数理解析研究所講究録
第 1704 巻 2010 年 172-186

172

Remarks to study about distribution of DNA-knot
by use of Jones polynomial of topological invariant
Isamu Ohnishi0

Takashi Yoshino 1

Department of Mathematical and Life Sciences,
Graduate School of Science, Hiroshima University

Abstract
In this paper, a finer mechanism of transcription process of gene as a cost-saving and resourcesaving transportation system of codewords information is investigated by studying distributionof the DNA knots by use of topological invariants to phage stmctural operation as one ofinteresting examples. This work is just a finer mathematical analysis to the mechanism by use of
topological invariants, but better understanding makes better using, and moreover, it is known that
there are some cases in which some pieces of global topological information is very important fortranscription process. To use the topological invariants has another important point, because it
makes the simulation time be shorter and will be able to make simulation itself be precise even in
the case of big crossing number. We use two types of topological invariants: Alexander polynomials
and Jones polynomials. Generally speaking, Alexander polynomials are easy to use, but have lower
separating ability. On the other haiid, Jones polynomials are somewhat difficult to utilize, but havefiner separating ability. We compare the results gotten by both ways of simulations to discuss aboutboth advantages and disadvantage to suggest that they should be used according to the objectives.
Moreover, these results are compared with the actual biochemical experimental results to make aninterpretation to what happens in the cell, and we will state perspectives in the future.
$\lambda$

1$s$Introduction

Due to the central dogma of molecular biology, the transcription-translation process based on informations on DNA is essential for decision making in a cell, but nowaday, it is well-known that there only arefewer genes than expected on a DNA, compared to the diversity $ot$ results of expression of genes. For
example, in human-being, before the genome project, it was said that there must be more than 100,000
genes, although only 22,000 approximately genes have existed actually. Several devices for transport of
inforrnations on DNA make it possible and, as a consequence, it saves costs about the transcriptiontranslation system. Analysing the mechanism is not only important for progress of molecular biology,
but also it has a possibility for the transportation system of complex informations, at least in nano-scale,to be improved very well by use of the mechanism. For the purpose, as the first step, it is nescessary
for the biological system to be analysed and to be understood in details experimentally as well as mathematically. When the transcription and translation occur, genomes are variously modified, and it isbecause of the diversity of expression. Typical operations are “conservative site specific recombination“and ”transposition“. These two operations are responsible for a lot of events in a cell.
“Conservative site specific recombination“ occurs at defined sequence elements in a DNA. Recombinase
proteins recognize these sequence elements and act to containing the recombination sites. Three types ofrearrangements are common: DNA insertion, DNA deletion, and DNA inversion. These rearrangements$0$

isamu-o@math.sci.hiroshima .ac.jp$arrow u$

1 m083450@hiroehima.u.ac.jp

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have many functions, including insertion of a viral genome into that of the host cell during infection, rosolving DNA multimers, and altering gene expression. There are two families of conservative site-specificrecombinases. Both families cleave DNA using a protein-DNA covalent intermediate. Tkansposition is aclass of recombination that moves mobile genetic elements, so called transposons, to new genomic sites.
There are three major classes of transposons. Tkansposons existin the genomes of all organisms, where
they can constitute a huge distribution of the total DNA sequence. They are a major cause of mutations
and genome rearrangements. TMansposition is often regurated to help ensure that transposons do notcause too much of a dismption to the genome of the host cell. Control of transposon copy number andregulation of the choice of new insertion sites are commonly observed. (See [7] for the details about thebiological point of views.)
As in the above summary, by use of very clever spontaneous mechanisms in nano-level in a cell, thecost-saving effective transport system is established and realized. A kind of leaning of distribution for
DNA is often produced by such kind of operations of the enzyme of nucleic acid. If this kind of leaning
can be calculated experimantally and it can be interpreted theoretically, then it should be useful as anindex about degree of the operations: How much was it operated by the enzyme ? If so, it also is an
important index in order to understand the control and the reguration. We pick up topological type of
DNA knot operated by macro-phage in E. coli (See in [2]). As in Procaryote, there is creature with ringtype DNA. Distribution of topologiCal type of DNA knot and its changing give us an index of the degreeof the operations. In this paper, calculation method and its result are presented by use of topologicalinvariants of knot: Alexander polynomials and the Jones polynomials. Both are well known as the
topological invariants for the knots and each quantity has its advantages and its disadvantages. We
calculate the Distribution of topological type of DNA knots in both ways, and consider about meaningof these. Moreover we compare them to the biochemical experimental results, and make a study about
how much it can be operated to give a theoretical result.
Before stating our method and result, we revise the structure of DNA and some examples of nucleic
acid enzyme changing DNA structure:

1.1 Topological Structures of a DNA
The structure of DNA is a double helix, much like a ladder that is twisted into a spiral shape. In 1953
the double-helix structure of a DNA was proposed by J. D. Watson and F. H. Crick with techniqueto build a molecular model. The double helix is the ideal model that J. D. Watson and F. H. Crickarrived at from many studies about a DNA and seven next important characteristics are emphasized in
the structure.
1. The double helix is formed by two polynucleotide.
2. A purine and a pyrimidine ring orient it in the inside of the double helix.
3. The base in complementary relations is bound by hydrogen bond.
4. There is 10.4 base pair per spiral 1 round per minute.
5. As for two polynucleotide of the double helix, a direction is reverse (reverse parallel) each.
6. Double helix has two kinds of grooves, the major groove and the minor groove.
7. The double helix is clockwise twining (right handed).

