A Supersymmetry and Quantum Cryptosystem with Path Integral Approach in Biology

Abstract

The source of cancerous mutations and the relationship to telomeres is explained in an alternative way. We define the smallest subunit in the genetic code as a loop braid group element. The loop braid group is suitable to be defined as a configuration space in the process of converting the information written in the DNA into the structure of a folded protein. This smallest subunit, or a flying ring in our definition, is a representation of 8-spinor field in the supermanifold of the genetic code. The image of spectral analysis from the tensor correlation of mutation genes as our biological system is produced. We apply the loop braid group for biology and authentication in quantum cryptography to understand the cell cocycle and division mechanism of telomerase aging. A quantum biological cryptosystem is used to detect cancer signatures in 36 genotypes of the bone ALX1 cancer gene. The loop braid group with the RSA algorithm is applied for the calculation of public and private keys as cancer signatures in genes. The key role of this approach is the use of the Chern–Simons current and then the fiber bundle representation of the genetic code that allows a quantization procedure.

1. Introduction

For a long time, scientists have been trying to understand the source of self and nonself authentication keys [1] for cancer cells and the immune system’s recognition cells. It might be related to information encryption [2] inside 64 codons and a ${\left(CG\right)}_n$ repeated sequence in the non-protein-coding DNA as key pairs [3] of a biological cipher in the cancer quantum public-key cryptosystem—some information from the genus of an ancient organism. This information is a biological Chern–Simons current [4,5] induced from an adaptive behavior field in the genetic code. Some questions arise from a new theory of cell biology. What is the smallest subunit used for a representations of the structures of the folded proteins in a living organism? Is it possible to quantize such a structure by adopting standard procedures of theoretical physics? We try to show that it can be done by a new way of unifying triplet states of DNA, RNA, and protein. The new approach of the Chern–Simons current in the genotype with a mathematical model of the hidden state in a genome [6] is trying to answer some questions. The suppersymetry approach [7] was also used for the model of graphene wormhole and for the computation of the Chern–Simons current in a Josephson junction of superconductor states in the graphene. Additionally, the applications of neural networks for density estimation [8] outperformed parametric and non-parametric approaches.

The information in a genetic code can serve as the public key encryption used for authentication of the correct creation of the structure of a folded protein. The loop braid group [9] is suitable for defining the configuration space of this undivided small subunit in the genetic code as a so-called flying ring between the DNA and RNA. We redefine this transition as an anyon state in the process of protein folding over underlying Hopf fibration in Kolmogorov space [10] of biological time series data. As opposed to protein chemistry, in this paper, by "protein folding" we mean the entire process of translating the DNA sequence into the three-dimensional structure of a folded protein.

Recently, scientists applied the loop braid group to studying the anyon [11] in topological quantum computer architecture [12]. We borrow this theory to redefine a new model of the asymmetric key between public and private keys in protein–protein interactions as a new moduli state space model in quantum cryptography for biology. The anyon in biology can be associated with a phase transition of DNA–RNA–protein with a quaternionic state with two fermion pairs in the genetic code. Nowadays, a topological quantum cryptography is the most secure cryptography concept with the highest impact on computer security, business, and society, but it is rarely used in biology. Experimental quantum cryptography narrows the gap and it approaches the implementation of theoretical knowledge outside the laboratories, e.g., by the investigation of quantum key distribution with finite decoy states [13,14].

In the loop structure of telomeres [7], there exist layers of G-quadruplex (G4) [15]. Their structures are similar to the structure with the loop braid group in the standard model [16,17]. The G letter (guanine) can be suitably chosen as the hidden time scale in a biological clock. The nature of the telomerase enzyme is that it reacts like a retrotransposon hidden in transition state in the non-protein-coding DNA area. The loop braid group’s operation is the composition of braids representing the translation of an amino acid being the disjoint component operator that is convenient for representing G-quadruplex in 4-dimensional coordinates of spacetime curvature in a protein folding. We investigate why cancer and the adaptive behavior of the immune system are involved with a hidden feedback control mechanism in the trash DNA-like status.

The numbers of four letters in the DNA alphabet cannot be completely separated like transition levels of energy-momentum of gene expression. The natural process of transcription and reversed transcription in extended central dogma [18] we can describe by providing definitions of the relative natures of the genes involved with three objects: DNA, RNA, and protein. These three things can not be separated from one another as these three molecules are interconnected into a 4-dimensional coordinate system of living things. We can unify them as a loop braid group in the genetic code. Based on the analogies from the theory of general relativity (e.g., a definition of connection), we combine some more properties of the braid group and Khovanov cohomology [19,20] to them. We redefine the loop braid group by adding some extra properties to them for visualizing the folding structure in the amino acids [21]. We can apply the anyon [22] states of the pair between DNA and RNA as braid groups in anyon–geneonic pair states in the genetic code. The codon can be related to a table of public keys which can release noncoding RNA [23] as the private key for correctly translating a DNA sequence into a folded protein for the specific docking site in the antigen–antibody immune system.

It is the empirical fact that only 2% of human genetic material is expressed, but another 98% does not change into any proteins. This material is called the non-protein-coding DNA area, or in analogy with physics, the dark matter of the genetic code [24]. Recently, experiments confirmed that this DNA area is very important to the development of the brain in humans, and reproductive systems. It is possible that this area has an important part in controlling protein formation and acts like a big center for issuing instructions and verifying cryptography in the process of decoding genes. Additionally, the noncoding genome can generate or form three-dimensional structures in such a way as to generate protein molecular systems associated with disease (cancer). Our purpose is to build suitable new methodology for detection of such secondary and tertiary DNA structures in noncoding regions.

The paper is organized as follows. In Section 2 we define the concept of using a loop braid group for the genetic code. We discuss the source of translating the DNA sequence into a folded protein by using loop braid group operation. In Section 3 we define new asymmetric cryptography for biology, as a source of the public and private keys in the system of protein–protein interactions. We apply the moduli state space model to RSA algorithm to calculate public and private keys in asymmetric cryptography for biology for samples of 36 genes as a new signature. We use tensor correlation for gene expression to generate the image of a geneonic spectral sequence for a mutation gene signature in Section 4 and an empirical analysis with some concrete cases is developed. For brevity and easier understanding of the issue, we are using the term cancer gene for all genetic mutations and the term cancer signature defines all patterns for known types of damage and abnormalities. In the last section we conclude and discuss all results, giving future perspectives of this approach. Some technical aspects of our model are discussed in Appendix A and Appendix B. Amino acid sequences are given in Appendix C and Appendix D; the cipher text for the encryption of a gene is reported in Appendix E.

2. Algebraic Construction of G-Quadruplex in Telomere

In each cell division, the telomeres (see Figure 1a,b) are shortened [25] and the total length of DNA becomes shorter. As the result of the shorter biological clock from cell division, living things die. In order to understand the cell cocycle and the division mechanism of telomerase aging, we are looking on the source of cancer as the source of age acceleration and we relate it to the telomere shortening mechanism [26,27] (also see Figure 2 for detail of the biological clock), which is a source of braid group operation [28]—so-called self-diffeomorphism—in the genetic code. In the next part we are also trying to explain the real source and state of cancer as damage to the biological clock which can induce age acceleration [29]. The age acceleration is a relative measurement between the chronological clock and the biological clock in the telomere [30]. Up to now, scientists understood telomeres and telomerase as remnants of ancient viruses that rely on DNA in the chromosomes of living organisms. The telomere is composed of the repeated sequence of ${\left(TTAGGG\right)}_{d{t}^{\ast }}$ where $1000\le d{t}^{\ast }\le 2000$. The size of the duplicate sequence at the end of this open chromosome is amplified by six braids caused by six superspaces in time series data of organisms. It is naturally selected by the evolution of transgenic reproduction in all possible directions of the transcription process. The G alphabet might be suitable for the hidden time scale in the biological clock.

