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

8/3/2019 Irwin Tobias, David Swigon and Bernard D. Coleman- Elastic stability of DNA configurations. I. General theory

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Elastic stability of DNA configurations. I. General theory

Irwin Tobias,1,* David Swigon,2, and Bernard D. Coleman2,1Department of Chemistry, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854

2Department of Mechanics and Materials Science, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854

Received 16 July 1999

Results are presented in the theory of the elastic rod model for DNA, among which are criteria enabling one

to determine whether a calculated equilibrium configuration of a DNA segment is stable in the sense that itgives a local minimum to the sum of the segments elastic energy and the potential of forces acting on it. Thederived stability criteria are applicable to plasmids and to linear segments subject to strong anchoring endconditions. Their utility is illustrated with an example from the theory of configurations of the extranucleoso-mal loop of a DNA miniplasmid in a mononucleosome, with emphasis placed on the influence that nicking andligation on one hand, and changes in the ratio of elastic coefficients on the other, have on the stability ofequilibrium configurations. In that example, the configurations studied are calculated using an extension of themethod of explicit solutions to cases in which the elastic rod modeling a DNA segment is considered impen-etrable, and hence excluded volume effects and forces arising from self-contact are taken into account.

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

I. INTRODUCTION

In this paper we derive various necessary conditions andsufficient conditions for the elastic stability of equilibriumconfigurations of DNA segments subject to the constraintsthat can arise from the presence of bound proteins and thetopology of the segment. The results obtained hold in thetheory of the commonly employed elastic rod model whichtreats a DNA segment as an inextensible rod with elasticproperties characterized by two elastic constants, the flexuralrigidity A and the torsional rigidity C. In that theory, theconfiguration of a DNA segment is determined once oneknows the curve C representing the duplex axis and the den-sity of the excess twist or overtwisting about C. The

total energy of the segment is taken to be the sum of twoterms. One term, , is determined by C and accounts for theelastic bending energy and the possible presence of longrange forces having a potential depending on C. The other, T, is the twisting energy:

T. 1

In conventional units,

TC

2 0L

s 2ds, 2

with L the length of the segment and s the arc-length param-

eter along C. When long range forces are absent, reduces tothe bending energy, B , which is determined by the curva-ture ofC:

BA

2 0L

s 2ds . 3

Excluded volume effects and the possibility that the segment

makes contact with itself are taken into account by treating aDNA segment as an impenetrable rod with circular crosssection.

So as to give the reader an idea of the nature of our theoryof stability, we now state several definitions and give a sum-mary of our principal results. The precise meaning of someof the terms employed in this Introduction will be clarifiedlater in the paper.

An equilibrium configuration, i.e., a configuration forwhich the first variation of vanishes for variations in con-figuration compatible with the imposed constraints, is herecalled stable if it gives a strict local minimum to in theclass of configurations compatible with the constraints. The

meaning of the term strict local minimum is discussed inSec. II. The imposed constraints include those that followfrom the assumption that the DNA segment, which we treatas an impenetrable rod, either is a plasmid i.e., a segmentthat is closed in the sense that each of its two DNA strands,and hence C, forms a closed curve or is subject to stronganchoring end conditions.

There is a topological invariant L, called excess link,which is meaningful for plasmids but can be defined also foropen segments also called linear segments that are subjectto strong anchoring end conditions. In both cases, L isrelated to the total excess twist Tof the segment and anappropriately defined writhe W by the relation

LWT, 4

which for plasmids is equivalent to a well-known result ofCalugareanu 1 and White 2. Of course, Tin turns isrelated as follows to in radians per unit length:

T1

20L

ds . 5

The writhe W, on the other hand, like in Eq. 1, is deter-mined by the curve C. The fact that , which is minimized

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

PHYSICAL REVIEW E JANUARY 2000VOLUME 61, NUMBER 1

PRE 611063-651X/2000/611/74712 /$15.00 747 2000 The American Physical Society


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by a stable configuration, and L, which is preserved invariations, each can be expressed as a sum of a term depend-ing only on C, and a term depending only on , plays animportant role in the derivation of stability criteria.

In our treatment of necessary conditions for stability, wefollow up on ideas given in a paper of Le Bret 3 and con-sider cases in which the configuration whose stability is be-

ing investigated is a member of a one-parameter family E ofequilibrium configurations of segments that differ only intheir values of L, a parameter that can be taken to varycontinuously. See the discussion at the beginning of Sec.III. We show that if the configuration is stable, then theslope of the graph of L versus W for E is not negative.Thus, the relation dLE/dW0, which we call the E con-dition, is a necessary condition for stability. This conditionwas obtained in a different form by Le Bret 3 for the im-portant special case of a protein-free plasmid without longrange forces affecting .

An equilibrium configuration in E for which the excesstwist density vanishes, remains an equilibrium configu-

ration when the segment is nicked, i.e., when one of the twoDNA strands is severed, an operation that eliminates the seg-ments ability to support a torsional moment. We show thatif a configuration with 0, when considered a configu-ration of an intact segment, is stable, with dLE/dW1, itdoes not remain stable after nicking. In other words, the re-lation dLE/dW1, called here the n condition, is a neces-sary condition for an equilibrium configuration in E to bestable both before and after nicking. In Sec. IV in the dis-cussion of Fig. 4, we give an example of a case in whichthere are three values of L that give rise to equilibriumconfigurations with 0 with all three stable in fact, glo-bally stable while the segment remains intact, but such that

one becomes unstable and two remain stable when the seg-ment is nicked.Since an equilibrium configuration of a segment is stable

only if the corresponding configuration of each of the sub-segments is stable when subject to appropriate constraints,one can obtain a strengthened form of the Econdition, whichwe call the condition: In order for an equilibrium configu-ration of a segment of length L to be stable, it is necessarythat, for each between 0 and L, there holds ()0, where() is, by definition, the slope of the graph ofL versus Wfor the family of equilibrium configurations of the subseg-ment for which 0s, when the subsegment is subject tothe strong anchoring end conditions that would be imposed

on it if its complementary subsegment of length L

wereheld rigid.The importance of the condition lies in the fact that

families of equilibrium configurations can contain configura-tions that obey the E condition but not the condition, andhence are unstable. Examples in which such is the case aregiven in the accompanying paper on supercoiled configura-tions of plasmids 4, here referred to as paper II.

