non-linear dynamic intertwining of rods with self-contact

International Journal of Non-Linear Mechanics 43 (2008) 65–73www.elsevier.com/locate/nlm

Non-linear dynamic intertwining of rods with self-contact

Sachin Goyala, N.C. Perkinsb,∗, Christopher L. Leec

aApplied Ocean Physics and Engineering, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USAbDepartment of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA

cDepartment of Mechanical Engineering, Olin College, Needham, MA 02492, USA

Received 30 January 2007; received in revised form 25 September 2007; accepted 2 October 2007

Abstract

Twisted marine cables on the sea floor can form highly contorted three-dimensional loops that resemble tangles. Such tangles or ‘hockles’are topologically equivalent to the plectomenes that form in supercoiled DNA molecules. The dynamic evolution of these intertwined loopsis studied herein using a computational rod model that explicitly accounts for dynamic self-contact. Numerical solutions are presented for anillustrative example of a long rod subjected to increasing twist at one end. The solutions reveal the dynamic evolution of the rod from an initiallystraight state, through a buckled state in the approximate form of a helix, through the dynamic collapse of this helix into a near-planar loopwith one site of self-contact, and the subsequent intertwining of this loop with multiple sites of self-contact. This evolution is controlled by thedynamic conversion of torsional strain energy to bending strain energy or, alternatively, by the dynamic conversion of twist (Tw) to writhe (Wr).� 2007 Elsevier Ltd. All rights reserved.

Keywords: Rod dynamics; Self-contact; Intertwining; DNA supercoiling; Cable hockling

1. Introduction

Cables laid upon the sea floor may form loops and tanglesas illustrated in Fig. 1. The loops, sometimes referred to ashockles, may cause localized damage and, in the case of fiberoptic cables, may also prevent signal transmission. These highlynon-linear deformations are initiated by a combination of lowtension or compression (i.e., cable slack) and residual torsionsufficient to induce torsional buckling of the cable. Tanglesevolve from a subsequent dynamic collapse of the buckled cableinto highly non-linear and intertwined configurations with self-contact.

The looped and tangled forms of marine cables are topo-logically equivalent to the ‘plectonemic supercoiling’ of longDNA molecules as illustrated in Fig. 2 (refer to [1,2]). Fig. 2depicts a DNA molecule on three different length scales as re-produced from [3,4]. The smallest length scale (far left) showsa segment of the familiar ‘double-helix’ which has a diameter

∗ Corresponding author.E-mail addresses: [email protected] (S. Goyal), [email protected]

(N.C. Perkins), [email protected] (C.L. Lee).

0020-7462/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2007.10.004

of approximately 2 nanometers (nm). One complete helical turnis depicted here and this extends over a length of approximately3 nm.

On an intermediate spatial scale (middle of Fig. 2), the dou-ble helix now appears as a long and slender DNA molecule thatmight be realized when considering tens to hundreds of helicalturns (approximately tens to hundreds of nm). Two idealized‘long-length scale structures’ of DNA are illustrated to the farright in Fig. 2. Here, the exceedingly long DNA molecule maycontain thousands to millions of helical turns and behave as avery flexible filament with lengths ranging from micron to mil-limeter scales or even longer. The long-length scale curving andtwisting of this flexible molecule is referred to as supercoiling.Two generic types of supercoils are illustrated. A plectonemicsupercoil leads to an interwound structure where the moleculewraps upon itself with many sites of apparent ‘self-contact’. Bycontrast, a solenoidal supercoil possesses no self-contact andforms a secondary helical structure resembling a coiled springor a telephone cord.

Often with the aid of proteins, DNA must supercoil forseveral key reasons. First, supercoiling provides an organizedmeans to compact the very long molecule (by as much as 105)

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66 S. Goyal et al. / International Journal of Non-Linear Mechanics 43 (2008) 65–73

within the small confines of the cell nucleus. An unorganizedcompaction would hopelessly tangle the molecule and render ituseless as the medium for storing genetic information. Second,supercoiling plays important roles in the transcription, regula-tion and repair of genes. For instance, specific regulatory pro-teins are known to aid or to hinder the formation of simpleloops of DNA which in turn regulate gene activity; refer, forexample, to Schleif [5] and Semsey et al. [6].

