symmetric equilibria of a thin elastic rod with self-contactslcvm · symmetric equilibria of a thin...

Symmetric equilibria of a thin elastic rod

with self-contacts

By E. L. Starostin

The Bernoulli Institute, Swiss Federal Institute of Technology, Lausanne

CH-1015, Switzerland

The thin elastic rod is a conventional model for the mesoscale structure of DNA.The symmetric multi-leafed closed equilibria with one multiple self-contact pointare examined. The contact force is assumed to be pointwise and frictionless. Thebifurcation diagram is presented for first branches of two- and three-leafed families.

Keywords: thin elastic rod, equilibrium, boundary value problem, self-contact.

1. Introduction

The thin elastic rod was proposed as a model for DNA as early as in late 70-s(Benham 1977). The model was elaborated on the basis of Kirchoff’s theory oflinear elastic rods. The elastic properties of the rod are characterized by threestiffness coefficients: two bending and one torsional. It is possible to estimate theireffective values for the DNA rod from the results of experiments.

According to Kirchhoff’s kinetic analogy, the equations of equilibrium governinga thin elastic rod under the forces and moments only acting at its ends, are formallyanalogous to the equations of motion of a gyrostat in a gravitational field. Thesymmetric case of the two equal bending stiffnesses is an analogue of the gyrostatwith two equal moments of inertia. In 1969 Ilyukhin derived the equations describingthe coordinates of the centreline in a special cylindrical reference frame. Theirexplicit expressions as functions of the arc length include the elliptic integrals andthe Jacobian elliptic functions. Dichmann et al. (1996) developed the Hamiltonianformulation of the problem to compute the stationary states also in the case whenno closed-form expression exists for the general solution of the initial value problem.

A sufficiently long segment of DNA is often in contact with itself or with thebending protein that justifies the formulation of a boundary value problem (BVP)for a loop shape.

An interesting special case is a circular (i.e. smoothly closed) DNA because, onthe one hand, the applied constraint may be efficiently controlled by varying the pa-rameters of the DNA’s double helix and, on the other hand, their spatial structurehas been extensively studied experimentally. The solutions of the correspondingBVP were obtained by Le Bret (1984) and classified by Starostin (1996). A param-eter continuation was used starting from a (multicovered) circle. With the changeof the parameter (the effective twist density), the shape of equilibrium evolves sothat the centreline coils around a toroid. For every type of the coil, there exists acritical value of the parameter when the centreline has one point of self-crossing.The shape then looks like a propeller with multiple blades.

Article submitted to Royal Society TEX Paper

2 E. L. Starostin

Another way to obtain the same configuration is to consider first a more generalBVP for a closed rod when the tangents to the centreline have different orientationsat the ends. Such equilibria are called simple loops here. In some particular case, thisBVP can be solved analytically and a one-parameter family of solutions is obtained(Starostin 2000). As the parameter varies, the simple loop evolves from the circleto the “semi-figure-8” configuration, which is attained for the critical value of theelliptic modulus. Among these solutions a countable set of loops may be selectedto make a smoothly closed propeller-like configurations having one contact pointbut not self-interacting. The figure-8 represents the first shape in the series thatcorresponds to the “resonant” values of the parameters. The subsequent equilibriaare essentially three-dimensional.

The further evolution of these symmetric shapes is computed taking into accountthe additional contact force in the multiple self-contact point. For every pair ofthe contacting rod fragments, this pointwise frictionless force is assumed to beorthogonal to both tangents. This model was adopted by Le Bret (1984) as well asby Julicher (1994) who concentrated on two-leafed shapes. More recently, Coleman& Swigon (2000) carried out a study of unknotted equilibria in the finite thicknessrod model.

We build the configurations with the self-contact force by assembling them fromthe known simple loop elements. The key part is played by the symmetry propertiesof the solutions and the arrangement of their proper orientations. The result deliversnon-self-penetrating conformations of the rod. We restrict our consideration to therods of zero thickness.

Each shape consists of an integer number of identical simple loops, parametersof which satisfy certain conditions. In the general case, the simple loops constitutea number of surfaces in a 3D parameter space. Finding a solution of only onealgebraic equation is needed to define the parameters of a particular loop. By solvingan additional algebraic equation it is possible to compute a countable set of one-parameter families of symmetric solutions with self-contact. A bifurcation diagramis presented for the first branches of the configurations with two and three leaves.It complements the diagrams obtained earlier for the self-penetrating model.

2. Model and equations

The thin elastic rod is assumed to be inextensible, unshearable and uniform. Itscentreline is parametrized with the arc length s. The points on the centreline aredescribed by their radius vector r(s) with respect to an origin O. We assume that

the tangent vector t(s) = dr(s)ds exists everywhere. We define the orthonormal triad

e1(s) ≡ t(s), e2(s), e3(s), the unit vectors e2, e3 directing along the principal axesof the cross section of the rod.

