Theory of sequence-dependent DNA elasticity

Bernard D. Coleman∗

Department of Mechanics and Materials ScienceRutgers, The State University of New Jersey

Piscataway, New Jersey 08854

Wilma K. Olson† and David Swigon‡

Department of Chemistry and Chemical BiologyRutgers, The State University of New Jersey

Piscataway, New Jersey 08854(Dated: January 28, 2003)

The elastic properties of a molecule of duplex DNA are strongly dependent on nucleotide sequence.In the theory developed here the contribution ψn of the n-th base-pair step to the elastic energyis assumed to be given by a function ψn of six kinematical variables, called tilt, roll, twist, shift,slide, and rise, that describe the relative orientation and displacement of the n-th and (n + 1)-thbase pairs. The sequence-dependence of elastic properties is determined when one specifies the wayψn depends on the nucleotides of the two base pairs of the n-th step. Among the items discussedare the symmetry relations imposed on ψn by the complementarity of base pairs, i.e., of A to T andC to G, the antiparallel nature of the DNA sugar-phosphate chains, and the requirement that ψn

be independent of the choice of the direction of increasing n. Variational equations of mechanicalequilibrium are here derived without special assumptions about the form of the functions ψn, andnumerical solutions of those equations are shown for illustrative cases in which ψn is, for eachn, a quadratic form and the DNA forms a closed, 150 base-pair, minicircle that can be called aDNA o-ring because it has a nearly circular stress-free configuration. Examples are given of non-circular equilibrium configurations of naked DNA o-rings and of cases in which interaction withligands induces changes in configuration that are markedly different from those undergone by aminicircle of intrinsically straight DNA. When a minicircle of intrinsically straight DNA interactswith an intercalating agent that upon binding to DNA causes local reduction of intrinsic twist, theconfiguration that minimizes elastic energy depends on the number of intercalated molecules, butis independent of the spatial distribution of those molecules along the minicircle. In contrast, hereit is shown here that the configuration and elastic energy of a DNA o-ring can depend strongly onthe spatial distribution of the intercalated molecules. As others have observed in calculations forKirchhoff rods with intrinsic curvature, an o-ring that has its intrinsic twist reduced at a single base-pair step can undergo large deformations with localized untwisting and bending at remote steps,even when the amount α of twist reduction is less than the amount required to induce supercoiling inrings of intrinsically straight DNA. We here find that the presence in the functions ψn of cross-termscoupling twist to roll can amplify the configurational changes induced by local untwisting to thepoint where there can be a value of α at which a first-order transition occurs between two distinctstable non-circular configurations with equal elastic energy.

I. INTRODUCTION

Although there are cases in which one can treat a DNAmolecule as an idealized homogeneous elastic rod that isstraight when stress-free and has uniform elastic prop-erties, it is known that the genetic information in DNAdetermines not only the amino acid sequences of encodedproteins and RNA but also the geometry and deformabil-ity of DNA at a local level, i.e., at the level of base-pairsteps. Recent advances in structural biochemistry haveyielded information about the dependence on base-pairsequence of the structure and elastic response of DNA. In

∗Electronic address: bcoleman@stokes.rutgers.edu†Electronic address: olson@rutchem.rutgers.edu‡Electronic address: swigon@jove.rutgers.edu; Author to whomcorrespondence should be addressed.

this paper we present a method of taking such informa-tion into account when one calculates and determines thestability of equilibrium configurations of DNA segmentssubject to preassigned end conditions.

In a stress-free state, the local bend, twist, and stretchof duplex (double helical) DNA varies from one base-pairstep to another. Some steps are wedge shaped with incli-nation (i.e., bending) strongly dependent on nucleotidecomposition and occasionally as large as 5◦. Moreover,there are steps that are sites of over- or under-twisting aslarge as 3.5◦, and there are steps at which the barycenterof a base pair is displaced relative to that of its predeces-sor and gives rise to a local shear that can be as large as0.65A (or 20% of the elevation of a base-pair relative toits predecessor) [1–6].

The chemical architecture of DNA results also in me-chanical couplings between kinematical variables (suchas those of Fig. 1) that characterize the relative displace-ment and orientation of adjacent base pairs. Observed

2

FIG. 1: Schematic representation of kinematical variables de-scribing the relative orientation and displacement of succes-sive base pairs. Tilt and roll measure local bending, twistmeasures rotation in the base-pair plane, shift and slide localshearing, and rise local stretch. For precise definitions, seeRefs. 8–10 and Eqs. (2.5a)–(2.7).

correlations in the values of the kinematical variables forcrystals of DNA oligomers have led to the proposal thatsuch couplings should appear as cross-terms in consti-tutive equations relating the elastic energy (or, equiva-lently, local forces and moments) to the kinematical vari-ables [6].

Recent research on the derivation of energy functionsthat govern the structure and deformation of individ-ual base-pair steps in crystals of pure DNA and DNA-protein complexes [5–7] makes it possible to develop mod-els for DNA elasticity that take into account a strongdependence of intrinsic structure and elastic moduli onnucleotide sequence and have, of necessity, a discreterather than a continuum nature with constitutive rela-tions showing cross-terms that couple twisting to bend-ing. In this paper we construct a theory of a general classof discrete models in which the elastic energy Ψ of a DNAsegment is assumed to be the sum of the interaction en-ergies of adjacent base pairs. For each n, ψn, the energyresulting from deformation of the n-th (dimer) base-pairstep, is assumed to be given by a function ψn of six inde-pendent kinematical variables that determine the relativeorientation and displacement of the n-th and (n + 1)-thbase pairs; these are the three angular variables (tilt, roll,twist) and the three displacement variables (shift, slide,rise) that are illustrated schematically in Fig. 1 and aredefined precisely in Section II.

The underlying mathematical assumptions of the the-ory are stated in Section II along with the equations thatgovern configurations in mechanical equilibrium. Thoseequations are obtained by setting equal to zero the firstvariation of the total energy E, which we take to be a sumof the elastic energy Ψ =

∑n ψn and the potential energy

P of external forces and moments. Our derivation of theequilibrium equations does not require special assump-tions about the form of the functions ψn. The symmetryrelations imposed on the ψn by the complementarity ofadenine to thymine and guanine to cytosine, the antipar-allel nature of the DNA sugar-phosphate chains, and therequirement that those functions be independent of thechoice of the direction of increasing n are presented at thesame level of generality as our derivations of the equilib-

rium equations. Also shown in Section II are the formsthat the symmetry relations take when ψn is quadraticin appropriate kinematical variables. Details of deriva-tions of the equations of equilibrium and of the symmetryrelations for ψn are given in Section III.

The principal results of the paper are those given inSections II and III, namely (i) the formulation of thetheory, (ii) the analysis of the symmetry restrictions im-posed on ψn by the structure of commonly found formsof duplex DNA including the B, A, and C forms, (iii)the derivation of the variational equations of equilibriumwhich requires the attainment of a computationally man-ageable expression for the matrix inverse of the gradientof the orthogonal matrix [Dn

ij ] relating bases embeddedin the n-th and (n + 1)-th base pairs to the parameterstilt, roll, and twist (see Eq. (A.1) and Eqs. (3.7), (3.9)–(3.11)), and (iv) the development of an efficient compu-tational method (of order O(N), with N the number ofbase pairs) for solving the equilibrium equations and de-termining the stability of computed solutions.

To illustrate the way in which the theory can be usedto explore the influence of nucleotide sequence on con-figurations of DNA subject to constraints such as ringclosure and ligand induced changes in local structure, wegive, in Section IV, a few examples in which the the-ory is employed to calculate equilibrium configurationsof closed minicircles which, because they have nearly cir-cular stress-free configurations, are called DNA o-rings.1Discussions are given of ways in which equilibrium con-figurations of naked DNA and DNA that is interactingwith ligands can, under appropriate circumstances, showa behavior markedly different from that predicted by thefamiliar idealized rod model.2 Among the new topicstreated here is the influence of the cross-terms couplingtwist to roll on the configurational changes induced bylocal untwisting.

