# Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Algebraic Geometrical Models of Secondary Structures

#### 2.1. The Gieseking Manifold $m000$

#### 2.2. The Hypercartographic Group ${\mathcal{H}}_{2}^{+}$

#### 2.3. Fundamental Groups of 3-Manifolds

## 3. Secondary Structure with $\alpha $ Helices: Drosophila Melanogaster Histone H3 (PDB 6PWE_1)

#### 3.1. The Primary (Linear) Structure

- IVFSNVK–T-TLVKPKSE
**MARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRP****GTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVM****ALQEASEAYLVGLFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA**-ADTALTCR-SASVLYNRSFS

#### 3.2. The Secondary Structure

- CCCCCCCCCCCCCCCCCHHHHCHHHHCCCCCCCCCCCCCCCCCCCCHHHHHHHCCCCC
- CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCHHHHHHHHHHHHCC
- CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCHHHHHHHHHHHHCCC
- CCCCCCCCCCCCCCCCCCCCHHHHHCCCCCCCCCCCCCCCCCCHHHHHHHHHHHHHCC

- HHHHHCCCCHHHHHHHHHHHCCCCCCCCHHHHHHHHHHHHHHHHHHHHHHHHCHHHH
- CCHHHCCCHHHHHHHHHHHHCCCCCCCCHHHHHHHHHHHHHHHHHHHHHHHHHHHHC
- HHHHHHHHHHHHHHHHHHHHHCCCCCCCHHHHHHHHHHHHHHHHHHHHHHHHHHHHC
- HHHHHHHHHHHHHHHHHHCCCCCCCCCCHHHHHHHHHHHHHHHHHHHHHHHHHHHHC

- CCCCCCHHHHHHHHHHCCCCC
- CCCCCCHHHHHHHHHHCCCCC
- CCCCCCHHHHHHHHHHCCCCC
- CCCCCCHHHHHHHHHHHCCCC

## 4. Secondary Structures with $\mathbf{\alpha}$ Helices and $\mathbf{\beta}$ Sheets: Myelin P2, Carbonic Anhydrase and the Lsm 1-7 Complex

#### 4.1. Myelin P2 for Homo Sapiens (PDB 2WUT)

- GMSNKFLGTWKLVSSENFDDYMKALGVGLATRKLGNLAKPTVIISKKGDIITIRTESTFKN
- CCCHHCCEEEEEEEECCHHHHHHHCCCCHHHHHHHHHCCCEEEEEEECCEEEEEEECCCC
- CCCHHCCEEEEEECCCCHHHHHHHCCCCHHHHHHHHHCCCEEEEEEECCEEEEEEECCCC
- CCCCCCEEEEEEEEECCHHHHHHHHCCCHHHHHHHHCCCCEEEEEEECCEEEEEEECCCC
- CCCCCCEEEEEEEEECCHHHHHHHCCCCHHHHHHHHCCCCEEEEEEECCEEEEEEECCCC

- TEISFKLGQEFEETTADNRKTKSIVTLQRGSLNQVQRWDGKETTIKRKLVNGKMVAECKM
- CCCHHCCEEEEEEEECCHHHHHHHCCCCHHHHHHHHHCCCEEEEEEECCEEEEEEECCCC
- EEEEEEEECCEEEEECCCCCEEEEEEEEECCEEEEEEECCCCEEEEEEEEECCEEEEEEEE
- EEEEEEECCCEEEEECCCCCEEEEEEEEECCEEEEEEECCCCCEEEEEEEECCEEEEEEEE
- EEEEEEECCCEEEEECCCCCEEEEEEEEECCEEEEEEECCCCCEEEEEEEECCEEEEEEEE

