**2. Spinor fields in biological systems**

One of the present problems in genetic engineering is the prediction of biological gene variation and the representation of corresponding genetic code. This issue emerges in the plotting graphs related to the connection curvature of a docking processes. The docking process is important in the genes of the protein structure and could be adopted instead of using a very long alphabet notation as the string sequence and the comparison of the sequences of docking. From this point of view, methods of quantum field theory, general relativity, and related tools can be of high interest. The equilibrium between the supersymmetry and the mirror symmetry of the left-handed and right-handed DNA, RNA, nucleic and amino acid molecules can be explained by anti-de Sitter (AdS) correspondence in the Yang-Mills theory and the Chern-Simon currents in biology as the curvature of the spectrum in genetic code of the protein curvature.

Today, a genetical structure is studied by standard alphabet codes *A*, *T*, *C*, *G*, and *U* as a sequence of strings for the representation of genetic code for various organisms without any exact definition of a new time series of genetic code [17] in contrast to standard time series modeling. With this representation [18], it is very difficult to calculate the genetic variation [19] and to perform calculations within a framework of self-consistent mathematical theory [20], namely, in the context of string theory and M- and G-theories [21, 22].

There are still attempts to perform empirical data analysis of the genetic variation [23] and to detect the pattern matching over the gene sequence by using algorithm over a standard alphabet code as their time series representation. It seems that one main problem in this field is how can we predict the genetic variation and the gene structure in the viral particle and other organisms, or in the context of new representation, the question is how we can explain the intuition behind a definition of new time series data of gene, e.g., involved in the Batalin-Vilkovisky cohomology of DNA and the viral gene structure. The Chern-Simons current and the anomaly over a superspace of cell membrane can be applied to diagnose new gene diseases, the cloning technology or the gene therapy in medicine. Moreover, a presented method can be improved also in view of describing a useless trash area of DNA, which is considered as unknown part of human genome.

can define a mutual genetic code as passive or dual hidden states ½ � *<sup>s</sup>*<sup>1</sup> <sup>∗</sup> *,* …*, s*½ � <sup>8</sup> <sup>∗</sup> and

*jπαi* 2

The reversed transcription process of the gene expression is defined by a moduli state space model of a coupling spinor field between the gene of a viral particle and

<sup>¼</sup> <sup>1</sup>*; <sup>q</sup>*<sup>j</sup>

One can define *Sp*ð Þ!<sup>1</sup> *<sup>S</sup>*<sup>7</sup> ! <sup>H</sup>*P*<sup>1</sup> as a Hopf fibration of eight states of the

of the genetic code of the space of a viral RNA *Xt* and a space of host cell DNA, *Yt*. If <sup>U</sup>½ � *<sup>A</sup> <sup>α</sup>* <sup>⊂</sup> <sup>H</sup>*P*<sup>1</sup> is a chart of local coordinate in a manifold of genetic code over *Xt=Yt*, where ½ � *A <sup>α</sup>* is defined over the right-hand isomer genetic code f g *A; T;C; G*

mirror symmetry of a genetic code f g *NA; NT; NC; NG* . We have a cycle and a cocycle of an orbifold as a trivialization over the tangent of the living organism manifold, so-called codon and anticodon U*<sup>i</sup>* ∩ U*<sup>j</sup>* ∩ U*k*. Let ð Þ M*; g* be a living organism manifold with <sup>M</sup> <sup>¼</sup> <sup>H</sup>*P*<sup>1</sup> for a living organism with the Riemannian metric

*(a) The site of docking between CCR5 Δ*32 *co-receptor in host T-cell membrane and V3 loop in HIV viral gp120. The cell membrane has a mirror symmetry structure of D-brane for outer layer. The anti-D-brane structure of the cell membrane is an inner layer of phospholipid. (b) The knot 4*<sup>1</sup> *model of a short exact sequence*

*DNA q* <sup>∗</sup> j *RNA*

� �

h i <sup>þ</sup> ½ � <sup>0</sup> **<sup>i</sup>** <sup>þ</sup> ½ � <sup>0</sup> **<sup>j</sup>** <sup>þ</sup> ½ � <sup>0</sup> **<sup>k</sup>***,*

h i**<sup>i</sup>** <sup>þ</sup> ½ � <sup>0</sup> **<sup>j</sup>** <sup>þ</sup> ½ � <sup>0</sup> **<sup>k</sup>***,*

� �**<sup>j</sup>** <sup>þ</sup> ½ � <sup>0</sup> **<sup>k</sup>***,*

<sup>¼</sup> *<sup>q</sup>*<sup>j</sup>

M> over a tangent manifold and a cotangent manifold

� �**k***:*

*DNA q* <sup>∗</sup> j *RNA ;* 1 � �*:* (31)

(30)

*<sup>p</sup>* M states

*<sup>α</sup>* , with the

�*jπα<sup>i</sup>* <sup>2</sup>

active eight states ½ � *s*<sup>1</sup> *,* …*, s*½ � <sup>8</sup> for the spinor field in the genetic code by

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology*

½ � *<sup>C</sup> tRNA* <sup>≔</sup> ½ � *NG mRNA* <sup>≔</sup> ½ �þ <sup>0</sup> ½ � <sup>0</sup> **<sup>i</sup>** <sup>þ</sup> *ejπα<sup>i</sup>*

*m,n*¼1*;*2*;*3*;*4

(for the simplicity we use a symbol G also for U) with their dual ½ � *<sup>A</sup>* <sup>∗</sup>

genetic code ½ � *<sup>s</sup>*<sup>1</sup> *, s*½ � <sup>2</sup> *,* …½ � *<sup>s</sup>*<sup>8</sup> <sup>∈</sup>*S*<sup>7</sup> <sup>¼</sup> *Tp*M, denoted by ½ � *<sup>s</sup>*<sup>1</sup> <sup>∗</sup> *, s*½ � <sup>2</sup> <sup>∗</sup> *,* …½ � *<sup>s</sup>*<sup>8</sup> <sup>∗</sup> <sup>∈</sup>*<sup>T</sup>* <sup>∗</sup>

½ � *<sup>G</sup> tRNA* <sup>≔</sup> ½ � *NC mRNA* <sup>≔</sup> ½ �þ <sup>0</sup> ½ � <sup>0</sup> **<sup>i</sup>** <sup>þ</sup> ½ � <sup>0</sup> **<sup>j</sup>** <sup>þ</sup> *<sup>e</sup>j*2*πα<sup>i</sup>*

½ � *A tRNA* ≔ ½ � *NU mRNA* ¼ *e*

*DOI: http://dx.doi.org/10.5772/intechopen.88486*

*e*2*π* <sup>4</sup>*niα e*2*π* <sup>4</sup>*mjβ* � �

M*, T* <sup>∗</sup> ½ � *A <sup>α</sup>*

the host cell (**Figure 3**)

tensor *gij* <sup>¼</sup> <sup>&</sup>lt;*T*½ � *<sup>A</sup>* ½ � *<sup>A</sup> <sup>α</sup>*

**Figure 3.**

**103**

*of hidden eight states in genetic code.*

<sup>H</sup>*P*<sup>1</sup> <sup>¼</sup> *Xt=Yt*<sup>∍</sup> <sup>1</sup>*;*

½ � *U tRNA* ≔ ½ � *NA mRNA* ≔ ½ �þ 0 *e*

On the other side, another approach based on the usage of a spinor field in the Kolmogorov space of the time series data [24] over the genetic code can represent the gene structure as the ghost and anti-ghost fields of the codon and anticodon. This can be achieved in the frameworks of supersymmetry [25] and G-theory [22]. Results of the works show that all calculations over the codon can be assumed as a new superspace of the time series representation of the gene structure [24].

In [26–28], we introduced a new representation of the genetic code in the time series using a modeling by strings and D-branes. By applying a spinor field to a superspace in time series data [29], the method allows us to develop supersymmetry for living organisms. In particular, it is possible to control the anomalies in the codon and anticodon ghost fields and construct an algebraic approach for the trash DNA.

