 Open Access
 Authors : Subhamoy Singha Roy, Sayan Mukherjee, Sitabra Das, Sayan Das, Wriju Sadhukhan, Rahul Mondal
 Paper ID : IJERTCONV9IS11062
 Volume & Issue : NCETER – 2021 (Volume 09 – Issue 11)
 Published (First Online): 16072021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
The Theoretical Analysis of the Phenomenon of Solitons in DNA
Subhamoy Singha Roy Department of Physics, JIS College of Engineering(Autonomous), Kalyani, Nadia 741235, India Sayan Das Department of Computer Science and Engineering, JIS College of Engineering (Autonomous), Kalyani, Nadia 741235, India 
Sayan Mukherjee Department of Computer Science and Engineering, JIS College of Engineering (Autonomous), Kalyani, Nadia 741235, India Wriju Sadhukhan Department of Civil Engineering, JIS College of Engineering (Autonomous), Kalyani, Nadia – 741235, India 
Sitabra Das Department of Electrical Engineering, JIS College of Engineering (Autonomous), Kalyani, Nadia 741235, India Rahul Mondal 5Department of Electronics and Communication Engineering, JIS College of Engineering (Autonomous), Kalyani, Nadia 741235, India 
AbstractWe have introduced that a DNA supercoil can be considered as a quantum spin system such that spins are located

THEORETICAL BACKGROUND
on the axis formulate an antiferromagnetic chain. These spins
A change in the linking number from
L k as a result of
can be connected with
SU 2 gauge field currents when gauge 0 o
fields recline on the links. We have expressed bending (curvature) and twisting (torsion) with regard to these gauge fields. In fact, the topological property acting as the linking number can be borrowed from the ChernSimons topology affiliated with a quantum spin. The current study additionally shows that DNA loops in the supercoil execute topological objects like solitons.
KeywordsDNA supercoil; antiferromagnetic chain; solitons; ChernSimons topology

