bernard d. coleman, david swigon and irwin tobias- elastic stability of dna configurations. ii....

8/3/2019 Bernard D. Coleman, David Swigon and Irwin Tobias- Elastic stability of DNA configurations. II. Supercoiled plasmids

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Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact

Bernard D. Coleman,1,* David Swigon,1, and Irwin Tobias2,1Department of Mechanics and Materials Science Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854

2Department of Chemistry, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854

Received 16 July 1999

Configurations of protein-free DNA miniplasmids are calculated with the effects of impenetrability and

self-contact forces taken into account by using exact solutions of Kirchhoffs equations of equilibrium forelastic rods of circular cross section. Bifurcation diagrams are presented as graphs of excess link, L, versuswrithe, W, and the stability criteria derived in paper I of this series are employed in a search for regions of suchdiagrams that correspond to configurations that are stable, in the sense that they give local minima to elasticenergy. Primary bifurcation branches that originate at circular configurations are composed of configurationswith Dm symmetry (m2,3, . . . ). Among the results obtained are the following. i There are configurationswith C2 symmetry forming secondary bifurcation branches which emerge from the primary branch with m3, and bifurcation of such secondary branches gives rise to tertiary branches of configurations withoutsymmetry. ii Whether or not self-contact occurs, a noncircular configuration in the primary branch with m2, called branch , is stable when for it the derivative dL/dW, computed along that branch, is strictlypositive. iii For configurations not in , the condition dL/dW0 is not sufficient for stability; in fact, eachnonplanar contact-free configuration that is in a branch other than is unstable. A rule relating the number ofpoints of self-contact and the occurrence of intervals of such contact to the magnitude of L, which in paperI was found to hold for segments of DNA subject to strong anchoring end conditions, is here observed to holdfor computed configurations of protein-free miniplasmids.

PACS numbers: 87.10.e, 46.70.Hg, 02.40.k, 46.32.x

I. INTRODUCTION

In paper I of this series on the theory of the elastic rodmodel for DNA 1, we derived several criteria for the elasticstability of a calculated equilibrium configuration of a DNAsegment that is either a plasmid i.e., a closed ring or alinear segment subject to strong anchoring end conditions.We here apply the criteria to a classical problem: analysis of

the stability of supercoiled configurations of protein-freeplasmids. See, e.g., Le Bret 2 and Julicher 3.Our results hold for the theory of the commonly em-

ployed elastic rod model which treats a DNA segment as inintrinsically straight, homogeneous, inextensible rod withelastic properties that are characterized by two elastic con-stants, the flexural rigidity A and the torsional rigidity C.Hence, the configuration Zof a DNA segment is determinedonce one has specified the curve C representing the duplexaxis and the density of the excess twist about C. Theelastic energy of the segment is the sum of a bendingenergy B which depends on the curvature of C and atwisting energy T which depends on . Thus,

B T, 1

and, when , B , and T are expressed in units of A/Lwith L the length of the segment,

BL

2 0L

s 2ds , TL

2 0L

s 2ds , 2

where

C/A . 3

In earlier studies of the elastic rod model cf. Refs.4 6, explicit and exact solutions of Kirchhoffs equa-tions were employed to calculate equilibrium configurationsfree from points of self-contact. The configurations and bi-furcation diagrams shown here and in paper I were calcu-lated using generalizations of those explicit solutions forcases in which excluded volume effects and forces arisingfrom self-contact must be taken into account.

We are concerned with plasmids, i.e., segments for whichboth DNA strands form closed curves. The excess link in aplasmid, L, is a topological constant, which, by a nowfamiliar result 7,8, obeys the equation

LWT, 4

in which,

T1

20L

ds , 5

and W is the writhe of the closed curve C. There are severalequivalent definitions of writhe; one, due to Fuller 9, wasmentioned in paper I. For a sufficiently smooth curve C, onemay write see, e.g., the introductory survey 10

*Electronic address: [email protected] address: [email protected] address: [email protected]

PHYSICAL REVIEW E JANUARY 2000VOLUME 61, NUMBER 1

PRE 611063-651X/2000/611/75912 /$15.00 759 2000 The American Physical Society


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W1

40L

0

L t s t s* x s x s*x s x s*3

ds ds*, 6

in which t(s) is the unit tangent vector for C, and x(s) is thelocation in space of the point on C with arc-length parameters.

We confine our attention to knot-free plasmids which are

modeled as elastic, but impenetrable, closed rodsi.e., rings

for which the cross sections are circular with an invariantdiameter D. The two important parameters for the calcula-tions we present are and

dD/L . 7

We assume that no external forces act on the plasmidunder consideration, and that disjoint subsegments of theplasmid can interact only through contact. We assume fur-ther that when such contact occurs, the contact forces arenormal to the surfaces of segments involved and momentsare not exerted at points of contact; hence changes in con-figuration do no work against the contact forces.

A configuration is called an equilibrium configuration if, the first variation of, vanishes for each variation Zin configuration that is admissible in the sense that it is com-patible with the imposed constraints, which include the re-quirement that such topological properties as the value ofL and the knot-free state ofC be preserved.

In the units employed for Eqs. 1 and 2, those equationsimply that the resultant moment on the cross section witharc-length parameter s is

M s tdt

ds t. 8

In an equilibrium configuration of the plasmid, at values ofsother than those characterizing points of self-contact, M(s)and the resultant force F(s) obey the equations

dF

ds0,

dM

dsFt. 9

These two balance equations, with M as in Eq. 8, yield asystem of equations for C and which can be solved interms of elliptic functions and integrals for a subsegmentbetween points of self-contact see, e.g., Refs. 4 and 6.Each such solution is determined by six solution parameters.A plasmid with n points of contact has 2n contact-free sub-

segments, and hence its configuration is determined when12n solution parameters are specified. The condition of pre-assigned L and equations rendering precise geometric con-straints at contact points and laws of balance of forces andmoments at those points i.e., Eqs. 44 and 45 of paper Iyield 12n equations which can be solved to calculate thesolution parameters 11. In this manner we obtain the con-stant value of and a precise analytic representation ofCfor equilibrium configurations in which self-contact occurs ata finite number of points. The closed form expressions for Cyield formulas that greatly facilitate calculations both of theelastic energy cf. Ref. 5 and of the integral along C ofthe geometric torsion cf. Ref. 6. Once the torsion integralis known, the writhe W of C is determined to within an

integer, called the self-link cf. Refs. 7 and 12. In thecases considered here, that integer is not difficult to evaluatefor details see Refs. 6 and 11.

