Xia Hua, Diana Nguyen, Barath Raghavan, Javier Arsuaga and Mariel Vazquez- Random state transitions of knots: a first step towards modeling unknotting by type II topoisomerases

8/3/2019 Xia Hua, Diana Nguyen, Barath Raghavan, Javier Arsuaga and Mariel Vazquez- Random state transitions of knots: a

1/25

Random state transitions of knots: a first step towards modeling

unknotting by type II topoisomerases

Xia Hua2, Diana Nguyen3, Barath Raghavan4, Javier Arsuaga5, and Mariel Vazquez1

Xia Hua: [email protected]; Diana Nguyen: [email protected]; Barath Raghavan: [email protected];Javier Arsuaga: [email protected]; Mariel Vazquez: [email protected]

1 Mathematics Department, San Francisco State University, San Francisco, CA

2 Mathematics Department, The University of Hong Kong, Hong Kong

3 David Geffen School of Medicine, UCLA, Los Angeles, CA

4 Computer Science and Engineering Department, University of California at San Diego, San Diego,

CA

5 Mathematics Department and Center for Computing in the Life Sciences, San Francisco StateUniversity, San Francisco, CA

Abstract

Type II topoisomerases are enzymes that change the topology of DNA by performing strand-passage.

In particular, they unknot knotted DNA very efficiently. Motivated by this experimental observation,

we investigate transition probabilities between knots. We use the BFACF algorithm to generate

ensembles of polygons inZ3 of fixed knot type. We introduce a novel strand-passage algorithm which

generates a Markov chain in knot space. The entries of the corresponding transition probability matrix

determine state-transitions in knot space and can track the evolution of different knots after repeated

strand-passage events. We outline future applications of this work to DNA unknotting.

Keywords

Topoisomerase II; DNA knots; polymer models; BFACF algorithm; random knotting

1 Introduction

Biological systems pose a wealth of problems that give rise to very interesting mathematical

questions. We are interested in the process of DNA unknotting by type II topoisomerases (topo

II). Motivated by this biological problem, we study state transitions within knot space for all

prime knots with 8 or fewer crossings and fixed length.

We model each knot of type K and lengthL as a polygonal chain in the simple cubic lattice

(Z3). We explore the space of configurations of type K and lengthL e using the Monte CarloBFACF algorithm, first proposed by Berg and Foerster [4], and Argao de Carvalho, Caracciolo

and Frlich [2]. The BFACF algorithm generates a reducible Markov Chain whose ergodicity

Correspondence to: Mariel Vazquez, [email protected] .

Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers

we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting

proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could

affect the content, and all legal disclaimers that apply to the journal pertain.

NIH Public AccessAuthor ManuscriptTopol Appl. Author manuscript; available in PMC 2009 November 17.

Published in final edited form as:

Topol Appl. 2007 April 1; 154(7): 13811397. doi:10.1016/j.topol.2006.05.010.

NIH-PAAu

thorManuscript

NIH-PAAuthorManuscript

NIH-PAAuthorM

anuscript


2/25

classes are the knot types [30][36]. We generate a list of Dowker-Thistlethwaite (DT) codes

[18][26] for each knot type K by taking multiple projections in random directions for each

knotted configuration (section 4). We perform a random-strand-passage simulation on the

obtained lists of DT codes, and at each step we use the HOMFLY polynomial for knot

identification (section 5). For fixed L and e this process results in a transition probability matrix

PLe which models random transitions in knot space. We generate PLe for several values of

L and e, and find their corresponding steady state distribution vectors. We compare our results

to those of Flammini et al. [21] and use our simulation to verify and improve, results on minimalnumber of sticks needed to realize a knot inZ3 [35]. Finally, in section 7 we outline limitations

of our method and future directions.

2 Biological motivation

Certain enzymes such as site-specific recombinases and topoisomerases are able to change the

topology of circular DNA molecules by changing their knotting, linking and supercoiling levels

[37,63,11,27,29,61,48]. The mechanisms of several site-specific recombinases have been

successfully studied using knot theory and low-dimensional topology to investigate the

topological changes performed by these enzymes [19,13,55,8]. We here study state-transitions

within knot space, a problem motivated by the action of DNA unknotting by type II

topoisomerases (topo II).

Random cyclization of linear DNA results in DNA circles that are knotted with some

probability. This probability depends on the length and effective helical diameter of the DNA

chain [45,44,28]. For instance, for DNA molecules 10.5 kilobases long free in solution, the

knotting probability is about 0.03, with a predominance of trefoils. Furthermore, Arsuaga et

al. [3] showed that knots are the most likely configuration of DNA molecules in confined

volumes. Confinement of DNA inside P4 viral capsids increased the knotting probability to

0.95. However, in vitro studies have shown that DNA knots are known to inhibit important

cellular processes such as transcription and chromatin assembly [40,32]. These findings

suggest that DNA knots are undesirable for the cell [46,16]. The cell is able to solve these

topological obstructions (i.e. DNA knots) by means of some enzymes called Type II

topoisomerases.

The molecular mechanism of topo II is fairly well understood [37,39,5,6,49]: topo II traps twosegments of the DNA molecule, cuts one of the segments, performs a strand-passage and reseals

the broken segment. It has been shown that topo II unknots and unlinks DNA very efficiently

[43], i.e. it does it beyond expected steady-state levels obtained by performing random strand-

passage. A number of models have been proposed to explain the efficiency of topology

simplification by topo II [7,21,38,52,53,60,62,50]. Our long-term goal is to find an accurate

model for the mechanism of unknotting by topo II. The first step towards this goal is to model

DNA unknotting by random strand-passage. In this paper we set the theoretical framework for

such modeling by studying the effects of random strand-passage on polygonal knots with

crossing number Cr(K) < 9 [31,21]. Flammini et al. [21] investigated transition probabilities

within knot space by modeling single phantom crossings using a modified crank-shaft

algorithm on freely-jointed polygonal chainsR3. In our approach we represent knots as

polygonal chains in the simple cubic lattice (Z3) and strand-passage is done at the Dowker code

level. A comparison with [21] is done in section 6.