Like the tangling of marine cables above, the intertwining ofDNA is inherently a non-linear dynamic process controlled bystructural properties (e.g., elasticity) and applied forces (e.g.,protein interactions). Rod theory provides a useful frameworkto explore the dynamics of intertwining of long filament-likestructures such as cables and DNA molecules, as described, forexample in Goyal [7]. The mechanics of intertwining immedi-ately invokes formulations for self-contact in rod theory whichremain a significant challenge as emphasized recently in [8,9].

The inclusion of self-contact in equilibrium formula-tions of rod theory has been treated in [8–16]. In particular,Chouaieb et al. [8] evaluate helical equilibria where self-contact

Fig. 1. Low tension cable forming loops and (intertwined) tangles on the seafloor.

Fig. 2. DNA shown on three length scales. Smallest scale (left) shows a single helical repeat of the double-helix structure (sugar-phosphate chains andbase-pairs). Intermediate scale (middle) suggests how many consecutive helical repeats form the very long and slender DNA molecule. Largest scale (right)shows how the molecule ultimately curves and twists in forming supercoils (plectonemic or solenoidal). (Courtesy: Branden and Tooze (�) 1999 FromIntroduction to Protein Structure by Carl Branden & John Tooze. Reproduced by permission of Garland Science/Taylor & Francis, LLC) [3] and Lehningeret al. (Copied with permission from W.H. Freeman) [4]).

is accounted for by imposing bounds on helical curvatureand torsion. The formation of self-contact in the equilibriagenerated from ‘closed’ or ‘circular’ rods (e.g., representativeof DNA plasmids) is examined in Coleman et al. [12,14–16]using numerical energy minimization. The mathematical ex-istence of such solutions is deduced in Gonzalez et al. [13]by careful formulation of the geometric excluded volume con-straint on self-intersection. The excluded volume constraint isformulated in terms of rod centerline curvature in Schurichtand Mosel [10] and appended via Lagrange multiplier to theEuler–Lagrange equation for rod equilibrium with self-contact.The analysis of ‘open’ rods (e.g., rods that do not close uponthemselves) requires consideration of two sets of boundaryconditions through which loads may also be applied. A nu-merical study of the self-contacting equilibria of ‘open’ rodsreveals the bifurcations generated by varying compression ortension and twist applied at the boundaries; refer to Colemanet al. [16] and Heijden et al. [11]. A recent extension in Hei-jden et al. [9] considers cases where the rod is constrained tolie on the surface of a cylinder. Open questions regarding theanalysis of rods with self-contact are emphasized in Heijdenet al. [9] by the lament “We are still far from understandinganalytically the solutions of the Euler–Lagrange equations forgeneral contact situations. Even if we limit ourselves to globalminimizers of an appropriate energy functional, we can provelittle about the form of solutions as soon as contact is taken intoaccount.”

In contrast to the equilibrium formulations above, very fewdynamical formulations of rod theory have been proposed thatincorporate self-contact. Nevertheless, such formulations en-able one to explore the dynamic evolution of self-contactingstates and possible dynamic transitions between them. For in-stance, the slow twisting of the filament treated in Goyal et al.[17] ultimately induces a sudden dynamic collapse of an inter-mediate helical loop into an intertwined form. An approximatedynamical formulation is also presented in Klapper [18] whereinertial effects are ignored in favor of dissipation and stiffnesseffects.

S. Goyal et al. / International Journal of Non-Linear Mechanics 43 (2008) 65–73 67

In this paper, we revisit the slow twisting of a filament [17]with the objective to develop a fundamental understanding ofthe dynamic evolution of its intertwined states. In particular,we describe how intertwined states result from a sudden col-lapse of helically looped states through a rapid conversion oftorsional to bending strain energy. The remainder of this paperis organized as follows. Sections 2 and 3 summarize a compu-tational dynamic rod model that incorporates self-contact (referto Goyal [7]). Section 4 presents an illustrative example of anon-homogeneous rod subject to pure torsion. Results highlightthe dynamic evolution from straight to looped to intertwinedstates following a dramatic collapse to self-contact. We closein Section 5 with conclusions.

2. Computational rod model—a summary

The rod segment illustrated in Fig. 3 is a thin (one-dimensional) element that may undergo two-axis bending andtorsion in forming a three-dimensional space curve. This curverepresents the rod centerline which, in the context of double-stranded DNA, represents the helical axis of the duplex. Wedevelop the dynamical model by employing the classical ap-proximations of Kirchhoff and Clebsch [19] as detailed inGoyal [7]. A summary is provided here.