Figure 1. (a) T-loop in a telomere as a loop braid group in a genetic code. (b) Noncoding RNA in a telomerase. It is an example of reverse transcription from RNA to DNA in extended central dogma. (c) The upper picture demonstrates an active state of gene expression with an open switch in a telomere braid group element. The bottom picture shows an inactive state of gene expression with a closed switch in a telomere braid group element. (d) The picture represents a flying ring between two rings.

Figure 2. The left picture (a) shows a vierbien diagram of gene expression in a eukaryotic cell. The immature mRNA is composed of intron and exon states. The intron state is an analogy of the public key in gene expression and the exon state with private key. (b) The genetic code is produced from the rotation of biological clock as a signal generation represented by circular disk in the middle of right diagram.

Definition 1.

Let $d\in D$ be a superspace of DNA, $r\in R$ be a superspace of RNA, and $p\in P$ be a superspace of protein. We have a Wigner ray [31] of biological time series data of living organism $x_t\in X_t$ defined by $p:D\to R$, $(d,r)\mapsto p(d,r)$

$$W_{X_t}⟨d,r⟩:=⟨\lambda d,\lambda r⟩=\lambda ⟨d,r⟩=⟨d,r⟩.$$

(1)

The protein structure can be considered a transition state. It is a partition function with $d\in \left[d\right]$, an equivalent class of homotopic paths of gene expression in DNA, $\left[d\right]\in [D,S^2]$.

Definition 2.

Let $t=(t_1,t_2,t_3,t_4)$ be a biological clock time scale in four corners of G-quadruplex. We denote $t_G=t_1$, $t_{G^{\ast }}=t_2$, $t_{NG}=t_3$, $t_{N{G}^{\ast }}=t_4$ to be a possible pair of times in four corners of G-quadraplex.

There are two types of cell cocycles (Figure 1c) in the loop structure of G-quadruplex of telomere and in telomerase enzymes; we call them $\beta$- and $\alpha$- cocycles. They are associated with T-loop and D-loop in G-quadruplex (Figure 2). We visualize the gene expression as an anyon transition state with behavior field ${\beta }_i$. It is the cocycles of the flying ring in geneon wave function $\mathsf{\Psi }({\beta }_1,{\beta }_2,\cdots ,{\beta }_n)={\mathrm{e}}^{\mathrm{i}{\sum }_{i=1}^n{\beta }_i}={\mathrm{e}}^{\mathrm{i}{\beta }_1}{\mathrm{e}}^{\mathrm{i}{\beta }_2}\cdots {\mathrm{e}}^{\mathrm{i}{\beta }_n}$ (see Figure 3). The sequence of a gene in a protein is separable and can be processed by permutation as the mutation in the new order of the gene with transposon ${\mathsf{\Psi }}^{\mathrm{transposon}}\left({\beta }_i\right)$ and retrotransposon states ${\mathsf{\Psi }}^{\mathrm{retrotransposon}}\left({\beta }_i\right)$. The dark line in the middle of the plot is DNA with the transcription process. We define a DNA flying ring as an orbit of geneonic transition in superspace of DNA $d\in D:=S^2$, for the RNA flying ring is $r\in R:=S^2$. The trajectory of intersection between these two rings induces a loop braid group in genetic codes in $(3+3)$ extra dimensions. The fiber bundle along the trajectory is a folding structure of the secondary protein in a superspace $p\in P$. The upper layer is a starting point for flying; the lower layer is the passive layer of protein folding in the feedback loop. The yRNA and intron are just a result of the Reidemeister move for loop braid group in three trajectories of $(d,r,p)$ (Figure 4b).

Figure 3. We define gene expression as a cocycle with behavior field ${\beta }_i$ in geneonic wave function (a). It is equivalent tothe exon state in mRNA. In the right picture (b) is a loop braid group diagram for extended central dogma. There exist six superspaces in Kolmogorov space of biological time series data. The trajectory of $d\in D$ means DNA, $r\in R$ means RNA, and $p\in P$ means a protein. The upper layer is an active layer of gene expression with a flying ring along the trajectory as a loop braid element. The Reidemeister move switches state of gene expression between two loop braid group elements in genetic code.

Figure 4. (a) The picture shows a loop braid group structure in a telomere. The passive layer of a telomerase protein at the end of the chromosome is an element of superspace $p^{\ast }\in P^{\ast }$. Inside the layer of telomerase there exists a flying ring of RNA $r\in R$ and DNA $d\in D$. When the cell divides the loop braid group element, the genetic code will fly to the active side of a histone protein $p\in P$. The four circular rings inside the loop braid group are T-loop and D-loop structures in a telomere. (b) The picture shows gene expression from DNA to mRNA with a stem-loop structure. It is an example of a flying ring in loop braid group of a superspace of DNA and RNA.

This interaction of D-brane in a living organism is a coupling in loop space in biological time series data with parasitism state between a host cell of the living organism. One state is used for shortening the telomere, and the other reversed transcription state is used for an adenovirus to prolong the length of telomere during cell division. It is a real function of ncRNA embedded in telomerase. They both have G-quadruplex stabilizers. We can represent the T-loop and D-loop in the end-point of the chromosome by using the loop braid group for biology. The loop braid group for biology is not just a representation for DNA, RNA, and protein transitions (Figure 2). It can be used also to define the intrinsic source of the energy-momentum tensor with undivided properties in genotypes of underlying behavior fields as a flying ring (Figure 1d) in the genetic code.

2.1. Anyon in Biology and Configuration Phase Space for Protein Folding

The permutation of an alphabet code from an evolutional process induces a mutation and it appears as an element of a symmetric group ${\mathsf{\Omega }}_n^{BIO}$.

Definition 3.

Let $G={\mathsf{\Omega }}_n^{BIO}$ and $L{B}_n^{BIO}\to {\mathsf{\Omega }}_n^{BIO}$. The permutation of alphabet code in mutation induced a symmetric group as group operation. Let $G\times X_t\to X_t$ be a loop braid group operation over Kolmogorov space in time series data $X_t$, as configuration space in the process of converting the DNA sequence into a folded protein

$${\mathsf{\Omega }}_n^{BIO}=⟨{\omega }_1,\dots ,{\omega }_{n-1}|{\omega }_i{\omega }_{i+1}{\omega }_i={\omega }_{i+1}{\omega }_i{\omega }_{i+1},{\omega }_i{\omega }_j={\omega }_j{\omega }_i,|i-j|\ge 2,{\omega }_i^2=1⟩.$$

(2)

Definition 4.

