Writhe of DNA induced by a terminal twist

Bulletin of Mathematical Biology 67 (2005) 197–209

www.elsevier.com/locate/ybulm

Writhe of DNA induced by a terminal twist

Kai Hu

Department of Applied Mathematics, National DongHwa University, Shoufeng, Hualien 97441, Taiwan,Provinceof China

Received 13 November 2003; accepted 6 May 2004

To my mother for her70th birthday

Abstract

This paper considers the three-dimensional structure of B-form DNA. The molecule may be openor covalently closed. For the former, its two ends are not allowed to move or rotate freely in spaceunless the molecule is under the influence of rigid body motions of the ambient space. Implied bythe elastic rod model for DNA, the molecule writhes immediately when subject to a terminal twistas long as its axis is none of the following curves: lines, circular arcs, circular helices. This result isremarkably different from well-known results about DNA of other conformations. For example, if aDNA is regarded as an elastic rod whose axis is a circle, then it has no induced writhe when subjectto a terminal twist until the latter meets a critical extent.© 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved.

1. Introduction

The elastic rod model for DNA uses the configurations of isotropic, elastic thin rods tounravel the three-dimensional structure of B-form DNA. According to the Watson–Crickmodel of DNA (Watson and Crick, 1953), any rod mimicking the molecule is intrinsicallylinear, that is, the undeformed state of the rod has a linear axis. Furthermore, the state hasa uniform twist which is approximately 1.848 rad/nm (the reader may also seeWestcottet al. (1995)for other details of how a DNA molecule is regarded as a rod). Although thiselastic rod model simplifies some features of DNA, it has nevertheless contributed much to

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0092-8240/$30 © 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.bulm.2004.05.008

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198 K. Hu / Bulletin of Mathematical Biology 67 (2005) 197–209

understanding of DNA by imitating diverse observed conformations of the molecule andby predicting DNA structure with satisfactory accuracy. An enormous amount of researchhas emerged from this viewpoint, for example,Benham (1977, 1983, 1985, 1987, 1989),Le Bret (1979), Wadati and Tsuru (1986), Hao and Olson (1989), Coleman et al. (1993),Shi and Hearst (1994), Tobias et al. (1994, 1996), Coleman et al. (1995), Olson (1996),Fain et al. (1997), Swigon et al. (1998), Fain and Rudnick (1999), Coleman and Swigon(2000), Qian and White (unpublished)and the references therein.

Supercoiled DNA has been largely observed in biological systems and extensivelystudied; seeBauer et al. (1980)for a review. It may result from an introduction of terminaltwist into a covalently closed circular DNA. By cutting the DNA, rotating one of the cutends about the DNA axis, and then resealing the cut, the linking number of the moleculechanges. According to White’s formula (White, 1969), such a change converts as a singlechange of the twisting number or writhing number of the molecule, or both. Using theelastic rod model of DNA, the detail of the conversion becomes clear: the change of thelinking number of the DNA converts into a single change of the twisting number before theterminal twist reaches a critical extent (Zajac, 1962; Le Bret, 1979; Benham, 1989; Fainet al., 1997; Qian and White, 1998, unpublished; Fain and Rudnick, 1999). This criticaltwist depends on some elasticity parameters of the DNA and torsional deviation of theDNA from its relaxed state (Zajac, 1962; Le Bret, 1979; Benham, 1989; Fain et al., 1997;Qian and White, 1998, unpublished; Fain and Rudnick, 1999). A concrete example of thecritical twist for a covalently closed circular DNA of 2100 bp can be found inBauer et al.(1993).

Using the elastic rod model for DNA, I study the outcome of introducing a terminal twistinto other closed DNA molecules in this paper. In fact, any DNA which can be regardedas a clamped-end rod in a biological system, such as a DNA–protein complex, is an objectof study. Roughly speaking, a clamped-end rod is a rod whose two ends cannot move orrotate freely in space except if the rod is under the influence of rigid body motions of theambient space. So a closed rod may be regarded as a clamped-end rod as well since it isobtained from the undeformed state by deforming the latter in some specific way and thengluing the ends [recall, the undeformed state is a linear rod]. Introducing a terminal twistto a clamped-end rod, or referring to a clamped-end rod subject to a terminal twist, meansthat one of the ends of the rod is unclamped initially, then rotated about the local tangentto the rod axis, and finally reclamped, cf.Bauer et al. (1993).

