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Performance for mathematical model of DNA supercoil.


The recent publications of the genome book and subsequent revelations have not only falsified earlier myths but have in the way thrown open fresh challenges to the scientific community. While no body can deny that a gene clad future is on anvil, one can also clearly discern the fact that a gene clad future is on anvil, one can also clearly discern the fact that playing god is fraught with dire consequences. While new research has given the scientific community supreme power, it has also put an enormous sense of responsibility on their shoulders. The findings of the human genome project have shattered many myths. For example, it was perceived earlier according to one gene-one function theory that a complete human genome map would make it easy to diagnose a particular disease. Now it has been revealed that the genes work in coiled rather than in isolation and hence this coil has to be studied. De-oxyribo nucleic acid (DNA) is the genetic material rather than a protein. Evidence for that statement first came from simple experiments and observations with the bacteria and viruses. The bacterium Diplococus pneumonia causes pneumonia in mammals. In 1928 Frederick Griffith, British medical scientist, found that there are two strains of D. Pneumonia, one that form smooth colonies (s-strain) protected by a capsule when grown on a suitable medium in Petri-dishes. Supercoiled: "Supercoiling" is an abstract mathematical property, and represents the sum of what are termed "twist" and "writhe". "Supercoil" is seldom used as a noun with reference to DNA topology. It is the combination of twists and writhes that impart the supercoiling, and these occur in response to a change in the linking number. A coiled structure is at a higher energy (less stable). When the linking number is reduced in closed circular DNA, the molecule supercoils by minimizing twisting and bending. Double stranded circular (or linear) DNA can have tertiary or higher order structure. Superhelicity is therefore sometimes referred to as DNA's tertiary structure. Supercoils refer to the DNA structure in which double-stranded circular DNA twists around each other. This is termed supercoiling, supertwisting or superhelicity--meaning the coiling of a coil, also understood in terms of knots. Only topological closed domains (such as a covalently closed circle) can undergo supercoiling. A linear molecule can have topological domains as long as there is a region of the DNA bounded by constraints on the rotation of the DNA double helix. Eukaryotic DNAs in association with nuclear proteins acquire superhelical conformation in chromosomes. Adding a twist to the DNA (as catalyzed by an enzyme), imposes a strain. A DNA segment so strained that is closed into a circle would then convert into a figure of eight (or its topological equivalents) the simplest supercoil. This is the shape that a circular DNA assumes to accommodate one too many or one too few helical twists. For each additional helical twist that is accommodated, the lobes will show one more rotation about their axis. Such superhelicity results in more compact structures. In any other naturally found geometry, the DNA is either under- or overwound. Its helical axis does not lie in a plane or on the surface of a sphere because of writhing and twisting of it. This is the physical solution to the potential (torsional) energy minimization problem. Supercoiling can therefore be: negative (right-handed): Supercoils formed by deficit in link are called negative supercoils. They result from underwinding, unwinding or subtractive twisting of the DNA helix (due to a deficit in link). All naturally occurring double stranded DNAs are negatively supercoiled. Negative supercoiling facilitates DNA-strand separation during replication, recombination and transcription. All the naturally occurring double stranded DNAs are negatively supercoiled (including bacterial and viral circular duplex DNAs). Positive (left-handed): Supercoils formed by an increase in link are called positive supercoils. They result from tighter winding or overwinding of the DNA helix (due to an increase in link) resulting in extra helical twists. The two lobes of the figure of eight then appear rotated clockwise with respect to each other. This would compact DNA as effectively as negative supercoiling, but would make strand separation much more difficult. In non-dividing eukaryotic cells, chromosomal DNA is wrapped around a nucleosome core which consists of highly basic proteins called histones. The DNA is wrapped around the nucleosome in a left-handed solenoidal arrangement. This negative supercoiling is one of the forms taken up by underwound DNA. Writhing: Global contortions of circular DNA are described as "writhe". The writhing number describes the supertwisting or supercoiling of the helix in space. It is the number of turns that the duplex axis makes about the superhelix axis. Writhe describes the supercoiling, the coiling of the DNA coil. It is a measure of the DNA's superhelicity (supercoiling) and can be positive or negative. Twisting: Twist is the number of helical turns in the DNA, i.e., the complete revolutions that one polynucleotide strand makes about the duplex axis in the particular conformation under consideration that is the number of bases per turn of the helix. Twist is altered by deformation and is a local phenomenon. The total twist is the sum of all of the local twists. Twist is a measure of deformation due to a twisting motion. Linking number: This is a topological property that determines the degree of supercoiling; It defines the number of times a strand of DNA winds in the right-handed direction around the helix axis when the axis is constrained to lie in a plane. It is the number of times that one DNA strand crosses about the other when the DNA is made to lie flat on a plane. Relaxed: Circular DNA without any superhelical twist is known as a relaxed molecule. DNA in its relaxed (ideal) state usually assumes the B configuration. In a relaxed double-helical segment of DNA, the two strands twist around the helical axis once every 10.6 base pairs of sequence. Relaxed, closed circular DNA is defined as DNA which has no supercoils when constrained to lie flat in a plan. The following structures are consistent with the relaxed state: (a) Linear DNA (either straight or curved) (b) Closed circular DNA, provided its axis lies in a plane or on the surface of a sphere. Supercoiling is thus vital to two major functions. It helps pack large circular rings of DNA into a small space by making the rings highly compact. It also helps in the unwinding of DNA required for its replication and transcription. Supercoiled DNA is thus the biological active form. The normal biological functioning of DNA occurs only if it is in the proper topological state. Boles et at (1990) studied structure of DNA supercoil. Schlick (1995) developed a supercoiled DNA model with its analytical approach. Stump et al (1998) give a new idea about writhe number and its applications. Wasserman et al (1988) reported about the helical repeat of double strands nature in DNA supercoiling. The purpose of this work is highlighting the application of mechanical theory to a problem in determining macro-molecular shape of DNA.


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).


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


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


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



[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




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]'.





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.



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.



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).


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:

** ICFAI Tech, ICFAI University, Dehradun (Uttaranchal), India Email:
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Author:Kumar, Anil; Kumar, Narendra
Publication:Bio Science Research Bulletin -Biological Sciences
Date:Jul 1, 2006
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