Let $d=(d_1,d_2,\cdots ,d_n)\in D$ and $r=(r_1$, $r_2$, $r_3$, $r_4,\cdots ,r_n)\in R$. We have an anyon in biology defined by the wave function of gene expression ${\mathsf{\Psi }}_i^d{\mathsf{\Psi }}_i(d_1,d_2,\cdots ,d_n)\in X_t$ with behavior field $({\alpha }_i,{\beta }_i)$ as the cocycle in adaptive behavior in the immune system.

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_i(d_1,d_2,\cdots ,d_n)& ={\mathrm{e}}^{2\mathrm{i}\pi {\beta }_j}{\mathsf{\Psi }}_i(r_1,r_2,\cdots ,r_n),\hfill \\ \hfill {\mathsf{\Psi }}_i(d_1^{\ast },d_2^{\ast },\cdots ,d_n^{\ast })& ={\mathrm{e}}^{-2\mathrm{i}\pi {\alpha }_j}{\mathsf{\Psi }}_i(r_1^{\ast },r_2^{\ast },\cdots ,r_n^{\ast }).\hfill \end{array}$$

(3)

It is used to define the configuration space in the wave function of the transition state for gene expression by ${\mathsf{\Psi }}_i(d_1,d_2,\cdots ,d_n)\in X_t$, where anyon fields are induced from the behavior of immune system. In DNA there is a pair of genotypes in the form of a pair of alleles $({\alpha }_i,{\beta }_i)$; it is an adaptive behavior of cocycle of gene expression for undocking behavior in the protein–protein docking system.

2.2. Circular Artin Braid Group Representation for Spinor Field in Genetic Code

In this section, we assume that all genetic code cannot be completely separated. The biological clock in telomere length is parametrized by a hidden state of the number of $d{t}^{\ast }:=\left[G\right]$ alphabet in a repeated pattern of the telomere ${\left(TTAGGG\right)}_{n=d{t}^{\ast }}$.

The element is Grothendieck topology over an adjoint cofunctor. It is a self-diffeomorphism $\xi :B_3^c\to B_3^c$. The loop braid generator for $B_3^c$ is a quaternionic field in genetic code. We define their explicit forms and their permutations over the symmetric group by a chosen basis in Clifford algebra.

Definition 5.

Let

$${\sigma }_D\to \left[\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right],{\sigma }_R\to \left[-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right],{\sigma }_P\to \left[\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right],$$

and ${\sigma }_{D^{\ast }}={\sigma }_D^{-1}$, ${\sigma }_{R^{\ast }}={\sigma }_R^{-1}$, ${\sigma }_{P^{\ast }}={\sigma }_P^{-1}$. We have

$$\begin{array}{c}{\sigma }_{D^{\ast }}\to \left[-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right],{\sigma }_{R^{\ast }}\to \left[\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right],{\sigma }_{P^{\ast }}\to \left[-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right];\end{array}$$

therefore, one can write eight bases for spinor field in the genetic code in braid form as follows.

$$\begin{array}{ccc}\hfill {\sigma }^{\left[G\right]}& :=& [0,0,0]{\sigma }_R^{-1}{\sigma }_D={\sigma }_R^{-1}{\sigma }_D,\hfill \\ \hfill {\sigma }^{\left[A\right]}& :=& [0,0,1]{\sigma }_R^{-1}{\sigma }_D=\left({\sigma }_D{\sigma }_R\right)\left({\sigma }_R^{-1}{\sigma }_D\right),\hfill \\ \hfill {\sigma }^{\left[U\right]}& :=& [0,1,0]{\sigma }_R^{-1}{\sigma }_D=\left({\sigma }_P{\sigma }_D\right)\left({\sigma }_R^{-1}{\sigma }_D\right),\hfill \\ \hfill {\sigma }^{\left[C\right]}& :=& [1,0,0]{\sigma }_R^{-1}{\sigma }_D=\left({\sigma }_R{\sigma }_P\right)\left({\sigma }_R^{-1}{\sigma }_D\right),\hfill \\ \hfill {\sigma }^{\left[NA\right]}& :=& [0,1,1]{\sigma }_R^{-1}{\sigma }_D=\left({\sigma }_R{\sigma }_P\right)\left({\sigma }_P{\sigma }_D\right)\left({\sigma }_R^{-1}{\sigma }_D\right),\hfill \\ \hfill {\sigma }^{\left[NU\right]}& :=& [1,1,0]{\sigma }_R^{-1}{\sigma }_D=\left({\sigma }_D{\sigma }_R\right)\left({\sigma }_R{\sigma }_P\right){\sigma }_R^{-1}{\sigma }_D,\hfill \\ \hfill {\sigma }^{\left[NC\right]}& :=& [1,0,1]{\sigma }_R^{-1}{\sigma }_D=\left({\sigma }_P{\sigma }_D\right)\left({\sigma }_D{\sigma }_R\right)\left({\sigma }_R^{-1}{\sigma }_D\right),\hfill \\ \hfill {\sigma }^{\left[NG\right]}& :=& [1,1,1]{\sigma }_R^{-1}{\sigma }_D=\left({\sigma }_R{\sigma }_P\right)\left({\sigma }_P{\sigma }_D\right)\left({\sigma }_D{\sigma }_R\right)\left({\sigma }_R^{-1}{\sigma }_D\right),\hfill \end{array}$$

One can also use $\theta =2\pi s$ with spin quantum number s to be an integer number for the retrotransposon, so that

$${\mathrm{e}}^{\mathrm{i}\theta }={\mathrm{e}}^{2\mathrm{i}\pi s}={(-1)}^{2s}|{\psi }_2{\psi }_1⟩.$$

(4)

The geneon and retrotransposon statistics operators are $-1$ and $+1$ respectively. In 2-dimensional position space, the abelian anyonic statistics operators ${\mathrm{e}}^{\mathrm{i}\theta }$ are 1-dimensional representations of eight loop braid elements ${\sigma }_1,{\sigma }_1^2,{\sigma }_1^3,\cdots ,{\sigma }_1^8$ in circular Artin braid group $B_2^3$ acting on the space of wave functions (Figure 5).

Figure 5. (a) The picture shows a biological Artin braid element ${\sigma }_1\in B_2$ in complex plane $\mathbb{C}$. There are eight equivalent classes span by eight orders of ${\sigma }_1$. It is a representation of 8 states in 3 genetic codes in codon as braid element in $({\sigma }_1^{\left[A_{\mu }\right]},{\sigma }_1^{\left[A_{\mu }\right]},{\sigma }_1^{\left[A_{\mu }\right]})\in B_2^3$. The red color line represents the curvature from the physiology of biological time series data and blue color represents the active and passive behavior field layers. (b) The picture shows a member of loop braid group. We have three circles $S^1$. They are a source of closed 3-balls $B_3^c$, the structure group of the affine transform of 3 behavioral fields in genetic code ${\sigma }_i^{\left[A_{\mu }\right]}\in B_2$ action on an affine fiber bundle of the behavioral field in the genetic code ${\sigma }^{{\omega }^{\pm }}$, ${\sigma }^{i{m}^{\pm }}$, ${\sigma }^{m^{\pm }}\in B_3^c$. The affine group is as a loop braid group in genetic code by $B_2^3⋊B_3^c$.

2.3. Classification of the Loop Braid Group in Genetic Code

One can classify three types of loop braid group operations as representations of an anyon for protein folding. For a two dimensional representation of D-brane in the loop braid group for the genetic code, one can define the abelian anyon for biology in (2 + 1) dimensions. The extra dimensions are used to represent the homotopy path of protein folding.

