Coupling of twist and writhe in short DNA loopsShlomi Medalion, Shay M. Rappaport, and Yitzhak Rabin Citation: J. Chem. Phys. 132, 045101 (2010); doi: 10.1063/1.3298878 View online: http://dx.doi.org/10.1063/1.3298878 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v132/i4 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Coupling of twist and writhe in short DNA loopsShlomi Medalion,1,a� Shay M. Rappaport,1 and Yitzhak Rabin1,2

1Department of Physics and Institute of Nanotechnology and Advanced Materials,Bar-Ilan University, Ramat-Gan 52900, Israel2Department of Biomedical Engineering, Northwestern University, Evanston, Illinois 60208, USAand James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA

�Received 30 November 2009; accepted 3 January 2010; published online 27 January 2010�

While bending and twist can be treated as independent degrees of freedom for linear DNAmolecules, the loop closure constraint introduces a coupling between these variables in circularDNA. We performed Monte Carlo simulations of wormlike rods with both bending and twist rigidityin order to study the coupling between the writhe and twist distributions for various DNA lengths.We find that for sufficiently short DNA, the writhe distribution differs from that of a model withbending energy only. We show that the factorization approximation introduced by previousresearchers coincides, within numerical accuracy, with our simulation results, and conclude that theclosure constraint is fully accounted for by the White–Fuller relation. Experimental tests of ourresults for short DNA plasmids are proposed. © 2010 American Institute of Physics.�doi:10.1063/1.3298878�

I. INTRODUCTION

The first generation of single molecule experiments onthe stretching of double stranded DNA �dsDNA� �Ref. 1� wassuccessfully described �both qualitatively and quantitatively�by the wormlike chain �WLC� model of semiflexible poly-mers, which incorporates the combined effects of bendingelasticity and thermal fluctuations.2 Next generation experi-ments combined stretching and torsional deformations of in-dividual DNA molecules.3,4 Since torque affects the twistdegrees of freedom that cannot be described by the WLCmodel, different variants of the wormlike rod �WLR� model�rodlike chain in the terminology of Ref. 5�, which combinesbending and twist elasticity were introduced.6–11 Eventhough the WLR model contains both bending and twist de-grees of freedom, the corresponding elastic energy can bewritten as the sum of bending and twist energies and, in theabsence of constraints that couple these degrees of freedom,the partition function of a linear �open-ended� WLR can befactorized into a product of bending and twist contributions.Thus, if one associates the bending degrees of freedom withthe conformations of the centerline of a linear WLR and thetwist degrees of freedom with the rotations of the cross sec-tion about this centerline, one concludes that twist rigiditydoes not affect statistical properties of the linear polymersuch as its radius of gyration, mean square end to end dis-tance and tangent-tangent correlation function.

The decoupling between bending and twist degrees offreedom breaks down even in the absence of external forcesif the polymer is a closed �circular� loop.12,13 The latter sce-nario is important in the case of dsDNA, which can formboth short �plasmids� and long �bacterial chromosomes�loops. Yet, all computer simulations of the distribution of thelinking number �Lk� of DNA loops to date, were based on

the factorization approximation in which one performs aMetropolis Monte Carlo �MC� simulation to generate thewrithe �Wr� distributions of a wormlike chain14 �recall thatLk is a property of two linked curves and cannot be definedfor a single WLC�. One then assumes that twist �Tw� is givenby a Gaussian distribution for a linear polymer and uses theWhite–Fuller relation,15,16 Lk=Wr+Tw, in order to expressthe twist distribution in terms of Lk−Wr. Finally one multi-plies the above writhe and twist distributions and sums overall values of writhe in order to obtain the linking numberdistribution.

In this work we perform MC simulations of ideal �i.e.,excluded volume is neglected� WLRs with bending and twistrigidity. We focus on the case where topology is not con-served �due to the action of topoisomerases and gyrases� andthe segments of our phantom WLR can cross each other inthe course of a MC move and thus change both the linkingnumber and the knot index. Notice that since in our case thelinking number is calculated explicitly for all conformationssampled, our phantom rod algorithm is self-consistent in thesense that Lk changes by �2 each time two segments cross�a fixed value of Lk is kept regardless of crossings during thesimulation in Refs. 17 and 18�. Finally, for reasons of sim-plicity we assume that our WLR has no intrinsic twist/linking number as this only introduces a trivial shift,�Lk→Lk �and �Tw→Tw� in the corresponding distribu-tions for dsDNA �which has intrinsic twist defined by thewinding of the double helix�.