II. CONSTITUTIVE RELATIONS ANDVARIATIONAL EQUATIONS

In the present theory the base pairs are taken to be flatobjects that are represented by rectangles. The configu-ration of a DNA segment with N+1 base pairs, and henceN base pair steps, is specified by giving, for each n, boththe location xn of the center of the rectangle Bn that rep-resents the n-th base pair and a right-handed orthonor-mal triad dn

1 ,dn2 ,d

n3 that is embedded in the base pair.

(See Fig. 2.) The polygonal curve C formed by the Nline segments connecting the spatial points x1, ...,xN+1

is called the axial curve.Each DNA nucleotide in a base pair belongs to one

1 In engineering practice the term “o-ring” is commonly used forelastic rings that are circular when stress-free.

2 See also the discussion given in Refs. 11–19.

3

FIG. 2: Drawing of the model employed for a base-pair stepshowing the vectors rn, dn

i , dn+1j . Each nucleotide base of a

pair is covalently bonded (at its darkened corner) to one ofthe two sugar-phosphate chains. The arrows give the 5′–3′

directions of those chains. The gray-shaded long edges are inthe minor groove of the DNA.

of the two strands that make up the duplex molecule.In a strand the nucleotide bases are laterally attached(through covalent bonds) to a sugar-phosphate chain.The chains of the two strands are antiparallel in the sensethat increasing n corresponds to advance in the 5′–3′ di-rection on one strand and in the 3′–5′ direction on theother.3

We assume that the elastic energy Ψ of a configurationis the sum over n of the energy ψn of interaction of then-th and (n+1)-th base pairs, and that ψn is a functionof the relative orientation and displacement of the twoneighboring base pairs, i.e., of the components of thevectors dn+1

j and rn = xn+1 − xn with respect to thebasis dn

i . As the 3 × 3 matrix [Dnij] with entries Dn

ij =dn

i · dn+1j is orthogonal, it can be parameterized by a

triplet (θn1 , θ

n2 , θ

n3 ) of angles, with θn

1 the tilt, θn2 the roll,

and θn3 the twist of Bn+1 relative to Bn. Hence, in the

formula

Ψ =N∑

n=1

ψn, (2.1)

the elastic energy ψn of n-th base-pair step is given by aconstitutive relation of the type,

ψn = ψn(θn1 , θ

n2 , θ

n3 , r

n1 , r

n2 , r

n3 ), rn

i = dni · rn. (2.2)

3 In our numbering of base pairs the choice of the direction ofincreasing n is arbitrary. Although it is common to number thebase pairs so that increasing n corresponds to advance in the 5′–3′ direction of the strand that (in discussions of transcription) iscalled the sequence strand, we do not impose such a conventionhere. Because we study the implications of the requirement thatthe energy ψn be independent of the choice of the direction ofincreasing n, it is necessary for us to consider transformationsthat change that directionwhile leaving invariant the orientationand spatial position of the base pairs.

It is often useful to write this relation in the form4

ψn = ψn(θn1 , θ

n2 , θ

n3 , ρ

n1 , ρ

n2 , ρ

n3 ), (2.3)

where the displacement variables ρn1 , ρ

n2 , ρ

n3 , called shift,

slide, and rise are given by coordinate transformationfunctions, ρk, that are independent of n:

ρnk = ρk(θn

1 , θn2 , θ

n3 , r

n1 , r

n2 , r

n3 ). (2.4)

As shown in Fig. 2, the vectors dni are defined so that

dn3 is perpendicular to Bn with dn

3 · rn > 0; dn2 is paral-

lel to the long edges of Bn and points toward the shortedge containing the corner of Bn bonded to a deoxyribosegroup in the sugar-phosphate chain for which n increasesin the 5′–3′ direction; dn

1 is parallel to the short edges ofBn and points toward the major groove of the DNA, i.e.,dn

1 = dn2 × dn

3 .5Each drawing in Fig. 1 illustrates one of the kinemati-

cal variables θn1 , θ

n2 , θ

n3 , ρ

n1 , ρ

n2 , ρ

n3 for a case in which that

variable has a positive value and the others (with theexception of ρn

3 ) are set equal to zero. That figure isintended to be suggestive rather than precise. To givea precise definition of θn

i and ρni for cases in which sev-

eral of these variables are not zero, we follow a procedure(introduced by Zhurkin, Lysov, & Ivanov [9] and furtherdeveloped by El Hassan & Calladine [8]) in which oneemploys the Euler-angle system ζn, κn, ηn for which6

Dnij = dn

i · dn+1j = Zik(ζn) Ykl(κn) Zlj(ηn), (2.5a)

[Yij(α)] =

cosα 0 sinα

0 1 0− sinα 0 cosα

, (2.5b)

[Zij(α)] =

cosα − sinα 0

sinα cosα 00 0 1

(2.5c)

and defines θn1 , θ

n2 , θ

n3 by the relations:

ζn = 12θ

n3 − γn , ηn = 1

2θn3 + γn, (2.6a)

κn =√

(θn1 )2 + (θn

2 )2, tan γn =θn1

θn2

. (2.6b)

4 The superposed symbols, , , etc. here serve to emphasize thedistinction between a value of a quantity and a function relatingthe quantity to other quantities.

5 In general, xn, the center of the rectangle Bn, is not the barycen-ter (center of mass) of the corresponding base pair. Here, as inRef. 20, xn, dn

1 , and dn3 are obtained by the following construc-

tion: Let �2 be the straight line connecting the C1′ atoms of the

deoxyribose groups corresponding to the n-th base pair, and let�1 be the line that passes at a right angle through the midpointof �2 and intersects the straight line containing the C8 atom ofthe purine base and the C6 atom of the complementary pyrimi-dine base. The point xn is taken to lie at the intersection of �1and �2; the vector d

n1 is parallel to �1 and points in the direction

of the major groove; dn3 is perpendicular to both �1 and �2 with

dn3 · rn > 0.

6 We use the Einstein summation convention for subscripts (butnot superscripts).

4

As in Ref. 8, we here use Eqs. (2.5)–(2.6) to define ρni in

terms of rni and θn

i :

ρni = Zij(−γn) Yjk(−1

2κn) Zkl(−ζn) rn

l . (2.7)

This last equation renders explicit the functions ρk inEq. (2.4) and implies that the numbers ρn

i are the com-ponents of rn with respect to an orthonormal basis dn

i ,which is called the mid-basis for step n and is defined bythe relation

dni · dn

j = Zik(ζn) Ykl(12κ

n) Zlj(γn). (2.8)

In Section III (i.e., in the discussion containing Eqs. (3.4)and (3.8)) we render precise the following remark: Thepresent definition of θn

i and ρni is such that, for each

configuration of a DNA segment, a change in the choice ofthe direction of increasing n leaves θ2 , θ3, ρ2, ρ3 invariantbut changes the sign of θ1 and ρ1.

Several computer programs have been developed to re-duce atomic coordinate data (of the type available forDNA crystal structures in the Nucleic Acid Database[21]) to variables θn

i , ρni which, like those just defined, are

compatible with a convention [10] known as the “Cam-bridge Accord of 1988”. For a survey see Ref. 22.