Definition 6.

The loop braid group, $L{B}_n^{BIO}$, for a genetic code has three types of generators, $\sigma =({\sigma }_D,{\sigma }_R,{\sigma }_P)$, $\rho =({\sigma }^{\left[A\right]},{\sigma }^{\left[U\right]},{\sigma }^{\left[C\right]},{\sigma }^{\left[G\right]})$, and $\tau =({\sigma }^{\omega },{\sigma }^{im},{\sigma }^m)$. The generators $\left\{{\sigma }_i,{\rho }_i\in B_2^3|i=1,\cdots ,n-1\right\}$ and $\left\{{\tau }_i\in B_3^c|i=1,\cdots ,n\right\}$ fulfill following relations

$$\begin{array}{cc}\hfill & {\tau }_i{\tau }_j={\tau }_j{\tau }_i,i\ne j\hfill \\ \hfill & {\tau }_i^2=1,i=1,\cdots ,n\hfill \\ \hfill & {\sigma }_i{\tau }_j={\tau }_j{\sigma }_i,|i-j|>1\hfill \\ \hfill & {\rho }_i{\tau }_j={\tau }_i{\rho }_j,|i-j|>1\hfill \\ \hfill & {\tau }_i{\rho }_i={\rho }_i{\tau }_{i+1},i=1,\cdots ,n-1\hfill \\ \hfill & {\tau }_i{\sigma }_i={\sigma }_i{\tau }_{i+1},i=1,\cdots ,n-1\hfill \\ \hfill & {\tau }_{i+1}{\sigma }_i={\rho }_i{\sigma }_i^{-1}{\rho }_i{\tau }_i,i=1,\cdots ,n-1\hfill \end{array}$$

(5)

Definition 7.

Let G be a group operation of the genotype. Let ρ be a representation for gene translation as an anyon. We have $\rho :G\to U\left(1\right)=S^2$ as a representation in ${\mathsf{\Omega }}_n^{BIO}$ and in $L{B}_n^{BIO}$. In order to visualize 3D folding structure of the protein, we define three types of loop braid group operations in biology. All loop elements of the representation of amino acids arose from group operations over the superspace of time series data. These tree types are a translation, reflection, and rotation; ${\rho }^{T-I}$, ${\rho }^{T-II}$, and ${\rho }^{T-III}$. For the translation as a string of amino acids, we have the anyon T-I for biology:

$$\begin{array}{ccc}\hfill {\rho }^{T-I}\left({\sigma }_i\right){\rho }^{T-I}\left({\sigma }_{i+1}\right){\rho }^{T-I}\left({\sigma }_i\right)& =& {\rho }^{T-I}\left({\sigma }_{i+1}\right){\rho }^{T-I}\left({\sigma }_i\right){\rho }^{T-I}\left({\sigma }_{i+1}\right),\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\rho }^{T-I}\left({\sigma }_{i+1}\right){\rho }^{T-I}\left({\sigma }_i\right){\rho }^{T-I}\left({\sigma }_{i+1}\right)& =& {\rho }^{T-I}\left({\sigma }_{i+1}\right){\rho }^{T-I}\left({\sigma }_{i+1}\right){\rho }^{T-I}\left({\sigma }_i\right),\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\rho }^{T-I}\left({\sigma }_i\right)& =& {\rho }^{T-I}\left({\sigma }_{i+1}\right).\hfill \end{array}$$

(8)

Considering an analogy with the action of the symmetric group by permutations, one can find a natural action of the braid group on n-tuples of objects or on the n-folded tensor product that involves some twistors. For this purpose let us consider an arbitrary group G and let X be the set of all n-tuples of elements of G whose product is the identity element of G. The kernel of the homomorphism $L{B}_n^{BIO}\to {\mathsf{\Omega }}_n^{BIO}$ is a subgroup of $L{B}_n^{BIO}$ called pure loop braid group for biology on n strands and it is denoted as $L{P}_n^{BIO}$. In the pure braid, the beginning and the end of each strand are in the same position. Pure braid groups fit into a short exact sequence

$$1\to L{F}_{n-1}\to L{P}_n^{BIO}\to L{P}_{n-1}^{BIO}\to 1.$$

(9)

As is obvious, the sequence splits and therefore pure braid groups are realized as iterated semi-direct products of free groups. The braid group $B_3$ is the universal central extension of the modular group $PSL(2,\mathbb{Z})$.

Definition 8.

Let ${\mathcal{O}}_D$, ${\mathcal{O}}_R$, ${\mathcal{O}}_P$ be active layers over the superspace of DNA, RNA, and protein. Let ${\mathcal{O}}_{D^{\ast }}$, ${\mathcal{O}}_{R^{\ast }}$, ${\mathcal{O}}_{P^{\ast }}$ be passive layers. We define a braid group in genetic code by a curvature inside DNA, RNA, or protein structure. It is a source of an acceleration of biological clock in the epigenetic code. We let ${\sigma }_D$, ${\sigma }_R$, ${\sigma }_P$, ${\sigma }_{D^{\ast }}$, ${\sigma }_{R^{\ast }}$, ${\sigma }_{P^{\ast }}$, and ${\sigma }^{\left[A_{\mu }\right]}$, $\left[A_{\mu }\right]=\left[A\right],\left[U\right],\left[C\right],\left[G\right],\left[NA\right],\left[NU\right],\left[NC\right],\left[NG\right]$ be loop braid group elements in the genetic code. They are the circular Artin braid groups for the genetic code. One can define

$${\mathsf{\Psi }}^R={\sigma }_D{\sigma }_R{\sigma }_D,{\mathsf{\Psi }}^P={\sigma }_D{\sigma }_R,{\mathsf{\Psi }}^{P^{\ast }}={\sigma }_R{\sigma }_D{\sigma }_R.$$

(10)

In that way, a braid group operation gives ${\mathsf{\Psi }}^P$

$${\sigma }_D{\mathsf{\Psi }}_P{{\sigma }_D}^{-1}={\sigma }_R{\mathsf{\Psi }}^P{{\sigma }_R}^{-1}={\mathsf{\Psi }}^P$$

(11)

which implies that ${\mathsf{\Psi }}^P$ is in the center of $B_3$. It is a wave function of protein transition anyon state. If $G=L{B}_n^{BIO}$ acts on $X_t$

$${\sigma }_i^{D^{\ast }}{\mathsf{\Psi }}_i(d_1,\dots ,d_{i-1},d_i,d_{i+1},\dots ,d_n)={\mathsf{\Psi }}_i(d_1,\dots ,d_{i-1},d_{i+1},d_{i+1}^{-1}{d}_i{d}_{i+1},d_{i+2},\dots ,d_n).$$

(12)

If $G=L{B}_n^{BIO}$ acts on $Y_t$, one gets

$${\sigma }_i^{R^{\ast }}{\mathsf{\Psi }}_i(r_1,\dots ,r_{i-1},r_i,r_{i+1},\dots ,r_n)={\mathsf{\Psi }}_i(r_1,\dots ,r_{i-1},r_{i+1},r_{i+1}^{-1}{r}_i{r}_{i+1},r_{i+2},\dots ,r_n).$$

(13)

Additionally, if $G=L{B}_n^{BIO}$ acts on $P_t=X_t/Y_t$

$${\sigma }_i^{P^{\ast }}{\mathsf{\Psi }}_i(p_1,\dots ,p_{i-1},p_i,p_{i+1},\dots ,p_n)={\mathsf{\Psi }}_i(p_1,\dots ,p_{i-1},p_{i+1},p_{i+1}^{-1}{p}_i{p}_{i+1},p_{i+2},\dots ,p_n).$$

(14)

From above, it follows that the elements $d_i$ and $d_{i+1}$ exchange places in DNA strand by an analogy with genetic variation. One can check that the braid group relations are satisfied and this formula indeed defines a group action of $L{B}_n^{BIO}$ on $X_t$.