Traditionally, circular DNA molecules have been modeled in the computer as polygonal chains

and their behavior in solution has been accurately simulated using Monte Carlo simulations in

3-space [58,56,59,57,54,22,42,15]. We here consider those models in the simple cubic lattice.

We work inZ3 because it has a number of advantages overR3. There is extensive analytical

work that investigates the knotting behavior of lattice chains. There is a remarkably solid

Hua et al. Page 2

Topol Appl. Author manuscript; available in PMC 2009 November 17.

NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


3/25

mathematical foundation for the development of algorithms that generate equilibrium

distributions of self-avoiding polygons inZ3 [30]. Also, polygons inZ3 inherently have volume

exclusion (which can be used to model the effective diameter of DNA). However a simplistic

lattice model is not well-suited for DNA studies. In order to apply our results to DNA we will

need to adjust our theoretical model as proposed in [54]; we leave a detailed biological

application of our work to a future publication. In section 7, we discuss steps needed to achieve

accurate modeling of DNA in solution as a polygonal chain inZ3, including excluded volume

effects, as well as future topological biases as those suggested by fig. 1 to approximate topo IIaction.

3 Basic Definitions

A knot is a proper embedding of a circle in 3-space. Here we let the knot, Kbe the image of

the circle under the embedding. Two knots K1 and K2 are topologically equivalent if they are

related by an ambient isotopy. A knot diagram is a 2-dimensional projection of the knot where

there are only finitely many multiple points, all multiple points are double and can be resolved

into over and undercrossings. Knots can be studied through their knot diagrams [41]. Polygonal

knot diagrams have the extra constraint that no vertex is mapped to a double point.

The DT code of a knot diagram is an integer vector from which the knot diagram and the knot

type can be recovered [18]. The DT code of the diagram is obtained by selecting a starting

point and an orientation. The knot diagram is followed along the orientation and each crossing

is labeled with consecutive numbers. At the end of the process, each crossing is labeled with

one even and one odd number. The numbers are stored in a 2 n matrix where the first row

lists the odd numbers in lexicographical order and the second row lists the corresponding even

numbers. This second row forms the DT code with the convention that an entry is negative if

it was assigned to a crossing while traversing the knot on the understrand.

The DT code representation is used in computational studies due to its simplicity. In particular,

DT codes can be used to compute most polynomial invariants of knots and links (e.g.[20,26]).

Furthermore, note that a change in sign in one of the DT code entries corresponds to a change

of crossing in the knot diagram (from and undercrossing to an overcrossing), and therefore to

a strand-passage in the 3-dimensional configuration. This key feature of the DT code, added

to its computational advantages, make it particularly attractive for our study.

We aim to generate lists of DT codes for each knot type which will be representative of a

polygonal chain as it samples its state space. To this end we first generate equilibrium

distributions of knotted polygons using Markov chains. Here we introduce Markov chains and

leave the details of our work to sections 46.

A discrete time Markov process is a family of random variables {Xn: n } where theXntake values in a subset ofS . S is called the state space and each element ofS,Xn, is called

a state, or configuration, of the process. A Markov process is an homogeneous Markov chain

ifP(Xn = i |X0,X1 Xn1) = P(Xn = i |Xn1) and P (Xn = i |Xn1 =j) = P (X1 = i |X0 =j) =

pij. Thepijs are called transition probabilities and the square matrix whose entries are the

transition probabilities is called the transition matrix which we denote by P. The evolution of

the Markov chain is given by the n-step transition matrix P(Xn = i | P0 =j). By the Chapman-Kolmogrov equations P(Xn = i | P0 =j) = Pn (i.e. the evolution of the chain is determined by

repeatedly multiplying P with itself). If we denote P(Xn = i) by , and , then it is

a well known fact that un = u0Pn. Therefore the evolution of the chain is determined by P and

u0. In other words, Markov chains are determined by the probabilities of their states [25].

Hua et al. Page 3


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


4/25

A statej is accessible from state i ifpij > 0. Accessibility defines an equivalence relation. A

Markov Chain is said to be irreducible is there is only one equivalence class. Otherwise it is

said to be reducible. Theperiodof a state d(i) is the . A Markov chain is aperiodic

if the gcd{d(i) | i S} = 1. A state ispositive recurrentif the probability of going back to the

same state on a finite number of iterations is non-zero. A Markov chain is irreducible if

for some n and all i,j. A Markov chain is ergodic if it is irreducible, positive recurrent,

and aperiodic.

For ergodic Markov chains the ( ) exists. A vector u is called stationary distribution

of the chain ifu = {ui | i S} is such that ui 0 and j(uj) = 1 and uj = uipij. If the state space

S is finite, then the stationary distribution can be obtained analytically. In this case u is obtained

by computing the left eigenvector ofP corresponding to the largest eigenvalue, which is 1 (all

the other real eigenvalues are strictly smaller than 1).

