DNA is a double stranded melecular consisting of double helix phosphate backbone .... [5] A. Vologodskii, Unlinking of supercoiled DNA catenanes by Type IIA ...
K NOTS A ND DNA R EPLICATION Abdul Mohamad and Tsukasa Yashiro The University of Nizwa, Sultan Qaboos University
Abstract
DNA Replication
Watson and Click discovered the double helix structure in 1953. They mentioned that the DNA replication is done by semiconservative process. This process however left a long standing problem; that is, how the replicated DNAs are split into new cells in living organisms. This problem can be interpreted into a placement problem in topology. Knot theory is one of tools to tuckle this problem. In this research we try to find a possible topological model for replication process so that we can explain the splitting process of daughter DNAs.
DNA replication produces a pair of copies of the original DNA. This process is done by the semiconservative process; that is each strand of the double helix is preserved in the daughter DNAs.
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Ori
Fork
Introduction A knot is a closed circle in 3-space without self-intersections. Knots are everywhere in our world from DNA to a ring of the planet Saturn. DNA is a double stranded melecular consisting of double helix phosphate backbone and four bases. A cell in a living organism will be divided into two new cell. To do this the DNA in the cell must be copied into two daughter DNAs, called a replication. DNA is a very long thin string and it must be packed into a very small nuclear. The double helix structure of DNA and its length leads mainly two problems:
Fork
The replication starts some specific place called ori where the helix is relaxed and forks are generated. Then replication is done in the fork parts. However, if we continue this process, as the original DNA is linked, the daughter DNAs must be linked. Some mechanism must be inserted in the boxes with ?s.
Copy of DNA
1. When it is replicated, the DNA is twisted and thus supercoiling is introduced and it is an obstruction of the replication process. 2. When it is replicated, the daughter DNAs are linked, and as it is very long, it is difficult to split into two separate parts.
When DNA is replicated and daughter DNAs A and B are created, then we suppose that they preserve following properties.
Obviously, these problems are resolved in our living cells, however we do not have any clear explanation of this process [3][4][5]. Our research focuses on this problem and from topological view, we propose a topological model of replication that resolves the problem.
1. The sequence of base (genetic information) 2. topological shape (supercoilings) 3. positions of oris. Semiconservative replication process guarantees to preserve the first property. Thus our model must preserve the second and the third properties.
Cell Division and Splitting Daughter DNAs
Topological Model
In the micro world a knot appears in DNA (Deoxyribonucleic Acid) which has double helix structure. A cell is divided periodically. There are mainly four stages to split cell into two newly copied daughter cells. In a eukryaotic cell, there exists a nuclear of the cell.
To relax the helix, a rotation around the double strands axis is needed. Then we assume such rotation exists. This rotation however creates the negative supercoilings.
During the replication as the DNA has the double helix structure, the daughter DNAs must be linked. However, it is not known how the daughter DNAs are split into two cells. DNA has the double helix structure; that is, two strands are linked each other. If we oriented the two strand in the same direction, we can define the linking number If two strands are linked, then we can define the signature for each crossing:
Inside an eye (left), negative supercoilings are produced, and between eyes (right), positive supercoilings are produced. The positive supercoilings can be cancelled by existing negative supercoilings. Therefore, we propose the following DNA topological model.
ori
+1
ori
ori
−1
Suppose that a link L has only two components k1 and k2. Let L+ be the number of positive crossings between k1 and k2, and let L− be the number of negative crossings. The linking number L is defined by A
1 L(k1, k2) = (L+ + L−). (1) 2 For a DNA, it is a two component link, thus we simply write L = L(k1, k2) and it satisfy the following equation: L = T + W,
(2)
where T is the number of twistings along the double helix DNA and W is the writhe, the number of crossings of supercoilings.
Supercoiling and Linking Number
The left diagram is the topological formulation of DNA. It is a union of negatively super coiled DNA. Each super coiled loop has an ori. The replication process starts from the ori. Inside eye, negatively super coiled DNA strands are produced. This process produces positive super coils outside the eyes but they are cancelled by the negative supercoilings in the topological formulation. The right diagram shows the daughter DNAs are produced and at some places like A they are linked. This linking could be released by enzymes.
Further Study
The following diagram that a torus model of a circular DNA. The orientation of the torus is an orientation along the standard longitude. When we twist a part of the torus A against B , a supercoiling will be created. This diagram shows that one negative crossing of the supercoiling is equivalent to −1 writhe and it gives one full twist to the double strand.
There are some issues in our topological model to be resolved:
z
1. The number of supercoilings in the topological formulation is far less than the twisting number of double helix. This may be resolved by the existence of histones. A
C1
2. The source of energy of the rotational movements. This may be explained by molecular motors.
A
C1
C2
References
y
C2 C1
B x
3. There is no effects of enzymes such as topoisomerases.
[1] A. Kornberg and T. A. Baker, DNA replication, Second Edition, Uiversity Science Book (2005).
C2
B 1 2
+
1 2
=
Full twist
For the torus, fixing B part, rotate A around the z -axis π radian. Then the torus is fully twisted around the longitudal axis. This shows that each negative supercoiling crossing will give one full twist in the double strand DNA so that the equation (2) holds.
[2] K. Murasugi, Knot theory & its applications, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008. [3] R. R. Sinden, DNA structure and function, Academic Press, (1994). [4] A. Vologodskii, DNA upercoiling helps to unlink sister douplexes after replication, Bioessay, (2010); 32 (1), 9-12. [5] A. Vologodskii, Unlinking of supercoiled DNA catenanes by Type IIA topoisomerases, Biophysical Journal, Vol. 101, (2011), 1403-1411.