3. Affine Loop Braid Group in Public Key Cryptosystem of Protein Folding

Topological cryptography for biology releases a self versus non-self protein–protein authentication as a public key, a non-protein-coding piece of DNA. An example of such an area of DNA is mitochondrial DNA (miDNA). The genetic codes of tRNA in miDNA and in the chromosome are the same codes but in a different places. The methyl transfer to AdoMet [32] of tRNA represents the role of knot protein as the RSA algorithm in public key biological cryptosystem. The encryption of the moduli state of a protein comes from 20 modulus states in tRNA. The real function of tRNA is a hash function for encrypting the digital signature of biological information.

3.1. Public Key Biological Cryptosystem in Protein–Protein Docking

The system of protein–protein interaction in an immune system can be realized as a public key biological cryptosystem. It is the main application of the loop braid group relations in self and non-self protein–protein authentication in a cryptosystem of antigen-antibody recognition.

Secret information is encoded in the genetic code and immune system with some help from y-RNA. There exists a normal form for the elements of $L{B}_n^{BIO}$ in the terms of generators ${\sigma }_1^{\left[A_{\mu }\right]},\cdots ,{\sigma }_n^{\left[A_{\mu }\right]}\in L{B}_n^{BIO}$. It induces the application of loop braid groups in the genetic code to pubic key biological cryptosystem. In order to explain the authentication mechanism of the digital signature in the system of protein–protein interaction, we have to recall the natural phenomena in the immune system. It is a classification between self and non-self cell death program in apoptosis. Unlike the use of a two-factor authentication certificate, the user needs to know the digital signature to authenticate the self and non-self certificate when a cancer cell or virus attacks the immune system. There must be a private key associated with the noncoding y-RNA to transfer the certificate through a nuclear pore, so the immune system will be able to access the defense authentication system.

We have left and right loop braid groups for biology denoted by $L{B}_l^{BIO,L}$ and $L{B}_r^{BIO,R}$. The elements of $L{B}_l^{BIO,L}$ and $L{B}_r^{BIO,R}$ are generated by the generators ${\sigma }_1^{\left[A_{\mu }\right]}$, ${\sigma }_2^{\left[A_{\mu }\right]},\cdots$, ${\sigma }_{l-1}^{\left[A_{\mu }\right]}$ and the generators ${\sigma }_{l+1}^{\left[A_{\mu }\right]}$, ${\sigma }_{l+2}^{\left[A_{\mu }\right]},\cdots$, ${\sigma }_{l+r-1}^{\left[A_{\mu }\right]}$ respectively. We have commutative properties between left and right supersymmetry of loop braid group as an asymmetric cryptosystem for the authentication public and private keys in the RSA algorithm for biology (see Figure 6c), for any ${\mathsf{\Psi }}^a\in L{B}_l^{BIO,L}$ and ${\mathsf{\Psi }}^b\in L{B}_r^{BIO,R}$ we have ${\mathsf{\Psi }}^a{\mathsf{\Psi }}^b={\mathsf{\Psi }}^b{\mathsf{\Psi }}^a$.

Figure 6. (a) The picture shows eight anyon smallest subunits in biology. It is a representation of eight spinor fields in the genetic code of living organisms. We define them as a flying ring in the unification between three elements from DNA, RNA, and protein. (b) The picture shows three layers of DNA, RNA, and protein in the asymmetric cryptosystem model for biology. (c) The picture demonstrates the encryption and decryption modules in the public key cryptosystem. The non-protein-coding DNA part is a source of the private key. The telomere is a biological clock subunit; it is equivalent to the modulus part in a moduli state space model.

Let a one-way function of the gene expression be defined by

$$\begin{array}{c}\xi :L{B}_l^{BIO,Left}\times L{B}_{l+r}\to L{B}_{l+r}^{BIO,Right}\times L{B}_{l+r}^{BIO},\\ \xi ({\mathsf{\Psi }}^a,{\sigma }^x)=({\mathsf{\Psi }}^a{\sigma }^x{{\mathsf{\Psi }}^a}^{-1},{\sigma }^x).\end{array}$$

(15)

It is a one-way function to carry public-private keys given by a pair $({\mathsf{\Psi }}^a,{\sigma }^x)$. Now we define key agreement in the protein docking system between the proteins A and B with the help of the braid group version of the Diffie–Hellman key agreement in a quantum cryptosystem for biology. We have the following steps.

• Preparatory step: A suitable pair of integers $(l,r)$ are chosen, and a sufficiently complicated $(l+r)$-loop braid ${\sigma }^x\in L{B}_{l+r}^{BIO}$ is selected and published through the gene expression.
• Key agreement: A key is shared through the protein folding by performing the following steps each time a shared key is required:
  • Protein A chooses a random secret loop braid ${\mathsf{\Psi }}^a\in L{B}_l^{BIO,L}$ and sends ${\sigma }^{y_1}={\mathsf{\Psi }}^a{\sigma }^x{\mathsf{\Psi }}^{a^{-1}}$ to B.
  • Protein B chooses a random secret loop braid ${\mathsf{\Psi }}^b\in L{B}_r^{BIO,R}$ and sends ${\sigma }^{y_2}={\mathsf{\Psi }}^b{\sigma }^x{\mathsf{\Psi }}^{b^{-1}}$ to A.
  • Protein A receives ${\sigma }^{y_2}$ and computes the shared key ${\mathsf{\Psi }}^K={\mathsf{\Psi }}^a{\sigma }^{y_2}{\mathsf{\Psi }}^{a^{-1}}$.
  • Protein B receives ${\sigma }^{y_1}$ and computes the shared key ${\mathsf{\Psi }}^K={\mathsf{\Psi }}^b{\sigma }^{y_1}{\mathsf{\Psi }}^{b^{-1}}$.
  Since ${\mathsf{\Psi }}^a{\mathsf{\Psi }}^b={\mathsf{\Psi }}^b{\mathsf{\Psi }}^a$, we have

  $${\mathsf{\Psi }}^a{\sigma }^{y_2}{{\mathsf{\Psi }}^a}^{-1}={\mathsf{\Psi }}^a\left({\mathsf{\Psi }}^b{\sigma }^x{{\mathsf{\Psi }}^b}^{-1}\right){{\mathsf{\Psi }}^a}^{-1}={\mathsf{\Psi }}^b\left({\mathsf{\Psi }}^a{\sigma }^x{{\mathsf{\Psi }}^a}^{-1}\right){{\mathsf{\Psi }}^b}^{-1}={\mathsf{\Psi }}^b{\sigma }^{y_1}{{\mathsf{\Psi }}^b}^{-1}.$$

  (16)
  Thus both proteins A and B obtain the same loop braid group element with the same curvature. Therefore they can dock to each other; otherwise the system will not be in equilibrium and the moduli state space for the gene expression with control equation will recursively loop back to send the key again.

