Bernard D. Coleman, Wilma K. Olson and David Swigon- Theory of sequence-dependent DNA elasticity

8/3/2019 Bernard D. Coleman, Wilma K. Olson and David Swigon- Theory of sequence-dependent DNA elasticity

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Theory of sequence-dependent DNA elasticity

Bernard D. Colemana)

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

Wilma K. Olsonb) and David Swigonc)

Department of Chemistry and Chemical Biology, Rutgers, The State University of New Jersey,Piscataway, New Jersey 08854

Received 21 October 2002; accepted 17 January 2003The elastic properties of a molecule of duplex DNA are strongly dependent on nucleotide sequence.In the theory developed here the contribution n of the nth base-pair step to the elastic energy isassumed to be given by a function n of six kinematical variables, called tilt, roll, twist, shift, slide,and rise, that describe the relative orientation and displacement of the nth and (n1)th base pairs.The sequence dependence of elastic properties is determined when one specifies the way n dependson the nucleotides of the two base pairs of the nth step. Among the items discussed are the symmetryrelations imposed on n by the complementarity of bases, i.e., of A to T and C to G, the antiparallelnature of the DNA sugarphosphate chains, and the requirement that n be independent of thechoice of the direction of increasing n. Variational equations of mechanical equilibrium are herederived without special assumptions about the form of the functions n, and numerical solutions ofthose equations are shown for illustrative cases in which n is, for each n, a quadratic form and the

DNA forms a closed, 150 base-pair, minicircle that can be called a DNA o-ring because it has anearly circular stress-free configuration. Examples are given of noncircular equilibriumconfigurations of naked DNA o-rings and of cases in which the interaction with ligands induceschanges in configuration that are markedly different from those undergone by a minicircle ofintrinsically straight DNA. When a minicircle of intrinsically straight DNA interacts with anintercalating agent that upon binding to DNA causes a local reduction of intrinsic twist, theconfiguration that minimizes elastic energy depends on the number of intercalated molecules, but isindependent of the spatial distribution of those molecules along the minicircle. In contrast, it isshown here that the configuration and elastic energy of a DNA o-ring can depend strongly on thespatial 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 singlebase-pair step can undergo large deformations with localized untwisting and bending at remotesteps, even when the amount of twist reduction is less than the amount required to inducesupercoiling in rings of intrinsically straight DNA. We here find that the presence in the functionsn of cross-terms coupling twist to roll can amplify the configurational changes induced by localuntwisting to the point where there can be a value of at which a first-order transition occursbetween two distinct stable noncircular configurations with equal elastic energy. 2003 American Institute of Physics. DOI: 10.1063/1.1559690

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 properties,

it is known that the genetic information in DNA determinesnot only the amino acid sequences of encoded proteins andRNA but also the geometry and deformability of DNA at alocal level, i.e., at the level of base-pair steps. Recent ad-vances in structural biochemistry have yielded informationabout the dependence on base-pair sequence of the structure

and elastic response of DNA. In this paper we present amethod of taking such information into account when onecalculates and determines the stability of equilibrium con-figurations of DNA segments subject to preassigned end con-

ditions.In a stress-free state, the local bend, twist, and stretch ofduplex double helical DNA varies from one base-pair stepto another. Some steps are wedge shaped with inclinationi.e., bending strongly dependent on nucleotide compositionand occasionally as large as 5. Moreover, there are stepsthat are sites of over- or undertwisting as large as 3.5, andthere are steps at which the barycenter of a base pair is dis-placed relative to that of its predecessor and gives rise to a

aElectronic mail: [email protected] mail: [email protected] to whom correspondence should be addressed. Electronic mail:

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JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 15 15 APRIL 2003

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local shear that can be as large as 0.65 or 20% of theelevation of a base pair relative to its predecessor.1 6

The chemical architecture of DNA results also in me-chanical couplings between kinematical variables such asthose of Fig. 1 that characterize the relative displacementand orientation of adjacent base pairs. Observed correlationsin the values of the kinematical variables for crystals of DNAoligomers have led to the proposal that such couplingsshould appear as cross-terms in constitutive equations relat-ing the elastic energy or, equivalently, local forces and mo-ments to the kinematical variables.6

Recent research on the derivation of energy functionsthat govern the structure and deformation of individual base-pair steps in crystals of pure DNA and DNAproteincomplexes5 7 makes it possible to develop models for DNAelasticity that take into account a strong dependence of in-trinsic structure and elastic moduli on nucleotide sequenceand have, of necessity, a discrete rather than a continuumnature with constitutive relations showing cross-terms that

couple twisting to bending. In this paper we construct atheory of a general class of discrete models in which theelastic energy of a DNA segment is assumed to be the sumof the interaction energies of adjacent base pairs. For each n,n, the energy resulting from deformation of the nth dimerbase-pair step, is assumed to be given by a function ofn ofsix independent kinematical variables that determine therelative orientation and displacement of the nth and (n1)thbase pairs; these are the three angular variables tilt, roll,twist and the three displacement variables shift, slide, risethat are illustrated schematically in Fig. 1 and are definedprecisely in Sec. II.

The underlying mathematical assumptions of the theory

are stated in Sec. II along with the equations that governconfigurations in mechanical equilibrium. Those equationsare obtained by setting equal to zero the first variation of thetotal energy E, which we take to be a sum of the elasticenergy n

n and the potential energy P of externalforces and moments. Our derivation of the equilibrium equa-tions does not require special assumptions about the form ofthe functions n. The symmetry relations imposed on the n

by the complementarity of adenine to thymine and guanineto cytosine, the antiparallel nature of the DNA sugarphosphate chains, and the requirement that those functionsare independent of the choice of the direction of increasing nare presented at the same level of generality as our deriva-

tions of the equilibrium equations. Also shown in Sec. II arethe forms that the symmetry relations take when n is qua-dratic in appropriate kinematical variables. Details of deriva-tions of the equations of equilibrium and of the symmetryrelations for n are given in Sec. III.

The principal results of the paper are those given inSecs. II and III, namely i the formulation of the theory, iithe analysis of the symmetry restrictions imposed on n by

the structure of commonly found forms of duplex DNA in-cluding the B, A, and C forms, iii the derivation of thevariational equations of equilibrium that requires the attain-ment of a computationally manageable expression for thematrix inverse of the gradient of the orthogonal matrix D i j

n,

relating bases embedded in the nth and (n1)th base pairsto the parameters tilt, roll, and twist see Eq. A1 and Eqs.3.7, 3.93.11, and iv the development of an efficientcomputational method of order O(N), with N the number ofbase pairs for solving the equilibrium equations and deter-mining the stability of computed solutions.

To illustrate the way in which the theory can be used toexplore the influence of a nucleotide sequence on configura-

tions of DNA subject to constraints such as ring closure andligand-induced changes in local structure, we give, in Sec.IV, a few examples in which the theory is employed to cal-culate equilibrium configurations of closed minicircles,which, because they have nearly circular stress-free configu-rations, are called DNA o-rings.8 Discussions are given ofways in which equilibrium configurations of naked DNA andDNA that is interacting with ligands can, under appropriatecircumstances, show a behavior markedly different from thatpredicted by the familiar idealized rod model.9 Among thenew topics treated here is the influence of the cross-termscoupling twist to roll on the configurational changes inducedby local untwisting.