When a double-stranded DNA of the double-helix stmcture is twisted more or it is untied, the double
helix is more twisted as a whole. These are called the supercoiled structure (a positively supercoiled
DNA is the former and a negatively supercoiled DNA is the latter, respectively).
The ringed supercoiled DNA closed by covalent bond has ‘the twist“ and “the writhe”. “The twist”
is the spiral number of revolutions that one chain surrounds the chain of the other, that is the numberof times that one chain completely coils the chain of the other. “The twist” in a right-handed spiral isdefined as plus. “The writhe“ is a number that the double strand coils oneself or the double strand iscoiled cylindrically like the cord of the telephone. The total value of (the twist” and ‘the writh cannot
be changed unless we cut off the DNA. We consider of the topoisomerase for an example as an enzyme$\iota$$\iota$$e^{)}$

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changing topological structure of the DNA.

1.2 Topoisomerase
We here make an introduce to Topoisomerase as an important example of enzyme of nucleic acid which
is changing the DNA topology while it works. The topoisomerase is the enzyme which can change
structures of the DNA by cutting the single strand or the double strand of the DNA temporarily.
The topoisomerase has two types. Type I topoisomerases cut the single strand of the DNA temporarily
and connect it again after dipping the other single strand. Type II topoisomerases cut the double strand
temporarily and close after dipping another double strand which is not cut.
Both a prokaryote and organism with a nudeus have type I and II topoisomerase which can remove thesupercoiled structure from the DNA. Prokaryotes possess two structurally similar type II topoisomerases,
Gyrase and Topo IV, which differ in their function. Gyrase generates negative supercoils, but Topo IV
relaxes positive supercoils. In addition, it has reported that Gyrase is strongly bound with DNA, but
Topo IV is weak [1].
The global structure of DNA strongly ties to the expression pattem of DNA by the property of TopoIV and Gyrase, since the different DNA stmcture tends to produce different DNA modification evenunder the same local condition. For example, when Gyrase was added to the positively supercoiled
DNA, Gyrase unties DNA since Gyrase generates a negatively supercoils, but when Gyrase was added
to the negatively supercoiled DNA, Gyrase twists DNA more. Thus, the global structure of the DNA is
important for an enzyme.
It is known that the global topological structures of DNA tend to affect the expression of transcripts.
DNA is affected by other $nucJeic$ acids and enzymes in the process of DNA transcription. The global
structure of DNA strongly ties to the expression pattem of DNA, since the different DNA structure tends
to produce different DNA modification even under the same local condition. Then it has been reported
that the global topological structures of DNA influences expression of transcripts by performing various
topological changes in genetic recombination $[?]$ . In this paper we describe the topological structures of
DNA in a phage.

1.3 Structures of aA$\lambda$$\lambda$

phage

phage is a virus which infects bacteria. It is well known that it has tails and a regular icosahedralhead(see Fig. 1). The phage is the simplest system to study a basic process of life. The genome is small
generally and the genome reproduce only after invading a host cell (a bacteria cell). Then a gene isexpressed. A genome tends to be rearranged while infection.

Because the system is simple, the phage was used very well at the molecular biological dawn. Actually,
the phage was indispensable for the development of this field. The system is also most suitable for astudy of the DNA replication, the gene expression and basic structures of the recombination today.
Furthermore, a phage is important as a vector of the DNA recombination, and it is used to examine theactivity inducting variation of various compounds.
Phages usually pack genomes (generally, DNA) to a mantle consisting of subunits of the protein. A
subunit has a part making head structure (packing genomes) and an another part making tail struc,ture.
The phage particle is glued to the outside of the bacteria host cell with a tail and injects the phage
genome to the cell. Because each phage glues to a specific molecule (usually, protein) in the cell surface,
only a cell having a coping acceptor is infected with the phage.
A phage has a double-stranded DNA controlling a genetic information in the head. The linear DNA
packed in the head of phages forms various types of knots. As the part of knots, Fig.2 shows knots ,$3_{1},4_{1},5_{1},5_{2},6_{1},6_{2},6_{3},7_{1},7_{2},7_{3},7_{4},7_{5},7_{6}$ and.
It is the important step of the DNA studies to investigate the Distribution of various types of knots
and clarify the reasons of the phenomena.
$0_{1}$$7_{7}$

175$6_{1}\ovalbox{\tt\small REJECT} 6_{2}\ovalbox{\tt\small REJECT} 6_{3}\ovalbox{\tt\small REJECT} 7_{1}\ovalbox{\tt\small REJECT}^{\gamma_{2}}\ovalbox{\tt\small REJECT}$$7_{3}\ovalbox{\tt\small REJECT}^{7_{4}}\ovalbox{\tt\small REJECT}^{\gamma_{5}}\ovalbox{\tt\small REJECT} 7_{\epsilon_{\ovalbox{\tt\small REJECT}^{7_{7}}\ovalbox{\tt\small REJECT}}}$