INTRODUCTION

The presence of supercoiled DNA has been confirmed in experiments earlier and it was originated that in vivo chromosomal BDNA molecules consist of topological domains including supercoiling can occur [13]. DNA molecules from prokaryotes (cells without nuclear membranes) frequently adopt the interwound structures which are called plectonemic supercoils. In eukaryotes (cells with nuclei and other organelles with their own internal membranes) chromosomal DNA molecules are also known as arranged into topological independent loops [25]. Statistical mechanics of supercoiled DNA has been examined by several authors [6]. At length scale of thousands of base pairs DNA is formed into topologically selfsufficient loops. There are position in vivo when topological constraints induce supercoiling. DNA loops in a supercoil may perform as a topological object such as a soliton (skyrmion) which is accomplished when we execute DNA as a spin system. In fact, DNA loops in a supercoil when strained by a change in
twist of the ends induce a deviation from the planar circle configuration compare to a spin texture and symbolize a alteration of the spin system from the ground state when spin excitations occur. These excitations approach the solitons
explained by the nonlinear model.
Fig: 1(a) A diagram configuration BDNA Chain. (b) A diagram depiction of DNA as an anisotropic fixed spin double helix string model.
c
the linking number due to variation of twisting rate from W0
We can depict a twocomponent spinor as 1
c
with
compare to the formation of a spin texture when a DNA molecule is treated as a spin system.
2
e
e
c1 cos 1 2
i1
2
(1)
which equals 2 for the hedgehog skyrmion with 1.
Now we may write the nonlinear model Lagrangian in terms of the SU 2 matrices U as [10]
1
1
c sin1
i1
e
e
2 2
(2)
G m2 16 U U 1 32 2 UU , UU 2
(8)
In terms of the spin system we can consider the ground state
u
wave function depicting the DNA supercoil with linking number L0ko
where M is a constant having dimension of mass and is a dimensionless parameter, , being space time indices.
0 c1 c2
c2 c1
(3)
Taking the spin variable
Z UZ0
with
1
Z0 0
and
i j
i j j i
USU 2
where j and i communicate to the spin sites. When the
linking numeral deviates from L0ko owed to divergence of the twisting speed from 0 , the consequential skyrmion state is described through
The reliance may be incorporated through m and
where these parameters are taken as functions of .
For a indistinct loop we can believe the radius of the loop R1
c
as a function, R1(,) corresponding to the core radius of
D 2k
(4)
the Soliton. We can define the core size of the Soliton. such
c 0
that R R 1 where R is the size of the Soliton
k 1k
1 0 0
where the spin texture is incorporated within the mechanism
with minimum energy. The stationary nonlinear model
1
1
c and
k
c and 0 1[7] . If a smooth and
2
2
k
Lagrangian corresponding to eqn. (8) gives increase to the
energy integral as
monotonical function f is defined with f 0 0 and
i j
f then the skyrmion state can be written as
(9)
() cos( f () )er sin( f () )e
where i, j 1,2,3 are special indices.
To calculate the energy we take the Skyrme ansatz
(5)
where er and e
are the foundation vectors. The dimension
U (x) exp(iI (r) .x
(10)
x
x
of a skyrmion is resolute by the function
f
and
where
are Pauli matrices, x
and I (0)
f
describes the hedgehog skyrmion with spin in the
and I (r) 0 as
r
r
. We explicitly write
radial path r [8].
U cos I (r) i .x sin I (r)
(11)
The skyrmion state ( ) is controlled by the
with
R
R
R
R
relation ( ) 1. The quantum state for the skyrmion
r 2
r 2
( ) can be written as
cos I (r) 1
1
1 1
and
2
2
r r
sin f (k )
e ik
sin I (r) 2
R 1 R
(12)
D
2 2
1 1
f ( )
i 0
(6)
The energy integral becomes
k cos
k 2 e k 2
H (R ) 4 2m2 R I
2 2 I
2 R
(13)
where D is the normalization steady and f controls the
1 1 1 2 1
where
size of the skyrmion. From equ. (5) and (6) it is seen that
0 1 is determined from f and pedals the
1
2 2 2
(14)
size of the skyrmion [9] . Certainly we can define
I1 dx sin
0
I (r) x
I
x
3.0
2arctan (7)
and
I 1
dx sin4 I (r)
x2 sin2 I (r) I
x2 1.5
along Zaxis. In fig: 1(a) A diagram formation BDNA Chain. (b) A diagram representation of DNA as an anisotropic fixed spin double helix string model.
2
0
(15) (15)
with x
r . This gives the look of energy
R1
H (R ) 12 2m2 R 3 2 2 R
1 1 1(16)
The smallest amount of energy H (R1) is found from the relation
H
R1
12 2m2 3 2
2 R 2 0
1
1
(17)
which gives for Hmin the size as
R0 1 2m
and the energy
(18)
min 0
min 0
H H (R ) 12
(19)
It is wellknown that the coupling parameters m and are functions of such that in the limit 0, m() 0 and () 0 but m is fixed. When we take
R1 R0 1
we have
Fig.2. Skyrmion energy as a function of
R0 (1 k
In fig.2.we have designed the Skyrmion energy as a function
1
1
H (R ) 6 2m 1 1 1 (20)
of R0 (1 k ). From our analysis it seems that when the
Now we note that the parameter totally gives a measure of the hilarity associated with twisting strain into the loop given by L0k Lk0 .In fact in the simplest form we can
take k where k is a constant. So form the
relation R1 R0 (1 ) R0 (1 k ) , we can measure the energy of a DNA loop as a function of . It is noted that
the relation R R (1 ) gives a nonzero size for
long linear chromosomal DNA molecules are organized into loops, these topological independent loops appear as solitons. Solitons are nonlinear excitations which can travel as coherent solitary waves. The present analysis suggests that soliton excitations may well exist in DNA chains which is reliable with the observations of Englander et.al.[12] The linking number related with a supercoil is given by the
topological charge of the loops. As the skyrmion (soliton) showing a loop is designated by the nonlinear – model in terms of SU 2 gauge fields, the topological charge of a
1 0
loop is given by the winding number of the mapping of the 3
1( 0) when R0
is infinite. Definitely, it has been
space manifold into the group manifold SU2 S 3 which
found that for 0.02 the negligeable free energy state
resembles to homotropy 3 (SU2) 3 (S ) Z
3
has R1 P demonstrating that no reliable stable
where Z signifies the set of numbers [13]. When DNA loops
supercoiled state exists for small . For
0.02
the
supercoil, the linking number is given by an integer resolute
plectonemic free energy shows a minimum value for finite R1
and P which indicates that we have a stable supercoiled state. It seems that can be varied through unevenly –
0.1 to 0.1 as elsewhere these confines the double helix is unstable [11]. These observations are found to be reliable with this skyrmion model also Skyrmion energy circulate
by this homotropy group so that number of superhelix loops.
.
L0k nZ
where n is the
III. DISCUSSION
A significant result of our examination is that a DNA loop can be considered as topological object showed as a skyrmion (soliton) which arises due to the excitation of spins caused by the variation of the twisting rate from leading to additional (deficit) of linking number. The spin consistency is determined
by the twist parameterized by the quantity .The energy of the skyromion showing a DNA loop depends on the radius which is resolute by the parameter The linking number of a DNA molecule when ordered into loops is associated to the topological charge of a skyrmion showing a loop
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