4 Knots in Z3 and the BFACF Algorithm

The simple cubic latticeZ3 is a graph whose vertices are the points inR3 with integer

coordinates, and whose edges connect each pair of vertices, which are a unit distance apart. A

self-avoiding polygon inZ3 (SAP) is a connected subgraph of the lattice with all vertices of

degree 2. SAPs inZ3 may be knotted. Knotting of SAPs has been extensively studied both from

a theoretical and a computational point of view [47,51]. In our study, we generate SAPs in

Z3 using the BFACF algorithm [4,2].

The BFACF algorithm was first applied to self-avoiding walks in [1,2] and has been used to

measure a number of properties of polygons inZ3 such as the writhe of knots and links, the

radius of gyration of specific knot types, and the minimal number of sticks needed to realize

a knot type inZ3 [3335].

The BFACF algorithm is a dynamic Monte Carlo method that operates on paths inZdand

samples along the realization of a Markov Chain [30]. In 3-D, the Monte Carlo method

generates a Markov Chain that is reducible and whose ergodicity classes are the knot types

[30,36]. More specifically the algorithm samples from the Gibbs distribution with one

adjustable parameterz, thefugacity per bond, where 0 zzc, and where is the

connective constant forZ3. The connective constant is estimated at = 4.6834066 0.0002 in

the cubic lattice, and therefore 0 z 0.2134. We apply BFACF to unrooted SAPs inZ3 as

explained in [30,35], and produce a set of 3-dimensional conformations representative of each

knot type with 8 or less crossings.

The Markov Chain is generated as follows. BFACF selects an edge at a random location and

performs one of the three elementary moves shown in fig. 2. Move 1 changes the length of the

polygon by (2), move 2 by (2), and move 3 leaves the length fixed. Each of these moves is

selected with a probability given byp(2),p(2),p(0), respectively. For each edge there are

four possible lattice directions to choose from, and not all three moves are possible. If the

probability of the four possibilities adds up to q < 1 then we choose to leave the walk unchanged

with probability 1 q. The BFACF algorithm is well defined and is reversible with respect to

the Gibbs distribution if and only if the following constraints are satisfied:

Hua et al. Page 4


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


5/25

Furthermore, large values ofp(2),p(0), andp(+2) would minimize the probability of null

transitions (1 q) and thus improve the algorithms efficiency. It was shown by Caracciolo,

Pelissetto, and Sokal that the following choice is close to optimal [10]. Wherez is an adjustable

parameter calledfugacity parameter. Varyingz results in a change in the mean length of the

sampled conformations. It is proved in [30] that in 3-dimensions the best choices for the

probabilities are:

The algorithm changes the length of the polygon as illustrated in fig. 3. Varying the fugacity

parameterz results in a change in the mean length of the sample confirmation. The mean length

is finite forz


6/25

5 DT Code Representations

The BFACF algorithm produced a list of chain confirmations for each prime knot Kwith length

in the interval [Le,L + e] and 8 or less crossings. Each knotted polygon sampled along the

realization of the Markov Chain was projected in 50100 random directions. The

corresponding knot diagrams and associated DT codes were computed. We thus obtained a list

of DT codes for each knot type K.

For non-trivial knots, the lists of DT codes were further processed as follows. A high percentage

of Reidemeister moves of types I and II (RI, RII; illustrated in fig. 4 as 1 and 2, respectively)

were identified and simplified. The simplification is motivated by our future goal of modeling

circular DNA chains. In the experiments that we wish to model the DNA is usually in relaxed

form and therefore presents a limited number of RI loops. For the RII moves, it can always be

assumed that a rigid rotation in 3-space may remove the RII move in the projection without

altering the configuration. Failure to remove RI and RII moves resulted in DT codes with many

entries, which may not represent the reality of the knotted DNA. After simplification, the set

of DT codes were stored for each knot. In this way, proper weights were assigned to each of

the DT codes associated to the configurations along the realization of the Markov Chain defined

by the BCFAF algorithm. Fig. 5 shows the weights for different DT code lengths associated

to a polygonal knot of type 61 and length 56 4 segments inZ3.

Removing RI and RII moves for unknotted configurations results in a set of projections with

no crossings which do not realistically represent the 3-dimensional configuration. Therefore,

in the case of the unknot we opted to not remove RI or RII moves.

The lists of DT codes produced by BFACF for a knot Kmay include some DT codes of length

much larger than the crossing number ofK. This is true even after removing RI and RII moves.

For each knot type K, the probability of occurrence of large DT codes decreases with the number

of entries (i.e. with the crossing number of the chosen projection; see fig. 5). For large DT

codes 1-crossing changes rarely dramatically increase the crossing number of the resulting

knot. In future work we expect to expand our algorithm to knots with 9 crossings. The study

for 10 or more crossings becomes combinatorially intractable. We therefore choose to restrict

our state space to the set of all prime knots with 8 or less crossings. Occasionally one strand-

passage (described in the next section) results in a knot with crossing number greater than 9,in such case we reject the configuration.

6 Strand-Passage Simulation

Once a representative sample of DT codes for each knot type K(and length rangeL e) was

found, we proceeded to estimate the transition probabilities by single strand-passage between

knot types. Transitions between states in knot space define a Markov Chain, whose behavior

is given by the transition probability matrix PLe as described below.

For each knot type K, we computed the transition probabilities after one strand-passage as

follows. We selected 30,000 samples (or 60,000 depending on the knot) at random from the

list of DT codes for K. A single strand-passage was performed at a crossing selected at random

on each DT code by simply changing the sign of the crossing. Each strand-passage led to a

new knot type K2, which was identified by computing the HOMFLY polynomial [23] with an

implementation based on the algorithm of Gouesbet et al. [24]. The HOMFLY polynomial

distinguishes between a knot and its mirror image. For each length and knot type we repeated

the simulation taking 30,000 random samples several times and computed mean and standard

deviation for each transition probability. The standard deviation was typically 10% of the

mean. Occasionally we increased the sample size, however shorter the sample sizes correspond

Hua et al. Page 6


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


7/25

to faster HOMFLY calculations which have a great impact on the overall efficiency of the

simulation.