The "gravitational" analogy of the Chern-Simons currents in a gravitational physics, emanating from a system of DNA-RNA transcriptions, could have interesting counterparts also in biology. A representation of codons in human genome, derived from the Chern-Simons currents, can be useful in biology to explain the source of connections over protein-docking states. In this perspective, adopting cohomology in biology can be useful as a new modeling tool for plotting genes with spinor field in time series data. Especially, the junk area of DNA, with repeated inactive genes, can be represented by the Chern-Simons currents with extended structures of knot states in a Laurent polynomial of knots.

Further we discuss the role of spinors in the time series of genetic code. We can denote <sup>H</sup>*P*<sup>1</sup> as a quaternionic projective space and *<sup>H</sup>*0ð Þ *Xt* as a pointed space of DNA alphabet sequence with ½ � *A* , ½ � *T* , ½ � *C* , and ½ � *G* as an equivalent class of ½ � *A , T*½ �*, C*½ �*, G*½ �∈ *H*0ð Þ *Xt,DNA* ≔ Φ*i*ð Þ *Xt* a ghost field with parity two with *H*0ð Þ¼ *Xt H*<sup>0</sup> *Xt* ð Þ *;* ∗ where ∗ ¼ f g ∗ is a pointed space. We define an equivalent class of DNA–RNA translation processes by using the notation of a master equation for an interaction between the viral RNA and the host cell DNA by *xt; yt* � � <sup>¼</sup> f g *DNA; RNA* . The whole state space model of the viral replication cycle, embedded in the host cell, is denoted by *Xt,DNA=Yt,RNA* <sup>¼</sup> *Zt,GENE* <sup>¼</sup> <sup>H</sup>*=*<sup>H</sup> <sup>¼</sup> <sup>H</sup>*P*<sup>1</sup> as a moduli state space model with the definition of genetic code as an equivalent class of the map *α<sup>t</sup>* : *Xt,DNA* ! f g ½ � *A ;* ½ � *T ;* ½ � *C ;* ½ � *G* ⊂ H; the host cell gene alphabet is defined by a hidden state space *Xt* with the gene *β<sup>i</sup>*

$$\begin{aligned} [A]\_{DNA} &:= \left[e^{\frac{i\boldsymbol{\theta}\_i}{2}}\right] + [\mathbf{0}]\mathbf{i} + [\mathbf{0}]\mathbf{j} + [\mathbf{0}]\mathbf{k}, \\ [T]\_{DNA} &:= [\mathbf{0}] + \left[e^{\frac{-i\boldsymbol{\theta}\_i}{2}}\right] \mathbf{i} + [\mathbf{0}]\mathbf{j} + [\mathbf{0}]\mathbf{k}, \\ [C]\_{DNA} &:= [\mathbf{0}] + [\mathbf{0}]\mathbf{i} + \left[e^{i\boldsymbol{\pi}\boldsymbol{\beta}\_i}\right] \mathbf{j} + [\mathbf{0}]\mathbf{k}, \\ [G]\_{DNA} &:= [\mathbf{0}] + [\mathbf{0}]\mathbf{i} + [\mathbf{0}]\mathbf{j} + \left[e^{i2\boldsymbol{\pi}\boldsymbol{\beta}\_i}\right] \mathbf{k}. \end{aligned} \tag{29}$$

In the retroviral RNA of the observed state space *Yt,RNA* is a span by gene *α<sup>i</sup>* with the anti-ghost field Φ*i,*þð Þ *Yt,RNA* of viral particle. We define a pair of ghost and antighost field genes by a middle hidden state in mRNA and ribosomal EPA state in codon and anticodon state as the ghost and anti-ghost fields in the genetic code. One can define a mutual genetic code as passive or dual hidden states ½ � *<sup>s</sup>*<sup>1</sup> <sup>∗</sup> *,* …*, s*½ � <sup>8</sup> <sup>∗</sup> and active eight states ½ � *s*<sup>1</sup> *,* …*, s*½ � <sup>8</sup> for the spinor field in the genetic code by

$$\begin{aligned} [A]\_{tRNA} &:= [NU]\_{mRNA} = \left[e^{\frac{jna\_i}{2}}\right] + [\mathbf{0}]\mathbf{i} + [\mathbf{0}]\mathbf{j} + [\mathbf{0}]\mathbf{k}, \\ [U]\_{tRNA} &:= [NA]\_{mRNA} := [\mathbf{0}] + \left[e^{\frac{-jna\_i}{2}}\right] \mathbf{i} + [\mathbf{0}]\mathbf{j} + [\mathbf{0}]\mathbf{k}, \\ [C]\_{tRNA} &:= [NG]\_{mRNA} := [\mathbf{0}] + [\mathbf{0}]\mathbf{i} + \left[e^{jna\_i}\right] \mathbf{j} + [\mathbf{0}]\mathbf{k}, \\ [G]\_{tRNA} &:= [NC]\_{mRNA} := [\mathbf{0}] + [\mathbf{0}]\mathbf{i} + [\mathbf{0}]\mathbf{j} + \left[e^{j2na\_i}\right] \mathbf{k}. \end{aligned} \tag{30}$$

The reversed transcription process of the gene expression is defined by a moduli state space model of a coupling spinor field between the gene of a viral particle and the host cell (**Figure 3**)

$$\mathbb{HP}^1 = X\_t / Y\_t \oplus \left[ \mathbf{1}, \frac{e^{2\xi\_t^{\mathrm{q}\_{\mathrm{q}}}}}{e^{2\xi\_{\mathrm{q}} \mathrm{q}}} \right]\_{m, n = 1, 2, 3, 4} = \left[ \mathbf{1}, \frac{q \,|\_{\mathrm{DNA}}}{q^\* \,|\_{\mathrm{RNA}}} \right] = \left[ \frac{q \,|\_{\mathrm{RNA}}}{q^\* \,|\_{\mathrm{RNA}}}, \mathbf{1} \right]. \tag{31}$$

One can define *Sp*ð Þ!<sup>1</sup> *<sup>S</sup>*<sup>7</sup> ! <sup>H</sup>*P*<sup>1</sup> as a Hopf fibration of eight states of the genetic code ½ � *<sup>s</sup>*<sup>1</sup> *, s*½ � <sup>2</sup> *,* …½ � *<sup>s</sup>*<sup>8</sup> <sup>∈</sup>*S*<sup>7</sup> <sup>¼</sup> *Tp*M, denoted by ½ � *<sup>s</sup>*<sup>1</sup> <sup>∗</sup> *, s*½ � <sup>2</sup> <sup>∗</sup> *,* …½ � *<sup>s</sup>*<sup>8</sup> <sup>∗</sup> <sup>∈</sup>*<sup>T</sup>* <sup>∗</sup> *<sup>p</sup>* M states of the genetic code of the space of a viral RNA *Xt* and a space of host cell DNA, *Yt*.