As in paper I, we here call equilibrium configuration Z#

stable if it gives a strict local minimum to in the class ofconfigurations compatible with the constraints. In otherwords, Z# is stable if and only if it has a neighborhood Nsuch that (Z)(Z#) for each configuration Z in N that

is not equivalent to Z# and is accessible from Z# by a ho-motopy compatible with the constraints. Lack of equiva-lence means that the two configurations, Z# and Z, differ indistribution of twist density or are such that the correspond-ing curves, C # and C, are not congruent, or both.

As we remarked in paper I, our definition of stability, as itrequires that have a strict local minimum, differs from aconcept of stability often used in physics see, e.g., Ref. 3which, in the present subject, would require to have aglobal minimum, i.e., to not exceed its minimum value forany other configuration that may be reached by an arbitrarilylarge variation compatible with the constraints. A configura-tion that is stable according to our definition but does not

give a global minimum to the appropriate energy would becalled metastable in other contexts.When a configuration Z# is a member of a one-parameter

family E of equilibrium configurations Z for which one cantake L, T, and to be given by functions LE, TE,and E ofW, as in bifurcation diagrams presented here andin paper I, Z# is stable only if the slope of the graph of Lversus W for E is not negative at Z#. Thus the relation

d LE/dW0, 10

which we call the E condition, is a necessary but not suffi-cient condition for stability. This condition, derived in pa-per I under assumptions more general than the present, was

obtained in a different form in a seminal paper by Le Bret2.

On a cautionary note we mention, as we did in paper I,that there are exceptional families of equilibrium configura-tions for which L is not determined by W, because, forthem, equilibrium is maintained when L is changed with Ckept constant. Such is the case for those configurations of aplasmid in which C is a true circle, i.e., the configurationsthat form the trivial branch, which is labeled in thebifurcation diagram shown in Fig. 1 below.

An equilibrium configuration in E for which the excesstwist density vanishes remains an equilibrium configu-ration when the plasmid is nicked, i.e., when one of its two

DNA strands is severed. In paper I it is shown that astrengthened form of the relation 10, namely,

d LE/dW1, 11

called the n condition, is a necessary condition for an equi-librium configuration in E with 0 to be stable bothbefore and after nicking.

In order for an equilibrium configuration Z# to be stable,it is necessary that, for each between 0 and L, there hold()0, where () is the slope of the graph of L versusW for the family of equilibrium configurations of the plas-mid that contains Z# and is subject to the additional condi-tion that the subsegment with sL be held rigid. For the

760 PRE 61BERNARD D. COLEMAN, DAVID SWIGON, AND IRWIN TOBIAS


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formal definition of () see Eq. 29 of paper I. In thepresent paper we shall show that this condition, which wasderived in paper I and there called the condition, can fur-nish a practical method of demonstrating that certain con-figurations that obey the E condition are in fact unstable.

In order for an equilibrium configuration to be stable it isnecessary that the curve C representing the duplex axis give astrict local minimum to the bending energy B in the classof curves that obey constraints and have the same writhe asC. This Wcondition, like the condition, can yield a methodof further testing the stability of configurations known toobey the E condition.

Arguments given in paper I show that strengthened formsof the E condition and the W condition can be combined toobtain a condition sufficientfor stability, called the S condi-tion, which in the present context may be stated as follows:An equilibrium configuration Z# in E is stable ifdLE/dW0 at Z# and, in addition, curve C(Z#) has aneighborhood N such that for each Z* in the family E withwrithe W* close to W#, there holds B(C )BC(W*)for every closed curve C in N that has writhe W* and is not

congruent to C(W*).In the next section of the paper we present calculated

equilibrium configurations for plasmids and applications ofthe criteria just stated. The configurations and bifurcationdiagrams shown were obtained using the generalized methodof explicit solution. The basic parameters in our theory aredD/L and C/A . For the cross-sectional diameter D ofDNA we employed 20 , and we chose L to be the length ofa segment for which the number N of base pairs bp is 359i.e., L3593.4). When and the configuration or,equivalently, the solution parameters are known, the equa-tions of the theory enable us to calculate , B , and T inunits ofA/L cf. Eqs. 1920 of Ref. 5, which are easily

evaluated expressions for , B , and T in terms of solu-tion parameters. To express our reported values of andB in kcal/mol, we needed a value for A; we chose A2.0581012 erg nm, which corresponds to a persistencelength of 500 at 298 K cf. Ref. 13.

Although we have in hand an easily applied rule for trans-forming the bifurcation diagram presented as a graph ofLversus W for one value ofinto the bifurcation diagram foranother value of Eq. 40 of paper I, and we do describethe way in which the stability of equilibrium configurationsdepends on , we have chosen to present diagrams with set equal to 1.5, a value that corresponds to the very high endof the range of experimental results for C 14,15, because it

is only at high values of that there are ranges of L

inwhich stable nonplanar contact-free configurations occur cf.Julicher 3. The low end of the range of experimental de-terminations of C yields 0.7 for 16,17. Section II con-cludes with a figure showing how the class of stable configu-rations for 1.5 differs from that for 0.7.