Our working set consisted of all prime knots with 8 or less crossings. We rejected all strand-

passages taking a knot Kto a knot outside the working set, which happened only a small

percentage of the time for small knots. For example, knots with 6 or less crossings resulted in

a HOMFLY polynomial outside the working set < 1.1% of the time for mean lengthL = 44, 6. The first observation is that the diagonal values in our

matrix were approximately 10 times smaller than those in [21], ours being lower. For examplefor mean length 100 the probability of going from 31 to 31 was 0.04, while in [21] it was 0.34;

for 41 we obtained 0.045 versus 0.26 in [21]; for 51 we obtained 0.021 versus 0.23 in [21]. This

discrepancy can be attributed to the fact that we remove RI moves prior to performing strand-

passage. Intuitively a knot with RI loops will have a much higher probability to stay in the

same knot type as one for which RI moves have been removed. Preliminary runs performed

without removing RI moves prior to strand-passage gave the expected 10-fold increase in the

diagonal values. Despite the discrepancies we observed a remarkable agreement in the 1-step

transition probabilities of certain knots with length in [96, 104]. Fig. 11 shows the comparison

for the 61 knot. The 1-step transition probabilities for knots with 6 crossings all showed

remarkable agreement with the data of [21]. This suggests that despite their qualitative

differences (R3vs Z3; strand-passage in 3 dimensions vs strand-passage at the DT code level),

both models are able to consistently descrbe transitions of certain polymer chains by single

strand-passage. The differences are reflected in full at the steady-state. Greater discrepanciesarise when comparing n-step transitions as n goes to infinity. The knotting probabilities at the

steady state in our lattice model are much higher than those detected experimentally, and those

estimated using polymer models inR3. This is partially explained by the removal of

Reidemeister moves used in our model, and by the inherent rigidity of the lattice. This problems

can be overcome (work in preparation) by developing a model for DNA analogous to that of

[54].

7 Applications to Minimal Lattice Knots and the Strand-passage Metric

In order to validate our implementation of BFACF, we computed the length of minimal lattice

knots for each knot type with 8 crossings or less inZ3 and compared them to [34]. We

reproduced, and sometimes improved, the lower bounds of [34] (table 1). Our results mostly

agree with those previously given in [35]. In a few instances (i.e. 83, 88, 810, 812) the minimumnumber of segments is less than the one previously estimated (table 2 and [35].

The strand-passage metric, defined by Darcy [12], computes the minimal number of strand-

passage steps (over all projections) needed to go from one knot type to another. One interesting

question is how often the paths determined by the strand-passage metric are traveled when

random strand-passage is performed on knot populations. Preliminary results show that with

our method we can remove some ambiguities from the existing strand-passage metric table.

For example we see 2-step transitions between knots 41 and 51, while in the original table it

was unclear whether the strand-passage metric between these two knots was two or three.

Likewise we observe 3-step transitions between 41 and 71, versus 3 4 on the table. We leave

detailed analysis of the strand-passage metric to a future publication.

8 Conclusions and Future Directions

Motivated by the problem of DNA unknotting by topoII we have here created a framework

based in knot theory and Monte Carlo simulations to study state transitions in knot space. In

our study the state space is the set of all polygonal chainsZ3 of lengthL e and knot type K,

where all Kare prime knots with 8 or less crossings. We performed simulations for lengths in

the intervals [40,48], [52,60], [70, 80], and [96, 104]. We determine the transition probabilities

Hua et al. Page 8


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


9/25

between states. Our method uses the BFACF algorithm to generate ensembles of polygons

with identical knot type and mean lengthL. For each Kwe generate a list of configurations of

type Kand length in the chosen interval, representative of the knot moving in space. By taking

random projections, we produce a list of DT codes for each knot. We introduce a new algorithm

to simulate strand-passage, which produces a table of transition probabilities (table 1).

Our strand-passage algorithm is very fast but somewhat simplistic. For instance, accuracy of

the results depends on the faithfulness of the list of DT-codes to represent each knot type(guaranteed in this case by the properties of the BFACF algorithm). Ensuring that the transition

from the 3-dimensional object to the DT representation keeps the conformational information

is a challenge. Intuitively two strands that are far apart in space will produce a crossing in a

projection with low probability, and therefore by taking a sufficient number of 3-dimensional

conformations for each knot type, and by taking sufficient random projections, we guarantee

that DT-codes with such crossings have very small weight in the strand-passage simulation.

This approach serves as a framework to model the non-random unknotting behavior of topo

II. The representation of the computational data shown in Figure 7 can be compared directly

to the experimental results of J. Mann (Zechiedrich lab, Baylor College of Medicine) where a

population ofn-crossing twist knots (for fixed n) is incubated with topo II and the products are

harvested after a single strand-passage. We aim to compare our simulation results with their

data when it becomes available. In order to do this our model needs to be improved torealistically simulate DNA. Our first challenge is to simulate long enough polygons inZ3 with

the bending properties of DNA [54] and reproduce computational and experimental data on

random knotting probabilities [45,28]. The stationary state in our simulations currently results

in knotting probabilities considerably larger than those experimentally observed for DNA in

solution. The knotting probabilities that we observe ranged from 0.12 for mean length 44, to

0.18 for mean length 100. As the length increases the knotting probability increases which

indicates a much higher level of knotting that that observed when modeling random knotting

inR3, or that for DNA in solution (0.97 for polygonal chains of 33 Kuhn lengths, or 10.5kb

DNA plasmids) [45,44,28].