The function ψn of Eq. (2.3) depends on which nu-cleotide bases are in the n-th and (n+1)-th base pairs.To discuss the restrictions imposed on ψn by the comple-mentarity of A to T and G to C, the antiparallel direc-tions of the sugar-phosphate chains, and the assumptionthat ψn is independent of the composition of base pairsother than the n-th and (n+1)-th, let us write ψXY forψn, where XY, with X in the n-th and Y in the (n+1)-thbase pair, is a sequence of two nucleotide bases attachedto the chain for which n increases in the 5′–3′ direction.In this notation, for the complement of the base X, onewrites X (i.e., A = T, G = C). The complement of a se-quence of bases is the sequence of complementary basesin the reverse order: the complement of XY is Y X (e.g.,the complement of AC is GT), because X and Y are at-tached at the locations n and n+1 to the chain for whichn decreases in the 5′–3′ direction. In Section III it isshown, that when θn

i , ρni are defined as here, each func-

tion ψXY determines its complementary function ψY X bythe relation

ψXY(θn1 , θ

n2 , θ

n3 , ρ

n1 , ρ

n2 , ρ

n3 ) =

ψY X(−θn1 , θ

n2 , θ

n3 ,−ρn

1 , ρn2 , ρ

n3 ). (2.9)

It follows that, if XY is self-complementary, i.e., ifXY = Y X, then ψXY is an even function of thepair (θn

1 , ρn1 ) for each fixed value of the quadruple

(θn2 , θ

n3 , ρ

n2 , ρ

n3 ).

We define an equilibrium configuration to be one forwhich the first variation of the total energy, i.e.,

E = Ψ + P, (2.10)

vanishes for all variations in configuration that are ad-missible in the sense that they are compatible with the

imposed constraints. (Here P is the potential energy ofexternal forces and moments.) The resulting variationalequations, which are derived in Section III, can be writ-ten

fn−fn−1 = ϕn,

mn−mn−1 = fn×rn +µn, (2 ≤ n ≤ N), (2.11)

where, for each n, fn and mn are the force and momentthat the (n+1)-st base pair exerts on the n-th base pairand ϕn and µn are the external force and moment actingon the n-th base pair.7 The vectors fn and mn are theanalogues of the vectors f (s∗) and m(s∗) that, in Kirch-hoff’s theory of elastic rods, are interpreted as the resul-tant force and moment of the contact forces (i.e., Piolastresses) exerted on the cross-section with the arc-lengthcoordinate s∗ by the material with arc-length coordinatess > s∗. The components of fn and mn with respect tothe local basis dn

i are given by the relations

fni =

∂ψn

∂rni

, mni = Γn

ij

∂ψn

∂θnj

. (2.12)

The numbers Γnij form a 3 × 3 matrix [Γn

ij] =[Γij(θn

1 , θn2 , θ

n3 )] that depends on the definitions employed

for the kinematical variables θni . When those variables

are as in Eqs. (2.5)–(2.6), [Γnij] is given by Eq. (A.1). The

chain rule permits us to write Eqs. (2.12) in the forms,

fni =

∂ψn

∂ρnj

∂ρj

∂rni

, mni = Γn

ij

(∂ψn

∂θnj

+∂ψn

∂ρnk

∂ρk

∂θnj

),

(2.13)in which ψn and ρk are as in Eqs. (2.3) and (2.4). Whenθni and ρn

i are defined by Eqs. (2.5)–(2.8), for ∂ρj/∂rni

and ∂ρk/∂θnj we have

∂ρj

∂rni

= Zjk(−γn) Ykl(−12κ

n) Zli(−ζn), (2.14)

∂ρk

∂θnj

= jΛnkl ρ

nl , (2.15)

with jΛnkl as in Eqs. (A.2)–(A.4).

If a DNA segment with N+1 base pairs is subject tostrong anchoring end conditions at both ends, then (x1 ,d1

i ) and (xN+1 , dN+1i ) have preassigned values. A closed

(or “circularized”) DNA segment of N base pairs (bp)can be considered an N+1 bp segment that is subject to

7 Such an external force and moment can arise when the n-th basepair is in contact with a protein or with a base pair sequentiallydistant from it. When electrostatic forces are taken into explicitaccount, P in Eq. (2.10) will include the potential energy of theinteraction of charged residues.

5

the constraint that its terminal base-pairs coincide, i.e.,have the same location and orientation, and hence:

x1 = xN+1 , d1i = dN+1

i , (i = 1, 2, 3). (2.16)

A necessary condition for an equilibrium configurationto be stable is that there be no admissible variation inconfiguration for which δ2E, the second variation of E, isnegative. As a configuration is described in the presenttheory by giving a finite number of variables, namelythe 6N numbers, θn

1 , θn2 , θ

n3 , ρ

n1 , ρ

n2 , ρ

n3 , we can say that a

sufficient condition for stability is that δ2E be positive forall admissible variations. A configuration that is stablebut does not give a global minimum to E can be calledmetastable.

For calculations of the type reported in Section IV,ϕn = µn = 0, i.e., P is set equal to zero.8 TheEqs. (2.11) with fn and mn given by Eqs. (2.12) thenare a system E of 6N − 6 nonlinear algebraic equationsfor the unknown coordinates θn

1 , θn2 , θ

n3 , ρ

n1 , ρ

n2 , ρ

n3 . That

system is weakly coupled in the sense that for each n,2 ≤ n ≤ N , the Eqs. (2.11) are a set of 6 equationswhich are solved numerically using a Newton-Raphsonprocedure to obtain (θn

i , ρni ) for given (θn−1

i , ρn−1i ). This

allows one to solve E efficiently (using O(N) computa-tional steps) by a recursive algorithm in which, for eachn, (θn

i , ρni ) is obtained as a function of (θ1

i , ρ1i ). Equi-

librium configurations of segments subject to specifiedend conditions are calculated using an iterative schemein which (θ1

i , ρ1i ) is repeatedly adjusted until the appro-

priate relations, e.g., Eq. (2.16), hold.

The case in which ψn is given by a quadratic form

To describe quadratic energy functions of the type em-ployed for calculations presented in Section III, we writeθni and ρn

i for the values of θni and ρn

i in a stress-free statewith ψn = 0, and put

∆θni = θn

i − θni , ∆ρn

i = ρni − ρn

i . (2.17)

When |∆θni | and |∆ρn

i | are small, the assumption thatψn is three-times differentiable implies that, to within anerror O(δ3) with

δ = maxi=1,2,3

{|∆θni | , |∆ρn

i |} , (2.18)

ψn can be taken to be a quadratic function,

ψn = 12F

nij(∆θn

i )(∆θnj ) + Gn

ij(∆θni )(∆ρn

j )

+ 12Hn

ij(∆ρni )(∆ρn

j ), (2.19)

8 The configurations discussed in Section IV may be consideredappropriate to cases in which the ionic strength of the solution issufficiently high (≥ 1M) that the Debye length is significantly lessthan the distance between charged sites and the employed valuesof elastic moduli and intrinsic kinematical parameters have beenadjusted to account for the electrostatic interaction of adjacentbase pairs.

in which F nij, Gn

ij, and Hnij are constants with F n

ij = F nji,

Hnij = Hn

ji, (i, j = 1, 2, 3).9 If one wishes, one may writethe right-hand side of Eq. (2.19) as a quadratic formin (∆θn

1 ,∆θn2 ,∆θn

3 , νn1 , ν

n2 , ν

n3 ), where the dimensionless

numbers

νni = ∆ρn

i /ρn3 (2.20)

are such that νn3 can be interpreted as a stretching strain

and νn1 and νn

2 as shearing strains.It follows from Eq. (2.9) that if we replace F n

ij by FXYij ,

etc., and let Z stand for F , G, or H , then in Eq. (2.19)

θXYi = Qij θ

Y Xj , ρXY

i = Qij ρY Xj , (2.21a)

ZXYij = QikZ

Y Xkl Qlj , (2.21b)

where

−Q11 = Q22 = Q33 = 1, Qij = 0 when i = j.(2.22)

Equations (2.21) imply that, when XY = AT, GC, TA,or CG,10

FXY12 = FXY

13 = 0, HXY12 = HXY

13 = 0, (2.23a)GXY

12 = GXY13 = GXY

21 = GXY31 = 0, (2.23b)

θXY1 = ρXY

1 = 0. (2.23c)

Estimates from X-ray crystal structure data of valuesof the elastic moduli FXY

ij , GXYij , HXY

ij and the intrinsicparameters θXY

i , ρXYi for each distinct combination of a

base pair and its successor [5, 6] confirm that these mod-uli and intrinsic parameters are strongly dependent onthe nucleotide composition. If we focus attention on theangular variables θXY

i , we can give as follows a rough

9 Packer, Dauncy, and Hunter [23] have used a computationalmodel to calculate the potential energy surface for base-pair stepsas a function of two principal degrees of freedom, slide and shift,and obtained results that suggest that the presence of close lo-cal minima of energy (of the type observed in crystal structuresof DNA oligomers in Ref. 5) can limit the range of applicabil-ity of quadratic approximations for ψn. Their calculations fortetramer steps [24] further suggest that one should not blandlyassume that interactions of non-adjacent base pairs need not betaken into account. However, there are not as yet a sufficientnumber of high-resolution crystallographic structures availableto treat all 136 tetramer steps in the analysis of the sequence-dependent properties of DNA and to answer the question, posedin Ref. 25, of whether the base-pair neighbors adjacent to a dimerstep have a strong influence on its elastic properties.