In Sec. II we discuss the geometry and the elastic energyof the WLR model, define the writhe, twist and linking num-bers associated with the conformation of the WLR andpresent the White–Fuller relation between these numbers. InSec. III we present the Gaussian twist distribution of a linearWLR and introduce the factorization approximation whichallows one to express the writhe and the twist distributions asa sum over linking numbers of products of the writhe distri-a�Electronic mail: shlomed.uni@gmail.com.

THE JOURNAL OF CHEMICAL PHYSICS 132, 045101 �2010�

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bution of circular WLC and the Gaussian twist distributionof linear WLR. The details of the simulation, including thepivot and the twist moves, are discussed in Sec. IV. In Sec. Vwe present the simulated probability distributions of writheand twist for closed WLR and compare them to those ofclosed WLC �only writhe distributions can be defined in thiscase� and to the distributions calculated using the factoriza-tion approximation. We find that the factorization approxi-mation yields results that are indistinguishable from those ofWLR simulations, suggesting that it may in fact be exact �weare not aware of an analytical proof of this equality�. We alsofind that the WLR distributions deviate significantly from theWLC distributions for �a� DNA loops of intermediate length�of the order of ten persistence lengths� and �b� for WLRwith large twist rigidity. In Sec. VI we summarize our re-sults. The connection between local twist density and Eulerangles is discussed in the Appendix.

II. THE WORMLIKE ROD MODEL

In the WLC model of DNA, the macromolecule is de-scribed as a space curve r�s� whose conformation is com-pletely defined by the tangent t3�s�=dr�s� /ds to the curve ateach point s �s is the length measured along the contour ofthe curve and thus the tangent t3�s� is a unit vector�. En-semble averages are performed by weighting each conforma-tion of such a curve by an appropriate Boltzmann factorthat contains the bending energy associated with the localcurvature ��s�= �dt3�s� /ds�= �d2r�s� /ds2�. In the frameworkof linear theory of elasticity this energy can be written asEWLC= �kBTlp /2��0

Lds���s��2, where lp is the persistencelength associated with bending rigidity, L is the chain length,kB is the Boltzmann constant, and T is the absolute tempera-ture �kBTlp is the bending modulus�. The simplest generali-zation of the above model which captures the effect of twistas well as of bending, is the WLR model that describes semi-flexible polymers with finite cross section and both bendingand twist rigidities.10,19 In this model the conformation of thepolymer is given by the orientation of the triad of orthogonalunit vectors �t1�s� ; t2�s� ; t3�s�� at each point s along the cen-terline �the line that runs along the symmetry axis of theWLR�, where t3 is the tangent to the centerline and the vec-tor t1 lies in the cross-sectional plane �the third unit vector iscompletely defined in terms of the other two, t2= t3� t1�. Forexample, in the case of a ribbon with elliptical cross section,the vectors t1�s� and t2�s� are associated with the symmetryaxes of the ellipse �this choice becomes degenerate in thecase of a WLR with circular cross section�. The elastic en-ergy of a WLR is given by

EWLR

kBT=

0

L

ds1

2lp���s��2 +

1

2lT��3�s��2� , �1�

where lT is the twist persistence length and �3�s� is the localrate of twist about the t3 axis, defined as20,21

�3�s� = t3 · �t1 � dt1/ds� . �2�

In the discrete version of the WLR model, the chain is de-scribed by a succession of N segments of length �s each,such that its contour length is given by L=N�s. The nth

segment is represented by a triad of unit tangent vectors�t1,n ; t2,n ; t3,n� such that the relation between two neighboringframes: tk,n and tk,n+1 �k=1,2 ,3�, can be expressed in termsof the Euler angles �, �, and � �Appendix�, where � is as-sociated with curvature and the sum of the two other anglesis the twist angle. Scaling all lengths by �s, the energy of thediscrete WLR takes the form

EWLR

kBT= �

n=1

N 1

2l̃p��̃n�2 +

1

2l̃T��̃3,n�2� , �3�

where l̃p and l̃T are the dimensionless bending and twist per-sistence lengths respectively, and �̃3,n is the twist angle �seeAppendix and Fig. 6�. The dimensionless curvature �̃n canbe defined in terms of the angle between the nth and then+1th segments,22 as �̃n

2=2�1−cos �n�. Notice that since�̃3,n depends only on the orientations of the cross sectionsof the nth and the n+1th segments, and since �̃n dependsonly of the relative orientation of the tangents to these seg-ments t3,n and t3,n+1, the total energy is given by the sum ofcontributions of N−1 overlapping dimers,�1,2� , �2,3� , . . . �N−1,N� �with �̃N�0�.