An elastica is a rod at equilibrium, that is,a rod physically realizing a solution toEuler–Lagrange equations derived from the elastic energy functional (see the next sectionfor details). From the viewpoint of the elastic rod model, a conformation of DNA observedin the laboratory is the configuration of some elastica.

The following is a criterion on whether two distinct elasticas have the same axis.

Main Result Two distinct isotropic, intrinsically linear elasticas have the same axis ifand only if the latter is aline, a circular arc or a circular helix.

Once an elastica is subject to a terminal twist, the latter is retained inside the rodobtained from the elastica by introducing the terminal twist. The reason for this is thatthe ends of the rod are clamped. So the rod has to reach a new equilibrium configuration.If the axis of the new configuration is the same as the axis of the initial elastica, then there

K. Hu / Bulletin of Mathematical Biology 67 (2005) 197–209 199

are two distinct elasticas with the same axis.Therefore, the Main Result implies that anyelastica subject to a terminal twist shall writhe instantly if its axis is neither a line, a circulararc nor a circular helix.

Remark 1. The effect of introducing a terminal twist to an elastica whose axis is oneof the curves to which the Main Result refers is similar to what has been mentioned inthe second paragraph of this section. The detail can be found in various research workssuchas,Love (1944)for linear axes,Zajac (1962), Le Bret (1979), Benham (1989), Baueret al. (1993), Fain et al. (1997), Qian and White (1998)andFain and Rudnick (1999)forclosed circular axes,Qian and White (unpublished)for open circular axes,Goriely andTabor (1997)for helical axes. Especially, the formula of the critical twist given byQianand White (unpublished)also covers the cases of linear and closed circular axes.

My approach in proving the Main Result is elementary: deriving formulas of the axiscurvature and axis geometric torsion of an elastica, then using the formulas to check if thereis any difference between two elasticas with the same aforementioned geometric quantities.Note that a curve is uniquely determined by its curvature and geometric torsion up to rigidbody motions ofR3, seee.g.do Carmo (1976); furthermorethe motions do not alter thephysical properties of a rod.

Shi and Hearst (1994)presented such formulas explicitly in terms of Jacobian ellipticfunctions. However, strictly speaking, any proof of the Main Result that makes use of Shiand Hearst’s formulas should be considered incomplete. The reason is that the authorsdescribed a rod using the so-called Frenét frame which exists along the rod axis only if therod has nowhere zero axis curvature, see e.g.do Carmo (1976). Although Shi and Hearstremarked on how to generalize their discussion to any rod whose axis curvature has finitelymany zeros, see Appendix B ofShi and Hearst (1994), no single reason why the axis of ageneric elastica must be such a curve has ever been given.

This paper is organized as follows. In the next section, we describe each rod by theso-called directors instead of the Frenét frame. The directors are three orthonormal vectorsnaturally associated with the rod axis even if the latter is degenerate, namely, the axiscurvature is zero somewhere. From the regularity of the solutions to the Euler–LagrangeEqs. (3)–(5), one may conclude that the axis curvature of an elastica has at most finitelymany zeros ifit is not identically zero, seeProposition 1. In addition to the MainResult, several properties of elasticas are proved inSection 3: Proposition 5characterizesthose elasticas with constant axis curvature;Proposition 7gives anecessary condition ofknotted elasticas;Propositions 8and9 are examples of how initial data used to solve theEuler–Lagrange equations for an elastica affect the geometry of the axis of the elastica.Proposition 8is about the extrema of the axis curvature, andProposition 9relates to thenumber of degenerate points on the axis. InSection 4, from the Main Result, I study theeffect of introducing a terminal twist into a DNA molecule. The final section includes adiscussion on the outcome of intrinsicallyO-ring DNA (Marini et al., 1982; Ulanovskyet al., 1986) when subject to a terminal twist.