3.2. Moduli State Space Model for Protein–Protein Interaction

The protein–protein interaction can be realized as a share key agreement in the biological cryptosystem, if we consider the central dogma as the short exact sequence

$$D\stackrel{\beta }{\to }R\stackrel{\alpha }{\to }P,x_{t_G}\mapsto y_{t_{NG}}\mapsto y_{t_{NG}}/x_{t_G}$$

(17)

The moduli state space model for protein–protein interaction with G-quadruplex, $t_G=t_{NG}$, in the D-brane plane, comes from this short exact sequence:

$$\begin{array}{cc}\hfill y_{t_{NG}}& =\beta x_{t_G}+{ϵ}_{t_G},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill y_{t_{NG}}& =\alpha y_{t_{NG}}/x_{t_G}+{\mu }_{t_{NG}}.\hfill \end{array}$$

(19)

If we approximate this equation with the gene expression at first time period $t_1=t_{1_G}$ with a short exact sequence, then ${\beta }_1$ is the first genotype at time period $t_1=t_{1_G}$ in the equilibrium moduli state of the gene expression:

$$y_1/x_{1_G}={\beta }_1.$$

(20)

We have a recursive loop of reversed transcription of jumping gene of retrotransposon transition state $y_t$ with repeated geneon state $x_{t_G}$ by

$$y_t/x_{t_G}=y_{t-1}+{\mu }_t,$$

(21)

so we have

$$y_t=x_{t_G}{y}_{t-1}+{\mu }_t.$$

(22)

For no genetic variation, ${\mu }_t=0$, we have a recursive loop of repeated jumping gene of retrotransposon in the non-protein-coding DNA as

$$y_t=x_{t_G}{y}_{t-1}.$$

(23)

The initial value equation of moduli state space model for the non-protein-coding DNA with starting state of retrotransposon $y_0$ is

$$y_1=x_{t_G}{y}_0.$$

(24)

Therefore we have

$$\begin{array}{cc}\hfill y_2& =x_{t_G}{y}_1,\hfill \\ \hfill y_3& =x_{t_G}{y}_2,\hfill \\ \hfill y_4& =x_{t_G}{y}_3,\hfill \\ \hfill & \cdots \hfill \\ \hfill y_N& =x_{t_G}{y}_{N-1}.\hfill \end{array}$$

(25)

We have moduli state space model for DNA, ${\mathsf{\Psi }}^d={\mathsf{\Psi }}^d(d_1,d_2,\cdots ,d_n)$, RNA, ${\mathsf{\Psi }}^r={\mathsf{\Psi }}^r(r_1,r_2,\cdots ,r_n)$, and protein, ${\mathsf{\Psi }}^p={\mathsf{\Psi }}^p(p_1,p_2,\cdots ,p_n)$

$$\begin{array}{cc}\hfill \mathsf{\Psi }& \equiv {\mathsf{\Psi }}^p\mathrm{mod}{\mathsf{\Psi }}^d,\hfill \\ \hfill \mathsf{\Psi }& \equiv {\mathsf{\Psi }}^d\mathrm{mod}{\mathsf{\Psi }}^r,\hfill \\ \hfill \mathsf{\Psi }& \equiv {\mathsf{\Psi }}^r\mathrm{mod}{\mathsf{\Psi }}^p.\hfill \end{array}$$

(26)

Let $N_1=\frac{{\mathsf{\Psi }}^p{\mathsf{\Psi }}^d{\mathsf{\Psi }}^r}{{\mathsf{\Psi }}^p}$, $N_2=\frac{{\mathsf{\Psi }}^p{\mathsf{\Psi }}^d{\mathsf{\Psi }}^r}{{\mathsf{\Psi }}^d}$, $N_3=\frac{{\mathsf{\Psi }}^p{\mathsf{\Psi }}^d{\mathsf{\Psi }}^r}{{\mathsf{\Psi }}^r}$, so we have

$$\begin{array}{cc}\hfill N_1{N}_2\mathsf{\Psi }& \equiv 1\mathrm{mod}{\mathsf{\Psi }}^r,\hfill \\ \hfill N_1{N}_3\mathsf{\Psi }& \equiv 1\mathrm{mod}{\mathsf{\Psi }}^d,\hfill \\ \hfill N_2{N}_3\mathsf{\Psi }& \equiv 1\mathrm{mod}{\mathsf{\Psi }}^p.\hfill \end{array}$$

The definitions of public and private keys in a biological cryptosystem are the following.

Definition 9.

Let the protein A span by the loop braid group $L{B}_l^{BIO,L}$ and the protein B span by the loop braid group $L{B}_r^{BIO,R}$. Let a hash function for biology be $H_{RNA}^{BIO}:L{B}_{n=l+r}^{BIO}$$\to {(\mathbb{Z}/2)}^k$ from loop braid group in the genetic code to a biological message space. This message space specifies the spinor state of spin up or down over the fiber loop space in the genetic code. The public key biological cryptosystem is composed of three modules

• Key generation module: Protein A and B choose a sufficiently complicated genotype in the loop braid group in genetic code ${\sigma }^x{\sigma }_i^{\left[A_{\mu }\right]}\in L{B}_{l+r}^{BIO}$ and a hash with a fixed size of 20 amino acids (modulo 20) while transmitting the public key over a two protein docking system. The left protein chooses a genotype with representation in loop braid group in the genetic code $L{B}_l^{BIO}$. The public key is $({\sigma }^x,{\sigma }^y)$ where ${\sigma }^y={\mathsf{\Psi }}^a{\sigma }^x{{\mathsf{\Psi }}^a}^{-1}$.
• Encryption module: Given a biological message authentication code by y-RNA, tRNA, and ncRNA as a hash function with spinor state $m\in {(\mathbb{Z}/2)}^k$ and the public key $({\sigma }^x,{\sigma }^y)$, protein B chooses a loop braid group ${\mathsf{\Psi }}^b\in L{B}_r$ at a rate of random mutation ${ϵ}_t$. Biological cipher state is $({\mathsf{\Psi }}^c,{\mathsf{\Psi }}^d)$ where ${\mathsf{\Psi }}^c={\mathsf{\Psi }}^b{\sigma }^x{{\mathsf{\Psi }}^b}^{-1}$, ${\mathsf{\Psi }}^d=H_{RNA}\left({\mathsf{\Psi }}^b{\sigma }^x{{\mathsf{\Psi }}^b}^{-1}\right)\oplus m$.
• Decryption module: Given the biological cipher state $({\mathsf{\Psi }}^c,{\mathsf{\Psi }}^d)$ and the private key ${\mathsf{\Psi }}^a$, compute the element $m=H_{RNA}\left({\mathsf{\Psi }}^a{\mathsf{\Psi }}^c{{\mathsf{\Psi }}^a}^{-1}\right)\oplus d$.

