# 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

- Bartlett, S.D. Powered by magic. Nature
**2014**, 510, 345–347. [Google Scholar] [CrossRef] [PubMed] - Planat, M.; Gedik, Z. Magic informationally complete POVMs with permutations. R. Soc. Open Sci.
**2017**, 4, 170387. [Google Scholar] [CrossRef][Green Version] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Group geometrical axioms for magic states of quantum computing. Mathematics
**2019**, 7, 948. [Google Scholar] [CrossRef][Green Version] - Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Complete quantum information in the DNA genetic code. Symmetry
**2020**, 12, 1993. [Google Scholar] [CrossRef] - Planat, M.; Chester, D.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Finite groups for the Kummer surface: The genetic code and quantum gravity. Quantum Rep.
**2021**, 3, 68–79. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Informationally complete characters for quark and lepton mixings. Symmetry
**2020**, 12, 1000. [Google Scholar] [CrossRef] - The Protein Data Bank. Available online: https://pdb101.rcsb.org/ (accessed on 1 January 2021).
- Dang, Y.; Gao, J.; Wang, J.; Heffernan, R.; Hanson, J.; Paliwal, K.; Zhou, Y. Sixty-five years of the long march in protein secondary structure prediction: The final strech? Brief. Bioinform.
**2018**, 19, 482–494. [Google Scholar] - Pauling, L.; Corey, R.B.; Branson, H.R. The structure of proteins: Two hydrogen-bonded helical configurations of the polypeptide chain. Proc. Natl. Acad. Sci. USA
**1951**, 37, 205–211. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pauling, L.; Corey, R.B. Configurations of polypeptide chains with favored orientations around single bonds: Two new pleated sheets. Proc. Natl. Acad. Sci. USA
**1951**, 37, 729–740. [Google Scholar] [CrossRef] [PubMed][Green Version] - Adams, C.C. The noncompact hyperbolic 3-manifold of minimal volume. Proc. Am. Math. Soc.
**1987**, 4, 100. [Google Scholar] - Grothendieck, A. Sketch of a Programme, Written in 1984 and Reprinted with Translation in L. Schneps ans P. Lochak eds, Geometric Galois Actions 1. Around Grothendieck’s Esquisse d’un Programme, 2. The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups (Cambridge University Press, 1997); (b) The Grothendieck Theory of Dessins d’Enfants, Schneps, L., Lochak, P., Eds. (Cambridge Univ. Press, 1994). Available online: https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/EsquisseEng.pdf (accessed on 1 January 2021).
- Lando, S.K.; Zvonkin, A.K. Graphs on Surfaces and Their Applications; Springer: Berlin, Germany, 2004. [Google Scholar]
- Jones, G.; Singerman, D. Maps, hypermaps and triangle groups. In Geometric Galois Actions 1. Around Grothendieck’s Esquisse d’un Programme; Schneps, L., Lochak, P., Eds.; Cambridge University Press: Cambridge, UK, 1994; pp. 115–145. [Google Scholar]
- Planat, M.; Giorgetti, A.; Holweck, F.; Saniga, M. Quantum contextual finite geometries from dessins d’enfants. Int. J. Geom. Mod. Phys.
**2015**, 12, 1550067. [Google Scholar] [CrossRef][Green Version] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Universal quantum computing and three-manifolds, Universal quantum computing and three-manifolds. Symmetry
**2018**, 10, 773. [Google Scholar] [CrossRef][Green Version] - Thurston, W.P. Three-Dimensional Geometry and Topology; Princeton University Press: Princeton, NJ, USA, 1997; Volume 1. [Google Scholar]
- Adams, C.C. The newest inductee in the number hall of fame. Math. Mag.
**1998**, 71, 341–349. [Google Scholar] [CrossRef] - Milnor, J. Hyperbolic geometry: The first 150 years. Bull. Am. Math. Soc.
**1982**, 6, 9–24. [Google Scholar] [CrossRef][Green Version] - Culler, M.; Dunfield, N.M.; Goerner, M.; Weeks, J.R. SnapPy, a Computer Program for Studying the Geometry and Topology of 3-Manifolds. Available online: http://snappy.math.uic.edu/ (accessed on 1 January 2021).