II. CONSTITUTIVE RELATIONS AND VARIATIONALEQUATIONS

In the present theory the base pairs are taken to be flatobjects that are represented by rectangles. The configurationof a DNA segment with N1 base pairs, and hence N basepair steps, is specified by giving, for each n, both the locationx

n of the center of the rectangle Bn that represents the nthbase pair and a right-handed orthonormal triad d1

n , d2n , d3

n

that is embedded in the base pair see Fig. 2. The polygonalcurve C formed by the N line segments connecting the spatial

points x1, . . . ,xN

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

the two strands that make up the duplex molecule. In astrand, the nucleotide bases are laterally attached throughcovalent bonds to a sugarphosphate chain. The chains ofthe two strands are antiparallel in the sense that increasing ncorresponds to advance in the 5 3 direction on one strandand in the 3 5 direction on the other.10

We assume that the elastic energy of a configurationis the sum over n of the energy n of interaction of the nthand (n1)th base pairs, and that n is a function of therelative orientation and displacement of the two neighboringbase pairs, i.e., of the components of the vectors dj

n1 and

FIG. 1. Schematic representation of kinematical variables describing therelative orientation and displacement of successive base pairs. Tilt and rollmeasure local bending, twist measures rotation in the base-pair plane, shiftand slide local shearing, and rise local stretch. For precise definitions, seeRefs. 13, 14, and 17 and Eqs. 2.5a2.7.

7128 J. Chem. Phys., Vol. 118, No. 15, 15 April 2003 Coleman, Olson, and Swigon



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rnx

n1x

n with respect to the basis din . As the 33 ma-

trix D i jn

with entries D i jn

din

"djn1

is orthogonal, it can beparametrized by a triplet (1n ,2

n ,3n) of angles, with 1

n thetilt, 2

n the roll, and 3n the twist ofBn1 relative to Bn.

Hence, in the formula

n1

N

n, 2.1

the elastic energy n of the nth base-pair step is given by aconstitutive relation of the type

nn1n ,2

n ,3n ,r1

n ,r2n ,r3

n, rindi

n"r

n. 2.2

It is often useful to write this relation in the form

11

nn1n ,2

n ,3n ,1

n ,2n ,3

n, 2.3

where the displacement variables 1n ,2

n ,3n , called shift,

slide, and rise are given by coordinate transformation func-tions, k , that are independent ofn:

knk1

n ,2n ,3

n ,r1n ,r2

n ,r3n . 2.4

As shown in Fig. 2, the vectors din are defined so that d3

n

is perpendicular to Bn with d3nrn0; d2

n is parallel to thelong edges ofBn and points toward the short edge contain-ing the corner ofBn bonded to a deoxyribose group in the

sugarphosphate chain for which n increases in the 5 3direction; d1n is parallel to the short edges ofBn and points

toward the major groove of the DNA, i.e., d1nd2

nd3

n .12

Each drawing in Fig. 1 illustrates one of the kinematicalvariables 1

n ,2n ,3

n ,1n ,2

n ,3n for a case in which that vari-

able has a positive value and the others with the exceptionof 3

n) are set equal to zero. That figure is intended to besuggestive rather than precise. To give a precise definition ofi

n and in for cases in which several of these variables are

not zero, we follow a procedure introduced by Zhurkin,Lysov, and Ivanov13 and further developed by El Hassan andCalladine14 in which one employs the Euler-angle systemn, n, n for which15

D i jndi

n"dj

n1Zi j

n Ykl n Zl j

n, 2.5a

Yi jcos 0 sin

0 1 0

sin 0 cos , 2.5b

Zi jcos sin 0

sin cos 0

0 0 1 , 2.5cand defines 1

n ,2n ,3

n by the relations

n1

23

nn, n

1

23

nn, 2.6a

n1n 22

n2, tan n1

n

2n

. 2.6b

As in Ref. 14, we here use Eqs. 2.52.6 to define in in

terms ofrin and in :

inZi j

nYjk12

nZkl n rl

n . 2.7

This last equation renders explicit the functions k in Eq.2.4 and implies that the numbers i

n are the components ofr

n with respect to an orthonormal basis d in , which is called

the mid-basis for step n and is defined by the relation

din"dj

nZik

n Ykl 12

n Zl jn . 2.8

In Sec. III i.e., in the discussion containing Eqs. 3.4 and3.8 we render precise the following remark: The present

definition ofin and in is such that, for each configuration ofa DNA segment, a change in the choice of the direction ofincreasing n leaves 2 ,3 ,2 ,3 invariant but changes thesign of1 and 1 .

Several computer programs have been developed to re-duce atomic coordinate data of the type available for DNAcrystal structures in the Nucleic Acid Database16 to vari-ables i

n , in , which, like those just defined, are compatible

with a convention17 known as the Cambridge Accord of1988. For a survey see Ref. 18.

The function n of Eq. 2.3 depends on which nucle-otide bases are in the nth and (n1)th base pairs. To discuss

the restrictions imposed on n

by the complementarity of Ato T and G to C, the antiparallel directions of the sugarphosphate chains, and the assumption that n is independentof the composition of base pairs other than the nth and (n1)th, let us write XY for n, where XY, with X in the nthand Y in the (n1)th base pair, is a sequence of two nucle-otide bases attached to the chain for which n increases in the5 3 direction. In this notation, for the complement of thebase X, one writes X i.e., AT,GC). The complement ofa sequence of bases is the sequence of complementary basesin the reverse order: the complement of XY is YX e.g., thecomplement of AC is GT, because X and Y are attached atthe locations n and n1 to the chain for which n decreases

FIG. 2. Drawing of the model employed for a base-pair step showing thevectors rn, di

n , djn1. Each nucleotide base of a pair is covalently bonded at

its darkened corner to one of the two sugarphosphate chains. The arrowsgive the 53 directions of those chains. The gray-shaded long edges are inthe minor groove of the DNA.

7129J. Chem. Phys., Vol. 118, No. 15, 15 April 2003 Theory of sequence-dependent DNA elasticity



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in the 5 3 direction. In Sec. III it is shown that when in ,

in are defined as here, each function XY determines its

complementary function YX by the relation

XY1n ,2

n ,3n ,1

n ,2n ,3

n

YX 1

n ,2n ,3

n ,1n ,2

n ,3n . 2.9

It follows that, if XY is self-complementary, i.e., if XYYX, then XY is an even function of the pair (1

n ,1n) for

each fixed value of the quadruple (2n ,3

n ,2n ,3

n).We define an equilibrium configuration to be one for

which the first variation of the total energy, i.e.,

EP, 2.10

vanishes for all variations in configuration that are admis-sible in the sense that they are compatible with the imposedconstraints. Here P is the potential energy of external forcesand moments. The resulting variational equations, which arederived in Sec. III, can be written as

fnf

n1

n, mnmn1fnr nn 2nN,2.11

where, for each n, fn and mn are the force and moment thatthe (n1) th base pair exerts on the nth base pair and n andn are the external force and moment acting on the nth basepair.19 The vectors fn and mn are the analogs of the vectorsf(s*) and m(s*) that, in Kirchhoffs theory of elastic rods,are interpreted as the resultant force and moment of the con-tact forces i.e., Piola stresses exerted on the cross sectionwith the arc-length coordinate s* by the material with arc-length coordinates ss*. The components of fn and mn

with respect to the local basis din are given by the relations

fin

n

rin

, m in i j

nn

jn

. 2.12

The numbers i jn form a 33 matrix i j

n

i j(1n ,2

n ,3n) that depends on the definitions employed

for the kinematical variables in . When those variables are as

in Eqs. 2.52.6, i jn

is given by Eq. A1. The chainrule permits us to write Eqs. 2.12 in the forms

fin

n

jn

j

rin

, m in i j

n nj

n

n

kn

k

jn , 2.13

in which n and k are as in Eqs. 2.3 and 2.4. When in

and in are defined by Eqs. 2.52.8, for j /ri

n andk/j

n , we have

j

rinZjk

nYkl 12 n Zli n, 2.14

k

jnj kl

nl

n , 2.15

with jkln as in Eqs. A2A4.

If a DNA segment with N1 base pairs is subject tostrong anchoring end conditions at both ends, then ( x1,di

1)and (xN1,di

N1) have preassigned values. A closed or cir-cularized DNA segment of N base pairs bp can be con-sidered an N1 bp segment that is subject to the constraintthat its terminal base pairs coincide, i.e., have the same lo-cation and orientation, and hence

x1x

N1

, di1di

N1

i1,2,3 . 2.16A necessary condition for an equilibrium configuration

to be stable is that there be no admissible variation in con-figuration for which 2E, the second variation ofE, is nega-tive. As a configuration is described in the present theory bygiving a finite number of variables, namely the 6N numbers,1

n ,2n ,3

n ,1n ,2

n ,3n , we can say that a sufficient condition

for stability is that 2E be positive for all admissible varia-tions. A configuration that is stable but does not give a globalminimum to E can be called metastable.