The vector ω is the angular velocity of rotation of the triad e1, e2, e3 as it movesalong the centreline with the unit velocity. It may be represented as a sum of three

components ω =3∑i=1

ωiei, ω1 is the twist of the rod.

The equilibrium state of the rod is described in the principal reference frame by

Article submitted to Royal Society

Symmetric equilibria of elastic rod 3

the balance equations (Love 1927)

dF

ds+ ω × F + f = 0 ,

dM

ds+ ω ×M + t× F + m = 0. (2.1)

Here F(s) denotes the resultant of the internal forces acting on the cross sectionand M(s) the net moment of these forces. We assume that F(σ) (M(σ)) is theforce (moment) with which one part of the rod (s > σ) acts on the other part.In equation (2.1), f(s) and m(s) are the densities of external forces and momentsapplied to the rod.

The first case we consider is one when no such external forces and momentsact, i.e. we put f = 0,m = 0. Then the first equation (2.1) implies F = const. inthe absolute space. Let γ be the unit vector in the direction of F. We may write

F = Pγ, P = ‖F‖ ≥ 0, ‖γ‖ = 1, γ =3∑i=1

γiei.

The Hooke constitutive relation completes the equilibrium equations

Mi =

3∑

j=1

Bij(ωj − ω0j ), M =

3∑

i=1

Miei,

ω0j , j = 1, 2, 3, are the twist and curvatures of the rod in its relaxed, non-deformed

state. We assume that the rod is naturally straight (i.e. ω02 = ω0

3 = 0) but it maybe intrinsically twisted. The rod is supposed to be symmetric and we put Bij = 0for i 6= j, Bii ≡ Bi, B2 = B3.

Under the above assumptions we come, after normalization, to the equilibriumequations

dω1

ds= 0,

dω2

ds+ ω3(d− ω1)−

p

2γ3 = 0,

dω3

ds− ω2(d− ω1) +

p

2γ2 = 0, (2.2)

where b = B1/B2, p = 2P/B2, d = b(ω1 − ω01).

Equations (2.2) allow for the first integrals

ω1 = ω∗1 = const., (2.3)

γ1d+ ω2γ2 + ω3γ3 = l = const., (2.4)

ω22 + ω2

3 + pγ1 = h = const. (2.5)

Equation (2.3) implies immediately that d = b(ω∗1 − ω01) = const.

We choose an absolute coordinate system Oξηζ fixed in space such that thedirection of the ζ-axis is opposite to γ (figure 1). The three Euler angles ψ, φ, θare chosen to describe rotation of the principal triad with respect to the absolutecoordinates. Then we have in these coordinates −γ = (cos θ, sin θ sinφ, sin θ cosφ)T

and

ω1 =dφ

ds+

dψ

dscos θ, ω2 =

dψ

dssin θ sinφ+

dθ

dscosφ,

ω3 =dψ

dssin θ cosφ− dθ

dssinφ.


4 E. L. Starostin

-F

M

te

e

e

1

2

3

γ ρ

θ

α

ξ

η

ζ

Figure 1. To the description of the shape of the rod.

The integrals (2.4) and (2.5) take the form

γ1d− (1− γ21)

dψ

ds= l, (2.6)

(1− γ21)

(dψ

ds

)2

+

(dθ

ds

)2

+ pγ1 = h. (2.7)

After eliminating the derivative dψds from equations (2.6) and (2.7), we obtain

(dγ1

ds

)2

= f(γ1), f(γ1) = (h− pγ1)(1− γ21)− (l − γ1d)

2. (2.8)

We are here interested in the case of non-zero end force, when the cubic poly-nomial f(γ1) may be written as

f(γ1) = p(γ1 − g1)(γ1 − g2)(γ1 − g3), −1 ≤ g1 ≤ g2 ≤ 1 ≤ g3. (2.9)

Since |γ1| ≤ 1, a real solution of equation (2.8) is possible only in the intervalg1 ≤ γ1 ≤ g2.

Equation (2.8) may be integrated to find

γ1 = g1 + (g2 − g1) sn2 Ω(s− s0), (2.10)

where Ω2 = p(g3 − g1)/4, sn is the Jacobi elliptic sine of the modulus k, k2 =(g2 − g1)/(g3 − g1).

It is convenient to express the shape of the centreline in space in the cylindricalcoordinates ρ, α, ζ (see the solution of problem 5 on pp. 87–88 in (Landau & Lifshitz1970)): ζ = ρ cosα, η = ρ sinα. The equations for the centreline in these coordinateswere first obtained in the general form by Ilyukhin as early as in 1969 (see also hismonograph (Ilyukhin 1979)):

ρ =2√d2 − l2 + h− pγ1

p, (2.11)

dα

ds=

p(lγ1 − d)2(d2 − l2 + h− pγ1)

, (2.12)

dζ

ds= −γ1. (2.13)



The last two equations may be readily integrated to get

α− α0 = − l2(s− s0) +

l(d2 − l2 + h)− pd2Ω(d2 − l2 + h− pg1)

Π(Ω(s− s0), n, k), (2.14)

n =p(g2 − g1)

d2 − l2 + h− pg1,

ζ − ζ0 = −g3(s− s0) + 2

√g3 − g1p

E(Ω(s− s0), k), (2.15)

where E(u, k) =∫ u0

dn2 w dw, Π(u, n, k) =∫ u0

dw1−n sn2 w

are the incomplete ellipticintegrals of the second and third kind, respectively.