- Fominikh, E.; Garoufalidis, S.; Goerner, M.; Tarkaev, V.; Vesnin, A. A census of tetrahedral hyperbolic manifolds. Exp. Math.
**2016**, 25, 466–481. [Google Scholar] [CrossRef][Green Version] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Quantum computing, Seifert surfaces and singular fibers. Quantum Rep.
**2019**, 1, 12–22. [Google Scholar] [CrossRef][Green Version] - Jones, D.T. Protein secondary structure prediction based on position-specific scoring matrices. J. Mol. Biol.
**1999**, 292, 195–202. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mirabello, C.; Pollastri, G. Porter, PaleAle 4.0: High-accuracy prediction of protein secondary structure and relative solvent accessibility. Bioinformatics
**2013**, 29, 2056–2058. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kelley, L.A.; Mezulis, S.; Yates, C.M.; Wass, M.N.; Sternberg, M.J.E. The Phyre2 web portal for protein modeling, prediction and analysis. Nat. Protoc.
**2015**, 10, 845–858. [Google Scholar] [CrossRef][Green Version] - Wang, S.; Sun, S.; Li, Z.; Zhang, R.; Xu, J. Accurate de novo prediction of protein contact map by ultra-deep learning model. PLoS Comput. Biol.
**2017**, 13, e1005324. [Google Scholar] [CrossRef][Green Version] - Genbank. Available online: https://www.ncbi.nlm.nih.gov/genbank/ (accessed on 1 January 2021).
- Nucleic Acid Sequence “Massager”. Available online: http://biomodel.uah.es/en/lab/cybertory/analysis/massager.htm (accessed on 1 January 2021).
- Translate. Available online: https://web.expasy.org/translate/ (accessed on 1 January 2021).
- Dutta, S.; Akey, I.V.; Dingwall, C.; Hartman, K.H.; Laue, T.; Nolte, R.T.; Head, J.F.; Akey, C.W. The crystal structure of nucleoplasmin-core: Implications for histone binding and nucleosome assembly. Mol. Cell
**2001**, 8, 841–853. [Google Scholar] [CrossRef] - Sauter, C.; Basquin, J.; Suck, D. Sm-Like proteins in eubacteria: The crystal structure of the Hfq protein from Escherichia coli. Nucleic Acids Res.
**2003**, 31, 4091. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lührmann, W.C.L. Spliceosome, structure and function. Cold Spring Harb. Perspect. Biol.
**2011**, 3, a003707. [Google Scholar] - Bosma, W.; Cannon, J.J.; Fieker, C.; Steel, A. (Eds.) Handbook of Magma Functions, 2.23th ed.; 2017; p. 5914. Available online: http://magma.maths.usyd.edu.au/magma/ (accessed on 10 April 2021).
- Tozzi, A.; Peters, J.F.; Fingelkurts, A.A.; Marijuàn, P.C. Brain Projective Reality: Novel Clothes for the Emperor. Reply to comments on “Topodynamics of metastable brains” by Tozzi et al. Phys. Life Rev.
**2017**, 21, 46–55. [Google Scholar] [CrossRef] - Irwin, K.; Amaral, M.; Chester, D. The Self-Simulation hypothesis interpretation of quantum mechanics. Entropy
**2020**, 22, 247. [Google Scholar] [CrossRef][Green Version] - Jones, G.A. Maps on surfaces and Galois groups. Math. Slovaca
**1997**, 47, 1–33. [Google Scholar] - Planat, M. Geometry of contextuality from Grothendieck’s coset space. Quantum Inf. Process.
**2015**, 14, 2563–2575. [Google Scholar] [CrossRef] - Koch, R.M.; Ramgoolam, S. From matrix models and quantum fields to Hurwitz space and the absolute Galois group. arXiv
**2010**, arXiv:1002.1634. [Google Scholar] - Aspinwall, P.S. K
_{3}surfaces and string duality. In Fields, Strings and Duality, TASI 1996; Efthimiou, C., Greene, B., Eds.; World Scientific: Singapore, 1997; pp. 421–540. [Google Scholar]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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