A rewinding model for replicons with DNA-links

A double strand DNA has a double helical structure and it is modeled by a thin long twisted ribbon fixed at the both ends. A DNA-link is a topological model of such a DNA segment in the nuclear of a eukaryotic cell. In the cell cycle, the DNA is replicated and distributed into new cells. The complicated replication process follows the semi-conservative scheme in which each backbone string is preserved in the replicated DNA. This is interpreted in terms of splitting process of the DNA-link. In order to split the DNA-link, unknotting operations are required. This paper presents a recursive unknotting operations, which efficiently reduce the number of twistings.


I. INTRODUCTION
It is known that DNA is a polymer consisting of a set of base pairs and sugar-phosphate backbones (Watson-Click model (1953), see [2] [11]). The backbones form a double-helix structure with opposite directions induced from the ordered pair of 5 and 3 . The 360 • -rotation of the helical strings is joined by about 10.5 base pairs. Therefore, it is natural that the double strand DNA (ds-DNA) is modeled by a long, thin strip twisted around the centre curve of the strip (see Figure 1, also [2] [11]). The boundary curves of the strip form a double helix which correspond to the backbones of DNA [2] [11]. A full twist of the ds-DNA is interpreted as the 360 • -rotation of the strip about the centre curve (see Figure 1).
The replication of DNA has been studied since the helical double strand structure was discovered (see [1][3] [4][10] [12]). In the interphase of the 10.5bp eukaryotic cell cycle, there is a substructure of nucleus called the nuclear matrix (NM) (see [8] [9]). It is believed that the ds-DNA is organized as a set of loop-shaped structures, called replicons. Each replicon has a specific site in which the replication starts. This is called a replication origin or simply, origin.
It is observed that two ends of the replicon (loop) are anchored at NM [8][9] [14]. Thus it forms a pair of twisted strings fixed at the ends of the boundary of a 3-dimensional ball, which is a two-component tangle (see [6] for a definition). (see Figure 2 (c)); in this paper, we call this a DNA-link (see Section III for the definition).
In the DNA replication process, the sequence of base pairs along each backbone acts as a template to reproduce the base pairs in the synthesized DNA. This means that each backbone with base pairs is preserved in the synthesized DNA. This is called the semi-conservative scheme, we in-terpreted this scheme in terms of DNA-link in Lemma III.1.
As the replication process follows the semiconservative scheme, the DNA-link must be split at the end. Topologically, it should be done by unknotting operations. Biologically, it is believed that enzymes topoisomerases TopoIA, TopoIB and TopoII are responsible for this operation. Our model uses only TopoIA. This paper is organised as such: In Section II knots, links and linking numbers are introduced. In Section III, DNA-link is defined and the semiconservative scheme is interpreted in terms of DNA-link (Lemma III.1). In order to rewind the ds-DNA it is natural to require the minimal energy and maximal length of rewound segment. We obtained Theorem III.1. Section IV describes a topological model for rewinding replicon. Our model is equipped with a recursive unknotting operations. This model can reduce the twisting number of the DNA-link efficiently. The recursive unknotting operation resolve the unsoloved problem in [5].

II. KNOTS AND LINKS
A link is a disjoint union of circles embedded in R 3 ; K 1 , K 2 , . . . , K n denoted by L(K 1 , K 2 , . . . , K n ). Each K i i = 1, 2, . . . , n is called a component of the link L and the link L is called an n-component link. A 1-component link is called a knot. A knot is trivial if there is an embedded disc in R 3 bounded by the knot. A link is trivial if the link consists of mutually split trivial link components. A link diagram D L of a link L is a projected image of the orthogonal projection (x 1 , x 2 , x 3 ) → (x 1 , x 2 ) with crossing information at the crossings (see [6] for details).
There are three types of local moves on link diagrams depicted in Figure 4, called Reidemeister moves.
Two links L 1 and L 2 are equivalent if the diagram D L1 is deformed into the diagram D L2 by a finite sequence of three types of Reidemeister moves shown in Figure 4 [6] Example II.1. The diagrams in Figure 5 are equivalent. The left diagram is deformed into the right by applying R2 move.

A. Split link
Let A and B be compact disjoint subsets of R 3 . The pair {A, B} is said to be split if there exists an embedded sphere S in 3-space such that S bounds one of the subsets from the other. A 2-component link is called a split link if the components are split. A link L is said to be oriented if each component is oriented. A link diagram is the image of L in the plane R 2 under the orthogonal projection (x 1 , x 2 , x 3 ) → (x 1 , x 2 ) with crossing information; that is, at a crossing formed by two short arcs, one arc is upper and the other lower (see Figure 3). A link diagram of a link L is denoted by D L .

B. Linking number
Let L be an oriented link and let D L be a link diagram of L. At a crossing point of D L , there are two types of crossings formed by short subarcs of D L ; positive and negative crossings (see Figure 6).