10 O’Hern et al. [26] developed a model of DNA elasticity in whichbase-pairs are assumed to be rigid plates separated by a homo-geneous isotropic elastic material, and showed that their modelimplies that, for each n, ψn is a quadratic function obeying therelations seen in Eqs. (2.23). We here show that the Eqs. (2.23)hold in our more general model when XY = YX. The availableexperimental data [5, 6] strongly indicate that Eqs. (2.23) holdwhen and only when XY = YX. A DNA sequence for whichXY = YX at every base-pair step must be either of the type. . .ATATAT. . . or . . .GCGCGC. . . .

6

summary of the nature of the sequence-dependence ofmoduli and intrinsic parameters of the B-form of DNA:11

It follows from Eqs. (2.21) that the intrinsic tilt θXY1

obeys the relation θXY1 = −θY X

1 , and the available dataindicate that its largest value is ∼ 1.5◦. The data indicatethat the intrinsic roll θXY

2 is positive with its largest value∼ 5◦ and thus more than 3 times the largest value of θXY

1 .The smallest value of θXY

2 is ∼ 0.5◦. The ratio of the tiltmodulus FXY

11 to the roll modulus FXY22 varies in the range

∼ 1 to ∼ 5, and the ratio of the twist modulus FXY33 to

the harmonic mean AXY = 2/[(1/FXY11 )+(1/FXY

22 )] of thetilt and roll moduli varies in the range ∼ 0.5 to ∼ 1.8.The twist-roll coupling modulus F23, in units of kT/deg2,varies in the range ∼ 0.00 to ∼ 0.03.

In Ref. 27, Gonzales & Maddocks develop a theorythat relates the moduli F n

ij, Gnij, H

nij and the intrinsic pa-

rameters θni , ρ

ni to statistical averages and correlations

between θni and ρn

i for ensembles of configurations ob-tainable from all-atom molecular dynamics simulations.

III. DERIVATION OF THE PRINCIPALRESULTS

Symmetry relations

The underlying assumptions behind Eq. (2.2) is thatψn in Eq. (2.1) is given by a function ψn of rn andthe two right-handed orthonormal triads dn

1 ,dn2 ,d

n3 and

dn+11 ,dn+1

2 ,dn+13 ; thus

ψn = ψn(dni ,d

n+1j , rn), (i, j = 1, 2, 3). (3.1)

The principle of material frame indifference requires thatψn obey, for each proper orthogonal second-order tensorJ , the relation

ψn(dni ,d

n+1j , rn) = ψn(Jdn

i ,Jdn+1j ,Jrn) (3.2)

as an identity. It follows from this last relationand Cauchy’s representation theorem for scalar-valuedisotropic functions of lists of vectors that there is a func-tion such that12

ψn = ¯ψn(Dnij , r

ni ); (3.3)

this relation is equivalent to Eq. (2.2).Now, the functions ψn and ¯ψn are assumed to be de-

termined by the (ordered) pair XY of nucleotide baseslocated at the n-th and (n+1)-th positions on the sugar-phosphate chain for which n increases in the 5′–3′ di-rection. Hence for ψn and ¯ψn we may write ψXY and

11 The values and ranges we give for the moduli and parametersare based on data reported in Ref. 6.

12 In this application of Cauchy’s representation theorem, we makeuse of the fact that dn+1

j · rn, as it equals Dnjir

ni , is known once

Dnij and r

ni are known for i, j = 1, 2,3.

¯ψXY. To derive restrictions that the antiparallel arrange-ment of the oriented sugar-phosphate chains of duplexDNA impose on these functions, we note that, as thecomplements of A and G are A = T and G = C, the 16ordered pairs XY separate into 2 classes. One class con-tains the 4 self-complementary pairs {AT, GC, TA, CG},each of which, because it has Y = X, obeys the relationXY = Y X. The remaining class of 12 is made up of 6 setsof complementary pairs, (i.e., pairs {X1Y1, X2Y2} withX2Y2 = Y1 X1), namely, {AA,TT}, {AG, CT}, {AC,GT}, {GA, TC}, {GG, CC}, {TG, CA}.

Under the transformation from dni ,d

n+1j , rn to

∗dmi , ∗d

m+1j , ∗r

m, where

∗dm1 = dn+1

1 , ∗dm+11 = dn

1 , (3.4a)

∗dm2 = −dn+1

2 , ∗dm+12 = −dn

2 , (3.4b)

∗dm3 = −dn+1

3 , ∗dm+13 = −dn

3 , (3.4c)

∗rm = −rn, (3.4d)

and m = N − n + 1, ψn remains unchanged, i.e., ∗ψm =

ψn, because the rectangles Bn, although they are renum-bered in reverse order, keep their previous spatial posi-tions and orientations. The transformation takes the vec-tors dn

2 ,dn+12 , into vectors ∗d

m2 , ∗d

m+12 , that again point

toward the short edges that have corners attached to thechain for which n increases in the 5′–3′ direction (whichis the one for which n decreased in that direction be-fore the transformation). As the transformation replacesXY with its complement Y X while leaving ψn invari-ant, the functions ψXY and ψY X must obey the relation,ψXY(dn

i ,dn+1j , rn) = ψY X(∗dm

i , ∗dm+1j , ∗r

m), i.e.,

ψXY(dn1 ,d

n2 ,d

n3 ,d

n+11 ,dn+1

2 ,dn+13 , rn) =

ψY X(dn+11 ,−dn+1

2 ,−dn+13 ,dn

1 ,−dn2 ,−dn

3 ,−rn), (3.5)

or, equivalently,

¯ψXY(Dnij , r

ni ) = ¯ψY X(QikD

nlkQjl, QilD

nklr

ni ), (3.6)

where Qij is as in Eq. (2.22). If the pair XY isself-complementary, i.e., is AT, GC, TA, or CG, thenY X = XY, and Eq. (3.5) becomes an identity for thefunction ψXY .