For open �linear� polymers, the partition function of theWLR model can be calculated using Eq. �1� �or Eq. �3� forthe discrete case�

Z = de−EWLR/�kBT�, �4�

where d represents integration over all possible configura-tions. The ensemble average of any conformation-dependentfunctional A�� is given by

�A��� =1

Z dA��e−EWLR/�kBT�. �5�

Because the energy consists of independent contributions ofdifferent dimers and there is no coupling between curvatureand twist, for long enough chains �neglecting chain end ef-fects which introduce corrections of order 1 /N� the partitionfunction can be factorized into a product of segmental parti-tion functions, Z=N, where

= d� sin���e−�1/2�l̃p�̃n2 d�̃3,ne−�1/2�l̃T��̃3,n�2

. �6�

Using Eq. �6� it is straightforward to calculate mean values

such as ��̃n2�=2 / l̃p and ��̃3,n

2 �=1 / l̃T.When the chain is closed into a loop �i.e., its ends are

connected together�, the closure constraint r�0�=r�L� can bewritten as �0

Lds�dr�s� /ds�=�0Ldst3�s�=0 or, in its discrete

version, �n=1N t3,n=0. This constraint introduces a nonlocal

coupling between all the segments of the chain and the fac-torization of the partition function breaks down. Anotherconstraint, which couples the curvature and twist in a non-trivial way, is that the t1,n vector must be periodic, i.e.,t1,n= t1,n+N, which is equivalent to saying that the cross sec-tion must execute an integer number of revolutions about the

045101-2 Medalion, Rappaport, and Rabin J. Chem. Phys. 132, 045101 �2010�

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centerline as one traverses the closed rod. This integer,known as the linking number Lk, is a topological invariantthat does not change under thermal fluctuations and elasticdeformations. It was shown15,16 that Lk can be represented asthe sum of two quantities: writhe �Wr� and twist �Tw�

Lk = Wr + Tw . �7�

For closed continuous curves, Wr and Tw are defined as20,21

Wr =1

4�� � dsds��t3�s� � t3�s��� ·

r�s� − r�s���r�s� − r�s���3

, �8�

and

Tw =1

2�� ds�3�s� =

1

2�� dst3 · �t1 � dt1/ds� . �9�

Note that while the definition of twist applies equally well toclosed and to open WLRs, writhe is defined only for closedcurves. The discrete representation of twist is:

Tw =1

2��

n

�̃3,n, �10�

and of the writhe is

Wr =1

4��n,m

Wn,m, �11�

where Wn,m is defined by replacing the continuous variables sand s� by the discrete segments labels, n and m in the inte-grand, Eq. �8�. A complete description of the discrete writheand the algorithm for calculating it, can be found in Ref. 14.Note that even though Eq. �7� was originally defined forcontinuous curves, it is expected to apply for discrete curvesas well, at least as long as their thickness is small comparedwith the persistence length �the latter condition is alwayssatisfied in our simulation�. We checked this condition bydirectly calculating the writhe, twist and linking numbers ofthe sampled configurations and found that Eq. �7� was al-ways satisfied within our numerical accuracy.

Inspection of Eqs. �8� and �9� shows that writhe dependson the conformation of the centerline �and thus on the cur-vature� only, and that twist depends both on the centerlinet3�s� and on the orientation of the cross section t1�s� withrespect to it. The condition, Eq. �7�, that for a given topologyof the closed rod the sum of twist and writhe �Lk� is invariantunder any elastic deformations of the WLR, introduces anonlocal coupling between the curvature ��s� and the twistrate �3�s� and the partition function can no longer be factor-ized into bending and twist contributions �such separation ispossible for open WLR�. Nevertheless, one suspects that forsufficiently long closed DNA molecules, Eq. �7� has littleeffect on the possible values of local quantities such as ��s�and �3�s�, and that they become effectively decoupled in thislimit. For shorter chains, the coupling becomes importantand the study of its effects, the foremost of which is theeffect of twist on the distribution of writhe, is the main goalof the present paper.