Fig. 6. Crossings with signs
Let L(K 1 , K 2 ) be an oriented DNA-link with distinct link components K 1 and K 2 . Let C(D L ) be the set of crossings of the diagram D L . The linking number is defined by where c is a crossing of the link diagram, ε(c) is the sign ±1 according to the diagrams in Figure 6, and also Let K be a knot and let D K be a knot diagram. The total sum of signs w(D K ): is called a writhe of K.
Note II.1. The linking number does not depend on the choice of the diagram of L (see [6]). If a 2-component link is split, then the linking number between the components is of course zero but the converse is not always true (see [6]). The writhe depends on the choice of diagram (see [6]).

C. Unknotting operations
There is an operation to exchange the over arc and the under arc, called an unknotting operation (see Figure 7). For every non-trivial knot K, it is

III. DNA AS A 2-COMPONENT LINK
For a linear DNA, the length of the double strands is much longer than its diameter and also in the eukaryotic cells, the ends of a linear segment of DNA are fixed at the nuclear matrix. Therefore, it is possible to view the double strand DNA as a special 2-component link.

A. DNA-link
A ds-DNA can be modeled as the boundary components {S 1 , S 2 } of a long thin twisted strip with the centre curve γ. We write this as L = L(S 1 , S 2 ; γ), where S 1 and S 2 represent the single strands of the DNA. We call L a DNA-link. Let L 0 = L 0 (S 1 , S 2 ; γ 0 ) a DNA-link. After the DNA is replicated and distributed into two daughter cells, there are two identical DNA-links representing daughter DNAs, where S 1 and S 2 are single strands (templates) inherited from the original DNA.S 1 andS 2 represent synthesized single strands from S 1 and S 2 respectively and γ 1 and γ 2 are centre curves of the strips for L 1 and L 2 respectively.
From this observation, the authors described in [5] that the semi-conservative scheme is interpreted in terms of DNA-links as: Lemma III.1 ( [5]). The semi-conservative scheme is interpreted in terms of links: a deformation of the DNA-link L 0 (S 1 , S 2 ; γ) into a split link {S 1 , S 2 }, where S 1 and S 2 are inherited single strands from S 1 and S 2 respectively.
The complicated replication process can be interpreted in terms of DNA-links. As it is shown by Lemma III.1, the semi-conservative scheme is described as follows: the DNA-link L 0 is deformed into the split 2-component link {S 1 , S 2 }, where S i is obtained from S i (i = 1, 2) by applying unknotting operations to L 0 . White proved in [13] the following formula of the linking number lk(S 1 , S 2 ): Lemma III.2 (White [13]). For a DNA modeled by the DNA-link L(S 1 , S 2 ; γ), the following formula of the linking number holds: where Tw(S 1 , S 2 ) is the number of full-twists of the curves {S 1 , S 2 } along the centre curve γ and Wr(γ) is the writhe of γ.
Suppose lk(S 1 , S 2 ) = 0. Consider a diagram D L of L. Since γ is trivial, we can modify γ into a trivial circle γ so that Wr(γ ) = 0. Let L(S 1 , S 2 ; γ) be the modified link. Note that this modification does not change the linking number. Then lk(S 1 , S 2 ) = Tw(S 1 , S 2 ) + Wr(γ) = Tw(S 1 , S 2 ) = 0 The diagram D L is depicted in Figure 9.
Since the writhe Wr(γ ) = 0, the twisting number Tw(S 1 , S 2 ) is cancelled by the twistings obtained from the writhe Wr(γ). Thus the diagram gives split trivial circles S 1 and S 2 .

B. Efficient unknotting operations
If an unknotting operation is applied to one crossing of the DNA-link L(S 1 , S 2 ; γ) with Tw(S 1 , S 2 ) = m, then the resulting link has the number of twistings m−1. Consider the following operation. When the DNA-link is modified and moved so that it is partially rewound, some energy is used. The operation is expressed by a finite number of Reidemeister moves (see Figure 4). Thus it is possible that the necessary energy can be indicated by the number of steps consisting of those local moves. Of course, the number of local moves depends on the diagram. As these moves always occur around the center curve of ds-DNA, we can assume that the link L is linearly projected onto the plane can be used for describing this operation. Under this situation, the Reidemeister moves of R2 are used as shown in Figure 10. Once some R2 are applied along the center curve, count this as one step.
Then we have the following theorem. Fig. 9. Since lk(S 1 , S 2 ) = 0 and Wr(γ ) = 0, Tw(S1, S2) is cancelled by the opposite twistings induced from the writhe. The negative twistings come from the writhe Wr(γ). Proof: Let µ(k) denote the number of the steps of the deformation which is a function of k, expressed by where the function is the ceiling function. On the other hand, the outcome of the operation is indicated as the length of the rewound segment of the DNA. This can be counted as the number of crossings eliminated by the operation. Let ν(k) be the number of the eliminated crossings, expressed by An efficient operation should minimize µ and maximize ν. In order to see the efficiency of the operation, take the ratio: ν µ It is easy to see that this ratio is 2 or 4. Therefore, it can be justified to take the values µ = 1 and ν = 4 to make the operation efficient. This means k = 2. This is the required result.