Another necessary condition for the stability of an equi-librium configuration is that the curve C, representing theduplex axis, give a local minimum to in the class of curvesthat obey the same geometric and topological constraints,and have the same writhe as C. We call this condition the Wcondition.

We show further that strengthened forms of the E condi-tion and the W condition, taken together, yield a conditionsufficient for stability, called the S condition, which we usein the discussion of the example treated in Sec. IV.

Three of the necessary conditions for stability that wepresent, i.e., the E, n, and conditions, relate the stability ofan equilibrium configuration to graphs of L versus W.Hence, application of these conditions requires an efficient

method of finding equilibrium configurations of specifiedL and calculating their writhe.

When B , exact closed form solutions of the equi-librium equations can be derived, and adjustment of integra-tion constants yields explicit representations for equilibriumconfigurations compatible with the imposed constraints5 7. In previous applications of such a procedure to DNAsegments, attention was restricted to configurations free fromself-contact. We recently derived a generalization of the pro-cedure to cases, such as the present, in which the cross sec-tions are circular and impenetrable; that generalizedmethod of explicit solutions is used here to calculate theconfigurations of DNA segments of a type called extranu-

cleosomal loops see, e.g., Sec. IV, Fig. 1, and Refs. 7and 8.Adjustment of integration constants to obtain configura-

tions with prescribed L requires, by Eq. 4, repeated cal-culations ofW, which can be a delicate and time-consumingmatter. However, once the explicit expression for C is avail-able, a closed form relation can be obtained for the integralalong C of the geometric torsion 7, and it follows fromobservations of Calugareanu 1 and Pohl 9 that W differsfrom the torsion integral by an integer that we have foundnot difficult to evaluate. Thus, writhe calculations were not amajor difficulty in the present research. In addition, there isnow available an easily evaluated algebraic formula relating

the elastic energy of an equilibrium configuration directly tothe integration constants 6. Without the new explicit repre-sentations of solutions and computational methods for WandB based on these representations, precise calculations ofconfigurations and detailed analysis of their stability wouldbe far more difficult to perform.

In earlier work on closed form solutions of the equilib-rium equations 6, we have given examples of cases inwhich a nicked segment of DNA subject to strong anchoringend conditions can have two equilibrium configurations withone stable and the other not according to the present defini-tion of stability. The conclusions about the stability ofnicked configurations made in Ref. 6 were reached usingcriteria derived in this paper.

II. DEFINITIONS, ASSUMPTIONS, AND PRELIMINARY

RESULTS

As we remarked in the Introduction, in the present theorythe configuration Z of a segment of DNA of length L isspecified by giving i the space curve C traced out by theduplex axis and ii the excess twist density as a functionof the distance s along C; is the difference between thetwist density in the present state and in the torsionally re-laxed, stress-free state.

For a plasmid, C is a closed curve. It has been known forsome time that Eq. 4 holds for a plasmid, i.e., that the sum

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


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of the excess twist Tand the writhe W is a constant Lthat is topological in the sense that it is the same for allconfigurations that the plasmid can attain, at fixed tempera-ture and chemical composition of the medium, without cut-ting one or both of its two strands. This conclusion followsfrom the now familiar theorem that, for a plasmid, L, theGauss linking number for a DNA strand and the duplex axisC, obeys the relation 1,2 LWT, in which Tis the totaltwist of the strand about C in the present configuration, andW is the writhe ofC. If we write T0 for the value ofTwhenthe DNA segment is relaxed i.e., stress-free, put LLT0 , and note that TTT0 , then Eq. 4 is the same asthe relation LWT. Although L is an integer and is inde-pendent of the temperature and chemical composition of themedium, T0 and L are not.

There are several equivalent ways of defining the writheof a closed curve see, e.g., the exposition of White 10.One way 11 is to average, over all orientations of a plane,the sum of the signed self-crossings in the projection of thecurve on the plane and set Wequal to that average. A circle,or more generally, a plane curve that does not cross itself,

has zero writhe, and a flat figure eight has a writhe of mag-nitude one.

There are important examples of open segments of DNA,for which one can define a writhe Wand excess link L, insuch a way that Eq. 4 holds again with L constant for anappropriate class of deformations of the segment. This canbe done when the location and the orientation of the basepairs at each end of the open segment are specified. Such asegment is said to be subject to strong anchoring end condi-tions; for it the endpoints of both the duplex axis C and thetwo DNA strands, and also the tangents to C at its ends,remain fixed during variations in configuration. One canshow that if one joins the ends ofC with a fixed curve C*,

identifies W in Eq. 4 with the writhe of the closed curveformed by C and C*, and again takes T to be the totalexcess twist of the segment, then WTwill remain con-stant as the configuration of the segment is varied, provided,of course, that the deformation is not such that C passesthrough C*. The constant L will then depend on how thecurve C* is chosen.

The extranucleosomal loops of mononucleosomes treatedin Ref. 7 and here in Sec. IV are examples of open seg-ments for which there is precisely one natural choice for C*,and it is clear that, when that choice is made, the class ofadmissible variations is such that C does not cross C* 12.

Since we are treating DNA segments as inextensible, ho-

mogeneous, kinematically symmetric, intrinsically straight,elastic rods obeying the classical theory of Kirchhoff, theelastic energy of a segment is a sum,

B T, 6

in which, by Eqs. 2 and 3, the bending energy B and thetwisting energy T in units ofA/L are

BL

2 0L

s 2ds, TL

2 0L

s 2ds , 7

with

C/A . 8

Our treatment of equilibrium states and their stability isconfined to DNA segments that either are closed or are openand subject to strong anchoring end conditions. The theorywe develop is sufficiently general to allow for the possibilitythat a segment is subject to conservative forces that act alongits length, such as the long-range electrostatic forces that can

arise from its interaction with itself or with stationary ob- jects. We assume, however, that the potential P of suchforces is a function ofC alone and hence is independent of. We write for the sum of the potential P and theelastic energy :

ZC T , CPCBC.9

Our theory does not require the assumption that the seg-ment be free from self-contact or from contact with station-ary rigid bodies, but we do suppose that: i when such con-tact occurs, changes in configuration do no work against the

forces and moments exerted at a contact point, ii the con-tact forces are normal to the surface of the segment, and iiithe moment exerted at a contact point has no componentalong the tangent to C and hence does not affect (s).