As the tangent spaces to the manifold M of proper or-thogonal 3 × 3 matrices are of dimension 3, each pointof M has a neighborhood in which M has a smooth 3-coordinate parametrization of the type discussed in Sec-tion II, and we can write

Dnij = Dij(θn

1 , θn2 , θ

n3 ), (3.7)

where the functions Dij are independent of n and locallyone-to-one with continuously differentiable inverses. Interms of the kinematical variables θn

i and ρni defined in

Eqs. (2.5)–(2.7), the Eqs. (3.4) take the form

(∗θm1 , ∗θ

m2 , ∗θ

m3 , ∗ρ

m1 , ∗ρ

m2 , ∗ρ

m3 ) =

(−θn1 , θ

n2 , θ

n3 ,−ρn

1 , ρn2 , ρ

n3 ), (3.8)

7

and hence Eq. (3.5) is equivalent to the symmetry rela-tion Eq. (2.9)

For use below, we note that because the 3 matrices[lV n

ij ], (l = 1, 2, 3), with components

lVnij = Djk

∂Dik

∂θnl

, (3.9)

are skew in the sense that lVn

ij = −lVn

ji for all i, j, thereare numbers Ξn

kl, (k, l = 1, 2, 3), such that13

lVnij = εijkΞn

kl. (3.10)

The local invertibility of the function [Dij] implies thatthe columns of the matrix [Ξn

kl] form a (not necessarilyorthogonal) basis for the tangent space of M at the pointwith coordinates θn

1 , θn2 , θ

n3 , and hence [Ξn

kl] has a matrixinverse [Ξn

kl]−1. For the transpose of [Ξn

kl]−1 we write

[Γnkl]:

ΓnkiΞ

nkj = δij . (3.11)

Equations of equilibrium

A configuration Z of a DNA segment with N+1base pairs is specified by giving the (N+1)-tuple{(dn

1 ,dn2 ,d

n3 ,x

n)}N+1n=1 , of ordered sets (dn

1 ,dn2 ,d

n3 ,x

n) inwhich dn

1 ,dn2 ,d

n3 is a right-handed orthonormal basis and

xn a point in space. The total energy E of the segmentis given by Eq. (2.10) in which P = P (Z) is the potentialenergy of the external forces and moments acting on thesegment. A configuration Z# is said to be in equilibriumif for each one-parameter family

s → Z(s), Z(0) = Z#, −ε < s < ε, (3.12)

of configurations compatible with the imposed con-straints (e.g., the strong anchoring end conditions or theclosure conditions of Eq. (2.16)) there holds E′(0) = 0,i.e., in view of Eqs. (2.1) and (2.10),

N∑n=1

(ψn)′(0) + P ′(0) = 0; (3.13)

here primes indicate derivatives with respect to s.To derive the variational equations implied by this def-

inition, we employ Eq. (2.2) which yields

(ψn)′ =∂ψn

∂θni

(θni )′ +

∂ψn

∂rni

(rni )′ (3.14)

where

(rni )′ = (dn

i · rn)′, (3.15)

13 εijk is the permutation symbol of Levi-Civita.

and, in view of Eq. (3.7), (θni )′ obeys the relation

∂Dik

∂θnl

(θnl )′ = (Dn

ij)′ = (dni · dn+1

j )′. (3.16)

As the vectors dn1 ,d

n2 ,d

n3 are orthonormal, there is a vec-

tor wn such that

(dni )′ = wn × dn

i . (3.17)

By Eqs. (3.9)–(3.11), (3.15), and (3.17),

(rni )′ = dn

i · [(rn)′ − wn × rn], (3.18a)(θn

i )′ = Γnjid

nj · (wn+1 − wn), (3.18b)

P ′ =N∑

n=1

[(dn

i ×∂P

∂dni

)·wn +

∂P

∂xn·(xn)′

], (3.18c)

and, if we put

fn = fni dn

i , fni =

∂ψn

∂rni

, (3.19a)

mn = mni dn

i , mni = Γn

ij

∂ψn

∂θnj

, (3.19b)

ϕn =∂P

∂xn, (3.19c)

µn = dni × ∂P

∂dni

, (3.19d)

and write vn = (xn)′, then Eqs. (3.14) and (3.18c), re-spectively, can be cast into the forms

(ψn)′ = mn ·(wn+1−wn

)+fn ·

(vn+1−vn−wn×rn

),

(3.20)

P ′ =N∑

n=1

(µn · wn + ϕn · vn) . (3.21)

In the last two equations, although end conditions af-fect the choice of the vectors w1, v1,wN+1, and vN+1,the vectors wn and vn can be selected arbitrarily forn = 2, . . . , N . Hence, the relation (3.13) holds for theconfiguration Z# if and only if that configuration obeysthe following variational equations,

fn − fn−1 = ϕn,

mn − mn−1 = fn×rn + µn, (2 ≤ n ≤ N), (3.22)

and natural end conditions,(m1 − f 1 × r1 − µ1

)· w1 − (mN + µN+1) · wN+1

+(f 1 − ϕ1) · v1 − (fN + ϕN+1) · vN+1 = 0. (3.23)

The relations (3.22) are the analogues in the presenttheory of the equations of balance of moments and forcesin Kirchhoff’s (continuum) theory of elastic rods [28–31]and several recent generalizations of that theory (vid.,e.g., the treatise of Antman [32], Section VIII).

8

An equilibrium configuration Z# is stable if, for eachone-parameter family of configurations that obeys (3.12)and is compatible with the imposed constraints, thereholds, in addition to Eq. (3.13),

0 < E′′(0) =N∑

n=1

(ψn)′′(0) + P ′′(0). (3.24)

In practice, to verify stability we employ Eqs. (3.13)–(3.18) to express E′′(0) as a quadratic form Q onthe 6(N+1)-dimensional space of the components ofw1, v1, . . . ,wN+1, vN+1 with respect to one fixed ba-sis, e.g., d1

1,d12,d

13. If λ, the smallest proper number of

Q, is less than zero, the configuration Z# is unstable.If λ > 0, Z# is stable. This test was used to verifystatements made in Section IV about the stability of cal-culated configurations.

IV. EXAMPLES OF EQUILIBRIUMCONFIGURATIONS

Much of the research on DNA elasticity employs an ide-alized elastic rod model, based on the assumption thata duplex DNA molecule is an intrinsically straight, ho-mogeneous, elastic rod obeying the special case of Kirch-hoff’s general theory in which there are just two elasticconstants: a twisting modulus and a bending modulus.Several recent studies (e.g., Refs. 11–19) of the influenceof intrinsic curvature on the elastic response of DNA arebased on Kirchhoff’s general theory14 in which a rod hastwo bending moduli and, in addition, the possible pres-ence of intrinsic curvature and torsion is taken into ac-count. The constitutive relation for the elastic energy ofa rod obeying that theory does not contain cross-termsdirectly coupling the twisting moment to the local cur-vature. It is known that in a materially homogeneousregion of a rod with intrinsic curvature the density ofexcess twist is, in general, not uniform in equilibrium.In Ref. 12 it is shown that the equations of equilibriumappropriate to Kirchhoff’s theory imply that when in-trinsic curvature is present the derivative of twist withrespect to distance along the rod axis is a function of lo-cal curvature. See also papers of Bauer, Lund, & White[11, 14, 16] and Garrivier & Fourcade [19] and the calcu-lations therein of the effect of intrinsic curvature on theshape and twist density profile of o-rings [11, 14, 16] andplectonemic loops [19].

We here give examples of calculated equilibrium config-urations of DNA minicircles obeying the theory presentedin Section II. In order to obtain insight into the implica-tions of our theory, we deliberately treat cases in which

14 For a statement and a discussion of the assumptions behindKirchhoff’s theory of rods [28, 29] see Ref. 30.

known results for intrinsically curved rods obeying Kirch-hoff’s theory can yield physical intuition as to what toexpect. We therefore treat minicircles that may be called“DNA o-rings” because they are intrinsically bent in sucha way that in stress-free configurations their axial curvesare nearly circular. The helical nature of DNA makes itimpossible to form a stress-free o-ring with the intrinsickinematical variables the same at each base-pair step, forif they were the same, the bendings occurring in the twohalves of each helical repeat length, being then of equalmagnitudes but in opposite directions, would cancel. Forthis reason, the minicircles we treat are composed of 10bp helical repeat units in which 5 bp form an intrinsicallystraight segment and the remaining 5 bp form an intrin-sically curved segment.15 (The highly curved sequencesfound in the kinetoplast DNA from the Trypanosomati-dae Crithidia fasiculata and Leishmania tarentolae haveon one strand stretches of 4 to 6 adenine bases that areseparated by stretches of comparable length rich in gua-nine and cytosine bases [35–37].)16

In analogy with the familiar assumptions in the con-tinuum theories of DNA elasticity, in our examples ψn istaken to be a quadratic function of the kinematical vari-ables. Results are presented for two types of DNA mini-circles: For those of type I, the matrix of the quadraticform characterizing ψn is diagonal for each n. For thoseof type II, each helical repeat length contains 5 base-pairsteps for which the off-diagonal modulus associated withcoupling between roll and twist is not zero and hencethe minicircle is expected to show chiral phenomena notpredicted by Kirchhoff’s classical theory of rods. Theconfigurations shown were obtained by solution of theequations of mechanical equilibrium.