If <sup>U</sup>½ � *<sup>A</sup> <sup>α</sup>* <sup>⊂</sup> <sup>H</sup>*P*<sup>1</sup> is a chart of local coordinate in a manifold of genetic code over *Xt=Yt*, where ½ � *A <sup>α</sup>* is defined over the right-hand isomer genetic code f g *A; T;C; G* (for the simplicity we use a symbol G also for U) with their dual ½ � *<sup>A</sup>* <sup>∗</sup> *<sup>α</sup>* , with the mirror symmetry of a genetic code f g *NA; NT; NC; NG* . We have a cycle and a cocycle of an orbifold as a trivialization over the tangent of the living organism manifold, so-called codon and anticodon U*<sup>i</sup>* ∩ U*<sup>j</sup>* ∩ U*k*. Let ð Þ M*; g* be a living organism manifold with <sup>M</sup> <sup>¼</sup> <sup>H</sup>*P*<sup>1</sup> for a living organism with the Riemannian metric tensor *gij* <sup>¼</sup> <sup>&</sup>lt;*T*½ � *<sup>A</sup>* ½ � *<sup>A</sup> <sup>α</sup>* M*, T* <sup>∗</sup> ½ � *A <sup>α</sup>* M> over a tangent manifold and a cotangent manifold

#### **Figure 3.**

that one main problem in this field is how can we predict the genetic variation and the gene structure in the viral particle and other organisms, or in the context of new representation, the question is how we can explain the intuition behind a definition of new time series data of gene, e.g., involved in the Batalin-Vilkovisky cohomology of DNA and the viral gene structure. The Chern-Simons current and the anomaly over a superspace of cell membrane can be applied to diagnose new gene diseases, the cloning technology or the gene therapy in medicine. Moreover, a presented method can be improved also in view of describing a useless trash area of DNA,

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

On the other side, another approach based on the usage of a spinor field in the Kolmogorov space of the time series data [24] over the genetic code can represent the gene structure as the ghost and anti-ghost fields of the codon and anticodon. This can be achieved in the frameworks of supersymmetry [25] and G-theory [22]. Results of the works show that all calculations over the codon can be assumed as a new superspace of the time series representation of the gene structure [24].

In [26–28], we introduced a new representation of the genetic code in the time series using a modeling by strings and D-branes. By applying a spinor field to a superspace in time series data [29], the method allows us to develop supersymmetry for living organisms. In particular, it is possible to control the anomalies in the codon and anticodon ghost fields and construct an algebraic approach for the trash DNA. The "gravitational" analogy of the Chern-Simons currents in a gravitational physics, emanating from a system of DNA-RNA transcriptions, could have interesting counterparts also in biology. A representation of codons in human genome, derived from the Chern-Simons currents, can be useful in biology to explain the source of connections over protein-docking states. In this perspective, adopting cohomology in biology can be useful as a new modeling tool for plotting genes with spinor field in time series data. Especially, the junk area of DNA, with repeated inactive genes, can be represented by the Chern-Simons currents with extended

Further we discuss the role of spinors in the time series of genetic code. We can denote <sup>H</sup>*P*<sup>1</sup> as a quaternionic projective space and *<sup>H</sup>*0ð Þ *Xt* as a pointed space of DNA alphabet sequence with ½ � *A* , ½ � *T* , ½ � *C* , and ½ � *G* as an equivalent class of ½ � *A , T*½ �*, C*½ �*, G*½ �∈ *H*0ð Þ *Xt,DNA* ≔ Φ*i*ð Þ *Xt* a ghost field with parity two with

*H*0ð Þ¼ *Xt H*<sup>0</sup> *Xt* ð Þ *;* ∗ where ∗ ¼ f g ∗ is a pointed space. We define an equivalent class of DNA–RNA translation processes by using the notation of a master equation

f g *DNA; RNA* . The whole state space model of the viral replication cycle, embedded in the host cell, is denoted by *Xt,DNA=Yt,RNA* <sup>¼</sup> *Zt,GENE* <sup>¼</sup> <sup>H</sup>*=*<sup>H</sup> <sup>¼</sup> <sup>H</sup>*P*<sup>1</sup> as a moduli state space model with the definition of genetic code as an equivalent class of the map *α<sup>t</sup>* : *Xt,DNA* ! f g ½ � *A ;* ½ � *T ;* ½ � *C ;* ½ � *G* ⊂ H; the host cell gene alphabet is defined by a

> �*iπβ<sup>i</sup>* <sup>2</sup> h i

In the retroviral RNA of the observed state space *Yt,RNA* is a span by gene *α<sup>i</sup>* with the anti-ghost field Φ*i,*þð Þ *Yt,RNA* of viral particle. We define a pair of ghost and antighost field genes by a middle hidden state in mRNA and ribosomal EPA state in codon and anticodon state as the ghost and anti-ghost fields in the genetic code. One

½ � *<sup>G</sup> DNA* <sup>≔</sup> ½ �þ <sup>0</sup> ½ � <sup>0</sup> **<sup>i</sup>** <sup>þ</sup> ½ � <sup>0</sup> **<sup>j</sup>** <sup>þ</sup> *<sup>e</sup><sup>i</sup>*2*πβ<sup>i</sup>*

þ ½ � 0 **i** þ ½ � 0 **j** þ ½ � 0 **k***,*

**i** þ ½ � 0 **j** þ ½ � 0 **k***,*

� �**<sup>j</sup>** <sup>þ</sup> ½ � <sup>0</sup> **<sup>k</sup>***,*

� �**k***:*

� � <sup>¼</sup>

(29)

for an interaction between the viral RNA and the host cell DNA by *xt; yt*

*iπβi* 2 h i

½ � *<sup>C</sup> DNA* <sup>≔</sup> ½ �þ <sup>0</sup> ½ � <sup>0</sup> **<sup>i</sup>** <sup>þ</sup> *<sup>e</sup><sup>i</sup>πβ<sup>i</sup>*

which is considered as unknown part of human genome.

structures of knot states in a Laurent polynomial of knots.

½ � *A DNA* ≔ *e*

½ � *T DNA* ≔ ½ �þ 0 *e*

hidden state space *Xt* with the gene *β<sup>i</sup>*

**102**

*(a) The site of docking between CCR5 Δ*32 *co-receptor in host T-cell membrane and V3 loop in HIV viral gp120. The cell membrane has a mirror symmetry structure of D-brane for outer layer. The anti-D-brane structure of the cell membrane is an inner layer of phospholipid. (b) The knot 4*<sup>1</sup> *model of a short exact sequence of hidden eight states in genetic code.*

*T*½ � *<sup>A</sup> <sup>α</sup>* <sup>M</sup> <sup>¼</sup> *<sup>S</sup>*<sup>7</sup> with *gij* defined as a tensor behavior field transformation between 64 states in a codon as the distance in a space of the genetic code with *i, j* ¼ 1*;* 2*;* 3*,* …8. The active states in genetic code are denoted as ½ � *s*<sup>1</sup> *, s*½ � <sup>2</sup> *,* ⋯½ � *s*<sup>8</sup> , and the hidden states inside eight states or passive states are denoted as *s* <sup>∗</sup> 1 � �*, s* <sup>∗</sup> 1 � �*,* ⋯ *s* <sup>∗</sup> 8 � �. The dual part of a genotype *s* <sup>∗</sup> *i* � � is defined by a tangent of a manifold of the genetic code with the Jacobian *J* ¼ ffiffiffiffi *gij* p , where the metrics is *gij* ¼ < *si, sj* � � <sup>∗</sup> > � . A smallest state in gene is defined as a pair of a genetic code by a classical notation *A* � *T*, *C* � *G* with the coordinates ð Þ *A; T* , ð Þ *C; G* . One can define a superstate in a pair of genes as ghost and anti-ghost fields in the genetic code with the supersymmetry of D-isomer to L-isomer from the right-hand D-state in the light of the polarization in a nucleic acid as *si* ½ �j<sup>Φ</sup>*<sup>i</sup>* to a left-hand light L-state of an isomer of a light polarization, denoted as *si* ½ �j <sup>∗</sup> Φþ *i*

For an equilibrium state of evolution of an organism, we have no change of

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology*

*Sp*ð Þ!<sup>1</sup> *<sup>S</sup>*<sup>7</sup> ! <sup>H</sup>*P*<sup>1</sup> of a genetic code over a supermanifold of a viral particle <sup>A</sup>. We can use an exact sequence of the sheave cohomology O*Xt* with the chart over the supermanifold defined by homogeneous coordinates of H*P*<sup>1</sup> for viral gene *Xt* along the host cell gene O*Yt* , while the virus attachment to the host cell is defined by the