It may be shown that equations (2.11) and (2.12) take a simpler form in theparticular case when n = 1. The cylindrical coordinates then still may be defined,namely,

ρ =4Ωk

p| cn Ω(s− s0)|, (2.16)

α− α0 = − l2(s− s0)±

±πbΩ(s−s0)+K

2 Kc, if Ω(s− s0) 6= (2j + 1) K,

π2 + πj, if Ω(s− s0) = (2j + 1) K, j = 0,±1,±2, . . . ,

(2.17)

where bxc signifies the greatest integer not greater than x, K = K(k) is the completeelliptic integral of the first kind. The ζ-coordinate can be found from equation(2.15). Clearly, the centreline intersects the ζ-axis.

The last case was specifically considered by Shi & Hearst (1994, Appendix C)though taking the limit was not carried out correctly and, as a result, their expres-sion for the polar angle is different from equation (2.17).

3. Smoothly closed configurations with self-intersection

We start our consideration of closed configurations with those of a special kind.Let the rod loop begin and end in the origin of the absolute reference frame thatwas introduced above. Suppose that the length of the loop is S = 1

m0, where m0 is

an integer parameter (this causes no loss of generality, since the proper scaling ofparameters may compensate for the fixation of the rod length). The following BVPmay be formulated:

1) ρ(s1) = ρ

(s1 +

1

m0

)= 0, 2) ζ(s1) = ζ

(s1 +

1

m0

). (3.1)

Taking into account the symmetry property of the function γ1(s) with respectto the s0: γ1(s0− s) = γ1(s0 + s), we put s1 = s0 − 1

2m0to satisfy the first equality

in equation (3.1,1). Equation (2.15), together with equation (3.1,2) and with thedefinition of Ω, implies

g3 =8m0Ω

pE

(Ω

2m0, k

), g1 =

4Ω

p

[2m0 E

(Ω

2m0, k

)− Ω

]. (3.2)


6 E. L. Starostin

Using the expression for the modulus k, we can obtain the equation for theremaining root

g2 =4Ω

p

[2m0 E

(Ω

2m0, k

)− (1− k2)Ω

]. (3.3)

To express the parameters p, l, d and h as functions of the normalized rootsgi = pgi, i = 1, 2, 3, we make use of the form of the function f in equation (2.8).After some algebra, we come to the expression for p: p2

1,2 = 2l2G2−B±√−f(G2),

where B = g1g2 + g2g3 + g3g1 and G2 = g1 + (g2 − g1) sn2(

Ω2m0

). Clearly, g1 ≤

G2 ≤ g2 and −f(G2) ≤ 0. The only possibility to have the parameter p real is toput G2 = g2, which implies Ω = 2 Km0 (we are here interested in the minimalvalue for Ω). We have for the parameters then

p2 = g1g2 + g2g3 − g3g1, l2 = g1 + g3, d2 =g22l

2

p2, h = g1 + g2 + g3 − d2. (3.4)

The values g1, g2, g3 can be found as functions of m0 and k from equations (3.2),(3.3)

g1 = 16m20 K(k)[E(k)−K(k)], (3.5)

g2 = 16m20 K(k)[E(k)− (1− k2) K(k)], (3.6)

g3 = 16m20 K(k) E(k), (3.7)

where E(k) denotes the complete elliptic integral of the second kind. Equation(3.4), together with equations (3.5–3.7), gives us the analytical expressions of theparameters p, l, d, h as functions of k and m0. The modulus k varies between zeroand the maximal value k∗ ≈ 0.9089085575, which is the root of the equation K(k) =2 E(k). It may be obtained as an equation for the extremal values of the roots g1 andg3: g1 = −g3. Since g1 ≥ −1 and g3 ≥ 1, the last equation implies both g1 = −1 andg3 = 1. For k = k∗, we obtain from equation (3.4) l = 0 and d = 0. The centrelineshape is (one half of) the planar figure-8 with self-contact at the origin:

ξ =k

2 K(k)cn((4s− 1) K(k)), ζ =

1

2 K(k)Z((4s− 1) K(k)), 0 ≤ s ≤ 1

2

(we choose m0 = 2 to make the length of the whole figure-8 equal 1); Z(u) =

E(u, k)− E(k)u/K(k) is the Jacobian Zeta function.The figure-8 gives us the simplest example of a smoothly closed configuration

with self-contact in the origin. The shape consists of two identical loops, one of whichis turned through the angle π around the ζ-axis with respect to the other. Gener-ally, there exists a countable set of more complex spatial equilibria, containing m0

identical loops arranged accordingly around the ζ-axis. Each such solution servesas a limit for a couple of one-parametric continuous families of smoothly closedconfigurations of two (generally, different) knot types. The other limiting point is a(multicovered) circle. In the self-penetrating rod model, a correspondence betweenknot types is established through the self-intersecting solutions. Each family con-tains only one self-contact shape. Details of the classification of equilibrium shapescan be found in (Starostin 1996).



