *β*-sheet 2) of 3NVH.pdb can be obtained from A chain (i.e., *β*-sheet 1) by

(47)

and other chains can be got by

(48)

(49)

18. In **Figure 18** we see that G(H) chains (i.e., *β*-sheet 2) of 3NVE.pdb can be obtained from A(B) chains (i.e., *β*-sheet 1) by

(50)

and other chains can be got by

(51)

(52)

19. In **Figure 19** we see that H chain (i.e., *β*-sheet 2) of 3NVF.pdb can be obtained from A chain (i.e., *β*-sheet 1) by

(53)

and other chains can be got by

(54)

(55)

20. **Figure 20**.

21. In **Figure 21**, we see that the D chain (i.e., *β*-sheet 2) of 2OL9.pdb can be obtained from A Chain (i.e., *β*-sheet 1) using the mathematical formula

(56)

and other chains can be got by

(57)

(58)

22. In **Figure 22**, we see that the D chain (i.e., *β*-sheet 2) of 3FVA. pdb can be obtained from A Chain (i.e., *β*-sheet 1) using the mathematical formula

(59)

and other chains can be got by

(60)

(61)

23. **Figure 23**.

24. **Figure 24**.

All the above X-ray crystallography structures show to us, in the PrP(119-231) structured region, there are many segments of **peptides ** which can formulate into **amyloid fibrils**. The author’s theoretical calculations [5,6] (**Figures 25** and **26**, where -26 kcal/mol is the threshold energy of amyloid fibril formation: if energy is less than -26 kcal/mol it will have amyloid fibril forming property [7]) also show this point.

Last, seeing **Figure 26**, we may think there must be a laboratory X-ray crystal *β*-type structure in PrP(184-192) segment that will be produced by a X-ray lab in the near future.

In conclusion, this short article described about the comparison study of theoretical calculations and X-ray **crystallography **study of prion protein cross-beta structures. The author has very good prediction of theoretical calculations using mathematical formulas of unstructured region and structured region of prion proteins in detail well demonstrated. Moreover, the X-ray crystallography structures showed, in the PrP(119–231) structured region, there are many segments of peptides which can formulate into amyloid fibrils. The author’s theoretical calculations for **Figures 25** and **26** were shown threshold energy as -26 kcal/mol, which supports to the amyloid fibril formation. Even if the threshold energy is less than -26 kcal/mol also they will have amyloid fibril forming property. This is a good piece of work useful for the medicinal chemists. The author presented the results very well in the article, and provided an introduction of the structures to understand the significance in the biological use of amyloid fibrils [8-14].

This research was supported by a Victorian Life Sciences Computation Initiative (VLSCI) grant numbered FED0001 on its Peak Computing Facility at the University of Melbourne, an initiative of the Victorian Government (Australia). The author also acknowledges the anonymous referees for their numerous insightful comments and references [8-14] offered.

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**Posted on**Sep 29 2016 at 8:00 pm

This is a good piece of work useful for the medicinal chemists. Authors have very good prediction of theoretical calculations using mathematical formulas of unstructured region and structured region of prion proteins in detail well demonstrated.

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