Mills field of a behavioral field of the genetic code as a connection over ð Þ *A<sup>α</sup>* ∈f½ � *A* ,

with an anti self dual field over the gene of the host cell *DNA*, ∗ *Fμν*. A current *J* of the connection between the fields is defined by the Chern-Pontryagin density for the interaction of behavioral fields. The current varies from the curvature of docking between the behavior of the curvature over amino acid *k* in *Xt* and its dual curvature in *Yt* while docking <sup>&</sup>lt;*Fμν* <sup>∗</sup> *<sup>F</sup>μν*>, where <sup>&</sup>lt; � <sup>&</sup>gt; is an average or an

*<sup>ν</sup>* � *<sup>∂</sup>β*ð Þ *<sup>A</sup><sup>α</sup> <sup>μ</sup>*

*<sup>ν</sup>* ≔ Γ*<sup>μ</sup>*

pling between two alphabets of two organisms, i.e., from DNA and viral RNA. The above curvature is also useful in other situations like t-RNA docking with DNA in interaction between two D-branes of DNA and RNA. In a gauge field theory of DNA and RNA genetic codes of translation process, it is the group action of Liealgebra one form. We have an adjoint representation of the genetic code as a translation process over the codon and anticodon of *t* � *RNA*. One needs to define a new value to measure the curvature in amino acids of protein structure, not in the tangent space of a genetic code, and one needs to define a curvature over the codon and anticodon to represent a curvature of proteins while they are docking. The new value needs to be unique for all states in the codon and to have a meaning of curvature of genetic code with connection over manifold of genetic code. To represent a spectrum of genetic variation as curvature in protein structure while docking, one can introduce a new quantity, so-called Chern-Simons current

One can translate a genetic code in a codon in three steps. The translation

supermanifold of living organism with action in three times. It generates a codon representation as an adjoint representation over gene expression, and it is a precise definition of genetic code with parity two of ghost field and anti-ghost field in the Chern-Simons current for the representation of a gene *Ai* with the current density

Let a knot serve as a representation of anticodon in t-RNA topological structure

translation as group action of transcription process; then the genetic code is an average expectation value of Wilson loop operator of coupling between hidden state of *xt* and *yt* twist D-brane and anti-D-brane over superspace of cell membrane, i.e.,

*<sup>μ</sup>* as a representation *R* of gauge group *G* of gene geometric

operator of group is given by a behavior matrix in Lie group, a group of

*ναβ* is the curvature over a tangent space of a genetic code

½ � *<sup>T</sup>* , ½ � *<sup>C</sup>* , ½ �g *<sup>G</sup>* <sup>⊂</sup> <sup>H</sup>*P*<sup>1</sup> with the Hopf fibration of viral RNA gene *<sup>F</sup>μν* <sup>≔</sup> ð Þ *<sup>A</sup><sup>α</sup> <sup>μ</sup>*

∇*<sup>g</sup> g* ¼ 0*:* (34)

*<sup>ν</sup>* <sup>¼</sup> <sup>Γ</sup>*<sup>μ</sup> αν*

*<sup>ν</sup> ,* (35)

*αν* <sup>¼</sup> *<sup>F</sup>μν* is the connection of the cou-

*<sup>μ</sup>* as a connection on a principle bundle

. We apply a supersymmetry AdS theory over a Yang-

*<sup>ν</sup>* þ *Aα; A<sup>β</sup>* � �*<sup>μ</sup>*

covariant derivative for the tensor field *g*

*DOI: http://dx.doi.org/10.5772/intechopen.88486*

We can define a *<sup>F</sup>μν* <sup>≔</sup> ð Þ *<sup>A</sup><sup>α</sup> <sup>ν</sup>*

coordinate in sheave O*Xt=Yt*

expectation operator.

where *R<sup>μ</sup>*

for biology.

for amino acid *μ* with *J*

*J Ai* <sup>¼</sup> <sup>Ð</sup>*<sup>t</sup> n <sup>t</sup>*<sup>1</sup> *dJAi* .

**105**

The connection between genes is

ð Þ *<sup>A</sup><sup>α</sup>* <sup>∈</sup> f g ½ � *<sup>A</sup> ;* ½ � *<sup>T</sup> ;* ½ � *<sup>C</sup> ;* ½ � *<sup>G</sup>* <sup>⊂</sup> <sup>H</sup>*P*<sup>1</sup> and ð Þ *<sup>A</sup><sup>α</sup> <sup>μ</sup>*

*Rμ*

*ναβ* <sup>¼</sup> *<sup>∂</sup><sup>α</sup> <sup>A</sup><sup>β</sup>*

� �*<sup>μ</sup>*

$$\begin{aligned} [s\_1] &= ([A], [T]^\*) \in T\_p \mathcal{M}, \qquad [p1] = [s\_1]^\* = [s\_1]^\* = ([A], [T]^\*)^\* \in T\_p^\* \mathcal{M} \\ [s\_2] &= ([A], [N\mathcal{A}]) \in T\_p \mathcal{M}, \qquad [s\_2]^\* = ([A], [N\mathcal{A}])^\* \in T\_p^\* \mathcal{M} \\ [s\_3] &= ([C], [G]^\*) \in T\_p \mathcal{M}, \qquad [s\_3]^\* = ([C], [G]^\*)^\* \in T\_p^\* \mathcal{M} \\ [s\_4] &= ([C], [N\mathcal{C}]) \in T\_p \mathcal{M}, \qquad [s\_4]^\* = ([C], [N\mathcal{C}])^\* \in T\_p^\* \mathcal{M} \\ [s\_5] &= ([T], [T]^\*) \in T\_p \mathcal{M}, \qquad [s\_5]^\* = ([T], [T]^\*)^\* \in T\_p^\* \mathcal{M} \\ [s\_6] &= ([T], [N\mathcal{A}]) \in T\_p \mathcal{M}, \qquad [s\_6]^\* = ([T], [N\mathcal{A}])^\* \in T\_p^\* \mathcal{M} \\ [s\_7] &= ([G], [G]^\*)^\* \in T\_p \mathcal{M}, \qquad [s\_7]^\* = ([G], [G])^\* \in T\_p \mathcal{M} \\ [s\_8] &= ([G], [N\mathcal{C}]) \in T\_p \mathcal{M}, \qquad [s\_8]^\* = ([G], [N\mathcal{C}])^\* \in T\_p \mathcal{M} \end{aligned} \tag{3.2}$$

As known from biochemistry, there exist only ½ �¼ *<sup>s</sup>*<sup>1</sup> ½ � *<sup>A</sup> ;* ½ � *<sup>T</sup>* <sup>∗</sup> ð Þ and ½ �¼ *<sup>s</sup>*<sup>3</sup> ½ � *<sup>C</sup> ;* ½ � *<sup>G</sup>* <sup>∗</sup> ð Þ observational states of the living organism in nature. However, in the theory of supersymmetry of *S*<sup>7</sup> Hopf fibration, there exist eight states of the ghost fields with six hidden states in the mirror symmetry. In each state, an orbifold of the living organism, other eight states denoted *p*1–*p*<sup>8</sup> exist. One can define all 64 states with the help of a notation for the Riemann tensor field *gij* ¼ < *si* ½ �*, sj* � � <sup>∗</sup> >, e.g., *<sup>g</sup>*<sup>11</sup> <sup>¼</sup> <sup>&</sup>lt; ½ � *<sup>s</sup>*<sup>1</sup> *, s*½ � <sup>1</sup> <sup>∗</sup> <sup>&</sup>gt;, and we denote a pair of states in a gene as ½ � *<sup>s</sup>*<sup>1</sup> *; <sup>p</sup>*<sup>1</sup> � � � � . It is a pair of the genetic code *<sup>g</sup>*<sup>11</sup> <sup>¼</sup> <sup>&</sup>lt; ½ � *<sup>A</sup> ;* ½ � *<sup>T</sup>* <sup>∗</sup> ð Þ*, A*½ �*; <sup>T</sup>*<sup>∗</sup> ð Þ ½ � <sup>&</sup>gt;; for the case of codon, we replace ½ � *T* with ½ � *U* . Finally, we have for *<sup>g</sup>*<sup>11</sup> <sup>¼</sup> <sup>&</sup>lt; ½ � *<sup>A</sup> , T*½ � <sup>∗</sup> ð � ½ � ½ Þ *, A*½ �*; <sup>T</sup>*<sup>∗</sup> ð Þ ½ � <sup>&</sup>gt; <sup>¼</sup> <sup>&</sup>lt; ½ � *<sup>A</sup> , A*½ �*; <sup>T</sup>*<sup>∗</sup> <sup>ð</sup> ð Þ ½ � <sup>&</sup>gt; since the states *<sup>T</sup>*<sup>∗</sup> ½ � are

hidden states, so one has a codon AAU for *g*11.