Acknowledgments

The authors would like to thank the referee for a careful revision of our manuscript which led to major improvements.We are particularly grateful to Miki Suga and Janella Slaga for their contributions to this project, to G. Gouesbet for

providing the algorithm and code for computing the HOM-FLY polynomial, and to S. Whittington for making the

auto-correlation code available to us. We acknowledge Kathleen Phu for her help type-setting the manuscript and

Mark Contois for his help with table 2. We also thank R. Sachs, D. W. Sumners, J. Roca, J. Mann, S. Whittington, C.

Soteros and A. Stasiak for their valuable comments. Research was supported through NSF grant DMS 9971169 and

SFSU Center for Computing in the Life Sciences grant program (J.A.). M.V. was supported through NASA

NSCOR04-0014-0017, a Focus-Avengoa research award and NIH-RIMI grant NMD000262 (M.V.). B.R. was

partially supported by a NSF graduate research fellowship. Other support came from a URAP summer stipend (D.N.)

and NIH R01-GM68423 (X.H.).

References

1. Argao de Carvalho C, Caracciolo S. A new Monte Carlo approach to the critical properties of self-

avoiding random walks. J Phys (Paris) 1983;44:323331.

2. Argao de Carvalho C, Caracciolo S, Frlich J. Polymers and g4 theory in four dimensions. Nucl PhysB 1983;215:209248.

3. Arsuaga J, Vazquez M, Trigueros S, Sumners de W, Roca J. Knotting probability of DNA molecules

confined in restricted volumes: DNA knotting in phage capsids. Proc Natl Acad Sci U S A

2002;99:53735377. [PubMed: 11959991]

4. Berg B, Foerster D. Random paths and random surfaces on a digital computer. Phys Lett B 1981;106

(4):323326.

Hua et al. Page 9


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


10/25

5. Berger JM, Gamblin SJ, Harrison SC, Wang JC. Structure and mechanism of DNA topoisomerase II.

Nature 1996;379:225232. [PubMed: 8538787]

6. Berger JM, Wang JC. Recent developments in DNA topoisomerase II structure and mechanism. Curr

Opin Struct Biol 1996;6:8490. [PubMed: 8696977]

7. Buck GR, Zechiedrich EL. DNA disentangling by type-2 topoisomerases. J Mol Biol 2004;340:933

939. [PubMed: 15236957]

8. Buck D, Verjovsky Marcotte C. Tangle solutions for a family of DNA-rearranging proteins. Math Proc

Cambridge Philos Soc 2005;139(1):5980.9. Calvo JA. Geometric knot spaces and polygonal isotopy. J Knot Theory Ramifications 2001;10:245

267.

10. Caracciolo S, Pelissetto A, Sokal AD. A general limitation on Monte Carlo algorithms of Metropolis

type. Phys Rev Lett 1994;72(2):179182. [PubMed: 10056079]

11. Champloux JJ. DNA topoisomerases: structures, function, and mechanism. Annu Rev Biochem

2001;70:369413. [PubMed: 11395412]

12. Darcy I. Rational Tangle Distances on Knots and Links. Math Proc Cambridge Philos Sco

2000;128:497510.

13. Darcy I. Biological distances on DNA knots and links: applications to XER recombination. Knots in

Hellas 1998;2DelphiJ Knot Theory Ramifications 2001;10(2):269294.

14. Darcy I. Solving unoriented tangle equations involving 4-plats. J Knot Theory Ramifications. to

appear

15. Deguchi T, Tsurasaki K. A statistical study of random knotting using the Vassiliev invariants, randomknotting and linking. J Knot Theory Ramifications 1994;3:321353.

16. Deibler RW, Rahmati S, Zechiedrich EL. Topoisomerase IV, alone, unknots DNA in E. coli. Genes

Dev 2001;15:748761. [PubMed: 11274059]

17. Diao Y. Minimal knotted polygons on the cubic lattice. J Knot Theory Ramifications 1993;2:413

425.

18. Dowker CH, Thistlethwaite MB. Classification of Knot Projections. Topol Appl 1983;16:1931.

19. Ernst C, Sumners DW. A calculus for rational tangles: applications to DNA recombination. Math

Proc Cambridge Philos Soc 1990;108(3):489515.

20. Ewing, B.; Millett, KC., editors. Progress in knot theory and related topics. Vol. 56. Travaux en Cours.

Paris: Hermann; 1997. Computational algorithms and the complexity of link polynomials.

21. Flammini A, Maritan A, Stasiak A. Simulations of action of DNA topoisomerases to investigate

boundaries and shapes of spaces of knots. Biophys J 2004;87:29682975. [PubMed: 15326026]

22. Frank-Kamenetskii MD, Lukashin AV, Vologodskii AV. Statistical mechanics and topology of

polymer chains. Nature 1975;258:398402. [PubMed: 1196369]

23. Freyd P, Yetter D, Hoste J, Lickorish WBR, Millett K, Oceanu A. A New Polynomial Invariant of

Knots and Links. Bull Amer Math Soc 1985;12:239246.

24. Gouesbet G, Meunier-Guttin-Cluzel S, Letellier C. Computer evaluation of Homfly polynomials by

using Gauss codes, with a skein-template algorithm. Appl Math Comput 1999;105(23):271289.