Fig. 2 The prime knots having below 7 crossing

Fig. 1 A bacteriophage

1.4 Results of biological experiments
The DNA inside the phage was examined types of knots which is made inside the phage and the probability by J. Arsuaga et. al. [2]. First, purified DNA was analyzed by two-dimensional gel electrophoresis.
By gel electrophoresis, we can distinguish some knot types with the number of the same crossing number.
To the direction of I of Fig. 1 by [2], DNA samples were run at 0. $8V/cm$ for $40h$ at room temperature.
After a 90’ rotation of the gel, the direction of II of Fig. 1 by [2] was run in the same eJectrophoresis buffer at 3.4 $V/cm$ for $4h$ at room temperature. This gel electrophoresis segregated tbe linear DNA
molecules and segregated knots with six and more crossings to two groups. Knot populations of low
crossing number are showed in Fig. 1 by [2]. The number of the right in Fig. 1 by [2] is indicated thecrossing number.
Next, the projection obtained by this experiment was compared with the projection of known knot
types to examine the knots corresponding DNA knots separated by electrophoresis (see [2]). The gelvelocity at low voltage of individual knot populations resolved by two dirnensional electrophoresis (Rightof Fig. 2 by [2] is compared with the gel velocity at low voltage of twist knots $(3_{1},4_{1},5_{2},6_{1}$ andof a $10-kb$ nicked plasmid (Center of Fig. 2 by [2]) and with known relative migration distances ofsome knot types (Left of Fig. 2 by [2]). By this comparison, some knot types of each population are
distinguished. In addition to the unambiguous knots
and , the knot population of five crossings
matched the migration of the knot . The knot
appeared to be negligible or absent. The knot
population of seven crossings matched the migration of the torus knot
rather than the twist knot .
Yet, we cannot identify this gel band as the knot , because other possible knot types of’ seven crossingscannot be excluded.
Then, several indicators led us to believe that the second arch of the gel ( $6’-9’$ of Fig. 1 by [2])consists of mainly composite knots (see Def.2.6 in Chapter 2). First, the arch starts at knot populations
containing six crossings, and no composite knots of fewer than six crossings exist. Second, the populationof six crossings matched the migration at low voltage of the granny knot (composite of aplus a ,indicated as), although the square knot (the other possible composite of six crossings,)
cannot be excluded. Third, consistent with the low amount of knots, the size of the seven-crossing
subpopulation is also reduced; thus, any composite seven-crossing knot is$(or -3_{1}\# 4_{1})$. Theincreased gel velocity at high voltage of composite knots relative to prime knots (see Def.2.5 in Chapter
2 of the same crossing number likely reflects distinct flexibility properties of the composites during
electrophoresis.
Densitometer readings confirmed the apparent scarcity of the knot of four crossings (knot ) relative
to the other knot populations in the main arch of the gel. Fig.3 shows the results of the Distribution of
knots
and
by biological experiments. However, the scarcity of the knot
relative to the$)$$7_{2})$$3_{1}$$5_{1}$$4_{1}$$5_{2}$$7_{1}$$7_{2}$$7_{I}$$3_{1}$$3_{1}\# 3_{1}$$3_{1}$$3_{1}\neq-3_{1}$$4_{1}$$3_{1}\# 4_{1}$$)$$4_{1}$$3_{1},4_{1)}5_{1}$$5_{2}$$4_{1}$

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knot of three crossings (knot ) and to other knot populations is enhanced if we make the correction for
DNA molecules plausibly knotted outside the viral capsid. In such a case, one can predict that 38% of
the total number of observed knots and 75% of the observed knots are formed by random knottingin free solution.
$3_{1}$$4_{1}$$3_{1}$$*1-1-*u$$1*A\cdot 1L1\cdot\sim r\cdot urightarrow$

Fig. 3 The results of the Distribution of knots
and 52 by biological experiments [2]$3_{1},4_{1},5_{1}$Fig. 4 Quantification of the individual knot populations of three to nine crossings

1.5 Outline of this study
To study the geometry of the DNA inside a phage, we simulate Distribution of the various types ofknots and compared to those of biological experiments.
First, we simulate umder the assumption that DNA knots inside the phage are generated at random.
With Mersenne Twister we randomly make the knot topology having crossings $(n=3,4, \cdots, 8)$ . This
process is repeated 1, 000, 000 times to obtain 1, 000, 000 knots. Then we classified these knots with theinvariant for the knot and estimate the Distribution of the knots. These trials are repeated 100 times
and we consider the mean value of all trials as the conclusive Distribution of the above-mentioned typesof knots.
Next, we examined ‘the writhe value” $Wr$ at each crossing. After eliminating matrices having thesmall value of the total amount $\sum Wr$ , we performed the same simulation on the remaining matrices.
after considering 1‘the writhe value” were similar to the
The Distribution of the knots
andbiological results.
$\lambda$$n$$\iota$$3_{1},4_{1)}5_{1}$2$5_{2}$

Preparations from Topology

Def. 2.1 (Alexander matrix) Let be a group and let : $Garrow G/[G, G]$ be the Abel map. The imageof Jacobian by the Abel map is called Alexander matrix of .
$G$

$A(G, \alpha)$$\alpha$$G$$\alpha$Def. 2.2 (Alexander polynomial) The g. c. d. of determinant of $(n-1)\cross(n-1)$ submatrix of $nxn$
Alexander matrix is called Alexander polynomial.