2.1. Rod kinematics, constitutive law and energy

Consider the infinitesimal element of a Kirchhoff rod shownin Fig. 3. The three-dimensional curve R(s, t) formed by thecenterline is parameterized by the arc length coordinate s andtime t . The body-fixed frame {ai} at each cross-section is em-ployed to describe the orientation of the cross-section with re-spect to the inertial frame {ei}. The angular velocity �(s, t) ofthe cross-section is defined as the rotation of the body-fixedframe {ai} per unit time relative to the inertial frame {ei} andsatisfies(

�ai

�t

)ei

= � × ai , (1)

e2e3

e1 R (s, t)

�s

f + �f�1

�3

�2q + �q

Q

t

S

F

f

q

Fig. 3. Free body diagram of an infinitesimal element of a Kirchhoff rod.

where the subscript specifies the reference frame relative towhich the derivative has been taken. We also define a ‘curvatureand twist vector’ �(s, t) as the rotation of the body-fixed frame{ai} per unit arc length relative to the inertial frame {ei} whichsatisfies(

�ai

�s

){ei }

= � × ai . (2)

In a stress-free state, the rod conforms to its natural ge-ometry defined by �0(s). The difference {�(s, t) − �0(s)} re-sults in an internal moment q(s, t) at each cross-section of therod. The relationship between the change in curvature/twist{�(s, t) − �0(s)} and the restoring moment q(s, t) is governedby a constitutive law for bending and torsion. While many gen-eralizations of the constitutive law are discussed in Goyal [7],in this study we employ the linear elastic law

q(s, t) = B(s)(�(s, t) − �0(s)), (3)

where B(s) is a positive definite stiffness tensor that is a pre-scribed function of position s. As demonstrated in carefullycontrolled, laboratory-scale tests by Heijden et al. [11], the lin-ear elastic material law assumed above is capable of capturingthe bifurcation behaviors responsible for the looping and inter-twining of thin rods. While these experiments were conductedusing one type of material (nickel titanium alloy nitinol), otherapplications may require the specification of non-linear mate-rial laws. Such generalizations can be readily accounted for inthe formulation discussed in Goyal [7]. The resulting strain en-ergy density is therefore

Se(s, t) = 12 (�(s, t) − �0(s))

TB(s)(�(s, t) − �0(s)). (4)

We further employ a diagonalized form of B(s) by choosing{ai} to coincide with the ‘principal torsion–flexure axes’ ofthe cross-section (refer to Love [19]). In particular, a1 and a2are in the plane of the cross-section and are aligned with theprincipal flexure axes while a3 is normal to the cross-sectionand coincides with the tangent t . The resulting diagonal formof the stiffness tensor B(s) is

B(s) =[

A1(s) 0 00 A2(s) 00 0 C(s)

], (5)

where A1(s) and A2(s) are bending stiffnesses about the prin-cipal flexure axes along a1 and a2, respectively, and C(s) is thetorsional stiffness about principal torsional or ‘tangent’ axis a3.Furthermore, in the results that follow, the rod is assumed to beisotropic1 but non-homogeneous (i.e., A1(s)=A2(s)=A(s)).The resultant of the stress-distribution at any cross-section notonly results in a net internal moment q(s, t), but also net tensileand shear forces f (s, t) which remain unknown in this formu-lation.

1 The rod is assumed to have circular cross-section in this study withaxi-symmetric bending stiffness.


The kinetic energy of the rod depends upon the centerlinevelocity �(s, t) and the cross-section angular velocity �(s, t).Let m(s) denote the mass of the rod per unit arc length andI (s) denote the tensor of principal mass moments of inertia perunit arc length. Then the rod kinetic energy density is

Ke(s, t) = 12�(s, t)TI (s)�(s, t) + 1

2�(s, t)Tm(s)�(s, t). (6)

We choose the vectors �(s, t), �(s, t), �(s, t) and f (s, t) asfour unknown field variables in the formulation below. Thekinematical quantities �(s, t), �(s, t) and �(s, t) can be readilyintegrated to compute the rod configuration R(s, t) and thecross-section orientation as given by {ai(s, t)}; refer to Fig. 3and to Goyal [7].