The minicircles we consider have N = 150 bp and are

15 Base-pair level models of DNA deformability constructed to ac-count for results of measurements of cyclization kinetics andgel electrophoretic mobility of short DNA segments suggest thathighly curved DNA is made up of two types of duplex structurewithin each helical repeat length: the canonical B-type duplexand a perturbed form in which the base pairs are inclined withrespect to the duplex axis [33, 34] as they are in an A-type orC-type duplex.

16 Electron micrographs of a 219 bp segment with such a se-quence found in C. fasiculata indicate that the segment hasthe form of a complete circle [38], and DNA segments withhigher intrinsic curvature have been synthesized. Koo et al.[39] have given strong evidence, based a study of cycliza-tion kinetics, to the effect that a 210 bp DNA segment withthe sequence (AAAAAACGGCAAAAAACGGGC)n can forma complete circle when stress free. Previously, Ulanovskyet al. [40] had showed that DNA segments with sequences(AAAAAATATATAAAAATCTCT)n can be closed to formminicircles with 105, 126, 147, and 168 bp, i.e., with n = 5,6,7, 8.They found that the maximum value of the ratio of concentra-tions at equilibriumof circular and noncircular products occurredwhen the sequence length was either 126 or 147 bp, and they sug-gested that such an observation indicates that it may be possibleto obtain stress-free minicircles with as few as 137 bp. See alsothe review of Crothers et al. [41].

9

composed of 15 repetitions of a 10 bp unit in which thefirst 5 bp have

(θn1 , θ

n2 , θ

n3 ) = (0, 0, 36◦),

(ρn1 , ρ

n2 , ρ

n3 ) = (0, 0, 3.4 A), (4.1)

and hence form an ideal Watson-Crick DNA segmentwhile the remaining 5 bp have17

(θn1 , θ

n2 , θ

n3 ) = (0, 7.413◦, 35.568◦),

(ρn1 , ρ

n2 , ρ

n3 ) = (0, 0, 3.4 A), (4.2)

and hence form an intrinsically curved segment. (Forthe calculations we report, the kinematical variables θn

1 ,θn2 , θn

3 , ρn1 , ρn

2 , ρn3 that are schematically illustrated in

Figure 1 are taken to be those defined in Ref. 8 andEqs. (2.5)–(2.7).) A minicircle with this choice of N andthe indicated parameters θn

i , ρni has a linking number L

of 15, and in its stress-free configuration its axial curve Chas the appearance of a polygon of 15 sides with roundedvertices that is sufficiently close to a circle to permit usto refer to the minicircle as an o-ring. For both I andII, the moduli Gn

ij and Hnij are here taken to be indepen-

dent of n with Gij = 0 for all i, j, with Hij = 0 for i = j,and with H11, H22, H33 large enough to suppress defor-mations that change the displacement variables ρn

i . Thetwo minicircles differ in the value of the coupling coeffi-cients F n

ij, i = j; for both, F11, F22, F33 are independentof n and such that F11 = F22 = 4.27×10−2 kT/deg2; theratio

ω = F33/F11 (4.3)

was set equal to 0.7 for certain calculations and 1.4 forothers.18

Minicircle I

Minicircle I is such that, for i = j and n = 1, . . . , 150,

F nij = 0, (4.4)

and hence Eq. (2.19) does not contain off-diagonal termscoupling the kinematical variables.

17 So as to obtain an o-ring of small size, i.e., with N = 150 bp,while making the simplifying assumption that the intrinsic shearsρn1 and ρ

n2 are everywhere zero, we chose θ

n2 in Eq. (4.2) to be

larger than the available estimates of the upper bound for stress-free values of the roll.

18 The values of F11 and F22 were chosen so that the bending rigid-ity 2/[(1/F11) + (1/F22)] of each minicircle equals that of nat-urally straight DNA with a persistence length of 500 A. Thechoice ω = 0.7 is then compatible with measurements of flu-orescence anisotropy of dyes intercalated in open segments ofDNA (see Ref. 42). Measurements of topoisomer distributionsfor miniplasmids [43] suggest that ω should be close to 1.4. (Seealso Bouchiat and Mezard [44] who state that single moleculestretching experiments [45] yield a value of 1.6 for ω.)

FIG. 3: Equilibrium configurations of the 150 bp o-ring, Mini-circle I, for the case ω = 0.7. Shown are the stress-free config-uration A that gives a global minimum to Ψ, the unstable ev-erted ring configuration B, the metastable chair configuration,and the unstable configurations D and E; F is an unstable so-lution of the equilibrium equations that is incompatible withthe assumed impenetrability of DNA. Configuration D lies ata saddle point in the energy landscape for configurations ofthe o-ring. Here, as in Figs. 5, 7, 9, and 11, configurationsare drawn as tubes of diameter 20 A. The elastic energy Ψ isgiven in units of kT .

Even when this minicircle is kept free of bound pro-teins and other chemical agents, it has multiple equilib-rium configurations, several of which (each with L = 15)are shown in Fig. 3 where ω = 0.7. In that figure, A isthe stress-free, nearly circular, minimum energy configu-ration. The “everted ring” configuration B has the axialcurve close to a circle (as does A) and may be described,to a good approximation, as a configuration obtainedfrom A by rotating each base pair in its plane by onehalf turn (as in the motion called the “smoke-ring rota-tion”). It has been known for a long time19 that evertedring configurations of o-rings are unstable. The “chair”configuration C is familiar from experience with elasticrings and was treated by Charitat and Fourcade [15] us-ing Kirchhoff’s theory. The stability of chair configu-rations depends on ω. We find that (i) when ω < 1,as it is in the present example, the configuration C ismetastable; (ii) C is unstable when ω > 1; and (iii) whenω = 1, there is a one-parameter family of distinct chairconfigurations of equal energies.20 The configurations Dand E, which to our knowledge have not been discussedbefore, have less symmetry than C and are unstable; in

19 See, e.g., the treatise of Thomson and Tait [46]. A modern discus-sion containing exact solutions of the nonlinear dynamical equa-tions governing the oscillatory motion of an inextensible elasticrod (obeying Kirchhoff’s theory) with uniformly distributed in-trinsic curvature and torsion is given in Ref. 13.

20 The (approximate) analysis of Ref. 15 yielded the value ω = 0.9as the critical value of ω for loss of stability.

10

FIG. 4: The excess twist ∆θn3 (in degrees) as a function of n

for selected configurations of Fig. 3. The points on the graphfor the chair configuration C that are labeled α, β, γ, δ havethe geometric locations indicated in Fig. 3 and correspond tothe apexes (α,γ) of the chair and the midpoints (β, δ) of itssides.

the present example, D lies at a mountain pass for atransition path from C to the globally stable configura-tion A. The item labeled F, although it corresponds to asolution of Eqs. (2.11), (2.12), and (2.17), is not a phys-ically admissible configuration, because it does not obeythe condition of impenetrability of base pairs.