We say that a configuration Z# characterizes a state ofmechanical equilibrium, or, for short, is an equilibrium con-figuration, if, the first variation of, vanishes for everyvariation Z in configuration that is admissible in the sensethat it is compatible with the imposed constraints, includingin the case of an open segment of DNA the strong anchor-ing end conditions and for both closed segments and opensegments the constraint that the topological properties of thesegment among which are the knot type ofC and the valueofL) be preserved.

The relation

0, 10

or, equivalently,

T , 11

holds for each admissible variation Z from an equilibriumconfiguration Z#. Any variation from Z# that does not alterC and preserves Tis admissible, and for it Eq. 11 reducesto T0. Hence, a familiar argument tells us that, for anequilibrium configuration, is constant along the segment

and the second of Eqs. 7 reduces toT2

2T2. 12

Because it is easy to show that of all twist density functions (s) with a given value of T, that for which (s)2T/L minimizes T , from this point on we shall con-fine our attention to variations in configuration that keep spatially uniform and therefore are such that

T42T T. 13

This restriction will have no effect on the theory of the sta-bility of equilibrium.

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The requirement that each variation in configuration pre-serve L will be of particular importance in the treatment ofthe stability of equilibrium states given in the next section ofthe paper. For example, the variation in (C ) due to a varia-tion in C, which at first sight appears difficult to calculatedirectly, can be evaluated for an equilibrium state by notingthat Eqs. 11 and 13 and the relation

W T0 14

yield

42T T42TW. 15

This last relation implies that, for an equilibrium configura-tion,

if W0, then BP . 16

We consider two configurations to be equivalent if the

corresponding excess twist density functions are thesame and the corresponding duplex axes C are congruent. Forexample, all the configurations obtained by rotating a con-figuration of a closed segment in which C is a true circle and is constant along C are equivalent see also Ref. 13IIIB. Moreover, hypothetical traveling wave motions inwhich a closed segment attains equivalent equilibrium con-figurations of a figure-eight type are not unfamiliar in rodtheory see, e.g., Ref. 14, Fig. 4.

We call an equilibrium configuration Z# stable if has astrict local minimum at Z# in the sense that Z# has a neigh-borhood 15 N such that (Z)(Z#) for each Z in Nthat is not equivalent to Z#, and can be reached from Z# bya homotopy i.e., an appropriately regular one-parameterfamily of configurations that are compatible with the im-posed constraints.

This definition of stability, which requires that have astrict local minimum, differs from an often used definitionrequiring to have a global minimum. See, e.g., the paperof Julicher on supercoiled configurations of plasmids 16.When global minimization is used to define stability, a con-figuration that is stable according to our definition may beonly metastable. Global minimizers of are of impor-tance in many subjects, among which are the theory of to-poisomer distributions in miniplasmids with bound proteinscf. 7. Our experience has indicated that conditions forlocal stability of the type we present here can facilitate the

search for global minimizers of. For an open segment, theanswer to the question of whether a given local minimizer of is a global minimizer depends in general upon the choiceof the curve C* employed to define W and L.

For an equilibrium configuration Z# to be stable it isnecessary, but not sufficient, that the second variation in be nonnegative for each small admissible variation from Z#:

22 T0. 17

In the next section we shall use the apparatus assembledhere to obtain useful conditions for the stability of equilib-rium states.

III. CONDITIONS FOR STABILITY

The topological invariant L can be altered. In the caseof a plasmid, the cutting and subsequent ligation of a singleDNA strand can result in a change in L by an integralvalue that depends on the relative number of full rotationsmade in the plane of the cut before ligation. The correspond-ing process for an open DNA segment is rotation about the

tangents at the ends and can result in nonintegral changes inL. Nonintegral changes in L can result also fromchanges in the twist associated with the torsionally relaxedDNA due, for example, to changes in temperature or solventcomposition. Criteria to be derived here for stability of cal-culated configurations refer to virtual processes in which Lvaries continuously with the elastic coefficients A and C andthe potential energy function P held fixed.

Let E be a smooth one-parameter family of equilibriumconfigurations corresponding to values of L in an openinterval I which is sufficiently small that for each L in Ithere is a unique Z, and hence a unique value ofW. Let Jbethe interval of values ofW so obtained. The present discus-sion is confined to equilibrium families E for which one cantake L, T, , and to be given by functions, LE, TE,E, E, ofW. With the exception of families that we shallspecify below, this can be done for all W in J other thanthose that correspond to places where the graph ofWversusL for E has a turning point with dW/dL0. A value ofWthat is singular in this sense need not have a neighborhoodin which L is a single-valued function ofW.

That there are exceptional families of equilibrium con-figurations for which L is not determined by W is a con-sequence of the fact that a segment of DNA that is in equi-librium and is such that C is a piece of a helix, an arc of acircle, or a straight line remains in equilibrium whenever Tand hence, L) is changed with C and hence W kept

constant. The most important of the exceptional families ofequilibrium configurations, namely the circular configura-tions of a protein-free plasmid, will be discussed in paper II,which deals with bifurcation diagrams for such plasmids.

Let Z# be the equilibrium configuration in the family Ewhich gives a nonsingular value W# to the writhe. For eachadmissible variation Z taking Z# into a configuration forwhich the curve C equals the duplex axis of a configurationthat is both in E and near to Z#, there holds

dE

dWWW#

W, 18

2 d

2E

dW2WW#

W2. 19

It follows from Eqs. 15 and 18 that, along the family E,the derivative of with respect to the writhe of C is, towithin the constant factor 42, the total excess twist17:

dE

dW42TE. 20

Equations 19, 20, and 4 yield



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242 dL

E

dW1 W2. 21

Since we are considering variations that keep the excesstwist density uniform along C, we have, by Eq. 13,

2T4

2 T2, 22

and hence, by Eq. 21,

242

dLE

dWW2. 23

If the equilibrium configuration Z# is stable, the secondvariation of is nonnegative for each admissible variationfrom that configuration. Thus, a necessary condition for Z#

to be stable is that at the point in E where WW#,

dLE

dW0. 24

This condition on a derivative along E, called the E condi-tion, is far from sufficient for stability. In paper II we presentexamples of configurations of plasmids that obey the condi-tion, but for which there are admissible variations that vio-late the following, also necessary, condition for an equilib-rium configuration Z# to be stable:

If Z is such that W0, then 2 0. 25

That this, the W condition, is necessary for the stability ofZ# may be verified by using the relation 17 and noting that

2T0 for those admissible variations Z that leave the same and change C without changing W.