For calculations of the type reported in Sec. IV, n

n0, i.e., P is set equal to zero.20 The Eqs. 2.11 withf

n

andm

n

given by Eqs. 2.12 then are a system E of 6N6 nonlinear algebraic equations for the unknown coordi-nates 1

n ,2n ,3

n ,1n ,2

n ,3n . That system is weakly coupled

in the sense that for each n, 2 nN, Eqs. 2.11 are a set ofsix equations that are solved numerically using a NewtonRaphson procedure to obtain (i

n ,in) for given

(in1,i

n1). This allows one to solve E efficiently usingO(N) computational steps by a recursive algorithm inwhich, for each n, (i

n ,in) is obtained as a function of

(i1,i

1). Equilibrium configurations of segments subject tospecified end conditions are calculated using an iterativescheme in which (i

1,i1) is repeatedly adjusted until the ap-

propriate relations, e.g., Eq. 2.16, hold.

The case in which n is given by a quadratic form

To describe quadratic energy functions of the type em-ployed for calculations presented in Sec. III, we write i

n andi

n for the values ofin and i

n in a stress-free state with n

0, and put

ini

ni

n , ini

ni

n . 2.17

When in and i

n are small, the assumption that n isthree-times differentiable implies that, to within an errorO(3) with

maxi1,2,3

in,in, 2.18

n can be taken to be a quadratic function,

n12 Fi j

ni

n jnG i j

ni

n jn

12 Hi j

n i

n jn , 2.19

in which Fi jn , G i j

n , and Hi jn are constants with Fi j

nFji

n ,Hi j

nHji

n , (i,j1,2,3).21 If one wishes, one may write theright-hand side of Eq. 2.19 as a quadratic form in(1

n ,2n ,3

n ,1n ,2

n ,3n), where the dimensionless

numbers,




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ini

n/3n , 2.20

are such that 3n can be interpreted as a stretching strain and

1n and 2

n as shearing strains.It follows from Eq. 2.9 that if we replace Fi j

n by Fi jXY ,

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

iXYQ i j

jYX , i

XYQ i jj

YX , 2.21a

Zi jXYQ ikZkl

YXQ l j , 2.21b

where

Q11Q22Q331, Q i j0, when ij . 2.22

Equations 2.21 imply that, when XYAT, GC, TA, orCG,22

F12XYF13

XY0, H12

XYH13

XY0, 2.23a

G12

XYG

13

XYG

21

XYG

31

XY0,

1

XY

1

XY0. 2.23b

Estimates from x-ray crystal structure data of values ofthe elastic moduli Fi j

XY , G i jXY , Hi j

XY and the intrinsic param-eters i

XY ,iXY for each distinct combination of a base pair

and its successor5,6 confirm that these moduli and intrinsicparameters are strongly dependent on the nucleotide compo-sition. If we focus attention on the angular variables i

XY , wecan give as follows a rough summary of the nature of thesequence dependence of moduli and intrinsic parameters ofthe B form of DNA.23 It follows from Eqs. 2.21 that the

intrinsic tilt 1XY obeys the relation 1

XY1

YX , and theavailable data indicate that its largest value is 1.5. The

data indicate that the intrinsic roll

2

XY

is positive with itslargest value 5 and thus more than three times the largestvalue of1

XY . The smallest value of2XY is 0.5. The ratio

of the tilt modulus F11XY to the roll modulus F22

XY varies in therange 1 to 5, and the ratio of the twist modulus F33

XY tothe harmonic mean AXY2/(1/F11

XY)(1/F22XY) of the tilt

and roll moduli varies in the range 0.5 to 1.8. The twistroll coupling modulus F23 , in units ofkT/deg

2, varies in therange 0.00 to 0.03.

In Ref. 24, Gonzales and Maddocks develop a theorythat relates the moduli Fi j

n ,G i jn ,Hi j

n and the intrinsic param-eters i

n ,in to statistical averages and correlations between

in and i

n for ensembles of configurations obtainable from

all-atom molecular dynamics simulations.

III. DERIVATION OF THE PRINCIPAL RESULTS

A. Symmetry relations

The underlying assumption behind Eq. 2.2 is that n inEq. 2.1 is given by a function n ofrn and the two right-handed orthonormal triads d1

n ,d2n ,d3

n and d1n1,d2

n1,d3n1;

thus

nndin ,dj

n1,rn i,j1,2,3 . 3.1

The principle of material frame indifference requires that n

obey, for each proper orthogonal second-order tensor J, therelation

ndin ,dj

n1,rn nJdin,Jdj

n1,Jrn , 3.2

as an identity. It follows from this last relation and Cauchysrepresentation theorem for scalar-valued isotropic functions

of lists of vectors that there is a function such that

25

nnD i jn ,ri

n; 3.3

this relation is equivalent to Eq. 2.2.Now, the functions n and n are assumed to be deter-

mined by the ordered pair XY of nucleotide bases locatedat the nth and (n1)th positions on the sugarphosphatechain for which n increases in the 5 3 direction. Hence,for n and n we may write XY and XY. To derive restric-tions that the antiparallel arrangement of the oriented sugarphosphate chains of duplex DNA impose on these functions,we note that, as the complements of A and G are AT and

GC, the 16 ordered pairs XY separate into two classes.One class contains the four self-complementary pairs AT,

GC, TA, CG, each of which, because it has YX , obeys therelation XYYX . The remaining class of twelve is made upof six sets of complementary pairs i.e., pairs X1 Y1 ,X2Y2with X2Y2Y 1X 1), namely, AA,TT, AG,CT, AC,GT,GA,TC, GG,CC, TG,CA.

We consider now the unique transformation thatchanges the direction of increasing n while leaving invariantthe orientation and spatial position of base pairs. Under sucha transformation di

n ,djn1,rn is taken into *di

m ,*djm1,*r

m,where mNn1 and

*d1md1

n1, *d1m1d1

n , 3.4a

*d2md2

n1, *d2m1d2

n , 3.4b

*d3md3

n1, *d3m1d3

n , 3.4c

*rm r

n, 3.4d

n remains unchanged, i.e., *mn, because the rect-

angles Bn, although they are renumbered in reverse order,keep their previous spatial positions and orientations. Thetransformation takes the vectors d2

n ,d2n1, into vectors

*d2m

,*d2m1

, that again point toward the short edges thathave corners attached to the chain for which n increases inthe 5 3 direction which is the one for which n decreasedin that direction before the transformation. As the transfor-mation replaces XY with its complement YX while leaving

n invariant, the functions XY and YX must obey the re-

lation, XY(din ,dj

n1,rn)YX (*di

m ,*djm1,*r

m), i.e.,

XYd1n ,d2

n ,d3n ,d1

n1,d2n1,d3

n1,rn

YXd1

n1,d2n1,d3

n1,d1n,d2

n,d3n,rn, 3.5

or, equivalently,




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XYD i jn ,ri

n YX Q ikD lk

nQjl ,Q ilD kl

nri

n , 3.6

where Q i j is as in Eq. 2.22. If the pair XY is self-complementary, i.e., is AT, GC, TA, or CG, then YXXY,and Eq. 3.5 becomes an identity for the function XY.

As the tangent spaces to the manifold M of proper or-thogonal 33 matrices are of dimension 3, each point ofMhas a neighborhood in which M has a smooth three-

coordinate parametrization of the type discussed in Sec. II,and we can write

D i jnD i j1

n ,2n ,3

n, 3.7

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

n and in defined in Eqs.