Here, as in paper I, our calculations are intended to illus-trate a method of investigating the elastic stability of con-figurations, not to argue that specific values of or D areappropriate for the ratio of the elastic moduli or the effectivediameter of DNA.

The results we present in Sec. II have been obtained withrigor and precision in the theory under consideration, namelythat in which a DNA segment is modeled as a homogeneous

rod obeying the special case of Kirchhoffs constitutive rela-tions in which the rod is assumed to be not only both inex-tensible and kinematically symmetric, but also of circularcross section. Without dwelling on the obvious limitations ofsuch a model as a representation of a true DNA segment, weshould like to make two cautionary remarks about the appli-cability of our results that touch on matters other than thelack of homogeneity and axial symmetry in actual DNA seg-

ments:i Since the value ofN employed in our calculations cor-

responds to approximately 2.5 persistence lengths under nor-mal experimental conditions, we do not claim that the con-figurations we calculate would be free from fluctuations, orthat for them one can set differences in equal to differ-ences in the free energy of supercoiling. For calculations ofthe free energy of supercoiling for miniplasmids see, e.g.,Refs. 1822.

ii The calculations we report do not account for the ef-fects of electrostatic repulsion. To do so in an approximateway, one may consider replacing the average geometricvalue ofD by an effective value, say Df, with f1 see,

e.g., Refs. 23 and 24. When electrostatic effects are ab-sent, calculations of the type we have performed for plas-mids of size N and geometric cross-sectional diameter Dare applicable to plasmids of size Nfwith diameter Df. Equi-librium configurations for specified values of L remain es-sentially the same under such rescaling; i.e., curve C under-goes a similarity transformation with scale factor f, theexcess twist Tas well as the writhe W remains invariant,and the elastic energy in units of kcal/mole changes tof/f. One may hope that if one chooses f judiciously,one can obtain, from calculations that ignore electrostaticforces, useful, albeit approximate, values of properties of aminiplasmid of size Nf and geometric diameter D that is

subject to electrostatic effects.In Appendix A we discuss an illustrative example whichshows that, once one is able to find all the equilibrium con-figurations in a given region of the (L, W plane, or,equivalently, in the , L) plane, one can find lowerbounds for energy barriers for the various pathways bywhich a transition at fixed L from one locally stable stateto another stable state can be realized. We remark there thatbefore the estimate of activation energy which that exampleyields can be accepted with confidence, it will be necessaryto resolve an open problem mentioned in the last paragraphof Sec. II.

II. CONFIGURATIONS AND BIFURCATION DIAGRAMS

For each pair (N,L), a plasmid has at least one, andusually several, equilibrium configurations see, e.g., Refs.2 and 3. As L varies at fixed N, these configurationsvary and form families of equilibrium configurations. Wefocus our attention on groups of such families, also calledbranches, that contain configurations with three or fewerpoints of maximum curvature and that are connected to abranch, called the trivial branch, which is made up of theconfigurations for which C is a circle. In the figures thattrivial branch is labeled and is shown as a dotted line.Branches and , which originate at a bifurcation point ofthe trivial branch, are shown as heavy solid lines and are

PRE 61 761ELASTIC STABILITY OF DNA . . . . II. . . .


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called primary bifurcation branches. Branches that originateat bifurcation points of a primary branch and hence are con-nected to by a single primary branch are called secondarybifurcation branches. In the figures the secondary brancheshave labels that bear subscripts, e.g., I , II ,..., and areshown as light solid curves. A tertiary bifurcation branch,II* , that originates at a bifurcation point of the secondarybranch II is also shown as a light solid line see Fig. 1.

Since the protein-free and knot-free plasmids we con-

sider have a symmetry such that for equilibrium configura-tion the transformation LL takes W into W andleaves unchanged, we take L to be positive.

When L is less than a critical value, L*, a miniplas-mid has just one equilibrium configuration and it belongs to. In the present case (N359, 1.5), L*1.029 andthere are two equilibrium configurations with LL*:one is in , and the other, with W*0.927, has a single pointof self-contact and corresponds to the point A1 on branch .

Each point of for which

L1m21, m2,3,4, . . . , 12

is a bifurcation point at which a family of solutions withW0 intersects , i.e., at which a primary branch originates25. We call the integer m in Eq. 12 the index of theprimary branch. The first such bifurcation of occurs atLL1) and gives rise to branch which has anindex of 2; the second occurs at LL12& andgives rise to the branch which has an index of 3. We notethat L* is the smaller of the two numbers L andLA1, where A1 is the configuration of minimum writhewhich i lies in the branch and ii is such that the plasmidhas a point of self-contact.

A list of possible symmetry groups of contact-free equi-librium configurations of closed elastic rods with L0 wasgiven by Domokos 26. His analysis rests on the assump-

tion that the force vector F is independent of s, an assump-

tion valid when self-contact does not occur. We find that foreach m2, the symmetry group of all configurations in theprimary branch with index m is the dihedral group D m oforder 2m . Hence, whether or not self-contact is present, thecurve C for a configuration on the primary branch of index mhas a single m-fold symmetry axis that is perpendicular tothe plane Pcontaining the 2m points at which the curvature of C has a local extremum i.e., a maximum or a mini-mum. Each of the m lines that intersect the m-fold symmetryaxis and pass through two extrema of is a twofold symme-try axis.

It is clear that on the trivial branch there holdsdW/dL0, and L, T, and are not given by func-

tions ofW. Although the theory of the E condition and condition is not directly applicable to configurations in , onecan show that those circular configurations are stable whenL is less than L, and unstable when L is greater thanL. A formal proof of this assertion is given in AppendixB.