Note that self-contact of elastic rods has been neglected in this paper because it mayintroduce technical complications. I would like to refer the reader toShi and Hearst(1994), Coleman et al. (1995)andColeman and Swigon (2000)on the issue, includingits implications on DNA structure.


2. The squared axis curvature of an elastica

In this paper, all isotropic elastic rods are assumed to have the following features.First, in addition to being cylindrical and looking slender, the configuration of each rod iscompletely determined by an immersed curve,called the axis, and a preferred unit normalvector defined along the curve. Second, there is a rodRu that denotes the undeformedstate of the rods; the axis ofRu is a line, and, without loss of generality, the preferred unitnormal vector ofRu may be assumed to be constant. Finally, any rod other thanRu isconsidered a result of deformingRu inextensibly and unshearably.

Clearly, the axis and the preferred unit normal vector describe overall geometry andtwist of a rod, respectively. The inextensibility condition mentioned in the last paragraphimplies that all the rods have the same length, call it l , and thus the same arc lengthparameter, call its.

For a given rod, letd3 be the unit tangent vector to the rod axis,d1 the preferred unitnormal vector of the rod, andd2 = d3 × d1. These three orthonormal vectors are called thedirectors. Thebending curvatures u1, u2 and thetwisting density u3 of the rod are definedas follows:

u1 = −dd3

ds· d2, u2 = dd3

ds· d1 and u3 = dd1

ds· d2, (1)

where the dot between ddi /ds anddj denotes the standard inner product ofR3. Because

the axis curvaturek is the norm of dd3/ds, it is obvious thatk =√

u21 + u2

2. In terms oftheseui , theelastic energy of the rod can be written as

1

2

∫ l

0ρ(u2

1 + u22)+ ρu2

3 ds, (2)

where ρ and ρ measure the bending stiffness and the twisting stiffness of the rod,respectively. For DNA these parameters canbe obtained from experimental data, see e.g.Hagerman (1988).

Let C and thedi denote the axis and the directors of a rod, respectively. The so-calledclamped-end boundary conditionsof the rod may be expressed as follows:C(0) andC(l )are respectively the origin and some point ofR

3; d1(0), d2(0) andd3(0) are the unit vectorsin the direction of the positivex-, y- andz-axis, respectively, andd1(l ), d2(l ), d3(l ) aresome orthonormal vectors.

With respect to the elastic energy functional (2), an elastica R is an equilibriumrod among rods obtained by perturbingR slightly without altering its clamped-endboundary conditions. So the bending curvatures and the twisting density of an elastica,together with someconstant vectorλ, satisfy the following Euler–Lagrange equations(cf. Dichmann et al. (1996)):

ρu1 + (ρ − ρ)u2u3 = λ · d2, (3)

ρu2 + (ρ − ρ)u1u3 = −λ · d1, (4)

ρu3 = 0, (5)

where a function withn dots above it denotes thenth derivative of the function with respectto s. As can be easily derived from (5), the twisting density of an elastica must be constant.


Given an elastica, if one writesλ = −λ1d1 + λ2d2 + λ3d3, then through (3) and (4) itis clear that

λ1 = ρu2 + (ρ − ρ)u1u3,

λ2 = ρu1 + (ρ − ρ)u2u3.

Becauseλ is constant, dλ = 0 whichgives

λ1 + λ2u3 − λ3u2 = 0, (6)

−λ1u3 + λ2 − λ3u1 = 0, (7)

λ1u2 + λ2u1 + λ3 = 0. (8)

Using (8), it is easy to recover the so-calledKirchhoff–Clebsch conservation law, seee.g.Love (1944), stating that

ρ

2(u2

1 + u22)+ ρ

2u2

3 + λ3,

called thetotal energy density, is a constant function defined along the axis of the elastica.For a function f defined on[0, l ] and a numberc, the set{s ∈ [0, l ] : f (s) = c} is

called thepre-image of c under f.