If Γ*<sup>k</sup> ij* <sup>≔</sup> ½ � *<sup>A</sup> <sup>k</sup>* is a connection over a tangent of a manifold of *Xt=Yt* with <sup>M</sup> <sup>¼</sup> <sup>H</sup>*P*<sup>1</sup> of the genetic code *<sup>k</sup>*∈f g *<sup>A</sup>; <sup>T</sup>;C; <sup>G</sup>* , one can denote <sup>Γ</sup>*<sup>k</sup> μν* ≔ *Fμν* as the behavioral Yang-Mills field with its dual in AdS theory of supersymmetry with <sup>∗</sup> *<sup>F</sup>μν* <sup>¼</sup> *<sup>F</sup>μν*. It is a behavior of a protein folding inducing a curvature between a viral glycoprotein and a host cell receptor. The behavior of a Yang-Mills field is an interaction field between the behavior of virus and host cells which can survive by a change of curvature of the protein during the evolution. It is a connection in the sense of an evolutional field not in the sense of traditional gravitational field as usual

$$
\Gamma^k\_{ij} = \frac{1}{2} \mathbf{g}^{kl} \left( \partial\_{\bar{k}} \mathbf{g}\_{jk} + \partial\_{\bar{k}} \mathbf{g}\_{jk} - \partial\_{\bar{k}} \mathbf{g}\_{jk} \right), \tag{33}
$$

where *gij* ¼ < *si* ½ �*, sj* � � <sup>∗</sup> >. *Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology DOI: http://dx.doi.org/10.5772/intechopen.88486*

For an equilibrium state of evolution of an organism, we have no change of covariant derivative for the tensor field *g*

$$\nabla\_{\mathbf{g}} \mathbf{g} = \mathbf{0}.\tag{34}$$

We can define a *<sup>F</sup>μν* <sup>≔</sup> ð Þ *<sup>A</sup><sup>α</sup> <sup>ν</sup> <sup>μ</sup>* as a connection on a principle bundle *Sp*ð Þ!<sup>1</sup> *<sup>S</sup>*<sup>7</sup> ! <sup>H</sup>*P*<sup>1</sup> of a genetic code over a supermanifold of a viral particle <sup>A</sup>. We can use an exact sequence of the sheave cohomology O*Xt* with the chart over the supermanifold defined by homogeneous coordinates of H*P*<sup>1</sup> for viral gene *Xt* along the host cell gene O*Yt* , while the virus attachment to the host cell is defined by the coordinate in sheave O*Xt=Yt* . We apply a supersymmetry AdS theory over a Yang-Mills field of a behavioral field of the genetic code as a connection over ð Þ *A<sup>α</sup>* ∈f½ � *A* , ½ � *<sup>T</sup>* , ½ � *<sup>C</sup>* , ½ �g *<sup>G</sup>* <sup>⊂</sup> <sup>H</sup>*P*<sup>1</sup> with the Hopf fibration of viral RNA gene *<sup>F</sup>μν* <sup>≔</sup> ð Þ *<sup>A</sup><sup>α</sup> <sup>μ</sup> <sup>ν</sup>* <sup>¼</sup> <sup>Γ</sup>*<sup>μ</sup> αν* with an anti self dual field over the gene of the host cell *DNA*, ∗ *Fμν*. A current *J* of the connection between the fields is defined by the Chern-Pontryagin density for the interaction of behavioral fields. The current varies from the curvature of docking between the behavior of the curvature over amino acid *k* in *Xt* and its dual curvature in *Yt* while docking <sup>&</sup>lt;*Fμν* <sup>∗</sup> *<sup>F</sup>μν*>, where <sup>&</sup>lt; � <sup>&</sup>gt; is an average or an expectation operator.

The connection between genes is

*T*½ � *<sup>A</sup> <sup>α</sup>*

a genotype *s* <sup>∗</sup>

as *si* ½ �j <sup>∗</sup> Φþ *i*

Jacobian *J* ¼ ffiffiffiffi

*i*

*gij*

<sup>M</sup> <sup>¼</sup> *<sup>S</sup>*<sup>7</sup> with *gij* defined as a tensor behavior field transformation between 64 states in a codon as the distance in a space of the genetic code with *i, j* ¼ 1*;* 2*;* 3*,* …8. The active states in genetic code are denoted as ½ � *s*<sup>1</sup> *, s*½ � <sup>2</sup> *,* ⋯½ � *s*<sup>8</sup> , and the hidden states

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

defined as a pair of a genetic code by a classical notation *A* � *T*, *C* � *G* with the coordinates ð Þ *A; T* , ð Þ *C; G* . One can define a superstate in a pair of genes as ghost and anti-ghost fields in the genetic code with the supersymmetry of D-isomer to L-isomer from the right-hand D-state in the light of the polarization in a nucleic acid as *si* ½ �j<sup>Φ</sup>*<sup>i</sup>* to a left-hand light L-state of an isomer of a light polarization, denoted

½ �¼ *<sup>s</sup>*<sup>1</sup> ½ � *<sup>A</sup> ;* ½ � *<sup>T</sup>* <sup>∗</sup> ð Þ<sup>∈</sup> *Tp*M*, p*½ �¼ <sup>1</sup> ½ � *<sup>s</sup>*<sup>1</sup> <sup>∗</sup> <sup>¼</sup> ½ � *<sup>s</sup>*<sup>11</sup> <sup>∗</sup> <sup>¼</sup> ½ � *<sup>A</sup> ;* ½ � *<sup>T</sup>* <sup>∗</sup> ð Þ <sup>∗</sup> <sup>∈</sup>*<sup>T</sup>* <sup>∗</sup>

½ �¼ *<sup>s</sup>*<sup>2</sup> ð Þ ½ � *<sup>A</sup> ;* ½ � *NA* <sup>∈</sup>*Tp*M*, s*½ � <sup>2</sup> <sup>∗</sup> <sup>¼</sup> ð Þ ½ � *<sup>A</sup> ;* ½ � *NA* <sup>∗</sup> <sup>∈</sup>*T*<sup>∗</sup>

½ �¼ *<sup>s</sup>*<sup>3</sup> ½ � *<sup>C</sup> ;* ½ � *<sup>G</sup>* <sup>∗</sup> ð Þ∈*Tp*M*, s*½ � <sup>3</sup> <sup>∗</sup> <sup>¼</sup> ½ � *<sup>C</sup> ;* ½ � *<sup>G</sup>* <sup>∗</sup> ð Þ <sup>∗</sup> <sup>∈</sup> *<sup>T</sup>*<sup>∗</sup>

½ �¼ *<sup>s</sup>*<sup>4</sup> ð Þ ½ � *<sup>C</sup> ;* ½ � *NC* <sup>∈</sup> *Tp*M*, s*½ � <sup>4</sup> <sup>∗</sup> <sup>¼</sup> ð Þ ½ � *<sup>C</sup> ;* ½ � *NC* <sup>∗</sup> <sup>∈</sup> *<sup>T</sup>*<sup>∗</sup>