Depending upon the application, the rod may also interactwith numerous external field forces including those producedby gravity, a surrounding fluid medium, electrostatic forces,contact with other bodies or with the rod itself, etc. The resul-tant of these external forces and moments per unit length is de-noted by F(s, t, . . .) and Q(s, t, . . .), respectively. In general,these quantities may be functionally dependent on the kinemat-ical quantities �(s, t), �(s, t) and �(s, t) in addition to the rodconfiguration R(s, t).

We next specify the four field equations required to solve forthe four vector unknowns {�, �, �, f }. In the field equations, weemploy partial derivatives of all quantities relative to the body-fixed frame {ai} and recall the following relations to the partialderivatives relative to the inertial frame for a vector quantity �(refer to Greenwood [20]):

(��

�t

){ai }

=(

��

�t

){ei }

− � × � and(��

�s

){ai }

=(

��

�s

){ei }

− � × �. (7)

For notational convenience, henceforth, we drop the subscriptfor the body-fixed frame.

2.2. Equations of motion

The balance law for linear momentum of the infinitesimalelement shown in Fig. 3 becomes

�f

�s+ � × f = m

(��

�t+ � × �

)− F (8)

and that for angular momentum becomes

�q

�s+ � × q = I

��

�t+ � × I� + f × t − Q. (9)

Here, t (s, t) is the unit tangent vector along the centerline(directed towards increasing arc length s) and the internal mo-ment q(s, t) = B(s)(�(s, t) − �0(s)) upon substitution of theconstitutive law Eq. (3).

2.3. Constraints and summary

The above formulation is completed with the addition oftwo vector constraints. The first enforces inextensibility andunshearability which take the form

��

�s+ � × � = � × t . (10)

The second follows from continuity requirements for � and� in the form of the compatibility constraint

��

�s+ � × � = ��

�t. (11)

Detailed derivations of these constraints are provided inGoyal [7].

The four vector equations (8)–(11) in the four vector un-knowns {�, �, �, f } result in a 12th order system of non-linearpartial differential equations in space and time. They are com-pactly written as

M(Y, s, t)�Y

�t+ K(Y, s, t)

�Y

�s+ F(Y, s, t) = 0, (12)

where Y (s, t) = {�, �, �, f } and the operators M, K and F aredescribed in Goyal et al. [7,21]. These equations are not in-tegrable in general and thus we pursue a numerical solutionas detailed in Goyal et al. [7,21]. In particular, we discretizethe equations above by employing a finite difference algorithmusing the generalized-� method (refer to Chung and Hulbert[22]) in both space and time. Doing so yields a method thatis unconditionally stable and second-order accurate. A singlenumerical parameter can be varied to control maximum nu-merical dissipation. The difference equations so obtained areimplicit and their solution must satisfy the rod boundary con-ditions. The boundary conditions are satisfied using a shootingmethod in conjunction with Newton–Raphson iteration. In ad-dition, this formulation also incorporates the forces generatedby self-contact which, being central to the objective of this pa-per, we describe in some detail below.

3. Numerical formulation of dynamic self-contact

A numerical formulation of self-contact begins with first de-termining the likely sites where self-contact exists or will soonoccur. An efficient search strategy for these sites [7] is as fol-lows. Consider two remote segments of the discretized rod thatare approaching contact as shown in Fig. 4. The lower segmentcontains three spatial grid points denoted as 1, 2 and 3 whilethe upper segment contains one grid point denoted as 4. Gridpoint 4 is likely to interact with the grid point 2 as the twosegments approach each other. We introduce a screening aper-ture of angle � formed by a pair of conical surfaces centeredat each grid point (illustrated at grid point 2 in Fig. 4). We usethis aperture to efficiently search for only those points that maypotentially interact through self-contact. This aperture specif-ically excludes non-physical ‘contact’ forces between nearbygrid points on the same segment (such as 1, 2 and 3 in lower


Fig. 4. Two remote segments of a rod approaching contact. A screeningaperture is defined by a pair of conical surfaces constructed at each grid point.This aperture leads to an efficient numerical search for regions of self-contact.

segment). The aperture reduces to the plane of the rod cross-section as � → 0◦, and it expands to the entire space as � →180◦.