- KGVVCTRIYEKV
- CCEEEEEEEEEC
- CCEEEEEEEEEC
- CCEEEEEEEEEC
- CCEEEEEEEEEC

#### 4.2. The 3-Fold Symmetric Complex for Gamma-Carbonic Anhydrase (PDB 1QRE)

#### 4.3. The Hfq Protein Complex of Escherichia coli (PDB 1HK9)

- GAMAKGQSLQDPFLNALRRERVPVSIYLVNGIKLQGQIESFDQFVILLKNTVSQMVYKHAISTVVPSRPVSHHSCCCCCCCCCHHHHHHHHHHCCCCEEEEEECCCEEEEEEEECCCEEEEEECCCEEEEEEEEEEEEEECCCCCCCCCCCCCCCCHHHHHHHHHHHCCCCEEEEEECCEEEEEEEEEECEEEEEEECCCEEEEEEEEEEEEECCCCCCCCCCCCCCCCCCHHHHHHHHHHCCCEEEEEEECCEEEEEEEEEECCEEEEEECCCCEEEEEEEEEEEEECCEEEECCCCCCCCCCCCHHHHHHHHHCCCCEEEEECCCCEEEEEEEEECCCEEEEEECCCEEEEEEEEEEEEECCCCCCCC

#### 4.4. Other n-Fold Symmetric Complexes

#### 4.4.1. The 5-Fold Symmetric H2A-H2B Complex in Nucleoplasmin (PDB 2XQL)

#### 4.4.2. The 5-Fold Symmetric Acetylcholine Receptor (PDB 2BG9)

#### 4.4.3. The 7-Fold Symmetric Lsm 1-7 Complex in the Spliceosome (PDB 4M75)

**M**.

#### 4.4.4. Encoding a Protein with the Characters of the Finite Group ${G}_{7}$

## 5. The 8-Fold Symmetric Histone Complex of the Nucleosome: 3WKJ in the Protein Data Bank

## 6. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**A picture of the secondary structure of myelin P2 in homo sapiens (PDB 2WUT) as predicted from PHYRE2.

**Figure 3.**(

**a**) A picture of the structure of carbonic anhydrase (PDB 1QRE), (

**b**) A picture of the structure of Hfq protein complex of Escherichia coli (PDB 1HK9).

**Figure 4.**(

**a**) the nucleoplasmin H2A-H2B: 2XQL in the protein databank, (

**b**) the acetylcholine receptor: 2BG9 in the protein databank, (

**c**) the Lsm 1-7 complex in the spliceosome: 4M75 in the protein databank.

**Figure 5.**(

**a**) The structure of a nucleosome consists of a DNA double helix wound around eight histone proteins. There are eight periods (as shown in the picture) so that the two helices meet at 16 points. They map to the 16 double points of the Kummer surface. (

**b**) A section at constant ${x}_{4}$ of the Kummer surface for the group ${G}_{8}$.

**Table 1.**The d-coverings ($d=1\dots 10$) of the Gieseking manifold $m000$. The corresponding 3-manifolds (3-man) are identified thanks to SnapPy. The finite group P organizing the cosets of the index d fundamental group is given. It is shared by almost all subgroups (see lacking P) of the free group associated to the PORTER model of secondary structures of histone H3 (PDB; 6PWE_1). Some extra groups appear in the PORTER model (see extra P).

Index | 1 | 2 | 3 | 4 | 5 |

3-man | m000 | K4a1, ooct02_00001 | ntet03_00000 | m206, otet04_00002 | m407, ntet05_00007 |

m204, ntet04_00000 | m405, ncube01_00001 | ||||

P | (1,1) | (2,1) | (3,1) | (4,1) | (5,1) |

(12,3) | (20,3) | ||||

Index | 6 | 7 | 8 | 9 | 10 |

3-man | s961, otet06_00003 | y886, ntet07_00000 | t12839, otet06_00007 | ||

x252, ntet06_00004 | t12840, otet08_00002 | ||||

ntet06_00005 | ntet08_00002 | ||||

P | (6,2) | (7,1) | (8,1) | (9,1) | (10,2) |

(12,3) | (24,3) $\times 2$ | ||||

(24,13) | (24,13) | ||||

(96,70), (192,201) | (9,1), (648,705) | (10,2), (20,3), ${G}_{14400}$ | |||

lacking P | (72,39) | (320,1635) | |||

extra P | ${A}_{8}$, ${S}_{8}$ | (216,53), ${A}_{9}$, ${S}_{9}$ | ${S}_{10}$, ${G}_{7200}$ |