Proposition 1. Let v denote the squared axis curvature of an elastica. Then, eitherv is aconstant or the pre-image of any real number underv is a finitesubset of[0, l ].Proof. Suppose the total energy density and the twisting density of the elastica arec1 andc3, respectively. Because the bending curvaturesu1, u2 of the elastica satisfy (7) and (8),the transpose of(u1,u2, u1, u2) is a solution to the following system of first-order ordinarydifferential equations:

x1

x2

y1

y2

=

y1y2

−1

2x1(x

21 + x2

2)−δ

ρy2 − c2

ρx1

−1

2x2(x

21 + x2

2)+δ

ρy1 − c2

ρx2

, (9)

whereδ = −(2ρ − ρ)c3 andc2 = −c1 + (3ρ − 2ρ)c23/2. The right-hand side of the last

equation is a real analytic, vector-valued function ofx1, x2, y1 andy2. Sobothu1 andu2are real analytic due to Cauchy’s theorem on regularity of solutions to ordinary differentialequations, see e.g.Birkhoff and Rota (1969). As a result, v is real analytic, too. Hence,the pre-image of any real number underv is a finite subset of[0, l ], providedv is notconstant. �

Proposition 2. If there is an elastica whose bending curvatures satisfy u1 = u2 = 0 andu1 = u2 = 0 at some point s0 of [0, l ], thenits axis must be a line.

Proof. Because of the hypothesis, the bending curvatures of the elastica and their firstderivatives constitute a solution to an initial value problem consisting of (9), x1 = x2 = 0


andy1 = y2 = 0 ats0. Meanwhile, it is obvious that the initial value problem has the trivialsolution. So these two solutions are identical in an interval containings0 due to Cauchy’stheorem onuniqueness of solutions to ordinary differential equations, see e.g.Birkhoff andRota (1969). In particular,u1 andu2 are zero in the interval, and, consequently, so is thesquared axis curvaturev. Using the last proposition,v is equal to zero on[0, l ]. Hence, theaxis of the elastica is a line. �

Derived from the lasttwo rowequations of (9), we have

ρ(u1u2 − u2u1)+ δ

2(u2

1 + u22) = c4, (10)

ρ((u1)2 + (u2)

2)+ ρ

4(u2

1 + u22)

2 + c2(u21 + u2

2) = c5, (11)

wherec4 andc5 are constants of integration which correlate to each other since they canbe determined by the values ofu1, u2, u1 andu2 at any points0 of [0, l ].Proposition 3. Suppose the axis curvature of an elastica attains its maximum M at sM . IfM > 0, then

c5 = c24

ρM2− δ

ρc4 + ρ

4M4 +

(δ2

4ρ+ c2

)M2 + ρ(v(sM ))

2

4M2.

Otherwise, c4 = c5 = 0.

Proof. BecauseM > 0, the axis of the elastica admitsthe Frenét frame in an intervalI containingsM . Thus one may writed3 = T , d1 = N cosθ + B sinθ and d2 =−N sinθ + B cosθ in I , whereT , N and B denote the unit tangent vector, the principalnormal vector and the binormal vector to the axis, respectively, andθ is the angle measuredfrom N to d1 with respect to the orientation induced byT on the normal plane spanned byN andB. Therefore,

u1 = k sinθ, u2 = k cosθ and u3 = τ + θ (12)

in I , whereτ denotes the axis geometric torsion. Using (12), we rewrite (10) and (11) as

ρk2θ + δ

2k2 = c4 (13)

and

ρ((k)2 + k2(θ )2)+ ρ

4k4 + c2k2 = c5,

respectively. In particular,

ρM2θ (sM )+ δ

2M2 = c4, (14)

ρ(v(sM ))2

4M2+ ρM2(θ (sM ))

2 + ρ

4M4 + c2M2 = c5. (15)

Substituting (14) for θ (sM ) of (15), one obtains the relation betweenc4 andc5 proposed inthe proposition.


If M = 0, then the bending curvatures are zero. Using (10) and (11), one obtainsc4 = c5 = 0. �

The next equation is derived from the last two row equations of (9), and (10) and (11):

v = −3

2v2 + c6v + c7, (16)

wherec6 = −(4ρc2 + δ2)/ρ2 andc7 = 2(ρc5 + δc4)/ρ2.