Def. 2.3 (Trivial knot) Let
be a knot. If there exists, then $K$ is called the trivial knot.
$K\subset S^{3}$$K\subset S^{2}$for a two-dimensional sphere$S^{2}\subset S^{3}$

Def. 2.4 (Connected sum of knot) Let
be knots and let $B_{1}=S^{3}-intN(x_{1})$ ,$B_{2}=S^{3}-intN(x_{2})$ for points $x_{1}\in K_{1},$ $x_{2}\in K_{2}$ . If we set$(i=1,2)$ , thenis a
is a one-dimensional sphere in .
two-dimensional sphere and$K_{1}\subset S^{3},$$K_{2}\subset S^{3}$$S_{i}=\partial B_{1}$$K_{i}\cap S_{i}$$S_{i}$$S_{i}$

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Then we identify

withand identify
withby a homeomorphic mapping, and we connect
and.
The obtained knot
is called the connected sum of, we use the notation
and.
Conversely, it is called that $K$ is decomposed
and.
Def. 2.5 (Prime knot) For the arbitrary decomposition $K=K_{1}\# K_{2}$ of the knot $K$ , if at least one oforis the trivial knot, then $K$ is called the prime knot. However, we define that the trivial knot is
not the prime knot.
$S_{1}$$K_{1}\cap S_{1}$$S_{2}$$S_{1})arrow(S_{2}, K_{2}\cap S_{2})$$K_{2}\cap S_{2}$$(B_{1}, K_{1}\cap B_{1})$$(S_{1},$ $K_{1}\cap$$(B_{2}, K_{2}\cap B_{2})$$K\subset S^{3}$$K_{1}$$K_{1}$$K_{2}$$K_{1}\# K_{2}$$K_{2}$$K_{1}$$K_{2}$

Def. 2.6 (Composite knot) If there exists a decomposition $K=K_{1}\# K_{2}$ of the knot $K$ such thatis the non-trivial knot which is not prime, then $K$ is called the composite knot.

$K_{1}$

and$K_{2}$

Def. 2.7 (Skein relation) Let be the nonzero complex number and let $L+,$that only one part crosses like a Fig.5 and the other part is totally same.
Then
$t$

$L_{-}$

and$L_{0}$

be three knots$tV_{L_{+}}(t)-t^{-1}V_{L_{-}}(t)=-(t^{1/2}-t^{-1/2})V_{Lo}(t)$is called Skein relation.

.

$L_{-}$$L_{\{}$

Fig. 5 Three knots $L+,$

$L$

-and

$L_{0}$

Def. 2.8 (Jones polynomial) The polynomial
such that Skein relation hold is called Jones polynomial. However, Jones polynomial of the trivial knot of one component defines 1.
$V_{L}(t)$Def. 2.9 (Bracket polynomial) Thc polynomial $<D>$ of one variable
rules to the link diagram $D$ is called Bracket polynomial:
(i)
(ii)
(iii)

$A$

obtained by applying next three$<U>=1$ ;$<DU>=-(A^{2}+A^{-2})<D>$ ;$<\backslash ,’>=A<\wedge\vee>+A^{-1}<$ $)$ $(>$$<^{\backslash }\Gamma\backslash >=A<)(>+A^{-1}<\wedge>$;;

where is a trivial knot, $DU$ is an union with a trivial knot and we apply (iii) locally at each crossingof the link diagram.
Then for the link diagram of an arbitrary knot with the direction, Jones polynomial
has arelation with Bracket polynomial $<D>$ by
$U$

$D$

$L$

$V_{L}(t)$$V_{L}(A^{-4})=(-A^{3})^{-\omega(D)}<D>$ ,where$\omega(D)$is the sum of “ the writhe value“ at each crossing of the link diagram $D$ (see Section 3.1.1).

It is known that Alexander polynomial and Jones polynomial, which are the invariant for knots, iseffectively used for the classification of knots based on the topology. We can classify all prime knots
having below 8 crossing by a combination of the value of $\Delta(-2)$ and $\Delta(-3)$ when we use Alexanderpolynomial. Then we can distinguish the mirror images of knots and knots having more crossings
when we use Jones polynomial.
$\Delta(t)$178
3 Simulations
3.1 Alexander polynomial
3.1.1 Elements of AIexander matrix
We shall project the knot on a place along an arbitrarily chosen axis, while drawing breaks at the crossing
points in the part of the curve that lies below. Now the projection of the knot amounts to the set ofsegments of curves, which are called the generators. Let us fix arbitrarily the direction of passage of thegenerators and number them, having selected arbitrarily the first generator. The crossing that separates
the kth and $(k+1)th$ generators will be called the kth crossing. The crossings are of two types (seeFig.6). The writhe value $Wr$ is $Wr=-1$ at type I crossing and $Wr=1$ at type II crossing. The
elements of Alexander matrix $A(t)$ are defined as follows [3].
1. When $i=k$ or $i=k+1$ , independently of the type of crossing: $a_{kk}=-1,$ $a_{kk+1}=1$ .
2. When $i\neq k,$ $i\neq k+1$ , for a type I crossing: $a_{kk}=1,$ $a_{kk+1}=-t,$ $a_{k:}=t-1$ .
for a type II crossing: $a_{kk}=-t,$ $a_{kk+1}=1,$ $a_{ki}=t-1$ .
3. All the elements except
andare zero.
$a_{kk},$

Here

$k$

$a_{kk+1}$$a_{ki}$is the number of crossing and is the number of the overpassing generator. Then these relationunder the condition of the substitution $n+1arrow 1$ .
$i$ships hold for all $k=1,$$\cdots,$ $n$