**Table 2.**The models of the secondary structure for protein H3 of drosophila melanogaster and the cardinality list of d-coverings (alias conjugacy classes of subgroups) of the associated fundamental group. ${T}_{1}$ is the trefoil knot, ${K}_{0}$ is the figure-of-eight knot, the 0-surgery on ${K}_{0}$ is the Akbulut manifold ${\mathsf{\Sigma}}_{Y}$, ${\tilde{E}}_{8}$ is the singular fiber of type II* and $m000$ is the Gieseking manifold. One restricts to two-generator groups since histone H3 only consists of sections with $\alpha $ helices and coils. Observe that the series of cardinalities for the secondary structure of H3 fits the series of the Gieseking manifolds up to the first 7 indices. Bold characters are for partial sequences matching the cardinality sequence for subgroups of the fundamental group of Gieseking manifold $m000$.

Protein | Model | ${\mathit{\eta}}_{\mathit{d}}\left(\mathit{T}\right)$ |
---|---|---|

H3 (6PWE_1) | PSIPRED | [1,1,1,1,2, 2,1,3,5,5 .,.,.,.,.] |

H3 | PHYRE2 | [1,1,1,1,3, 4,1,5,10,10 .,.,.,.,.] |

H3 | PORTER | [1,1,1,2,2, 3,1,12,6,5 .,.,.,.,.] |

H3 | RAPTORX | [1,1,1,1,2, 1,1,2,3,3 .,.,.,.,.] |

m000 | Gieseking | [1,1,1,2,2, 3,1,4,3,5, 4,14,1,5,10] |

${T}_{1}$ | trefoil | [1,1,2,3,2, 8,7,10,18,28, 27,88,134,171,354] |

${K}_{0}$ | figure-of-eight | [1,1,1,2,4, 11,9,10,11,38, 26,62,39,89,228] |

${K}_{0}$(0,1) | ${\mathsf{\Sigma}}_{Y}$ | [1,1,1,2,2, 5,1,2,2,4, 3,17,1,1,2] |

${\tilde{E}}_{8}$ | singular fiber II* | [1,1,2,2,1, 5,3,2,4,1, 1,12,3,3,4] |

**Table 3.**A few proteins, the software used for determining their secondary structure and the cardinality list of d-coverings (alias conjugacy classes of subgroups of index d) of the associated group. One takes proteins that contain sections with $\alpha $ helices, $\beta $ sheets and coils. The groups obtained by mapping the appropriate characters of ${G}_{7}=(336,118)$ and ${G}_{8}=(384,5589)$ to amino acids are also considered. Bold characters are for partial sequences matching the sequence of the hypercartographic group ${\mathcal{H}}_{2}^{+}$.

Protein | aa | Model | ${\mathit{\eta}}_{\mathit{d}}\left(\mathit{T}\right)$ |
---|---|---|---|