Because ${\mathsf{\Psi }}^a$ and ${\mathsf{\Psi }}^b$ commute to each other we can recover the information that transmission from DNA to protein by encryption process in RNA by ${\mathsf{\Psi }}^a{\mathsf{\Psi }}^c{{\mathsf{\Psi }}^a}^{-1}={\mathsf{\Psi }}^a{\mathsf{\Psi }}^b{\sigma }^x{{\mathsf{\Psi }}^b}^{-1}{{\mathsf{\Psi }}^a}^{-1}={\mathsf{\Psi }}^b{\mathsf{\Psi }}^a{\sigma }^x{{\mathsf{\Psi }}^a}^{-1}{{\mathsf{\Psi }}^b}^{-1}={\mathsf{\Psi }}^b{\sigma }^y{{\mathsf{\Psi }}^b}^{-1}$, and therefore we have for $H_{RNA}\left({\mathsf{\Psi }}^a{\mathsf{\Psi }}^c{{\mathsf{\Psi }}^a}^{-1}\right)\oplus d=m$, the decryption gets the original message in loop braid group element m.

4. Empirical Analysis of Cancer Gene Signature

Our empirical analysis is based on a novel algorithm based on $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(n\right)$ in time series prediction introduced in [10]. The topological approach to computation on DNA time series data was successfully applied in the empirical mode decomposition of the Chern–Simons current related to genetic variations of viral glycoprotein gene and host T-cell receptor gene [4].

We also calculated the Chern–Simons current in the genotype. We used the genetic code of telomerase RNA component TERC from GenBank [33] (see also Appendix C) and the genetic code of ncRNA in telomerase (Appendix D). The results of the plots for the Chern–Simons current in the genetic code of PD1 and ncRNA are presented in Figure 7 and Figure 8. The plots are the Chern–Simons currents in genetic code, for different values of n; they represent the wave functions of retrotransposon states in ncRNA. Higher values of n yield smoother curves; they have parabolic shapes with concave or convex curvature, as one can see from the ncRNA or protein PD1 plots. Similar plot results for the Chern–Simons current can be used to detect a cancer signature, as can be seen in Figure 9.

Figure 7. (a) The plot of $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(10\right)$ of the Chern–Simons current in the genetic code of TERC gene in telomerase with 1069 alphabet codes (Appendix C). (b) The plot of $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(1\right)$ of the Chern–Simons current in PD1 protein. (c) The plot of $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(7\right)$ of the trend of the Chern–Simons current in PD1 protein.

Figure 8. (a) The plot of retrotransposon state ncRNA in telomerase. It is $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(8\right)$ of the Chern–Simons current in the genetic code of ncRNA. (b) The plot of $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(1\right)$ of telomerase ncRNA. (c) The plot of $((\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(4\right)$ of telomerase ncRNA.

Figure 9. (a) The plot of the result for computation $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(1\right)$ of the Chern–Simons current in the genetic code of first gene in bone cancer, so-called ALX1 gene, for a cancer gene (red) and normal gene (blue). (b) The plot of the result for computation of $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(8\right)$ of the Chern–Simons current in ALX1 gene. We can notice the difference between two genes by the shift of plotted line.

For that purpose we downloaded data of 36 samples of cancer genes (Table 1) from the online database Catalogue of Somatic Mutations in Cancer (COSMIC) with sample name PD7301a and COSMIC sample ID COSS1540693 [34]; the tumour location is in the bone; the pelvis (chondrosarcoma; central); see also Appendix E for the AX1 gene. In Figure 9, one can see the difference in the calculation of the Chern–Simons current in the ALX1 gene for cancer (red) and normal (blue) gene versions; the change is better visible in the computation of $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(8\right)$. In Figure 10 and Figure 11 are presented results for the tensor correlation of the Chern–Simons current from $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(1\right)$ and $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(6\right)$ between normal (blue) and cancer (red) genes in selected 28 samples of bone cancer. In the region of amino acids 700 and 1000, the mutation area is noticeable. In Figure 12 and Figure 13 are plots of the $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(1\right)$ and $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(8\right)$ of the Chern–Simons current in the genetic code for mutation for samples 33–36 presented in Table 1. Next are presented the images of the spectrum of tensor correlation of 28 cancer genes, computed for the amino acid number 531 (Figure 14) and the amino acid number 9980 (Figure 15). The computation uses canonical tensor correlation structure from $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(1\right))$ to $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(6\right)$ from 28 genes listed in Table 1 with matrix size $6\times$ 19,244. The number 19,244 comes from total sum of all amino acids in 28 samples. One can see the differences in the color patterns. The visualization of comparative difference between the cancer and normal genes can help in building the databases and the classification of different types of genes in the future.

Table 1. The table shows the result of a public key and the private key calculation for 36 cancer genes compared with 28 normal genes. The result of computation is calculated by the RSA algorithm over genetic codes.

Protein No. (Sample No.)	Gene Name	Public Key (Normal)	Public Key (Cancer)	Private Key (Normal)	Private Key (Cancer)	Modulus (Normal)	Modulus (Cancer)	Size
$p_1\left(1\right)$	ALX1	7	5	43	2297	341	3953	326
$p_2\left(2\right)$	ANO2(1)	3	7	467	1003	781	2449	998
$p_3$	ANO2(2)	−	5	−	437	−	2291	998
$p_4$	ANO2(3)	−	11	−	191	−	2201	999
$p_5\left(3\right)$	ATAD2	5	5	749	389	1027	703	1390
$p_6\left(4\right)$	C20orf103	3	5	1531	2165	2407	5561	280
$p_7\left(5\right)$	C6orf170	11	3	611	387	6887	649	1257
$p_8\left(6\right)$	CTNNA2	3	3	2515	1203	3901	1909	905
$p_9\left(7\right)$	DENND1B	5	3	1037	2347	1387	3649	396
$p_{10}\left(8\right)$	ERBB3(1)	5	5	5117	845	6557	1139	1342
$p_{11}$	ERBB3(2)	−	11	−	2291	−	4331	1342
$p_{12}\left(9\right)$	ESYT2	3	3	3171	323	4897	529	893
$p_{13}\left(10\right)$	F2RL1	5	5	1037	1037	1369	2701	397
$p_{14}\left(11\right)$	FAF1	5	7	1373	823	7031	1037	650
$p_{15}\left(12\right)$	GPR75	7	5	343	1373	1271	1817	540
$p_{16}\left(13\right)$	IDH1	5	3	53	2843	299	4399	414
$p_{17}\left(14\right)$	ISX	3	3	2707	4811	4189	7387	245
$p_{18}\left(15\right)$	MAGEB5	3	5	2707	749	4189	1343	275
$p_{19}\left(16\right)$	MAML3(1)	5	11	1109	611	1943	6887	1138
$p_{20}$	MAML3(2)	−	7	−	2263	−	2747	1138
$p_{21}\left(17\right)$	PTPN5(1)	5	3	2765	675	3589	1081	565
$p_{22}$	PTPN5(2)	−	3	−	299	−	493	565
$p_{23}\left(18\right)$	Q6ZVS6 HUMAN	3	3	763	4107	1219	6319	177
$p_{24}\left(19\right)$	RAD23A	3	5	1227	2297	1927	3953	363
$p_{25}\left(20\right)$	RGS7	5	7	1217	2263	6241	5429	495
$p_{26}\left(21\right)$	SETD4	5	7	281	103	1501	209	440
$p_{27}\left(22\right)$	SGCD	11	5	191	1997	2201	5141	290
$p_{28}\left(23\right)$	SMAD1	3	3	547	1531	913	2407	465
$p_{29}\left(24\right)$	SRCAP	5	5	317	1901	1691	3293	3230
$p_{30}\left(25\right)$	SULF1	11	5	1031	845	1349	1157	871
$p_{31}\left(26\right)$	THOC4	3	3	851	675	1357	1081	257
$p_{32}\left(27\right)$	UGT3A2	5	7	1037	2263	1843	2759	523
$p_{33}\left(28\right)$	XAGE3	3	5	1307	1037	2059	1369	111
$p_{34}$	ZNF253	−	5	−	1373	−	3551	549
$p_{35}$	ZNF90(1)	−	3	−	491	−	799	195
$p_{36}$	ZNF90(2)	−	7	−	823	−	1517	601

Figure 10. The picture shows the tensor correlation of the Chern–Simons current from $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(1\right))$ for normal (blue) and cancer (red) genes in 28 selected samples of bone cancer.