The simplestv is a constant taking on the value of a root of the quadratic polynomial onthe right-hand side of (16). Next, we assumev is not constant for the rest of this section.Multiplying 2v to (16) and thenintegrating, one obtains

(v)2 + v3 − c6v2 + c8v + c9 = 0, (17)

wherec8 = −2c7 andc9 is a constant of integration. In fact,c9 = 4c24/ρ

2 if one evaluates(17) at sM by making use ofProposition 3.

Remark 2. For the reader’s reference, in the case ofk being twice continuouslydifferentiable, (4.1a), (4.1b) and (4.6) ofShi and Hearst (1994)are, respectively, (16), (13)and (17) of this paper withQ = ρc3/ρ, c = c6/4, J = −c4/ρ andB = c7/4.

Let P(x) = −x3 + c6x2 − c8x − c9. Because(v)2 = P(v) andv is not constant, thecubic polynomialP(x) should be positive somewhere in[0,∞[. Furthermore, by virtue ofP(x) → ±∞ asx → ∓∞ and P(0) = −c9 ≤ 0, the polynomial has three real roots;among them the number of positive roots is either one or two, and there are at least twonon-negative roots. Thus one may writeP(x) = (x−c)(x−b)(a−x)with c ≤ 0 ≤ b < a.It is clear thatb ≤ m2 < M2 ≤ a wherem denotes the minimum ofk. Define the followingthreenumbers:

α =√

a − b

a − c, β2 = a − b

aand γ =

√a − c

2.

Obviously, 0< α2 ≤ β2 ≤ 1.The solution to (17) can be written as

v = a(1 − β2sn2(γ s − ν, α)), (18)

where sn(w, ζ ) is a Jacobian elliptic function with the modulusζ , seee.g. Byrd andFriedman (1954), andν is anumber given by

ν =

0 if v0 = a,K if v0 = b andα < 1,

sn−1

(√a − v0

a − b, α

)if b < v0 < a andv0 > 0,

−sn−1

(√a − v0

a − b, α

)if b < v0 < a andv0 < 0.


In the definition ofν, v0 and v0 denote the prescribed values ofv and v at s = 0,respectively, and

K =∫ π/2

0

dϑ√1 − α2 sin2ϑ

.

Remark 3. Note thatα = 1 if and only if b = c = 0. Because sn(w,1) = tanhw,v = a(1 − tanh2(γ s − ν)) whenα = 1. Moreover,v is never zero. So the conditionα < 1 is emphasized in the second case ofν. The reader may consult (122.09) ofByrd andFriedman (1954)for other Jacobian elliptic functions with the unit modulus, and (132.01)for Jacobi’s inverse elliptic functions with unit modulus as well.

3. More properties of elasticas

For a given elasticaR, let DR denote the set of all degenerate points on the axis ofR,namely,DR = {s ∈ [0, l ] : k(s) = 0} wherek is the axis curvature ofR. By virtue ofProposition 1, DR is either a finite subset of[0, l ] or the whole interval[0, l ].

Clearly, the equations shown in (12) are valid in the complement ofDR . So (13) yields

θ = c4

ρv− δ

2ρ(19)

which gives

τ = − c4

ρv+ ρc3

2ρ(20)

in [0, l ] \ DR .It is easy to conclude the following three properties about elasticas from the last two

equations. Hence, the proofs are omitted.

Proposition 4. LetR be an elastica whose axis is not a line. If it has zero c4, thenits axisgeometric torsion is equal toρc3/2ρ in [0, l ] \ DR .

Note that a special case of zeroc4 is DR �= ∅. Thus, the preceding proposition also assertsthat the axis geometric torsion of an elastica is almost constant if the axis is degenerate.

Proposition 5. Let R be an elastica. If its axis curvature is constant, then so is its axisgeometric torsion. The converse holds if the axis geometric torsion is a constant other thanρc3/2ρ.

The first half of the last proposition characterizes the axis of any elastica with constant axiscurvature. That is, the axis of such an elastica must be a line, a circular arc or a circularhelix because these are the only curves whose curvature and geometric torsion are bothconstant, see e.g.do Carmo (1976).

A rod is called planar if its axis is so. Furthermore, if the axis geometric torsion isdefined, then it has to be zero.