myelin P2 (2WUT) | 133 | PSIPRED | [1, 3, 13, 84, 336, 4216] |

2WUT | PHYRE2 | [1, 3, 7, 26, 164, 10,669] | |

2WUT | PORTER | [1, 3, 7, 26, 135, 871] | |

2WUT | RAPTORX | [1, 3, 10, 59, 348, 2899] | |

. | (336,118) | [1, 3, 7, 30, 122, 991] | |

. | (384,5589) | [1, 3, 7, 34, 130, 999] | |

carbonic anhydrase (1QRE_1) | 247 | PSIPRED | [1, 3, 10, 43, 135, 1071] |

1QRE_1 | PHYRE2 | [1, 3, 7, 26, 149, 1085] | |

1QRE_1 | PORTER | [1, 3, 7, 26, 415, 4382] | |

1QRE_1 | RAPTORX | [1, 3, 10, 35, 106, 804] | |

. | (336,118) | [1, 3, 7, 30, 150, 883] | |

. | (384,5589) | [1, 3, 10, 47, 148, 1015] | |

protein Hfq (1HK9_1) | 74 | PSIPRED | [1, 7, 17, 114, 1145, 14,275] |

1HK9_1 | PHYRE2 | [1, 7, 14, 149, 1458, 21,756] | |

1HK9_1 | PORTER | [1, 3, 7, 26, 97, 624, 4163, 34,470] | |

1HK9_1 | RAPTORX | [1, 3, 10, 51, 162, 1434] | |

. | (336,118) | [1, 3, 7, 26, 134, 912] | |

. | (384,5589) | [1, 3, 7, 34, 146, 894] | |

H2A-H2B (2XQL_1) | 91 | PHYRE2 | [1, 3, 7, 26, 103, 688] |

2XQL_1 | RAPTORX | [1, 3, 7, 26, 165, 2272] | |

. | (336,118) | [1, 3, 7, 26, 130, 943] | |

. | (384,5589) | [1, 3, 7, 26, 136, 967] | |

acetylcholin receptor (2BG9_1) | 370 | PSIPRED | [1, 3, 10, 35, 151, 1023] |

2BG9_1 | PHYRE2 | [1, 7, 11, 92,288, 2087] | |

2BG9_1 | PORTER | [1, 7, 11, 92, 239, 2058] | |

2BG9_1 | RAPTORX | [1, 3, 7, 34, 169, 1432] | |

. | $(336,118)$ | [1, 3, 10, 47, 124, 1026] | |

. | $(384,5589)$ | [1, 3, 7, 30, 140, 931] | |

Lsm 1-7 complex (4M75_1) | 144 | PSIPRED | [1, 3, 16, 81, 184, 1800] |

4M75_1 | PHYRE2 | [1, 7, 14, 201, 705, 8850] | |

4M75_1 | PORTER | [1, 3, 7, 26, 139, 1118] | |

4M75_1 | RAPTORX | [1, 3, 7, 26, 125, 747] | |

. | $(336,118)$ | [1, 3, 7, 34, 145, 948] | |

. | $(384,5589)$ | [1, 3, 10, 35, 135, 975] | |

${\mathcal{H}}_{2}^{+}$ | na | oriented hypermaps | [1, 3, 7, 26, 97, 624, 4163, 34,470] |

ooct02_00017 | 3-manifold | [1, 3, 7, 26, 40, 231] | |

ooct02_00006 | 3-manifold | [1, 3, 10, 43, 112, 802] | |

noct02_00024 | 3-manifold | [1, 3, 10, 43, 117, 804] | |

ooct02_00009 | 3-manifold | [1,3,7,30,105, 649] | |

ooct04_00001 | 3-manifold | [1, 3, 7, 34, 43, 240, 254] | |

L7a1 | 3-manifold link | [1, 3, 7, 34, 75, 377, 807] | |

ooct03_00019 | 3-manifold | [1, 7, 11, 85, 95, 240, 492] |

**Table 4.**For the group ${G}_{7}:=(336,118)\cong {\mathbb{Z}}_{7}\u22ca2O$, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 29-dimensional Pauli group, the order of a group element in the class, the angles involved in the character and a good assignment to an amino acid according to its polar requirement value. All characters are informationally complete except for the trivial character and the one assigned to

**M**. The entries involved in the characters are ${z}_{1}=2cos(2\pi /7)$, ${z}_{2}=2{z}_{1}$, ${z}_{3}=-6cos(\pi /7)$, ${z}_{4}=\sqrt{2}$, and ${z}_{5}=2cos(2\pi /21)$ featuring the angles $2\pi /8$ (in ${z}_{4}$), $2\pi /7$ and $2\pi /21$.

(336,118) | dimension | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathbb{Z}}_{7}\u22ca({\mathbb{Z}}_{2}.{S}_{4})$ | d-dit, d = 29 | 29 | 785 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ |

$\cong {\mathbb{Z}}_{7}\u22ca2O$ | amino acid | . | M | W | C | F | Y | . | . | H | Q |

order | 1 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | |

char | Cte | Cte | Cte | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}$ | ${z}_{4}$ | ${z}_{4}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | |

polar req. | . | 5.3 | 5.2 | 4.8 | 5.0 | 5.4 | . | . | 8.4 | 8.6 | |

(336,118) | dimension | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 |

d-dit, d = 29 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

amino acid | N | K | E | D | I | Stop | . | . | . | . | |

order | 14 | 14 | 14 | 21 | 21 | 21 | 21 | 21 | 21 | 21 | |

char | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | ${z}_{1,5}$ | Cte | Cte | Cte | ${z}_{1,2}$ | ${z}_{1,2}$ | ${z}_{1,2}$ | |