In Fig. 2 we give graphs ofB versus Wfor the branchesof the bifurcation diagram of Fig. 1. The utility of suchgraphs for investigation of the stability of calculated equilib-rium configurations of plasmids was noted by Le Bret 2.Arguments given in paper I imply that here, on each non-trivial branch,

dBE/dW42TE. 13

This relation, which follows from Eq. 20 of paper I withP0 see also Ref. 9, Eq. 5, tells us that the data in Fig.2 determine L as a function, LE, ofW, i.e., Fig. 1 can bereconstructed from Fig. 2.

As the configurations in have zero writhe and bendingenergy 22 in units ofA/L), the trivial branch reduces to asingle point in Fig. 2. It follows from Eqs. 12 and 13 thatthe primary bifurcation branches correspond to curves in Fig.2 with initial slopes given by

dBE

dWW0

42m21. 14

FIG. 1. Bifurcation diagram for a protein-free DNA plasmidwith N359 and 1.5 drawn as a plot of L versus W. Shownas a dotted vertical line is the trivial branch ; two branches, and, resulting from bifurcation ofare shown as heavy solid curves;

four branches, I , II , III , and IV , arising from secondary bi-furcations of, and one tertiary branch II* emerging from II , areshown as light solid curves.

FIG. 2. Graphs of bending energy B versus W for branches ofthe bifurcation diagram of Fig. 1.



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A configuration that minimizes B in the class of equi-librium configurations with a given writhe also minimizesB in the class of all configurations with the same writhe.Hence, the calculations that gave us Fig. 2 tell us that foreach writhe the global minimum of

B

at that writhe isattained by a configuration in branch .

Details of the graph of L versus W for branch areshown in Fig. 3a. For n0, 1, 2, and 3, the configurationswith n points of self-contact correspond to points on thatgraph between An and An1 and form smooth families ofequilibrium configurations. As we know that the branch isa locus of configurations that minimize B at fixed W, wecan apply the S condition and assert that a configuration in is stable if and only if it obeys the E condition. AlthoughdL/dWsuffers a jump at the points An, in the present casethe right-hand and left-hand derivatives are positive at A2

and A3, and hence the configurations corresponding to thosepoints are stable. At A1, dL/dW has left- and right-hand

derivatives that have opposite signs; at the point X, shown asa solid circle in Fig. 2, the derivative dL/dWis continuous

but changes sign. As the branch traverses the points A1 andX, the corresponding configurations either gain or lose sta-bility; such points are called points of exchange of stability.Since, in the present case, L is a strictly increasing func-tion ofWbetween A0 and X, and between A1 and A4, but isa decreasing function between X and A1, the configurationsin with W either between its values at A0 and X, or be-tween its values at A1 and A4 are stable, and those with Wbetween its values at X and A1 are unstable.

The configuration A0 at which branches off from ,the configuration X which lies in between the circularconfiguration A0 and the figure 8 configuration A1 andwhich has dL/dW0), and the configurations A1, . . . ,A4

are shown in Fig. 4. Each of these configurations has D2symmetry, i.e., has three twofold symmetry axes, two ofwhich lie in the plane P. As L increases, the number ofself-contact points in a configuration on the branch in-creases in the sequence 1, 2, 3, until L attains its value at

A4. For the values ofand D/L employed here, that valueofL is 2.521. A configuration with L near to, but greater

FIG. 3. Graphs of L versus W, and ver-sus L, for branches and . The configurations

corresponding to points and are shown inFig. 4.

FIG. 4. As explained in the text, configurations at X and A1 are points of exchange of stability, and for each n, An is the configurationof smallest writhe in branch with n points of self-contact. Here, as in Figs. 6, 8, 9, and 11, the following conventions are employed: thetop row shows the projection of C on plane Pwith the twofold symmetry axes in P, drawn as dashed lines. The bottom row shows theplasmid depicted as a tube of diameter 20 viewed at an angle of 75 to T. The scale is constant in each row, but is reduced in the top row. is given in kcal/mol.



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than, 2.521 has two isolated points and an interval of self-contact; for such a configuration, attains its minimum atthe two values of s corresponding to the midpoint of theinterval of self-contact.

The value ofW corresponding to point X is sensitive to

. Had we here put 1.4, as we did in a discussion ofmononucleosomes in paper I, the interval between A0 and Xwould have been too small to show clearly in Figs. 3a and3b. Recent calculations discussed below show that when1.375 i.e., when 118 ), dL/dW0 at point A

0 onbranch , and hence points A0 and X then coincide. Thus, allof the nonplanar contact-free configurations on branch areunstable if1.375. In addition, we have found that when1.993, points X and A1 coincide, which implies that allthe configurations on branch are stable if1.993. Be-cause configurations A0 and X do not show self-contact, butthe configuration A1 does, the value offor which X and A0

coincide is independent of the plasmid size and cross-

sectional diameter D, but the value ofat which X and A1

coincide depends on the ratio dD/L .With the exception of Ref. 2, the literature on bifurca-

tion branches in the theory of the elastic rod model forprotein-free plasmids deals primarily with branch . Somerelevant recent papers are those of Julicher 3, Yang et al.27, and Westcott et al. 28. Julicher 3 modeled a plasmidas an impenetrable rod with zero cross-sectional diameterand considered a configuration stable only if it gives a globalminimum to at fixed L. If one identifies the configura-tions that Julicher refers to as interwound with those in that show two or more points of self-contact, and takes intoaccount differences in assumptions about the diameter of

cross-sections, then his , L

)-phase diagram and observa-tions about the dependence on WofB for configurations inbecome compatible with the results shown here in Figs. 2,3b, and 12. Yang et al. 27 and Westcott et al. 28 presentnot stability analyses but numerical calculations. Yang et al.27 developed a finite element method in which self-contactis taken into account by introducing a penalty function.Westcott et al. 28 employed a finite difference method andaccounted for the excluded-volume effect that arises from aDebye-Huckeltype electrostatic repulsion. Each of the citedpapers contains examples of calculated configurations withevident D2 symmetry.