½ �¼ *<sup>s</sup>*<sup>5</sup> ½ � *<sup>T</sup> ;* ½ � *<sup>T</sup>* <sup>∗</sup> ð Þ∈*Tp*M*, s*½ � <sup>5</sup> <sup>∗</sup> <sup>¼</sup> ½ � *<sup>T</sup> ;* ½ � *<sup>T</sup>* <sup>∗</sup> ð Þ <sup>∗</sup> <sup>∈</sup>*T*<sup>∗</sup>

½ �¼ *<sup>s</sup>*<sup>6</sup> ð Þ ½ � *<sup>T</sup> ;* ½ � *NA* <sup>∈</sup>*Tp*M*, s*½ � <sup>6</sup> <sup>∗</sup> <sup>¼</sup> ð Þ ½ � *<sup>T</sup> ;* ½ � *NA* <sup>∗</sup> <sup>∈</sup>*T*<sup>∗</sup>

½ �¼ *<sup>s</sup>*<sup>7</sup> ½ � *<sup>G</sup> ;* ½ � *<sup>G</sup>* <sup>∗</sup> ð Þ∈*Tp*M*, s*½ � <sup>7</sup> <sup>∗</sup> <sup>¼</sup> ½ � *<sup>G</sup> ;* ½ � *<sup>G</sup>* <sup>∗</sup> ð Þ <sup>∗</sup> <sup>∈</sup>*Tp*<sup>M</sup> ½ �¼ *<sup>s</sup>*<sup>8</sup> ð Þ ½ � *<sup>G</sup> ;* ½ � *NC* <sup>∈</sup> *Tp*M*, s*½ � <sup>8</sup> <sup>∗</sup> <sup>¼</sup> ð Þ ½ � *<sup>G</sup> ;* ½ � *NC* <sup>∗</sup> <sup>∈</sup> *Tp*<sup>M</sup>

As known from biochemistry, there exist only ½ �¼ *<sup>s</sup>*<sup>1</sup> ½ � *<sup>A</sup> ;* ½ � *<sup>T</sup>* <sup>∗</sup> ð Þ and

states with the help of a notation for the Riemann tensor field *gij* ¼ < *si* ½ �*, sj*

e.g., *<sup>g</sup>*<sup>11</sup> <sup>¼</sup> <sup>&</sup>lt; ½ � *<sup>s</sup>*<sup>1</sup> *, s*½ � <sup>1</sup> <sup>∗</sup> <sup>&</sup>gt;, and we denote a pair of states in a gene as ½ � *<sup>s</sup>*<sup>1</sup> *; <sup>p</sup>*<sup>1</sup>

evolutional field not in the sense of traditional gravitational field as usual

replace ½ � *T* with ½ � *U* . Finally, we have for

If Γ*<sup>k</sup>*

**104**

where *gij* ¼ < *si* ½ �*, sj*

hidden states, so one has a codon AAU for *g*11.

of the genetic code *<sup>k</sup>*∈f g *<sup>A</sup>; <sup>T</sup>;C; <sup>G</sup>* , one can denote <sup>Γ</sup>*<sup>k</sup>*

Γ*k ij* <sup>¼</sup> <sup>1</sup> 2

� � <sup>∗</sup> >.

pair of the genetic code *<sup>g</sup>*<sup>11</sup> <sup>¼</sup> <sup>&</sup>lt; ½ � *<sup>A</sup> ;* ½ � *<sup>T</sup>* <sup>∗</sup> ð Þ*, A*½ �*; <sup>T</sup>*<sup>∗</sup> ð Þ ½ � <sup>&</sup>gt;; for the case of codon, we

*<sup>g</sup>*<sup>11</sup> <sup>¼</sup> <sup>&</sup>lt; ½ � *<sup>A</sup> , T*½ � <sup>∗</sup> ð � ½ � ½ Þ *, A*½ �*; <sup>T</sup>*<sup>∗</sup> ð Þ ½ � <sup>&</sup>gt; <sup>¼</sup> <sup>&</sup>lt; ½ � *<sup>A</sup> , A*½ �*; <sup>T</sup>*<sup>∗</sup> <sup>ð</sup> ð Þ ½ � <sup>&</sup>gt; since the states *<sup>T</sup>*<sup>∗</sup> ½ � are

Yang-Mills field with its dual in AdS theory of supersymmetry with <sup>∗</sup> *<sup>F</sup>μν* <sup>¼</sup> *<sup>F</sup>μν*. It is a behavior of a protein folding inducing a curvature between a viral glycoprotein and a host cell receptor. The behavior of a Yang-Mills field is an interaction field between the behavior of virus and host cells which can survive by a change of curvature of the protein during the evolution. It is a connection in the sense of an

> *<sup>g</sup>kl <sup>∂</sup>jgjk* <sup>þ</sup> *<sup>∂</sup>igjk* � *<sup>∂</sup>kgjk* � �

*ij* <sup>≔</sup> ½ � *<sup>A</sup> <sup>k</sup>* is a connection over a tangent of a manifold of *Xt=Yt* with <sup>M</sup> <sup>¼</sup> <sup>H</sup>*P*<sup>1</sup>

½ �¼ *<sup>s</sup>*<sup>3</sup> ½ � *<sup>C</sup> ;* ½ � *<sup>G</sup>* <sup>∗</sup> ð Þ observational states of the living organism in nature. However, in the theory of supersymmetry of *S*<sup>7</sup> Hopf fibration, there exist eight states of the ghost fields with six hidden states in the mirror symmetry. In each state, an orbifold of the living organism, other eight states denoted *p*1–*p*<sup>8</sup> exist. One can define all 64

� � is defined by a tangent of a manifold of the genetic code with the

1 � �*, s* <sup>∗</sup> 1 � �*,* ⋯ *s* <sup>∗</sup> 8

� � <sup>∗</sup>

� �. The dual part of

*<sup>p</sup>* M

(32)

� � <sup>∗</sup> >,

� � � � . It is a

*μν* ≔ *Fμν* as the behavioral

*,* (33)

> � . A smallest state in gene is

*<sup>p</sup>* M

*<sup>p</sup>* M

*<sup>p</sup>* M

*<sup>p</sup>* M

*<sup>p</sup>* M

inside eight states or passive states are denoted as *s* <sup>∗</sup>

p , where the metrics is *gij* ¼ < *si, sj*

$$R^{\mu}\_{\nu a \beta} = \partial\_a \left( A\_{\beta} \right)^{\mu}\_{\nu} - \partial\_{\beta} (A\_a)^{\mu}\_{\nu} + \left[ A\_a, A\_{\beta} \right]^{\mu}\_{\nu} \tag{35}$$

where *R<sup>μ</sup> ναβ* is the curvature over a tangent space of a genetic code ð Þ *<sup>A</sup><sup>α</sup>* <sup>∈</sup> f g ½ � *<sup>A</sup> ;* ½ � *<sup>T</sup> ;* ½ � *<sup>C</sup> ;* ½ � *<sup>G</sup>* <sup>⊂</sup> <sup>H</sup>*P*<sup>1</sup> and ð Þ *<sup>A</sup><sup>α</sup> <sup>μ</sup> <sup>ν</sup>* ≔ Γ*<sup>μ</sup> αν* <sup>¼</sup> *<sup>F</sup>μν* is the connection of the coupling between two alphabets of two organisms, i.e., from DNA and viral RNA. The above curvature is also useful in other situations like t-RNA docking with DNA in interaction between two D-branes of DNA and RNA. In a gauge field theory of DNA and RNA genetic codes of translation process, it is the group action of Liealgebra one form. We have an adjoint representation of the genetic code as a translation process over the codon and anticodon of *t* � *RNA*. One needs to define a new value to measure the curvature in amino acids of protein structure, not in the tangent space of a genetic code, and one needs to define a curvature over the codon and anticodon to represent a curvature of proteins while they are docking. The new value needs to be unique for all states in the codon and to have a meaning of curvature of genetic code with connection over manifold of genetic code. To represent a spectrum of genetic variation as curvature in protein structure while docking, one can introduce a new quantity, so-called Chern-Simons current for biology.