Table 1. The values of the elliptic modulus k for various m0-leafed roses.

m0 2 3 4 5 6

m1 = 1 0.9089085575 0.8909131369 0.8617750161 0.8315214520 0.8024216633

m1 = 2 0.8909131369 0.9089085575 0.9028843308 0.8909131369

m0 7 8 9 10 11

m1 = 1 0.7751554280 0.7498694109 0.7265052907 0.7049273686 0.6849768529

m1 = 2 0.8767597839 0.8617750161 0.8465855842 0.8315214520 0.8167660048

The particular self-crossing symmetric solutions were called roses and their exis-tence was shown by Le Bret (1984) though no closed form expression was obtainedfor them. Now we compute the values of parameters that allow for assembling therose-like shapes from the loop solutions. From equations (2.16), (2.17), and (2.15),we can find the shape of the loop of length 1/m0

ρ =8m0kK(k)

pcn((2m0s− 1) K(k)),

α = − l2

(s− 1

2m0

),

ζ =8m0 K(k)

pZ((2m0s− 1) K(k)), 0 ≤ s ≤ 1

m0.

The rotation angle between the projections of the tangents at the initial (t0)and the end (t1) points of the loop on the ξη-plane is ∆α± π mod 2π and

∆α = α1 − α0 = − l

2m0. (3.8)

Now suppose that we have m0 copies of the loop, each rotated through theangle (j − 1)(∆α ± π), j = 1, . . . ,m0, around the ζ-axis. The projection of theend tangent vector of the j-th loop then coincides with the projection of the initialtangent of the (j + 1)-th loop (j = 1, . . . ,m0 − 1). We may require that the endtangent of the last m0-th loop be aligned with the initial tangent of the 1-st loop,i.e. m0(∆α + π sgnm1) = 2πm1, m1 = ±1,±2, . . . .

Substituting ∆α from equation (3.8) into the last equation, we obtain l =2π(m0 sgnm1 − 2m1). Comparing this expression with the second equation (3.4),where the values of the roots from equations (3.5-3.7) are substituted, yields anequation for the elliptic modulus

π

∣∣∣∣1− 2|m1|m0

∣∣∣∣ = 2√

K(k)(2 E(k)−K(k)). (3.9)

The solutions of this equation are given in table 1 for some values of m0 andm1. Figure 2 corresponds to the solution of equation (3.9) for m1 = −2.

For each element loop (leaf), the angle between the tangent at the initial pointand the ζ-axis is equal to one between the tangent at the end point and the sameaxis. This follows from the symmetry of the loop and from equations (2.13), (2.10).

Therefore, not only projections of the tangent vectors coincide but also the tan-gents themselves. This is sufficient for that such an assembly results in a smoothlyclosed solution for a rod of length 1, since the rotational symmetry and identity


8 E. L. Starostin

t

t

N

ζξ

η

1

0

0.08

0.06

0.04

0.02

-0.020.02

0.02

0.01

-0.01

-0.02

Figure 2. The loop solution for m0 = 5, m1 = −2. The projection of the curve on theξη-plane is shown.

of all element loops provide for both the constancy of the internal force and thecontinuity of the internal moment. Clearly, the identical solution is obtained byapplying equations (2.11), (2.15) and (2.14) to the entire unit length of the curve.Note that although the solutions described have a point of self-contact, there is nointeraction between the remote fragments.

The inclination of each loop (which proves to be somewhat flat) to the ξη-planemay be estimated by using equation (2.13). At the initial and end points of eachloop the angle between the unit vector N = t0×t1

‖t0×t1‖and the ζ-axis can be found

from

cos ν = N · eζ =ρ′0 sin l

4m0√(g2p

)2

+ ρ′02 sin2 l

4m0

, ρ′0 =16m2

0kK2(k)√

1− k2

p

(figure 3, curve 1). At the middle point of the loop, where the polar radius ρ takeson its maximal value, the angle between the tangent and the ζ-axis satisfies theequation cos θm = −g1/p (figure 3, curve 2). Curve 3 is simply the sum ν + θmand its difference from π/2 indicates non-flatness of the loops. For k = 0, the loopentirely lies in the plane ζ = 0 and its shape is a circle of radius 1

2πm0.

With increase of the modulus k, the loop becomes “steeper” and, for k = k∗,its plane is orthogonal to the ξη-plane. The angle β is one between t0 and −t1 andit decreases monotonously from 180 for k = 0 to βmin ≈ 81.41982142 for k = k∗

(figure 3, right).An example of a shape consisting of 5 congruent loops is shown in figure 4.