There are ranges of with upper bounds depending ond for which there are values of L where either i both

branch and branch contain stable equilibrium configura-tions with the same L, or ii branch contains two suchstable configurations. The values of, N, and D which yieldFig. 3 are such that both i and ii can occur for separateranges ofL). We have found that for the present value of

d, i occurs when 1.707 and LA1LLA0L, and ii occurs when 1.3751.993 andmaxL(A1), LLL(X). When 1.993, foreach L there is precisely one stable equilibrium configura-tion in the union of branches and .

Perhaps of greater importance than the observations justmade, at least for those concerned with topoisomer distribu-tions, is the fact that there is a range of which is 1.600 for the present value of d such that for a smallinterval of values of L bounded above by L there arenot only two stable configurations, one in with a singlepoint of self-contact, i.e., a figure 8, and the other in ,but, in addition, the figure 8 configuration has lowerelas-

tic energy than the circular configuration. See also Le Bret2, Tsuru and Wadati 29, and Julicher 3.The graph of L versus W for , which branches off

from at B0 and has index 3, is shown in Figs. 1 and 5 a.The configurations in have D3 symmetry. Those betweenB0 and B1 are contact-free, those between B1 and B2 havethree points of self-contact, and those with L greater thanbut close to its value 4.653 at B2 have six points of self-contact. See Fig. 6.

For each branch that bifurcates from including branch there is an interval of values ofW containing W0 forwhich the corresponding equilibrium configurations arecontact-free. In fact, all nontrivial contact-free configura-

tions are in primary bifurcation branches.

We have foundthat the contact-free equilibrium configurations in primarybranches with index m2 do not obey the condition andhence cannot be stable, even if they obey the E condition.See, for example, the graph of () versus /L shown inFig. 7 for the configuration U, which does obey the E con-dition. It follows that a protein-free plasmid can have non-planar stable equilibrium configurations that are contact-freeonly if1.375, and if such configurations occur, they mustbe in . This conclusion is stronger than a result of Le Bret2 to the effect that for such a plasmid to have stable, non-planar, contact-free configurations it is necessary that 1.

Let m be the value of for which the branch that bifur-cates from with index m is such that dL/dW0, at the

FIG. 5. Graphs ofL versus Wand versusL for branches , , I , II , and II* . Theconfigurations corresponding to points , , and are shown in Figs. 6, 8, and 9. Branch is

shown as a heavy solid curve with kinks at B1and B2) and branches I , II , and II* as lightcurves with kinks at BI

2 and BII2 ).



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point where W0. If m , then there are contact-freeconfigurations in the branch with index m obeying the Econdition, but ifm there are no such contact-free con-figurations in the branch. Applying the explicit solutionmethod for contact-free equilibrium solutions 4 6 to an

analysis of solutions near to the circular solutions with L kas in Eq. 12, we have found that, to within eight significantfigures for m ,

m13/2m2. 15

In particular, 2118 and, as stated above, if

118 , the

interval of points in corresponding to stable equilibriumconfigurations is not empty. For m2, if m , thebranch with index m will have an interval with contact-freeequilibrium configurations that obey the E condition, but, aswe have noted, are unstable.

Since the value of that we are using exceeds 376 ,

there is an interval of writhe values, which in the present

case is 0W0.900, for which the configurations in arecontact-free and have dL/dW0. For 0.900W1.948where 1.948 is the writhe of B1), dL/dW0. Because allthe contact-free configurations in are unstable, the point inwhere W0.900 is not a point of exchange of stability.

We have remarked that the contact-free configurations in fail to obey the condition. One also can show that thoseconfigurations do not obey the W condition by constructingadmissible variations with W0 that lower B . Consider,for example, the configuration labeled U in Fig. 5 which has

W0.5, L1.975, and is shown in Fig. 8, where q0 de-notes the distance 32.8 nm between the two points of ex-trema of curvature ofC that lie on the twofold axes of sym-metry for U. We have constructed a one-parameter set H ofconfigurations U of the plasmid such that each configura-tion in H yields 0 for each variation in C obeying theconstraints that i C has a twofold axis of symmetry, ii Wstays at its value in the configuration U, iii the distancebetween the points on the symmetry axis isq(1)q0 . The members of H with 0.05, 0.2, and0.668 are shown in the figure. Clearly, U is in H with 0, and the variation (Z) that takes U into U

has W0. In the present case, BU

BU for each 0, nomatter how small. The configuration U* corresponding to0.668) is an equilibrium configuration of the plasmid andit lies in branch .

We now turn to the configurations in with self-contact.Such configurations have loops. For rods and ropes the con-cept of a loop is intuitive. DNA segments, such as the sub-

segment Df of a miniplasmid in a mononucleosome cf. pa-per I, that are subject to constraints that keep endpoints inproximity are often called loops, even if free from self-contact. Here, when we call a subsegment of a protein-freeplasmid a loop, we presuppose that its ends are in contact. Atsufficiently large values of the writhe, a loop shows self-contact not only at its end points, but also in its interior. It isin agreement with current usage to call a loop with more thanone self-contact a plectonemic loop, or, for short, a plec-toneme.

FIG. 6. Selected configura-tions in the branch . Bn denotesthe configuration of smallestwrithe in with 3n points of self-contact. Shown are B0, B1, andB2. The configurations at points Fand G would remain in equilib-rium if the plasmid were nicked; Fis unstable; G is stable and wouldremain so after nicking. P is theconfiguration at the point of sec-ondary bifurcation.

FIG. 7. Graphs ofversus /L for configura-tions U and G in branch that, as seen in Fig. 5,obey the Econdition with dL/dWstrictly posi-tive. The plasmid is contact-free in configurationU and has three points of self-contact in G. Inboth cases, as 0. For G, ()0 forall 0L; for U, () vanishes at two valuesof , and there are two singular values of marked with vertical dotted lines at which from the left and from the right.