2.52.7, the Eqs. 3.4 take the form

*1m ,*2

m ,*3m ,*1

m ,*2m ,*3

m

1n ,2

n ,3n ,1

n ,2n ,3

n , 3.8

and hence Eq. 3.5 is equivalent to the symmetry relationEq. 2.9.

For use below, we note that because the three matrices lVi j

n , (l1,2,3), with components

lVi jnDjk

D ik

ln

, 3.9

are skew in the sense that lVi jn lVji

n for all i ,j , there arenumbers kl

n (k,l1,2,3), such that26

*Vi jni jkkl

n . 3.10

The local invertibility of the function D i j implies that thecolumns of the matrix kl

n form a not necessarily orthogo-

nal basis for the tangent space ofM at the point with co-ordinates 1

n ,2n ,3

n , and hence kln

has a matrix inversekl

n1. For the transpose of kl

n1 we write kl

n:

kin k j

ni j . 3.11

B. Equations of equilibrium

A configuration Z of a DNA segment with N1 basepairs is specified by giving the (N1)-tuple(d

1

n ,d2

n ,d3

n ,xn)n

1

N1, of ordered sets (d1

n ,d2

n ,d3

n ,xn) inwhich d1

n ,d2n ,d3

n is a right-handed orthonormal basis and xn apoint in space. The total energy Eof the segment is given byEq. 2.10, in which PP(Z) is the potential energy of theexternal forces and moments acting on the segment. A con-figuration Z# is said to be in equilibrium if for each one-parameter family,

sZ s , Z0 Z#, s, 3.12

of configurations compatible with the imposed constraintse.g., the strong anchoring end conditions or the closure con-ditions of Eq. 2.16 there holds E(0)0, i.e., in view ofEqs. 2.1 and 2.10,

n1

N

n0P00; 3.13

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

nition, we employ Eq. 2.2, which yields

nn

in in

n

rin rin , 3.14

where

rin

din"r

n , 3.15

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

D ik

ln

ln

D i jn

din"dj

n1. 3.16

As the vectors d1n ,d2

n ,d3n are orthonormal, there is a vector

wn

such that

din

wndin . 3.17

By Eqs. 3.93.11, 3.15, and 3.17,

rindi

n"rn wnrn , 3.18a

inji

ndj

n"wn1wn, 3.18b

Pn1

N

dinPdi

n "wn Pxn xn , 3.18cand, if we put

fnfi

ndi

n , fin

n

rin

, 3.19a

mnm i

ndi

n , m in i j

nn

jn

, 3.19b

n

P

xn, 3.19c

ndin

P

din

, 3.19d

and write vn(xn), then Eqs. 3.14 and 3.18c, respec-tively, can be cast into the forms

nmn"wn1wn fn"vn1vnwnrn ,3.20

Pn1

N

n"wnn"vn . 3.21




7/14

In the last two equations, although end conditions affect thechoice of the vectors w1, v1, wN1, and vN1, the vectorsw

n and vn can be selected arbitrarily for n2, . . . ,N.Hence, the relation 3.13 holds for the configuration Z# ifand only if that configuration obeys the following variationalequations:

fnf

n1

n, mnmn1fnrnn

2nN, 3.22

and natural end conditions,

m1f1r11"w1mNN1"wN1f11"v1

fNN1"vN10. 3.23

The relations 3.22 are the analogs in the present theoryof the equations of balance of moments and forces in Kirch-hoffs continuum theory of elastic rods2730 and several re-cent generalization of that theory vid., e.g., the treatise ofAntman,31 Sec. VIII.

An equilibrium configuration Z# is stable if, for each

one-parameter family of configurations that obeys 3.12 andis compatible with the imposed constraints, there holds, inaddition to Eq. 3.13,

0E0 n1

N

n 0P 0. 3.24

In practice, to verify stability we employ Eqs. 3.133.18to express E(0) as a quadratic form Q on the6(N1)-dimensional space of the components ofw1,v1, . . . ,wN1,vN1 with respect to one fixed basis, e.g.,d1

1,d21,d3

1. If, the smallest proper number ofQ, is less thanzero, the configuration Z# is unstable. If 0, Z is stable.

This test was used to verify statements made in Sec. IV aboutthe stability of calculated configurations.

IV. EXAMPLES OF EQUILIBRIUM CONFIGURATIONS

Much of the research on DNA elasticity employs an ide-alized elastic rod model, based on the assumption that a du-plex DNA molecule is an intrinsically straight, homoge-neous, elastic rod obeying the special case of Kirchhoffsgeneral theory in which there are just two elastic constants: atwisting modulus and a bending modulus. Several recentstudies e.g., Refs. 32 40 of the influence of intrinsic cur-vature on the elastic response of DNA are based on Kirch-

hoffs general theory41 in which a rod has two bendingmoduli and, in addition, the possible presence of intrinsiccurvature and torsion is taken into account. The constitutiverelation for the elastic energy of a rod obeying that theorydoes not contain cross-terms directly coupling the twistingmoment to the local curvature. It is known that in a materi-ally homogeneous region of a rod with intrinsic curvature thedensity of excess twist is, in general, not uniform in equilib-rium. In Ref. 33 it is shown that the equations of equilibriumappropriate to Kirchhoffs theory imply that when intrinsiccurvature is present the derivative of twist with respect todistance along the rod axis is a function of local curvature.See also the papers of Bauer, Lund, and White32,35,37 and

Garrivier and Fourcade40 and the calculations therein of theeffect of intrinsic curvature on the shape and twist densityprofile of o-rings32,35,37 and plectonemic loops.40

Here we give examples of calculated equilibrium con-figurations of DNA minicircles obeying the theory presentedin Sec. II. In order to obtain insight into the implication ofour theory, we deliberately treat cases in which known re-sults for intrinsically curved rods obeying Kirchhoffs theory

can yield physical intuition as to what to expect. We there-fore treat minicircles that may be called DNA o-rings be-cause they are intrinsically bent in such a way that in stress-free configurations their axial curves are nearly circular. Thehelical nature of DNA makes it impossible to form a stress-free o-ring with the intrinsic kinematical variables the sameat each base-pair step, for if they were the same, the bend-ings occurring in the two halves of each helical repeat length,being then of equal magnitudes, but in opposite directionswould cancel. For this reason, the minicircles we treat arecomposed of 10 bp helical repeat units in which 5 bp form anintrinsically straight segment and the remaining 5 bp form anintrinsically curved segment.42 The highly curved sequences

found in the kinetoplast DNA from the TrypanosomatidaeCrithidia fasiculata and Leishmania tarentolae have on onestrand stretches of four to six adenine bases that are sepa-rated by stretches of comparable length rich in guanine andcytosine bases4346 .

In analogy with the familiar assumptions in the con-tinuum theories of DNA elasticity, in our examples n istaken to be a quadratic function of the kinematical variables.Results are presented for two types of DNA minicircles: Forthose of type I, the matrix of the quadratic form characteriz-ing n is diagonal for each n. For those of type II, eachhelical repeat length contains five base-pair steps for whichthe off-diagonal modulus associated with coupling betweenroll and twist is not zero and hence the minicircle is expectedto show chiral phenomena not predicted by Kirchhoffs clas-sical theory of rods. The configurations shown were obtainedby a solution of the equations of mechanical equilibrium.

The minicircles we consider have N150 bp and arecomposed of 15 repetitions of a 10 bp unit in which the first5 bp have

1n ,2

n ,3n 0,0,36 , 1

n ,2n ,3

n 0,0,3.4 ,4.1

and hence form an ideal WatsonCrick DNA segment while

the remaining 5 bp have

47

1n ,2

n ,3n

0,7.413,35.568 ,4.2

1n ,2

n ,3n 0,0,3.4 ,

and hence form an intrinsically curved segment. For thecalculations we report, the kinematical variables1

n ,2n ,3

n ,1n ,2

n ,3n that are schematically illustrated in Fig.

