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^{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

**Tel:**+61392145596/+61353276335

**E-mail:**[email protected] (or) [email protected]

**Received date:** May 14, 2016; **Accepted date:** May 25, 2016; **Published date:** May 30, 2016

**Citation: **Zhang J (2016) Mathematical Formulas for Some Cross-Β Structures of Human Aβ Protein. Med chem (Los Angeles) 6:349-355. doi:10.4172/2161-0444.1000369

**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.

**Visit for more related articles at** Medicinal Chemistry

For amyloid fibril cross-β structures of Aβ human protein, we find theoretical calculations are agreeing with laboratory X-ray crystallography experiments. This mini article summarized mathematical formulas of amyloid fibril cross-β structures of segments of human Aβ protein. These formulas are accurate and correct some data in the **Protein Data Bank** (PDB). However, more mathematical formulas for core Chains AB (PDB IDs: 3OW9, 3PZZ, 2OKZ), or ABGH (PDB IDs: 3OVJ, 2Y3J, 2Y3K, 2Y3L) (or ABCD (PDB ID: 2ONA)) are still needed to present and optimize.

Human Aβ protein; Cross-β structures; Theoretical calculations; Experimental laboratories; Mathematical formulas

A general presentation of Alzheimer's disease, its increase in prevalence in modern post-industrial societies as the average life expectancy increases steadily and the burden it represents for healthcare systems worldwide, the progress achieved in recent decades in understanding the molecular pathology of the disease could be referred in the review articles [1,2], and the paradox that in spite of an abundant literature in this field including tens of thousands of published articles there is to date no single radical treatment effective in stopping progression of disease or reverting neuronal damage it produces. The disease results in form either overproduction of amyloid-β 42 in hereditary cases or an impeded clearance of this peptide in sporadic cases, mainly due to a less performant apolipoprotein E (ApoE4) isoform. Misfolding of amyloid-β 42 monomers in antiparallel β sheet (that can further oligomerize forming less selective transmembrane pores) leads to intracellular calcium inflow triggering excessive reactive oxygen species at mitochondrial level, activation of caspases, tau protein hyperphosphorylation and aggregation into neurofibrillary tangles with microtubule disorganization. Therefore understanding the mechanisms of amyloid-β 42 misfolding is of vital importance and this was the aim of the present study.

The material of this article is taken from the webpage of Ref. [3]. The software tools used to infer the coordinate transformation formulae and to compute the average free energy per residue in β-sheet conformation are JMol (www.jmol.org), VMD (www.ks.uiuc.edu/Research/vmd/). The procedures used to perform these computations and to derive the formulae are the observations of the PDB files of Ref. [3] and FORTRAN codes written by the author. The optimization or refinement of the formulas is obtained by the Amber package (ambermd.org).

In Ref. [3], the author showed that theoretical calculation can predict the segments of PrP protein which can form amyloid fibrils after misfolding, leading thus to propagation of misfolded PrP proteins to adjacent normal PrP proteins, and ultimately to a number of vulnerable areas of the brain. Quite interestingly, such a mechanism (“misfolding / propagation” or “transconformation”) appears to occur not only in Prion diseases, but also in other neurodegenerative diseases [3] (for example Alzheimer’s disease discussed in the below). This brings a new link between structural biology and human disorders, and may be quite useful in translational medicine [3]. This paper will mathematically study the cross-β structures of human Aβ protein for Alzheimer’s disease.

In the use of the above sequence, we can predict the segments of
Aβ protein which can form amyloid fibrils after misfolding (**Figure 1**).
From Figure 1, we may see that the regions containing residues 15–23
and 29–42 have a strong propensity to form amyloid fibrils [4]. This
observation agrees with the results of laboratory X-ray crystallography
experiments [5,6]. For readers’ conveniences to know thirteen Aβ
amyloid fibril structures (**Table 1**), in the below we illuminate 13
colourful and beautiful pictures and give their accurate mathematical
formulas to describe these structures.

**Figure 1:** Fibril-forming prediction of Aβ(1–42). The boundary between the fibril-forming and non-forming sections was set at the energy threshold of -27 KCal/mol (where using the energy threshold –27 KCal/molis corresponding to the maximum accuracy - the minimum P-value). The black horizontal bars indicate regions which have been found or suggested to form fibrils in experiments. The computations of total energy and its components for each structure generated in this study could be using a simple algorithm (e.g., NAMD Energy in VMD).

Aβ segment | PDB ID | Class of the cross-β |
---|---|---|

Aβ(16-21) KLVFFA(form 1)01(form 2)02(form 3)03(orange G)04 |
2Y2A 3OW9 2Y29 3OVJ |
Class 7[6,7] Class 7[6,7] Class 7[6,7] Class 8[8,7](P 21 21 21) |

Aβ(27-32) NKGAII (interface A)05(interface B)06 |
3Q2X 3Q2X |
Class 1[6,7] Class 1[6,7] |

Aβ(29-34) GAIIGL07 |
3PZZ | Class 6[6,7] |

Aβ(30-35) AIIGLM08 |
2Y3J | Class 2[6,7] |

Aβ(35-40) MVGGVV(form 1)09(form 2)10 |
2ONA 2OKZ |
Class 8[7] Class 8[7] |

Aβ(35-42) MVGGVVIA (form 1)11(form 2)12 |
2Y3K 2Y3L |
Class 2[6,7] Class 7[6,7] |

Aβ(37-42) GGVVIA 13 |
2ONV | Class 4[7] |

**Table 1:** Some cross-β structures of human Aβ segments.

In the below, to show the beautiful of symmetry of mathematics, we will always use the Pair of Sheets of the amyloid fibril structures.