TheW

condition, like the E condition, is not, by itself,sufficient for stability. However, strengthened forms of thetwo conditions can be combined in the following way toobtain a sufficient condition which we call the S condition:

Let Z# be in a family E of equilibrium configurationssuch that i the curve C(Z#) corresponding to Z# has in anappropriate space of curves a neighborhood N with theproperty that for each Z* in E with writhe W* close to thenonsingular writhe W# of Z#, there holds (C )CE(Z*) for every C in N that has writhe W*, is com-patible with the boundary or closure conditions imposed onthe segment, and is not congruent to C(Z*), and ii therelation 24 holds with replaced by .

When such is the case, (Z)(Z#) for each configu-ration Zthat is close but not equivalent to Z#, and hence Z#is stable.

To prove the last assertion we let Z be an admissiblevariation taking Z# into a configuration Z which is notequivalent to Z# but is such that C(Z) is in N. We canconsider Z to be the result of the successive application oftwo admissible variations, (Z)1 and (Z)2 , where (Z)1 isa variation of the type employed in the discussion containingEqs. 1823, and (Z)2 is a variation in which Tandhence W is held fixed while C changes in such a way thatthe configuration Z is attained. We then employ argumentssimilar to others in this section to show that neither (Z)1nor (Z)2 decreases the energy .

To be specific: the variation (Z)1 takes Z# into a con-

figuration Z1 for which the duplex axis, C1 , equals that of aconfiguration in Enear to Z# with writhe equal to the writheofC(Z). In view of Eq. 23, the hypothesis ii tells us that,when Z1 is not equivalent to Z

#, the second variation in corresponding to (Z)1 is positive, and hence

Z1Z#. 26

Since the variation (Z)2 does not change W or T, thechange it induces in equals the change in . In view of thehypothesis i, C1 gives to a strict minimum in the class ofall curves in N with writhe equal to that of C(Z1), andhence, when Z is not equivalent to Z1 ,

ZZ1. 27

As Z and Z# are not equivalent to each other, they cannotboth be equivalent to Z1 . Thus, in view of Eqs. 26 and27, (Z)(Z#) and the equilibrium configuration Z#

must be stable.

The utility of the condition, which we now render pre-cise, is based on the fact that in order for an equilibriumconfiguration of a segment to be stable, the correspondingconfiguration of each subsegment must be stable when thesubsegment is subject to appropriate constraints.

We continue to take it for granted that the DNA segmentunder consideration has the following properties: i thecurve C is smooth in the sense that the spatial positions ofpoints on C and the tangent vectors to C are continuous func-tions of the arc-length distance s, and ii unless we stateotherwise as in a discussion of nicked DNA, the two DNAstrands are continuous structures. For a DNA segment D thatis in a given equilibrium configuration Z# and is either

closed or subject to strong anchoring end conditions, we mayconsider, for each with 0L, the subsegment D ofDthat corresponds to values ofs between 0 and , and we mayimagine cases in which D is subject to the geometric con-straints including end conditions and topological restric-tions that would be imposed on it if its complementary sub-segment for which sL) were held rigid. When thecomplement ofD is held rigid, the writhe W(Z) and theexcess link L(Z) for a configuration Z of the subsegmentD can be defined by setting W(Z) equal to W(Z), andL(Z) equal to L(Z), where Z is the configuration ofD.Moreover, if Z is the equilibrium configuration Z#, thenZ

# , the corresponding configuration ofD , is an equilib-rium configuration for the geometric constraints imposed onD , and each admissible variation Z in the configurationofD then corresponds to a unique admissible variation Zin the configuration of D. The constraints, including endconditions, imposed on D are a consequence of the smooth-ness assumptions i and ii the fact that the configuration ofthe rigid complement ofD is specified once Z

# is given.IfZ# is stable, then, for each between 0 and L, Z

# is stableand hence obeys the E condition, which implies that the fol-lowing assertion is true: For an equilibrium configuration Z#

ofD to be stable it is necessary that the condition hold, i.e.,that for each with 0L,

0, 28



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where

dLE

dWWW #

, 29

for the one-parameter family E of equilibrium configura-tions ofD that contains Z

# as a nonsingular point. In theexceptional cases in which is such that Z

# corresponds toa bifurcation point for D , and hence the family E is notunique, () should be taken to be the minimum, over allE , of the value of the right-hand side of Eq. 29.

It is clear that a configuration that obeys the conditionalso obeys the E condition. Although, for the examples offamilies E that we present for miniplasmids in mononucleo-somes in the final section of this paper, the configurationsthat obey the E condition also obey the condition, such isnot always the case. In a subsequent paper we shall show thatthere are branches of the bifurcation diagram for protein-freeminiplasmids that contain configurations at which the in-equality, dLE/dW0, holds, but which are nonethelessunstable because they do not obey the condition.

We say that a segment of DNA is nickedif one of its twoDNA strands has been severed. Once this has happened, Cand can be varied independently, and the concepts oflinking number and excess link are no longer applicable.Thus, we say that a variation Z in the configuration of anicked segment is admissible if it is compatible with all theconstraints that would be imposed if the segment were notnicked, with the single exception of the constraint that Zpreserve the sum of Tand W.

As in the case of a segment that is not nicked, a configu-ration Z# of a nicked segment is called an equilibrium con-figuration if 0 for each admissible variation Z fromZ#. Because variations in C and for a nicked segment

are independent, for an equilibrium configuration Z#

of sucha segment,

s 0 for all s. 30

It follows from this last relation and Eqs. 11 and 7 that inorder for Z# to be in equilibrium, it is not only necessary butalso sufficient that

0 31

for each admissible variation Z.In general, in order for Z# to be stable, it is necessary that

2 0 for each admissible variation. For a nicked segment,

in view of Eqs. 7 and 30, it is clear that the stability of anequilibrium configuration requires that

2 0. 32

In fact, as T attains its minimum value when 0, anequilibrium state Z# of a nicked segment is stable if andonly if(C) has a strict local minimum at C #, in the sensethat (C)(C #) for all C corresponding to configurationsthat are in a neighborhood ofZ# and are not equivalent toZ#.