1 are taken to be those defined in Ref. 14 and Eqs. 2.52.7. A minicircle with this choice of N and the indicatedparameters i

n ,in has a linking number L of 15, and in its

stress-free configuration its axial curve C has the appearance




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of a polygon of 15 sides with rounded vertices that is suffi-ciently close to a circle to permit us to refer to the minicircleas an o-ring. For both I and II, the moduli G i j

n and Hi jn are

here taken to be independent of n with G i j0 for all i,j ,with Hi j0 for ij , and with H11 ,H22 ,H33 large enough tosuppress deformations that change the displacement vari-ables i

n . The two minicircles differ in the value of the cou-pling coefficients Fi j

n , ij ; for both, F11 ,F22 ,F33 are inde-pendent of n and such that F11F224.2710

2kT/deg2;the ratio

F33/F11 , 4.3

was set equal to 0.7 for certain calculations and 1.4 forothers.48

A. Minicircle I

Minicircle I is such that, for ij and n1, . . . ,150,

Fi jn0, 4.4

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

Even when this minicircle is kept free of bound proteinsand other chemical agents, it has multiple equilibrium con-figurations, several of which each with L15) are shown inFig. 3, where 0.7. In that figure, A is the stress-free,nearly circular, minimum energy configuration. The evertedring configuration B has the axial curve close to a circle asdoes A and may be described, to a good approximation, as aconfiguration obtained from A by rotating each base pair inits plane by one half-turn as in the motion called thesmoke-ring rotation. It has been known for a long time49

that everted ring configurations of o-rings are unstable. Thechair configuration C is familiar from experience withelastic rings and was treated by Charitat and Fourcade36 us-

ing Kirchhoffs theory. The stability of chair configurationsdepends on . We find that i when 1, as it is in thepresent example, the configuration C is metastable; ii C isunstable when 1; and iii when 1, there is a one-parameter family of distinct chair configurations of equalenergies.50 The configurations D and E, which to our knowl-edge have not been discussed before, have less symmetrythan C and are unstable; in the present example, D lies at amountain pass for a transition path from C to the globallystable configuration A. The item labeled F, although it corre-sponds to a solution of Eqs. 2.11, 2.12, and 2.17, is nota physically admissible configuration, because it does notobey the condition of impenetrability of base pairs.

Figure 4 contains graphs of excess twist 3n versus n

for selected configurations of Fig. 3 and shows that although3

n is zero for the globally stable circular configuration Aand approximately zero for the unstable circular configura-tion B, such is not the case for the chair configuration C. Forthat configuration, 3

n vanishes near the two apexes ofthe chair labeled and , and attains its remarkably largemaximum near the centers of the sides of the chair and. In Fig. 4 and also in Figs. 6, 8, and 10 it can be seen thatthe excess twist 3

n is constant within each of the intrinsi-cally straight segments in which Eq. 4.1 holds. This localconstancy is a consequence of the fact that in the intrinsically

straight segments of Minicircle I the quadratic constitutivefunctions n are free of cross-terms coupling twist to otherkinematical variables. In the intrinsically curved segmentsobeying Eq. 4.2, regardless of whether cross-terms arepresent in n, 3

n is generally nonmonotone, and is con-stant only if the o-ring is stress-free. Arguments of the typethat led to Eq. 28 of Ref. 33 can be employed here to showthat if 1

n in an intrinsically curved segment changes signbetween n and n1, then 3

n has an extremum at n orn1.

It has been known for a long time that one can inducenegative supercoiling in closed DNA by reacting it with anintercalating agent, such as ethidium bromide EtBr, that

FIG. 3. Equilibrium configurations of the 150 bp o-ring, Minicircle I, for thecase 0.7. Shown are the stress-free configuration A that gives a globalminimum to , the unstable everted ring configuration B, the metastablechair configuration, and the unstable configurations D and E; F is an un-stable solution of the equilibrium equations that is incompatible with theassumed impenetrability of DNA. Configuration D lies at a saddle point inthe energy landscape for configurations of the o-ring. Here, as in Figs. 5, 7,9, and 11, configurations are drawn as tubes of diameter 20 . The elastic

energy is given in units ofkT.

FIG. 4. The excess twist 3n in degrees as a function of n for selected

configurations of Fig. 3. The points on the graph for the chair configurationC that are labeled , , , have the geometric locations indicated in Fig. 3and correspond to the apexes , of the chair and the midpoints , of itssides.




9/14

binds to DNA between base pairs and causes a local reduc-tion of the intrinsic twist of the double helix.51 If, whilemaintaining the other assumptions we have made aboutMinicircle I, we were to assume that Eqs. 4.1 hold for 1n150, which would make the minicircle not only homo-geneous in n but also one that was formed from intrinsicallystraight DNA with 1

n2n0), 52 then the equilibrium

shape and elastic energy of the minicircle would depend on

the number of molecules of EtBr intercalated in it but wouldbe essentially independent of their distribution along C.53 Wehave found, however, that, because Minicircle I is an o-ringformed from DNA that is intrinsically curved in accord withEqs. 4.1 and 4.2, its equilibrium shape and elastic energyare strongly dependent on both the number and the distribu-tion along C of the bound molecules.54

Figure 5 contains results of calculations in which tenidentical molecules are assumed to be intercalated at speci-fied base-pair steps of Minicircle I, with each such moleculereducing the intrinsic twist 3

n by 30 and doubling the in-trinsic rise 3

n at the step at which it is intercalated, as is thecase for EtBr.55 In the discussion that follows we assume that

the intercalating agent obeys the principle of nearest neigh-bor exclusion, requiring that two bound molecules be sepa-rated by at least one base-pair step. There is evidence to theeffect that this exclusion principle does hold for small inter-calating ligands.56,57 If at any given instant the distributionof the ten intercalated molecules is spatially uniform,Minicircle I retains its circular configuration. When thosemolecules are distributed in such a way that they occupyevery other base pair step within two antipodally placed sub-segments of equal length, the o-ring folds into one of thefigure-8 configurations B and B*. When the bound mol-ecules are concentrated in a short subsegment but yet obeythe principle of nearest neighbor exclusion, the o-ring again

deforms out of plane and, in the case of1.4, folds into aconfiguration of the clamshell type, here labeled C*. Inthe present case we obtain the result that, for both values of the distribution into two equal and antipodally placed sub-segments, seen in B and B*, is the distribution that mini-mizes over all distributions of ten intercalatedmolecules.58

The values of given in Fig. 5 tell us that, if we assumethat EtBr binding is independent of nucleotide composition,in the case of dynamical equilibrium the most probable spa-tial distribution of bound molecules will be close to that seenin B and B*. Of course, the number of bound molecules is afluctuating quantity that depends on the concentration ofEtBr and DNA in solution. A complete analysis of curvature-

induced binding cooperativity of EtBr molecules leading tothe prediction of Scatchard plots of EtBr binding activitywould be a computationally intensive task that has not yetbeen undertaken. However, the preliminary results shownhere do suggest that, for DNA subject to appropriate endconditions, such as those of the Eqs. 2.16 that characterizering closure, the spatial distribution of EtBr can be expectedto be sensitive to assumed intrinsic curvature.

Under the present assumptions about the o-ring and theintercalating agent, although 3

n is positive for all n whenthe intercalating molecules are equally spaced along thelength of the minicircle, if those molecules are concentratedinto a short enough subsegment S, as they are in the mini-

mum energy configurations C and C*, 3n

becomes nega-tive in a neighborhood of the antipode of the midpoint SseeFig. 6.

To investigate the effect of the localization of changes inthe intrinsic twist, we have calculated configurations forcases in which a single molecule of an agent is bound to ano-ring at one base-pair step, say that with n1, and reducesthe intrinsic twist of the DNA at that site by a large positiveamount , i.e., induces the transformation 3

13

1 while

leaving other intrinsic variables unchanged. In a study of theeffect of intrinsic curvature on configurations of DNAminicircles obeying Kirchhoffs theory in which, as is thecase for Minicircle I, the constitutive relations do not contain

FIG. 5. Influence of the spatial distribution of intercalated molecules on theminimum elastic energy configuration of Minicircle I. Shown are calculatedresults for cases in which ten molecules are intercalated into the minicirclewith each inducing the local change 3

n3

n30, 3

n3

n3.4 6.8 .