25. Grimmett, GR.; Stirzaker, DR. Probability and Random Processes. Oxford Science; Oxford: 1982.

26. Hoste J, Thistlethwaite M, Weeks J. The first 1,701,935 knots. Math Intelligencer 1998;20:3348.

27. Hsieh T. Knotting of the circular duplex DNA by type II DNA topoisomerase from Drosophilia

melanogaster. J Biol Chem 1983;258:84138420. [PubMed: 6305983]

28. Klenin KV, Vologodskii AV, Anshelevich VV, Dykhne AM, Frank-Kamenetskii MD. Effect of

excluded volume on topological properties of circular DNA. J Biomol Struct Dyn 1988;5 5:1173

1185. [PubMed: 3271506]29. Liu LF, Davis JL, Calendar R. Novel topologically knotted DNA from bacteriophage P4 capsids:

studies with DNA topoisomerases. Nucleic Acids Res 1981;9:39793989. [PubMed: 6272191]

30. Madras, N.; Slade, G. Probability and its Applications. Birkhauser; Boston: 1996. The Self-Avoiding

Walk.

31. Millett, KC. Knots in Hellas 98. Vol. 24. World Scientific; 1998. Monte Carlo explorations of

polygonal knot spaces.

Hua et al. Page 10


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


11/25

32. Portugal J, Rodriguez-Campos A. T7 RNA polymerase cannot transcribe through a highly knotted

DNA template. Nucleic Acids Res 1996;24:48904894. [PubMed: 9016657]

33. Van Rensburg EJ, Orlandini E, Sumners DW, Tesi MC, Whittington SG. The writhe of knots in the

cubic lattice. J Knot Theory Ramifications 1997;6:3144.

34. Van Rensburg EJ, Promislow SD. Minimal knots in the cubic lattice. J Knot Theory Ramifications

1995;4:115130.

35. van Rensburg, EJ. Ideal Knots. Vol. 19. World Scientific; Singapore: 1998. Minimal lattice knots.

36. van Rensburg EJ, Whittington SG. The BFACF algorithm and knotted polygons. J Phys A: Math Gen1991;24:55535567.

37. Roca J. The mechanisms of DNA topoisomerases. Trends Biochem Sci 1995;20:156160. [PubMed:

7770916]

38. Roca J. Varying levels of positive and negative supercoiling differently affect the efficiency with

which topoisomerase II catenates and decatenates DNA. J Mol Biol 2001;305:441450. [PubMed:

11152602]

39. Roca J, Berger JM, Harrison SC, Wang JC. DNA transport by a type II topoisomerase: direct evidence

for a two-gate mechanism. Proc Natl Acad Sci U S A 1996;93:40574062. [PubMed: 8633016]

40. Rodriguez-Campos A. DNA knotting abolishes in vitro chromatin assembly. J Biol Chem

1996;271:1415014155. [PubMed: 8662864]

41. Rolfsen D. Knots and Links. Publish or Parish. 1976

42. Rybenkov VV, Cozzarelli NR, Vologodskii AV. Probability of DNA knotting and the effective

diameter of the DNA double helix. Proc Natl Acad Sci U S A 1993;90:53075311. [PubMed:8506378]

43. Rybenkov VV, Ullsperger C, Vologodskii AV, Cozzarelli NR. Simplification of DNA topology below

equilibrium values by type II topoisomerases. Science 1997;277:690693. [PubMed: 9235892]

44. Shaw SY, Wang JC. DNA knot formation in aqueous solutions. J Knot Theory Ramifications

1994;3:287298.

45. Shaw SY, Wang JC. Knotting of a DNA chain during ring closure. Science 1993;260:533536.

[PubMed: 8475384]

46. Shishido K, Ishii S, Komiyama N. The presence of the region on pBR322 that encodes resistance to

tetracycline is responsible for high levels of plasmid DNA knotting in Escherichia coli DNA

topoisomerase I deletion mutant. Nucleic Acids Res 1989;17:97499759. [PubMed: 2557587]

47. Soteros CE, Sumners DW, Whittington SG. Entanglement complexity of graphs inZ3. Math Proc

Cambridge Philos Soc 1992;111

48. Stark WM, Sherratt DJ, Boocock MR. Site-specific recombination by Tn3 resolvase: topologicalchanges in the forward and reverse reactions. Cell 1989;58:779790. [PubMed: 2548736]

49. Stasiak A. DNA topology: feeling the pulse of a topoisomerase. Curr Biol 2000;10:R526528.

[PubMed: 10898993]

50. Stone MD, Bryant Z, Crisona NJ, Smith SB, Vologodskii A, Bustamante C, Cozzarelli NR. Chirality

sensing by Escherichia coli topoisomerase IV and the mechanism of type II topoisomerases. Proc

Natl Acad Sci U S A 2003 Jul 22;100(15):86548659. [PubMed: 12857958]

51. Sumners DW, Whittington SG. Knots in Self-Avoiding Walks. J Phys A: Math Gen 1988;21:1689

1694.

52. Szafron, M. Masters thesis. University of Saskatoon; 2000. Monte Carlo Simulations of Strand-

passage in Unknotted Self-Avoiding Polygons.

53. Trigueros S, Salceda J, Bermudez I, Fernandez X, Roca J. Asymmetric removal of supercoils suggests

how topoisomerase II simplifies DNA topology. J Mol Biol 2004;335:723731. [PubMed: 14687569]

54. Tesi MC, Janse Van Rensburg EJ, Orlandini E, Sumners DW, Whittington SG. Knotting and

supercoiling in circular DNA: A model incorporating the effect of added salt. Physical Review E.

1994

55. Vazquez M, Sumners DW. Tangle analysis of Gin site-specific recombination. Math Proc Cambridge

Philos Soc 2004;136(3):565582.