4. Configurations with non-zero contact force

It is interesting to follow the evolution of the shapes with a multi-contact point tak-ing into account the contact forces. For the slender rod, the region of self-interactionis idealized to be a point. Therefore, we assume that the contact forces are pointwiseand frictionless, i.e. they act only in the normal plane in the point of the contact.We keep on studying the symmetric configurations only.



k

1

2

3

0.80.60.40.2

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

k0.80.60.40.2

3

2.8

2.6

2.4

2.2

2

1.8

1.6

1.4

β

Figure 3. The angles ν (1), θm (2) and their sum (3) (left) and the angle β (right) vs. theelliptic modulus k.

Figure 4. The 5-leaf rose; m0 = 5, m1 = −2. No contact forces. Axonometric, top andside views.

(a) Non-smoothly closed loops

As in the forceless case, the whole closed shape is assembled from a numberof identical leaves. Each closed loops is now a solution of the more general BVP,in which we release the constraint that the ends of the rod should coincide withthe origin (or that the force should direct along the moment at the ends of theloop). This new BVP can be considered as a generalization of the problem forsmoothly closed shape in which we retain only the closedness condition. In otherwords, the tangent vector is allowed to jump in one point of the closed centreline.It is convenient to scale the problem such that the length of the loop be 1. We have


10 E. L. Starostin

three boundary conditions

1) ρ(s1) = ρ(s1 + 1), 2) α(s1) = α(s1 + 1) mod 2π, 3) ζ(s1) = ζ(s1 + 1). (4.1)

The function ρ(s), as defined by equation (2.11) with (2.10), has the propertyρ(s0 + s) = ρ(s0 − s), ∀s. In this paper, we restrict ourselves with the choice s1 =s0 − 1

2 which guaranties that the first boundary condition of (4.1) is satisfied. Thethird condition implies equations (3.2) and (3.3) for m0 = 1. We may also write

ρ =4Ωk

p

√1

n− sn2 Ω(s− s0).

The remaining boundary condition for α may be given the following form

(2πm+

l

2

)Ω(g2 − g1) = [l(g2 + g1(n− 1))− nd]Π

(Ω

2, n, k

), (4.2)

where m ∈ Z is the number of turns around the ζ-axis.We express the parameters l and d in terms of p and gi as

d =1

2[σ+

√−f(−1) + σ−

√−f(1)], l =

1

2[σ+

√−f(−1)− σ−

√−f(1)],

where σ± = ±1 and f is the polynomial from (2.9). The parameter n can also berepresented as n = p(g2 − g1)/[p(g2 + g3)− l2].

Thus, we can insert in equation (4.2) the parameters d, l and n as functions ofgi which in turn depend on Ω, k and p. As a result, in order to satisfy all threeboundary conditions, we need to solve one equation (4.2) treated as an equation forunknowns Ω, k, p. Generically, it serves as the constraint that defines a countable setof surfaces in space Ω, k, p, ordered by m. Every point on these surfaces representsa loop.

The tangent vectors at the beginning and at the end of the loop are t0 =(ρ′0, ρ0α

′0, ζ

′0)T , t1 = (−ρ′0, ρ0α

′0, ζ

′0)T (here the subscript 0 (resp., 1) indicates the

initial (resp., final) point of the loop). We rotated the reference frame so thatα(s1) = 0. We also used the fact that, as seen from equations (2.12)–(2.13), α′(s0 +s) = α′(s0 − s) and ζ ′(s0 + s) = ζ ′(s0 − s).

Now compute the common normal vector N = 1√(ζ′

0)2+ρ2

0(α′

0)2

(0,−ζ ′0, ρ0α′0)T .

The normalized moment M = M

B2at the beginning of the loop is M = (0,− p2ρ0,

−l)T and we introduce the angle µ that specifies the direction of the moment µ =arctan

(pρ02l

). Thus, the unit vector in that direction will be M = −(0, sinµ, cosµ)T .

(b) Assembling multi-leafed shapes

Here we describe the procedure of assembling the closed multi-loop shape. Toget started, let us project the tangent vectors t0 and t1 onto the plane orthogonalto M: t0⊥ = t0 − (t0 ·M)M and t1⊥ = t1 − (t1 ·M)M.

At the first stage we find a simple loop that satisfies the condition α∗ =∠(t0⊥, t1⊥), where α∗ = 2πm1

m0, m0 > 1 and m1 are coprime integers. For example,

α∗ = ±π for the figure-8-like configurations and α∗ = ± 2π3 for the trefoils.



At the second stage, we rotate the loop around the ξ-axis through the angleµ to make M = −eζ2, ζ2 the new ζ-axis, and translate the starting point to theorigin. Thus, the coordinate transformation is given by

ξ2η2ζ2

=

1 0 0

0 cosµ − sinµ

0 sinµ cosµ

ξ

η

ζ

−

ρ0

0

0

.