III. WRITHE AND TWIST DISTRIBUTIONS

For long enough rods �N� l̃p , l̃T� the constraints intro-duced by the closure of the WLR can be neglected and thedistribution of the twist can be approximated by the corre-sponding probability distribution of an open rod,

P�0��Tw� e−Tw2/2�T2. �12�

The variance of this distribution is given by

�T2 = �Tw2� =

1

�2��2�n,m

��̃3,n�̃3,m� =1

�2��2

N

l̃T

, �13�

where we used the independence of the local twist angles

��̃3,n�̃3,m�=�n,m / l̃T �based on the equipartition theorem forthe energy in Eq. �3��. Since writhe can only be defined forclosed curves and its calculation using Eq. �8� is rathercomplicated, the exact analytical form of the writhe distribu-tion is unknown, even in the absence of twist �i.e., forclosed wormlike chains�. Nevertheless, in the long rodlimit, application of the central limit theorem suggests thatthis distribution approaches a Gaussian, i.e., PWLC�Wr�→exp�−Wr2 /2�wr

2 �, where the variance �wr2 N / l̃p.23 This

conjecture is indeed confirmed by computer simulations �notshown�. For shorter WLR, an approximate expression for thewrithe distribution can be constructed in the followingmanner.17 First, we assume that writhe does not depend onthe twist degrees of freedom and calculate the writhe distri-bution PWLC�Wr� of a closed wormlike chain, using MCsimulations with statistical weights determined by Eq. �3��with l̃T=0�. We then express the conditional probability ofobserving writhe Wr for a given linking number Lk, as theproduct of the writhe distribution of a circular WLC andthe twist distribution of an open rod �Eq. �12�� in whichwe express twist in terms of linking number and writhe,Tw=Lk−Wr �Eq. �7��,

PWLR�Wr�Lk� � PWLC�Wr�P�0��Lk − Wr� . �14�

In the case of a phantom WLR which can cross itself andtherefore, change the linking number, the writhe distributionis given by summing Eq. �14� over all values of Lk

PWLR�Wr� � Pap�Wr� � PWLC�Wr��Lk

P�0��Lk − Wr� .

�15�

Similarly, an approximate expression for the twist distribu-tion of a phantom WLR is given by

PWLR�Tw� � Pap�Tw� � P�0��Tw��Lk

PWLC�Lk − Tw� .

�16�

In the following we will show that the factorization approxi-mation that leads to Eqs. �15� and �16� is numerically exact,in the sense that it gives writhe and twist distributions thatare indistinguishable from the corresponding distributionsobtained by direct MC simulations of phantom WLRs for the

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entire range of twist and bending rigidities and rod lengthsstudied. For convenience, the definitions of the different dis-tribution functions are summarized in Table I.

IV. THE SIMULATION

We performed MC simulations of phantom WLRs ofvarious lengths, without excluded volume. The algorithm forcalculating PWLR includes two moves whose goal is to main-tain ergodicity for a closed chain. First, we used the “pivot”algorithm originally designed to simulate closed wormlikechains.14 In each move we randomly chose an axis �pivot�that connects points along the discrete chain and rigidlyrotate the section of the chain between these two points bya random angle around the pivot �Fig. 1�. The angle ofrotation was chosen with uniform probability in the interval��−� /2,� /2��.

Even though the twist angles are also changed during thepivot step, in order to maintain ergodicity one needs to re-move the resulting correlation between the rotation of thecenterline and the rotation about the centerline �Fig. 1�. Thisis achieved by introducing local twist moves that consist ofchoosing a random segment n along the chain and rotatingthe t1,n vector around the t3,n vector of this segment by ran-dom angle in the interval �−� /2,� /2� �see Fig. 2�. We ac-cept or reject each step according to the Metropolis MCrules, using the energy given in Eq. �3�.

Note that in Refs. 17 and 18 the actual MC simulationinvolved wormlike chains for which the linking number andthe twist are undefined and therefore the nominal value ofthe linking number was chosen once and for all in the begin-ning of the simulation run and was not allowed to changeeven during chain crossings. In our simulation of the WLR

model, both Lk and Tw are well defined and therefore thetwist, writhe and linking number of each configuration canbe computed. Pivot moves change Lk �upon crossing ofphantom chain segments� by �2 and thus, changing the link-ing number from even to odd can only be done by local twistmoves.