Proposition 6. LetR be an elastica with nonzero twisting density. Then it is planar if andonly if k = √

2c4/ρc3.


As a corollary, if a planar elastica has nonconstant axis curvature, then its twisting densitymust be zero. Moreover, it has zeroc4 by (20).

An elastica is calledknottedif its axis is a nontrivial knot.

Proposition 7. Let R be a knotted elastica with total energy density c1 and twistingdensity c3. Then

l 2

ρ

(c1 − ρ

2c2

3

)> 8π2.

Proof. It is easy to observe that an elastica has a closed axis if and only if the integral ofthe associatedλ3 over[0, l ] is zero. For the elasticaR, by integrating

c1 = ρ

2(u2

1 + u22)+

ρ

2c2

3 + λ3

over[0, l ], one obtains

2l

ρ

(c1 − ρ

2c2

3

)=∫ l

0v ds.

Using Schwarz inequality, see e.g.Buck (1978), one has∫ l

0v ds =

∫ l

0k2 ds ≥ 1

l

(∫ l

0k ds

)2

.

The equality holds if and only ifk is constant, therefore, the axis ofR is a circle byProposition 5. Because a circle is a trivial knot, one may assume that the last inequality isstrict. By virtue ofR being knotted, a result of Milnor’s (Milnor, 1950) implies∫ l

0k ds> 4π.

Hence, the inequality proposed in the proposition is obtained.�

The following is a necessary condition forv to possess at least two critical points.

Proposition 8. Assume the notations used in the last paragraph of the preceding section.If v hasat least two critical points, then K≤ γ l for v0 = a,b; K +ν ≤ γ l for b < v0 < aandv0 > 0; 2K + ν ≤ γ l for b < v0 < a andv0 < 0.

Proof. Note that the modulusα occurring in (18) is not unit, otherwise the correspondingv has at most one critical point. Through direct computations, the first two zeros ofv are0, K

γif v0 is eithera or b; ν

γ, K+ν

γif b < v0 < a andv0 > 0; K+ν

γ, 2K+ν

γif b < v0 < a

andv0 < 0. So the proof is completed by asking the larger zero to be less than or equal tol for each case. �

According to (17), s0 is a critical point ofv if and only if v(s0) equals eithera or b. If vhas at least two critical points, then botha andb must be attained. So, the last propositionmay be considered as a criterion of whether botha andb are the extrema ofv.

In terms of initial data used to solve (9) for an elastica, the following propositionestimates the number of degenerate points on the elastica axis.


Proposition 9. SupposeR is an elastica whose axis has p degenerate points, where p isa positive integer. Write DR = {0 ≤ z1 < z2 < · · · < zp ≤ l }. Define an integer n whichequals p− 1 if 0 ∈ DR , and p otherwise. If n> 0, then

(2n − 1)K + ν

γ≤ l .

Proof. BecauseDR is finite and nonempty,β2 = 1 andthusv = a cn2(γ s − ν, α) [notethatα < 1, seeRemark 3]. It is easy to obtain the general form of the zeros of cn(γ s−ν, α):

(2 j + 1)K + ν

γ

for j ∈ Z. z1 corresponds toj = 1 if the former is zero, andj = 0 otherwise. So,

zp = (2n − 1)K + ν

γ.

Hence, the proof is complete. �

Finally, the Main Result is proved as follows.

Proof of the Main Result. Through a series of computations, one may conclude that ifthe axis of a rod is one of the curves mentioned in the Main Result, then the rod is anelastica as long as its twisting density is constant. Hence, there are indeed distinct elasticaswith the same axis if that axis is a line, a circular arc or a circular helix. Next, I show thatthere is no other possibility.