IV. A REWINDING MODEL FOR REPLICONS
It is believed that the replication process starts from a specific domain called a replicon which has a looped shape (see Figure 2 (a)). The replicon (loop) is modeled by the 2-component tangle (link) shown in Figure 2 (c). We described a topological model for DNA replication in [5] but it requires some over twist absorbing mechanisms which are unknown. If the process described in this section is used in the replication model, then the over twisting absorbing mechanism is not needed. The model in this section can be a solution for the problem unsolved in [5].
A. Estimate of the writhe From Lemmas III.1, III.2 and III.3, in order to split the DNA-link L 0 (S 1 , S 2 ; γ 0 ), it is necessary to eliminate the linking number lk(S 1 , S 2 ). The linking number can be estimated from the length of the DNA segment of the replicon. Let γ0 be the length of γ 0 in terms of base pairs and let b be the number of base pairs within one full twist of the double strands; b is approximately 10.5. Then the linking number is given by: The DNA in eukaryotic cell has a form called nucleosome, where the DNA is wrapped into two negative supercoils around a histone octamer (see [11] for details). The DNA string around the histone octamer forms a double loop with Wr = −2 (see Figure 11). The total writhe in the replicon 197bp Fig. 11. The nucleosome has the length about 197bpand its writhe −2.
induced from nucleosomes in it is calculated as follows. Each nucleosome has the writhe −2 with 147 bp and a linear piece of DNA called a linker which has 50 bp (see [11] for details). Therefore, we may assume that a nucleosome appears at least every 197 base pairs. If we compare the number of full twists within the 197 base pairs: 197 10.5 = 18.76 (full twists.) (10) The ratio between the writhe of a nucleosome and the number of full twists of the nucleosome is: This means that the ratio of the number of fully twisted supercoils is formed by nucleosomes; that is, the negative writhe Wr(γ) induced from the nucleosomes is at most about 10.7%. Therefore, by Lemma III.2, the equation lk(S 1 , S 2 ) = Tw(S 1 , S 2 ) + Wr(γ) implies that about 90% of Tw(S 1 , S 2 ) should be reduced by some operations.
For instance, it is estimated in [7] that the length of the replicon (loop) is about 100Kbp. Then the number of full-twists in one replicon (loop) is given by: The length for each nucleosome is about 197bp. Suppose that for every 197bp, there is one nucleosome. Then the number of nucleosomes is Each nucleosome has the writhe −2. Therefore, Wr(γ 0 ) is estimated as −1K.
This observation implies that about 8.5K of fulltwisting should be reduced by some operations.

B. Recursive unknotting operations
The enzymes TopoIA and TopoII act as unknotting operators to single and double strands respectively.
Theorem III.1 suggests the following operation. TopoIA is activated at the adjacent two crossings as shown in Figure 13.
This modification releases exactly two and a half twists. This decreases the twisting number 2.5. This relaxation does not require the counter rotation. This is different from the topological model proposed in [5]. The next problem to solve is to know how many times TopoIA should be applied to change the crossings during the replication in the replicon. From (12), the number of full twists in the replicon is about 9500 denoted by Tw 0 . Thus the number of places on the replicon where the TopoIA is activated is: The resulting relaxed DNA has now about 3800 crossings. For each replicon, about 3800 of activations of TopoIA will be needed to reduce the linking number. After activating TopoIA to one place, the linking number is reduced from 2.5 to 0.5. From this observation, if TopoIA is pairwisely activated at every 2.5 full-twists, the twistings will be reduced to 20% of the twistings Tw 0 of the replicon. It is denoted by Tw 1 .
Tw 1 = 0.5 × 3800 ≈ 1900 If TopoIA is applied again at every 2.5 full-twists of the relaxed DNA again, the twistings will be reduced to 4% of Tw 0 denoted by Tw 2 .
Tw 2 = 1900 2.5 × 0.5 = 380 This means that after applying the unknotting operation twice, only 380 twisings remain. Therefore, the unknotting operation gives Tw 2 which is 4% of Tw 0 . From Section IV-A, the writhe induced from the nucleosomes is at most 10% of Tw 0 . The twisting number Tw 2 is less than half of 10%. Therefore, from the formula (3): and Lemma III.3, the link is very close to be split.

V. CONCLUSION
In this paper, we studied a possible topological model for a replicon equipped with unknotting operations. It is proved that the model is efficient to make the linking number of the replicon close to zero. However, it remains that the existence of this kind of mechanism is unknown. This should be examined through experiments.