The Theoretical Analysis of the Phenomenon of Solitons in DNA

The Theoretical Analysis of the Phenomenon of Solitons in DNA

AbstractWe have introduced that a DNA supercoil can be considered as a quantum spin system such that spins are located

1. 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 Chern-Simons 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; Chern-Simons topology

   1. INTRODUCTION

The presence of supercoiled DNA has been confirmed in experiments earlier and it was originated that in vivo chromosomal B-DNA molecules consist of topological domains including supercoiling can occur [1-3]. 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 [2-5]. 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 self-sufficient 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 B-DNA 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 two-component 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 Z-axis. In fig: 1(a) A diagram formation B-DNA 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

(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 well-known 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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