Hua et al. Page 11


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


12/25

56. Vologodskii AV, Anshelevich VV, Lukashin AV, Frank-Kamenetskii MD. Statistical mechanics of

supercoils and the torsional stiffness of the DNA double helix. Nature 1979;280:294298. [PubMed:

460401]

57. Vologodskii AV, Cozzarelli NR. Monte Carlo analysis of the conformation of DNA catenanes. J Mol

Biol 1993;232:11301140. [PubMed: 8371271]

58. Vologodskii AV, Frank-Kamenetskii MD. Modeling supercoiled DNA. Methods Enzymol

1992;211:467480. [PubMed: 1406321]

59. Vologodskii AV, Levene SD, Klenin K, Frank-Kamenetskii MD, Cozzarelli NR. Conformational andthermodynamic properties of supercoiled DNA. J Mol biol 1992;227:12241243. [PubMed:

1433295]

60. Vologodskii AV, Zhang W, Rybenkov VV, Podtelezhnikov AA, Subramanian D, Griffth JD,

Cozzarelli NR. Mechanism of topology simplification by type II DNA topoisomerases. Proc Natl

Acad Sci U S A 2001;98:30453049. [PubMed: 11248029]

61. Wasserman SR, Cozzarelli NR. Biochemical topology: applications to DNA recombination and

replication. Science 1986;232:951960. [PubMed: 3010458]

62. Yan J, Magnasco MO, Marko JF. A kinetic proofreading mechanism for disentanglement of DNA

by topoisomerases. Nature 1999;401:932935. [PubMed: 10553912]

63. Zechiedrich EL, Khodursky AB, Cozzarelli NR. Topoisomerase IV, not gyrase, decatenates products

of site-specific recombination in Escherichia coli. Genes Dev 1997;11:25802592. [PubMed:

9334322]

Hua et al. Page 12


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


13/25

Figure 1.

Strand-passage on twist knots.

Hua et al. Page 13


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


14/25

Figure 2.

BFACF moves.

Hua et al. Page 14


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


15/25

Figure 3.

Conformation lengths for the 61 knot after BFACF runs. The fugacity parameterz was chosen

so that the mean chain length would be 56. The band bounded by red lines determine the range

of sampling ([52,60]).

Hua et al. Page 15


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


16/25

Figure 4.

Reidemeister Moves I and II denoted as 1 and 2, respectively.

Hua et al. Page 16


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


17/25

Figure 5.

Distribution of the lengths of the Dowker codes corresponding to the 61 knot. The data was

obtained by performing the BFACF algorithm with z= 0.153, for 100 million iterations,

sampling every 15,000 iterations, binning for chain lengths in [52,60]. The BFACF was

followed by 100 projections in random directions, and RI and RII move removal.

Hua et al. Page 17


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


18/25

Figure 6.

Directed graph illustrating a modified transition matrix P where the results for a knot and its

mirror image were averaged. The nodes correspond to a knot type and its mirror image, the

edges correspond to one strand-passage between two states. The data presented here correspond

to those on table 2. The knot length was in the interval [96, 104]. Only prime knots with 7 or

less crossings are shown, and they are organized by crossing number along the y axis and by

writhe alon the x-axis.

Hua et al. Page 18


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


19/25

Figure 7.

Unknotting of the knot 61 with length in the interval [96, 104] by random strand-passage.

Hua et al. Page 19


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


20/25

Figure 8.

One-step transitions (left) and unknotting after 8 steps (right) of the 61 knot as a function of

the chains length.

Hua et al. Page 20


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


21/25

Figure 9.

Knotting distribution at the steady state for knots with length in [96, 104].

Hua et al. Page 21


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


22/25

Figure 10.

Knotting distributions at the steady state as a function of chain length.

Hua et al. Page 22


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


23/25

Figure 11.

Comparison of the knot 61 transitions with results of [21].

Hua et al. Page 23


NIH-PAA

uthorManuscript


NIH-PAAuthor

Manuscript


24/25

NIH-PA

AuthorManuscript

NIH-PAAuthorManuscr

ipt

NIH-PAAuth

orManuscript

Hua et al. Page 24

Table

1

Minimumlength

ofknotsinZ3;ComparisonwithvanR

ensburgsresults[34](seetext).Inprovementsarehighlightedinbold.