Fig. 6 The type I crossing (left) and the type II crossing (right)

3.1.2 Generation of Alexander matrix
First, we fix the dimension of Alexander matrix $f\dot{r}om3-8$ with Mersenne Twister at random. Second,
we fix the element
matrix $A(-2),$ $A(-3)$ $(n=3,4, \cdots, 8)$ substituted$(k=1,2, –, n)$ of$t=-2,$ $t=-3$ with Mersenne Twister as to satisfy above list. When $i\neq k$ and $i\neq k+1$ , we take with
Mersenne Twister at random ( must not take and $k+1$ ). This work is repeated from $k=1$ to $k=n$
and we make thematrix corresponding to the random knot.
$n$$n\cross n$$a_{kk}$

$i$

$i$

$k$

$n\cross n$

3.1.3 Reidemeister moves
By Reidemeister moves, we untie the part which can untie of the knot. Reidemeister moves I, II (Fig.7)
are the next operation [3].
I. If the kth row contains only two nonzero elements $a_{kk}=-1$ and $a_{kk+1}=1$ , then we should
add the column to the column $k+1$ . Then we delete the kth column and the kth row andrenumber the rows and column afresh.
II. If in two adjacent rows of the matrix having the numbers and $k+1$ elements having thevalue $t-1$ lie in a single column and elements equal to $t-1$ are lacking in the $(k+1)$ th
$k$

$k$

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column, then we should add the th column to the $(k+2)$ th, and then delete the kth and$(k+1)$ th rows and columns and renumber all the rows andcolumns afresh.
$k$

The above list I and II correspond Reidemeister moves I and II, respectively.

$\Vert$$\Vert$–Fig. 7 Reidemeister moves I (left), II (right)

3.1.4 Classification of knots
We estimate the value $\Delta(-2),$ $\Delta(-3)$ of the determinant of the $(n-1)\cross(n-1)$ submatrix and classify
knots by the combination of the value of the determinant. Here, we removed all rows and columns fromthe first row and column to the nth row and column one by one. Then we treated as a knot whenthe value $\Delta(-2),$ $\Delta(-3)$ of all determinants was equal or different
times,
times $(m$ is an
integer).
$\pm 2^{m}$

3.1.5$\pm 3^{m}$

Reliability of the simulation

We gave 1, 000, 000 matrices at random and counted the number which generated each knot and estimatedthe Distribution. These trials were repeated 100 times and we considered the mean value of all trials as
the conclusive Distribution of the types of knots.

3.1.6 Addition ofwrithe“After the simulation of the random matrices, we employed “the writhe value” in the simulation. we
examined “the writhe value $Wr$ “ at each crossing. After eliminating matrices having the small value ofthe total amount $\sum Wr$ , we performed the same simulation on the remaining matrices.
$\downarrow\prime the$3.2

Jones polynomial

3.2.1 Recurrent algorithm of Bracket polynomial
Because we can find.Jones polynomial from Bracket polynomial if we find “the writhe value” of the knot,
we constitute algorithm by Bracket polynomial [4].
For an arbitrary link diagram $D$ , we paint with two colors of the black and white so that adjacentdomains do not become the same color. Here, infinite domain is the white. Next, we choose one point ineach black domain. If two black domains are adjacent, we link the point that we chose of each domain
and we establish the sign of the edge based on Fig.8. Thus, the plane graph with the sign is obtained.
We define thatis the graph which deleted the edge from the graph and $S/e$ is the graph which
contracted the edge from (that is the edge is condensed and the top of both ends correspond one
point). Then we define that the loop is the edge where both ends are the same tops and the bridge isthe edge where the connected component of the graph increases one when the edge is deleted.
It is known that the contraction /deletion formula of the edge holds for graph with the sign. For
example, when the edge linking two adjacent black domains is positive, the operation which tie two
black domains and the operation which divide two black domains is equivalent to the contraction andthe deletion, respectively.
$S$$S\backslash e$

$e$

$e$

$S$

$S$

$e$

$S$

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$+$

–

Fig. 8 The sign in two adjacent black domains$<S>=A<S/e>+A^{-1}<S\backslash e>$

Then when the edge is the positive bridge,
is disconnectedness (that is the number of thecomponent of the knot corresponding toincreases one), but
and $S/e$ is the isomorphism, andthe following relations hold between them.
$e$$S\backslash e$$S\backslash e$$S\backslash e$$<S\backslash e>=(-A^{2}-A^{-2})<S/e>$

Therefore, they hold the following relations when the edge$e$is the positive bridge.

$<S>=A<S/e>+A^{-1}<S\backslash e>$$=-A^{-3}<S\backslash e>$

It is similar in the case of the negative bridge or the loop too, therefore they hold the following recurrence
relations.

$<S>=$$\{\begin{array}{ll}-A^{-3}<S/e> if e is the positive bridge-A^{3}<S/e> if e is the negative bridge-A^{3}\backslash <S\backslash e> if e is the positive loop\end{array}$$-A^{-3}<S\backslash e>$$A<S/e>+A^{-1}<6\backslash e>$$A<S\backslash e>+A^{-1}<S/e>$

if
if
if

$e$

$e$

$e$

is the negative loop
is positive (otherwise)
is negative (otherwise)

Then when consists of one edge (the positive bridge, the negative bridge, the negative loop), Bracket polynomials hold the following relations.
$S$$L^{+}$$C^{+}$$C^{-}$, the positive loop$L^{-}$$<C^{+}>=-A^{-3},$ $<C^{-}>=-A^{3},$ $<L^{+}>=-A^{3},$ $<L^{-}>=-A^{-3}$

Furthermore, when the edge set of$S$

is the empty, Bracket polynomial has 1 from 2.9 (i).