During simulation, the separation d between each pair of gridpoints is measured. A repulsive (contact) force is introducedbetween these grid points only if two conditions are met: (1) thedistance d is less than a specified tolerance and (2) the two gridpoints lie within each other’s screening aperture. This searchstrategy ensures that the contact forces are approximately nor-mal to the rod surfaces and also allows for sliding contact. Theinteraction force can in general be a function of d and d (theapproach speed) and it is included in the balance of linear mo-mentum Eq. (8) through the distributed force term F. Exam-ple interaction laws that can be employed include (attractive-repulsive) Lennard-Jones type (refer to, for example Schlick etal. [23]), (screened repulsion) Debye–Huckle type (refer to, forexample Schlick et al. [24]), general inverse-power laws (re-fer to, for example Klapper [18]), and idealized contact lawsfor two solids (refer to, for example Heijden et al. [11] andColeman et al. [12]). In the specific case of DNA, one mightintroduce a fictitious charged and cylindrical surface that cir-cumscribes the molecule to capture the repulsive effects of thenegatively charged backbone.

4. Results

The computational model above is used to explore the dy-namic evolution of an intertwined state induced by slowly in-creasing the twist applied to one end of an elastic rod. The nu-merical solutions reveal three major behaviors: (1) the torsional

Fig. 5. A non-homogenous rod subject to slowly increasing twist created byrotating the right end about the e2 (loading) axis. The right end is otherwiserestrained in rotation and translation. The left end is fully restrained in rotationand translation except that it is free to slide along the loading axis.

buckling of an initially straight rod into the approximate shapeof a helix, (2) the dramatic collapse of this helix to a near-planarloop with self-contact at a single point and (3) the subsequentintertwining of the loop with multiple sites of self-contact.

4.1. Illustrative example

Fig. 5 defines an illustrative example which consists of aninitially straight, linearly elastic rod subjected to monotonicallyincreasing twist at the right end at s = 0. This end cannotmove and it is otherwise constrained in rotation (no rotationabout the principal axes a1 and a2). The left end at s = L isfully restrained in rotation and cannot translate in the transverse(a1.a2) plane. This end, however, may translate along the e2axis. Constraining the ends in rotation conserves the number oftwisting turns added to the rod which is equivalent to conservingthe topological invariant called the ‘linking number’ definedin Section 4.4. By contrast, having any rotational freedom atany end might allow the rod to swivel at that end releasing thelinking number and resulting in distinctly different behavior.An insightful discussion of these and other boundary conditionsis provided in [11].

The material and geometric parameters that define the exam-ple are listed in Table 1 together with basic discretization pa-rameters used in the numerical algorithm; refer to Goyal et al.[7,21] for a complete description of the numerical parameters.The example rod has a circular cross-section which varies alongits length. In particular, the central portion of the rod (middle25%) is necked down to a smaller diameter that is 10% smallerthan the end regions. We have chosen this non-homogenousrod to illustrate both the generality of the computational modelas well as to promote torsional buckling and subsequent inter-twining within the (‘softer’) central portion. The small 10% re-duction in the diameter produces a significant (≈ 35%) reduc-tion in torsional stiffness (C(s) = GJ 3) and bending stiffness(A(s) = EJ 1,2) in the central portion.


Table 1Example rod properties and simulation parameters

Quantity Units (SI) Value/Formula

Young’s Modulus, E Pa 1.25 × 107

Shear Modulus, G Pa 5.0 × 106

Diameter, D m See Fig. 5Length, Lc m 1.0Rod density, c kg/m3 1500Fluid density, f kg/m3 1000Normal drag coefficient, Cn – 0.1Tangential drag coefficient, Ct – 0.01Temporal step, �t s 0.1Spatial step, �s m 0.001

Cross-section area m2 Ac = D2

4Mass/length kg/m m = cAc

Area moments of inertia (bending) m4 J1,2 = AcD2

16

Area moment of inertia (torsion) m4 J3 = AcD2

8Mass moment of inertia/length kg/m I = cJ

As a representative law for self-contact, we choose for thisexample the following form for the repulsive force:

|Fcontact| = cAc

(k1

(d − 0.5D)k2+ k3

dd|d|k4

), (13)

with example parameters: k1 = 10−7 m4/s2, k2 = 3, k3 = 10−6

and k4 = 1. This contact law is one of many possible that cap-ture both non-linear repulsion and dissipation. This examplecontact law retains an inverse power dependence on the sepa-ration parameter (d − 0.5D) common to (the repulsive part of)the Lennard-Jones and Debye–Huckle formulations as well asa similar inverse power law employed in [18]. The results thatfollow are rather insensitive to changes in the specific parame-ter values selected above. However, the stability of the numer-ical algorithm is sensitive to the functional form of the contactlaw for the following reason. As will be seen in the examples,the first approach to self-contact follows from a dramatic dy-namic collapse to contact and thus the contact force can buildvery rapidly. The addition of dissipation in (13), which does notalter the steady-state prediction of the looped and intertwinedstates, helps stabilize the numerical algorithm by limiting theotherwise destabilizing growth of the contact force.