polar req. | 10.0 | 10.1 | 12.5 | 13.0 | 10 | 15 | . | . | . | . | |

(336,118) | dimension | 4 | 4 | 4 | f 4 | 4 | 4 | 6 | 6 | 6 | |

d-dit, d = 29 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||

amino acid | V | P | T | A | G | . | L | S | R | ||

order | 28 | 28 | 28 | 42 | 42 | 42 | 42 | 42 | 42 | ||

char | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{2,5}$ | ${z}_{1,3}$ | ${z}_{1,3}$ | ${z}_{1,3}$ | ||

polar req. | 5.6 | 6.6 | 6.6 | 7.0 | 7.9 | . | 4.9 | 7.5 | 9.1 |

**Table 5.**For the group ${G}_{8}=(384,5589)\cong {\mathbb{Z}}_{8}\u22ca2O$, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 37-dimensional Pauli group and the entries involved in the characters. The notation is ${z}_{1}=-\sqrt{2}$, ${z}_{2}=2\sqrt{2}$, ${z}_{3}=3\sqrt{2}$, ${z}_{4}=-\sqrt{3}$ and ${z}_{5}=-2cos(5\pi /12)$. All characters having ${z}_{4}$ and ${z}_{5}$ in their entries are informationally complete and are at the origin of the Kummer surface. All characters having entries with ${z}_{2}$ or ${z}_{4}$ are also informationally complete. A good matching to the amino acids (ordered according to their polar requirement and simultaneously to the order of a group element) is given.

(384,5589) | dimension | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathbb{Z}}_{8}\u22ca(48,28)$ | d-dit, d = 37 | 37 | 1333 | 1333 | 1333 | 1361 | ${d}^{2}$ | ${d}^{2}$ | 1367 | ${d}^{2}$ | ${d}^{2}$. |

$\cong {\mathbb{Z}}_{8}\u22ca2O$ | amino acid | . | . | M | W | . | . | . | . | . | . |

char | Cte | Cte | Cte | Cte | Cte | Cte | Cte | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}$ | |

(384,5589) | dimension | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 |

d-dit, d = 37 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 1367 | |

amino acid | C | F | Y | H | Q | N | K | E | D | . | |

char | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}$ | ${z}_{4}$ | ${z}_{4}$ | ${z}_{1,4,5}$ | ${z}_{1,4,5}$ | ${z}_{1,4,5}$ | ${z}_{1,4,5}$ | Cte | |

(384,5589) | dimension | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

d-dit, d = 37 | ${d}^{2}$ | 1367 | 1367 | ${d}^{2}$ | 1367 | 1367 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

amino acid I | Stop | . | . | . | . | . | . | . | V | ||

char | Cte | Cte | Cte | Cte | Cte | Cte | ${z}_{1,2}$ | ${z}_{1,2}$ | ${z}_{4}$ | ${z}_{4}$ | |

(384,5589) | dimension | 4 | 4 | 4 | 4 | 6 | 6 | 6 | |||

d-dit, d = 37 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 701 | 1365 | 1365 | ||||

amino acid | P | T | A | G | L | S | R | ||||

char | ${z}_{2,4,5}$ | ${z}_{2,4,5}$ | ${z}_{2,4,5}$ | ${z}_{2,4,5}$ | Cte | ${z}_{1,3}$ | ${z}_{1,3}$ |

**Table 6.**The structure of the addition table for the 16 singular Jacobian points of the hyper-elliptic curves ${\mathcal{C}}_{8}$.

A | B | C | D |

B | A | D | C |

C | D | A | B |

D | C | B | A |

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**MDPI and ACS Style**

Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K.
Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface. *Symmetry* **2021**, *13*, 1146.
https://doi.org/10.3390/sym13071146

**AMA Style**

Planat M, Aschheim R, Amaral MM, Fang F, Irwin K.
Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface. *Symmetry*. 2021; 13(7):1146.
https://doi.org/10.3390/sym13071146

**Chicago/Turabian Style**

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, Fang Fang, and Klee Irwin.
2021. "Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface" *Symmetry* 13, no. 7: 1146.
https://doi.org/10.3390/sym13071146