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When L exceeds its value at B1, the configurations inthe branch have three congruent loops. Between B1 andB2, each such loop has one point of self-contact at its end-points. For W or, equivalently, for L) greater than itsvalue at B2, these loops are plectonemes.

Points B1 and P are points of secondary bifurcation. AtB1, the point on with the smallest L for which self-

contact occurs, branches I and II originate. Two otherbranches, III and IV , originate at P. Although, as we haveremarked, the configurations in have in accord with theknown properties of primary branches one threefold sym-metry axis and three twofold symmetry axes, those inIIV have a symmetry corresponding to group C2 of order2, i.e., have only a single twofold symmetry axis and it isunique. See Figs. 9 and 11. The configurations in I haveone loop which, as L increases above its value at BI

2, be-comes plectonemic. The configurations in II have two suchloops, which are congruent. The configurations in III andIV have three loops, of which one intersects the symmetryaxis and the other two are congruent. In III , the loop that

intersects the symmetry axis becomes plectonemic whenW

increases above its value at BIII2 . In IV , the two congruent

loops become plectonemic when Wincreases above its valueat BIV

2 . In each case that we have studied, the number ofself-contacts in a plectonemic loop increases in the sequence1, 2, 3, with an interval of points of self-contact occurring athigher L. Such is the case also for the extranucleosomalloop of miniplasmids in mononucleosomes see paper I andfor plectonemic loops in linear DNA segments subject totension and torsional moments 11.

When we say that I , II , III , and IV originate at asecondary bifurcation point, i.e., are secondary branchesthat branch off from the primary branch , we employ aterminology that follows a natural convention: if, as a param-eter e.g., L or W is varied, a primary branch enters apoint and several other branches exit from the point withprecisely one of the exiting branches having the symmetryproperties of the primary branch, we consider the branch thatexits with the original symmetry to be the continuation of theprimary branch and the others to be secondary branches thatoriginate at the bifurcation point.

The notation we used in Figs. 511 when we labeledpoints B0, B1, B2, BI

2, etc., has the following property: Theconfigurations in each branch a where a stands for , I ,II , III , IV) that show at least one loop with p self-contacts, but no loop with p1 contacts, correspond to the

points in a between Bap and Ba

p1. In this notation, pointsBI

1 and BII1 are the same as point B1 and hence are labeled B1,

and points BIII1 and BIV

1 are the same as point P.The secondary branch II has a bifurcation point S, at

which a tertiary branch, II* , originates. The configurationsin II* have no discernible symmetry; each contains two non-congruent loops. As W increases, one loop becomes plec-

tonemic, while the other appears to flatten or becomemore planar as well as tighten or increase in bendingenergy see Fig. 9.

As shown in Fig. 5a, the graphs of L versus W forbranches , I , II cross the line LW. The configura-

FIG. 8. The configurations U() are the resultof special deformations of U described in thetext that change the distance between points aand b from q0 to (1)q0 while keeping Wfixed. It turns out that this one-parameter set ofdeformations lowers B .

FIG. 9. Shown here are: BI2, the configuration of smallest writhe

in I with two points of self-contact; BII2 , the configuration of small-

est writhe in II with four points of self-contact; S, the configura-tion at the point of tertiary bifurcation ofII ; and BII*

3, the con-figuration of smallest writhe in II* with four points of self-contact.



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tions corresponding to those crossing points would remain inequilibrium if the plasmid were nicked. Such points onbranch are labeled F and G and marked in Fig. 5a. AtF, dL/dW1, and hence the corresponding configuration

is unstable. Since point G lies between B1 and P, the con-figuration corresponding to G obeys the condition see Fig.7 and, since it obeys the condition dL/dW1, if it isstable, it will remain so when the plasmid is nicked.

Analysis of the stability of configurations in branches ,IIV , and II* requires care. By using the condition, wecan show that not only the contact-free configurations in but also all the configurations with self-contact points thatare in I , II , or II* , or are in with W greater than itsvalue at P, or in III between P and BIII

2 , or in IV between Pand BIV

2 are unstable, whether or not they obey the E condi-tion. The remaining configurations, namely those in be-tween B1 and P, those in III with W greater than W(BIII

2 ),

and those in IV with W greater than W(BIV2 ), obey the condition. Verification that the S condition holds suffices toprove stability, but such verification usually is not easy, be-cause for equilibrium configurations other than those on thebranch it is a very difficult matter to prove that B has alocal minimum at fixed W.

Our experience indicates that whenever an equilibriumconfiguration Z# of a plasmid fails to obey the condition,we can find a counterexample showing that Z# does not givea strict local minimum to B at fixed W see, e.g., Fig. 8.As we have not been able to find such counterexamples for

configurations that do obey the condition, we conjecture,with as yet no formal proof, that satisfaction of the condi-tion is not only necessary, but also sufficient for stability ofan equilibrium configuration of a plasmid. Consequences of

this conjecture of sufficiency of the condition for stabilityof configurations of plasmids will be studied in Appendix A.We consider the development of methods of proving or dis-proving the validity of the conjecture to be a major openproblem in our subject. To give some idea of the importanceof the problem, we remark that, if the conjecture is correct,then configurations corresponding to points of the heavysolid curves in Fig. 12 are stable in the sense that they giveto strict local minima in the class of configurations withequal L, but, until the validity of the conjecture is estab-lished, we can assert with absolute certainty only that thesegments of branches which are drawn as light curves in thefigure are composed of unstable configurations and the seg-

ments of the branches and drawn as heavy curves arecomposed of stable configurations.

ACKNOWLEDGMENTS

This research was supported by the National ScienceFoundation under Grant No. DMS-97-05016 and the U. S.Public Health Service under Grant No. GM34809. D.S. ac-knowledges support from the Program in Mathematics andMolecular Biology at the Florida State University with fund-ing from the Burroughs Wellcome Fund Interfaces Program.