The sites of binding are shown as dark bands. In configurations A and A*,these sites are equally spaced along the axial curve; in B and B * theyoccupy every other base-pair step in two antipodally placed 11 bp subseg-ments of the minicircle; in C and C* they occupy every other base-pair step

in a single 21 bp subsegment. As indicated, 0.7 for A, B, C, and 1.4for A*, B*, C*.

FIG. 6. 3n as a function of n for the configurations of Fig. 5 for which

0.7. The symbols mark the sites at which the intercalating agent isbound.




10/14

cross-terms coupling curvature to the twisting moment.Bauer, Lund, and White32,35,37 found that a reduction of theexcess link by cutting at a single cross-section which isroughly equivalent to an increase in intrinsic twist at a singlebase pair step can lead to remarkably large deformationseven when the magnitude of that reduction is less than 1 i.e.,

360) and hence less than the amount required to in-duce supercoiling in intrinsically straight rods.59 Results forMinicircle I with equal to 1.4 and for rather extreme valuesof are shown in Figs. 7 and 8.60 As seen in Fig. 7, as increases the o-ring folds up, with the folding becoming ap-preciable for near 180, and at sufficiently high theo-ring attains a clamshell configuration. When, as here, thechange of intrinsic twist is confined to one base-pair step,configurations but not their energies are approximately in-dependent of. In Fig. 8 one sees that, as expected from theresults shown in Fig. 6, the excess twist is negative in aneighborhood of the antipode to the binding site.61 As theconstitutive equation for the energy of Minicircle I does notcontain off-diagonal terms directly coupling roll or tilt totwist, if the twisting agent were one that raises 3

1 i.e., thathas 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.

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

In the case of Minicircle II, for those values of n at

which Eq. 4.1 holds,

F12n ,F23

n ,F13n 0,0,0 , 4.5

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

F12n ,F23

n ,F13n 0, 45 F22

n ,0. 4.6

Thus, in each repeated 10 bp unit, the first 5 bp form anintrinsically straight segment and do not show coupling be-tween kinematical variables, while the remaining 5 bp forman intrinsically curved segment and are such that the twistroll coupling modulus F23

n is positive and hence raising the

roll tends to lower the twist, i.e., an increase in 2n

results ina twisting moment that tends to decrease 3n . The positivity

of F23n is compatible with the coupling of parameters ob-

served in high-resolution DNA crystal structures.4,6 The cal-culations shown here for Minicircle II Figs. 912 weredone with 1.4.

FIG. 7. Configurations of Minicircle I that minimize when the minicircleis bound to an untwisting agent supposed capable of lowering 3

1 by 180 in

the case of A*, 240 in the case of B*, and 300 which is equivalent to thetotal untwisting induced by the intercalation of ten molecules of EtBr in thecase of C*. The binding site for the agent is indicated by a triangle. Here, asin all subsequent figures, 1.4.

FIG. 8. Graphs of3n vs n for the configurations of Fig. 7. The untwisting

agent is bound at the base-pair step marked by for which n1.

FIG. 9. The configurations of Minicircle II that minimize when 31 is

changed to 31 with at the indicated values.

FIG. 10. Graphs of3n vs n for the configuration B* of Minicircle I see

Fig. 7 and the configuration B of Minicircle II see Fig. 9. The untwistingagent is bound at the base-pair step marked by , for which n1.




11/14

Although Minicircles I and II both have minimum en-ergy configurations that are stress-free and nearly circular,the effect of a twisting agent on II is dependent on whetherthe agent reduces or increases the intrinsic twist of DNA. If,subject to the transformation 3

13

1, Minicircle II like

I shows appreciable folding when exceeds 180. How-ever, as one sees in Fig. 9, in contrast to what the theoryyields for I, the minimum energy configurations of II for and are not mirror images. The graphs of 3

n versus nin Fig. 10 for the configurations of Minicircles I and II with

240 tell us that in the 5 bp subsegments of II in whichcoupling is present 3n is much lower than it is in the cor-

responding subsegments of I, which is a consequence of thefact that in those subsegments 2

n i.e., roll is larger than itsstress-free value. The presence of coupling here induces un-twisting by as much as 2.7, with the most untwisting at-tained at the base-pair steps with n49 and 106, which arelocated roughly 1/3 of the circumference away from thebinding site of the untwisting agent.

The graphs of Fig. 11 showing the dependence on ofthe writhe W, which is a measure of the folding of theo-ring,62,63 illustrate well the difference in the response of thetwo minicircles to the binding of a twisting agent. While the

response of I is nearly antisymmetric, in the sense that theconfigurations corresponding to andare close to mirror

images and hence W()2W(0)W(), the response ofII is chiral and such that, for negative i.e., for an increasein intrinsic twist, W changes monotonically with , but foreach positive in the interval 211.9220.7 there arethree equilibrium configurations, with two stable and oneunstable. When 215.0, the two stable configurations la-beled S0,S1) have equal elastic energy Fig. 12. As seen inFig. 13, the configurations with preassigned (S0) thatgive a global minimum to form a continuous one-

parameter family of configurations that terminates at S0 andis not connected to the family of minimum energy configu-rations with (S1). At 215.0 infinitesimal changesin result in first-order transitions between the stable equi-librium states S0 and S1. The unstable configuration U with(U)(S0)(S1) lies at a mountain pass for these tran-sitions. The graph of versus seen in Fig. 13 tells us thatthere is an essentially negligible energy barrier, namely 0.13kT, for the transitions S0S1 and S1S0 between theopen configuration S0 and the partially folded configu-rations S1, both of which give to its global minimum inthe class of all configurations that are accessible withoutchanges in .

FIG. 11. Graphs of the writhe Wvs for minimum energy configurations ofMinicircle I light curve and Minicircle II heavy curve bound to a twistingagent. In both cases 1.4. The points A, B, A, B, C correspondto the configurations of Fig. 9, and the points S0, S1, U to the configurationsof Fig. 12. The configurations at which has its global minimum for thespecified lie in the regions drawn as solid curves light or heavy; there areconfigurations on the heavy dashed curve that are not even local minimizersof see Figs. 12 and 13. The marks the largest for which theminimum energy configuration of each minicircle is free from self-contact.

FIG. 12. The stable configurations S0, S1 and the unstable configuration Uof Minicircle II of Fig. 11 with 215.0.

FIG. 13. Graphs of W vs and vs for equilibrium configurations ofMinicircle II with in a neighborhood of 215.0. S0 and S1 are points ofexchange of global stability, U determines the energy barrier for the transi-tion S0S1. Stability of configurations is indicated as follows: globallystable, -- locally stable, unstable.




12/14

ACKNOWLEDGMENTS

The authors thank A. R. Srinivasan for discussions andmaterial help in the design of the here employed naturallydiscrete model for a DNA o-ring. This research was sup-ported by the National Science Foundation under Grants No.

DMS-97-05016 and No. DMS-02-02668 and by the U.S.Public Health Service under Grant No. GM34809.