Then, we make (m0−1) additional congruent copies of the loop and rotate j-thloop around Oζ2 through the angle (j − 1)α∗, j = 1, . . . ,m0. By construction, theentire shape is closed such that the tangent vector is continuous everywhere. Sinceall the moment vectors are aligned along the same axis, the end moment M of (j−1)-th loop is balanced with the starting moment of the j-th loop. Geometrically, itmeans that the curvature and torsion of the centreline are continuous at this point.

The force acting on each loop may be represented as the sum of two componentsF = FMM + FNN. Since all the loops are congruent and they have the same ζ2-axis, the first components compensate each other for any joined pair of loops. Thesecond components are realized through the normal contact force between the twoends of the simple loop. In other words, in order to balance the forces, we allow foradditional contact forces that are directed along the common normal N.

The equation for α∗ may be given the form

sin2 α∗

2=

4(g2 − g1)2Ω2 sn2 Ω2 cn2 Ω

2 dn2 Ω2 (d2 + h− pγ10)

(d2 − l2 + h− pγ10)(h− pγ10), (4.3)

where γ10 = g1 + (g2 − g1) sn2 Ω2 .

If the shape contains only two loops (m0 = 2), then the joining condition sim-plifies to M ·N = 0, which is

2γ10(g1 + g2 + g3 − γ10)− g1g2 − g2g3 − g3g1 − 1 = 0.

Thus, the problem of finding a multi-leafed configuration is reduced to the so-lution of a system of two algebraic equations (4.2) and (4.3) with respect to theparameters Ω, k and p. In the next section, we present some results obtained bymeans of the parameter continuation method.

5. Solutions and the bifurcation diagram

We begin our analysis with planar loops with α ≡ const. The last condition issatisfied if l = d = 0 which implies that the polynomial f(γ1) has ±1 as its rootsand either g1 = −1, g2 = 1, g3 = h

p≥ 1 or g1 = −1, g2 = h

p≤ 1, g3 = 1. In the first

case we have

(2− k2)Ω = 4 E

(Ω

2, k

), (5.1)

which gives us the curve Λ in the parameter plane (Ω, k) (figure 5). Similarly, theother condition implies

Ω = 4 E

(Ω

2, k

), (5.2)


12 E. L. Starostin

which is the curve Γ in figure 5. Both curves Λ and Γ intersect in the point G whereg1 = −1, g2 = g3 = 1, k = 1 and Ω can be found from the equation Ω = 4 tanh Ω

2 ,Ω ≈ 3.830. In addition, it may be shown that as Ω → ∞, the elliptic modulus k,defined by the equation (5.2), approaches k∗.

0.75

0.8

0.85

0.9

0.95

1

4 5 6 7

G

ka’

b’

a, c’d’e’

f’

b

g’

h’

i’

c de

f

Ω

Λ

Γ

Ξ

Υ

0.87

0.88

0.89

0.9

0.91

0.92

4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2

k

a’

b’

a, c’

d’

e’

b

Ω

Γ

Ξ

Figure 5. The bifurcation diagram for the first branches of the two-leafed (boxesa− b− c− d− e− f) and three-leafed (diamonds a′ − b′ − c′ − d′ − e′ − f ′ − g′ − h′ − i′)shapes. Right: a magnified view near the “figure-8” point.

The no-contact-force solutions that are analyzed earlier in this paper, lie on thecurve Ξ given by the equation Ω = 2 K(k). The point a ≡ c′ of intersection of Γand Ξ corresponds to the planar figure-8 shape (with k = k∗) and the point e′ tothe self-intersecting trefoil.

Following the assembling procedure described in the previous section, we havecomputed the branches of solutions with non-zero contact force by means of pa-rameter continuation. The curve a− f represents the evolution of shapes made upof two loops (figure 6). It starts at point a, the planar figure-8, and ends at anotherplanar shape (point f) having collinear tangents to the centreline in the contactpoint. The point c lies on the curve Υ where g3 = 1. In figure 6, F is a unit vectoralong the direction of the force needed to form a single loop (without the contactforce).

The evolution of the three-leafed shapes is more complicated (figure 7). Thepoint a′ corresponds to a planar shape and the curve a′ − b′ − c′ connects it to thefigure-8 point c′. Note how close the curve a′−b′−c′ is to the border Γ. The shape c′

is made up of three loops which are exactly the same as the elements of the planarfigure-8, but, in contrast to the latter, the three-leafed shape has non-zero contactforce. This fact was first noticed by Le Bret (1984) who computed only the partc′ − d′ − e′ of the three-leafed shapes with self-contact. The shape without contactforce (point e′) is the solution of the model with self-penetration. It separates theunknotted and knotted equilibria and actually represents two topologically differentconfigurations: the unknot and the trefoil knot. Thus, a′ − d′ → e′ are unknottedshapes while e′ ← f ′ − i′ are trefoil knots. (Of course, if we allow for attracting