3.2.2 Generation of the random plane graph with the signFirst, we fix the number of the top from 2–9 with Mersenne Twister at random and also fix the numberof the edge from 3–9 at random. Next, we choose tops linked each edges at random and give the signfor each edges. For the sign of each edges, we tried three kinds; at random, only plus and only minus.
Thus, we generate the random plane graph with the sign.

3.2.3 Classification of knots
We evaluate the recurrence algorithm of Bracket polynomial for the given graph. Then we classify knots
by the polynomial which is finally obtained. Here, because we considered the difference of the value of“the writhe value” of the knot, we also treat as the knot when the polynomial which is finally obtained
is different $(-A)^{3m}$ times ( $m$ is an integer).

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3.2.4 Reliability of the simulation
We give 100, 000 matrices at random and counted the number which generate each knot andestimatethe Distribution. These trials are repeated 100 times and we consider the mean value of all trials
as theconclusive Distribution of the types of knots.
3.2.5 Decrease of the BDD size
Since Bracket polynomial is the same for 2-isomorphic graphs, we may represent 2-isomorphic
minorsamong them by one of these members. However, testing the 2-isomorphism of arbitrary graphs is hardin general, and finding a112-isomorphic minors may be difficult.
The 2-isomorphism between two graphs whose edges have an identity map can be determined inlineartime [5]. For this reason, we may restrict ourselves just to find 2-isomorphic minors whose correspondingedges have the same order in the original graph . By modification, the expansion tree
becomes anacyclic graph (edges are directed $hom$ a parent to a child). This acyclic graph has a single
source (theoriginal graph ) and the m-th level may be regarded as a single sink. We call this acyclic
graph theBDD of all spanning trees of a graph with respect to the edge ordering. This is because
this acyclic graph has strong connection with the BDD (Binary Decision Diagram) representing
all thespanning tres of a graph. The size of this BDD is defined to be the number of its nodes. The
width ofthis BDD is defined to be the maximum among the numbers of nodes at each level.
In graph with the sign, any signs of remaining edges don’t change even if we contract ordeletethe edge . Therefore, the isomorphic minor which is also the same order of edges is the isomorphism
including the sign of each edge. That is, we can unify all isomorphic minor which is also the same orderof edges and calculate it.
We can reduce the number of minor if we take the order of calculating edges and tops well. First we
decide the first top . The top
is chosen as follows.
$S$

$S$

$S$

$e_{1},$$e_{2)}\cdots,$ $e_{m}$

$S$

$e$

$p_{1}$$p_{1}$

Next we decide the top . We choose the top
which is worked above 1.. 2. for tops connected tothe top . Furthermore we choose also the top
which is worked above 1., 2. for tops connected tothe top . If the top
isn’t connected to the top , we choose the top
which is worked above 1. 2.
for tops connected to the top . We choose after the top
equally. Thus we decide the order of tops
and calculate the graph from the edge which is connected to small number tops. In this way
we wereable to calculate Jones polynomial up to the maximum 20 crossings. However, in this algorithm,
thecalculation is finished only Jones polynomial up to the maximum 16 crossings because the calculation istaken quite a long time. Before decreasing BDD size, for crossings, the number of minor was need 2
theoretically, but by our simulation, the number of minor was decreased 44, 52, 100, 80, 109, 169 for 11,
12, 13, 14, 15, 16 crossings respectively. However, we may make better algorithm if we see
to ref.[4].
$p_{2}$$p_{2}$$p_{1}$$p_{1}$$p_{3}$$p_{3}$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$n$$’\iota$4

Results, Discussions and Concluding remarks

4.1

Comparison of the results by Alexander polynomial and those by Jones polynomial

When knots are generated at random, Fig.9 shows the results by Alexander
and Fig. 11 shows
those by Jones polynomial ( -suffix in Fig.11 represents the mirror images of the knots). In both results,
the distribution of the knot
is the highest, the distribution of is the second highest,is the third
andis the fourth. In the group of the knots having 6 crossing, the distribution of the knot
is thehighest, the distribution of 63 is the second andis the third. In this way, the distribution of each knot
obtains the almost same tendency, even if any topological invariant is used. Furthermore, by the results
by Jones polynomial, the population of each knot possesses the approximately same as theone of themirror images at each knot group having the same number of the crossing.
$polynomi_{C}\sqrt{}$$*$$3_{1}$$4_{1}$$5_{2}$$5_{1}$$6_{2}$$6_{1}$

182

Fig. 9 Distribution distinguished knots generated at random by Alexander polynomial