APPENDIX: EXPLICIT EXPRESSIONS FOR MATRICESIN EQS. 2.122.15

For the matrix i jn

of Eq. 2.12, the Eqs. 2.52.6and 3.93.11 yield

i jn

1n

nsin n

2n cos n

2 tan 12 n

2n

nsin n

1n cos n

2 tan 12 ntan 12 n cos n

1n

ncos n

2n sin n

2 tan 12 n2

n

ncos n

1n sin n

2 tan 12 ntan 12 n sin n

2n

2

1n

21

. A1For each j, the numbers jkl

n of Eq. 2.15 are the compo-nents of a skew matrix jkl

n. When j1,

112n

2n1cos 12 n

n 2,

1

13

n

1n2

n2 sin 12 nn2n 3

,

123n

1

2 2

n

n

2 2sin 12 nn2n

. A2

When j2,

212n

1ncos 12 n 1

n

2,

213n 1

n

n

2 n2 sin 12 n 2n

1

2,

223n

1n2

nn2 sin 12 n 2n 3

. A3

When j3,

312n

1

2cos 12 n , 313n

1n

2nsin 12 n ,

323n

2n

2nsin 12 n . A4

1 C. R. Calladine and H. R. Drew, J. Mol. Biol. 178, 773 1984.2 A. Bolshoy, P. McNamara, R. E. Harrington, and E. N. Trifonov, Proc.

Natl. Acad. Sci. U.S.A. 88, 2312 1991.3 L. Nekludova and C. O. Pabo, Proc. Natl. Acad. Sci. U.S.A. 91, 6948

1994.4 Z. Shakked, G. Guzikevich-Guerstein, F. Frolow, D. Rabinovich, A.

Joachimiak, and P. B. Sigler, Nature London 368, 469 1994.5 A. A. Gorin, V. B. Zhurkin, and W. K. Olson, J. Mol. Biol. 247, 34 1995.6 W. K. Olson, A. A. Gorin, X.-J. Lu, L. M. Hock, and V. B. Zhurkin, Proc.

Natl. Acad. Sci. U.S.A. 95, 11163 1998.7 W. K. Olson, N. L. Marky, R. L. Jernigan, and V. B. Zhurkin, J. Mol. Biol.232, 530 1993.

8 In engineering practice the term o-ring is commonly used for elasticrings that are circular when stress-free.

9 See also the discussion given in Refs. 3240.10 In our numbering of base pairs the choice of the direction of increasing n

is arbitrary. Although it is common to number the base pairs so that in-creasing n corresponds to advance in the 5 3 direction of the strandthat in discussions of transcription is called the sequence strand, we donot impose such a convention here. Because we study the implications ofthe requirement that the energy n be independent of the choice of thedirection of increasing n, it is necessary for us to consider transformationsthat change that direction while leaving invariant the orientation and spa-tial position of the base pairs.

11 The superposed symbols, , , etc., here serve to emphasize the distinc-tion between a value of a quantity and a function relating the quantity toother quantities.

12 In general, xn, the center of the rectangleBn, is not the barycenter centerof mass of the corresponding base pair. Here, as in Ref. 64, xn, d1

n , andd3

n are obtained by the following construction: Let l2 be the straight lineconnecting the C1 atoms of the deoxyribose groups corresponding to thenth pair base, and let l1 be the line that passes at a right angle through themidpoint of l2 and intersects the straight line containing the C8 atom ofthe purine base and the C6 atom of the complementary pyrimidine base.The point xn is taken to lie at the intersection of l1 and l2 ; the vector d1

n




13/14

is parallel to l1 and points in the direction of the major groove; d3n is

perpendicular to both l1 and l2 with d3n"r

n0.

13 V. B. Zhurkin, Y. P. Lysov, and V. I. Ivanov, Nucleic Acids Res. 6, 10811979.

14 M. A. El Hassan and C. R. Calladine, J. Mol. Biol. 251, 648 1995.15 We use the Einstein summation convention for subscripts but not super-

scripts.16 H. Berman, W. Olson, D. Beveridge, J. Westbrook, A. Gelbin, T. Demeny,

S. Hsieh, A. Srinivasan, and B. Schneider, Biophys. J. 63, 751 1992.17 R. E. Dickerson, M. Bansal, C. R. Calladine, S. Diekmann, W. N. Hunter,

O. Kennard, E. von Kitzing, R. Lavery, H. C. M. Nelson, W. K. Olsonet al., EMBO J. 8, 1 1988.

18 X.-J. Lu, M. Babcock, and W. Olson, J. Biomol. Struct. Dyn. 16, 8331999.

19 Such an external force and moment can arise when the nth base pair is incontact with a protein or with a base pair sequentially distant from it.When electrostatic forces are taken into explicit account, P in Eq. 2.10will include the potential energy of the interaction of charged residues.

20 The configurations discussed in Sec. IV may be considered appropriate tocases in which the ionic strength of the solution is sufficiently high 1M that the Debye length is significantly less than the distance betweencharged sites and the employed values of elastic moduli and intrinsickinematical parameters have been adjusted to account for the electrostaticinteraction of adjacent base pairs.

21 Packer, Dauncy, and Hunter Ref. 65 have used a computational model tocalculate the potential energy surface for base-pair steps as a function oftwo principal degrees of freedom, slide, and shift, and obtained results thatsuggest that the presence of close local minima of energy of the typeobserved in crystal structures of DNA oligomers in Ref. 5 can limit therange of applicability of quadratic approximations for n. Their calcula-tions for tetramer steps Ref. 66 further suggest that one should notblandly assume that interactions of nonadjacent base pairs need not betaken into account. However, there are not as yet a sufficient number ofhigh-resolution crystallographic structures available to treat all 136 tet-ramer steps in the analysis of the sequence-dependent properties of DNAand to answer the question, posed in Ref. 67, of whether the base-pairneighbors adjacent to a dimer step have a strong influence on its elasticproperties.

22 OHern et al. Ref. 68 developed a model of DNA elasticity in whichbase pairs are assumed to be rigid plates separated by a homogeneousisotropic elastic material, and showed that their model implies that, foreach n, n is a quadratic function obeying the relations seen in Eqs. 2.23.We here show that the Eqs. 2.23 hold in our more general model whenXYYX . The available experimental data Refs. 5 and 6 strongly indi-cate that Eqs. 2.23 hold when and only when XYYX . A DNA se-quence for which XYYX at every base-pair step must be either of thetype ATATAT or GCGCGC.

23 The values and ranges we give for the moduli and parameters are based ondata reported in Ref. 6.

24 O. Gonzales and J. H. Maddocks, Theor. Chem. Acc. 106, 76 2001.25 In this application of Cauchys representation theorem, we make use of the

fact that dnn1

"rn , as it equals Djin ri

n , is known once D ijn and ri

n are knownfor i ,j1,2,3.

26ij k is the permutation symbol of Levi-Civita.27 G. Kirchhoff, J. Reine. Angew. Math. 56, 285 1859.28 G. Kirchhoff, Vorlesungen uber Mathematische Physik, Mechanik, Vorl.

28 Teubner, Leipzig, 1876.29 E. H. Dill, Arch. Hist. Exact Sci. 44, 1 1992.30 B. D. Coleman, E. H. Dill, M. Lembo, Z. Lu, and I. Tobias, Arch. Ration.

Mech. Anal. 121, 339 1993.31 S. S. Antman, Nonlinear Problems of Elasticity Springer-Verlag, New

York, 1995.32 W. R. Bauer, R. A. Lund, and J. H. White, Proc. Natl. Acad. Sci. U.S.A.90, 833 1993.

33 I. Tobias and W. Olson, Biopolymers 33, 639 1993.34 I. Tobias, B. D. Coleman, and M. Lembo, J. Chem. Phys. 105, 2517

1996.35 J. H. White, R. A. Lund, and W. R. Bauer, Biopolymers 38, 235 1996.36 T. Charitat and B. Fourcade, Eur. Phys. J. B 1, 333 1998.37 J. H. White, R. A. Lund, and W. R. Bauer, Biopolymers 49, 605 1999.38 P. B. Fuhrer, R. S. Manning, and J. H. Maddocks, Biophys. J. 79, 116

2000.39 R. S. Manning and K. A. Hoffman, J. Elast. 62, 1 2001.

40 D. Garrivier and B. Fourcade, Europhys. Lett. 49, 390 2000.41 For a statement and a discussion of the assumptions behind Kirchhoffs

theory of rods Refs. 27 and 28 see Ref. 29.42 Base-pair level modes of DNA deformability constructed to account for

results of measurements of cyclization kinetics and gel electrophoreticmobility of short DNA segments suggest that highly curved DNA is madeup of two types of duplex structure within each helical repeat length: thecanonical B-type duplex and a perturbed form in which the base pairs areinclined with respect to the duplex axis Refs. 69 and 70 as they are in anA-type or C-type duplex.