V. RESULTS AND ANALYSIS

In order to examine the effect of twist on the spatialconformation of WLR, we first simulated the probability dis-tribution of writhe for the WLC model PWLC�Wr�, using onlypivot moves and weighting them by the bending energy only

�Eq. �3� with l̃T=0�. We then used this distribution and Eq.�15� to compute the factorization approximation for thewrithe distribution of WLR, Pap�Wr�. Finally, we simulatedthe writhe distribution of WLR, PWLR�Wr�, using both pivotand segment twist moves with MC weights determined bythe full energy, Eq. �3�. We ran the simulation for dsDNAparameters,17 where lp=50 nm, lT=74 nm, and �s=10 nm�the value �s=10 nm, which sets the length scale, was keptin all the simulations reported in the present work�. Wesimulated short �150 nm�, intermediate �300 nm�, and long�2000 nm� dsDNA rings. As expected, in the long chainlimit, the writhe distribution becomes independent of thetwist degrees of freedom and the three distributions�PWLC�Wr�, Pap�Wr�, and PWLR�Wr�� coincide within ournumerical accuracy and approach a single Gaussian-shapedcurve peaked about Wr=0 �Fig. 3�a��.

For shorter circular DNA molecules �150 and 300 nm�,twist effects on the writhe distribution become noticeableand while the WLR simulation and the approximation yieldnumerically indistinguishable distributions, they are bothnarrower than the WLC distribution �Figs. 3�b� and 3�c��. Amore careful examination of these figures reveals that thedeviations from the WLC distribution �and therefore, the ef-fects of twist� are more pronounced for intermediate DNAlengths than for very long or very short DNA circles. Noticethat in the short chain limit �L� lp�, thermal fluctuations be-come negligible and the lowest energy configuration �a pla-nar circle of vanishing writhe� dominates. In this limit, thewrithe distribution approaches a �-function for any finitevalue of twist rigidity, and therefore becomes independent oftwist. Since the three writhe distributions �PWLC, Pap, andPWLR� coincide in both the short and the long chain limits,one expects that significant differences between them mayarise only for intermediate DNA lengths, 1�L / lp�10, inagreement with Figs. 3.

In order to understand the way in which the writhe dis-tribution becomes independent of twist in the limit of long

TABLE I. Different distributions used throughout the paper.

Distribution Short description

P�0��Tw� Gaussian distribution of twistPWLC�Wr� Wr distribution derived from WLC simulationPWLR�Wr� Wr distribution derived from WLR simulationPWLR�Tw� Tw distribution derived from WLR simulationPWLR�Wr �Lk� Wr distribution for given Lk derived from WLR

simulationPap�Wr� Factorization approximation for Wr distribution of WLRPap�Tw� Factorization approximation for Tw distribution of WLR

FIG. 1. Sketch of a single pivot move. Two points along the chain arerandomly chosen and one side of the chain is rotated by a random angle withrespect to the pivot connecting the two chosen points �dashed line�

FIG. 2. A sketch of a single twist step: twisting the cross-section vectors t1,n

and t2,n by a random angle around the nth segment.

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chains, consider the linking number distribution of phantomchains. Since the widths of the writhe and the twist distribu-tions increase with the number of segments as N1/2, we ex-pect that the width of the linking number distribution alsoincreases as N1/2. Since Lk is always an integer, the sum overlinking numbers in Eq. �15�, �LkP

�0��Tw�=�LkP�0��Lk−Wr�,

is dominated by the first N1/2 integers and depends on Wr. Inthe limit N1/2�1, this sum can be replaced by an integralfrom −� to � and, since for finite values of Wr one can shiftvariables, Lk�=Lk−Wr, the integral over the normalizedGaussian yields unity and we recover Pap�Wr�= PWLC�Wr�.