In addition to the contact law above, the only other bodyforce considered is a dissipative force. As one example, weintroduce the viscous drag imparted by a surrounding fluidenvironment in the form of the standard Morison drag law [25].This distributed drag, which manifests itself in the balance oflinear momentum Eq. (8) through the distributed force term F,is computed as [21]

Fdrag = − 12fD{Cn|� × t |t × (� × t ) + Ct(� · t )|� · t |t},

(14)

Here, Cn is the normal (form) drag coefficient, Ct is the tan-gential (skin friction) drag coefficient, and f is fluid density.Example values of these parameters are reported in Table 1.

Any form of fluid and material dissipation would tend toslow the dynamic collapse of the helically deformed rod into

Fig. 6. Prescribed angular (twist) velocity at the right end (note: not to scale).

a loop with self-contact. Therefore, any quantitative predictionof the time scale of this collapse (e.g., for a cable in the oceanenvironment, or DNA in a biological buffer) would first requirean accurate characterization of the relevant dissipation mecha-nisms. In our example, we chose a Morison drag law to demon-strate how one may incorporate realistic hydrodynamic drag inthe context of an oceanographic cable.

4.2. Evolution of self-contact and intertwining

By increasing the rotation (twist) slowly at the right end,the internal torque eventually reaches the bifurcation conditionassociated with the classical torsional buckling of a straightrod (refer to Zachmann [26]). This rotation is generated byprescribing the angular velocity component �3 at the right endas shown in Fig. 6 (not to scale). In addition, the left end isallowed to translate freely during the first 30 s and is then heldfixed to control what would otherwise be an exceedingly rapidcollapse to self-contact as described in the following.

As the right end is initially twisted by a modest amount,the rod remains straight. There is an abrupt change, however,when the twist reaches the bifurcation value associated withthe Zachmann buckling condition [26] and the straight (trivial)configuration becomes unstable. This occurs at approximately16 s in this example. The computational model captures thisinitial instability as well as the subsequent non-linear motionthat leads to loop formation and ultimately to intertwining.

Fig. 7 illustrates four representative snapshots during the dy-namic evolution of an intertwined state. The geometry just af-ter initial buckling is approximately helical as can be observedin the uppermost snapshot (20 s). Notice that the rod center-line appears to make a single helical turn as predicted fromthe fundamental buckling mode of the linearized theory (re-fer to Zachmann [26]). The superimposed black stripe recordsthe computed twist distribution of the rod for this state whichexhibits nearly four complete turns; refer to the discussion oftwist and major topological transitions below. As this twist isincreased, the rod continuously deforms into a larger diameterhelical loop and the left end slides substantially to the right asshown by the second image (25 s).

Upon greater twist, the left end continues to slide towardsthe right end and the helical loop continues to rotate out of the


Time = 20 Sec

Time = 25 Sec

Time = 29 Sec

Time = 32 Sec

Strain EnergyDensity Se

(Joules/m)

2

1

0

x 10-4

Fig. 7. Snapshots at selected times during the transition from a buckledhelical form (time = 20 s) to an intertwined form (time = 32 s). Black stripesuperimposed on the first form illustrates the twist distribution. Color indicatesstrain energy density.

plane of this figure. Eventually the loop undergoes a loss ofstability followed by a rapid dynamic collapses into self-contactin forming a nearly planar loop. The collapsed loop is shownby the third image (which occurs at approximately 29 s).

The dynamic collapse can be anticipated from stability analy-ses of the equilibrium forms of a rod under similar loading con-ditions; refer to Lu and Perkins [27] and studies cited therein.The snapshot at 25 s shows the three-dimensional shape of therod just prior to dynamic collapse. Here, the apex of the loophas rotated approximately 90◦ about the vertical (e1) axis sothat the tangent at the apex is now orthogonal to the loading(e2) axis. This was the noted secondary bifurcation conditionin Lu and Perkins [27] at which the three-dimensional equilib-rium form loses stability.