Fig. 10 Enlarged picture of the part of 6 7
crossings of Fig.9

Fig. 11 Distribution distinguished knots generated at random by Jones polynomial

Fig. 12 Enlarged picture of the part of 6 7
crossings of Fig. 11$\sim$$\sim$

4.2 Comparison of the results of the simulation and those of biological experiments
About knots
and , we compare the results of the simulation by Alexander polynomial withtlie results of biochemical experiments. The distribution of knot types that emerges in biochemical
experiments done by Professors J. Arsuaga et. al. [2] is sliown in Fig.13. In the biochemical experiments
the distribution of the knot
is the highest and the Distribution ofis the second highest, while
are low (Fig.13). In contrast, the distribution of knots that are computationally generated atandrandom (Fig.14) is very different from the results of biochemical experiments. In Fig.14, the abundanceof is higher thanand .
However, by employing “the writhe value” in the simulation, the result is changed. The distributionof the knots
andafter considering (the writhe value” tends to be rather similar to thebiochemical result (Fig.16).
$3_{1},4_{1},5_{1}$$5_{2}$$3_{1}$$5_{2}$$4_{1}$$5_{1}$$3_{1)}4_{1},5_{1}$$5_{2}$$5_{2}$$5_{1}$$4_{1}$

183

Fig. 13 The results of biological experiments by
J. Arsuaga et. al. [2]Fig. 14 Distribution of the knots
generated at random

Fig. 15 Distribution of the knots
generated at random

Fig. 16 Distribution of the knots
after examining “the writhe value”$4_{1},5_{1}$and$5_{2}$$3_{1},4_{1},5_{1}$and$5_{2}$$4_{1},5_{1}$and$5_{2}$

4.3 The results in case of giving lean of the total sign on all the edges by Jones polynomials
When we give only the plus sign to all the edges, the population of the mirror images is higher in themost knots group having the same crossing number (Fig. 17). On the other hand, when we give only theminus sign, the inverse result happens (Fig. 18).
Furthermore, let us see how the distribution changing, according to varying the total sign of the edges.
We denote the absolute value of the total sign of all the edges as. For $|L|\geq 0$ , the distributionof the knots are decreasing with the change of the crossing number(Fig.19). However, for $|L|\geq 8$ , thepopulation of the knots with 4 crossings are rapidly decreased (Fig.20) and makes a hollow in the bar
graph. This also has the same tendency as in the biochemical experiment done by Professors J. Arsuaga
et. al. [2].
$|L|$

184
Un

$1r$

Fig. 17 Distribution with only plus for the signof all edges$\cdot$Fig. 18 Distribution with only minus for the signof all edges
ILI ’

Fig. 19 Distribution of the knots for $|L|\geq 0$

4.4$\cdot\cdot$

Fig. 20 Distribution of the knots for $|L|\geq 8$

Discussions

We conclude that DNA knots inside the phage are not generated randomly and they would have somebias related to “the writhe value”. This means that the enzyme of nucleic acids have an operation whichmakes DNA knots having such a kind of leaning in the cell. This is a piece of important information
about the total operation of the enzymes, and moreover, can be one of the keys by which we will solve
more precisely a piece of the mechanism producing the cost-saving and resource-saving transportation
system of the transcription-translation process in the cell. It is important partially because such kindof leaning may cause some obstacles for transcriptions, if the leaning is too heavy. Interestingly andmarvelously, it is very important that the topological invariants extract global information about DNA
knots mathematically. We hope that, if this direction of research is proceeded a little bit more andmore, then it is possible that such purely mathematical notions are more and more useful for this kind ofanalysis and may get a more advanced application, because several examples are known there in which
the global topological informations are remarkably important. Further, Jones polynomials can separatea kind of enantiomers (a kind of mirror image of knots) from the other, as we have done here. If a$\lambda$

185um$* 4\cdot\cdot$

Fig. 21 Enlarged picture of the part of 6–7
crossings at the maximum 9 crossings

Fig. 22 Enlarged picture of the part of 6 7
crossings at the maximum 16 crossings$\sim$

distribution separating such a kind of enantimers is obtained biochemically experimentally, this type ofsimulations may be compared with it to get something important about the enzyme’s operation. For
instance, our simulations indicates that both the one and the enantimer has the almost same ratio in
the distribution (this is because the initial state is randomly given.) If there is a remarkable leaning in$bioc1_{1}emica1$ experinlelits under some conditions, thenwc will conjecture that the enzyme has this kindof operation, which is maybe interesting dependingly upon cases.
While the crossing number is very low, these simulations have been done well as a tendency by useof both topological invariants. Especially, we can give the leaning of “the writhe valu explicitly, whenusing Alexander polynomials, so that the result is clearer than in the simulation of Jones polynomials
in this point of view. This is an advantage of Alexander polynomials, although Alexander polynomials
haslower resolution for knots than the Jones polynomials (See also [9]). For instance, AlexanderpolynomiaJs c\v{c}mnot separate some nontrivial knots from the trivial one. But, as we cho $ose$ the testing
population randomly a hundred times to test it and take the average distribution among a hundred
attempts, we can expect that the no-well separating objects are randomly scattered a kind of good way.
Therefore, Alexander polynomials have a piece of reason to make a simulation with these, if the crossing
number is less. Especially, it is used easily and takes a shorter time to spend than Jones polynomials.
On the other hands, we should not use Alexander polynomials in case of separating much more typesof knots. It is well-known that there exists a nontirvial knot which is not separated from the trivial knot,
and moreover it is known that there are several examples indicating low separating ability even in thecase of knots with few numbers of crossing. In the actual biochemical experiments, the types of knots
separated explicitly are from 3 to 7 (See also in Fig.13 of Professors J. Arsuaga et.al. [2]), but there
seems to be much more types of knots and much more quantity of knots, which cannot be separated by’this experiment and which are gotten into “thetypes. If we use Jones polynomials, instead, thentheoretically we can separate them exactly in the case of the crossing number less than or equal to nine(This is easily ensured by computer calculations, and we have also done it already).
In Jones polynomiaJs case, there is one difficult point how to give “the writhe value” explicitly insimulations. Here, although we cannot give the leaning of “the writhe value” explicitly, instead, we givethe bias of the total sign of all the edges in the scheme by which Jones polynomia]s can be calculated. By
this device, the knots are implicitly given the leaning, but we cannot measure how much it is explicitly.
The resulting bar graph also has the same tendency as Alexander polynomials, but the hollow is smaller
than in the case of Alexander polynomials. It seems that it is very difficult to give the leaning of“the writhe value” explicitly in the definition of the Jones polynomials so far. But in spite of such a
disadvantage, the Jones polynomials are a very powerful tool, which is much higher resolution of knots$e^{)}$$n111c^{\backslash }h$$other^{)}$