A closer examination of Fig. 3�c� reveals the presence of“shoulders” in the writhe distribution of WLR of intermedi-ate length. Since these shoulders are clearly associated withthe effects of twist on the conformations of a WLR, we de-cided to magnify these effects by examining the large twistrigidity limit in which the twist persistence length lT wastaken to be equal to the DNA length L �the bending persis-tence length was kept constant, lp=50 nm�. In Figs. 4�a� and

4�b� we show the writhe distributions, for intermediate andlong rods. Again, the factorization approximation, Eq. �15�yields results that are indistinguishable from the WLR simu-lation. While the WLC distributions are smooth Gaussian-like functions peaked about Wr=0, the WLR distributionscontain several peaks centered around integer values of Wrwhose magnitude decreases with increasing writhe. Thenumber of peaks increases and the differences between theheights of successive peaks decrease with increasing chainlength. In order to understand the physical mechanism thatgives rise to the peaks, note that if we insert lT=L in Eq.�13�, the width of the twist distribution P�0��Tw� becomesmuch smaller than unity ��T=1 /2��. Since Lk is an integer,and since Tw is much smaller than unity �it is narrowly dis-tributed about �Tw�=0�, the relation Wr=Lk−Tw forces theallowed values of writhe to be localized about Lk and sincethe latter are integers, this results in a writhe distribution thatis peaked about integer values of Wr. The number of peaksincreases with the rod length L since for sufficiently large N,the width of the linking number distribution is expected toincrease as N1/2. In the case of dsDNA with lp , lT�L, thewidth of the twist distribution is larger than unity and theoverlap between neighboring peaks results in the appearanceof shoulders in Fig. 3�c�.

Since the writhe of a particular conformation of a closedcurve can be obtained by calculating the number of crossingsin its projection on a plane, averaged over all possibleplanes,16 one may expect that integer values of Wr corre-spond to curves with a well defined number of crossings.24

Inspection of the shape of the simulated conformations with

−5 0 50

0.05

0.1

0.15

0.2

0.25

Wr

P(W

r)

(a)

−0.5 0 0.50

1

2

3

4

5

Wr

P(W

r)

(b)

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

Wr

P(W

r)

(c)

FIG. 3. Writhe distributions for lp=50 nm and lT=74 nm: PWLR�Wr��green dots�, Pap�Wr� �red circles�, and PWLC�Wr� �blue crosses�. The chainlengths are the following: �a� L=2000 nm, �b� L=150 nm, and �c�L=300 nm. Note the different scales on the x-axis in the above figures.

−4 −3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

Wr

P(W

r)

(a)

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

WrP

(Wr)

(b)

FIG. 4. Writhe distributions for lp=50 nm: PWLR�Wr� �green dots�, Pap�Wr��red circles�, and PWLC�Wr� �blue crosses�. �a� L= lT=2000 nm and �b�L= lT=300 nm.

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integer Wr shows that this is not the case and that there is noobvious distinction between curves of integer and nonintegerwrithe.

Turning now to the effect of bending rigidity on the twistdistributions of closed DNA, in Figs. 5�a� and 5�b� we com-pare the Gaussian approximation for the twist distribution,P�0��Tw�, Eq. �12�, with both the simulated twist distributionfor WLR and the distribution given by the decoupling ap-proximation, Eq. �16�. As in the case of writhe, the simulatedand the approximate twist distributions coincide within ournumerical accuracy. Furthermore, while both PWLR�Tw� andPap�Tw� approach P�0��Tw� in the limit lp , lT�L �Fig. 5�a��,for chains of intermediate length PWLR�Tw� and Pap�Tw� arenarrower than P�0��Tw�. Interestingly, although for writhedistributions deviations from WLC become most pronouncedin the limit of large twist rigidity, for twist distributions suchdeviations decrease with increasing twist rigidity �notshown�.

VI. DISCUSSION

Previous computer simulations of the topological prop-erties of polymers were based on the WLC model for whichone can define a knot index but not a linking number. Thisshortcoming was overcome by assuming that just like thecase of linear chains, the twist distribution of a closed poly-mer is Gaussian, and replacing twist by the difference be-tween linking number and writhe. One then calculates thelinking number distribution in the factorization approxima-tion by forming the product of the simulated writhe distribu-tion and the above twist distribution and summing overwrithe. Similarly, the factorization approximation can be

used to calculate twist effects on the writhe distribution ofclosed phantom chains, by forming the product of thewrithe and twist distributions and summing over all linkingnumbers.