Figure 4 contains graphs of excess twist ∆θn3 versus

n for selected configurations of Fig. 3 and shows thatalthough ∆θn

3 is zero for the globally stable circular con-figuration A and approximately zero for the unstable cir-cular configuration B, such is not the case for the chairconfiguration C. For that configuration, |∆θn

3 | vanishesnear the two “apexes” of the chair (labeled α and γ),and attains its (remarkably large) maximum near thecenters of the “sides” of the chair (β and δ). In Fig.4 (and also in Figs. 6, 8, and 10 below) it can beseen that the excess twist ∆θn

3 is constant within eachof the intrinsically straight segments in which Eq. (4.1)holds. This local constancy is a consequence of the factthat in the intrinsically straight segments of Minicircle Ithe quadratic constitutive functions ψn are free of cross-terms coupling twist to other kinematical variables. Inthe intrinsically curved segments obeying Eq. (4.2), re-gardless of whether cross-terms are present in ψn, ∆θn

3 isgenerally non-monotone and is constant only if the o-ringis stress-free. Arguments of the type that led to Eq. (28)of Ref. 12 can be employed here to show that if ∆θn

1 inan intrinsically curved segment changes sign between nand n + 1, then ∆θn

3 has an extremum at n or n + 1.It has been known for a long time that one can induce

negative supercoiling in closed DNA by reacting it withan intercalating agent, such as ethidium bromide (EtBr),that binds to DNA between base pairs and causes localreduction of the intrinsic twist of the double helix [47]. If,while maintaining the other assumptions we have madeabout Minicircle I, we were to assume that Eqs. (4.1) hold

FIG. 5: Influence of the spatial distribution of intercalatedmolecules on the minimum elastic energy configuration ofMinicircle I. Shown are calculated results for cases in which 10molecules are intercalated into the minicircle with each induc-ing the local change θn

3 → θn3 − 30◦, ρn

3 → ρn3 + 3.4A = 6.8A.

The sites of binding are shown as dark bands. In configura-tions A and A* these sites are equally spaced along the axialcurve; in B and B* they occupy every other base-pair step intwo antipodally placed 11 bp subsegments of the minicircle;in C and C* they occupy every other base-pair step in a single21 bp subsegment. As indicated, ω = 0.7 for A, B, C, andω = 1.4 for A*, B*, C*.

for 1 ≤ n ≤ 150, which would make the minicircle notonly homogeneous in n but also one that was formed fromintrinsically straight DNA (with θn

1 ≡ θn2 ≡ 0),21 then

the equilibrium shape and elastic energy of the minicirclewould depend on the number of molecules of EtBr inter-calated in it but would be essentially independent of theirdistribution along C.22 We have found, however, that, be-cause Minicircle I is an o-ring formed from DNA that isintrinsically curved in accord with Eqs. (4.1) and (4.2),its equilibrium shape and elastic energy are strongly de-pendent on both the number and the distribution alongC of the bound molecules.23

Figure 5 contains results of calculations in which 10identical molecules are assumed to be intercalated atspecified base-pair steps of Minicircle I, with each suchmolecule reducing the intrinsic twist θn

3 by 30◦ and dou-bling the intrinsic rise ρn

3 at the step at which it is in-tercalated, as is the case for EtBr [51]. In the discus-sion that follows we assume that the intercalating agent

21 Such a minicircle would be a discrete analogue of a closed ide-alized elastic rod of a type for which much is known about thesymmetry and stability of equilibrium states, with and withoutself-contact [48, 49].

22 We say “essentially independent,” for the intercalative bindingof EtBr changes not only the twist θn

3 but also the rise ρn3 [47].

23 The present theory is not intended to provide an interpretation ofan observed preference of EtBr for binding to pyrimidine-purinebase-pair steps [50].

11

FIG. 6: ∆θn3 as a function of n for the configurations of Fig. 5

for which ω = 0.7. The symbols ◦ mark the sites at which theintercalating agent is bound.

obeys the principle of nearest neighbor exclusion requir-ing that two bound molecules be separated by at least onebase-pair step. (There is evidence to the effect that thisexclusion principle does hold for small intercalating lig-ands [52, 53].) If at any given instant the distribution ofthe 10 intercalated molecules is spatially uniform, Mini-circle I retains its circular configuration. When thosemolecules are distributed in such a way that they oc-cupy every other base pair step within two antipodallyplaced subsegments of equal length, the o-ring folds intoone of the “figure-8” configurations B and B*. Whenthe bound molecules are concentrated in a short sub-segment (but yet obey the principle of nearest neighborexclusion), the o-ring again deforms out of plane and,in the case of ω = 1.4, folds into a configuration of the“clamshell” type, here labeled C*. In the present case weobtain the result that, for both values of ω the distribu-tion into two equal and antipodally placed subsegments,seen in B and B*, is the distribution that minimizes Ψover all distributions of 10 intercalated molecules.24

The values of Ψ given in Fig. 5 tell us that, if weassume that EtBr binding is independent of nucleotidecomposition, in the case of dynamical equilibrium themost probable spatial distribution of bound moleculeswill be close to that seen in B and B*. Of course, thenumber of bound molecules is a fluctuating quantity thatdepends on the concentration of EtBr and DNA in solu-tion. A complete analysis of curvature-induced bindingcooperativity of EtBr molecules leading to prediction of

24 As the o-ring we are considering here is composed of periodicallyrepeating units of length 10 bp, the minimum energy configura-tion for a given distribution of sites of intercalation would not beaffected if each such site were shifted by 10 bp in the directionof increasing n. Although a shift in the distribution by less than10 bp can affect the minimum energy configuration, our calcula-tions show that for Minicircle I the corresponding change in Ψdoes not exceed 0.15 kT .

FIG. 7: Configurations of Minicircle I that minimize Ψ whenthe minicircle is bound to an untwisting agent supposed capa-ble of lowering θ13 by 180◦ in the case of A*, 240◦ in the caseof B*, and 300◦ (which is equivalent to the total untwistinginduced by the intercalation of 10 molecules of EtBr) in thecase of C*. The binding site for the agent is indicated by atriangle. Here, as in all subsequent figures, ω = 1.4.

Scatchard plots of EtBr binding activity would be a com-putationally intensive task that has not yet been under-taken. However, the preliminary results shown here dosuggest that, for DNA subject to appropriate end condi-tions, such as those of the Eqs. (2.16) that characterizering closure, the spatial distribution of EtBr can be ex-pected to be sensitive to assumed intrinsic curvature.

Under the present assumptions about the o-ring andthe intercalating agent, although ∆θn

3 is positive for alln when the intercalating molecules are equally spacedalong the length of the minicircle, if those molecules areconcentrated into a short enough subsegment S, as theyare in the minimum energy configurations C and C*, ∆θn

3

becomes negative in a neighborhood of the antipode ofthe midpoint of S. (See Fig. 6.)

To investigate the effect of localization of changes in in-trinsic twist, we have calculated configurations for casesin which a single molecule of an agent is bound to ano-ring at one base pair step, say that with n = 1, andreduces the intrinsic twist of the DNA at that site by alarge positive amount α, i.e., induces the transformationθ13 → θ1

3 − α while leaving other intrinsic variables un-changed. In a study of the effect of intrinsic curvatureon configurations of DNA minicircles obeying Kirchhoff’stheory (in which, as is the case for Minicircle I, the con-stitutive relations do not contain cross-terms couplingcurvature to twisting moment) Bauer, Lund, & White[11, 14, 16] found that a reduction of the excess link bycutting at a single cross-section (which is roughly equiv-alent to an increase in intrinsic twist at a single basepair step) can lead to remarkably large deformations evenwhen the magnitude of that reduction is less than 1 (i.e.,|α| < 360◦) and hence less than the amount required toinduce supercoiling in intrinsically straight rods.25 Re-

25 It follows from the theory of ideal elastic rods that if the mini-circle were formed by closing naturally straight DNA, i.e., ifEq. (4.2) were replaced by Eq. (4.1), then the untwisting at asingle base-pair step by an angle of magnitude less than 360◦,as it corresponds to a change in excess link less than 1, wouldnot affect the shape of the minicircle, i.e., the axial curve wouldremain circular.