SupposeR andR are two distinct elasticas with the same axisC whose curvature maybe assumed nonconstant. IfC is not degenerate, by making use of (20) one has

− c4

ρv+ ρc3

2ρ= − c4

ρv+ ρc3

2ρ,

wherec3 and c3 are the twisting densities ofR and R, respectively, c4 and c4 are theconstant of integration found in (10) for R andR, respectively. Because the elasticas havethe same axis, their preferred unit normal vectors must differ by an angleψ which is notconstant. Thus,c3 andc3 are not equal since they differ byψ . From the lastequation, it iseasy to obtain

v = 2(c4 − c4)

ρ(c3 − c3),

meaning thatv is constant. Thus, a contradiction is obtained.If C is degenerate, letS = DR ∪ {0, l } and writeS= {0 = s0 < s1 < s2 < · · · < sn =

l }. ThenR andR must be different in]sr , sr+1[ for somer ∈ {0,1,2, . . . ,n − 1}. Sincethere is no degenerate point in]sr , sr+1[, the preceding argument concludes immediatelythatv is constant in]sr , sr+1[. By virtue ofProposition 1, v is actually constant in[0, l ].So a contradiction is obtained.

Therefore, two distinct elasticas cannot have the same axis whose curvature is notconstant. This completes the proof.�


4. Conclusion

Using the elastic rod model for DNA, this paper demonstrates the effect of introducinga terminal twistinto a covalently closed DNA or an open DNA whose two ends are notallowed to move or rotate in space unless the molecule is under the influence of rigid bodymotions of the ambient space. Implied by theMain Result, such a DNA writhes instantlyas long as the axis is neither a line, a circular arc nor a circular helix. This result is differentfrom the biological implication of some well-known results: a rod whose axis is linear orcircular does not writhe when subject to a terminal twist until the twist meets a criticalextent.

The twist induced writhe could bring some sequentially distant sites of DNA spatiallyclose. Such a conformation change has been observed widely in biological systems,for example, the initiation of transcription and DNA replication and also site-specificrecombination. The preceding conclusion merely states the qualitative aspect of the twistinduced DNA writhe. The quantitative one is not addressed here, though, questions likehow a terminal twist determines the locations of the sites which will be ultimately closestto one another, the spatial distance between any two such sites, and whether introducing aterminal twist is an energetically flavored method to make DNA supercoiled seem naturallyto be the next avenues of exploration. Not only may the answers be helpful when oneattempts to manipulate the tertiary structure of DNA but the questions themselves haveinherent intellectual interest.

5. Discussion

Aside from the DNA molecules studied in this paper, there are some that are notintrinsically linear. Indeed, examples of a DNA molecule which intrinsically forms a circlehave been reported (Marini et al., 1982; Ulanovsky et al., 1986). Using the elasticity theoryof intrinsically O-ring rods and a finite element method,Bauer et al. (1993)were able tostudy the result of introducing a terminal twist into such an intrinsicallyO-ring DNA. Incontrast to a covalently closed, circular, intrinsically linear DNA, an intrinsicallyO-ringDNA writhes immediately when subject to a terminal twist. The same phenomenon alsoexists for DNA with bends lying on the same plane (also seeBauer et al. (1993)). Similarstudies of DNA with intrinsic curvature can also be found inTobias and Olson (1993)andWhite et al. (1996).

The aforementioned result on intrinsicallyO-ring rods was verified analytically byTobias et al. (1996)andQian and White (1998). Additionally, some intrinsically curvedhelical rods were studied byGoriely and Tabor (1997). In essence, these works requiresolutions to the respective Euler–Lagrange equations. Unfortunately, the general form ofthe solutions has not been discovered yet. It seems very unlikely that the argument used inTobias et al. (1996), Goriely and Tabor (1997)or Qian and White (1998)can be extendedto other intrinsically curved elasticas. So it is necessary to seek a new argument whichhasnothing to do with solving differential equations, or perhaps as little as possible. Suchan argument is possible as shown inHu (2003), although the latter does not exceed thepreceding works in scope. Continuing study of this topic has already begun and will appearelsewhere.


Acknowledgements

I am grateful to Prof. Shiu-Yuen Cheng for calling my attention toLanger and Singer(1984), and to Prof. Hong Qian for sending me his unpublished joint work with Prof.James H. White (Qian and White, unpublished). I am also thankful to Prof. Li-Tien Chengfor his suggestions on writing. In addition, I am indebted to the referees for their valuablecomments in the revision of this paper. This work was partially supported by the NationalScience Council (R.O.C.), NSC-89-2115-M-259-004.

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