FIG. 10. Graphs ofL versus W, and ver-sus L, for branches , III , and IV . The con-

figurations corresponding to points are shownin Fig. 11.

FIG. 11. Selected configura-tions in branches III and IV .



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APPENDIX A:

TRANSITIONS BETWEEN CONFIGURATIONS

Figure 12 contains graphs of versus L for fixed val-ues ofN and D, and two values ofC/A near the extremepoints of the range ofthat has been reported as compatiblewith experiment. In each case there is a critical value Lc ofL such that each configuration in branch gives a globalminimum to when LLc at 1.5, Lc1.123; at0.7, Lc1.848) and there is an interval Jof values ofLLc for which there are two configurations that obeythe condition: AL in and BL in .

In this Appendix we discuss matters that can be regardedas extrapolations of results given in the text, and that suggestroutes by which one may seek to extend the present theory to

the point where it permits treatment of the kinetics of tran-sitions between locally stable configurations. If we assumethe validity of the conjecture of sufficiency of the conditionfor stability of equilibrium configurations of plasmids 30,then, for each L in J, the configuration BL gives a localminimum BL to , and it becomes meaningful to askquestions about the activation energy for a transitionfrom BL to AL. A familiar mountain pass theoremthen tells us that, for a given L in J, is no less than the

minimum #(L) of differences between BL and at unstable equilibrium configurations with the same L.We find that, as L varies over J, the maximum value, , of these lower bounds, #(L), is attained at LL3.282 for 1.5, and at L4.335 for 0.7.We here focus our attention on the transitions B(L)A(L).

For both 1.5 and 0.7, when LL, there arefour unstable equilibrium configurations close to B( L):one on each of the secondary branches, I , II , and two onthe tertiary branch II* . These are labeled BI, BII, BII1* , BII2*in Fig. 13. For 1.5 and 0.7, the lower bound of the heightof the energy barriers at L, i.e., #(L), isattained on two paths, one taking B to A through BII, and the

other taking B to A through BII1* . For 1.5, 3.48 kcal/mol or 5.6 kB T per molecule at T310K; for0.7, 3.29 kcal/mol or 5.3 kB T.) These, as well astransition paths from B to A involving the configurations BIand BII2* , and hence with energy barriers higher than #(L) albeit less than 1/2 kB T higher in the case ofthe path BBI2* A are depicted in Fig. 14.

The total decrease in for the transition BA at L is

FIG. 12. Graphs of versus L for the vari-ous primary and secondary branches discussed inthis paper. Light curves: configurations that donot obey the condition and hence are not stable.Heavy curves: configurations that obey the con-

dition. As the configurations of the branch forwhich L exceeds its value at A4 open circlehave intervals of self-contact, they were com-puted by an extension of the explicit-solutionmethod employed here and in paper I for configu-rations showing only a finite number of points ofself-contact.

FIG. 13. Graphs ofL versus Wand ver-sus L for 1.5. The configurations corre-sponding to points have LL3.282 andare shown in Fig. 14. The corresponding graphsfor 0.7 have the same structure with, ofcourse, a change in L and the values of atBI, BII, BII1* , BII2* , and A.



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19.4 kB T for 1.5 and 20.5 kB T for 0.7. Our calcu-lation of the configurations A( L) and their energies forcases in which self-contact occurs in intervals required anextension of the method used elsewhere in the paper. Aforthcoming paper will deal with explicit representations ofequilibrium configurations with intervals of self-contact.

The calculations summarized in Fig. 14, although givenfor illustrative purposes only, suggest the following conclu-sion: although the value L of L that maximizes thelower bound #(L) for the activation energy for a tran-sition from the branch is sensitive to the value of, thatmaximum, #(L), is not.

APPENDIX B: STABILITY OF CIRCULAR

CONFIGURATIONS OF MINIPLASMIDS

The fact that for each specified value of the writhe thebending energy B of a miniplasmid is minimized on branchyields a straightforward proof that an equilibrium configu-ration Z# in , i.e., with C # a circle, is stable if 0L(Z#)L, and is unstable if L(Z#)L,where, by Eq. 12, L1). That proof goes as fol-lows.

According to our definition, an equilibrium configurationZ# of a plasmid is stable if there is a neighborhood ofZ#

such that for each configuration Z in that neighborhood thathas L equal to L(Z#) and is not equivalent to Z#, thereholds (Z)(Z#). To prove that Z# is unstable, it suf-fices to show that in each neighborhood ofZ# there is aconfiguration Z with L equal to L(Z#) and (Z)(Z#).

Let us write L #, T#, W#, C #, # etc., for L(Z#),T(Z#), W(Z#), C(Z#), (Z#), etc. Of course, since C #

is a circle, W#0. In the units employed for B and T inEqs. 2, the bending energy of a circle is equal to 22;hence B

#22. Moreover, the bending energy of a circle is

strictly less than the bending energy of all noncircular closedcurves. Since is independent ofs in an equilibrium con-

figuration, Eqs. 2 and 4 yield I#22(T#)2

22(L #)2.We suppose first that 0L #L. Let Z be a con-

figuration that is not equivalent to Z# and is in an appropri-ately chosen neighborhood ofZ# with L(Z)L #, andlet Z be the equilibrium configuration in branch withW(Z)W(Z), and hence with B(Z)B(Z). In view ofthe fact that, of all configurations with a given value of T,that for which is constant minimizes T, we haveT (Z)2

2T(Z)2 and, by Eqs. 1 and 4,

Z #TZT#BZB

#

42L #WZBZ22

OWZ2. B1

Since C(Z) is a closed curve, B(Z)22. IfW(Z)0,

then, because L0, we have (Z) #. IfW(Z)0,then, we have L(Z)L # and T(Z)T#, and in or-der for Z to not be equivalent to Z#, it must be the case thatCZ is not a circle and B(Z)2

2, which again yields(Z) #. For the remaining possibility, i.e., W(Z)0,we note that as W#0, the neighborhood ofZ# in which Zlies can be chosen so that WZ is small and, by Eqs. 12and 14 with m2) and the facts that B(Z)22 andW(Z)W(Z), we have

BZ2242LWZOWZ2, B2

which, when combined with the relation B1, yields

Z #42 LL #WZOWZ2,B3

and hence shows that, if the neighborhood ofZ# is chosensmall enough, then (Z) #, even if W(Z)0; thiscompletes the proof that Z# is stable when L #L.