F0

0.2

–0.2

–0.1

–0.05

0

0.05

0.1

(a) Ω = 4.64, k = 0.909, p = 43.1

F M

0

–0.2

0.2

–0.050

–0.1

–0.05

0

0.05

0.1

(b) Ω = 4.83, k = 0.880, p = 47.9

MF

0

–0.2

0.2

–0.1

0

–0.1

0

0.1

(c) Ω = 5.62, k = 0.841, p = 66.5

MF

0

–0.2

0.2

–0.1

0

–0.1

0

(d) Ω = 6.05, k = 0.847, p = 74.8

M

F

0

–0.2

0.2

–0.1

00.1

–0.1

–0.05

0

(e) Ω = 6.22, k = 0.852, p = 78.0

F0

–0.2

0.2

–0.1

00.1

(f) Ω = 6.30, k = 0.855, p = 79.3

Figure 6. Two-leafed shapes, first branch. m = −1, σ+ = 1; σ− = −1 for (a)–(c) andσ− = 1 for (d)–(f).

contact forces, arbitrary topology may be attributed to any configuration.) Similarlyto the two-leaf case, the curve ends at a planar solution i′, but now three moreintersection points emerge.

The shape of the rod can be characterized by the writheWr . In the left figure 8,the bending energy is shown as a function of writhe for two first branches of thebifurcation diagram. The bending energy is essentially the value of h and it isnormalized so that to make the energy of the unit-length circle equal 1. In the rightfigure 8, another quantity is presented, namely, the difference between the actualtotal twist and its intrinsic value, multiplied by b:

b∆Tw =b

2π

S∫

0

(ω1 − ω01) ds =

Sd

2π.

To observe the homotopic evolution of shapes, we start with the single-coveredcircle and vary its twist difference. The branch 2.1a corresponds to the first-modeperturbations. When the writhe changes from 0 to 1, we have the warped circlesolution with no self-contact. Wr = 1 corresponds to the point a and then, as thewrithe increases, we follow the branch a−f up to the planar solution withWr = 2.

For every solution with positive twist difference ∆Tw and writheWr there existsa counterpart configuration with −∆Tw , −Wr and with the same elastic energy.Both solutions are simply mirror images of each other. In other words, they differin their chirality. Thus, the branch 2.1b represents these mirror-reflected solutions.It is interesting to note that if we formally shift the branch 2.1b to the right alongthe Wr -axis by 2 to 2.1b′, then 2.1a and 2.1b′ together appear as a pair of smoothcurves touching each other in the point a in the energy graph and intersecting inthe same point in the other graph, thus forming a χ-shaped curve.


14 E. L. Starostin

–0.2

–0.1

0

0.1

0.2

0.3

0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

(a) Ω = 5.12, k = 0.879, p = 52.5,g1 = −1, g3 = 1

–0.2

–0.1

0

0.1

0.2

0.3

0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

–0.050

0.05

(b) Ω = 4.90, k = 0.890, p = 48.0,m = −1, σ+ = 1, σ− = −1

–0.2

–0.1

0

0.10.2

0.3

0.4

–0.3

–0.2

–0.1

0

0.10.2

0.3

–0.1

0

0.1

(c) Ω = 4.64, k = 0.909, p = 43.1,g1 = −1, g3 = 1, n = 1

–0.2

–0.1

0

0.1

0.2

0.3

0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

–0.1

0

0.1

(d) Ω = 4.47, k = 0.906, p = 40.6,m = 0, σ+ = −1, σ− = 1

–0.2

–0.1

0

0.1

0.2

0.3

0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

–0.1

0

0.1

(e) Ω = 4.49, k = 0.891, p = 41.6,n = 1

–0.2

–0.1

0

0.1

0.2

0.3

0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

–0.1

0

0.1

(f) Ω = 5.33, k = 0.774, p = 69.0,m = 1, σ+ = −1, σ− = −1

–0.2

–0.1

0

0.1

0.2

0.3

0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

–0.1

0

0.1

(g) Ω = 6.48, k = 0.812, p = 93.7,m = 1, σ+ = −1, σ− = −1

–0.2

–0.1

0

0.1

0.2

0.3

0.4

–0.4

–0.2

0

0.2

0.4

(h) Ω = 7.28, k = 0.863, p = 106.7,m = 1, σ+ = −1, σ− = −1

–0.2

–0.10

0.10.2

0.3

0.4

–0.4

–0.2

0

0.2

0.4

(i) Ω = 7.34, k = 0.866, p = 107.7,g1 = −1, g3 = 1

Figure 7. Three-leafed shapes, first branch, (a)–(d) are unknotted, (f)–(i) are trefoilknots, (e) is a zero contact force configuration.