KnotTypesvanRensburgOurAlgorithmKnotTypesvanRensburgO

urAlgorithm

31

24

24

86

50

50

41

30

30

87

48

48

51

34

34

88

52

50

52

36

36

89

50

50

61

40

40

810

52

50

62

40

40

811

52

50

63

40

40

812

54

52

71

44

44

813

50

50

72

46

46

814

54

50

73

44

44

815

52

52

74

46

44

816

52

50

75

46

46

817

52

52

76

46

46

818

52

52

77

44

44

819

42

42

81

50

50

820

46

46

82

50

50

821

46

46

83

52

48

91

54

54

84

50

50

92

56

56

85

50

50

101

60



25/25

NIH-PA

AuthorManuscript

NIH-PAAuthorManuscr

ipt

NIH-PAAuth

orManuscript

Hua et al. Page 25

Table

2

knottypeunknot

31a

41

51a

52a

61a

62a

63

71a

72a

73a

74a

75a

76a

77a

81a

82a

83

84a

85a

86a

87a

88a

89

810a

811a

812

813a

814a

815a

816a

817a

818a

819a

820a

821a

unknot

0.8520.780.842

00.3120.2660.1680.33

00.242

0

0

00.1530.3250.214

0

0

0

0

00.1

37

00.266

00.141

00.2550.137

0

00.269

0

00.2580.515

31a

0.0610.04

00.3180.206

00.1270.245

0

0

00.0790.1

190.1180.08

00.072

0

0

00.1030.0

720.173

0

0

0

0

00.071

00.1310.1350.492

0

0

0

41

0.022

00.045

0

00.4430.339

0

0

0

0

0

00.2440.488

0

0

00.131

0

0

0

0

0

0

00.4370.1330.216

00.137

0

0

0

0

0

51a

00.027

00.0210.061

0

0

00.3420.0050.1560.0040.12

0

0

00.108

0

0

0

00.1

04

0

00.134

0

00.008

00.168

0

0

00.4440.0120.04

52a

0.0020.035

00.0980.018

0

0

0

00.2460.2010.3210.1

540.121

0

0

0

0

0

0

0

00.106

00.0880.108

00.1010.1110.2720.069

0

0

00.195

0

61a

0

00.027

0

00.0510.045

0

0

0

0

0

0

0

00.258

00.3470.141

00.14

0

0

0

00.1090.221

0

0

0

0

0

0

0

0

0

62a

00.0040.024

0

00.0540.013

0

0

0

0

0

0

0

0

00.217

00.1810.3660.141

0

00.282

00.139

0

00.106

0

00.221

0

0

00.181

63

00.007

0

0

0

0

00.024

0

0

0

0

0

0

0

0

0

0

0

0

00.3

650.287

00.357

0

00.267

0

00.445

0

0

00.232

0

71a

0

0

00.016

0

0

0

00.016

00.047

00.03

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

72a

0

0

0

00.011

0

0

0

00.0130.034

00.0

420.028

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

73a

0

0

00.0160.01

0

0

00.0870.0320.0140.084

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

74a

0

0

0

00.009

0

0

0

0

00.0480.012

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

75a

0

0

00.0160.008

0

0

00.0540.052

0

00.0

12

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00.035

0

0

76a

00.0010.002

00.01

0

0

0

00.03

0

0

00.011

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00.023

0

77a

0

00.002

0

0

0

0

0

0

0

0

0

0

00.011

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

81a

0

0

0

0

00.01

0

0

0

0

0

0

0

0

00.009

00.062

0

00.039

0

0

0

00.0250.052

0

0

0

0

0

0

0

0

0

82a

0

0

00.002

0

00.007

0

0

0

0

0

0

0

0

00.01

0

0

00.023

0

0

0

00.039

0

00.026

0

0

0

0

0

0

0

83

0

0

0

0

00.011

0

0

0

0

0

0

0

0

00.053

00.0190.075

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

84a

0

00.001

0

00.0060.005

0

0

0

0

0

0

0

0

0

00.0820.009

00.021

0

00.076

0

0

0

0

0

0

0

0

0

0

0

0

85a

0

0

0

0

0

00.004

0

0

0

0

0

0

0

0

0

0

0

00.0130.024

0

0

0

0

0

0

0

0

0

0

0

0

0

00.007

86a

0

0

0

0

00.0070.008

0

0

0

0

0

0

0

00.0530.023

00.0270.0820.009

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

87a

0

0

00.001

0

0

00.01

0

0

0

0

0

0

0

0

0

0

0

0

00.0

090.021

0

0

0

00.035

0

0

0

0

0

0

0

0

88a

0

0

0

00.001

0

00.015

0

0

0

0

00.003

0

0

0

0

0

0

00.0

260.034

00.044

0

00.011

0

0

0

0

0

00.007

0

89

0

0

0

0

0

00.01

0

0

0

0

0

0

0

0

0

0

00.078

0

0

0

00.02

0

0

0

0

0

0

0

0

0

0

0

0

810a

0

0

00.0010.001

0

00.007

0

0

0

0

0

0

0

0

0

0

0

0

0

00.024

00.013

0

0

0

0

0

0

0

0

00.005

0

811a

0

0

0

00.0010.0060.007

0

0

0

0

0

0

0

00.0210.045

0

0

0

0

0

0

0

00.01

0

0

0

0

0

0

0

0

0

0

812

0

00.001

0

00.014

0

0

0

0

0

0

0

0

00.051

0

0

0

0

0

0

0

0

0

00.018

0

0

0

0

0

0

0

0

0

813a

0

0

0

00.001

0

00.009

0

0

0

0

0

0

0

0

0

0

0

0

00.0

38

0

0

0

0

00.009

0

0

0

0

0

0

0

0

814a

0

0

0

00.001

00.006

0

0

0

0

0

0

0

0

00.025

0

0

0

0

0

0

0

0

0

0

00.01

0

0

0

0

0

0

0

815a

0

0

00.001

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00.014

0

0

00.002

0

0

816a

0

0

0

0

0

0

00.004

0

0

0

0

0

00.002

0

0

0

0

0

0

0

0

0

0

0

00.008

0

00.009

0

0

0

00.001

817a

0

0

0

0

0

00.002

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00.01

0

0

0

0

818a

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00.008

0

0

0

819a

0

0

00

.01

0

0

0

0

0

0

0

00.0

22

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00.046

0

0

00.019

0

0

820a

0

0

0

00.006

0

00.033

0

0

0

0

00.021

0

0

0

0

0

0

0

0

0

00.042

0

0

0

0

0

0

0

0

00.013

0

821a

0

0

00.001

0

00.017

0

0

0

0

0

0

0

0

0

0

0

00.04

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00.013


Documents

Xia Hua, Diana Nguyen, Barath Raghavan, Javier Arsuaga and Mariel Vazquez- Random state transitions of knots: a first step towards modeling unknotting by type II topoisomerases