The collapsed loop, however, is very sensitive to the increas-ing twist and rapidly continues to rotate about the vertical (e1)

axis leading to intertwined forms with multiple sites of self-contact. A snapshot of a fully intertwined loop is illustrated atthe bottom of Fig. 7 (32 s). The strain energy density (colorscale in Fig. 7) reveals that the strain energy becomes highlylocalized to the apex of the intertwined loop where the curva-ture is greatest. The decomposition of this strain energy intobending and torsional components provides significant insightinto the dynamic evolution of an intertwined state as discussednext.

4.3. Energetic transitions

Fig. 8 summarizes the energetics of this process by illustrat-ing how the bending and torsional strain energy componentscontribute to the total strain energy. Starting at time zero, theinitially straight rod remains straight and the applied twist sim-ply increases the torsional strain energy. This elementary, pure-twisting of the straight rod ceases at approximately 18 s with thefirst bifurcation due to torsional buckling (refer to Zachmann[26]). The torsional strain energy achieves its maximum at thisstate and immediately thereafter the rod buckles into a three-

1.5

1

0.5

0

0 5 10 15 20 25 35

Time (seconds)

BendingEnergy

TorsionalEnergy

Total StrainEnergy

Work

x 10-4

Ene

rgy

(Jou

les)

(a)

(b)(d)

(c)

Fig. 8. The bending, torsional and total strain energy during the dynamicevolution of an intertwined state. The work done by the applied twist is alsoreported.

dimensional form resembling a shallow helix (a). This transitionis accompanied by a conversion of torsional to bending strainenergy. This conversion is dynamic and markedly increases asthe rod is twisted further while developing a distinctive loop(b). The apex of this loop rotates further out of plane during thisstage. Just prior to 29 s the apex becomes nearly orthogonal tothe loading axis (original axis of the straight rod) which marksthe secondary bifurcation [27] that generates an extremely fastdynamic collapse to self-contact. The resulting loop with self-contact is nearly planar (c). During this secondary bifurcation,the rod loses both torsional and bending strain energies untilself-contact and, thereafter intertwining begins. As intertwin-ing advances (d), the torsional strain energy continues to de-crease while the bending strain energy increases once more. Inaddition, the bending strain energy becomes localized to theapex of the loop due to the significant and increasing curva-ture developed there; refer also to snapshot at 32 s in Fig. 7.In the case of DNA forming plectonemes, such localized strainenergy might possibly be the forerunner of the non-linear ‘kink-ing’ of the molecule as proposed recently in Wiggins et al. [28].Fig. 8 also illustrates the total strain energy and the work doneby twisting the right boundary. The energy difference betweenthe work done and the total strain energy derives from the sig-nificant kinetic energy during this process as well as the dissi-pation developed from the included fluid drag.

We emphasize that the results above are insensitive to anyreasonable changes in the parameters in the contact law (13).Consider that the maximum dynamic contact force generatedin this example is of the order of 10−3 N and that the corre-sponding contact energy density remains an order of magnitudesmaller than the rod strain energy density Se. Thus, when inte-grated over the length of the rod, the contribution of the contactenergy remains negligible relative to the total rod strain energy.That said, for very long (hence flexible) DNA filaments withrelatively large sub-domains in close proximity, the energy of


self-interaction (electrostatic potential) may indeed grow to aconsiderable portion of the structural deformation energy of themolecule. In these instances, the equilibrium conformation ofintertwined DNA would be materially affected by the electro-static interaction.

4.4. Topological transitions

It is interesting to observe that the topological changes forthe example rod above are also exhibited by DNA during su-percoiling. As discussed in Calladine et al. [1], the above con-version of torsional strain energy to bending strain energy forDNA is more frequently described topologically as the con-version of twist to writhe. We explore this conversion in theabove example after briefly reviewing the definitions for twistand writhe.

