# Performance for mathematical model of DNA supercoil.

INTRODUCTION

MATHEMATICAL FORMATION OF MODEL

The model under our study is shown geometrically in fig. (1a,b). Here we are consider straight elastic rod of length's' and radius 'b', which is bent in circle initially. Ends of rod are twisted relative to one another up to an integral number of times. To formulate the model, we are taking theory of elasticity using three-dimensional formulation of stress and strain within the rod, which centralize the effective force, moments and curvature. This provides a simplification to general theory of elasticity in that it reduces the problem to a manageable set of ordinary differential equations. There are many constraints for the theory but primary being extended aspect ratio in the shape and the size of the body within our constraints, the strands are assumed to have the shape of a uniform helix which is held apart by the electrical repulsion of the Changed strands. The vector path of the rod centre-line in these regions is

S(r) = s cos (r sin[alpha])i + s sin(r sin[alpha])j + r cos[alpha]k (1)

where [alpha] is helix angle, s is the standoff distance between the strands and 'r' is the arc length coordinate along the strands. Fig. (2) shows the directional relation among tangential tension force(T), shear force([F.sub.s]), tangential twisting moment(Q), bending moment vector ([M.sub.b]). These quantities are computed by differentiating equation (1) and the results given by Love (1927).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

All these quantities are dimensionless which have been normalised by the rod of radius b and bending rigidity [EM.sub.s]. The helix is an exact solution to the equations of motion and provides a simple formulation of a complicated model. Here all the mechanical quantities in the supercoil are described in terms of helix angle and standoff distance s. The next ingredient in the model is a description of the end loops. Detailed descriptions of this construction investigated by Stump et al. (1998). It shows that the shape depends on just the helix angle and standoff distance's'. With the help of Stump et al (1998) we find the half length of the end loop is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

We know that electric repulsive force which keeps the strands in the molecule apart is required in DNA supercoilling. Experiments are generally conducted in a salt mixture. In this way that the Debye-Huckel model takes place of classical Coulomb electrostatic potential. Electrostatic potential replaced and Debye-Hucle model taking the form

U = q/4[pi][epsilon]p [e.sup.-(p/d)] (4)

where q is the charge, d is the debye distance nearly equal to the rod radius for the DNA model , p is the distance from the charge. Since debye distance nearly equal to radius of DNA and supercoil assumed to be in ionic solution hence the effect of repulsion between the two strands at a distance s can be expressed in a dimensionless term as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

with condition that when [beta] = 0 all the phosphate groups have been neutralised and there is no repulsion force and when [beta] = 1 phosphate group still carry their charge.

We have a relation among L, [L.sub.T] and n using Stump et al (1998)

2L/cos [alpha] + 2sL'([alpha])n = [L.sub.T] (6)

We can obtain L' from equation (3)

Here we are going to link the results obtained by Fuller (1978). Once the ends of the rod are joined together, the centre line of the strand and the inscribed reference line on the surface form a set of interlaced lines. The integral interlacement numbers known as linking number (Ln). The linking number is equal to the number of times that the surface line spirals about the strands centre line

[L.sub.n] = -i (7)

White (1969) shown that the linking number decomposed into two parts, which can be expressed:

[L.sub. n] = [T.sub.w] - [W.sub.s] (8)

and [T.sub.w] and [W.sub.s] can be defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

[T.sub.w] is twist number and [W.sub.s] is writh number of the rod.

In equation (9) the first term on the right hand side is the contribution to balanced ply and second term that to the end loop. The term [F.sub.loop] depends only on [alpha] and is given by the truncated series

[F.sub.loop] = -0.4356[[alpha].sup.1/2] + 4.5[alpha] - 9.8[[alpha].sup.2] - 86.3[[alpha].sup.5/2] + 50[[alpha].sup.3] (11)

Equation (3) came into existence due to electrostatic repulsion, since the supercoil is in equilibrium hence the opposite side force factor given by

[phi] = [sin.sup.4] [alpha]/[s.sup3] cos 2[alpha]

by combining equations (8), (9) and (10), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where k = 0.67 and the expression for [F.sub.loop] and L are function of [alpha] given by the equation (3) and (5) respectively and so we can solve 'F' and 'G' simultaneously for 's' and '[alpha]'.

[FIGURE 1A OMITTED]

[FIGURE 1B OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Fig.3, A graph of the number of cross overs nc (nodes) as a function of the absolute value of the linking deficit L.K. The solid line is for a stand off distance s =1, i.e. the strands are in contact. The dash line is for a stand off distance of s = 2. The dash line is for a stand off distance of s=2. The dot dash lines are the experimental results of Boles et al. (1990).

Fig. 4, In this graph the solid line is for a stand off distance of s = 1(the strands are in contact) and the dash line is for s = 2. The experimental measurements of Boles et al.(1990) are shown by the short dot-dash line at D/S = 0.45.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Fig.5, A graph of the stand off distance versus the linking deficit. [absolute value of [L.sub.n]] for increasing values of a. From left to right, a = 0.1, 0.3, 0.5, 0.7, 0.9 and 1 and 1.1.

Fig. 6, A graph of the supercoling angle [beta] versus the linking deficit [absolute value of [L.sub.n]] for increasing values of a. From the bottom to the top , a = 0.1, 0.3, 0.5, 0.7, 0.9 and 1 and 1.1.

[FIGURE 6 OMITTED]

RESULT AND DISCUSSION

The shape of the supercoil is assumed as given in the figure 1, with single balanced ply segment and loops. To prescribed the standoff distance's' are considered, s = 1.2 and s = 2.4, where the value s = 1.2 correspond to a tightly coiled molecule. The results for s = 1.2 are very close linear and when fitted with the linear regression give slopes -0.94 and -0.95 for the long and short cases. In figure (3) We give the comparison of our results with Boles et al. (1990).

The effect of ionic concentration of the solution on the standoff distance is given by the graph (4). Graph (5), which shows that a large amount of ionic concentration correspond to a small amount of phosphate group, means the smaller value of parameter [beta]. The feature of graph (6) is, after certain value of [L.sub.n], 'a' decreases until the linking deficit is large to cause the strands in the DNA to contact. Where it increase almost linearly while as the stand of distance decreases rapidly with increasing [L.sub.n], the length of DNA in the balanced ply increases as given in graph (4).

CONCLUSION

The present mathematical model have reasonable merit over the models developed as before since this model has advantage over finite element and molecular dynamics model lie in the relation, simplicity over equations for determining the variable used in the model. Present model gives reasonable agreement with the experimental results of Boles et al. (1990). Model has sensitivity for ionic solution in there expressions as far so we hope the present model is more improved model than discussed model earlier in the region.

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Anil Kumar and Narendra Kumar *

Department of Mathematics , Dr. K.N. Modi Institute of Engineering & Technology , Modinagar (Ghaziabad) UP India. Email: dranilkumar06@yahoo.co.in

** ICFAI Tech, ICFAI University, Dehradun (Uttaranchal), India Email: narendra.ibs@gmail.com
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