FIG. 14. Transition diagram for configura-tions with 1.5 and LL3.282. Theconfiguration labeled A is that which minimizes in the class of configurations with LL,

B is a locally stable configuration in , and BI,BII, BII1* , BII2* are unstable equilibrium configu-rations. The numbers above the arrows indicatingthe direction of transitions give the correspondingincrements in in units ofkB T). An analogousdiagram holds for 0.7, which yields L

4.335, increments in as shown in parenthe-ses, and configurations B, BI, BII, BII1* , BII2* andA that with the scale and lines of view em-ployed are nearly indistinguishable from thoseshown here.



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Suppose now that L #L. For each positive number, let Z be the configuration in with writhe W, and letZ be the configuration with C(Z)C(Z) that has in-dependent ofs and such that L(Z)L # i.e., such thatT(Z)L #W. As W0, the configuration Z ap-proaches Z#. By Eq. 2, T(Z

)22(L #W)2,and the argument that gave us B2 here yields

BZ2242LWW2. B4

Hence,

Z #2212LW L #W2

221 L #2

42 L #LWOW2.

B5

Because L #L, the right-hand side of Eq. B4 isnegative, i.e., (Z) # whenever W is sufficientlysmall.

1 I. Tobias, D. Swigon, and B. D. Coleman, preceding paper,Phys. Rev. E 61, 747 2000.

2 M. Le Bret, Biopolymers 23, 1835 1984.3 F. Julicher, Phys. Rev. E 49, 2429 1994.4 I. Tobias, B. D. Coleman, and W. Olson, J. Chem. Phys. 101,

10 990 1994.5 B. D. Coleman, I. Tobias, and D. Swigon, J. Chem. Phys. 103,

9101 1995.6 D. Swigon, B. D. Coleman, and I. Tobias, Biophys. J. 74, 2515

1998.7 G. Calugareanu, Czechoslovak Math. J. 11, 588 1961.8 J. H. White, Am. J. Math. 91, 693 1969.9 F. B. Fuller, Proc. Natl. Acad. Sci. USA 68, 815 1971.

10 J. H. White, in Mathematical Methods for DNA SequencesCRC, Boca Raton, Florida, 1989, p. 225.

11 D. Swigon, Configurations with Self-Contact in the Theory ofthe Elastic Rod Model for DNA Rutgers University, NJ,1999.

12 W. F. Pohl, J. Math. Mech. 17, 975 1968.13 P. J. Hagerman, Annu. Rev. Biophys. Biophys. Chem. 17, 265

1988.14 D. S. Horowitz and J. C. Wang, J. Mol. Biol. 173, 75 1984.15 The values of C compatible with the data for miniplasmids

reported in Ref. 14 are centered about 1.4, which valuewas employed in the preceding paper.

16 J. M. Schurr, B. S. Fujimoto, P. Wu, and L. Song, in Topics inFluorescence Spectroscopy, Vol. 3: Biochemical Applications,edited by J. R. Lakowicz Plenum Press, New York, 1992.

17 As we remarked in the preceding paper, that the value ofCcandepend on the method of measurement is a matter of currentconcern. See, in particular, P. J. Heath, J. B. Clendenning, B.

S. Fujimoto, and J. M. Schurr, J. Mol. Biol. 260, 718 1996.18 J. Shimada and H. Yamakawa, Biopolymers 23, 853 1984.19 J. Shimada and H. Yamakawa, J. Mol. Biol. 184, 319 1985.20 M. D. Frank-Kamenetskii, A. V. Lukashin, V. V. Anshelevich,

and A. V. Vologodskii, J. Biomol. Struct. Dyn. 2, 1005 1985.21 J. A. Gebe, S. A. Allison, J. B. Clendenning, and J. M. Schurr,

Biophys. J. 68, 619 1995.22 I. Tobias, Biophys. J. 74, 2545 1998.23 D. Stigter, Biopolymers 16, 1435 1977.24 A. Vologodskii and N. Cozzarelli, Biopolymers 35, 289

1995.25 Linear analyses giving rise to Eq. 12 are familiar in the lit-

erature on classical mechanics. For the case m2 see, e.g., E.E. Zajac, J. Appl. Mech. 29, 136 1962. For a general disper-sion relation for vibrations of twisted rings implying Eq. 12,see B. D. Coleman, M. Lembo, and I. Tobias, Meccanica 31,565 1996.

26 G. Domokos, J. Nonlinear Sci. 5, 453 1995.27 Y. Yang, I. Tobias, and W. K. Olson, J. Chem. Phys. 98, 1673

1993.28 T. P. Westcott, I. Tobias, and W. K. Olson, J. Chem. Phys.

107, 3967 1997.29 H. Tsuru and M. Wadati, Biopolymers 25, 2083 1986.30 Our belief in the validity of this assumption, which we employ

only for configurations in , is based on two observations. iFor a configuration in obeying the E condition, we havefound perturbations that lower B at fixed Wonly when the condition is not obeyed. ii Our experience with manipulationof closed elastic rods indicates that, for appropriate L, con-figurations in can withstand gentle perturbations.


bernard d. coleman, david swigon and irwin tobias- elastic stability of dna configurations. ii....

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