12

FIG. 8: Graphs of ∆θn3 versus n for the configurations of

Fig. 7. The untwisting agent is bound at the base-pair stepmarked by ◦ for which n = 1.

sults for Minicircle I with ω equal to 1.4 and for ratherextreme values of α are shown in Figures 7 and 8.26 Asseen in Fig. 7, as α increases the o-ring folds up, withthe folding becoming appreciable for α near to 180◦, andat sufficiently high α the o-ring attains a clamshell con-figuration. When, as here, the change of intrinsic twistis confined to one base-pair step, configurations (but nottheir energies) are approximately independent of ω. InFig. 8 one sees that, as expected from the results shownin Fig. 6, the excess twist is negative in a neighborhoodof the antipode to the binding site.27 As the constitutiveequation for the energy of Minicircle I does not containoff-diagonal terms directly coupling roll or tilt to twist, ifthe twisting agent were one that raises θ1

3 (i.e., that hasα < 0), the resulting configurations would be essentiallymirror images of those shown in Fig. 7. See, however,Fig. 11 and comments made below about Minicircle II.

Minicircle II, an o-ring with off-diagonal termscoupling twist to roll

In the case of Minicircle II, for those values of n atwhich Eq. (4.1) holds,

(F n12, F

n23, F

n13, ) = (0, 0, 0) (4.5)

and, for the n at which Eq. (4.2) holds,

(F n12, F

n23, F

n13, ) = (0, 4

5Fn22, 0). (4.6)

26 There are protein structures that can induce a comparablechange in the magnitude of twist, but the changes they induceare generally distributed over several base-pairs steps. See, e.g.,Ref. 54 reporting a case in which an 8 bp long region is untwistedby 147◦ in a crystal structure of the human TFIIB-TBP proteincomplex bound to duplex DNA.

27 See also [11].

FIG. 9: The configurations of Minicircle II that minimize Ψwhen θ13 is changed to θ13 − α with α at the indicated values.

FIG. 10: Graphs of ∆θn3 versus n for the configuration B* of

Minicircle I (see Fig. 7) and the configuration B+ of MinicircleII (see Fig. 9). The untwisting agent is bound at the base-pairstep marked by ◦ for which n = 1.

Thus, in each repeated 10 bp unit, the first 5 bp forman intrinsically straight segment and do not show cou-pling between kinematical variables, while the remaining5 bp form an intrinsically curved segment and are suchthat the twist-roll coupling modulus F n

23 is positive andhence raising the roll tends to lower the twist, i.e., anincrease in θn

2 results in a twisting moment that tends todecrease θn

3 . (The positivity of F n23 is compatible with the

coupling of parameters observed in high-resolution DNAcrystal structures [4, 6].) The calculations shown herefor Minicircle II (Figs. 9–12) were done with ω = 1.4.

Although Minicircles I and II both have minimum en-ergy configurations that are stress-free and nearly circu-lar, the effect of a twisting agent on II is dependent onwhether the agent reduces or increases the intrinsic twistof DNA. If subject to the transformation θ1

3 → θ13 − α,

Minicircle II (like I) shows appreciable folding when |α|exceeds 180◦. However, as one sees in Fig. 9, in contrastto what the theory yields for I, the minimum energy con-figurations of II for α and −α are not mirror images. The

13

FIG. 11: Graphs of the writhe W versus α for minimum en-ergy configurations of Minicircle I (light curve) and MinicircleII (heavy curve) bound to a twisting agent. (In both casesω = 1.4.) The points A+,B+,A−,B−,C− correspond to theconfigurations of Fig. 9, and the points S0,S1, U to the con-figurations of Fig. 12. The configurations at which Ψ has itsglobal minimum for the specified α lie in the regions drawn assolid curves (light or heavy); there are configurations on theheavy dashed curve that are not even local minimizers of Ψ(see Figs. 12 and 13). The × marks the largest α for whichthe minimum energy configuration of each minicircle is freefrom self-contact.

FIG. 12: The stable configurations S0,S1 and the unstableconfiguration U of Minicircle II of Fig. 11 with α = 215.0◦.

graphs of ∆θn3 versus n in Fig. 10 for the configurations

of Minicircles I and II with α = 240◦ tell us that in the 5bp subsegments of II in which coupling is present ∆θn

3 ismuch lower than it is in the corresponding subsegmentsof I, which is a consequence of the fact that in those sub-segments θn

2 (i.e., roll) is larger than its stress-free value.The presence of coupling here induces untwisting by asmuch as 2.7◦, with the most untwisting attained at thebase-pair steps with n = 49 and 106 which are locatedroughly 1/3 of the circumference away from the bindingsite of the untwisting agent.

The graphs in Fig. 11 showing the dependence on αof the writhe W , which is a measure of the folding ofthe o-ring [55, 56], illustrate well the difference in theresponse of the two minicircles to the binding of a twist-ing agent. While the response of I is nearly antisym-metric, in the sense that the configurations correspond-ing to α and −α are close to mirror images and henceW (−α) ∼= 2W (0) − W (α), the response of II is chiraland such that, for negative α (i.e., for an increase in in-

FIG. 13: Graphs of W versus α and Ψ versus α for equilib-rium configurations of Minicircle II with α in a neighborhoodof 215.0◦. S0 and S1 are points of exchange of global stability,U determines the energy barrier for the transition S0 → S1.Stability of configurations is indicated as follows: (——) glob-ally stable, (– · – · –) locally stable, (· · · · ·) unstable.

trinsic twist), W changes monotonically with α, but foreach (positive) α in the interval 211.9◦ < α < 220.7◦there are three equilibrium configurations, with two sta-ble and one unstable. When α = 215.0◦, the two sta-ble configurations (labeled S0, S1) have equal elastic en-ergy (Fig. 12). As seen in Fig. 13, the configurationswith preassigned α ≤ α(S0) that give a global mini-mum to Ψ form a continuous one-parameter family ofconfigurations which terminates at S0 and is not con-nected to the family of minimum energy configurationswith α ≥ α(S1). At α = 215.0◦ infinitesimal changesin α result in first-order transitions between the stableequilibrium states S0 and S1. The unstable configurationU with α(U) = α(S0) = α(S1) lies at a mountain pass forthese transitions. The graph of Ψ versus α seen in Fig. 13tells us that there is an essentially negligible energy bar-rier, namely 0.13 kT , for the transitions S0 → S1 andS1 → S0 between the “open” configuration S0 and the“partially folded” configuration S1, both of which give toΨ its global minimum in the class of all configurationsthat are accessible without change in α.

14

Acknowledgments

We thank A.R. Srinivasan for discussions and materialhelp in the design of the here employed naturally discrete

model for a DNA o-ring. This research was supported bythe National Science Foundation under grants DMS-97-05016 and DMS-02-02668 and by the U.S. Public HealthService under grant GM34809.

APPENDIX

For the matrix [Γnij] of Eq. (2.12), the Eqs. (2.5)–(2.6) and (3.9)–(3.11) yield,

[Γnij] =

− θn

1κn sin ζn + θn

2 cos ζn

2 tan�

12 κn

� − θn2

κn sin ζn − θn1 cos ζn

2 tan�

12 κn

� tan(

12κn)

cos ζn

θn1

κn cos ζn + θn2 sin ζn

2 tan�

12

κn� θn

2κn cos ζn − θn

1 sin ζn

2 tan�

12

κn� tan

(12κ

n)

sin ζn

−θn22

θn12 1

(A.1)

For each j, the numbers jΛnkl of Eq. (2.15) are the components of a skew matrix [jΛn

kl]. When j = 1,

1Λn12 =

θn2

�1−cos

�12 κn

��

(κn)2 , 1Λn13 =

θn1 θn

2

�2 sin

�12 κn

�−κn

�

2(κn)3 , 1Λn23 = 1

2 +(

θn2

κn

)2 2 sin�

12κn

�−κn

2κn . (A.2)

When j = 2,

2Λn12 =

θn1

�cos

�12

κn�−1

�

(κn)2, 2Λn

13 =(

θn1

κn

)2 κn−2 sin�

12

κn�

2κn − 12, 2Λn

23 =θn1 θn

2

�κn−2 sin

�12

κn��

2(κn)3. (A.3)

When j = 3,

3Λn12 = 1

2 cos(

12κ

n), 3Λn

13 = − θn1

2κn sin(

12κ

n), 3Λn

23 = − θn2

2κn sin(

12κ

n). (A.4)

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