Twist (Tw) is a kinematical quantity representing the totalnumber of twisted turns along the rod centerline as computedfrom

Tw = 1

2

∫ LC

0(� · t ) ds. (15)

Writhe (Wr) is defined as the average number of crossoversof the rod centerline when observed over all possible views ofthe rod (refer to Calladine et al. [1]). For our initially straightconfiguration, Wr = 0. At the first self-contact shown by thesnapshot at 29 s in Fig. 7, Wr = 1. The writhe then continuesto increase to Wr = 2 for the intertwined state at 32 s in Fig. 7.The writhe is purely a function of the space curve defining therod centerline and it may also be positive or negative dependingon whether the crossing is right-handed or left-handed (referto Calladine et al. [1]). In our illustrative example, the sumTw + Wr equals the number of rotations of the right boundaryand this sum is called the Linking number Lk.2 Refer to Fuller[29] and White [30] for the proof of conservation of the Linkingnumber (Lk).

In our example, the initial twisting phase rapidly increasesLk from 0 to approximately 4, all in the form of twist, prior tothe first bifurcation (torsional buckling) as illustrated in Fig. 9.An additional increase in Lk (end rotation) of less than 1

2 (turn)produces all of the sudden transitions noted above. Followingthe first bifurcation, Wr increases from 0 to 1 at self-contact(29 s) and Tw correspondingly reduces so that the sum Wr+Twremains equal to Lk. Following the first self-contact, the loopcontinues to rotate as it intertwines. In doing so, every halfrotation of the loop establishes an additional contact site therebyincreasing Wr by 1 and reducing Tw by 1. At 32 s, Wr is slightlylarger than 2. Thus, we observe two crossovers in any threeorthogonal views of the snap-shot at 32 s shown in Fig. 7. Thereis a compensatory loss in Tw as shown in Fig. 9.

It should also be noted if self-contact is ignored, as has of-ten been done in some prior studies of the looping of rods, thenumerical solution for the rod may allow it to artificially ‘cutthrough itself’ leading to entirely different and non-physical

2 This is not true, in general, for other boundary conditions that allowrotations about the other axes (i.e., cases where �1,2 �= 0 at the boundaries).

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5

4

4.5

LinkingNumber,

Lk

Twist,Tw

Writhe,Wr

Time, t (s)

Fig. 9. Conversion of twist (Tw) to writhe (Wr) during loop formation andintertwining. The linking number Lk = Tw + Wr is equivalent to the numberof turns prescribed at the right end of the rod in this example.

results. Following each ‘cut’, both Wr and Lk are reduced dis-continuously by 2. Examples of this readily follow from thepresent computational formulation by simply eliminating thecontact force. However, doing so leads to non-physical discon-tinuous changes in Wr and Lk following artificial ‘cuts’ throughthe rod. Thus, modeling self-contact is fundamentally neces-sary when one endeavors to understand the pathway(s) leadingto the intertwined loops.

5. Conclusions

This paper summarizes a computational rod model that cap-tures the dynamical evolution of intertwined loops in rods undertorsion. A major feature is the explicit formulation of dynamicself-contact. An illustrative example is selected which revealsa fundamental understanding of how loops first form, then col-lapse, and then intertwine. This knowledge may also promotean understanding of how long cables form ‘hockles’ and howDNA molecules form plectonemic supercoils.

Numerical simulations reveal that an originally straight rodundergoes two bifurcations in succession as twist is added. Thefirst bifurcation is elementary and occurs at the (Zachmann)buckling condition where the trivial equilibrium becomes un-stable and the rod buckles into the approximate shape of a shal-low helix. Upon increasing twist, this helix grows in amplitudeto form a distinctive loop. In doing so, the apex of this loop con-tinues to rotate towards the out-of-plane direction. When theapex ultimately becomes orthogonal to the loading axis (axisof the original straight rod), the loop experiences a secondarybifurcation and a sudden dynamic collapse into a near-planarloop with self-contact. As twist is again added, the near-planarloop rotates upon itself becoming intertwined with multiplesites of self-contact. The energetics leading to the intertwinedform confirm the large exchange of torsional strain energy for


bending strain energy which becomes increasingly localized tothe apex of the loop. These transitions parallel the dynamicconversion of twist (Tw) to writhe (Wr) during this process.

Acknowledgments

The authors (NCP and SG) gratefully acknowledge the re-search support provided by the US Office of Naval Research(Grant N00014-00-1-0001), the Lawrence Livermore NationalLaboratories (Grant B 531085), and the US National Sci-ence Foundation (Grants CMS 0439574 and CMS 0510266).The author (CLL) performed work under the auspices ofthe US Department of Energy by the University of Califor-nia, Lawrence Livermore National Laboratory under contractNo.W-7405-ENG-48.

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non-linear dynamic intertwining of rods with self-contact

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