Let us now consider cases in which a family E of equi-librium configurations of an intact i.e., not nicked DNAsegment D contains a configuration Z# with T0 and

hence (s)0. Equation 15 with T0 then tells usthat Eq. 31 holds not only for all variations from Z# thatare admissible for D, but also for all variations admissiblefor a segment Dn obtained by nicking D without changingend conditions or closure properties. This follows from thefact that, because the curve C and the twist density in Dn

can be varied independently, for each admissible variation inthe configuration ofDn there is another that gives rise to thesame change in C, but for which the change in Tequals thenegative of the change in W, and which is admissible notonly for Dn but also for D. Thus, a configuration with (s)0 would remain in equilibrium ifD were nicked.

IfZ# is stable when considered a configuration ofD, it isnot necessarily true that Z# is also stable as a configurationofDn. For although, by Eqs. 17 and 22, stability ofZ# asa configuration ofD requires that, for all admissible varia-tions,

242W2, 33

stability ofZ# as a configuration ofDn requires that 2

0 for all such variations.Now, Eqs. 1821 hold for each variation Z that isadmissible for D and takes Z# into a configuration that hasan axial curve equal to that of a nearby configuration in E. Itfollows from Eq. 19 that when Zis such that W0 i.e.,when, as we assume, W# is not a singular value ofWfor thefamily E, the relation 2 0 can hold for Z only if

d2E

dW20, 34

and, by Eq. 20 or Eq. 21, we may conclude that in orderfor Z# to be a stable configuration ofDn it is necessary thatthe relation,

dLE

dW1 , 35

or, equivalently,

dLE

dT0, 36

hold at the point in E where WW #, i.e., where TTE(W #). This condition, called the n condition, tells usthat when the intact DNA segment D is nicked, a stableequilibrium configuration Z# in the family E with T#0

will remain in equilibrium, but will lose its stability, if

0dLE

dW1. 37

Using an argument analogous to the one that gave us theS condition for the stability of an intact segment, one canprove the following assertion: If a family E of equilibriumconfigurations of an intact segment contains a configurationZ# with T#0 and with W # nonsingular, and is such thati the curve C(Z#) has a neighborhood Nwith the propertythat, for each writhe W* close to W #, CE(W*) gives to astrict minimum in the class of all curves C that are in N, havewrithe W*, and are compatible with the boundary or closure



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conditions imposed on D, and ii the inequality 35 is strict,then (C)(C #) for each such curve C not congruent toC #, and hence Z# is stable and would remain stable if thesegment D were nicked.

It follows from Eqs. 13 and 20 that for a family E ofequilibrium configurations of an intact segment there holdsthe relation,

dE

dLE42TE42 LEW, 38

which tells us how the graphs of E versus LE and LE

versus W determine each other.If a point on the graph of E versus LE happens to

correspond to a configuration Z# with T#0, then thatpoint is a local minimum, a local maximum, or a point ofhorizontal inflexion. A necessary, but not sufficient, condi-tion for the configuration Z# with T#0 to be stablewhen the segment is nicked is that the corresponding point ofthe graph ofE versus LE be a local minimum.

A relation between free energy and average writhe analo-

gous to Eq. 38 holds even for segments with length suchthat fluctuations in configuration cannot be neglected. Forpertinent discussions of the statistical mechanics of protein-free DNA plasmids free from long range interactions, seeRefs. 18 and 19.

We close our discussion of the general theory with a rulefor constructing a graph ofLE versus W for a family E()of equilibrium configurations of a DNA segment when onehas such a graph for the corresponding family E(*) for asegment that is subject to the same constraints, has the samevalue ofA, but has a different value of C and hence of.

The first of Eq. 7 and the second of the Eq. 9 implythat both sides of Eq. 20 are independent of, and hence

TEW,,A *

TEW,*,A , 39

i.e.,

LEW,,A W*

LEW,*,A W. 40

As our notation indicates, TE and LE can be consideredfunctions ofW, , and A. If long range interactions are ab-sent, i.e., ifP0, then TE and LE are independent ofA.Equation 40 tells us that under a change in , the verticaldistance of a point on the graph ofLE versus W, from theline LEW, scales as 1. Of course, the location of thepoints that correspond to configurations with TE0, i.e.,the points where that graph crosses the line LEW, isindependent of the assumed value of.

IV. MINIPLASMIDS IN MONONUCLEOSOMES

A. Method for calculating configurations

This section contains examples of calculated equilibriumconfigurations to which results presented above are appli-cable. The configurations are calculated using explicit solu-tions obtained under the assumption that no external forcesact at a cross section, other than those that may arise from

contact of the cross section with others or with stationaryobjects. Hence the potential P is set equal to zero, we haveB , , and the variational equation characterizingan equilibrium configuration, i.e.,

BT0, 41

is equivalent to the field equations of Kirchhoffs theory of

inextensible and symmetric rods:dF

ds0,

dM

dsFt. 42

Here, t(s) is the unit tangent vector to C; F(s), the result ofthe internal forces acting on the cross section under consid-eration, is a reactive force not given by a constitutive rela-tion; and M(s), the result of moments of the internal forces,is given by a relation that, in the units employed for Eqs. 2and 3, can be written as

MAtdt

dsC t. 43

A DNA molecule is here approximated by a tube with circu-lar cross sections of diameter D. The number dD/L , likeA/C, is an important dimensionless parameter in ourtheory. We assume that the contact force f* i.e., the forceexerted on the cross section with ss* by a cross sectionwith ss**s*) is a reactive force concentrated at thepoint of contact that is normal to the surface of the tube atss*. Geometrical considerations yield

x s*x s** D, t s*x s*x s** 0,44

where x(s) is the position in space of the point on C with

arc-length parameter s. Because contact forces are assumedto be normal to the surface of the tube,

f*f*x s*x s**

D, 45a

F s*0 F s*0f*0,

M s*0 M s*00. 45b

The relations 42 and 43 can be considered differentialequations for x and . In our previous studies of the elasticrod model 5 7, explicit solutions of these equations were

employed to calculate equilibrium configurations free frompoints of self-contact. In the present research we use a gen-eralization of the explicit solution method to cases in whichself-contact occurs at isolated points. We make use of thefact that the configuration of each subsegment lying betweentwo consecutive points of contact is given by an explicitsolution that depends on six parameters which are related tointegration constants for the differential equations 42 and43. These parameters determine, among other things, theexcess twist density which turns out to be constant in sand the moduli of elliptic functions and integrals occurring inexpressions for the coordinates of x(s). A segment with npoints of contact has 2 n1 contact-free subsegments if it isopen, or 2n if it is closed, and hence its configuration is