One can translate a genetic code in a codon in three steps. The translation operator of group is given by a behavior matrix in Lie group, a group of supermanifold of living organism with action in three times. It generates a codon representation as an adjoint representation over gene expression, and it is a precise definition of genetic code with parity two of ghost field and anti-ghost field in the Chern-Simons current for the representation of a gene *Ai* with the current density *J Ai* <sup>¼</sup> <sup>Ð</sup>*<sup>t</sup> n <sup>t</sup>*<sup>1</sup> *dJAi* .

Let a knot serve as a representation of anticodon in t-RNA topological structure for amino acid *μ* with *J <sup>μ</sup>* as a representation *R* of gauge group *G* of gene geometric translation as group action of transcription process; then the genetic code is an average expectation value of Wilson loop operator of coupling between hidden state of *xt* and *yt* twist D-brane and anti-D-brane over superspace of cell membrane, i.e.,

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

$$\mathcal{W}(K,R) = \operatorname{Tr}\_{\mathbb{R}} P \exp\left(\oint\_{K} A\right). \tag{36}$$

*AdA* ¼ *A*2ð*A*2*A*<sup>2</sup> � *A*2*A*3Þ ¼ *A*2*A*2*A*<sup>2</sup> � *A*2*A*2*A*3. The minus sign represents a linear

*AαAβAγ*>

*U* ∧ *U* ∧ *C*

For a translation in reversed direction of antigen shift and drift in gene evolution theory, we can use the definition of the group action of reversed direction of time by the *CPT* theory for anti-ghost field in field of time series of antibody gene as

> *g<sup>i</sup> xt=yt* � �

Then a numerical representation for spinor field of curvature in the gene

Tr *A* ∧ *dA* þ

ffiffiffiffiffiffiffiffiffiffiffi 2 *k* þ 2

*t* � � <sup>¼</sup>

> 2 3

*<sup>k</sup>* <sup>þ</sup> <sup>2</sup> <sup>¼</sup>

sin *<sup>π</sup>*

The derivation of the Chern-Simons current can be done by a simple algorithm [30], i.e., the Chern-Simons current maps the string of genetic code into numerical values by explicit formulas. It can be used to plot the time series data directly into the superspace of gene expression. We transform the alphabet string values, which cannot be computed in the classical standard definition of genetic code, into the Chern-Simons current of time series data of genetic code with *k* ¼ 1*;* 2*;* 3*,* …64 over spinor field with ground field of real values. We think that the approach is more

In each cell division, the telomeres are shortened [31], and total length of DNA is

changing. As the result of shorter biological clock from cell division, the living things die. In order to understand cell cocycle and division mechanism of telomerase aging, one can explain the source of cancer as a source of age acceleration and its relationship to telomere shortening mechanism. It is a source of braid group operation [32] so-called self-diffeomorphism in the genetic code. The age acceleration is a relative measurement between the chronical clock and the biological clock in telomere. Up to now, scientists understand that a telomere and telomerase are the locations of ancient viruses that rely on DNA in the chromosomes of living organisms. Telomere is composed of the repeated sequence of ð Þ *TTAGGG dt* <sup>∗</sup> where

¼ ð Tr *H*<sup>3</sup> ð Þ <sup>M</sup> � �*:*

ð

*A* ∧ *A* ∧ *A* � � (44)

ð

Tr*H*<sup>3</sup> <sup>M</sup>*; <sup>g</sup>; <sup>F</sup>μν* ð Þ*:*

*D A*½ �*SCS :* (45)

(42)

(43)

combination of basis for codon. Therefore it follows

*<sup>A</sup>α∂βA<sup>γ</sup>* <sup>þ</sup>

<sup>6</sup> *<sup>U</sup>* <sup>∧</sup> *<sup>U</sup>* <sup>∧</sup> *<sup>U</sup>* � <sup>1</sup>

ð Þ <sup>M</sup> � � <sup>¼</sup> <sup>X</sup>

expression by the Chern-Simons action is defined as follows:

*<sup>k</sup>* ≃

suitable for computational programs used in data analyses.

**3. Circular Artin braid group representation for spinor field**

r

*SCS* <sup>¼</sup> *<sup>k</sup>* 4*π* ð

Amino <sup>¼</sup> *<sup>J</sup>*

1 3

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology*

� �

2

3

*i*¼1

*xt=yt* ! *<sup>β</sup>txt=αtyt* <sup>≃</sup> <sup>ϵ</sup> <sup>∗</sup>

1 2

*DOI: http://dx.doi.org/10.5772/intechopen.88486*

*J*

*Phe* <sup>¼</sup> <sup>ϵ</sup>*μαβα* <sup>&</sup>lt;

ð Tr *H*<sup>3</sup>

¼ *αtyt , g*<sup>3</sup>

where *k* ¼ 1*;* 2*;* 3…*n* are

**in genetic code**

**107**

*J*

¼ ð Tr <sup>5</sup>

*xt; yt* � � <sup>¼</sup>

The above term gives the asymmetric property of chiral molecule of DNA and RNA, twisted from the left hand to the right hand in a supersymmetry breaking as knot polynomial related to the connection *A*. By such a way, we can represent the genetic code as Laurent polynomials in variable *q* with integer coefficients, and for any knot *K* we have

$$J(K, q) = \sum\_{i=1}^{n} a\_n q^n. \tag{37}$$

By using the new parameter of knot *q*

$$q = e^{\frac{2x}{k+h}} \tag{38}$$

one can induce a spinor field for representation of genetic code, where *h* is the dual coexter number for group action of supersymmetry of gene expression *G*. It might be the source of evolution from the adaptive behavior derived from the environment. In the next, we set *h* ¼ 0 in our definition of the Chern-Simons current for biology for the simpler derivation of formulas.

A Chern-Simons current *J <sup>μ</sup>* for anomaly quantum system of codons can be also defined as the spectrum of curvature in the genetic code for gene evolution detection. Under the definition we mean a differential 3-forms in cohomology of spin fiber *S*<sup>3</sup> over the homotopy class *S*<sup>3</sup> *;Xt=Yt* � � in the codon of t-RNA molecule. A path integral of gene expression is defined by the Chern-Simons theory over knots of codon and anticodon: it is defined by the interaction between codon *Ai* and anticodon between DNA and RNA in the form of integral *Ai* <sup>þ</sup> *SCS* <sup>¼</sup> <sup>Ð</sup> D*Ai* exp ð Þ *iSCS*

$$\mathcal{S}\_{\text{CS}} = \frac{k}{4\pi} \int\_{W} \text{Tr}\left(A \wedge dA + \frac{2}{3} A \wedge A \wedge A\right) \tag{39}$$

and

$$\begin{split} J\left(q; K^{A\_i}, R\_i\right) &= \, \, \, \! \! \! \! \left(K\_i, R\_i\right) > \\ &= \, \, \, \mathrm{Tr}\_{\mathbb{R}} P \oint\_{K\_i} A > \, \, \frac{\int \mathcal{D}A\_i \, \exp\left(i \mathcal{S}\_{\mathrm{CS}}\right) \Pi\_i W(K\_i, R\_i)}{\int \mathcal{D}A\_i \, \exp\left(i \mathcal{S}\_{\mathrm{CS}}\right)} . \end{split} \tag{40}$$

The explicit definition of curvature over the connection of genetic code has also new meaning of the genetic spectrum current *J <sup>μ</sup>*, *<sup>μ</sup>* <sup>¼</sup> <sup>1</sup>*,* <sup>2</sup>⋯20 of the Chern-Simons current; it is generated from the representation of Lie group over manifold of a host cell.