186
than the Alexander‘s ones. This can separate even the mirror images from the knots. If biochemical
experiments get much stronger separation ability, then the Jones polynomials method is more and more
useful.
This kind of simulations’ results can be a kind of index of degree of operation of the enzyme of nucleic
acid. This is because the operation is repeated a lot of times, and the leaning is more and more. But
we must point out that the Jones polynomials cannot always separate two arbitrarily given knots. For(See [8]). furthermore, it is
has the exactly same polynomial as
example, it is known thatpolynomial constructed
thesameobtaining
knots
many
distinguishedknown that there are infinitely
little,the initial states areverybecauseaffectedbeitselfcanconcretely (See [6]). But the distribution
randomly.
emergies
unseparable
knots
kind
of
such
expectedthat
randomly,
it
is
that
so
given
Now, we mension how the simulation results agree with the biochemical experiments’ distribution ofand .
is much bigger comparably thanknot types. We make a glance at Fig. 13 to notice that
This suggest that it has a kind of leaning of “the Writhe value” of the knots. Our simulation ensured thisproperty theoretically. Namely, in both ways of simulations, if “the writhe value” is let biased more andmore, then the shape of distribution of the topological knot type is similar to the one of the experiment
more and more (See Fig. $s14,15$ , and 16).
When moving eyes to the finer part, we find some little differences of distributions between the biochemical experiments and our simulation results. For example, in the case of actual biochemical experiments,
is greater than in Fig.13, and on the other hand, the inverse is holds in the both cases of simulation
greater than the others is agree with the actual biochemical experidone by us in Fig.14, although
are too low in the whole distribution. One reason may be that thements. In another point, 6 , andmaximum crossing number of initial state is too small. The initail crossing number is only 11 maximallyso far in our simulation. If the initial crossing number is let much bigger, then the distribution is changing and expecting to be more similar shape of the bar graph. In fact, if the maximum values of initial
are increasing (Seecrossing number is changed kom 9 to 16, then ratios of distribution of 6., andpossiblethat
there may be someitisdisagreement,
first
hand,
the
for
Fig.22).
On the otherFig.21 andunknown biological mechanism. In the actual biochemical experiment, the 4-crossing knot population isis rather big. This is caused of some kind of biological actions. In the future, ifvery low, although
we make the simulation methods be better to calculate knots having more crossings in the shorter time,
we will make a finer interpretation and a precise prediction.
$\}$$8_{8}$$10_{129}$$4_{1}$$5_{1}$$5_{2}$$5_{2}$$4_{1}$$5_{1}$$7_{*}$$*$$7_{*}$$3_{1}$

References[1] G. Charvin, T. R. Strick, D. Bensimon and V. Croquette, Topoisomerase JV Bends and Overtwists
DNA upon Binding, Biophysical Journal, Vol.89, (2005), 384-392.
[2] J. Arsuaga, M. Vazquez, P. McGuirk, S. Trigueros, D. W. Sumners and J. Roca, DNA knots reveala chiral organization of DNA in phage capsids, PNAS, Vol.102, (2005), 9165-9169.
[3] M. D. Frank-Kamenetskii and A. V. Vologodskii, Topological aspects of the physics of polyrners: The$theo\eta$ and its biophysical applications, Sov.Phys.-Usp.(Engl.TYansl.), Vol.24, (1981), 679-696.
[4] K. Sekine, H. Imai and K. Imai, Computation of the.Jones Polynomial, rRansactions of JSIAM,
Vol.8, (1998), 341-354.
Polynomial of a Graph of Moderate Size,[5] K. Sekine, H. Imai and S. Tani, Computing theProceedings of the 6th International Symposium on Algorithms and Computation (ISAAC‘ 95),
Lecture Notes in Computer Science, Vol.1004, (1995), 224-233.
[6] T. Kanenobu, Infinitely many knots unth the same polynomial, Proc. Amer. Math. Soc. Vol.97,$\mathcal{I}htte$(1986), 158-161.

[7] J. D. Watson, T. A. Baker, S. P. Bell, A. Gann, M. Levine, and R. Losick, 1‘Molecular biology of thegene” (the sixth edition), Pearson, Benjamin Cummings.
[8] W. B. R. Lickorish, “An Introduction to Knot Theory”) Springer.
[9] T. Yoshino, and I. Ohnishi, Applied analysis to DNA knot by topological $invar\iota$ ants, Research reportof RIMS in Kyoto University, 1616, (2008), 181–194.