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determined when 12n6 parameters are specified if it isopen, or 12n if it is closed. The boundary conditions or, inthe case of a protein-free plasmid, the closure conditionsimposed on the segment, the geometric relations 44, thebalance equations 45 for the forces and moments at thepoints of self-contact, and the condition of preassigned Lyield 12n6 or 12n) equations which are solved to calcu-late the 12n6 or 12n) parameters. Once the parametersare fixed, the reactive force vectors F(s) and f(s*) are de-termined, and, in addition, the calculation of the elastic en-ergy and the writhe is facilitated by use of expressions givenin earlier work 6, Eqs. 1719, and 7, Appendix B.

For the present work on examples intended to be illustra-tive of possible applications of the theory developed in Sec.II and III, we have set D20 nm. The calculations we reportdo not account for the effects of electrostatic repulsion.

B. Mononucleosomes

In a recent paper 7, we presented calculations of con-

figurations that minimize the elastic energy of miniplasmidsin mononucleosomes with specified values of the plasmidsize N, the extent w of wrapping of DNA about the histonecore particle, the helical repeat h0

b of the bound DNA, andthe linking number L of the plasmid. There the emphasis wasplaced on calculations of equilibrium topoisomer distribu-tions for miniplasmids in mononucleosomes and the resultsobtained were for configurations free from self-contact. Herewe present calculated graphs ofL versus W and versusL for the extranucleosomal loop, for a range of L inwhich self-contact can occur.

A plasmid of length L or of size N in base pairs bp in amononucleosome is made up of i the nucleosomal DNA,

i.e., a bound segmentD b

with duplex axisC b

and length Lb

which is in contact with the core particle, and ii a freesegment D f the extranucleosomal loop with duplex axis C f

and length LfLLb. See Fig. 1. The segment D f can bethought of as an open segment of DNA subject to stronganchoring end conditions, that follow from the assumptionthat the duplex axis C, the tangent to C, and both DNAstrands are continuous at the places where D f and D b meet.We suppose that D b is rigid with the curve C b being a left-handed helix that, in accord with the structural informationrecently reported 20, has a pitch p of 2.39 nm and a diam-eter d of 8.36 nm.

For miniplasmids in mononucleosomes, values between1.4 and 1.8 have been observed for the wrap w, i.e., the

number of turns about the core particle made by D b 8.Expressions relating w, p, d, and L to such end conditions forC f as the distance between the endpoints ofC f and the ori-entation of the tangents to C f at those endpoints can be foundin our recent paper 7, where, in addition to calculatingequilibrium configurations for specified values of N, w, andL, we develop a multistate model in which w is allowed to

fluctuate. Study of the model has suggested that, in principle,DNA-histone binding energy can be obtained from compari-sons of measured and calculated topoisomer distributions.

So as to have specific examples of families E of equilib-rium configurations with varying L but fixed end condi-tions, we here put N359 and w1.7. We take the excesslink L of the free segment D f to be defined by Eq. 4 withW the writhe of the duplex axis of the miniplasmid i.e., ofthe closed curve C and with Tequal to the total excesstwist Tf ofD f. The curve C b plays the role of C*, thecurve employed to close the open curve called C in the dis-cussion of the ways of defining L for open segments sub-

ject to strong anchoring end conditions which appears in the

paragraph immediately preceding the one containing Eq.6. Here, that open curve is C f. With the present and natu-ral choice ofC* and the present definition of the excess linkL ofD f, the integral valued linking number of the plasmidL, is

LLTbT0f , 46

where Tb is the total twist in the segment D b, and T0f is the

total twist in D f when D f is in a torsionally relaxed statewith 0.

In the calculations we report, A was set equal to 2.0581012 ergnm which corresponds to a persistence lengthA/RT of 50.0 nm at T298K; cf. 21. For C/A we

FIG. 1. Drawing of a DNA miniplasmid in a mononucleosome,with the core particle shown schematically as a cylinder. The rela-tive lengths of the bound and free segments, D b and D f, corre-spond to a miniplasmid of 359 base pairs that is wrapped about thecore particle for 1.7 turns. Here, as elsewhere in the paper, DNA isdepicted as a tube of diameter 20 .

FIG. 2. Equilibrium configurations of DNA miniplasmids inmononucleosomes with N359 bp, w1.7 turns, and W as listedin the figure. In the configurations shown here,D f has one or morepoints of self-contact. The corresponding values of L and aregiven in Table I for 1.4 and 0.7; see Figs. 4 and 5 for thecorresponding graphs of L versus W and versus L.



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employed two values: 1.4 and 0.7. The choice 1.4 iscompatible with the measurements of topoisomer distribu-tion for miniplasmids 22, and appears appropriate forphysical applications of our calculations. The value 0.7is compatible with measurements of fluorescence polariza-

tion anisotropy of dyes intercalated in open segments ofDNA with N104 bp see 23; it is employed here to illus-trate a way in which stability can be sensitive to the param-eter . That the value ofcan depend on the topology andperhaps size of the segment under investigation is a matterof current interest and concern; see, e.g., 24. At the veryleast it suggests that one must be cautious in the interpreta-tion of quantitative results of calculations based on a theoryin which a DNA segment is modeled as a homogeneous rodobeying the special case of Kirchhoffs constitutive relations,in which the rod is assumed to be both inextensible andaxially symmetric.