43

J. C. Marini, S. D. Levene, D. M. Crothers, and P. T. Englund, Proc. Natl.Acad. Sci. U.S.A. 79, 7664 1982.44 S. Diekmann and J. C. Wang, J. Mol. Biol. 186, 1 1985.45 T. E. Haran, J. D. Kahn, and D. M. Crothers, J. Mol. Biol. 244, 135

1994.46 Electron micrographs of a 219 bp segment with such a sequence found in

C. fasiculata indicate that the segment has the form of a complete circleRef. 71 and DNA segments with higher intrinsic curvature have beensynthesized. Koo et al. Ref. 72 have given strong evidence, based on astudy of cyclization kinetics, to the effect that a 210 bp DNA segment withthe sequence AAAAAACGGCAAAAAACGGGCn can form a completecircle when stress-free. Previously, Ulanovsky et al. Ref. 73 had showedthat DNA segments with sequences (AAAAAATATATAAAAATCTCTncan be closed to form minicircles with 105, 126, 147, and 168 bp, i.e.,with n5, 6, 7, 8. They found that the maximum value of the ratio ofconcentrations at equilibrium of circular and noncircular products oc-

curred when the sequence length was either 126 or 147 bp, and theysuggested that such an observation indicates that it may be possible toobtain stress-free minicircles with as few as 137 bp. See also the review ofCrothers et al. Ref. 74.

47 So as to obtain an o-ring of small size, i.e., with N150 bp, while makingthe simplifying assumption that the intrinsic shears 1

n and 2n are every-

where zero, we chose 2n in Eq. 4.2 to be larger than the available

estimates of the upper bound for stress-free values of the roll.48 The values of F11 and F22 were chosen so that the bending rigidity

2/(1/F11)(1/F22) of each minicircle equals that of naturally straightDNA with a persistence length of 500 . The choice 0.7 is then com-patible with measurements of fluorescence anisotropy of dyes intercalatedin open segments of DNA see Ref. 75. Measurements of topoisomerdistributions for miniplasmids Ref. 76 suggest that should be close to1.4. See also Bouchiat and Mezard Ref. 77, who state that single mol-

ecule stretching experiments Ref. 78 yield a value of 1.6 for .49 See, e.g., the treatise of Thomson and Tait Ref. 79. A modern discussioncontaining exact solutions of the nonlinear dynamical equations governingthe oscillatory motion of an inextensible elastic rod obeying Kirchhoffstheory with uniformly distributed intrinsic curvature and torsion is givenin Ref. 34.

50 The approximate analysis of Ref. 36 yielded the value 0.9 as thecritical value of for loss of stability.

51 L. V. Crawford and M. J. Waring, J. Mol. Biol. 25, 23 1967.52 Such a minicircle would be a discrete analog of a closed idealized elastic

rod of a type for that much is known about the symmetry and stability ofequilibrium states, with and without self-contact Refs. 80 and 81.

53 We say essentially independent, for the intercalative binding of EtBrchanges not only the twist 3

n but also the rise 3n Ref. 51.

54 The present theory is not intended to provide an interpretation of an ob-served preference of EtBr for binding to pyrimidine-purine base-pair steps

Ref. 82.55 H. M. Sobell, C.-C. Tsai, S. C. Jain, and S. G. Gilbert, J. Mol. Biol. 114,333 1977.

56 L. S. Lerman, J. Mol. Biol. 3, 18 1961.57 P. J. Bond, R. Langridge, K. W. Jennette, and S. J. Lippard, Proc. Natl.

Acad. Sci. U.S.A. 72, 4825 1975.58 As the o-ring we are considering here is composed of periodically repeat-

ing units of length 10 bp, the minimum energy configuration for a givendistribution of sites of intercalation would not be affected if each such sitewere shifted by 10 bp in the direction of increasing n. Although a shift inthe distribution by less than 10 bp can affect the minimum energy con-figuration, our calculations show that for Minicircle I the correspondingchange in does not exceed 0.15 kT.

59 It follows from the theory of ideal elastic rods that if the minicircle wereformed by closing naturally straight DNA, i.e., if Eq. 4.2 were replacedby Eq. 4.1, then the untwisting at a single base-pair step by an angle of




14/14

magnitude less than 360, as it corresponds to a change in excess link lessthan 1, would not affect the shape of the minicircle, i.e., the axial curvewould remain circular.

60 There are protein structures that can induce a comparable change in themagnitude of twist, but the changes they induce are generally distributedover several base-pair steps. See, e.g., Ref. 83 reporting a case in which an8 bp long region is untwisted by 147 in a crystal structure of the humanTFIIB-TBP protein complex bound to duplex DNA.

61 See also Ref. 32.62 F. B. Fuller, Proc. Natl. Acad. Sci. U.S.A. 68, 815 1971.63

J. H. White, in Mathematical Models for DNA Sequences CRC, BocaRaton, FL, 1989, p. 225.64 W. K. Olson, X.-J. Lu, J. Westbrook et al., J. Mol. Biol. 313, 229 2001.65 M. J. Packer, M. P. Dauncey, and C. A. Hunter, J. Mol. Biol. 295, 55

1999.66 M. J. Packer, M. P. Dauncey, and C. A. Hunter, J. Mol. Biol. 295, 85

2000.67 K. Yanagi, G. G. Prive, and R. E. Dickerson, J. Mol. Biol. 217, 201

1991.68 C. S. OHern, R. D. Kamien, T. C. Lubensky, and P. Nelson, Eur. Phys. J.

B 1, 95 1998.69 W. K. Olson and V. B. Zhurkin, in Biological Structure and Dynamics,

Vol. 2, edited by R. H. Sharma and M. H. Sharma Adenine, Schenectady,NY, 1996, p. 341.

70 W. K. Olson and V. B. Zhurkin, Curr. Opin. Struct. Biol. 10, 286 2000.71 J. Griffith, M. Bleyman, C. A. Rauch, P. A. Kitchin, and P. T. Englund,

Cell 46, 717 1986.72 H. S. Koo, J. Drak, J. A. Rice, and D. M. Crothers, Biochemistry 29, 4227

1990.73 L. Ulanovsky, M. Bodner, E. N. Trifonov, and M. Choder, Proc. Natl.

Acad. Sci. U.S.A. 83, 862 1986.74 D. M. Crothers, J. Drak, J. D. Kahn, and S. D. Levene, Methods Enzymol.212, 3 1992.

75 J. M. Schurr, B. S. Fujimoto, P. Wu, and L. Song, in Topics in Fluores-cence Spectroscopy, Vol. 3: Biochemical Applications, edited by J. C. La-kowicz Plenum, New York, 1992, Chap. 4, p. 137.

76 D. S. Horowitz and J. C. Wang, J. Mol. Biol. 173, 75 1984.77 C. Bouchiat and M. Mezard, Phys. Rev. Lett. 80, 1556 1998.78 T. R. Strick, J.-F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette,

Science 271, 1835 1996.79 W. Thompson and P. G. Tait, Principles of Mechanics and Dynamics,

formerly Treatise on Natural Philosophy Dover, New York, 1962.80 B. D. Coleman, D. Swigon, and I. Tobias, Phys. Rev. E 61, 759 2000.81 B. D. Coleman and D. Swigon, J. Elast. 60, 171 2000.82 R. V. Kastrup, M. A. Young, and T. R. Krugh, Biochemistry 17, 4855

1978.83 F. T. F. Tsai and P. B. Sigler, EMBO J. 19, 25 2000.


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Bernard D. Coleman, Wilma K. Olson and David Swigon- Theory of sequence-dependent DNA elasticity