An example of our approach can serve as a case of phenylalanine (*Phe*), where UUU and UUC definition is

$$\begin{split} J^{\text{Ph}} &= \varepsilon^{\mu a \beta \alpha} < \frac{1}{2} A\_a \partial\_\beta A\_\gamma + \frac{1}{3} A\_a A\_\beta A\_\gamma > \\ &= \varepsilon^{\mu a \beta \alpha} < \frac{1}{2} (A\_2)\_\nu^{\mu} d(A\_2)\_\nu^{\mu} + \frac{1}{3} (A\_2)\_\nu^{\mu} (A\_2)\_\nu^{\mu} (A\_2)\_\nu^{\mu} > \end{split} \tag{41}$$

where we explicitly define the differential form of genetic code for *Phe* by *dA*<sup>2</sup> ¼ *A*2*A*<sup>2</sup> � *A*2*A*<sup>3</sup> ≔ *UU* � *UC*, so we have

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology DOI: http://dx.doi.org/10.5772/intechopen.88486*

*AdA* ¼ *A*2ð*A*2*A*<sup>2</sup> � *A*2*A*3Þ ¼ *A*2*A*2*A*<sup>2</sup> � *A*2*A*2*A*3. The minus sign represents a linear combination of basis for codon. Therefore it follows

$$\begin{split} \boldsymbol{J}^{\text{Phe}} &= \boldsymbol{\epsilon}^{\mu a \beta \alpha} < \frac{1}{2} \mathbf{A}\_{a} \partial\_{\beta} \mathbf{A}\_{\gamma} + \frac{1}{3} \mathbf{A}\_{a} \mathbf{A}\_{\beta} \mathbf{A}\_{\gamma} > \\ &= \int \text{Tr} \left( \frac{5}{6} \mathbf{U} \wedge \mathbf{U} \wedge \mathbf{U} - \frac{1}{2} \mathbf{U} \wedge \mathbf{U} \wedge \mathbf{C} \right) = \int \text{Tr} \left( \mathbf{H}^{3} (\mathcal{M}) \right). \end{split} \tag{42}$$

For a translation in reversed direction of antigen shift and drift in gene evolution theory, we can use the definition of the group action of reversed direction of time by the *CPT* theory for anti-ghost field in field of time series of antibody gene as

$$\begin{split} \left\{ \mathbf{x}\_{t}, \mathbf{y}\_{t} \right\} &= \int \text{Tr} \left( H^{3} (\mathcal{M}) \right) = \sum\_{i=1}^{3} \mathbf{g}^{i} \left( \mathbf{x}\_{t} / \mathbf{y}\_{t} \right) \\ &= a\_{t} \mathbf{y}\_{t}, \mathbf{g}^{3} \mathbf{x}\_{t} / \mathbf{y}\_{t} \to \beta\_{t} \mathbf{x}\_{t} / a\_{t} \mathbf{y}\_{t} \cong \left[ \mathbf{e}\_{t}^{\*} \right] = \int \text{Tr} H^{3} (\mathcal{M}, \mathbf{g}, F^{\mu \nu}) . \end{split} \tag{43}$$

Then a numerical representation for spinor field of curvature in the gene expression by the Chern-Simons action is defined as follows:

$$S\_{\rm CS} = \frac{k}{4\pi} \left[ \text{Tr} \left( A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right) \right] \tag{44}$$

where *k* ¼ 1*;* 2*;* 3…*n* are

*W K*ð Þ¼ *; <sup>R</sup>* Tr*RP* exp <sup>∮</sup> *<sup>K</sup><sup>A</sup>* � �*:* (36)

*anq<sup>n</sup>:* (37)

*<sup>k</sup>*þ*<sup>h</sup>* (38)

*<sup>μ</sup>* for anomaly quantum system of codons can be also

*A* ∧ *A* ∧ *A*

D*Ai* exp ð Þ *iSCS* Π*iW Ki* ð Þ *; Ri*

D*Ai* exp ð Þ *iSCS*

� � in the codon of t-RNA molecule. A path

D*Ai* exp ð Þ *iSCS*

*:* (40)

*<sup>μ</sup>*, *<sup>μ</sup>* <sup>¼</sup> <sup>1</sup>*,* <sup>2</sup>⋯20 of the Chern-Simons

(39)

(41)

The above term gives the asymmetric property of chiral molecule of DNA and RNA, twisted from the left hand to the right hand in a supersymmetry breaking as knot polynomial related to the connection *A*. By such a way, we can represent the genetic code as Laurent polynomials in variable *q* with integer coefficients, and for

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

*J K*ð Þ¼ *; <sup>q</sup>* <sup>X</sup>*<sup>n</sup>*

*q* ¼ *e* 2*π*

one can induce a spinor field for representation of genetic code, where *h* is the dual coexter number for group action of supersymmetry of gene expression *G*. It might be the source of evolution from the adaptive behavior derived from the environment. In the next, we set *h* ¼ 0 in our definition of the Chern-Simons

defined as the spectrum of curvature in the genetic code for gene evolution detection. Under the definition we mean a differential 3-forms in cohomology of spin

*;Xt=Yt*

integral of gene expression is defined by the Chern-Simons theory over knots of codon and anticodon: it is defined by the interaction between codon *Ai* and antico-

Tr *A* ∧ *dA* þ

2 3

Ð

1 3

*AαAβAγ*>

*<sup>ν</sup>* ð Þ *<sup>A</sup>*<sup>2</sup> *<sup>μ</sup>*

*<sup>ν</sup>* ð Þ *<sup>A</sup>*<sup>2</sup> *<sup>μ</sup> ν*>

The explicit definition of curvature over the connection of genetic code has also

current; it is generated from the representation of Lie group over manifold of a host

An example of our approach can serve as a case of phenylalanine (*Phe*), where

*<sup>ν</sup>d A*ð Þ<sup>2</sup> *<sup>μ</sup> <sup>ν</sup>* þ 1 3 ð Þ *<sup>A</sup>*<sup>2</sup> *<sup>μ</sup>*

*<sup>A</sup>α∂βA<sup>γ</sup>* <sup>þ</sup>

where we explicitly define the differential form of genetic code for *Phe* by

� �

*i*¼1

any knot *K* we have

By using the new parameter of knot *q*

A Chern-Simons current *J*

fiber *S*<sup>3</sup> over the homotopy class *S*<sup>3</sup>

and

cell.

**106**

*J q*;*KAi*

UUU and UUC definition is

*; Ri*

� � <sup>¼</sup> <sup>&</sup>lt;*W Ki* ð Þ *; Ri* <sup>&</sup>gt;

¼ <Tr*Ri*

new meaning of the genetic spectrum current *J*

*J*

*dA*<sup>2</sup> ¼ *A*2*A*<sup>2</sup> � *A*2*A*<sup>3</sup> ≔ *UU* � *UC*, so we have

*Phe* <sup>¼</sup> <sup>ϵ</sup>*μαβα* <sup>&</sup>lt;

<sup>¼</sup> <sup>ϵ</sup>*μαβα* <sup>&</sup>lt;

current for biology for the simpler derivation of formulas.

don between DNA and RNA in the form of integral *Ai* <sup>þ</sup> *SCS* <sup>¼</sup> <sup>Ð</sup>

*P*∮ *Ki*

*A*> ¼ Ð

1 2

1 2 ð Þ *<sup>A</sup>*<sup>2</sup> *<sup>μ</sup>*

*SCS* <sup>¼</sup> *<sup>k</sup>* 4*π* ð *W*

$$J^{\text{Amino}} = J^k \simeq \sqrt{\frac{2}{k+2}} \sin \frac{\pi}{k+2} = \int D[A]^{\text{Sci}}.\tag{45}$$

The derivation of the Chern-Simons current can be done by a simple algorithm [30], i.e., the Chern-Simons current maps the string of genetic code into numerical values by explicit formulas. It can be used to plot the time series data directly into the superspace of gene expression. We transform the alphabet string values, which cannot be computed in the classical standard definition of genetic code, into the Chern-Simons current of time series data of genetic code with *k* ¼ 1*;* 2*;* 3*,* …64 over spinor field with ground field of real values. We think that the approach is more suitable for computational programs used in data analyses.