C. Results for C/A1.4

We suppose first that 1.4. When has that value,each of the configurations seen in Figs. 2 and 3 gives aglobal minimum to the elastic energy ofD f for a value of

L that is listed in Table I and hence is not only an equilib-rium configuration but also a stable configuration. Theseconfigurations belong to a single one-parameter family E(1.4)of equilibrium configurations, whose members minimize at fixed L, and hence minimize B at fixed Win the classof configurations ofD f compatible with the given N and w.The solid curve in Fig. 4a is the graph ofL versus W for

that family. Because each configuration in the family isstable, each obeys the Econdition, and the slope of the graphis positive. The configurations ofD f corresponding to pointson the graph between A1 and B1 are free from self-contact;for 1n4, the configurations corresponding to points be-tween An and An1 or between Bn and Bn1 have n points ofself-contact. Our calculations show that when one point ofself-contact is present, the contact force varies nearly linearlywith L, increasing from 0 to 4.5 pN as L decreases fromits value at A1 to its value at A2, and from 0 to 8.9 pN as Lincreases from its value at B1 to its value at B2.) As we seein Fig. 4a, the graph ofL versus W is continuous, and atthe places where the derivative dL/dWsuffers a jump i.e.,

at the points A

n

and B

n

at which the number of self-contactsin D f increases, the left-hand and right-hand derivativesagree in sign and are positive.

There is a reason why the graphs presented in Fig. 4 areconfined to values ofL greater than L at A4 and less thanL at B4. For L in that range, D f has less than four self-contacts. Our numerical results strongly indicate that, for Loutside of that range, D f has not only discrete points but alsoregions of self-contact, and as a result, precise calculation ofC requires an extension of our present method that will begiven in subsequent publications.

In Fig. 4a, there are three points, C, D, E, at which thegraph ofL versus W intersects a line with a slope of 45passing through the origin. These points correspond to con-figurations which, because they have T0, would remainin equilibrium ifD f were nicked. These three configurations,like all other configurations in E(1.4) , minimize B in theclass of configurations that have the same writhe. The slope

FIG. 3. Equilibrium configurations of DNA miniplasmids inmononucleosomes with N359 bp, w1.7 turns, and with Lsuch that the configuration would remain in equilibrium ifD f werenicked. D f has one point of self-contact in E; it has none in C andD.

TABLE I. Calculated values ofW, B , L, and for the configurations shown in Figs. 2 and 3.

W

Bkcal/mol

1.4 0.7

L

kcal/molL

kcal/mol

A4 3.006 15.31 3.443 17.33 3.880 19.36A3 2.921 14.54 3.333 16.34 3.745 18.14

A2 2.551 11.66 2.890 12.88 3.228 14.09A1 2.053 7.89 2.347 8.81 2.641 9.72B1 0.456 9.69 0.560 9.81 0.664 9.92B2 0.233 12.76 0.560 13.89 0.887 15.02B3 0.708 16.31 1.109 18.01 1.511 19.72B4 0.809 17.19 1.237 19.13 1.665 21.08

C 1.627 5.94 1.627 5.94 1.627 5.94D 0.637 9.88 0.637 9.88 0.637 9.88E 0.307 9.50 0.307 9.50 0.307 9.50

S 1.240 7.72 0.664 9.47X 0.958 9.16 0.578 9.96



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dL/dW of the solid curve in Fig. 4a exceeds 1 at C andE and is less than 1 at D i.e., the n condition holds at C andE but not at D. Hence, even though each of these threepoints corresponds to a stable configuration of the intact seg-ment, if the segment D f were nicked, the configuration atpoint D would not be stable. Since the configurations atpoints C and E meet conditions i and ii given in theparagraph that follows relation 37, these two configurationswould be stable ifD f were nicked.

In Fig. 4b, we see that the graph of versus L has aglobal minimum at C, a local minimum at E, and a localmaximum at point D that corresponds to an equilibrium con-figuration that loses stability when nicked.

Remark: In our previous paper on mononucleosomes 7we put 1.4 and considered the set S(N,w) of integervalues of the linking number L for which the configurationofD f with minimum elastic energy does not show self-

contact, and we remarked that preliminary calculations hadstrongly indicated that, for a range of N in which can beidentified with the free energy G ofD f, when L is not inS(N,w) the configurations of D f compatible with L haveelastic energies high enough to preclude the occurrence of anobservable concentration of topoisomer L in an equilibriumdistribution of topoisomers in a mononucleosome with thespecified N and w. Careful calculations of the type that gaveus Fig. 4b confirm the validity of this property ofS(N,w)for 1.4, when 330N370 and 1.4w1.8.

D. Results for C/A0.7

Once we have in hand the graph of L versus W shownin Fig. 4a for E(1.4) , Eq. 40 yields, forthwith, correspond-ing graphs for families E() with other values of. Such a

graph for

0.7 is shown in Fig. 5a. For families E()related in this way, the curve C corresponding to a givenvalue ofW is independent of, and the calculated configu-rations seen in Figs. 2 and 3 are appropriate to each suchfamily, with L and dependent on in accord with Eqs.40 and 38. As each configuration in E(1.4) minimizes Bat fixed W, so also does each configuration in E() for arbi-trary .

Returning now to Fig. 5a, we note that as each configu-ration in E(0.7) is a minimizer of B at fixed W, we canapply the S condition and assert that a configuration in E(0.7)is stable if dL/dW is strictly positive. The derivativedL/dW suffers a jump at points An and Bn. In the present

case, both the right-hand and left-hand derivatives are posi-tive at A1, A2, A3, B2, and B3, and hence the configurationscorresponding to those points are stable. At B1, the left-handand right-hand derivatives do not agree in sign; at point X,shown as a solid circle in Fig. 5, the derivative dL/dW iscontinuous but changes sign. As the branch traverses pointsX and B1, the corresponding configurations either gain orlose stability; such points are called the points of exchangeof stability. As L is here a strictly increasing function ofW between A4 and X and between B1 and B4, but is a de-

FIG. 4. Graphs of L versus W, and ver-sus L, for equilibrium configurations with1.4 and end conditions corresponding to

N359 bp and w1.7 turns. Here, as in Fig. 5below, the symbol indicates that the configu-ration (An or Bn) is such that the number ofpoints of self-contact changes with L, and indicates that the configuration C, D, or E issuch that T0. The dashed line in a is thegraph ofLW.

FIG. 5. Graphs of L versus W, and ver-sus L, for equilibrium configurations with0.7, N359 bp and w1.7 turns. The sym-bol indicates a point of exchange of stabilitywith dL/dW continuous and equal to 0. Thepoint B1 is a point of exchange of stability withdL/dW discontinuous. In a, the horizontallightface solid line with L0.664 passesthrough the point B1 and a point S correspondingto a stable configuration with lower elastic energythan B1.



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creasing function between X and B1, the configurations inE(0.7) with W either between its values at A

4 and X or be-tween its values at B1 and B4 are stable, and those with Wbetween its values at X and B1 are unstable.