Figure 11. (a) The picture shows the tensor correlation of the Chern–Simons current from $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(6\right)$, between normal (blue) and cancer (red) genes in 28 selected samples of bone cancer for the first 1000 amino acids. We can see that the signature of mutation appears between the amino acids 400 and 600. On the plot (b), if we zoom the region between amino acids 700 and 1000 we can also notice some signs of the mutation in that area.

Figure 12. (a–d) The plots of $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(1\right)$ of the Chern–Simons current in the genetic code of mutation in cancer gene for the samples 33–36 in Table 1.

Figure 13. (e–h) The plots of TREND of $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(8\right)$ of the Chern–Simons current in the genetic code regarding mutation in cancer genes for the samples 33–36 in Table 1.

Figure 14. (a) The image of the spectrum of tensor correlation of 28 cancer genes, computed at amino acid number 531. (b) The same image at the same position of amino acid which we computed from selected 28 normal reference genes. The computation uses canonical tensor correlation structure from $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(1\right))$ to $(\mathrm{ITD}-\mathrm{IMF}){\mathrm{chain}}_1\left(6\right)$ from 28 genes listed in Table 1. We used slide window with the size 6 because we had total samples of amino acids with matrix size $6\times$ 19,244. The number 19,244 comes from total sum of all amino acids in 28 samples.

Figure 15. (a) The image of the spectrum of geneonic transition state of 28 cancer genes at amino acid number 9980 from 19,244 amino acids in tensor correlation. (b) The image of normal 28 genes computed from the same position with slice window size of 6 amino acids. We can use the comparative difference between color patterns as the signature of cancer instead of using only a color diagram, as in [34].

5. Discussion and Conclusions

We use loop braid group to define the flying ring in the (3 + 3) extra dimensions of triplet state of transition between the DNA, RNA, and protein. These three molecules are interconnected by combining them into a four-dimensional coordinate system. We can unify them as a loop braid group in genetic code as the smallest subunit in the translating of the DNA sequence into a folded protein. The flying ring is just an 8-spinor field in the homogeneous coordinate of supermanifold in the genetic code.

The interaction between the D-brane of a telomere in the non-protein-coding DNA and the anti-D-brane of the telomerase enzyme with embedded noncoding RNA is the source of the biological clock and cell aging in a living organism. This process of the biological clock generator is the rotation of a homogeneous coordinate in the gene expression. The rotation of the clock produces a spinor field in time series data as a signal for cell transduction. It is the information inside codon which generates the public and private keys in quantum cryptography along the geneonic state. The signal of protein expression is expressed as immune receptors and signal transduction with the public and private keys. The protein state belongs to a superspace P. The gene expression in the eukaryotic cell with mRNA will have approximately eight exon states with ${\beta }_i$ cocycle geneonic states.

The other part of noncoding RNA in mRNA is an intron state, denoted as ${\alpha }_i$ cocycle of retrotransposon state in ncRNA space $R^{\ast }$. The function of intron state is to control the gene expression as public and private keys to deliver the sequence of mRNA out of the nuclear membrane with the help of y-RNA. The keys will match to each other outside the nuclear pore in ribosomes in the rough endoplasmic reticulum for the process of t-RNA to match the public and private keys inside ribosome for the translation process of producing the amino acid. It might be a real function of the non-protein-coding DNA. The acceleration of the life clock is caused by the imbalance of collisions between the telomere and telomerase by passing the energy-momentum tensor in protein–protein interactions.

In this work, a new theory of quantum cryptography for biology has been developed. In particular, we used the loop braid group for biology to visualize the G-quadruplex with six repeated sequences in a telomere. We plotted the Chern–Simons current in the genotypes of all samples to detect the physiology in the biological time series data. We found that the graph of cancer gene has a degree of nonstationary time series data more than the normal gene. The tensor correlation network for 36 cancer genes has been generated and the image of spectra of geneon state in a cancer gene has been produced. We perform RSA image encryption to see the private and public keys over the image of geneon. In our approach, quantum cryptography is applied in biology to find the pattern of mutation genes by using an image encryption method.

We think that this new methodology could evolve into the more practical visualization of genes in the future, instead of showing the sequence of amino acids. DNA (and genes) is permanently changing, it has a dynamic structure. The mutations on the gene level affect the genes, which subsequently can lead to production malfunctions of various proteins. Gene-level mutations are caused by different types of processes. It may be a swapping of one nucleotide for another one, a deleting or omitting of a specific nucleotide from the strand, or an inserting of an extra nucleotide into the replication strand. Besides the single codon substitutions, on the level of codons there exist other categories of mutations. The incorrect amino acids can be inserted into a protein molecule, the synthesis of a protein can be prematurely terminated, and the mutated codon can code the same amino acid as the non-mutated codon. As a consequence of DNA constant change, the difference between the mutated and normal versions of the gene changes over time. We can train our algorithm on the different versions of the genes, on normal genes with non-modified sequences of nucleotides or on their mutated versions as cancer genes, which code different structures than the normal ones.

With our model we are looking for the effective change of topology of proteins, i.e., the stable structures of noncoding DNA binding to genes; this procedure selects only certain structures that can be detected. We hope for the application in cancer genes, but at first the genes on which the measurements can be made must be identified by genetic experts. However, we provide the tool for the terminal measurements incorporated into more complex biophysical experiments. In the future, the application of the methodology can be developed into more practical automatic detection of mutated genes, either for medical purposes or for imaging production of the cancer genome project.

In conclusion, we applied a direct application of algebraic topological quantum cryptography and quantum field theory with loop braid group theory directly to perform the analysis of the ALX1 cancer gene for the detection of a cancer signature. We proposed and used a new theory of quantum public key encryption in a cancer cryptosystem. The result of the computation of the RSA algorithm for public and private keys in normal and cancer genes appeared to be significant for the biological cryptosystem. Finally, we used the Chern–Simons current to plot the tensor correlation and compute the image of spectra for cancer signature.

The relevant point of our approach is the fact that quantum field theory developed for physics, in particular, the Chern–Simons theory, is extremely useful for describing biological systems [35,36], despite the huge difference in orders of magnitude between fundamental particles and genetic code. However, our mathematical method can also be used not only for mapping and determination of the genomes of viruses and subsequent prediction of their mutations, but also in astrophysical and cosmological systems. Following symmetry in biological nature gives us a chance to also compute the structures of biological systems in our description of the mutation genes.