In the present work we carried out a MC simulation ofclosed loops of dsDNA modeled as phantom WLRs withbending and twist rigidity but without excluded volume andused it to obtain, directly, the writhe and twist distributions.Since for long WLR, the writhe distribution coincides withthat of WLC and the twist distribution coincides with theGaussian twist distribution of an open WLR, we concludethat the coupling between writhe and twist due to the closureconstraint becomes irrelevant in the long chain limit. Forshort �but not too short� DNA loops, the writhe distributionof WLR deviates significantly from that of WLC, and thetwist distribution is no longer Gaussian. However, eventhough for short chains the closure constraint introduces sig-nificant coupling between twist and writhe, this coupling canbe fully accounted for by the factorization approximationwhich yields numerically exact results. While we cannotprove this analytically, we postulate that as long as the elasticenergy can be written as a sum of bending and twist contri-butions, the coupling between bending and twist degrees offreedom introduced by the loop closure constraints is fullyaccounted for by the White–Fuller relation. This suggeststhat the factorization approximation holds even in the pres-ence of excluded volume interactions �such effects were ne-glected in our work�. We expect the factorization approxima-tion to break down when the elastic energy cannot beseparated into pure bending and twist terms as, for example,in the case of a closed WLR with asymmetric cross sectionand spontaneous curvature.25 It is unclear at present, whetherthis approximation holds in the presence of real topologicalconstraints, i.e., for nonphantom chains that cannot crosseach other, for which the writhe distribution depends on knottopology.26

The effects of twist on the writhe distribution are mag-nified in the limit of high twist rigidity. In this case, theGaussian twist distribution becomes very narrow and, as aconsequence, peaks around integer values of writhe appear inthe writhe distribution. For lower twist rigidity these peaksbecome wider and overlap to form shoulders about the cen-tral peak in the writhe distribution. It would be interesting totry to verify our prediction by experimental studies of thespatial conformation of dsDNA plasmids.

FIG. 6. The twist angle is �̃3,n=�2−�1, where �1 and �2 are the anglesbetween the ribbon vectors t1,n and t1,n+1, and the binormal relative to theplane of the corresponding segments.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

Tw

P(T

w)

(a)

−1 −0.5 0 0.5 10

0.5

1

1.5

2

Tw

P(T

w)

(b)

FIG. 5. Twist distributions for lp=50 nm and lT=74 nm: PWLR�Tw� �greendots�, Pap�Tw� �red circles�, and P�0��Tw� �blue line�. The chain lengths arethe following: �a� L=2000 nm and �b� L=300 nm.

045101-6 Medalion, Rappaport, and Rabin J. Chem. Phys. 132, 045101 �2010�

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ACKNOWLEDGMENTS

This work was supported by a grant from the Israel Sci-ence Foundation and by the Israel-France Research Net-works Program in Biophysics and Physical Biology. Y.R.’swork was partially supported by the Materials Research Sci-ence and Engineering Center program of the National Sci-ence Foundation �DMR-0520513� at Northwestern Univer-sity.

APPENDIX: EULER ANGLES AND TWIST

The total twist Tw of a discrete ribbon is defined by Eq.�10�. Each local twist angle �the discrete form of Eq. �2�� canbe expressed as

�̃3,n = �2 − �1, �A1�

where �1 and �2 are the angles of the t1 vectors of thenth and the �n+1�th segments relative to the plane definedby the two successive segments and the binormal bn

= �t3,n� t3,n+1��t3,n� t3,n+1�−1 �see Fig. 6�

�1 = sgn��bn � t1,n� · t3,n�arccos�bn · t1,n� ,

�A2��2 = sgn��bn � t1,n+1� · t3,n+1�arccos�bn · t1,n+1� .

Notice that although in principle �̃n,3 may be higher then �,in our simulation it is confined to the range: �−� ,��.

The local twist can also be defined as the sum of the twoEuler twist angles

�̃3,n = � + � . �A3�

These angles correspond to the Euler rotation that rotates thevector tn of the nth segment into the vector tn+1 of the�n+1�th segment

tn+1 = Atn. �A4�

Here,

tn = �t1,n

t2,n

t3,n� ,

and the Euler rotation matrix is given by the three Eulerangles �� ,� ,��

A = � cos � cos � − cos � sin � sin � sin � cos � + cos � cos � sin � sin � sin �

− cos � sin � − cos � sin � cos � − sin � sin � + cos � cos � cos � sin � cos �

sin � sin � − sin � cos � cos �� . �A5�

By using Eqs. �A2� and �A4� together with Eq. �A5� it can beshown that �1=−� and �2=�.

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045101-7 Coupling of twist and writhe J. Chem. Phys. 132, 045101 �2010�

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