Our calculations show that there is a critical value c ofsuch that dL/dW0 for all values ofWbetween 3.006W for A4) and 0.809 W for B4) when c. For c, dL/dW0, and there is a single value of W

namely 0.606 at which dL/dW0 the correspondingconfiguration is one for which D f is without self-contact.For c, there is at least one interval of values ofW inwhich the E condition does not hold, and hence the corre-sponding configurations are unstable. The number c and thevalues ofW that we have mentioned depend on the assumedvalues ofN and w. In the present case, in which N359bpand w1.7 turns, c0.860.

The configuration corresponding to each labeled point,An,Bn,C,D,E,X,S, of the graph in Fig. 5a is shown in Fig.2, 3, or 6. The corresponding values of L, , and B arelisted in Table I. Of course, and B are the total elasticenergy and the total bending energy of the segment D f

which, in the present case in which N

359, has length Lf

77.23 nm and therefore contains approximately 227 bp.We now give a detailed analysis of some properties of the

graphs shown in Fig. 5, not because they are immediatelyrelevant to experiment at the present time the value of 0.7for is not expected to be appropriate for miniplasmids, butbecause they illustrate well certain possible consequences ofa lack of monotonicity in a graph of L versus W for a

segment subject to strong anchoring end conditions. Thegraph in Fig. 5a has a local maximum at point X whereW0.958 and L0.578) and a local minimum at B1

where W0.456 and L0.664). For each L with0.664L0.578 there are three configurations inE(0.7) , two of which are stable with W either less than itsvalue at X or greater than its value at B1. For L0.664, there are two configurations in E

(0.7): the unstable

configuration B1 and the stable configuration S. As Fig. 5band the last column in Table I show, for the configuration S,which gives not only a local but also a global minimum to at fixed L, at S has a value approximately 0.45 kcal/mole less than its value at B1. The configurations S, X, andB1 are shown in Fig. 6. In the inset to Fig. 5b, an arrowpoints to the place where the graph of versus L crossesitself. At that point, L has a value namely 0.608 forwhich there are two configurations in E(0.7) with equal valuesof but distinct values ofW i.e., W1.139 and W

0.425). To each value ofW there corresponds a uniqueconfiguration in E(0.7) , and ifWW

or WW, then that

configuration not only is stable but gives a global minimumto in the class of all configurations with equal L.The rule that we employed to generate, from the graph of

L versus W for E(1.4) , corresponding graphs for familiesE() with different is such that the value ofW for whichLW is independent of. Although the stability of con-figurations of the intact segment corresponding to such val-ues ofW can depend on e.g., for the intact segment, theconfiguration D is stable when 1.4 but unstable when0.7), the stability of corresponding configurations ofnicked segments as it is governed by the dependence ofBon C is independent of.

ACKNOWLEDGMENTS

This research was supported by the U.S. Public HealthService under Grant No. GM34809 and the National ScienceFoundation under Grant No. DMS-97-05016. D.S. acknowl-edges support from the Program in Mathematics and Mo-lecular Biology at Florida State University, with fundingfrom the Burroughs Wellcome Fund Interfaces Program.

1 G. Calugareanu, Czechoslovak Math. J. 11, 588 1961.2 J. H. White, Am. J. Math. 91, 693 1969.

3 M. Le Bret, Biopolymers 23, 1835 1984.4 B. D. Coleman, D. Swigon, and I. Tobias, following paper,

Phys. Rev. E 61, 759 2000.5 I. Tobias, B. D. Coleman, and W. Olson, J. Chem. Phys. 101,

10990 1994.6 B. D. Coleman, I. Tobias, and D. Swigon, J. Chem. Phys. 103,

9101 1995.7 D. Swigon, B. D. Coleman, and I. Tobias, Biophys. J. 74, 2515

1998.8 Y. Zivanovic, I. Goulet, B. Revet, M. Le Bret, and A. Prunell,

J. Mol. Biol. 200, 267 1988; see also the appendix by M. LeBret, ibid. 200, 285 1988.

9 W. F. Pohl, J. Math. Mech. 17, 975 1968.

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

11 F. B. Fuller, Proc. Natl. Acad. Sci. USA 68, 815 1971.12 One might expect that in cases in which no particular choice

for C* is distinguished by matters of physical relevancy, thecondition that C and C* not cross each other in a variation mayplace significant but unnatural restrictions on the class ofvariations. But such is not the case, for we here employ aconcept of local, rather than global, stability, and crossing ofCand C* can be avoided by choosing sufficiently small the di-ameter of the neighborhood over which the configuration isvaried in the statement of the definition of stability.

13 I. Tobias, B. D. Coleman, and M. Lembo, J. Chem. Phys. 105,2517 1996.

14 B. D. Coleman and E. H. Dill, J. Acoust. Soc. Am. 91, 26631992.

FIG. 6. Equilibrium configurations corresponding to points S, X,and B1 of Fig. 5.



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15 For our present purpose, it is not essential to specify the topol-ogy on sets of configurations Zand curves C. Far more impor-tant to our treatment is the fact that is the sum of the twofunctionals, with one, , depending on C alone and the other,T, a positive definite quadratic functional of .

16 F. Julicher, Phys. Rev. E 49, 2429 1994.17 For the case in which P0, Eq. 20 goes back to a now

classical paper of Fuller Ref. 11.

18 J. A. Gebe, S. A. Allison, J. B. Clendenning, and J. M. Schurr,Biophys. J. 68, 619 1995.

19 I. Tobias, Biophys. J. 74, 2545 1998.

20 K. Luger, A. W. Mader, R. K. Richmond, D. F. Sargent, and T.J. Richmond, Nature London 389, 251 1997.

21 P. J. Hagerman, Annu. Rev. Biophys. Biophys. Chem. 17, 2651988.

22 D. S. Horowitz and J. C. Wang, J. Mol. Biol. 173, 75 1984.23 J. M. Schurr, B. S. Fujimoto, P. Wu, and L. Song, in Topics in

Fluorescence Spectroscopy, Vol. 3: Biochemical Applications,edited by J. R. Lakowicz Plenum Press, New York, 1992.

24 P. J. Heath, J. B. Clendenning, B. S. Fujimoto, and J. M.Schurr, J. Mol. Biol. 260, 718 1996.


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

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