2Y2A.pdb (Pair of Sheets)’s mathematical formulas are (**Figure 2**)

3OW9.pdb (Pair of Sheets)’s mathematical formulas are (**Figure 3**)

But the mathematical formula between Chains A and B should be found out.

2Y29.pdb (Pair of Sheets)’s mathematical formulas are (**Figure 4**)

3OVJ.pdb (Pair of Sheets)’s mathematical formulas are: applying
the following mathematical formulas to Chains A, B, G, H to get Chains
C, D, I, J and Chains E, F, K, L: (**Figure 5**)

But the mathematical formulas among Chains A, B, G, H should be found out.

2Y2A.pdb – interface A – face-to-face (Pair of Sheets)’s mathematical
formulas are (**Figure 6**)

2Y2A.pdb – interface B – back-to-back (Pair of Sheets)’s
mathematical formulas are (**Figure 7**)

3PZZ.pdb (Pair of Sheets)’s mathematical formulas are: applying
the following to Chains A, B: (**Figure 8**)

But the mathematical formula between Chain A and Chain B should be found out.

2Y3J.pdb (Pair of Sheets)’s mathematical formulas are: applying the
following mathematical formulas to Chains A, B, G, H to get Chains C,
D, I, J and Chains E, F, K, L: (**Figure 9**)

But the mathematical formulas among Chains A, B, G, H should be found out.

2ONA.pdb (Pair of Sheets)’s mathematical formulas are: applying
the following mathematical formulas to Chains A, B, C, D to get Chains
I, J, K, L and Chains E, F, G, H: (**Figure 10**)

But the mathematical formulas among Chains A, B, C, D should be found out.

2OKZ.pdb (Pair of Sheets)’s mathematical formulas are: applying
the following mathematical formula to Chain A (**Figure 11**)

applying the following mathematical formulas to Chain B

and applying the following mathematical formulas to Chains A, B, C, D to get Chains I, J, K, L and Chains E, F, G, H:

But the mathematical formula between Chain A and Chain B should be found out.

2Y3K.pdb (Pair of Sheets)’s mathematical formulas are: applying the following mathematical formulas to Chains A, B, G, H to get Chains
C, D, I, J and Chains E, F, K, L: (**Figure 12**)

But the mathematical formulas among Chains A, B, G, H should be found out.

2Y3L.pdb (Pair of Sheets)’s mathematical formulas are: applying the following mathematical formulas to Chains A, B, G, H to get Chains
C, D, I, J and Chains E, F, K, L: (**Figure 13**)

But the mathematical formulas among Chains A, B, G, H should be found out.

2ONV.pdb (Pair of Sheets)’s mathematical formulas are: applying
the following mathematical formulas to Chains A, B to get Chains CD,
EF, IJ, GH: (**Figure 14**)

and the mathematical formula between A, B Chains is:

This paper has presented very useful information for the development of human Aβ protein for Alzheimer's disease (some recent works about the development of human Aβ protein structures and inhibitors are introduced, for example, in Ref. [7-14]).

The paper explores the symmetry properties of several structures representing crystallized short segments of amyloid-β 42 retrieved from the PDB repository, defining for each of them formulae including translation and rotation of atom coordinates such as to generate an entire array of regular secondary structure of paired β-sheets starting from seed segments of the polypeptide. This article is mathematically interesting. More mathematical formulas for the core Chains AB (PDB IDs: 3OW9, 3PZZ, 2OKZ), or ABGH (PDB IDs: 3OVJ, 2Y3J, 2Y3K, 2Y3L) (or ABCD (PDB ID: 2ONA)) are still needed to present [5,6,15,16] and optimize. The formulas in this paper are accurate, correcting the misprinting in PDB bank. The influence of these formulas on the whole structure human Aβ protein could be further discussed based on those data correction.

The basic contribution of this paper is to show 13 colourful and
beautiful pictures (**Figures 2**-**14**) for Aβ (human) amyloid fibril
structures and presents conveniences to readers for a quick reference
Table (**Table 1**) and many accurate mathematical formulas.

Furthermore researches could be presenting some comments about the different structures included in this study and their relationship to the antiparallel β-sheet folding process of amyloid-β 42 monomers and subsequent propensity for oligomerization or fibrillation. We could indicate: whether the proposed structures may represent intermediate steps or local minima across the free energy landscape of β-amyloid folding; where would the folding process most likely start and how would it progress, which are the main energy barriers along the folding pathway; why amyloid-β 42 is so prone to form β-sheets and fibrils compared to amyloid-β 40, given it is only two residues longer at the C-terminal end; how does the β-sheet folding template propagate from one monomer to another; what is the relationship between X-ray crystallography-derived structures used in this study and NMR-derived structures of amyloid-β 42 in solution (e.g., 2BEG), and what is the likelihood that amyloid-β monomers/oligomers adopt these structures in realistic environments at physiological temperature; what is the propensity of these structures to form oligomeric transmembrane pores and thus be pathogenic in early stages of Alzheimer's disease (see, for example, Ref. [16]), etc.

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). VLSCI’s facilities and staff supports are gratefully acknowledged. Thanks also go to Landau M who let us know that the 3OVJ - steric zipper (perturbed by the small molecule orange G) belongs to Class 8. Last but not least, the author thanks the two anonymous referees for their numerous insightful comments, references offered, and furthermore research directions.

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