The second perturbation mode of the single-covered circle also has an initialcontact-free part withWr varying from 0 to some value slightly greater than 2 wherethe self-intersection occurs (the curve 3.1a). The geometrically identical solutionmay be reached starting from the double-covered circle along the curve 3.1b. On thediagram, this conformation corresponds to two points e′ representing topologicallydifferent shapes. Their writhe differs by 6. The non-zero contact force equilibriaare represented by the curves e′ − a′ (unknotted) and e′ − i′ (knotted). Again, theχ-diagram may be obtained by translating the branch 3.1b along abscissa axis by6 to make both points e′ coincide. The mirror-reflected shape branches of 3.1a and3.1.b are not drawn.

The two parts a′ − e′ and e′ − i′ complement the self-penetrating homotopy ofthe single-covered to the double-covered circle represented by the pieces of graphthat correspond to contact-free configurations (Starostin 1996).

The graph b∆Tw versus Wr is universal in the sense that, given a particularvalue of the ratio b, it is easy to find the intervals where the necessary condition



for stability is satisfied

d∆TwdWr

+ 1 ≥ 0.

This condition was obtained by Le Bret (1984); see also (Tobias et al. 2000). Inparticular, the above inequality is satisfied for the intervals on the branches in theleft graph of figure 8 where the derivative is non-negative. For the three-leafedequilibria these intervals correspond to both unknotted and knotted configurationswith self-contact around the point e′.

2

4

6

8

10

12

–6 –4 –2 0 2

i’

e’ e’ a’

a

f

Wr

E b

2.1a

3.1a

2.1b

3.1b

–2

–1

1

2

–6 –4 –2 2Wr

i’

e’a’

e’

c’fa

Twb2.1a

3.1a

2.1b

3.1b

∆

Figure 8. The normalized bending energy Eb = h/(4π2) (left) and the twist difference(right) versus the writhe for two- and three-leafed configurations.

6. Conclusion

A one-parameter family of solutions of the BVP for the simple elastic loop is found.They widen the set of the known analytically described configurations of a closedrod. Some of these solutions serve simultaneously as basic elements of the sym-metric multi-leafed shapes of the rod with one multiple self-contact point. Thefurther evolution of equilibria may be followed taking into account the pointwiseand frictionless contact forces in the point of multi-contact by using an assemblingprocedure. A bifurcation diagram is presented for first branches of two- and three-leafed configurations. It complements the diagram obtained earlier for the modelallowing for self-penetration of the rod. The solutions obtained may be used asbasic shapes by numerical computation of more complex configurations, describedby more elaborated models.

The calculations in this paper were carried out with the help of Maple program.

The research was carried out during the author’s stay at the University of Karlsruhe.The support from Alexander von Humboldt Foundation is gratefully acknowledged. Theauthor thanks Prof. J. Wittenburg for his attention to the work and Prof. J.H. Maddocksfor helpful discussions and support.


16 E. L. Starostin

References

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Coleman, B. D. & Swigon, D. 2000 Theory of supercoiled elastic rings with self-contactand its application to DNA plasmids. Journal of Elasticity 60, 173–221.

Dichmann, D. J., Li, Y. & Maddocks, J.H. 1996 Hamiltonian formulations and symme-tries in rod Mechanics. In Mathematical Approaches to Biomolecular Structure and

Dynamics, The IMA Volumes in Mathematics and its Applications (ed. J. P. Mesirov,K. Schulten, D. Sumners) 82, pp. 71–113. Springer.

Ilyukhin, A. A. 1969 On the deformation of the elastic curve. Mekhanika tverdogo tela 1.Kiev, 128–138 [in Russian].

Ilyukhin, A. A. 1979 Spatial problems of the nonlinear theory of elastic rods, pp. 28–30.Kiev: Naukova Dumka [in Russian].

Julicher, F. 1994 Supercoiling transitions of closed DNA. Phys. Rev. E 49 (3), 2429–2435.

Landau, L. D. & Lifshitz, E. M. 1970 Theory of elasticity, 2nd edn., pp. 87–88. Oxford:Pergamon.

Le Bret, M. 1984 Twist and writhing in short circular DNAs according to first-orderelasticity. Biopolymers 23, 1835–1867.

Love, A. E. H. 1927 A treatise on the mathematical theory of elasticity, 4th edn. London:Cambridge University Press.

Shi, Y. & Hearst, J. E. 1994 The Kirchhoff elastic rod, the nonlinear Schrodinger equation,and DNA supercoiling. J. Chem. Phys. 101, 5186 – 5200.

Starostin, E.L. 1996 Three-dimensional shapes of looped DNA. Meccanica 31 (3), 235–271.

Starostin, E.L. 2000 Closed loops of a thin elastic rod and its symmetric shapes with self-contacts. In Proc. 16th IMACS World Congress, Lausanne, Switzerland, 21–25 August

2000, 6 p.

Starostin, E.L. 2002 Equilibrium configurations of a thin elastic rod with self-contacts.PAMM, Proc. Appl. Math. Mech. 1, 137–138.

Tobias, I., Swigon, D. & Coleman, B.D. 2000 Elastic stability of DNA configurations. I.General theory. Phys. Rev. E 61 (1), 747–758.


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